Mathematics
American Standard Code for Information Interchange
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Augustin-Louis, Baron Cauchy
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Benoit B. Mandelbrot
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Bernhard Riemann
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Although first studied by Nathaniel Bowditch, they owe their name to a French mathematician who investigated them in the middle of the nineteenth century. They are produced by an intersection of two sinusoidal curves at right angles to each other. If the amplitudes and frequencies of the sinusoidals are the same, but their phases differ, ellipses are formed; however, the patterns become very intricate if frequencies and amplitudes are not equal. For ten points, name these patterns, most frequently associated with swinging pendulums.
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His distribution is one over the product of pi and one plus x squared. His theorem states that the integral over a closed curve of an analytic complex function is zero if there are no singular points within the curve. He developed two criteria to test for the convergence of infinite series. For 10 points, name this baron who introduced clear and rigorous methods into mathematics with three great treatises published in French between 1821 and 1828.
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Memory is stored one-dimensionally without segmentation, and there is no distinction between instructions and data. There are no floating-point numbers, and data are stored in binary format. There is no indirect or indexed addressing. Three registers are used for arithmetic, and alternates instruction and execution cycles. FTP, name this computer architecture, named for the mathematician who also established the standard formulation for quantum mechanics and proved the minimax theorem in game theory.
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The inventor of the 'silent' dog whistle, he also attempted to compile a map of human physical beauty in England. A pioneer in statistics, he created the first regression line and developed correlational calculus. He was the first person to realize that fingerprints were unique. FTP name this cousin of Charles Darwin whose belief that genius is inherited led him to found the eugenics movement.
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The inversion of this curve with respect to its focus is a cardioid, and with respect to its vertex the cissoid of Diocles. More familiarly, it's those points where the distances between a line and a point not on the line are equal. Rays parallel to the axis reflect from this curve to meet at the focus, a feature used in satellite dishes. For ten points, an example of what conic section is the curve y equals x squared.
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This thinker proposed the reformulation of the major laws of physics so that all calculations can use complex numbers, and put forward a new model of the universe based upon entities he calls "twistors". Author of Shadows of the Mind, he devised geometrical figures which cannot be constructed in 3-dimensions, as seen in several Escher lithographs, proposed the "cosmic censorship" hypothesis, and with Stephen Hawking developed the singularity theorems. FTP, name this English scientist, author of The Emperor's New Mind.
Abelian
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Groups
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PASCAL
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curvature (accept "sectional curvature" before the word "sectional")
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cycloid
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cycloid[s]
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cycloidÂ
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derivative
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derivatives
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determinant
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determinant [prompt on det]
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determinant of a matrix [accept det before mentioned]
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duplication (or doubling) of the cube or equivalents
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dynamic programming
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A useful formulation of this property states that any collection of sets satisfying the finite intersection condition will have nonempty intersection. In Euclidean space, the Heine-Borel theorem says this property is seen in closed, bounded subsets. For ten points, name this topological property which states that any open covering of a space contains a finite subcovering.
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After volunteering for the Naval Ordinance Computer project during WWII, this programmer joined a firm that would become the Univac division of Spersy Rand. When ordered to assist Howard Aiken at Harvard in building a computer, a moth few into the machine and led this programmer to coin the term "debugging". FTP, identify this programmer whose work for the pentagon led to the development of COBOL.
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Both the Fréchet derivative and the Gâteaux derivative are analogous versions of the derivative from differential calculus defined on these, and a non-separable polynomially reflexive subclass of them is named for Tsirelson. The space of continous functions defined on a closed interval that map to an n-dimensional Euclidean space is one of these, as are the sets of real and complex numbers. They are Hilbert spaces if they obey the parallelogram identity and the inner product that defines it as a Hilbert space is given by the polarization identity. FTP, name these spaces, one of the central objects of study in functional analysis, which are defined as being complete normed vector spaces and are named for a Polish mathematician.
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Equivalents include assuming three non-collinear points always lie on a circle. Or that the sum of the angles of a triangle is equal to two right angles. Unproven for over two millenia, in the 19th century it was realized that not assuming its truth led to the hyperbolic geometries of Gauss, Bolyai, and Lobachevsky. FTP, identify this axiom of Euclid.
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His name is given to a random variable with probability density function 1 over the quantity pi plus pi times x squared. His name also graces a theorem that finds the radius of convergence of a Taylor series in a complex variable with Hadamard. His name can also be found in a formula for evaluating integrals of analytic functions of complex variables, as well as an inequality that says the dot product of two vectors is less than or equal to the sum of the vectors' lengths. FTP, identify this mathematician who discovered this triangle inequality independent of Schwarz.
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In the plane, this mathematician's namesake measure involves finding the greatest lower bound of the areas of the coverings of a given set, where a covering consists of a finite union of rectangles. One result named for this mathematician and Hölder follows from Zassenhaus's Lemma via Schreier's Refinement Theorem and states that any two composition series in a group are isomorphic. Any matrix with coefficients from an algebraically closed field can be rewritten in terms of blocks containing a single eigenvalue on the main diagonal and ones on the superdiagonal, a block form name after this man. Another result of his states that any closed curve divides the plane into an inside and an outside. For 10 points, identify this French mathematician, namesake of a "normal" or "canonical" form of a matrix, as well as a curve theorem.
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In this language, the eighth through eleventh columns of the code are known as Area A, and the twelfth through seventy-second as Area B, while column seven contains page breaks and comment markers. A program written in this language has four divisions: identification, environment, data, and procedure. Currently, the 85 version is the standard, despite past problems caused by its six-digit date storage. FTP, name this language, developed in 1959 by Grace Hopper.
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It typically uses load and store commands to read and write to memory. Because of the increase in RAM storage, the fixed instruction length feature of this architecture has become more feasible. A large number of general purpose registers are found on these types of chips in order to store intermediate results in more complicated operations. FTP, name this architecture containing a small number of simple operations, an example of which is the PowerPC-based processor in Macintosh computers.
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It was developed from a product initially designed by Tim Paterson. A major flaw is that it was not written in reentrant code, and thus does not support multitasking, and the earliest versions did not implement a hierarchical directory structure. Version 1.1 allowed double-sided floppy disks, while version 4.0 supported hard disks greater than 32 megabytes. Identify this command-based operating system, a Microsoft product which, FTP, dominated the Intel 8086 (eighty eighty-six) processor family for more than a decade.
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One appropriate application of this technique is in the case of a bimolecular reaction, in which the reaction rate is characterized by R, times A minus Y, times B minus Y, where R, A, and B are all constants. For a case in which the numerator is 1 and the denominator is the polynomial x squared minus 8x plus 15, the proportion must be transformed into two ratios of numerical value A over x minus 5, and B over x minus 3, and then solved for A and B. FTP, identify this method of integration involving the splitting of complex polynomial fractions.
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One of his theorems in differential topology states that a subbundle of a manifold's tangent bundle is integrable if and only if its space of sections is closed under the Lie bracket. His namesake algebras are equipped with associative, nondegenerate bilinear forms. More famously, he gives his name to a field automorphism which for a field of prime characteristic p maps every element a to its pth power. He is perhaps most famous for a result which categorized all real associative finite-dimensional division algebras. FTP, name this German, whose namesake theorem shows that the only algebras of the above type are the reals, the complex numbers, and the quaternions.
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One theorem named for this man means that for any simply connected open proper subset of the complex plane, there exists a bijective analytic map from it to the open unit circle about the origin. Manifolds with a smooth inner product are named for him. He and Cauchy name a set of differential equations used in determining whether a function is holomorphic. He is also known for a function for which a namesake hypothesis states that all trivial non-zeros of the function have real part one-half. For 10 points, name this formulator of a zeta function and a method of integration which involves using his namesake sums.
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Recently the low-energy theory on a stack of M2 branes has been shown to be a doubled parity-invariant version of it. The Jones polynomial for a knot in a 3-manifold is given by a path integral in this theory of the holonomy of a connection around the knot, as shown by Edward Witten. The topological nature of the theory implies that it is associated with an integer level, and has been used to explain the exactness of the fractional quantum Hall effect. One of its namesakes is a Chinese differential geometer also known for a type of characteristic class for complex vector bundles. Its action is the integral of a three form and its equation of motion says the field strength is zero. For 10 points, what is this theory also named for an American mathematician who founded Renaissance Technologies?
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The Bell series of this function is equal to one minus x over 1 minus px, while the partial sums of this function show that its average order is equal to 3n divided by pi squared, a result which can be used to find the number of lattice points visible from the origin. The Dirichlet convolution of the identity function N and the Mobius function gives this function, and if a number has r distinct odd prime factors, 2 to the r divides this function of the number. Summing this function over the divisors of a number n gives n. It gives the number of elements in the group Z sub n star, and for a prime number p this function returns p minus 1. Also used in its namesake's generalization of Fermat's Little Theorem, for 10 points, name this function that returns the number of integers less than and relatively prime to the input.
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The Bogomolov-Miyaoka-Yau inequality implies the truth of this statement, as does a statement relating the minimal discriminant to a conductor of a certain type of curve called Szpiro's conjecture. One statement that leads to the proof of this statement is part of the Langlands program. The proof by Ken Ribet of the "epsilon conjecture," which stated that the Frey curve was not modular, led to a proof of a special case of the Taniyama-Shimura conjecture, whose truth implies the truth of this theorem. For ten points, identify this theorem which states that the equation "x to the n plus y to the n equals z to the n" has no nonzero integer solutions for integer n greater than two, eventually proved by Andrew Wile.
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The Lucas-Lehmer test can determine if one of them is of interest. All even perfect numbers are the product of a power of 2 and a prime one; for example, 28 is "2 squared times 7" and 496 is "2 to the fourth times 31". For ten points, what are these numbers named for a Minim friar which are of the form "2 to the nth minus one"?
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The man who proved this theorem has another important result concerning the isomorphism of semi-simple right Artinian rings to n x n [N by N] matrices over a division rings. William Hamilton's discovery of the quaternion in the 19th century provided an example of a skew field, which is a ring with non-commutative multiplication where every non-zero element has a multiplicative inverse. More than half a century passed until the question of the existence of finite skew fields was settled in the negative. Named for the Scotsman who derived it, FTP, name this theorem that specifically states that every finite division ring is a field.
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The p-adics are useful examples of these in number theory, and a theorem of Weierstrass and Hilbert shows that any attempt to generalize this concept to triplets of numbers is equivalent to the complex numbers. These are called Pythagorean if they contain the square root of one plus the square of any element, and perfect if all finite extensions are separable. The main theorem of Galois theory relates certain subsets of Galois extensions of these to the subgroups of the Galois group. The characteristic of one of these is the number of times you must add one to itself to get zero and must always be a prime, and a finite one of these must have order equal to a power of some prime. For 10 points, name this algebraic structure, examples of which include the reals and rationals but not the integers, defined as a set with two binary operators in which addition, multiplication, subtraction, and division are all well-defined and the distributive law is satisfied.
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The regular pattern expression matching function in Perl bears this property. It relies on the characteristic of a given n-variable Boolean formula in conjunctive normal form, which is known as satisfiability, and Cook's theorem indicates satisfiability has this property. Formally, it is defined in terms of reduction, the idea that if problem A is easier to solve than B, a relationship can be drawn between the two that defines this property for B. FTP, identify this property known to have no polynomial time solutions, famously including the traveling salesman problem.
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This concept was recently attacked by Mendelson, who used analogous concepts like Tarski's theorem and Weierstrass' theorem. An example that gives credibility to this theory is Ackermann's function, which is mu-recursive but not primitive recursive. In addition to recursiveness, another property required by this theory is lambda-definability. Generally considered intrinsically unprovable, this thesis relies on the computability of a Turing machine. FTP, identify this thesis that states any function regarded naturally as computable can be computed by a suitable Turing machine
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This holds that there exists an eigenform under the Hecke operators in modular form of weight two and the utility of its proof was increased greatly by Ken Ribet's 1986 proof of the epsilon conjecture. Given an elliptic curve with integral coefficients, this conjecture links the fields of topology and number theory and parts of it were first proven by Andrew Wiles. FTP name the conjecture used to proof Fermat's last Theorem and conventionally named for its Japanese co-formulators.
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This man created a theory of distribution based on marginal productivity, and, along with Daniel Bernoulli, he derived an equation for the torque on a thin elastic beam. He found all even perfect numbers and his phi-function is the number of smaller positive integers relatively prime to a number. With Lagrange, he derived the calculus of variations, while his angles specify the orientation of a rigid body. For ten points, name this Swiss mathematician and possible namesake of the number e.
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This man's summation is used to compute a possibly divergent series of complex numbers, and his limit theorem states that the limit assigned to one of his convergent summations exists and agrees with the sum. The introducer of elliptic functions, he gives his name to groups in which ab [A B] equals ba [B A] for all elements in the group. FTP. identify this mathematician who demonstrated in 1823 that there is no algebraic formula for the solution of a generic polynomial of the 5th degree.
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This person's namesake function is a generalization of length to integer-graded modules over graded rings. This thinker's namesake symbol returns 1 if a three-variable quadratic equation is solvable with its two p-adic inputs as coefficients. This is the first namesake of a type of operator useful in the study of Fredholm integral equations with symmetric kernels; the study of such equations is known as his and Schmidt's namesake theory and those operators are also bounded in his and Schmidt's namesake norm. Like Peano, this mathematician has a namesake curve of limiting Hausdorf dimension 2 which is his namesake square-filling curve. This mathematician names complete inner product spaces equipped with complete L-2 norms, all of which are Banach spaces. For 10 points, name this mathematician perhaps better known for posing twenty-three important problems in mathematics.
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This problem is a type of membership problem, and, by formulation as a dynetic decision problem, it is neurally solvable. Rice's theorem was proven as a result of this, and Chaitin's constant is also sometimes named as this problem's probability. This problem is sometimes stated as a form of Hilbert's second problem. The first proof of this problem's undecidability, as well as that of the Decision problem, was published by Turing in a paper where he defined the Turing machine. One method of proving a quality on this problem relies on Cantor's theorem and the diagonalization of the results on input of a sequence of Turing machines. For 10 points, name this problem of determining whether a program and a certain input will run forever or finish.
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This problem was first created as a toy in 1883, sold by Professor. Claus of the college of Li-Sou-Stian, which are just anagrams for its inventor, Eduard Lucas of Saint Louis. The Buneman-Levy algorithm gives a purely iterative solution to the problem, in which 64 disks of increasing size placed in a pile must be moved, with the conditions that a larger disk can never be placed on a smaller one, and disks must be moved one at a time. Supposedly involving a sacred ritual at the ancient temple of Benares in which the world will vanish when the task is completed, name this mathematical puzzle that, FTP, is a commonly used example of a problem easily solved by recursion.
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This two dimensional manifold is orientable and has Euler characteristic 0. It can be expressed as the product of two circles, but like the Klein bottle is more often viewed as a square with opposite sides identified. FTP, name this surface, which is more familiar to most people as the surface of a donut.
(Georg Friedrich) Bernhard Riemann
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(Johann Friedrich) Karl Gauss
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Ada
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Dirichlet
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Fibonacci numbers
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Gauss-Jordan elimination (or: Gauss-Jordan method or Gaussian elimination or Gaussian method)
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Gaussian random processes
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Genetic Algorithms (prompt on "GA")
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Georg Ferdinand Ludwig Philip Cantor
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Georg Ferdinand Ludwig Phillip Cantor
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Gödel's Incompleteness Theorem
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Gödel's Incompleteness Theorem (accept either) or Gödels Unvollständigkeitssatz
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Halting problem or HaltTM
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Hamiltonian Path Problem
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Hamiltonian cycle [prompt on Hamiltonian; do not accept "Hamiltonian path"]
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Hamiltonian function
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Hausdorff space [or T2 space before mentioned]
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Heine-Borel-Lebesgue theorem or property [prompt on partial answer]
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Hermann von Helmholtz
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Hero or Heron of Alexandria
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Hilbert spaces
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LISP (prompt on List Processing)
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Laplace Transform
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Laplacian (accept del-squared before it's mentioned)
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Laplacian [or Laplace operator]
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Latin squares [accept Greco-Latin Squares before "generalized"]
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Law of Large Numbers
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Lebesgue integral or Lebesgue integration [prompt on integral or integration until "namesake"]
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Leonard Euler
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Leonardo Pisano or Fibonacci (prompt on Liber Abaci or the Book of Calculation if someone is confused and says it prior to the asterisk)
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Leonhard Euler
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Leopold Kronecker
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Liouville's Theorem
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Lissajous curves or figures
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MS-DOS or PC-DOS or just DOS
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Mandelbrot set
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Maria Gaëtana Agnesi
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Markov chain
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Mersenne numbers (reluctantly accept Mersenne primes)
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Mersenne primes (do not accept "Mersenne numbers")
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Mobius strip (or band)
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Monte Carlo algorithms
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Monte Carlo methods
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Monte Carlo methods (accept "annealing" before *)
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NP-completeness
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Niels Henrik Abel
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Nikolay Lobachevsky
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Norbert Wiener
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Ordinary Differential Equations (prompt on "differential equations" before "one variable")
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Ordinary least squares regression (prompt on regression; accept OLS)
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PROLOG
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Pafnuty Chebyshev
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axiom of choice (accept Zorn's lemma or well ordering theorem)
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binary tree
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brachistochrone problem
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cache
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knots or links
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l'Hospital's rule
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lambda calculus
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least squares methods [accept LSRL]
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least-squares
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linear programming
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linked list
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logarithm
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logistic function
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maximum likelihood estimation method
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mean value theorem
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measurement or observation [accept experiment before *]
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metric (accept metrizable before "tensor")
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n-sphere or hypersphere or sphere in n-space or equivalents
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nephroid
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object-oriented programming languages (prompt on "programming languages" on the first sentence)
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octahedron (or octahedral)
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operator overloading
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optimization (accept minimization until "highest" or maximization until "lowest")
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ordinary differential equations (prompt on differential equation)
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parabola
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primality or testing for prime numbers or equivalents
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prime number theorem
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prime numbers [accept Mills theorem until "these numbers"]
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prime numbers [or primes]
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prisoner's dilemma
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production possibilities curve or production possibility curve
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quicksort
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quintic (or fifth-degree or other clear equivalents)
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quintic equations [accept fifth-degree polynomial or fifth-degree equation before mentioned; also accept quintic formula or quintic expression]
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random walk (prompt on "Markov chain")
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shortest path algorithms [accept with any of the following modifying the answer; all-pairs, single-source, or single-source single-destination]
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shortest-path algorithms [accept clear equivalents; prompt on “graph searchâ€, “tree searchâ€, or anything else involving searching]
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simple network management protocol (accept SNMP early)
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sine
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sorting
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trapezoid
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traveling salesman problem
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triangle
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universality
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variational principles
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vector space
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virtual
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 Fibonacci numbers or sequence
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 Goldbach's Strong Conjecture
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 garbage collection
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 groups
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 open [accept word forms like openness or open set etc.]
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 prime numbers
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Évariste Galois
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A bounded set is said to be this type of measurable if it is well approximated by simple sets. One of this man's namesake theorems claims that any closed curve in R-2 divides the plane into a bounded interior and unbounded exterior, which are complements. Another theorem bearing his name states that all composition series of a group have the same length. His namesake canonical basis consists of cyclic subsets, and he also names a form in which the diagonal of the matrix of the operator is zero if and only if the operator is nilpotent. For 10 points, name this mathematician whose namesake canonical form consists of eigenvalues on the diagonal and ones below the diagonal in each of the namesake boxes.
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A common way to implement one operation in them is named for a roulette wheel, and a proof that one of these can efficiently solve the Two-Armed Bandit problem relies on the Schema Theorem. Goldberg outlined the "simple" varieties of them, which contrast with the steady state type by replacing a generation at each iteration. Their optimality can sort of be explained with the much-maligned Building Block Hypothesis. They were first proposed by John Holland, and usually incorporate crossover and a fitness function to solve problems. For 10 points, name this type of algorithm that makes use of factors like mutation and inheritance to solve problems in a biologic fashion.
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A crude branch-and-bound backtracking algorithm can be developed from a similar algorithm for the Hamilton cycle problem, but the Lin-Kernighan algorithm, which relies on lambda-change, is probably the best heuristic known for this problem. It can be shown that the problem is inapproximable, as an epsilon-approximation algorithm would prove imply that there are polynomial-time algorithms for other NP-complete problems. FTP, identify this most famous example of an NP-complete problem, in which a certain businessman would like to find the shortest distance needed to tour a set of cities.
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A derivative identity for them can be obtained in terms of Chebyshev polynomials, and their generating function is "exponential of x over 2 times quantity t minus one over t." For small values of the argument, the ordinary one may be approximated as "x to the n over quantity 2 to the n times n factorial" and the nth modified one is related to the nth regular one by a factor of i to the n with multiplication of the argument by i. Hankel functions are complex linear combination of them, and they arise as solutions of Laplace's equation in cylindrical coordinates, so they're also known as cylinder functions. FTP, identify these special functions named for the German astronomer who discovered them and symbolized J and Y.
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A function with specified zeros and specified orders of those zeros that also has this property exists as a consequence of the Weierstrass Product Theorem. According to a theorem of Liouville, a function that is bounded and has this property everywhere must be constant. A function that has this property on a closed region may be expanded in a Laurent series within that region, and that function has this property if it satisfies the Cauchy-Riemann equations within that region. The real parts of functions with this property are harmonic, and a function may fail to have this property in the complex plane due to the presence of poles or branch cuts. For 10 points, identify this property of a complex-valued function, which is equivalent to being complex differentiable.
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A more generalized version of this algorithm uses an admissible heuristic function that can result in a faster run time but requires more memory, and is known as A star. Unlike the Bellman-Ford algorithm that finds the same solution, this algorithm does not work with negative-weight edges. Its worst case run time of big O of n2 when using ordinary linked lists can be improved in sparse cases by the use of various heaps as a priority queue when running its central extract-min function. For 10 points, name this greedy algorithm that finds the shortest path tree for a given graph, named for its Dutch developer.
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A notion of kernel and cokernel exists for abelian ones, which allows the definition of an exact sequence to be defined. Maps between them are called functors, and they consist of objects with morphisms between them. Introduced by Eilenberg and Maclane, these are, FTP, what very general algebraic structures that share their name with an unrelated topological object described by Baire's theorem and a 12-fold classification by Immanuel Kant?
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A proof of this lemma proceeds by applying the Cauchy mean value theorem to a relevant ratio, noting that two of the difference terms vanish, then shrinking the interval. A hypothesis in this theorem is that the relevant functions oscillate at most finitely often near the relevant point; this is because a derivative of the denominator function must exist and be other than 0. Though credited to its namesake's 1696 text, this rule was probably created by John Bernoulli. FTP, name this rule from calculus used to evaluate limits that result in indeterminate ratios using derivatives; a rule named for a French mathematician.
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A sequence can be related to a Newton series appearing in the Mellin transform of this man's namesake generating function using the cycle named for this man, Mellin and Newton. Solvation effects can be analyzed using an equation named for him and Boltzmann, while an equation named only for him is an inhomogeneous version of the Laplace equation. The Jacobi identity is one of the relations satisfied by his namesake bracket, and Fresnel's wave theory predicts his namesake spot. The lateral strain over the axial strain gives the ratio named for, FTP, this Frenchman whose distribution has mean equal to its variance and gives the probability of a certain number of independent events in a fixed period of time in terms of the average rate.
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A space admits this type of topology if and only if it is Hausdorff, regular and has a countably locally finite basis, by the Nagata-Smirnov theorem. This term also denotes a type of tensor which is preserved by the Levi-Civita connection and whose eigenvalues are characterized by the signature. Spaces of this type in which every Cauchy sequence converges are said to be complete, and these spaces are characterized by a namesake symmetric, non-negative function that returns 0 only if the input points are identical and satisfies the triangle inequality. FTP, name this type of space characterized by a function that gives a measure of distance between points.
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A technique using the 48-bit AGGR allows endpoint addresses to be stored in its header, enabling tunneling via its predecessor. Created by a team lead by Steve Deering at PARC, convention states that strings of zeros in these addresses can be replaced by a double colon and the last 32 bits can be written in decimal instead of hex. Although its predecessor's header is half as big and is limited to four billion nodes, address translation has slowed its adoption, even though its 128-bit address space allows for 3.4 times ten to the 38 nodes. China's Next Generation Internet is abandoning the fourth version of, FTP, what addressing protocol usually paired with TCP that is to replace IPv4?
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A theorem named for him states that for a polynomial of degree n with integral coefficients, there exists a maximum of n solutions to f of x equals 0 modulo p. In addition, this man proved that every positive integer can be represented as a sum of four squares. A theorem named for him states that the order of a subgroup must divide the order of the group, and he is also the namesake of a method of finding the optima of a function by introducing the parameter lambda. The namesake of an error function associated with Taylor series, for 10 points, identify this Frenchie, who also found a set of points where a tiny body remains stationary relative to two larger bodies.
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A value named for this man is proportional to the integral of Gaussian curvature over a closed surface. He is the namesake of an identity stating that “cosine theta plus sine theta times i†equals “e to the i times theta.†His namesake “characteristic†equals vertices minus edges plus faces. The limit of the difference between the harmonic series and the natural log is named for Mascheroni and this solver of the Bridges of Konigsberg problem. A value of about 2.718, the base of the natural log, is named e in his honor. For 10 points, name this Swiss mathematician.
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A variation of this algorithm which preserves the natural ordering does not perform a merge if an original node exists between two merged nodes and is named for Hu and Tucker. Faller, Gallagher, Knuth, and Vitter all worked on a modification to this algorithm which allowed it to operate in one pass, the so-called dynamic or adaptive form of it. The length limited version of this problem is reduced to the Coin-Collector problem in a generalization of it known as the Package-Merge Algorithm. It is more efficient than a similar technique named for Shannon and Fano, and the selection of the locally smallest two nodes at each step makes it a Greedy Algorithm. Using a binary tree to create prefix-free codes so that the most frequently used symbol has the smallest code, for 10 points, name this type of lossless data compression named for an MIT Computer Scientist.
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A velocity-space generalization of this lemma is used to take moments of the Boltzmann equation, because it trivially shows that the velocity-space integral of the distribution function under the namesake operation in velocity space vanishes. A special case of the generalized Stokes theorem for a Hodge-starred exterior derivative, it can be used to reduce by one the dimensionality of the set over which a scalar field is integrated as long as that scalar field can be expressed as a vector field under the namesake operation. FTP, name this fundamental theorem from vector calculus; a multi-dimensional analogue of integration by parts sometimes called Guass' theorem and named for the operation symbolized by del-dot.
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According to Neumark's theorem, one of these realizable actions can always be constructed for a given positive value operator. The resolution of the consistency problem associated with this action(*) requires that it be irreversible. Repeating this process very frequently results in a "freezing" of the system state, a phenomenon known as the quantum Zeno effect. Mathematically, this process can be thought of as the forced projection of the system state onto an eigenstate of the Hamiltonian, and according to the Heisenberg uncertainty principle, this process cannot be performed simultaneously with arbitrary accuracy for non-commuting observables. For 10 points, identify this action which in von Neumann's formalism of quantum mechanics results in the collapse of the wave function to an energy eigenstate.
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Adiutori argues that this person underpinned all of modern science with his annunciation of the principle of dimensional homogeneity. This mathematician is the namesake of an n-by-n matrix whose jk entry is e to the two pi i j k over n. The heat diffusion equation assumes a constitutive relation formulated by this scientist in The Analytical Theory of Heat; that relation, which says that the heat flux is proportional to the negative gradient of the temperature, is his namesake law. e to the i k x is the kernel of his namesake transform, which is a continuum limit of his namesake series. FTP, name this French scientist and mathematician in whose namesake series functions are decomposed into sine and cosine components.
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Aitken developed the “General†form of this method which overcomes the “HAC†problem. X prime X inverse X prime y is a closed-form expression to calculate a parameter obtained by these techniques. Non-linear problems utilizing this method can be solved using the Gauss-Newton algorithms, and the Gauss-Markov theorem says that the parameter obtained by one type of this method is the best linear unbiased estimator. The “ordinary†form of this technique is frequently employed in regression analysis to find best-fit lines. For 10 points, identify this method which involves minimizing the second power of the difference between an estimate and an observation.
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All finite field extensions are also field extensions of this type, as they are generated by a finite number of elements of this type. Elements of a field are said to be of this type over a subfield if they are roots of a nonzero polynomial with coefficients in the subfield. Equations of this type involve only the four basic operations, taking powers, and taking roots. FTP, identify this term, which describes all real numbers that are not transcendental.
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Along with Stuart Hameroff, he developed a hypothesis connecting human consciousness to quantum gravity effects in microtubules. His namesake objects mark the plane aperiodically, and it was proved that the rhomb form of those objects is three-colorable. His two versions of the censorship hypothesis stated that naked singularities do not exist in nature. Edward Witten used one of his formulations to connect them to S-matrices and named it twistor string theory. His name is attached to a pseudoinverse operation first discovered by E. H. Moore. FTP name this British mathematician, best known for some tiles and for developing a hypothesis about black holes along with fellow-Brit Stephen Hawking.
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Among its properties are the sifting property and the integral property, which defines the Heaviside unit step function. Often used in conjunction with Green's function, which determines the impulse response of a differential equation, this function is sometimes known as the unit impulse function. Created by its namesake to deal with the completeness relation for position and momentum eigenstates, FTP, identify this equation of quantum mechanics that deals with the action of heat flow over a very small region and named after the scientist that predicted antimatter.
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An algorithm of this type, developed by Matsumoto and Nishimura, failed the TestU01 Crush but passed the Diehard Tests for them. One of the best algorithms of this type hinges on the difficulty of solving the quadratic residuosity problem, and, like RSA, involves the product of two large prime numbers, the Blum Blum Shub algorithm. Another of these algorithms, which generally strive to prevent backtracking, was developed for the ENIAC by Johnny von Neummann and operated by taking a subsection of a squared number and then feeding the result back into the algorithm. Besides the middle-square method, another example is the Mersenne Twister, and they are often necessary for providing input to Monte Carlo algorithms. Often accepting a seed, for 10 points, name these algorithms which strive to produce numbers without patterns.
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An approximation of this kind is due to Robbins and Murno and optimizations of this kind include the Nelder-Mead method and simulated annealing. Resonances by this name occur due to interactions of perterbative signals with noise, while matrices of this kind are square, non-singular, and have entries between 0 and 1 inclusive. The calculus by this name includes Weiner processes; processes of this kind are mappings from a probability space to a state space and are exemplified by Markov chains. FTP, name this type of mathematical entity, opposite to deterministic and synonymous with random.
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An element of an algebraic closure system has this property if and only if it is finitely generated. The Arzela-Ascoli theorem gives a set of criteria for determining whether a subset of the set of continuous functions between two metric spaces has this property, and Tychonoff's theorem states that the topological product of any number of spaces with this property will also have it. A set is said to have this property if every open cover contains a finite subcover. For 10 points, identify this topological property that, when applied to subsets of Rn according to the Heine-Borel theorem, is equivalent to being closed and bounded.
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An example from the C++ language is defining the "less than less than" operator on an output stream to represent a data transfer --- not to represent a bitwise shift, as is the case when the operator is used with integers. For ten points, identify this practice of redefining standard operators to act on newly defined data types.
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An important 1968 paper illustrated how an integer-preserving form of this procedure may be implemented using the Sylvester identity. This procedure has the unfortunate property of instability unless pivoting is included in its operations. Originally mentioned by Liu Hui in The Nine Chapters on the Mathematical Art, it may be used to invert a matrix by augmenting with the identity, then reducing the unaugmented portion to the identity. It may generally be used to solve a matrix equation by appending augmenting the coefficient matrix by the inhomogeneous vector. FTP, name this algorithm for solving systems of linear equations named for one or two German mathematicians.
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An integral domain is a ring with no zero divisors that has this property. Division algebras where this holds are fields. The symmetric group S3 is the smallest group where this doesn't hold, and all cyclic groups have this property. Unlike addition and multiplication in fields, groups do not necessarily have this property, and when this holds for a group it is called abelian. Matrix multiplication does not have this property nor does subtraction of real numbers which has the anti-form of this property, but it holds for a related operation. For 10 points, name this property that when applied to addition means that A + B = B + A.
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Anders Angstrom employed this man's namesake resonance theory to conclude that an incandescent gas emits light of the same frequency which it absorbs. He gives his name to a set of polynomials which are functions of the Bernoulli numbers, and to a function which counts all the positive integers less than and relatively prime to the argument. The general equations of motion can be derived from a differential equation named for this man and Langrange. His namesake formula relates trigonometric functions to complex exponentials. FTP, name this mathematician whose namesake constant is the base of the natural logarithm.
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Any non-orientable surface contains one. It contains one boundary curve and cannot be embedded in the real plane. A torus can be cut into one with an even number of half-twists and a Klein bottle cut in half makes two of these. B.F. Goodrich supposedly patented a conveyor belt in, for ten points, what form, theorizing that its one-sidedness would ensure even wear.
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Applications of this type of data structure include parsing mathematical expressions and calling subroutines from parent programs, but NOT reading from an input stream or running processes in the order in which they were called. It can be implemented in order-one time in real life as a to-do box which is open at only one end, or in a program by a singly-linked list, since pushing and popping can all be done at the head. FTP, name this data structure which uses a last-in, first-out system.
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Applied to Fourier analysis, this technique minimizes the norm of a function and its Fourier series approximation. Trend analysis is an application of this method, whose general model involves writing the predictions as the sum of the observations matrix times the parameters, plus an error vector. The solution is a set of parameters for a curve assuming the observation matrix is fixed, and the predictions have a Gaussian distribution. Weighted and multiple regression varieties generalize this curve-fitting technique. FTP name this technique which minimizes the magnitude of residuals about a line, a namesake type of fitting.
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Applying an inversion to this figure gives the Dupin cyclide, and the systole and area of one of these figures is related by Loewner's inequality. The Weyl group can be defined as the normalizer of a group isomorphic to one of these figures, modulo the centralizer of said group. A compact Lie Group has a maximal one of these, and the standard one of these requires seven colors to color according to the Heawood conjecture. Double and triple ones of these have more than one handle, and N-dimensional ones are the direct product of a circle with itself n times. FTP, name this shape important in topology which is homeomorphic to the coffee cup and is shaped like a donut.
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Arising from the calculus of variations, the long way to solve it is to use the Euler-Lagrange differential equation, but one can use the Beltrami identity for a shortcut. The parametric equations are identical to its solution and Newton's Second Law comes into play if one considers the effect of kinetic friction. Greek for "shortest time," this famous problem was posed in the publication Acta Eruditorum by Johann Bernoulli and has a cycloid as its solution. FTP, name this classic problem that seeks to find the minimal time it takes for a body under the force of gravity to get from one point to another.
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As a young adult, he influenced the founding of the Turin Academy of Sciences and he was later director of mathematics at the Berlin Academy of Sciences. During the French Revolution he headed a committee to come up with a new system of weights and measures. He studied isoperimetrics, of which he formed an algorithm better than that of Euler. FTP, name this author of Mécanique analytique best known for his multipliers and gravitational points.
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At a banquet celebrating the 1831 acquittal of the "the nineteen," he offered a toast "To Louis-Philippe" while brandishing a jackknife; Alexander Dumas, hoping to avoid the scandal of an implicit assassination threat, escaped through an open window. Of his twice-lost magnum opus, Poisson claimed that its central idea "must have an external character" testable by "coefficients or... equations of a lesser degree." Unwittingly duplicating Abel's result, he proved that the general quintic equation could not be solved by radicals, and more generally that the automorphism group of an equation can be solved by radicals if and only if the equation can. FTP, name this mathematician who laid the groundwork for abstract algebra before dying in a duel in Paris at the age of 20.
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At the age of 28, he solved the famous Basel problem, and he discovered the law of quadratic reciprocity. His contributions to number theory include the invention of the totient function, which gives the number of positive integers less than and coprime to a positive integer n. He developed analytic number theory, where he developed the theory of hyperbolic functions, and he is also considered the father of graph theory with the solving of the famous Königsberg Bridge problem. FTP, name this most prolific mathematician in history, whose famous namesake formula connects complex numbers and trigonometry, and who was the namesake for the symbol of the natural log base.
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Between 1842 and 1846, he helped publicize Galois' as yet unrecognized works, by publishing them in a journal he founded. Most of his own work was in number theory, where he worked on quadratic reciprocity. The arithmetic function that bears his name is negative 1 at "n" if "n" is the product of an odd number of primes, and 1 otherwise. However, his more famous results are analytic, where he showed that any bounded complex function that is differentiable in the whole complex plane must be constant. FTP, name this French mathematician whose work on continued fractions led to the first proofs that certain numbers are transcendental.
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Brocard's problem asks which outputs of this function are one less than a perfect square, and Christian Kramp was the first to use the notation commonly used to denote this function. By Stirling's approximation, the natural log of this function on n is equal to n natural log of n all minus n. It is extended to all the complex numbers except the non-positive integers by the gamma function, and the Taylor series of e to the x features this function acting in the denominator of the polynomial terms. This function can be defined recursively as n times this function for n minus one. For 10 points, name this function denoted by an exclamation mark.
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By a special case of the projection-slice theorem, performing the Hankel transform is equivalent to performing this man's transform for axially symmetric functions along with Fourier's. The absolute value of the inner product of two vectors is bounded by an inequality named for this man, who sometimes names the formula for summation by parts. The equation of titration is derived from this man's equation, which equates f(h(x)) with f(x+1), but he is better known for an identity linking the Wronskian of a second order differential equation to the equation itself and for an identity with Ruffini. Better known for proving that the general form of a quintic polynomial is insolvable, for 10 points, name this mathematician who names groups which are commutative, a Norwegian.
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By using these to construct a branched cover, Ozsvath and Szabo construct invariants of them. Up to an equivalence given by the Markov moves, these can be represented by closed braids. Classes of them include satellite, torus, and alternating; the last is the case in which the crossings in some projection are alternately over and under. FTP identify the mathematical objects defined as closed, non-self-intersecting curves embedded in 3-space.
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Complexity class BPP problems are solved by these methods in polynomial time. Antithetic variates and control variates are common variance reduction techniques for its estimates, while low-discrepancy sequences are used in their "quasi" form. Lazzarini's choice of stopping time and needle lengths when attempting one named for Buffon gave an excessively accurate estimate of pi in just 3,408 tosses. The Gibbs sampler and the more general Metropolis-Hastings algorithm for calculating high-dimensional integrals, and random walk simulation of binomial options pricing models, are examples that make use of Markov chains. For 10 points, name these methods that can be converted into Las Vegas methods by continuing simulation until a sufficiently correct result is produced, named after a city in Monaco.
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Computationally equivalent to determining if a number is a quadratic residue modulo a prime, among the algorithms proposed to solve it are an elliptic curve method named for Lenstra. Pollard's rho algorithm for achieving this iterates a polynomial formula until it falls into a cycle, while Fermat's method uses trials to find an appropriate difference of two squares and generalizes to the continued fraction method. A fast method for it is the quadratic sieve, and it can be done in polynomial time by the quantum-computational Shor's algorithm. For ten points, identify this problem in NP and co-NP, whose difficulty insures the success of RSA encryption, and which consists of breaking an integer into a product of primes.
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Created as a solution to the Entscheidungsproblem, this theoretical construct consists of a head and a tape. The head may or may not alter its internal state and may or may not overwrite a symbol on the tape. Depending on the input, the head performs an operation such as moving to the right or left or halting the tape. FTP name this device, the basis for modern computers.
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Curry and Howard demonstrated its correspondence to intuitionistic logic and any simply-typed version of it can be interpreted in a Cartesian closed category. System F was a typed version of this system which added parametric polymorphism. Languages based on it may encounter both the upwards and downwards funarg problems. Its expressions are either variables, abstractions or applications. It can be implemented in Eiffel or Python, but it has the most similarity to functional programming languages such as Scheme and LISP. Equivalent to a Turing machine, FTP identify this system of formal computation developed by Kleene and Church, which is named for a Greek letter.
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Derived from the midpoint rule it should be chosen over its less-complicated analogue when the curvature of the function of "x" is high. Not very similar to its namesake's 3/8 rule, it approximates a quadratic as opposed to a straight line within each subinterval, differentiating it from the trapezoidal rule. Requiring the integer to be even, FTP, name this method for approximating integrals.
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Discovered in 1761 by its namesake, it was brought into its modern form by Laplace. It is not a Punnett Square, but is often used to approximate parental genotypes from available family data. Properly understood, it is the fundamental mathematical law governing logical inference--determining what degree of confidence one would have in a hypothesis based on the body of evidence available. FTP, name this theorem that allows one to apply quantitative reasoning to the scientific method, the fundamental theorem of inverse probability.
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Dragoon is a statically typed one, Python dynamically typed, and both typings are included in Eiffel. In these languages, self-contained collections of procedures and data structures can be used in new programs by assembling a set of the predefined structures. For ten points, name this class of programming languages which includes C++ and Java and is often abbreviated OO.
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Fermat accomplished the quadrature of every curve in its class except this one, for in this case, his formula yields division by zero. Gregorie de St. Vincent succeeded, and one of his students used the logarithm to compute the area under it. For ten points, name this curve, the only non- degenerate disconnected conic section, an example of which is the function "f of x equals 1 over x".
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Formulated by Lotfi Zadeh in 1965, it's employed in robotic control and pattern recognition software. Rejecting the law of the excluded middle, it instead asserts that variables --- tallness for instance --- have levels of truthfulness and uses "if x and y then z" rules to plan the system response. For ten points, name this type of logic whose name describes its imprecise view of quantities.
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Goldbach proved that no two of them share a common factor. It has been shown that an n-sided regular polygon can be constructed with a compass and straightedge if and only if n is a power of two or the product of a power of two and two distinct prime members of this sequence. As such, there are only five known prime ones, despite their namesake's assertion that all of them are prime. However, only the first twelve have been completely factored, and it is not known whether there are infinitely many composite ones. FTP, name this sequence of integers of the form 2 to the quantity (2 to the n) + 1, which bears the name of an amateur mathematician with well-known last and little theorems.
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Hadwiger's theorem characterizes this quantity as the measure for the finite union of compact convex sets that is homogeneous degree zero. For a chain complex, it is defined as the alternating sum of the ranks of the homology groups, and can be defined as the alternating sum of Betti numbers. The this quantity is proportional to the integral of the curvature of a closed Riemannian manifold according to the Gauss-Bonnet theorem. The only closed compact surfaces for which it is zero are the torus and the Klein bottle, and its namesake's formula states that it is equal to two for the surface of any convex polyhedron. For 10 points, name this topological invariant, denoted by the letter chi, which was defined by its Swiss namesake as vertices minus edges plus faces.
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Hausdorff's generalization of this statement relates it directly to the inaccessible cardinals axiom, which was used by Solovay to solve the related Lebesgue [luh-BEG] measurability problem. Freiling's axiom of symmetry or Woodin's omega-logical axiom imply its inconsistency with the common framework in which it arises. Cohen's 1966 work on Set Theory and this introduced the method of forcing to show that no contradiction is introduced by including its negation with the ZF axioms and the axiom of choice, thus showing its truth to be undecidable by normal Zermelo-Frankel set theory. FTP, name this subtle axiom of set theory that states that the power of the reals is aleph-one.
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He argued that the Royal Society was ineffective because it was controlled by a small clique in his Reflections on the Decline of Science in England, which led to the founding of the British Association for the Advancement of Science. After touring factories on the Continent he wrote On the Economy of Machinery and Manufactures, but he spent most of his life working on two machines of his own, one of which was meant to integrate difference equations. FTP, name this English thinker who failed to create a machine which could change in response to its own calculations, the so-called "analytical engine" which is regarded as the forerunner of modern computers.
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He found that the coordinates of a point on any algebraic curve can be expressed in terms of automorphic functions. Making use of Lobachevsky's non-Euclidean geometry, he issued pioneering studies of the geometric properties of functions defined by differential equations. His work on solving the orbiting three body problem led to the development of chaos theory, and he was heavily involved in the development of special relativity, introducing the notation used for Minkowski space. He is best remembered, though, for a statement about a three-dimensional surface that contains no holes, the solution to which is a Millenium Problem. FTP name this man whose conjecture says that every simply connected, closed three-manifold is homeomorphic to the three-sphere.
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He invented the discipline of proof theory in an attempt to formalize all of mathematics. His interest in formalism is evident in his earliest published work, where his proof of his Basis Theorem was criticized for being too purely abstract. In addition to making modern algebraic geometry possible with his Basis Theorem and Nullstellensatz, he properly rigorized Euclid's axioms for plane geometry and showed how various non-Euclidean geometries followed from different modifications of these axioms. FTP, name this mathematician whose address at the 1900 International Congress of Mathematicians stimulated much of the work of the 20th century.
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He lends his name to the eigenvalue equation for the Laplacian, to a type of resonator which works like thebody of a guitar, and to a pair of coils which creates a uniform magnetic field. His work on the propertiesof vortices led Lord Kelvin to theorize that ether is the only substance in the cosmos, and his interest in the senses led to his invention of the ophthalmoscope and to his influential 1863 book On the Sensation ofTone. FTP, name this scientist who identified the quantity "internal energy minus temperature timesentropy," which is denoted by a capital F and referred to as his "free energy."
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He told his most famous student not to study too much math, lest it interfere with her child birthing abilities. In 1838, he presented one of the first clear explanations of mathematical induction, a term which he coined, but his real contributions lie in the field of symbolic logic, where he invented notations that help prove prepositional equivalences. FTP, name this teacher of Ada Lovelace, one of whose two namesake laws states that the negation of the conjunction of two propositions is logically equivalent to the disjunction of their negations.
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Helmholtz separability depends on one of these due to Stäckel. These mathematical objects have a property known as n-linearity. There is a Sylvester identity and Cauchy theorem about them. A version of them is applicable to quadratic forms. The cross product can be generalized to n dimensions if it is treated as one of these operations performed on n minus one n-vectors, a fact often exploited in 3-space. Ratios of these functions can be used solve linear equations by Cramer's Rule. If this vanishes for a matrix, the matrix is called singular. FTP, what is this matrix function that, for a 2-by-2 matrix a b c d, is ad minus bc?
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His name denotes a problem which has been solved by Allan Gibbard and Simon Blackburn; also known as the "embedding problem," and co-named for Geach, it revolved around the phrase "telling lies is wrong." He also coined the "Julius Caesar problem," which states that Hume's Principle cannot distinguish between the number of Roman emperors and Caesar himself. In one work, he distinguishes between a trivial truth and a significant philosophical distinction and puts forth the idea of bedeutung as an aspect of meaning. For 10 points, name this author of Concept Script and "On Sense and Reference," who created a formal system of modern logic.
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Hurwitz's Theorem gives the bounds for the best approximation to an arbitrary one of these. Ones that have periodic continued fractions are known as quadratic surds, and Rivoal proved that the form gamma of 2n+1 can generate infinitely many of them. Topologically, the set of them is a Baire space but is not locally compact. The Erdos-Borwein constant is this, as is Gelfond's constant, but this property is not known for pi to the e power or for the Euler-Mascheroni constant. FTP name these numbers that were feared by the Pythagoreans and that are non-terminating and non-repeating
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Identities applicable to these are due to Gelin and Cesàro. d'Ocagne, and Catalan. They are a special case of the Lucas sequence, terms in which obey the same recurrence relation as these numbers. The ratio of successive ones of these converges to four over quantity one plus root five squared, which is one over the golden ratio squared. They are the sums of successive diagonals of Pascal's triangle and each one is equal to the sum of the two before it. FTP, name this set of numbers originally devised by considering the multiplication of rabbits and which are named for a Pisan mathematician.
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If a continuous function satisfies the Lipschitz condition, Picard's theorem applies to this type of expression involving that function. Numerically, they can be solved using the collocation or the Galerkin method, or by the more robust Runge-Kutta method. Second order linear ones are studied in Sturm-Liouville theory, while first-order ones can be solved exactly with the integrating factor method. One of these, with the form x-double-dot equals minus omega squared times x, governs the motion of harmonic oscillators. FTP, name these mathematical expressions which involve functions of one variable and their derivatives.
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If you multiply two of these values and find the Euler totient function of the product, you can create public and private keys symbolized d and e. Whether a given number is one of these values is tested using an algorithm named for Miller and Rabin. One naïve test for these values, named for Fermat, gives false positives called Carmichael numbers. The difficulty of factoring composite numbers composed of two of these numbers is the basis of RSA encryption. For 10 points, name these non-composite numbers with only two divisors, one and themselves.
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In 1799 an Italian built upon the work of Joseph Lagrange and used permutation groups to approach this mathematical result. Despite this one can achieve good numerical results, for instance with Newton-Raphson methods. It shows that an analog of Cardano's formula cannot exist in some cases. In 1824, it was shown that a certain group did not have a composition series with Abelian quotients, and hence was not solvable. The group of interest is called a Galois group. FTP, name this algebraic result proved by Niels Abel which demonstrates the inability to generalize the exact forms of solutions for polynomials above a certain degree.
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In 1999, McIlroy created an adversary for this algorithm that guarantees that it will run in worst-case time. Its worst-case runtime can be avoided by switching to heapsort after a certain recursion depth, a construction known as introsort. Like mergesort, it is easily parallelizable, and its runtime can be decreased by first selecting the median of the unsorted input list. It was invented by C.A.R. Hoare, and its second phase is the partition function, which splits the original list into lists of elements that are greater or less than the chosen pivot value. For 10 points, identify this divide-and-conquer sorting algorithm which runs in big O of n log n time, named for its speed.
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In 2000, an Indian mathematician used Eilenberg modules, Hall matchings, Riemann surfaces, Steiner systems, and a map of Madhya Pradesh to prove this theorem. In 1976 Appel and Haken pioneered the use of the computer in proofs in proving this conjecture. First proposed by Mobius in 1840, FTP, name this mathematical theorem that states that no more than a certain number of colors are needed to fill in a map.
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In April 2002, M. J. Dunwoody announced that he had produced a five-page proof of this famous mathematical problem included on the Clay Mathematics Institute's list of million dollar problems. Trivial for the case n=1, the n=2 case is a classic proof. The n=4 case won Freedman the Fields medal in 1986, and Smale proved it true for all n greater than or equal to 5. Stating that every compact n-manifold is homotopy-equivalent to the n-sphere if and only if it is homeomorphic to the n-sphere, FTP, identify this conjecture which takes its name from the student of Charles Hermite who originally proposed it.
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In Matlab, it is the percent sign. In FORTRAN, the letter C. In Pascal, enclosure between parentheses and asterisks. In Perl, the pound sign. In C++, double forward slash, or, as in C, enclosure between forward slashes and asterisks. FTP, what are these symbols in their respective languages?
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In Perl, the efficiency of a method for this process can be improved with a Schwartzian transform, which grew out of the Lisp idiom of decorate, do this, undecorate. David Cohen and William Gates both wrote papers about a version of this process that cannot be done on a von Neumann machine in constant time, known as the pancake type, while Knuth shuffle is used in the purposefully bad bogo type. An algorithm for this task is stable if it preserves the order of two equal elements, and if the range of the input elements is known, the counting or bucket varieties can be used. For 10 points, bubble and quick also name types of what task of arranging elements in a given array?
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In a regular Sturm-Liouville problem, the spectrum of these numbers is discrete and semi-infinite. There is a conjecture that, for some positive operator, the zeroes of the Riemann zeta function are in fact this type of value, whose magnitudes can be bound by the size of matrix elements using Schur's inequalities. Symbolized lambda, they can be found by setting to zero the determinant of the matrix minus lambda times the identity matrix or by finding the roots of the* characteristic equation. FTP, what are these values that reveal by how much matrix multiplication changes their corresponding eigenvectors?
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In contrast to the internal, or L1 type, the L2 type sits between the CPU and DRAM, but both L1 and L2 are made of high-speed SRAM for the memory type. This disk type actually uses main memory, but stores information in a memory buffer. The original Pentium processor had a 16 kilobyte one of these built into it. FTP, name this high-speed storage mechanism that keeps frequently accessed data in a location separate from the main memory for rapid recall.
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In group theory, his theorem states that any finite group whose order is divisible by a prime p contains an element of order p. He is known to students of complex analysis for his theorem stating that the integral over a closed contour of any holomorphic function without interior singularities is zero. Perhaps this Frenchman's most lasting result holds that the inner product of two vectors is less than or equal to the product of their lengths. FTP, who is this discoverer of a famous inequality named for he and Schwartz?
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In his youth he always carried a copy of Disquestiones Arithmaticae with him. Along with Legendre he gave the first proof of Fermat's Last theorem in the case of n equals 5. For his corrections of Cauchy, he is credited with developing the field of Fourier analysis. In 1837, he developed the modern definition of a function. FTP, name this successor to Gauss at Göttingen whose name is also sometimes given in association with the pigeonhole principle and whose namesake theorem deals with primes in arithmetic progression.
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In hyperbolic geometry, two figures with the “ultra†form of this property must share a perpendicular, and two geodesics with this property with respect to a third can intersect. Given a line and a point not on the line, Playfair's axiom states that only one figure with this property can be drawn. If two lines intersect a third line, and the interior angles on one side sum to less than 180 degrees, the lines will not have this property, according to Euclid's fifth postulate. For 10 points, name this term that identifies lines in a plane that do not intersect.
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In iron meteorites arranged in this kind of structure, treatment with weak acid can produce visible bands of kamacite bordered by taenite, called the Widmanstätten pattern. Pollock conjectured that every number is the sum of at most seven of this kind of number, which is defined as the sum of two consecutive pyramidal numbers. The gyrobifastigium [GUY-roh-bye-fass-TIH-gee-um] and tridiminished icosahedron have the same number of faces as this solid, which Plato associated with air and which has six vertices and twelve edges. FTP, give this term which is the most common structure and which is more frequently seen than trigonal prismatic, a polyhedron with eight faces.
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In one-space, these are generally continuous but have kinks where the source and field points coincide. These distributions may be used to create Friedholm integral equations from PDE's since their spatial convolution with the density function is equal to the unknown function, and those associated with self-adjoint operators have the useful property of symmetry, so that the source and field vectors may be interchanged. FTP, name these generalized functions equal to the Dirac delta function under the operation of an associated operator and derived from their namesake's second identity.
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In polar coordinates it is expressed by the equation "p equals a times 1 plus the cosine of theta," and its area can be found by "1.5 pi a squared." In the point-wise construction of it, let "O" be a fixed point of a circle "C" of diameter "a," and "Q" a variable point of "C." Lay off distance "a" along the secant "OQ," in both directions from "Q," and the locus of the two points thus obtained is this figure. Generated by a point of a circle that rolls without slipping on a fixed circle of the same diameter, FTP, name this curve that appears heart-shaped.
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In statistics, a restriction method which uses these objects minimizes a residual sum of squares which assesses a penalty for a function's curvature. When using the "periodic" type, the values of a function and its first two derivatives are forced to be equal at its endpoints, while using the more common "natural" type requires that the second derivative at the endpoints be zero. It is named in part for a strip of wood with fixed points used by draftsmen. FTP, name this type of curve-fitting technique commonly used to render smooth 3-D graphics, in which functions are approximated using piecewise third-degree polynomials.
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In the 13th chapter of this man's most famous book, we encounter the problems of the "four men who found a purse," the "five men who bought a horse," and a tree which lies partly underground. Those problems can be solved with the method "elchataym," which is also known as the method of double false position. Other chapters of his book discuss the alloying of money, the barter of commercial things, and companies made among parties, though it is better known for discussions of the "finding of square and cube roots" and the "multiplication of whole numbers." In the first chapter of his book, he explains how to use the "nine Indian figures" to write all numbers. This businessman (*) from Bugia wrote The Book of Squares, but his major work was a 1202 book that introduced algebra to Italy. FTP, name this author of the Liber Abaci, a mathematician best known for a namesake sequence which begins with the numbers 1, 1, and 2.
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Ingham proved it using Ramanujan's identity, and it is equivalent to the statement that the limit as x goes to infinity of theta of x over x or psi of x over x is one, where theta and psi are the Chebyshev functions. Hadamard's proof depends on the fact that the quantity one plus cosine theta squared is greater than or equal to zero, and follows from the fact that the Riemann zeta function has no roots of the form one plus i times t, where i is the square root of minus one. FTP, name this theorem which gives x over natural log x as an approximation to the number of non-composite integers less than or equal to x, which takes its name from those non-compposite integers.
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Integral to RSA encryption and provable via the Fundamental Theorem of Algebra, for twenty-five hundred its value is one thousand, for one thousand it is four hundred, for twelve it is four, for three it is two, and for one it is one. It can be given by the theorem that if p is prime and n is any positive integer, then this is p to the power of n minus one times p minus one. Defined as the number of positive integers less than or equal to a given number which are also co-prime to it, FTP, name this function developed by Euler.
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Internet Explorer renders it as a Cascade Style Sheet and Netscape is still experimenting with it. Developed by the W3C, it is a pared-down version of SGML, designed especially for Web documents. It allows designers to create their own customized tags, enabling the definition, transmission, validation, and interpretation of data between applications and between organizations. FTP identify this programming language, possibly the successor of HTML as the standard Web formatting specification if it is supported by future Web browsers.
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Invented by Tony Hoare in 1961, its advantages include its ability to operate in place and the velocity of its inner loop. Its effectiveness can be greatly increased by using various strategies to choose an efficient pivot. After a pivot is selected, all numbers less than the pivot are placed to the left, and all numbers larger than the pivot are placed to the right, with the algorithm then implemented recursively on the smaller lists. FTP, what is this computer science sorting algorithm named for its purported speed?
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It can be topologically abstracted as a connected graph with four vertices and seven edges, two pairs of which connect the same two sets of points. The fact that it had no solution was achieved by proving that no connected graph with more than two "odd" vertices could have a circuit that traverses every edge and concludes at its starting point without crossing any edge twice. It was the original conundrum that led to Euler's formulation of graph theory. FTP, identify this classic problem inspired by the crossings of the Pregel River in the namesake Prussian city.
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It contains such public member functions as preorder, inorder, and postorder traversals, which call its own recursive utility functions to perform the appropriate operations on the internal representation. This nonlinear, two dimensional data structure picks a root value and orders subsequent values either to a right branch or left branch depending on that value's relation to the root. FTP, identify this ordering mechanism from computer science with an arboreal name.
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It follows for large exponents if the Bogomolov-Miyaoki-Yau inequality is true, while if certain conditions are met by Wieferich primes the ABC conjecture proves it as well. A key idea in its proof was the replacement of elliptic curves with Galois representations, circumventing a clash between Selmer Groups and the Taniyama-Shimura conjecture, thus allowing for the use of the class formula. Proven by Andrew Wiles, FTP, what is this theorem stating that there are no integer solutions for the equation x to the n plus y to the n equals z to the n when n is greater than 2, famously proposed in the margins of a book by its namesake?
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It is built from 46 definitions of recursive functions, the last labeled "bew," short for beweisbar. Atomic elements are represented as odd integers, and the first n prime numbers are raised to the appropriate power to encode statements. The "bew" function, however, returns 1 if the input is a theorem consistent with Principia Mathematica or loops forever if it isn't. It gives, however, a negative result to Hilbert's tenth problem by affirming that the system of Whitehead and Russell -- or any comparable logical system -- is inconsistent. Stating that a system powerful enough to describe arithmetic cannot be both consistent and complete, for ten points, identify this theorem, proven in 1934 by an Austrian logician.
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It often arises near transverse homoclinic points, as in the Smale horseshoe. A Poincaré section or first-return map for this type of dynamics will exhibit a fractal structure. In examples such as the logistic map, as a parameter is increased a sequence of period doubling bifurcations characterized by the Feigenbaum constant leads to its onset. It is characterized by positive Lyapunov exponents, meaning that small errors in initial data grow quickly and make practical long-term predictions impossible. Devaney specified three requirements for it: topological transitivity, dense periodic orbits, and sensitivity to initial conditions. For 10 points, what is this physical phenomenon, exemplified by the weather, in which a deterministic system exhibits apparently random behavior?
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Matiyasevich showed that these can be defined by a Diophantine equation, leading to a proof of the insolubility of Hilbert's tenth problem. They have a generating function of "x over quantity 1 minus x minus x squared" and Euclid's algorithm for the greatest common factors has its worst run time when the entries are successive terms of these numbers, so, in that sense, the most irrational number is the golden mean, to which the ratio of successive members of these converges. FTP, name this sequence created to model multiplication of rabbits and having as its first few terms 1, 1, 2, 3, 5.
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Matsumoto and Nishimura developed a method of generating random numbers using these numbers in their namesake twister. These numbers were first studied due to their one-to-one correspondence with even perfect numbers. As of August of this year, there are only 42 known ones, and they are currently being searched for by the GIMPS project, which makes use of the Lucas-Lehmer algorithm. FTP, identify these special primes given by 2 to the power p minus one, where p itself is prime, and which take their name from the French mathematician Marin.
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Napoleon's theorem concerns the use of four of these figures to construct a fifth with special properties, and a similar method is used to locate the Fermat point for this shape. Nine significant locations on it are traversed by the nine-point circle, the center of which lies on an Euler line in this shape. Ceva's theorem concerning this shape states that lines drawn from a vertex to a side meet under certain conditions, and another statement about it allows its area to be determined from side length and semiperimeter alone. For 10 points, name this simple shape, whose area is calculable by Heron's formula and whose angles add up to 180 degrees.
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Novikov proved that the word problem for these objects is sometimes undecidable. Special types of these are called periodic or torsion ones, which are the subject of Burnside's problem. The Cayley graph is a method of representing these, and another one of these can be represented as a 194-by-194 character table, the Fischer-Greiss Monster. They may act on a set, usually as what may be called "symmetries" of that set, and functions from one to another that preserve their structure are called homomorphisms. Characterized by closure, associativity, and inverses, those that are commutative are also called abelian. FTP, name these algebraic structures, distinct from rings and fields.
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Official rounding modes for these include unbiased, towards zero, towards infinity and towards negative infinity. Some examples of these are known as normalized and denormalized numbers. Using these, multiplying infinity by zero results in NaN, or not a number. The IEEE 754 standard for these specifies them in single- and double-precision forms among others. They are specified by a sign bit, some exponent bits and some fraction bits. FTP, name these approximate representations of real numbers used in many computers, so named because the decimal place is not fixed.
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One form of this, due to Lindeberg and Feller, applies to the difference of variates. While it does not hold for general stochastic processes, there is a version for martingales. The rate behind this relation is determined by the Berry-Esséen theorem. It holds generally under the Lyapunov condition, but the conventional statement of it requires that the variance be finite and that the mean of the sampling distribution of means be the population mean. FTP, identify this theorem stating that the distribution of a sample of independent random variables approaches a normal distribution with sample size.
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One instance of this problem occurs with the soldering of parts to a circuit board by a robot. It can be solved using the elastic net method, where a circular route is mapped to the plane and iteratively stretched out towards all points on the plane to define a route. The complexity still rises exponentially with the number of points or "cities"in the problem. FTP, name this notorious NP-complete problem that is concerned with traversing all nodes in a network while keeping the path length to a minimum.
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One unproven generalization of this theorem is equivalent to the statement that chi of G is equal to or greater than the Hadwiger number of G. It is equivalent to the statement that no snark is planar. Another generalization of this theorem was proven by Ringel and Youngs and depends on the floor of an expression involving the square root of 1 plus 48g. It was proven by finding an unavoidable set of reducible configurations, using the method of discharging; a later proof improved on the quartic time algorithm used. It is a special case of the Heawood conjecture. For 10 points, name this theorem proved by Appel and Hankin via computer, which gives the minimum number of shades necessary to produce a map.
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Performing one type of this process requires the use of a "burn-in," (*) and several adaptive or recursive methods named after it involve importance sampling. The error term in the "integration" named after it decreases as one over square root of N and bootstrapping yields estimates of parameters in another type of this process. Peter Lepage invented a version of this technique that applies to particle physics which involves constructing a separable multi-dimensional weight function. Including the VEGAS algorithm and simulated annealing approaches such as the Metropolis-Hastings algorithm, this type of technique includes is stochastic, contrasting it with deterministic algorithms such as molecular dynamics. For 10 points, give the general term for these methods of Markov Chain, integration, or sampling, which generally involves randomly selecting a whole bunch of points and is named for a European gambling mecca.
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Prefetching in LDS can be accomplished using the jump type of these objects. A dispatch table consists of function ones. C allows arithmetic with these objects, which are used for the internal representation of arrays. When objects in memory are deallocated, these can dangle. A segmentation fault arises when these objects do not correspond to a valid location. Each node in a linked list contains two of these data types, one of which can be dereferenced to return the next node. References are safer ways to handle some of the functionality of these objects. For 10 point, name this type of variable which stores a memory address of another variable.
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Procedures of this type can provide polynomial time solutions to such problems as finding optimal places to insert line breaks in a body of text, finding the largest common substring of two strings, and the integer knapsack problem. They can provide vast improvements over exhaustive search algorithms by caching the answers to intermediate computations. FTP, name this type of algorithm that generally uses recurrence relations and the principle of optimality to solve optimization problems by first solving related subproblems, and which often works where greedy algorithms fail.
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Proving the namesake property in one of these data structures relies on the Dynamic Finger Conjecture, and data structures without their characteristic property can obtain it by running an algorithm named for Day, Stout and Warren. In one of these data structures, their characteristic property is maintained through operations like a zig-zag during searches. Because they are often difficult to implement, skip lists are sometimes used instead. Data structures without this property are called degenerate and approximate linked lists. Some of these data structures impose requirements that all nodes and leaves be the same color. Exemplified by AVL and Red-Black Trees, for 10 points, name these special types of binary search trees that often perform tree rotations to guarantee tree heights of at most log n.
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Purdue University's Louis De Branges de Bourcia claimed to have a proof for this in June 2004. It initially began with Euler's definition of the related function, which he evaluated only at real numbers and which he investigated at Bernoulli numbers to show that the sum of the squares of the inverses of the integers equals pi over two. Its namesake extended that function, the zeta function, over the complex plane and made, FTP, what hypothesis that all the non-trivial zeros of this function would lie along the critical line, having real part equal to one half?
give this seven-letter keyword from C++, which can also be refer to a type of "memory" in computers.
Pure functions of this sort give rise to abstract base classes that cannot be instantiated. Non-pure ones may be overridden within subclasses to provide polymorphism. It can also indicate whether multiple inheritance gives rise to duplicates of shared ancestor classes. For 10 points
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Recently, experimental realizations of this model have been achieved in optical lattices, first a bosonic analogue and later the original fermionic version in work of Köhl et al. An important limiting case was derived by Spalek in 1977, applying a projection operator to remove double-occupancy states. This is a large U limit of it, called the t-J model. It was introduced by its namesake in 1963 in a study that revealed a type of metal-insulator transition. Its Hamiltonian involves a hopping term that moves an electron from one site to another, and a potential energy term giving on-site Coulomb repulsion. For 10 points, what is this model of interacting electrons, introduced in 1963 by a British physicist who also invented a transformation named for him and Stratonovich?
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Related forms of this curve are described as curtate and prolate, but in the standard type the cusps are separated by 2 times pi times the radius. An early success of the calculus of variations was proof that it's the solution to the brachistochrone problem. For ten points, name this curve formed by a point on a circle rolling on a straight line.
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Related to the Baire category, Martin's axiom is a weaker version, but it was Gödel that demonstrated that it was compatible with ZFC theory while Paul Cohen proved that it is undecidable in ZFC. Originating with the diagonal argument of its main advocate, Georg Cantor, it can be stated in both "cardinal numbers" and "sets of reals" versions. The first of Hilbert's twenty-three unsolved problems, FTP, name this doubtful set theorem that states that any uncountable set of real numbers has the same cardinality as the entire set of real numbers.
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She retired from the Navy at the age of 80, with the rank of Rear Admiral, in order to join Digital Equipment Corporation. Influential in the development of COBOL, she programmed the early Mark I computer to solve complex partial differential equations. FTP, name this woman who discovered the first computer bug.
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Smith and Price called it "Hawk-Dove" in the 1973 Nature article "The logic of animal conflict." Bertrand Russell likened U.S. foreign policy under John Foster Dulles to this "sport...practiced by some youthful degenerates." It has multiple Nash equilibria because there is no dominant strategy, and it is similar to the Prisoner's Dilemma in that a smaller-payoff compromise solution is threatened by a tantalizing larger reward. The worst possible outcome occurs when neither side defects from this game, the most famous example of which involves two cars, either rushing towards each other or a cliff. FTP, name this game theory scenario in which the first to swerve loses.
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Sophie Kowalevski completed a theorem involving local existence and uniqueness of analytical partial differential equations initially proven in a special case by this man, whose name, along with that of Binet, is also appended to a formula generalizing the multiplicativity of the determinants of two matrices. If the partial sums of an infinite series become arbitrarily close to each other, said infinite series converges under a test named after this man, who is also the namesake of a set of differential equations for finding holomorphic functions in the complex plane with Riemann, as well as a theorem stating that the closed line integral of holomorphic functions in simply connected subsets of the complex plane is equal to zero. A theorem stating that the product of the length of two vectors is greater than their inner product is named after, FTP, which German mathematician, along with Hermann Schwarz?
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Spivak's definition requires that they admit a metric, but this is not really necessary for most purposes. The Nash embedding theorem states that those having metric tensors can be isometrically embedded in Euclidean space. They are typically required to be Hausdorff and second countable, but must always be locally Euclidean. The differentiable type has smooth transition maps, unlike the topological version, and the Riemannian type has a metric tensor. The Poincaré conjecture states that all simply connected closed examples of these in 3-space are homeomorphic to the 3-sphere. FTP, name these mathematical objects coverable by many open sets homeomorphic to Euclidean space; generalizations of surfaces.
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Substituting this mathematical object in for the inhomogeneous term in an integrodifferential equation yields the equation satisfied by the associated Green's function. This object is the fundamental example of the class of abstract sequentially continuous linear functionals on non-metric vector spaces of C-infinity-zero functions; such objects are known as singular distributions. Operation of convolution with a sequence approximating this functional converges in distribution to the identity operation. Therefore, the convolution of this object with any function f yield f of zero, a fact known as the sifting property. This distribution can be considered the generalized derivative of the step function. For 10 points, name this mathematical object that can be understood as a spike of unit area, zero width, and infinite height.
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The "Dedekind" version of this property states that any bounded, nonempty subset has a least upper bound, while the "Cauchy" version of this property states that if the terms of a sequence are eventually contained in an epsilon-neighborhood for every epsilon, then the sequence is convergent. In the real numbers, this property is equivalent to the Bolzano-Weierstrass Theorem, which states that every bounded sequence has a convergent subsequence, as well as the Nested Interval Property. FTP, name this property that is essential to the real number line as constructed from Dedekind cuts of rationals, indicating that nothing is left out.
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The Cartan-Hadamard theorem says that the universal cover of a manifold where this is nonpositive is diffeomorphic to Euclidean space; more precisely, it requires that all the sectional ones are nonpositive. For a connection on a principal bundle, viewed as a gauge field, it is the field strength two-form. For a torus, it is intrinsically zero, but there can be an extrinsic one from the embedding in a Euclidean space. A surface's Euler characteristic is related by the Gauss-Bonnet theorem to the integral of the Gaussian form of it over the surface. The Riemann tensor is one measure of it. For the sphere, it is positive, while for a saddle, it is negative. For 10 points, what is this quantity, measuring an obstruction to mapping a space isometrically onto flat Euclidean space?
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The Gauss-Bonnet theorem tells us that for a particular manifold this quantity is proportional to the integral of Gaussian Curvature. More generally it can be defined as the alternating sum of the ranks of corresponding homology groups, which are known as the Betti Numbers. Equal to 0 for the Torus and Klein bottle, this is-- for 10 points-- which mathematical quantity defined as Vertices minus Edges plus Faces, and named for a prolific Swiss Mathematician.
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The Miller-Jaeschke technique is only applicable for numbers less than 3 times ten to the fourteenth power, while Rabin's method is only probabilistic. In Lucas' technique, it is necessary to find a fixed primitive root, but Pocklington's gives more flexibility. The converse of Wilson's theorem is another test of what mathematical property, whose most simplistic test, FTP, is the sieve of Eratosthenes?
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The Rousopolos variety of these equations suffers the limitation that it does not implicitly cancel normalizations, so care must be taken that the substituted shape fields have the correct amplitude. Given a sourceful Sturm-Liouville-type differential equation, these relations may be found in general for the adjoint source moment of the forward field or the forward source moment of the adjoint field, which are equal, and work by generating a second-order approximate value from a first-order approximation to an eigenfunction, using the completeness of an operator's unknown eigenspectrum. FTP, identify these widely applicable principles, best known in the Schwinger type, which utilize the namesake type of calculus to generate approximate eigenvalues.
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The Smirnov metrization theorem asserts that a locally metrizable space must be a paracompact one of these in order to be metrizable. The real numbers in a topology with basis open intervals and open intervals minus the set of inverse positive integers is not regular, but it is this. Every simply ordered set is one of these spaces in the ordered topology. A space X is this if and only if the diagonal is closed in the product space X times X. In a space of this type, a sequence of points converges to at most one point. This class of topological space has the property that any two points have disjoint neighborhoods and thus possess finite subsets that are closed by the T2 separation axiom. For 10 points, what is this type of topological space, named for a mathematician also known for a concept of dimension often used for fractals?
name the statistical law proven by Jakob Bernoulli whose principle is often erroneously labeled the "law of averages," which states if the sequence x-sub-n of n random variables have finite mean µ, then sequence x-sub-n over n tends to infinity.
The asymptotic equipartition property is a consequence of the weak version of this law, which is often proven using Chebyshev's inequality and was defined by Ernst Fischer. The strong version, defined by Friedrich Riesz, implies the sample average converges almost surely to the mean µ [myou]. For 10 points
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The basics behind this computer language were developed by its inventor at a 1956 Dartmouth Summer Research Project, and its primary dialect, version 1.5, came out in 1965. Its significant features include the use of "atoms", reliance on recursion, garbage collection, and computation performed by applying functions to arguments rather than variables. Invented by John McCarthy, FTP, what is this language famous for its use in artificial intelligence whose name is short for "List Processing"?
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The cannonball problem asks when the number of cannonballs in a pyramidal stack can be one of these-the answer is only twice, when the number of cannonballs is either 1 or 4900. Lagrange proved that every positive integer can be expressed as the sum of four of these numbers, and the zeta function of two is defined as the sum of their reciprocals. They are always congruent to zero or one modulo four, and their last digit is always 1, 4, 9, 6, or 5. FTP, identify these numbers which are formed by raising an integer to the second power.
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The converse of this theorem is the basis of the Pratt certificate. Paul Zeitz' proof of this theorem relies on considering a bracelet consisting of beads with a fixed number of characters, then removing those with only one unique character. Another proof of it relies on using Lagrage's theorem on a multiplicative group G. Also the basis of the RSA encryption scheme, the Carmichael numbers notably foil the test named for this theorem, which can be generalized using Euler's totient function. Stating that a to the p is congruent to a modulo p, for ten points, identify this theorem of number theory named after a French mathematician, which is less famous than its namesake's last one.
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The double data rate version of this technology achieves greater bandwidth by transferring data on both the rising and falling edges of the clock signal. The current synchronous model uses an internal finite state machine to pipeline incoming commands and the speed of its operation is associated with the pipelining latency. By nature the device is volatile and the in the past its types include both single and dual in-line. With a most basic set-up storing each bit on a unique capacitor, FTP, name this type of memory which is one aspect associated with a computer's speed.
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The generating function for the number of weakly binary trees on a given number of nodes provides a sequence named for Etherington and this man. His first published paper in 1905 proved important results on hypercomplex numbers. By an investigation of skew fields with a finite number of elements, he proved his second namesake theorem, which states that a central-simple algebra is isomorphic to the algebra of all n by n algebras. FTP, identify this Scottish mathematician, whose first theorem states that a finite division ring is a field.
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The group of deck transformations of a universal cover of a topological space is isomorphic to this. When it is Abelianized, the result is the associated space's first homology group. It can be calculated by Van Kampen's theorem if its value for some subspaces that cover the space in question is known. Examples of it include the group Z of integers for a circle and Z*Z ["z star z"] for a torus, and any space for which it is trivial is said to be simply connected. FTP, name this topological construct, that is defined as the group of equivalence classes of loops in a space, and denoted "pi sub one."
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The lemma that is not Burnside's is actually due to this man and his namesake problem is to solve a general first-order ODE with an analytic inhomogeneous term. With Euler, this person is the namesake of equations of the form x-squared times y-double-prime, plus a times x times y-prime, plus b times y, equals 0. With Riemann, he names two PDE's satisfied by all holomorphic functions and the completeness of a space is defined by uniform convergence of his namesake sequences. He is the namesake, with Schwarz, of an inequality comparing the norms of two vectors to the magnitude of their inner product, and he made the original definition of the limit. FTP, name this prolific mathematician; a French military engineer.
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The most direct way to understand this concept in lambda calculus is via the fixed-point operator. The general form of this property is synonymous with computability; the Ackerman function has the general form of this property, but not the primitive kind. The McCarthy 91 Function makes calls of this kind if its input is less than 100. Because any positive integer factorial can easily be stated as the integer times the next-lowest integer factorial, many implementations of the factorial function have this property. FTP, name this property of algorithms that call themselves, that is often contrasted with iterativeness.
for 10 points - what are these entities that govern the transformation of quantities from one set of coordinates to another?
The most famous examples of these are related by the Bianchi identities and can be converted to matrix form using Petrov Notation. In the namesake contraction, unlike indices are made equal, resulting in a reduction of 2 in the rank and a value of zero for the Weyl type. Other types of this mathematical object include the Ricci type and the stress-energy type, but the best-known kind is probably the Riemann version, which defines the curvature of manifolds. Widely used in physics
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The non-linear Anscombe transform on this type of variable sets the variance of it to 1, and Dobinski's formula demonstrates that nth Bell number is the nth moment of this distribution. With PDF proportional to its characteristic value to the k over k factorial, the difference between two variables with this distribution follows the Skellam distribution, and it can be considered a limiting case of the binomial distribution. It has both mean and variance lambda, and von Bortkiewicz used it to model deaths by horse kick in the Prussian army, illustrating his proposed Law of Small Numbers. For 10 points, identify this probability distribution, used to model the number of low-probability events that occur in a specific time period.
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The nth of these at x times t to the n summed over n as a doubly-infinite Laurent [law-RAWNT] series produces the generating function e to the quantity x over two times the quantity t minus 1 over t. Because these functions are the zero-regular Froebenius solutions of the radial component of Laplace's equation in cylindrical coordinates, they are sometimes called cylindrical harmonics. The Hankel functions are a complex linear combination of these and their companion Neumann functions, which are also known as these of the second kind. FTP, name these mathematical functions symbolized J and named for the German astronomer and mathematician who first calculated a star's parallax.
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The strongest form of this lemma is implicit in the Henstock-Kurzweil method, which applies gauge summations to all forms of the inverse relationship it implies. It may be relaxed to Lebesgue's theorem for locally continuous curves. In multidimensional space it may be partially generalized to surface integrals on over manifolds in the form of the generalized Stokes' theorem. FTP, name this mathematical theorem that states that the integral of the derivative of function on an interval is equal to the change in the function on the interval.
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The surreal numbers have the universal embedding property for ordered varieties of these objects. A finite one with n elements exists if and only if n is a prime or a power of a prime. One is called "algebraically closed" if univariate polynomials whose coefficients are elements of it also has a solution in it. FTP, identify these mathematical structures, examples of which include the reals and the complex numbers but not the quaternions, and which are defined as rings wherein every element has a multiplicative inverse.
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The use of stacks to implement this was inaugurated by Edsger Dijkstra in his development of a complier for ALGOL 60 and is used in now almost all compiled languages. Though such efficient algorithms as BinarySearch and quicksort are usually written to use this, its use on most Von Neumann machines makes an algorithm's runtime and memory usage greater than those of an analogous algorithm utilizing only iteration. For ten points, name the feature that an algorithm that calls itself is said to utilize.
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The use of these over finite fields for a type of public-key cryptography was proposed by Koblitz and Miller. Adding a point at infinity allows one to define a group operation that maps two points on it to a third point on it such that all three points are collinear. It discriminant, proportional to 4 A cubed plus 27 B squared when its equation is y squared equals x cubed plus A x plus B, is zero when it is singular. Topologically, it is equivalent to a torus. A conjecture relating them to modular forms was proven in part by Andrew Wiles as part of his proof of Fermat's Last Theorem. For 10 points, what is this type of algebraic curve, of which the similarly-named conic section that looks like a squashed circle is not an example?
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The volume of a simplex in n dimensions can be given by the Cayley-Menger type of this function, and this function can be generalized to non-square entities using the Cauchy Binet formula. The linear independence of a set of solutions to a differential equation is given by a nonzero value for the Wronskian type of this, and changing variables when integrating a function over its domain involves the use of the Jacobian type of this function. One of these for a two-by-two matrix is used in Cramer's rule to find the solution to a system of two equations. For 10 points, name this scalar function of a square matrix, denoted by single or double bars around a matrix.
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These numbers can be obtained by the generating function “x†over quantity “1-x-x2â€, and a closed form expression of them is obtained from Binet's formula. The first two of these numbers are different from the first two Lucas numbers, but both sets satisfy the same recursion relation. Named for the author of the Book of the Abacus, these numbers' successive ratios approach 1.618, the golden ratio. They occur in the sums of diagonals in Pascal's Triangle and various places in nature such as the multiplication of rabbits. The nth of these is equal to the n minus 1st plus the n minus 2nd of them. For 10 points, name this sequence whose initial entries are 0, 1, 1, 2, 3, 5.
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This algorithm is often paired with OAEP, which makes it probabilistic by adding random oracles. Now implemented via the PCKS scheme, modern versions use the Carmichael function. One pair of values generated have a product congruent to 1 modulo lambda of n. One step involves computing the totient of the product of two numbers and finding an integer that is coprime to that result. Another step in this algorithm is to find the modular multiplicative inverse of e, and the efficacy of this algorithm is proven by Fermat's Little Theorem. Its secure because of the difficulty in factoring large primes. For 10 points, name this public key encryption algorithm pioneered at MIT by the three scientists whose initials give it its name.
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This can be determined in linear time with the algorithm of Williamson or with that of Lempel, Even, and Cederbaum and it is logically equivalent to the existence of a topological K-5 or K-3,3 subgraph. Simple graphs with this characteristic have a number of edges less than or equal to three times the number of vertices minus six. Duals are found only in graphs with this property, which is possessed by trees, cycles, and wheel graphs. FTP name the subject of a 1930 theorem by Kazimierz Kuratowski, the property of a graph that it can be rendered in two-space without having any edge cross?
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This computer scientist created Metafont to produce vector fonts, but it never caught on with font designers, perhaps due to its sixty parameters. His MMIX is a fictional 64 bit RISC architecture that will replace the MIX architecture, unusual for running in either binary or decimal, as the implementor chooses, and for having bytes of either six binary bits or two decimal digits in the fourth edition of his most famous work, which is currently unfinished, and includes a volume on "Fundamental Algorithms" and fascicles on combinatorial algorithms. Version numbers for Metafont and this man's more famous program approach e and pi, respectively. For 10 points, name this author of The Art of Computer Programming, for which he created the typesetting program TeX.
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This guarantees the existence of a Hamel basis for every vector space, thereby implying that any continuous function can be expressed as a linear combination of finitely many members of a certain set of continuous functions, and also gives rise to the Bararch-Tarsky paradox. Implying the existence of the namesake function on a given set, it is equivalent to the claim that the product of cardinal numbers is zero if and only if one of those numbers is zero. Paul Cohen demonstrated this hypothesis to be independent of the Zermelo-Frankel axioms in 1963. FTP, identify this axiom that states that, for any set of non-empty sets, there exists a set with exactly one element in common with each set.
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This man invented a resource-allocation procedure that makes a safe-state check before granting requests, the banker's algorithm. He developed a method for parsing equations given in infix notation, the shunting yard algorithm, which can output in a format he developed with F. L. Bauer. This developer of reverse Polish notation and semaphores also created an algorithm with a heuristic modification called "A " (a-star) that fails for negative edge weights. For 10 points, identify this formulator of the dining philosophers problem, hater of GOTO statements, and namesake of a shortest path-finding algorithm, a Dutch computer scientist.
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This mathematical method, attributed to Cauchy, is an intrinsic test as it only depends on the series being examined. Particularly effective when the terms of a series contain factorials of expressions, or expressions raised to the nth power, like the nth-root test, is it not conclusive when rho = 1. FTP, identify this convergence test for nonnegative infinite series, in which the limit of the quotient of successive terms is evaluated.
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This mathematical tool used in digital audio applications decomposes or separates a waveform or function into sinusoids of different frequencies which sum to the original waveform. It identifies or distinguishes the different frequency sinusoids and their respective amplitudes. If you have SETI installed on your PC, a fast version is performed a few times per second. FTP, identify this linear transform named after a French mathematician.
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This mathematician's "characteristic" is defined to be the number of vertices of a graph minus the number of edges plus the number of faces. He introduced the symbols of pi and sigma to mathematics, and his phi function tells the amount of numbers between 1 and n relatively prime to n. He solved the Seven Bridges of Konigsburg problem, but went completely blind by 1766. FTP, who was this Swiss mathematicisn whose namesake identity involves the number e?
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This number is an invariant for geometric objects belonging to a prescribed category that are equivalent in that category. For smooth manifolds, it is defined as the number of signed zeroes of a generic tangent vector-field with isolated zeroes. For topological manifolds, it is the alternating sum of dimensions of the homology groups for the homology with coefficients in a field. Defining a value that is, for polyhedra, the number of its vertices minus the number of its sides plus the number of its faces, FTP, identify this mathematical term named for an 18th-century Swiss mathematician.
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This operation is generally equivalent to the wedge product, which is also characterized by anticommutativity and multilinearity. The differential of this operation can be calculated by Jacobi's formula. If it equals zero, its associated object is singular; otherwise, its associated object is invertible. The Wronskian is a special case of this operation, which is sometimes evaluated using Sylvester's theorem. For the two-by-two matrix “a b c dâ€, it is equal to “a times d†minus “b times c.†For 10 points, name this operation performed on square matrices, abbreviated “det.â€
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Used by Cantor in lieu of Dedekind cuts for a constructive definition of the reals, they are not necessarily convergent, but their convergence is a necessary condition of completeness. Any convergent sequence in a metric space must be of this type. Named for a French Mathematician, FTP, identify these sequences such that for any epsilon greater than 0, two members of the sequence may be found which are separated by a distance less than epsilon.
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Used to prove that there is no general method for solving Diophantine equations, methods for computing them include Cassini's identity, Binet's formula, and determining how many ways there are to tile a 2-by-n strip with 1-by-2 rectangles. One may also expand their generating function: f-of-x equals x divided by 1 minus x minus x-squared, or sum the shallow diagonals of Pascal's triangle to compute, FTP, what sequence of numbers, whose first five members are 1, 1, 2, 3, 5?
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Utilizing the circle method, Hardy and Littlewood proved the weaker version of this statement, though their proof required the grand Riemann hypothesis to be true. Vinogradov provided a direct proof of its weak version, a result similar to Levy's statement. Chen Jing Run used a weighted sieve method that improved on the work of Brun and Selberg to give a new upper bound for this problem. This statement is equivalent to the claim that for every positive integer m, there exist primes p and q such that phi of p plus phi of q equals two times m, where phi is the totient function. The aforementioned weak version states that every odd number greater than 7 can be written as the sum of three odd primes. For 10 points, identify this unproven statement whose strong version states that every even number greater than 2 can be written as the sum of two primes.
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Variants of it include System V and BSD. Among its innovations was the idea of "pipes" that allow the results of one program to be used as the input for other programs. It began in the 1960s with work at Bell Labs, MIT, and General Electric to develop a reliable time-sharing operating system, giving rise to a 1971 edition by Thompson and Ritchie. Its source code was offered freely to colleges and universities but without any offer to support it from Bell Labs, leading to its rapid and divergent evolution. Originally based on a system called Multics, FTP, what is this operating system similar in nature to Linux?
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Versions of this device include the R-type and D-type, while one of the most versatile versions is called the J-K Master-Slave type. Since it is important that they operate synchronously, most types include a clock input, with groups often controlled by a single clock line. It will maintain its state indefinitely until an input pulse called a trigger causes it to switch its state from 1 to 0 or from 0 to 1. Also called a * bistable gate, FTP, what is this basic type of computer circuit that shares its name with a type of footwear?
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Volume 2 of Whitehead's Mathematical Works contains an incorrect proof and his namesake link to this problem. Rubinstein's theorem attempts to explain it by assuming the shape of the original problem is algorithmically decidable. Smale proved the cases for n greater or equal to seven in 1961, and Michael Freedman won the 1986 Fields Medal for proving the case for n=4. FTP, what is this conjecture that every simply connected closed n-manifold is homeomorphic to an n-sphere?
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Weyl's theorem can be used to show that the Dirchlet type of these are asymptotic, providing an answer to the Mark Kac-proposed question “can one hear the shape of a drum?â€In quantum mechanics, ladder operators alter these quantities by one, and they are taken to be 1 or -1 for the permutation operator. Always real for a Hermitian matrix, they are the roots of the characteristic polynomial, and for diagonal matrices, they are given by the entries of the diagonal. For 10 points, name these scalars that, when multiplying a matrix, are equal to the product of a vector with that matrix and which are usually denoted by the Greek letter lambda.
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What are known as the Tait-Bryan angles in aeronautics are elsewhere known for this person. An equation that calculates loading of beams is named after Bernoulli and this scientist. A formula that can evaluate series using integrals is named after Maclaurin and him. Initial value problems can be integrated approximately by his namesake iterative method. The limit of the difference between the harmonic series and the natural logarithm is a constant denoted gamma that is named after Mascheroni and him. FTP, name this mathematician whose namesake number is the base of the natural logarithm, e.
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When applied to charts, these programs can employ Earley's algorithm, and the CYK algorithm can be used to counter the local and global ambiguity problems these programs face. They can be created by programs such as Yacc, provided there is a valid Backus-Naur notation. When they employ a stack to push symbols through cycles of shifting followed by reduction, they are bottom up, and when they process input strings from left to right using leftmost derivations they are known as predictive or top-down. For 10 points, name these programs that typically map grammar symbols to data elements based on the rules of a particular grammar, and are present in compilers.
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When this function is divided by its input, a new function results that acts as the ideal reconstruction filter. That filter exhibits the Gibbs phenomenon, and this function is used in Fourier series representations of odd functions. In Euler's formula, this function represents the imaginary part and as such is multiplied by i. The Taylor series for this function consist of alternating terms of the form X to N where N is odd, divided by N factorial. Its derivative is cosine, a function that is this function shifted by 90 degrees. For 10 points, name this trigonometric function that for an angle is equal to the length of the opposite side divided by the hypotenuse.
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When this word is followed by "character," it is used to describe a family of sets that, according to Tukey's Lemma, has a maximal element with respect to the inclusion ordering. A set S satisfies Kuratowski's notion of this property if it is a member of the sub-semi-lattice generated by the empty set and the singletons in the power set of S. The set of all sets that are hereditarily this is denoted (V-sub-omega), and can be constructed recursively by beginning with the empty set, continually taking the power set of the last set in the sequence, and taking the union of every set in the sequence. A set satisfies Dedekind's notion of this property if it is not equinumerous to any of its proper subsets. FTP, what is this property which describes a set that has cardinality equal to that of a natural number?
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Wide sense stationarity implies strict sense stationarity for this class of processes, which is invariant under linear transformation and which has the property that uncorrelated variables are also independent. Unlike other processes, these are completely characterized by their mean and covariance functions. For 10 points, identify this class of random processes, whose one-dimensional probability distributions are characterized by the bell curve.
that result is also named for Mordell. He helped develop Ramsey theory and discovered an elementary proof of the prime number theorem, and believed that all beautiful proofs came from THE BOOK. FTP, name this prolific Hungarian mathematician, whose namesake number gives the publishing distance between him and a given mathematician.
With Faber and Lovasz, this mathematician names an unproved conjecture about a union of k copies of a k-clique intersecting at one vertex, and another result partly named for this man regards transforming a non-convex polygon into a convex one with a series of flips. He names the constant equal to the sum of the reciprocals of the Mersenne primes along with Borwein, and an inequality named for him relates the distances from a point to the vertices and edges of a triangle
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With Jacobi and Bellman, this man names an equation central to optimal control problems, and he names a theorem which implies that a square matrix A satisfies its own characteristic polynomial, along with Cayley. He names a cycle on a graph such that the path visits each vertex exactly once, and he is also the namesake of an operator which is the Legendre transform of the Lagrangian operator and gives the total energy of a system. Better known for the creation of a division ring where i squared, j squared, and k squared equal negative one, for 10 points, identify this Irish mathematician who developed the mathematics of quaternions.
whether the real part of any nontrivial zero of his eponymous zeta function has real part one-half. FTP, name this German mathematician whose namesake sums also approach his namesake integral.
With Lebesgue, this man's name denotes a lemma which states that if a function oscillates rapidly about zero, its integral will be small. With Cauchy, his name describes a set of equations necessary for a complex function to be holomorphic, and his namesake surface is a one-dimensional complex manifold. This man may be most famous for formulating one of the most well known unsolved problems in mathematics
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With the exception of the smallest of their kind, all known examples of these can be expressed as the sum of cubes of consecutive odd numbers. Euler proved that there is a one-to-one correspondence between Mersenne primes and even types of this number, and they form a subset of the triangular numbers. Numbers which do not fit this category are classified as deficient or abundant, and these numbers all yield 2n when they form the argument of the divisor function. It is unknown whether any odd ones exist. 6, 28, and 496 are the first three examples of, for 10 points, what type of number that equals the sum of its proper divisors?
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With this man's encouragement Leopold Crelle began publishing his world famous journal of mathematics in 1826. In competition with Karl Gustav Jacobi, before his death, this man rapidly developed the theory of elliptic functions. When he sent his famous proof to Gauss, Gauss incorrectly dismissed it. Earlier he had personally published that proof stating the impossibility of algebraically solving the general equation of the fifth degree. His namesake theorem states that there is a finite number, or genus, of independent integrals of algebraic functions. FTP, name this Norwegian mathematician who shares his surname with the second son of Adam and Eve.
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Within a constant it's the Laplacian of the function "one over distance" and is also the charge density of a point charge. Actually an operator which projects out a function's value at the origin, it's not really a function at all. For ten points, name this mathematical entity, which is zero everywhere except the origin and is designated by a Greek letter.
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Work on it began at the IBM under the direction of E. F. Codd. BETWEEN allows you match ranges of dates, strings, or numbers. The DROP command allows you delete a data structure, while the TRUNCATE command allows you to delete all the data within the structure. The Inner and Outer varieties of this language's most important command return either all matching elements, or non-matching elements. This language also contains commands like WHERE and SELECT, as well as the aforementioned JOIN. FTP, identify this computer language used to get data from a relational database.
Adrien-Marie Legendre
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Amalie Emmy Noether
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Archimedes
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August De Morgan
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August Ferdinand Möbius
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August Mobius
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Augustin Louis Cauchy
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Augustin-Louis Cauchy
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B-Tree [or B Minus Tree]
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Banach space
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Banach spaces
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Bayes' Theorem
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Bayesian Networks or Bayes Nets
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Bessel function (prompt on "cylinder function")
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Binomial theorem
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Bravais lattices
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Calculus of Variations or Variational Calculus
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Camille Jordan
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Camille Jordan
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Carmichael numbers
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Cauchy sequence
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Charles Babbage
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Chern-Simons theory
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Chinese Room Argument
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Chinese remainder theorem
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Church-Turing Thesis [accept in either order; prompt on only a single name]
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Church-Turing thesis
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Church-Turing thesis (accept either name until "Turing," Church-Turing hypothesis, Turing-Church thesis or hypothesis, etc.)
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Church-Turing thesis [accept in either order; prompt on a single name before "doubly"]
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Claude Shannon
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Common Business-Oriented Language (or COBOL)
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Compiler [or Compilation; accept Code Generation before "ROSE"]
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Completeness
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Cramer's Rule
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David Hilbert
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Dijkstra's Algorithm
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Dirac delta function [prompt on delta function]
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Domain Name System or Domain Name Service or DNS
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Donald Ervin Knuth
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Donald Ervin KnuthÂ
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Dynamic Programming
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Edsger Wybe Dijkstra
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Euler Characteristic
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Euler characteristic
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Euler characteristic [or Euler number]
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Euler characteristic or Euler number
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Euler gamma function
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Euler phi function [or Euler totient function]
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Euler-Poincaré characteristic or number
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Fast Fourier Transform (prompt on Fourier transform)
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Ferdinand Georg Frobenius
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Fermat numbers
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Fermat's Last Theorem
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Fermat's Last or Fermat's Great theorem
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Fermat's Little Theorem
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Fermat's Little Theorem (do not accept Fermat's Last Theorem)
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Fermat's last theorem
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Fibonacci numbers [or Fibonacci sequence]
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Fibonacci numbers or sequence
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Fibonacci numbers or the Fibonacci sequence
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Fields
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Four-Color Map Theorem (accept equivalents)
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Fourier Transform
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Fourier series
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Galois theory
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George Boole
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Gottlob Frege
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Grace Hopper
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Green's functions
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Guillaume de L'Hopital or L'Hospital [lo-PEE-tal]
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Hamiltonian systems
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Hausdorff space [accept T2 before mentioned]
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Hilbert's problems or the Mathematical Problems of David Hilbert
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Hubbard model
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Huffman Coding
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Intel x86 Architecture (or Intel x86 CPUs or Intel x86 Framework; prompt on "Pentium" or "Intel Pentium" before "Celeron")
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Internet Protocol version 6 (prompt on partial answer or on "IPng")
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JPEG or JPG (pronounced j-peg) or Joint Photographic Experts Group
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Jean-Baptiste-Joseph, Baron Fourier
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John "Johnny" von Nuemann [also accept Margittai Nuemann Janos Lajos]
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John Napier
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John von Neumann or Margittai Neumann János Lajos
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Joseph Henry Maclagan Wedderburn
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Joseph Liouville
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Joseph Louis Lagrange
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Joseph Louis Lagrange's theorem
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Joseph-Louis Lagrange
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Joseph-Louis Lagrange
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Jules Henri Poincaré
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Jules-Henri Poincaré
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Karl Jacobi
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Kernel
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Klein bottle (prompt on Franklin Graph before it is read)
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Knapsack problem
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Kurt Gödel
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Königsberg Bridge Problem
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LISP (prompt if "Scheme" is given as an early buzz)
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Pascal
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Paul (or Pal) Erdös
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Paul Erdos
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Pell's equation
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Peter Gustav Lejeune Dirichlet
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Phi-Function or Totient Function [prompt on "Euler's Function"]
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Poincare Conjecture
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Poincare conjecture
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Poisson Distribution
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Poisson distribution
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Prisoner's dilemma
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Pseudo Random Number Generators [or Pseudo RNGs]
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Pythagorean theorem
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Python
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RISC or Reduced Instruction Set Computing
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Richard Dedekind
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Riemann hypothesis
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Riemann zeta function [prompt on partial]
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Riemann(ian) surface
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SDRAM or Synchronous Dynamic Random Access Memory
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SMTP or simple mail transfer protocol
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Self-Balanced Binary Search Tree (or Self-Balancing Binary Search Tree or Height-Balanced Binary Search Tree; prompt on "Tree", "Binary Tree" or "Binary Search Tree")
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Simpson's Rule
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Simpson's rule
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Siméon Denis Poisson
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Siméon-Denis Poisson
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Sir Roger Penrose
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Solving a Rubik's Cube
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Square numbers or Squares
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Stack
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Stokes
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Stokes' Theorem
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Structure Query Language (pronounced "sequel")
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Taniyama-Shimura-Weil or Shimura-Taniyama-Weil Conjecture
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Tartaglia (accept Nicolo Fontana before it is given)
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TeX (pronounced "tech"; also accept LaTeX, pronounced "LA-tech")
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The Axiom of Choice [accept Zorn's Lemma or the well-ordering principle before each is read]
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The Four-Color map Theorem
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Torus
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Towers of Hanoi or Tower of Hanoi or equivalents
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Traveling Salesman problem
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Trees (do not accept or prompt on binary tree)
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Triangle Inequality [prompt on Cauchy-Schwarz Inequality before mentioned]
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Turing machines
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UNIX
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Virtual Memory
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Wedderburn's Theorem
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William Rowan Hamilton
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Zermelo's axiom of choice (accept ZFC, Zorn's lemma, trichotomy law, well-ordering principle, multiplicitave axiom, etc.)
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algebraic
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algebraic varieties
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cardioid
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category (the ordering of the first clues makes "group" wrong by the time "abelian" is mentioned)
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central limit theorem
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chaos
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chicken [accept Hawk-Dove really early]
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commutative [accept word forms; accept abelian before mentioned]
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compactness
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compactness (do not accept limit point compactness or sequential compactness)
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complex number
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continuity (accept equivalent forms)
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continuous [accept word forms]
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continuum hypothesis
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continuum hypothesis or CH
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convolution
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cosine
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cubic spline interpolation (accept natural cubic spline before "periodic")
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divergence theorem (accept Gauss' theorem before it's mentioned; accept Ostrogradsky's theorem or Ostrogradsky-Gauss theorem or Gauss-Ostrogradsky theorem)
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dynamic algorithms (or dynamic programming, I suppose)
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e or Euler's number
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eigenvalues (accept characteristic values until *, prompt afterwards)
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eigenvalues (or characteristic values)
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eigenvalues [or latent roots; proper values; characteristic roots; characteristic values]
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elliptic curve
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ethernet
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expert systems
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extensible markup language or XML
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extra dimensions (accept reasonable equivalents)
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factorial
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field
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finiteness
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first fundamental theorem of calculus
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flip-flop (accept bistable gate until *)
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floating-point numbers
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fluctuation-dissipation theorem
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four color problem
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four color theorem
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functional languages
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fundamental group
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fuzzy logic
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garbage collection
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generator [accept word forms, e.g. "generates," etc.]
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graphs
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graphs [accept graph theory]
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greedy algorthm
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group
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hairy ball theorem (prompt on "ball")
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hyperbola
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hyperbolic
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hyperbolic cosine or cosh
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ideal
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information theory (accept equivalents, e.g. informatics)
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inner product
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insolvability of the quintic (accept Abel-Ruffini theorem; also accept clear knowledge equivalents, so long as they refer to 5th degree polynomials)
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integer or prime factorization
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integers
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irrational numbers
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irreducibility (accept word forms)
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kernel
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knots
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manifolds
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mathematical induction
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parallel
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parsers (accept parsing programs)
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parsers [or parsing; prompt on "compiler"]
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partial fractions
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perfect number
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planar graphs
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planarity
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planarity (accept word forms, e.g., that the graph is planar)
...
pointers
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ratio test
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recursion (prompt on "recursive")
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recursiveness [accept word forms like recursive]
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red-black tree (or symmetric binary B-tree; prompt on "B-tree" or "tree")
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residue
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ring
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rings
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sample; accept "sampling" and "sample size"
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spiral
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stochastic [prompt on random or Markov]
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straightedge and compass [prompt on "ruler" for straightedge; accept answers in either order]
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surjectiveness (or onto)
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taking/finding the integral [accept word forms like integration; prompt on anti-derivative before read]
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tensors (accept Riemann tensor until "Weyl")
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the Poisson distribution [prompt on P-sub-lambda]
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the ordinary Bessel functions of the first kind [prompt on J]
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torus [accept word forms like toroidal before "objects"]
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 RSA encryption [or Rivest, Shamir and Adleman]
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 analyticity [or holomorphicity; accept complex differentiable before mention]
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 field [accept rational domain until mentioned, prompt on commutative division algebra until mentioned]
...
 four-color theorem [or Guthrie's problem]
...
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A Bellman equation is an condition associated with this technique, which is employed by the Cocke-Younger-Kasami algorithm that asks whether a string can be generated by a given context-free grammar. The Needlemna-Wunsch algorithm for nucleotide alignment also employs this technique, and this method is called stochastic if the immediate payoffs and the next state are known onyl as probabilities. An example of this method is modifying a recursive program to calculate the Fibonacci sequence so that the program stores each value of the sequence after calculating it and then reuses it if it is needed again instead of recalculating. For 10 points, name this method of problem solving, which aims to efficiently solve optimization problems which have overlapping sub-problems, and is often used to solve the traveling salesman problem and to check edit distance in spellcheckers.
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A Besov Space is one of these if the factors p and q are between one and infinity. The Schauder fixed point theorem involves convex subsets of them, and they are called prime if each of their complemented subspaces are isomorphic to them. Every Hilbert space with the inner product is one, although the converse is not true, and the simplest example is an n-dimensional Euclidean space with a distance function. For 10 points -- identify these complete vector spaces with a norm, named for a mathematician who did work on cutting spheres with Alfred Tarski.
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A consequence of it is the Boolean Prime Ideal Theorem, and Skolem used it to correct Lowenheim's Theorem on first-order models. It can be equated to one formulation of König's theorem, which relates the cardinality of certain sums to products, and it is used to decompose a hollow unit sphere into two identical spheres in the proof of the Banach-Tarski paradox. Equivalent to saying a partially ordered set in which every chain is bounded has a maximal element, it is independent of Zermelo-Frankel set theory. For 10 points, identify this equivalent to the well-ordering principle and Zorn's Lemma, which states that one can pick an element from each of an infinite collection of sets.
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A corollary of the Lyusternik theorem is that, at a certain point, the Frechet derivative of the map between Banach spaces has this property. According to the Fredholm alternative theorem, either the adjoint of a continuous linear operator with a closed range has a non-trivial kernel or the operator is of this type. More generally, a function is said to have this property if and only if the composition of it and its inverse is everywhere the identity function. Only applicable to functions whose domain spans the entire codomain, this is, FTP, what property, functions having which are bijective only if they are also injective?
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A generalization of it utilizing nonlinear techniques to address variational problems is known as Morse theory. Its fundamental equations include the Euler-Lagrange condition and the Beltrami identity it can be defined as the type of mathematics that seeks to find the minimum or maximum parameters for the relationship between two constrained variables. Developed by a host of 17th century mathematical luminaries, FTP, name this type of calculus used to solve the brachistochrone problem.
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A generalization of these operations to mappings between Banach spaces is named for Fréchet. Higher order ones can be calculated by Faa di Bruno's Formula, and they do not exist for any point on the Weierstrass Function. The entries of the Jacobian matrix are first order examples of these, as are the components of the gradient. When there are multiple variables, partial ones are used. This mathematical concept can be defined using limits. When there is a composition of functions, the chain rule can be used to compute them. For 10 points, name this operation that describes the change of a function as its input changes and which is the opposite of an integral.
what is this theory that seeks to analyze the transmission and processing of data?
A key concept in this discipline is the surprisal, which is defined as the negative log base 2 of the probability of encountering an item being examined, and is found in one of the field's major equations, in which the sum of the products of the surprisals and probabilities equals the uncertainty. Concepts like Landauer's Principle and Holevo bounds are concerned with connecting the subject to physics and mathematics. Pioneered by Harry Nyquist and R.V.L. Hartley, it saw its seminal development in the work of another Bell Labs employee, Claude Elwood Shannon. For 10 points
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A linear one in which p and q are equal exemplifies of a martingale, while a periodic radial one may be used to model loss of magnetic confinement of ions in a collisional torus. An open problem is to find a general formula for the number of finite non-intersecting ones in a plane. A given non-trivial one always eventually returns to its beginning in one- and two-space, but in 3-space only does so with probability about 0.34, which is known as Pólya's third constant of this kind. FTP, name this type of Markov chain in which a point takes a certain number of fixed-length "steps" in a probabilistic direction and which can be used to model such stochastic processes as Brownian motion and diffusion.
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A non-Abelian version of this theorem has been used to show the relationship between the Wilson loop and magnetic monopole in QCD. This theorem can be thought of as an expression of duality between the homology of chains and the de Rham cohomology, and it is used extensively in the proof of the Poincare-Hopf theorem. It is the real-plane generalization of Cauchy's integral theorem. The most general form of this theorem relates the integral of a differential form over a manifold boundary to the integral of the external derivative of that manifold, and it can also be reduced to the Ostrogradsky-Gauss theorem. FTP, name this theorem whose best known form relates the integral around a boundary of a vector field to the surface integral of the curl of that vector field, a three-dimensional analog of Green's theorem.
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A problem named for this man is equivalent to finding the number of free distributive lattices with n generators, but it is more commonly stated as determining the number of monotonic increasing Boolean functions of n variables. His number-theoretic psi function involves taking the product over all distinct prime factors p of n of (1 + 1/p), then multiplying by n. A type of integral domain named for him is an integrally closed Noetherian domain for which every nonzero prime ideal is also a maximal ideal. Another concept associated with this mathematician is a partition of the rationals into two nonempty subsets S1 and S2 such that all members of S1 are less than those of S2, and S1 has no greatest element. For 10 points, identify this mathematician who thus constructed the reals from the rationals with his namesake "cuts."
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A special case of this problem was solved using DNA oligonucleotides by Leonard Adelman. Other methods have included the Christofides algorithm and k-opt heuristic, but the first mathematical exposition of this problem was by Karl Menger, who knew that the naïve "nearest neighbor" heuristic solution was only optimal when the triangle inequality was satisfied. It can be solved exactly in some cases with the cutting plane method, and a specific instance based on 24,798 cities in Sweden has been solved, though an efficient general algorithm remains elusive. The bottleneck version is NP-hard, but the decision version of this problem can be shown to be NP-complete by a reduction to a Hamiltonian Path. FTP, identify this important problem in optimization, in which a hypothetical merchant must travel to each city once and then return home.
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A specific type of these entities x obey a relation in which x squared divides the x minus Legendre symbol x over fifth Fibonacci number. Another type satisfies the conditions of Wolstonholme's theorem for the fourth power, though all of them satisfy that result for the third power. If x is one of these and 2x plus 1 is also, they are named for Germain. In addition to the Wall-Sun-Sun ones, these numbers divide themselves minus one factorial plus one according to Wilson's theorem. Their count is asymptotic to x over the natural log of x, and Goldberg's hypothesis states that every even number can be expressed as the sum of two of these. Fermat and Mersenne give their names to varieties of, for 10 points, what kind of number whose only factors are one and itself?
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A theorem on this property that connects pointwise convergence of characteristic functions and convergence in distribution is named for Lévy. The Radon-Nikodym derivative exists when one measure is "absolutely" this with respect to another measure. The sequential form is strictly weaker in non-first-countable spaces than the topological form of this property, which requires that the preimage of open sets be open. The Lipschitz form of this property bounds the rate of change, and for a function, this property exists when the left and right limits at every point are equal to the value at that point. FTP, name this property that describes functions without "jumps."
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A variant of this algorithm used for selection can be made to run in worst-case time by a median-of-three killer sequence, though this algorithm can be used to find the smallest or largest few elements of an array effectively. In 1999, McIlroy created an adversary for this algorithm that guarantees that it will run in worst-case time, though a selection algorithm that finds an array's median will ensure a running time close to n log n. It works in three phases: choosing a pivot element, recursing on lesser elements, and recursing on elements greater than or equal to the pivot element. For 10 points, name this divide-and-conquer algorithm, noted for being faster than other n log n sorting algorithms in practice.
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Algebraic surfaces of this type include the Togliatti surface and equations of this type include the de Moivre equation. These equations may be solved by reducing them to the Bring form and applying Jacobi theta functions. Alternatively, after undergoing a Tschirnhausen transformation, the related icosahedral equation can be used to find a solution in terms of hypergeometric functions. Those and other methods are now known, but there can never be a general equation for them using only field and radical operations. FTP, name this class of algebraic equations whose study led to the development of Galois groups and Abel's impossibility theorem.
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Algebraic varieties have this property if they cannot be written as the union of other nonempty algebraic varieties. Individual elements of an integral domain have this property if they are not units, and this property holds when a prime number doesn't divide the first coefficient of a polynomial but does divide the rest, a condition known as Eisenstein's criterion. The pth cyclotomic polynomial always has this property for any prime number p, and if this property is not true over the rational numbers, it will not be true over the integers. For polynomials of degree two or three, this property will hold if and only if the polynomial has a zero. FTP, identify this property of polynomials, which states that any time the polynomial can be factored, one of the factors is a unit.
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All fundamental solutions of this result are generated from pairs of relatively prime integers which aren't both odd. Most proofs use either similarity arguments, the dissection of plane figures, or shears; the latter being the case in Proposition 47 of Euclid's Elements. For ten points, what theorem was demonstrated by Leonardo da Vinci and James Garfield, but was first proven 2500 years ago?
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Although M. L. Frankenheim was the first to attempt to enumerate them, he miscounted and came up with one too many of them, a problem corrected by their namesake. They are translation subsets of Fedorov groups and each one is given a Pearson symbol. The primitive one has ten Wyckoff points belonging to two separate tetrahedrally coordinated sets. Of the cubic-derived ones, one is triclinic, two are monoclinic, and four are orthorhombic including the simple, base-centered, body-centered, and face-centered. In all totaling 14 basic unit cells in three dimensions, FTP, identify this set of crystal lattices named after a Frenchman.
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Although a positive Lyapunov exponent is not required for this behavior, it is typical. It cannot occur in two-dimensional phase spaces but in three-dimensional flows such as the Lorenz equations used in an early weather simulation it can appear. For ten points, what term describes apparently random behavior in a deterministic system caused by sensitivity to initial conditions.
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An alternative version of this operation was conceived by and named for Percy Daniell, while Fatou's lemma states that this operation on a limit approaching infinity is less than or equal to the limit approaching infinity of this operation. This operator's namesake proved the dominated convergence theorem, which establishes conditions for which pointwise convergence commutes with this operation. The monotone convergence theorem does not work on this operation's predecessor, and that namesake's eponymous measure provides the grounding with which a function is integrated. For 10 points, name this type of integration, a generalization of the Riemann integral named for a Frenchman.
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An extension of this statement was refined by Wilfried Sieg to include only boundedness, locality, and determinancy, and Steven Wolfram expanded it to systems in the natural world. Tibor Rado introduced a function for which it doesn't apply known as the busy beaver, and it could be challenged by the existence of an oracle. One version of it grew out of Hilbert's decision problem, while another formulation stated that real world numerical problems could be expressed using the lambda calculus. For 10 points, name this doubly eponymous principle that says every computable function can be computed with one of its namesake's machines.
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An identity named for this man can be obtained by using only two, rather than four sets of elements in the Cauchy-Binet identity and gives a more general form of the Cauchy-Schwarz inequality. This man also names the remainder obtained in the Taylor series for a function. In physics, his namesake operator is equal to kinetic energy minus potential energy, laying down his theory of mechanics. Locations in an orbital system where centrifugal and gravitational forces balance are five points named for this man. For 10 points, name this mathematician whose namesake multipliers are used to maximize a function subject to constraints.
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Applying it in spherical coordinates to the function "one over r" gives a delta function. It is the operator appearing in the vector wave equation and Poisson's equation while in one Cartesian dimension it's just a second derivative. Equal to the divergence of the gradient is, for ten points, what scalar operator often written as "del-squared".
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As x approaches infinity, this function of x approaches e-to-the-x-power over two. It ranges up to infinity but never takes a value lower than one. It describes the shape of a hanging cable-the caternary-and it repeats itself every 2-pi-i radians in complex space. FTP, name this hyperbolic function, defined as one-half times (e to the x) plus (e to the minus x).
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Brent's method for doing this is sometimes called inverse parabolic interpolation. The Fletcher-Reeves algorithm and its superior cousin, the Polak-Ribiere algorithm, can also be used to do this, while the DFP and BFGS algorithms are examples of variable metric methods for doing this. Powell's algorithm is the prototype for direction set methods of doing this. This can also be done by annealing or through dynamic programming, whose "canonical problem" involves this. The easiest ways of doing it computationally are the golden section search and the simplex method. For 10 points, name this process which involves finding the highest or lowest point of a function.
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Capable of being bounded by Gresgorin discs, Jacques Strum first studied these numbers in a general setting. Often labelled by a lambda, their constituent set is referred to as the spectrum, and are solutions to the characteristic polynomial. Their sum equals the matrix's trace, and their product equals the matrix's determinant. FTP, what are these "values" which for some value of x and matrix A fulfill the equation A times x equals lambda x?
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Dagan, Golumbic, and Pinter defined the class of graphs that generalize interval and permutation graphs and correspond to this shape. Romberg's algorithm recursively uses a technique named for this shape that is the two points Newton-Cotes formula. That technique uses small line segments to approximate the definite integral. It's not a triangle, but the isosceles version of this shape has diagonals equal to the square root of the sum of the square of the leg length and the product of the bases. The area of this figure is equal to its height times the length of the central median, which is the average of the bases. For 10 points, name this quadrilateral with a pair of parallel sides.
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Damping constants beta one and beta two of these special cases of the harmonograph are equal to zero, and they remain open if the ratio of the periodicity constants from the parametric equation that describes them is irrational. Sometimes called Bowditch Curves, these patterns are produced naturally in tone analysis, which relies on a liquid medium and two tuning forks mounted at right angles to each other. FTP, name these figures created by the addition of harmonics along the x and y axes of a cathode ray oscilloscope, named for a Frenchman who depicted multidirectional vibrations.
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David Hilbert published a purported proof of this mathematical idea, but his proof was flawed. Godel proved that even the generalized form is consistent with the axioms of set theory. However, in 1963, Paul Cohen showed that negation of this hypothesis is also consistent, and the majority of set theorists now think that it is false. FTP, identify this hypothesis first conjectured by Cantor, which states that there can be no set with cardinality strictly between the cardinalities of the rational and real numbers.
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Delayed load scheduling is a variant on the Sethi-Ullman algorithm frequently used in simple implementations of one of their stages. One of these called ROSE produces a higher level output than usual, and up to three different systems are employed in the Canadian Cross version of one of these. Some of these perform strength reduction and rematerialization, the latter of which reduce calls to memory. Simpler ones execute in a single pass, and common optimizations in them include loop unrolling. Their traditional first step outputs a series of identifiers such as "Keyword" or "Integer" that are called tokens; that is the lexical analysis step. Self-hosting ones of these are called bootstrappers. Typically not performed by interpreted languages like Perl, Java uses a just-in-time one. For 10 points, name this operation that turns source code into a machine readable target language.
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During his later days at Gottingen, this man would travel to the lecture halls by bike, or by skis during the winter, and he was the calculus teacher of Max Van Laue. Mathematical results include an existence proof of Waring's problem, two proofs of Gordan's problem in invariant theory, and his "Nullstellensatz" theorem, which states that for every polynomial g vanishing on the ideal's set of vanishing points, some power of g is contained in the ideal. He published an important work on algebraic number theory in 1897, followed two years later by the Foundations of Geometry, but is most well known for a speech he gave at the International Congress of Mathematicians in Paris in 1900. FTP, identify this man who, in his paper, listed the 23 most significant unsolved problems in mathematics.
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During the French attack on Brescia in 1512 this man suffered a saber cut to the face, and while the story has it that a dog licked the wound clean and perhaps prevented a fatal infection, he nevertheless suffered a characteristic speech impediment for the rest of his life. He came to fame by defeating Antonio Fior in a mathematics contest using of an independently derived solution to certain kinds of cubic equations that he later confided to Girolamo Cardano, who broke his promise not to publish it, though he is perhaps better known for inventing the gunner's quadrant and for his discussion on falling bodies in his Nova Scientia. FTP name this man whose given name was Nicolo Fontana, the "Stammerer" who usually credited for inventing ballistics.
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Encapsulation can be accomplished by use of the opaque variety of this structure and segmentation faults are generally caused by access of a non-physical address by one. These explicitly allow manipulation of their contents, the namesake type of arithmetic. Iterators are an extension of these that are subject, usually, to postfix operators and a simple singly linked list object contains variable storage and one of these to the same type of object. As these do not contain literals but only the location of literals, they must be dereferenced before use, which is accomplished in C++ by the star operator. FTP, name these types of references that hold the memory address of something of interest, rather than the thing itself.
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Equations with a solution in the reals and in the p-adics for each prime p have a rational solution by the Hasse principle, an application of this statement. The decomposition of the finitely generated modules of a Dedekind domain into the direct sum of submodules is managed by this statement. The Good-Thomas algorithm for the fast Fourier transform uses this statement on the output, in conjunction with a mapping called Ruritanian on the input, to reindex data. An extension of this statement to principal ideal domains gives an isomorphism between a quotient ring and a product of quotient rings. For 10 points, name this statement that guarantees an integer solution to a set of congruence relations if the moduli are pairwise relatively prime.
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Equivalent to Tychonoff's Theorem, which states that any product of compact topological spaces is compact. It is also equivalent to the statement that any vector space over a field has a basis over that field. Cohen and Gödel showed that neither accepting it nor rejecting it leads to a contradiction. And Russell likened it to the ability to decide if a sock is left-footed or right-footed. FTP, name this mathematical assertion that is normally stated: for any collection of nonempty sets, we can choose a member from each set in that collection.
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Ernst Kummer showed that non-unique factorization produced a fatal flaw in the work of Cauchy and Lamé that could not be overcome for certain irregular primes. Gerhard Frey showed this problem's equivalence to the Taniyama-Shimura conjecture, which Ken Ribet proved in 1986, while three years earlier, Gerd Faltings showed it has a finite number of relatively prime solutions for n greater than or equal to 3. Finally, in 1995 Andrew Wiles proved, for ten points, what conjecture that there are no non-trivial positive integers such that x to the n plus y to the n equal z to the n for n bigger than 2?
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Euler defined a formula for performing this operation on trigonometric functions by rewriting trigonometric expressions in terms of “e to the i times x†and “e to the minus i times x.†One method for approximating this operation is by using Simpson's rule, which can be derived by using the trapezoidal rule. Another method of approximating this operation is by Riemann sums. For 10 points, name this mathematical operation, also known as the anti-derivative, which is defined as finding the area underneath a curve.
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Euler's study of the vibrations of a stretched membrane, and Daniel Bernoulli's studies of the oscillations of a chain suspended by one end led to earlier formulations of some of them. When Laplace's equation is formulated in cylindrical coordinates, these functions arise in the solution. Discovered by their namesake during a study of solutions of Kepler's equations, they are important in describing the deformation of elastic bodies, the diffraction of light, and the flow of heat or electricity through a solid cylinder. FTP, identify these functions named for their discoverer, a German astronomer and mathematician.
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Every connected one can be shown to be a manifold of constant curvature -1, 0, or 1. They are used to study sheaves of analytic functions and for the process of analytic continuation. The most trivial case is the complex plane itself. They arise naturally when examining how functions like the logarithm appear to become multi-valued in the complex plane. FTP, what are these eponymous surfaces named for the 19th century German mathematician who also lends his name to a type of sum often seen in introductory calculus classes?
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Extensions to this to make it better meet the Popek-Goldberg requirements include the Vanderpool project. A modern extension to this can run in both Long and Legacy modes. By default it disables the A20 line when started, which helps avoid crashes when running in Protected rather than Real mode. A successor to MMX called SSE added SIMD support to this system. The IOPL bit is present on systems of this type that are IA-32 compatible, while flags present on all of them include the Overflow and Carry flags. The company that standardized it attempted to depart from it with the Itanium project, and rivals to it include ARM and PowerPC. The platform on which the Celeron was built, for 10 points, name this 32 bit architecture powering Intel Pentium processors, which is slowly being supplanted by x64.
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Fibonacci heaps are used in Johnson's Algorithm to determine this, and an efficient way to determine it via a best-first method is A* [A-STAR]. One variety of this class of algorithms runs in N cubed time in the size of the graph and is the all-pairs approach, which is slower than the Bellman-Ford algorithm of this type. The simplest algorithm for determining this involves a single source and a single destination with nonnegative edge weights and is named for Dijkstra. For 10 points, name this class of algorithms which determines the minimum sum of the weights between two nodes of a graph, exemplified by the quickest way to get between two cities given the time spent on roads between them.
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First defined in 1905 by Pierre Fatou, it is now known by the name of the man who wrote the first computer program to view it in 1975. It can be shown that, if the modulus its recurrence becomes greater than 2, then it will certainly tend towards infinity. Representations of it are often shown in color, though technically only a black and white picture accurately portrays one, and even then all its complexity cannot be captured. Generated by the recurrence zn+1 = zn + c [z sub n plus one equals z sub n plus c], name, FTP, this set used to generate fractals.
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First described in a 1976 paper by Metcalfe and Boggs, it implements Carrier Sense Multiple Access with Collision Detect (CSMA/CD) technology. Its developers envisioned its segments as shared buses with decentralized control, in which each attached transceiver would be responsible for detecting packet collisions and implementing exponential backoff. Hosts are identified by globally unique 48-bit MAC addresses that are burned into each interface card. Described by IEEE standard 802.3, this is, FTP, what early local area networking technology still in use today?
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First proved by Brouwer in 1912, this theorem can be shown to be a consequence of the more general Poincare-Hopf theorem. A consequence of it is that any continuous function that maps a sphere into itself has either a fixed point or a point that maps onto its own antipodal point. One corollary of this theorem is that an any moment in time, there is at least one point on Earth where there is no wind, which therefore means that there must always be a cyclone on earth. FTP, name this theory of algebraic topology which states that given a vector field on the sphere there must be at least one point of the sphere for which the vector is zero, and which is usually visualized by thinking of a tennis ball.
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Fleischner's theorem states that the square of a two-connected graph has one of these, and Chvatal conjectured in 1973 that every t-tough graph has one of these for some integer t. Every graph with more than three vertices and a degree greater than half the number of vertices has one of these, a result known as Dirac's theorem. The task of finding these is incomplete for both directed and undirected graphs, and finding one of these for a graph with the minimum edge cost is the Traveling Salesman Problem. For 10 points, name this tour through a graph which starts and ends at the same vertex and contains every other vertex exactly once.
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For relative precision, its effective value is the coefficient of variation, while for population means, it is Z sigma squared over allowable difference squared. The model-based approach to obtaining it uses an unbiased predictor Z hat whose variance is a correction factor times the distribution variance over n for a mean estimator. Its namesake proportion is the fraction of successes of Bernoulli trials, and its space is the largest set in the sigma-algebra formed by the set of all possible events for a random variate. The finite population Central Limit theorem applies when its size goes to infinity. Markov chain Monte Carlo and simple random techniques are used to obtain, for ten points, what selection from a population that forms the basis of surveys?
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For stiff types of these, Richardsonian extrapolation algorithms far outperform the other general type. Methods for conquering them can be implicit, such as Adams-Moulton, or explicit, such as Adams-Bashforth. Some non-iterative solvers for these add a middle step called a modifier, however, these predictor-corrector methods are not self-starting and must be initiated, for example using Runge-Kutta [RUN-guh CUT-tuh] algorithms. They can be handled analytically using either D operators or integral transforms, both of which generally transform them into sets algebraic equations. FTP, identify this type of equation involving derivatives in only one dependent variable.
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GNU Guile is one language derived from it. It has a 1300 page draft standard, prompting the development of the Scheme dialect, a version stressing simplicity which is often taught in introductory CS classes and is specified in only fifty pages. Invented by John McCarthy at MIT, it is known for its ability to manipulate information contained in parentheses, including other programs, leading to its extensive use in the field of artificial intelligence. FTP, what is this programming language whose name is short for "list processing"?
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Gelfand's problem of describing all closed invariant subspaces of the integration operator may be solved using the Titchmarsh theorem of this type. This operation is often represented as a Toeplitz matrix, and the Dirichlet variety of this operation for a given integer n is defined using a sum over all divisors of n. Its discrete analog is naturally computed in "big O of N squared" with respect to resolution, but can be sped by switching to a circular domain of summation and using a namesake theorem to transform the problem to a product of fast Fourier transforms; this enables image processing applications like edge detection and modeling distortions such as blurring. For 10 points, name this binary operation that maps functions f and g to "integral f of x times g of quantity u minus x dx" and is denoted "f star g".
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He said, "Life is good for two things, learning mathematics and teaching mathematics." He was not just a pure mathematician, though; his 1835 Researches on the Movement of Projectiles in Air contained the first account of the effects of the Earth's rotation on motion. Named after him are a predicted "spot" in diffraction theory later verified by Arago, brackets in differential equations, an integral, and in the 1837 Researches on the Probability of Opinions, he defined a distribution that can be used as an alternative to the binomial distribution for very large samples. FTP, identify this French scientist who also discovered a ratio in elasticity which is constant for a given material.
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His conjecture says that if the determinant of the derivative of a polynomial mapping is one, its inverse exists and is a polynomial. One of the founders of elliptical theory with his four theta functions, his symbol in number theory is a product of Legendre symbols, and his name is also used for hypergeometric polynomials. FTP, identify this German whose identity for Lie algebras is a weaker version of associativity and whose matrix consists of the first partial derivatives of a vector-valued function.
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His father raised him to believe that Africans discovered America around 1000 CE. The recipient of the 1933 Bôcher Prize for his work on Tauberian theorems, this mathematician went on to codiscover the theory of the stationary time series with Kolmogorov. He gained a large degree of fame for developing a rigorous mathematical explanation of Brownian motion, but he is most famous for the discipline he founded that studies statistical methods of control in communications. FTP, identify this American mathematician and founder of cybernetics.
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His insistence on rigor in geometry led him to establish axioms of betweenness, congruence, continuity, parallelism, and incidence, discussed in his The Foundations of Geometry. His first major achievement was a proof of the finiteness theorem, and later accomplishments include a proof of Waring's theorem and a description of the infinite dimensional space used in functional analysis that now bears his name. Extending Kronecker's theorem, justifying Schubert's enumerative geometry, determining whether solutions to Lagrangians are always analytic, and the axiomatization of physics are four of the 23 eponymous problems posed by, FTP, which German mathematician in 1900 as the most important of the 20th century?
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His namesake points are the midpoints of the line segments that join a triangle's vertices to its orthocenter. His namesake squares are arrays composed of Latin Squares. His namesake characteristic is 0 only for the Klein bottle and the torus. Along with Lagrange, he names a differential equation involved in the calculus of variations. In 3-dimensional space, his rotation theorem says that any rotation can be described with the matrix formulation of his namesake angles. FTP, give this namesake of a formula that relates the vertices, edges and faces of a polyhedron and a constant equal to approximately .577, who also happens to be a Swiss mathematician.
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His namesake tetrahedrons circumscribe and inscribe each other. He introduced the null system in his book on statics, while his namesake net first appeared in a book which worked from the principle that a combination of weights at various points can be replaced by a single weight at the combination's center of gravity. That book on the "barycentric calculus" appeared in 1827, and was the basis of his later work on spherical trigonometry. He also published books on the path of Halley's comet and a notable treatise on celestial mechanics that didn't use advanced math, and in 1840 he asked how a kingdom could be divided into five parts such that each region would border on each of the others, a forerunner of the four-color problem. However, he is best known for an 1858 discovery found in an essay on the geometrical theory of polyhedrons. FTP, name this German mathematician who twisted a piece of paper and connected its ends to make a namesake strip.
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His work in set theory included proving the consistency of the Axiom of Choice by analyzing the inner model of constructible sets, and in philosophy, he formulated a symbolic version of Leibniz's ontological proof of God, though he is best known for his results in mathematical logic. He showed in his doctoral thesis that all true statements of first-order predicate calculus are provable, while his most famous result states that this is not the case for any logical system powerful enough to describe the Peano axioms of arithmetic. FTP, name this Eastern European mathematician, most well-known for his "Incompleteness Theorem."
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Hofstadter's objection to this focuses on the "A" property of letters and Bingsjord's anonymous straw-man points out its failure for mind-like entities. The Zuse-Fredkin hypothesis is a form of its "strong version," which compares the universe to the machine to which it applies. Ackermann's function lends credence to this idea, which resulted from attempts to prove the Entscheidungsproblem unsolvable. Its two most important requirements are that functions be recursive and lambda-definable. FTP, what is this thesis that states that all naturally computable effective algorithms may be carried out by a Turing machine?
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If two variates obey this, their difference obeys a Skellam distribution. It has moment-generating function e to the quantity lambda times quantity e to the t minus one, where lambda is the only parameter on which this distribution depends. Bortkiewicz called it the "law of small numbers" after noting that it models rates of rare events like radioactive decays or horse-kick deaths. This function gives as lambda to the n times e to the minus lambda over n factorial as the probability of n successes from a large number of the trials named for its French namesake. FTP, name this distribution usually symbolized P.
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In 1838, De Morgan coined the term for this technique and codified its use, although it had already been utilized to celebrated effect by Euler in his proof of Fermat's Little Theorem. The "transfinite" type can be used in arguments involving transfinite numbers, although an extra step is usually required to account for cases involving limit ordinals. Calculus students often encounter this type of argument for the first time in proofs of summation formulas. In it, the result is proven for a starting number, often 1, after which it is argued that the case involving the number n is true if one knows that the result is true for numbers less than n. FTP, what is proof technique whose name is also used in philosophy for the opposite of deduction?
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In 1963 Paul Cohen introduced the method of forcing, in his proof that this concept's generalized version implies its original formulation. Combining a proof by Gödel with Cohen's proof shows that it is independent of the axiom of choice. It is believed that to disprove it as a theorem would require axioms that guarantee the existence of sets having cardinalities different than those proven to exist. For 10 points, name this conjecture that can be stated as: there is no cardinality between an infinite set's cardinality and that of its power set.
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In 1980 Robert Axelrod held an experiment in which a number of computer programs opposed each other in this game, and determined that the best long-term strategy is to adopt a forgiving approach. Applicable to real-life problems such as the nuclear arms race and price wars, it was first formulated by Albert Tucker, and concerns two criminals who must each decide whether to confess to a robbery. FTP, what is this famous dilemma?
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In 1994, Blair Spearman and Kenneth Williams showed that they could be successfully dealt with only if two of their coefficients satisfied certain constraints. Three of their other terms can be eliminated before solving them by using Tschirnhaus transformations. Charles Hermite showed that they could be solved using elliptic functions, familiar from their use in solving lower-degree analogues of these equations. The proof of an important theorem concerning these equations depends on finding a subfield of a certain degree whose splitting field over the rationals has a Galois group isomorphic to a certain symmetric group. For 10 points, name these equations which can be solved if their corresponding Galois groups are solvable, and for which no general radical solution exists, as shown by the Abel-Ruffini theorem.
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In 1994, Leonard Adleman used a variety of molecular biological and genetic techniques to have DNA find a solution to this problem. Since it is NP-complete, the massive parallel computing power of a solution of DNA could be desirable to find a solution for problems with many vertices. To find a solution for a given directed graph, one must begin at the designated starting point, finish at the designated end point, and pass through each vertex exactly once. FTP, identify this problem of which the traveling salesman problem is a simplifie version and which is named for a 19th-century Irish mathematician.
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In an April fool's column, Martin Gardner claimed that "e to the pi radical 163" is one of these. The Eisenstein ones are proper supersets that are extended with complex numbers. They form an Abelian group when endowed with addition and a commutative ring when endowed with addition and multiplication, but they are not a field, as they lack multiplicative inverses. As a set, they have cardinality "aleph 0", although they cannot be enumerated monotonically. The floor and ceiling functions yield, for 10 points, what type or numbers represented with a bold "Z," the set that is exactly the union of the natural numbers, their additive inverses, and zero.
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In chemistry, they are used to describe molecular symmetries, and in physics they have led to new particles being discovered. In math the smallest has order 1, has only the element e, and only the finite simple ones have been classified. All of them contain an identity element and an inverse for every element in the set, while commutative ones are named after the man who generalized the Binomial Theorem. FTP, name this set with an associative binary operation under which the set is closed and whose best-known type might be the Abelian.
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In mathematics, this name is given to a theorem derived by using the Cauchy integral theorem to discard low- and high-order terms of the closed path integral of the series expansion of a meromorphic function. That theorem relates to this, its namesake object, equal to the first negative-power term of the Laurent series of a function about an isolated pole, or to one over two pi i times the integral of a function about a closed contour enclosing a single isolated pole. In chemistry, this term is contrasted with moiety and refers to the portion of a monomer present in a polymer chain, especially an amino acid in a protein chain. In number theory, this is another term for the principal part of a congruence. FTP, give this term from chemistry and from mathematics, in which it is synonymous with remainder, as in everyday usage.
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In order theory, this is a subset of a partially ordered set that is both lower and directed, and the dual notion is called a filter. Krull's theorem on these states that in a commutative unit ring, the height of every proper one generated by n elements is at most n. They were first proposed by Dedekind in Lectures on Number Theory to generalize the number of this type introduced by Kummer. The most general form of the Chinese Remainder Theorem concerns these when they are pairwise coprime, and they are the object in ring theory analogous to normal subgroups. In the ring of polynomials all of these are principal, which is to say generated by one element. FTP, name this mathematical object, defined in algebra as an additive subgroup of a ring that is stable under multiplication.
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In order to have finite limits on these objects from the categorical point of view, nilpotents must be allowed in the sheaf of rings. Grothendieck generalized them to obtain his theory of schemes, but lost some of their content in the process. These mathematical objects are reduced schemes of finite type over a field and kinds of these objects include the Abelian, Albanese, Chow, and the Brauer-Severi. FTP, name these sets of zeroes of non-empty sets of polynomial equations, the basic object of study in algebraic geometry.
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In pattern theory, an object of this name is characterized by having attributes and bonds. In coding theory, the Golay code is an example of this kind of matrix, while an integer lacking a digit addition variety of one of these objects is called a self number. In a Banach space, an infinitesimal one is defined as the limit of a linear operator T of s minus the identity divided by s as s goes to zero. A Lie algebra defines the commutation relations obeyed by representations of these objects, and finite-dimensional Hermitian matrices can be used to represent those of a compact Lie group. In quantum mechanics, the angular momentum operator has this relation to rotations, and a group is called cyclic if some element in it has this relation to the group. For 10 points, identify this term which in group theory identifies an element whose products with itself give every element in the group.
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In the standard implementation of it, the worst case runtime occurs when the input data is already in the desired order. The recursive, partitioning nature of this algorithm makes it susceptible to boosts from parallelization. It can be optimized using in-place memory allocation to use only Big O of log n additional memory, and its worst case runtime can be avoided by picking a random element of the list to use when subdividing the list into smaller lists. FTP, identify this algorithm that operates by moving elements along a pivot that runs in big O of n log n time with a small constant, named for its speedy organization.
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It can be used to simulate more complex data structures like mergeable heaps, and has a worst case search time on the order of the number of objects, since each object needs to be examined until a match is found, or until a null pointer is reached. Variations include also keeping a previous pointer in order to facilitate traversal, and the circular form, where the head object points to the tail and vice versa. FTP, identify this simple data structure, similar to an array, but which uses pointers to arrange the objects in a linear order.
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It is a special case of a conjecture that one half times the smallest integer less than the quantity seven plus the square root of 48 times the genus of an orientable space plus 1, close quantity, is equal to a certain number associated with the space. Originally known as Gurthrie's conjecture, a fallacious counter example was published as an April Fools day joke. This theorem showed that except in the case of the Klein bottle, the Heawood conjecture holds. Originally solved by reducing the problem to 1,936 cases that were checked by computer, for 10 points, identify this theorem, which states that any map in a plane can be filled in with a certain number of colors.
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It is a topological invariant which is related to the curvature of a surface, a local geometric quantity, through the Gauss-Bonnet theorem. It is computed as an alternating sum of Betti numbers. It is also related to the sum of indices of a vector field with isolated zeros, by the Poincare-Hopf theorem. For a surface of genus g, it is equal to two minus two g. FTP, what is this characteristic of a surface, which for polyhedra is equal to two and given by vertices minus edges plus faces?
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It is concave due to the law of increasing opportunity costs. Any point outside of it is unattainable, any point inside it is attainable but inefficient, and points on it are attainable and the most efficient. It can move to the right if resources increase in quantity or quality or technology advances, but it normally assumes full employment, fixed resources, fixed technology, and a two-product economy. FTP, name this curve which relates the maximum outputs of two goods that use similar resources.
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It is equivalent to the n=5 and n=6 cases of the Hadwiger Conjecture, and may be viewed as the genus zero case of the Heawood Conjecture, which gives upper bounds on chromatic numbers for graphs that can be embedded in various topological surfaces. In 1890, Heawood pointed out the error in Alfred Kempe's proposed proof of it by using an example with 18 regions. In 1976, Wolfgang Haken and Kenneth Appel proved it by combining Kempe's idea with a controversial use of computer analysis. FTP, what is this theorem stating that only a certain number of hues are needed to illustrate a map so that no two neighboring regions are shaded the same?
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It is especially useful in converting negative or fractional exponents into ordinary algebraic expressions from which the leading-order dependence may be determined. It can be written in the form "(n choose r) times a to the n minus r, times b to the r. Newton stated it in general form in 1676, and Bernoulli's later proof of it was published in 1713. It is often connected to Pascal's triangle, in which you can quickly look up the coefficients of any term in an expansion. FTP, name this theorem that allows for the expansion of its namesake algebraic expression without requiring the explicit multiplication of the binomial terms.
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It is often used instead of Wilson's Theorem in primality tests. It is sometimes rephrased using its Corollary which states that if A is an integer and p is prime, then A raised to the pth power equals A modulo p. Its more traditional form, however, states that for an integer A and prime p, A raised to the quantity p minus 1th power equals 1 modulo p. FTP, identify this theorem first proven by Euler but earlier stated by a French lawyer and amateur mathematician.
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It is the subject of a treatise by David Gregory. Euler showed that one revolved around its asymptote generates the only minimal surface of revolution. This term was first used by Huygens in a letter to Leibniz, where it was described as the evolute of the tractrix and the locus of the mid-point of the vertical line segment between the curves ex and e-x. Its Cartesian equation is given by y equals a times the cosine of x over a, which was found after Jacob Bernoulli issued a challenge to find the equation of the "chain curve." FTP, identify this curve that is a suspended flexible chain acted on by gravity.
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It is thought that Euler and Lagrange both explored rudimentary forms of this procedure. The techniques leading to the development of this procedure were explored in detail by Oliver Heaviside over a century after its namesake first conceived it. This procedure can be performed unilaterally and bilaterally but it can be reversed using a formula named for Bromwich. One way of arriving at its results is computing a generalized product through a convolution integral. This procedure gives rise to an operator that is linear and returns unique results. The transfer function obtained by this procedure can be identified as the impulse response of the system. One prominent application of this technique is solving differential equations with constant coefficients by turning differential equations into algebraic polynomial equations. For ten points, identify this transform named for a French mathematician whose unilateral is formally defined as the integral from zero to infinity of e to the negative s t times f of t with respect to t.
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It provides the only exception to the Heawood conjecture, as 6 colors are required to legally color its regions when the Franklin Graph is embedded on it. Its topology is equivalent to two cross-caps with coinciding boundaries, and one can cut it and produce either one or two Mobius strips. It is often thought of as the gluing of opposite ends of a rectangle while giving one pair a half-twist. It can be immersed in three-space, but an embedding of it requires four dimensions, as it must pass through itself without creating a hole. FTP, name this nonorientable, one-sided closed surface named for a German mathematician.
one regarded as a set of moving particles. If the Einstein summation has been used, then it can be defined as q, the generalized coordinate, times p, generalized momentum, minus the Lagrangian. Representing the sum of all kinetic and potential energies of the aforementioned dynamic system, FTP, identify this function named for the Irish mathematician who discovered it.
It was originally introduced to express the rate of change in time of the condition of a dynamic physical system
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It's equivalent to the binomial distribution when the average number of events is small and the possible number large. It's the distribution usually governing traffic patterns and radioactive decays, but as the average number of events increases it becomes a Gaussian. For ten points, what probability distribution is named for its French discoverer?
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Its "generalized" form is also known as the Richards function, and its derivative is the Hubbert curve. It is useful in Rasch modeling, in which the subject's ability to answer and the question's difficulty are equal at a point where the probability of a correct answer is one-half, while it's also used as an activation functions in neural nets. Functions of this form are relevant to autocatalytic reactions, where the product catalyzes its own formation, and it was first described by Verhulst in response to statements by Malthus. For 10 points, give the term for this function, such as e to the x over quantity one plus e to the x, the most common example of a sigmoid curve, typically applied to problems of population growth.
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Its complexity of n log 2n allows it to be used on large quantities of data, a significant improvement over the n2 complexity of its discrete counterpart. Its applications include quickly multiplying large polynomials and compression analysis, but wavelets are starting to replace it in practical applications because of the Gibbs phenomenon at sharp transitions. It is based on the discovery that it is possible to take any periodic function of time and resolve it into an equivalent infinite summation of sine and cosine waves. FTP, name this process of breaking a complex signal into an infinite series.
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Its developer was forced to flee to Derby after claiming to have "raised a devil" from a young girl. It has an error roughly equal to 1 over n to the fourth, and because of its construction, n is required to be even. If h represents the length of the n equal subintervals of the interval from a to b, then the formula for it reads 1/3 h times the sum of the values of the points of the partition with various coefficients, which with the exception of the first and last alternate between 4 and 2. FTP, what is this method for approximating definite integrals using parabolas, named for its British inventor?
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Its inventor, Guido van Rossum, claims that he was influenced primarily by ABC and Modula-3. It supports objected-oriented and generic programming styles, and is perhaps best known for its emphasis on readability, with indented blocks and a notable lack of semicolons compared to many other languages. FTP, identify this programming language with a name inspired by a 70s British comedy series rather than a type of snake.
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Its multi-bodied generalization is known as the free rider problem. One of its cases was the subject of a tournament hosted by Robert Axelrod at the University of Michigan consisting of computer-implemented personalities representing the players. Each algorithm attempted to maximize its payoff matrix, a tool useful to game theorists, with the winner being the simple program tit-for-tat, which cooperated with those that didn't betray it. FTP, name this problem of game theory that illustrates the difficulties that arise in noncooperative games by utilizing a deal offered to two potential criminals.
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Its transitional densities satisfy the Chapman-Kolmogorov equation, and if it is Gaussian is said to obey Doob's theorem. Its transition matrix is generally related by the element P sub i j, which represents the probability of transitioning from state i to state j. This statistical entity is best exemplified by the simple random walk. FTP, identify this concept named after a Russian mathematician defined as a set containing a finite number of states for which the probability of being in each future state depends only on the set's present state.
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JPEGs are compressed using the discrete transform of this name, which separates the image pixels into bands of similar importance. A correlation representing this function, also known as uncentered correlation, is used in pattern recognition and information retrieval to determine the similarity of two feature vectors via a normalized dot product. Its hyperbolic form describes the shape of a catenary, and for two vectors of unit length, taking a dot product will give you this function of the angle between them. The reciprocal of the secant function, FTP, name this trigonometric function, which, for an angle in a right triangle, equals the ratio between the adjacent side and the hypotenuse.
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Kenneth Bowles at UC-San Diego provided a version of this language that increased its popularity, and its developer won the Turing Award in 1985 for its invention. Important features include the BEGIN and END commands and although it wasn't initially released with object-oriented programming, in 1986 Apple released a version that did include the feature. It was developed to be suitable for teaching purposes in 1970 by Niklaus Wirth. FTP, identify this programming language named after a seventeenth century French mathematician
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Kenneth Bowles, a professor at UCSD, oversaw the development of a version of this computer language that allowed the underscore to be used in identifier names. The UCSD version also implemented the string variable type as a better alternative to packed character arrays. Other data structure include sets, arrays, and records, in FTP, what beginner computer programming language which encloses compound statements in begin-end blocks?
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Korselt's criterion can be used the check for them, and they form the only candidates for solutions to Lehmer's Totient problem. In order to be one of these, a number n must divide the denominator of n-1th Bernoulli number, and their general definition requires that n is one if and only if the n-th power-raising function from the ring Z-sub-n to itself is the identity. Alford, Granville, and Pomerance proved in 1994 that there exist infinitely many of these numbers, beginning with 561, 1105, and 1729. Squarefree and having at least three prime factors, these numbers satisfy the congruence of Fermat's Little Theorem for all choices of base. For ten points, identify these numbers that are often also called absolute pseudoprimes.
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Kummer showed that early attempts at proof of this depended on the faulty assumption that factorization is unique in all rings of integers or algebraic number fields. The correct argument uses L-functions, which are generalized zeta functions; a combination of Kolyvagin-Flach theory and Iwasawa theory; and Hecke algebras. In 1983, Ribet noted that a counterexample to it would imply the existence of a non-modular elliptic curve, contradicting a special case of the Taniyama-Shimura conjecture. In 1995, Taylor and Wiles proved that special case of that conjecture and thus proved, FTP, what centuries-old math problem named for a French margin-scribbler?
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Lemmas named for him include one that determines the angle that a tangent vector makes with a geodesic, while another states that a polynomial is irreducible over the integers if and only if it is irreducible over the rationals. His eponymous curvature is the product of the principal curvatures of a surface, while his eponymous function is its own Fourier transform and is important in probability theory. As a teenager he proved the law of quadratic reciprocity and the fundamental theorem of algebra. FTP, name this mathematician who also developed the method of least squares and the normal probability distribution.
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Like unit rings, properties of these mathematic objects include commutativity, associativity, the existence of unique additive and multiplicative identities, and a unique additive inverse for every element. Finite dimensional ones have a natural isomorphism with their double duals, are free, injective, projective, and flat as modules, and are isomorphic to a finite Cartesian product of fields. Zorn's Lemma can be invoked to show that a set of elements exists in any of them such that any element can be written uniquely as a linear combination of finitely many elements in that set. FTP, name these objects which are modules over a field, of which an example is three dimensional real space.
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Lindstrom's theorem states that first-order logic is the strongest logic having both this property and satisfying the Lowenheim-Skolem property. A method for finding a Hausdorff space with this property containing a given space is named for Stone and Cech. Equicontinuity is the main condition for determining whether a family of functions has this property by the Arzela-Ascoli theorem. Using the axiom of choice, it can be shown that an arbitrary product of spaces with this property has this property, a result known as Tychonoff's theorem. In Euclidean space, being closed and bounded is sufficient to show, according to the Heine-Borel theorem, that a set has this property. For 10 points, identify this property which holds of a set on which every open cover has a finite subcover.
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Nicholas of Cusa studied them while attempting to find the area of a circle by integration, Vincenzo Viviani found a method of constructing tangents with them, and Charles Bouvelles discussed them in 1501 as a mechanical means of squaring the circle. Pascal thought about them to take his mind off of a toothache, and later wrote a "History of" them. No matter where a particle is placed on an inverted one of these, it takes the same time to slide to the bottom, according to the Tautochrone problem. Also the solution to the Brachistochrone problem, this was named by Galileo, who suggested that arch bridges be built in the shape of this, and Huygens proposed that pendulum clocks swing in this kind of arc. FTP, name this curve traced out by a point on the circumference of a circle as the circle rolls along a straight line.
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Niklas, Sune, and Allan are moves in the Petrus System used for doing this, and that system was designed to be a replacement for the more layer based Fridrich Method employed by the likes of Tyson Mao. In Hellboy, Abe Sapien laments only accomplishing one third of this action, and a blind man in UHF repeatedly fails to do this. In the Pursuit of Happyness, Will Smith does this in a taxi to impress his future employer. Erik Akkersdijk holds the world record for doing this in the 2x2x2, 3x3x3, and 4x4x4 categories. For 10 points, name this action that involves isolating blue, green, yellow, red, orange, and white tiles on separate sides of a six sided solid.
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One analogy used to illuminate this principle involves a device capable of answering all questions, known as a Universal Truth Machine. If one writes a statement such as "The machine will never say that this sentence is true," and then asks the machine whether the statement is true, the fallibility of the device is demonstrated. It states that the propositions on which mathematics are based are unprovable, because it is possible, in any logical system using symbols, to construct an axiom that is neither provable nor disprovable within the same system. FTP, identify this proposition put forward in 1931 by an Austrian mathematician.
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One class of them correspond to irreducible root systems; that class has extended and affine varieties, the latter sort of which can describe Cartan matrices. The Hadwiger conjecture concerns these objects, as does the Robertson-Seymour theorem. Some functors have these objects as domains, and in that situation they are called quivers. The clique number of one of these objects is denoted by omega, and one class of them define a Weiner polynomial and may be represented by an adjacency matrix. One kind, called a forest, disallows cycles, while in quivers, loops are allowed. Forests are so called because they could be considered collections of trees, which are, for 10 points, a subset of what mathematical constructs composed of edges and vertices?
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One early method for this procedure was Fourier-Motzkin elimination, and the Hirsch conjecture provides an upper bound on the runtime of one method. Another method involves introducing slack variables and then manipulating the tableau through repeated pivoting. Karmarkar's algorithm and other interior point methods are among the fastest solutions, and the ellipsoid algorithm was the first polynomial time algorithm for it. The cutting-plane method can be used to solve the integer variety, which is NP-hard but which can sometimes be approximated by examining the dual. Originally developed by Kantorovich to optimize plywood production, FTP, name this problem, efficiently solved using the simplex method, which calls for the optimization of a linear objective function given linear constraints.
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One example is that the power law behaviors of the Ising ferromagnetic transition and the liquid-gas phase transition are identical, because in each case there is a scalar order parameter. Another example is that in one-dimensional maps that approach a chaos in a period-doubling cascade, the ratio of successive doublings has a fixed value called the Feigenbaum constant. In general it is a consequence of the renormalization group: roughly speaking, details of microscopic behavior are invisible at long distances. FTP, what is this term, which in computing is synonymous with Turing completeness?
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One example of these operations carries a subset of a topological space to the set of its accumulation points, and is named for Cantor and Bendixson. One of these operations defined on Banach spaces is named for Fréchet. Christoffel symbols can express the Levi-Civita connection, an example of the covariant type. Closed forms are those which vanish for the exterior type of this operator. The complex type must return the same value regardless of the direction of evaluation, giving the Cauchy-Riemann equations. For a vector-valued function, all of the first-order ones are collected in the Jacobian matrix. FTP, name this operation which satisfies the chain rule and whose inverse is, according to the Fundamental Theorem of Calculus, the integral.
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One example of this class of programs is the operator-precedence type which has difficulty processing symbols with multiple meanings. Top-down ones include the packrat and the LL algorithm while chart ones include the Earley and the CKY algorithms and are frequently used for natural languages. Handle pruning is a technique used in shift-reduce types of them. Some compilers use a namesake generator, such as GNU Bison, to create one of these for each code. These usually operate on the output of a lexical analyzer and themselves output a tree of tokens. FTP, identify these programs that analyze an input sequence and determine its grammatical structure.
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One mathematical result of this name may be generalized to the statement that every holomorphic function from a parabolic Riemann surface to a hyperbolic one must be constant. A generalized version of the physical result of this name can be derived for non-Hamiltonian systems by introducing a metric factor. A more specific mathematical result of this name states that a bounded, entire function in the complex plane is constant and has the fundamental theorem of algebra as a corollary, and is unrelated to the physical result. That physical theorem states that in a Hamiltonian system, the phase space density is conserved. FTP, give the common name of these theorems, whose namesake was a Frenchman whose work with Jacques Sturm resulted in a theory used to solve integral equations.
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One mechanism in this component can be implemented with either a Second-Chance Algorithm or an LRU replacement scheme. This entity needs to find a victim frame in cases where there are no free frames. It's not a cache, but Belady's Anomaly states that increasing the number of frames can lead to slower performance when implementing it with First In First Out frames. A translation lookaside buffer can improve the speed at which it translates addresses from page tables, and thrashing occurs when too much of it is allocated and a system spends all of its time retrieving pages. Its ability to provide programs with an address space creates an extra layer of security in operating systems, and provides an abstraction for non-contiguous memory regions. FTP, identify this representation of computer memory that often involves using the hard drive.
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One of his most influential papers contained sections on "Elements, Synchronism Neuron Analogy" and "The Binary Point". Ravi Lonberg implemented a random number generation technique he posited in the paper "Various Techniques Used in Connection with Random Digits", based on the theory of cellular automata he helped develop. With Fermi and Ulam he is credited with developing Monte Carlo algorithms. One of his most influential ideas, which had the unintended consequence of making buffer overflow attacks possible, was laid out in his "First Draft of a Report on the EDVAC". Also known for developing co-developing game theory with Oskar Morgenstern, for 10 points, name this Hungarian born mathematician who proposed storing programs and data in the same memory component, his namesake computer architecture.
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One of the developers of this algorithm collaborated with Teller and Fermi on the first nuclear reactor. In chemistry, the First Reaction method is a dynamic type of this algorithm, and is used to model molecular behavior. One version of this technique is used to determine solutions for complex Ising models. Simulated annealing is a heuristic approach to global optimization using this method, and the Metropolis-Hastings algorithm is used to simulate a Markov Chain. A Las Vegas algorithm is a special case in which the outcome is always correct. FTP, name this type of algorithm, which refers to another center of gambling and which uses random numbers to solve problems, which is most notably attached to a method of numerical integration.
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One of the terms in its name is the imaginary part of the generalized AC susceptibility. It was established by Callen and Welton in 1951, and it may be formulated by noting that the linear susceptibility of an order parameter is given by minus beta times the time derivative of the correlation function. The susceptibility term characterizes an effect that could be associated with friction, and the result holds assuming the external interaction can be treated as a small perturbation. On the other side of the equation, the correlation function characterizes the expectation value of statistical deviations of the perturbed quantity. FTP, what is this theorem relating two important quantities in statistical mechanics?
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One of them which involved Fuchsian systems and monodromy groups was addressed by Josip Premelj, but his work was shown to be flawed by Andrei Bolibruch. Another was addressed using the "p-adic" system developed by Helmut Hasse, while Montgomery and Zippin extended the work on one of them by Andrew Gleason. One of them considers the physical sciences, and mentions the kinetic theory of gases and the "laws of the motion of rigid bodies" as subjects which merit an axiomatic treatment. The first to be addressed was considered by Max Dehn, who showed that a regular tetrahedron and a cube of the same volume cannot be divided into identical piles of pieces. Some of them involved the "most general law of reciprocity in any field," "building up space from congruent polyhedra," and the extension of Kronecker's theorem on Abelian fields. The first of them asked about Cantor's problem of the cardinal number of the continuum. FTP, name this program for research which was announced at a lecture in 1900, in which a German professor suggested that his colleagues work on 23 questions.
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One of these constructs is called perfect if it is equal to its own commutator. The purpose of the Hölder program was to classify a certain type of these. Those of odd order are solvable according to the Feit-Thompson theorem. Any one of these of prime order is also cyclic. Symmetric types of this algebraic structure consist of permutations of a given number of elements. Examples of these include one that has a 194 by 194 character table created by Fisher and Greiss. One of these of order six is the smallest one of these that is non-commutative. They must be associative and have an identity, inverses and a zero element, and Abelian ones are commutative. For 10 points, name these algebraic structures that consist of a set and a binary operation performed on that set.
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One of these named after Dirichlet is obtained by integrating the number theoretic character over a ball, while one named after Poisson is used to find the values of a harmonic function in an open disk. A technique known as this type of polynomial method uses Chebyshev expansions to calculate quantities in condensed-matter physics, and the Green's function is an example of the integral type of this mathematical object, which appears in all integral transforms. In group theory, it is a normal subgroup that comprises the set of all elements mapped to the identity by a group homomorphism. For ten points, identify this mathematical object, which in linear algebra is identical to the null space of a linear transformation.
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One of these which performs an important action only when writing data to disk is used in the Reiser4 file system and is known as a dancing one. The Day-Stout-Warren algorithm runs in linear time when operating on these objects, of which the radix variety is used to store sets of strings. The tasks of spatial searching and the storage of database information usually use the R and B varieties respectively, while types known as scapegoat, splay, red-black, and AVL are all self-balancing and binary. For ten points, identify these directed acyclic graphs, a type of data structure which consists of nodes that each have one parent and possibly many children.
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One procedure that uses this technique calculates the target score of the second-batting team in an interrupted cricket match, while another is used for finding the maximum likely path in a Hidden Markov model. In addition to the Duckworth-Lewis method and the Viterbi algorithm, its also used to evaluate B-spline curves in De Boor's algorithm. The optimality condition for this procedure is given by Bellman's equation, and its more famously used in the Needleman-Wunsch sequence alignment algorithm and to find the longest common substring of a set of strings. This technique also relies on optimal substructure and recursion. FTP, name this programming technique that divides a problem into a set of subproblems.
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One representation of these objects is called the Fary embedding, and these objects also satisfy the MacLane and Schnyder criteria. Hopcroft and Tarjan formulated an efficient algorithm for checking for these objects, while another way of testing for them uses the Fraysseix-Rosenstiehl left-right algorithm. The Steinitz theorem couples them to a polyhedral representation, while the Whitney criterion states that their set must be closed under the matroid duality. According to Barnett's conjecture, every bipartite cubic one of these is Hamiltonian, and one of these must not contain either K-sub-5 or K-sub-3,3 as a minor, a result known as Kuratowski's theorem. For 10 points, identify these mathematical objects, which consist of vertices and edges that can be drawn on a flat surface.
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One statement of this theorem relates a quotient ring of a ring R over the intersection of its ideals to the product ring formed by multiplying the quotient rings of R over individual ideals. This theorem may be used for constructing Gödel numbering for sequences, allowing the proof of Gödel's incompleteness theorem, and it is central to solving problems of secret sharing by finding minimal combinations of shares to recover the whole. A more widely known application is representation of the private key in RSA. For 10 points, name this theorem which states that for two relatively prime integers r and s, there is an integer N such that N is congruent to a mod r and N is congruent to b mod s, where a and b are integers, named for the country in which it was formulated by Sun Zi.
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One theorem named for this man states that the group of units of a ring is finitely generated, and one function named for this man has a pole at s=1 if chi is a principal character. Another theorem named for this man states that there are an infinite number of rational numbers p over q within one over q squared of any given irrational number, a result known as his diophantine approximation theorem. His namesake statistical distribution is the conjugate prior of the multinomial distribution, and his original proof of his namesake theorem on arithmetic progressions of primes used his L-functions, which are also used to generalize the Riemann hypothesis. He gives his name to values that solutions of a partial differential equation, but not their derivatives, must take on a border, known as his namesake boundary conditions. For 10 points, name this French mathematician who also lends his name to the pigeonhole principle.
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One use of this adjective is in general relativity, where a spacetime is said to be globally this if it has a Cauchy surface. When applied to a knot, it means that the knot complement can be given a metric of constant curvature negative one. In the theory of partial differential equations, it refers to a large class of equations including the wave equation. It also describes a type of geometry studied by Lobachevsky and Bolyai. FTP, what is this adjective deriving from the conic section with two branches?
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One version of this statement made use of lambda definability, which is described as a function that can be calculated by a process of repeated substitution, and another was originally developed while attempting to prove that a posed problem for the predicate calculus was unsolvable. Copeland and Proudfoot wrote a controversial Scientific American article about it, in which they suggest the creation of an oracle. It says that the halting problem is undecidable, and though it may not be able to be proved, every realistic model discovered has proven to be equivalent. FTP name this thesis that says any effective computation can be translated into an equivalent computation by one of its namesake's eponymous machines.
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Operator algebras on objects of this type are known as W-star or von Neumann algebras and it was von Neumann who named these structures in a 1929 paper in Mathematische Annalen. Grover's algorithm requires two angles between three members of one of these corresponding to the state space of a quantum computer. Every one of these is also a Banach space, but the converse does not hold. They are inner product spaces such that the resulting norm makes them complete metric spaces. FTP, name these algebraic structures named for a mathematician who delivered a famous 1900 address to the Paris International Congress of Mathematics.
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Ordinary numerical integration causes systems of this type to deviate, so symplectic methods are often used instead. Systems of this type obey Liouville's Theorem, and their dynamics can be described by taking derivatives of a single function on phase space. Their evolutions are determined by the functions of the same name that express their energy in terms of momentum and positional coordinates. FTP, what are these dynamical systems named for the developer of quaternions?
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Proof of the elementary form of this result hinges on an argument that, by hypothesis, the greatest common divisor of the moduli of two congruences divides the difference of the bases of those congruences. That elementary form, may be easily generalized to an arbitrary system of congruences in certain cases. The earliest known use of this lemma is in the solution to Problem 26 of the Sun Tsu Suan Chang, in which the moduli of a set of three congruences are prime and, therefore, coprime, as this theorem requires. FTP, name this theorem from number theory that states that a unique solution exists for a system of simultaneous congruences with pairwise coprime moduli and that is named for its Asian country of origin.
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Rodriguez-Rivera et al created one algorithm for doing this for Geodesic's Great Circle which is both non-moving and reduces fragmentation. Another method for doing this portions off half of the target at any one time and is known as Cheney's Algorithm. The generational type of this process relies on the infant mortality principle to focus on recently created objects. Incremental forms of this process have fewer performance repercussions than stop-the-world techniques, and several implementations use reference counting. A feature of managed languages, they help prevent bugs related to dangling pointers and can be implemented with a mark-and-sweep algorithm. For 10 points, name this process in computer science in which data no longer being used is reclaimed in memory.
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Suppose that f is a piecewise smooth function. Then, for f, this converges to the arithmetic mean of the left hand and right hand limits of f at points where f is discontinuous. It can be understood by considering a 2n-dimensional functionspace with the set of the sines and cosines of the products of the integers between 0 and n and x as it basis; each of its namesake coefficients is then the dot product of one of the basis functions with a function f(x) and becomes an integral over the functionspace as n approaches infinity. First introduced in its namesake French mathematician's 1822 treatise The Theory of Heat, FTP, name the type of infinite trigonometric series that is invaluable in the study of partial differential equations.
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The Cohen-Macaulay variety of this mathematical construct contains the class of the Gorenstein variety, which includes all regular local types known as the Artinian type. Perhaps the most famous kinds are the Dedekind type, which is a commutative one that is necessarily a Noetherian one. Considered Abelian under addition and a semigroup under multiplication, FTP, identify this mathematical idea defined as the set of numbers that satisfy certain associative, commutative, and distributive conditions, and has a circular name.
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The Deutsch-Bobrow algorithm implements a "deferred" variety of one technique used to perform this action in run-time, and one problem with that algorithm is the issue of zero count table overflow. Christopher's algorithm was created to perform this function for FORTRAN, while Lins' algorithm is a lazy algorithm which peforms this function using a control set. The Deutsch-Schorr-Waite algorithm is an example of a pointer-reversal algorithm for doing it. A two-phase algorithm to perform this action was developed by McCarthy and is known as mark-and-sweep, while Unix-based systems employ the reference-counting method. For ten points, identify this action in which memory that is no longer in use is returned to the heap.
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The Gelfand-Neimark theorem deals with proving two of these kinds of spaces to be homeomorphic to one another. Only Tychonoff spaces can be embedded in these types of spaces, and every topological space can be expressed as the quotient of spaces of this type. Etale spaces are the classical example of spaces failing to meet this condition, and if a space is both compact and meets this condition, it is normal and all continuous maps from compact spaces to these spaces are homeomorphisms. For 10 points, name this type of space defined as every point having disjoint neighborhoods around any pair of points, alternately called T2 and named after a German dude.
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The Heine-Cantor theorem states that if there is compactness, this property is equivalent to its uniform variety. The Dirichlet function does not have this property anywhere. In Lesbegue integration, its absolute version holds for a function f on an interval if and only if there exists a function g such that the integral of g from the start of the interval to any point in the interval equals f. Its topological definition requires that the inverse of every open set be open. A function f of x has this property at a point b if the limit of f of x as x approaches b is f of b. For 10 points, name this property of some functions, sometimes described as the ability to be drawn without raising a pencil.
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The Hermitian variety of this operation is a complex-valued bilinear form in which reversing the order of arguments of this operation is equivalent to taking a complex conjugate of this operation's result. A common example of this operation for the space of square-integrable real-valued functions is the integral of the product of two such functions from minus infinity to plus infinity. Hilbert spaces are vector spaces on which this operation is defined such that the norm results in a complete metric space. For 10 points, identify this binary operation which maps two elements of a vector space to an element of the underlying scalar field, as exemplified by the dot product in Euclidean spaces.
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The Hopf-Rinow theorem can generalize this theorem to complete Riemann manifolds. Its first proof utilized a Lebesgue chain, and one of its namesakes originally linked it to uniform continuity. Veblen noted its logical equivalence to the statement that every cut on an ordered field F goes through some element, the Dedekind Cut Axiom. The Weierstrauss property provides equivalence to this statement if every sequence in the set has a convergent subsequence, though this is commonly stated as every cover having a finite subcover when applied to metric spaces, where this theorem holds for completeness and total boundedness. FTP, identify this doubly eponymous theorem which states that a subset of a Euclidean space is closed and bounded if and only if it is compact.
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The Landau problem of proving that a prime exists between n-squared and quantity "n plus one squared" is known as this man's namesake conjecture. Along with Gauss, this man lends his name to an algorithmic process that uses four parameters to quickly calculate the value of pi to a large number of decimal places. His namesake symbol names a construct modulo a prime number 'p' that is equal to one if an arbitrary number 'a' is a quadratic residue of p. When taken at an argument of cosine this man's namesake 'associate functions' form a component of spherical harmonics. His namesake transform involves finding the maximum value of 'x' times a derivative minus its antiderivative and can be used to obtain the Lagrangian from the Hamiltonian. FTP, identify this early 19th century French mathematician who should not be confused with Laplace or Laguerre.
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The Langlands program in mathematics relates to the attempt to understand this as applied to the algebraic closure of the rational numbers. For the polynomial x to the fourth plus three x squared plus five, it associates the dihedral group of order 8. It involves eponymous groups of automorphisms of a field that preserve a given subfield. The fundamental theorem of this subject establishes a one-to-one correspondence between subgroups of its namesake group and intermediate fields of a field extension. Although its most famous result was proved earlier by Abel, the establishment of this field led to a deeper understanding of the insolubility of the quintic equation by radicals. For 10 points, what is this branch of mathematics named for a French mathematician who was killed in a duel at age 20?
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The Nielsen type of these can be used to derive the Cesaro equation. One of these named after Cotes represents the solutions to a central orbit subject to an inverse-cubed force law, and a polygonal one may be generated from the ratios of the inradius to circumradius of a regular polygon. One of these objects named after Ulam is a graphical representation of the primes, while the Cornu type is a plot of the solutions of the Fresnel integrals. More famous are the Fermat and hyperbolic varieties, which correspond to special cases of the equation "r equals a times theta raised to the 1 over n power." For ten points, identify these mathematical curves, of which the most famous is the Archimedean.
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The Tetris Algorithm can be used to perform a range query on the universal type of these data structures, which make use of a Z Order Curve. A function which compares only the first k bits of data can be used to implement the space-saving prefix type of these data structures, while the regular type places elements to the left and right of a median node during the split portion of data insertion. A. A. Toptsis developed a variant of this data structure which increased the density of certain components within it to two thirds from another variant's one half, the so called star variety. In that latter variant, records are only stored at the leaf level, that is, the plus variety of this data structure, which is used to implement the NTFS file system. Developed by Ed McCreight and Rudolf Bayer, it is a more generalized form of a 2-3 tree. For 10 points, name this data structure, a type of self-balancing tree often used to implement databases.
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The cases in which these entities have a certain important property are limited by Kuratowski's theorem. Algorithms involving these mathematical objects include Prim's and Floyd's, which solve important problems like finding a minimal span. These objects may contain cycles named for Euler or Hamilton, finding which is one form of the Traveling Salesman Problem. A shortest path on one may be found by Djikstra's Algorithm. The theory named for them was inaugurated by the solution of the Konigsberg bridge problem. FTP, name these mathematical entities, sets of nodes and edges, which may be planar.
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The category of these objects has the property that every morphism is a monomorphism. The spectrum of one of these objects contains one point, and the p-adic numbers for a given p form one of these. These objects necessarily have zero Krull dimension. Those with a finite number of elements have order equal to a power of a prime. Their extensions have a one-to-one correspondence with their intermediates according to the fundamental theorem of Galois theory. A division algebra is one of these without multiplicative commutativity, and can be defined as a ring with multiplicative identity, inverse, and commutativity. For 10 points, name these algebriac structures that include the rationals and reals.
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The close distribution of its solutions is known as Lehmer's Phenomenon. It constrains the function f of z in the plane of an open disk with radius less than 1/4 according to the Voronin Universality Theorem. The eigenvalues of a suitable Hermitian operator are given by its nonimaginary solutions at one half plus i times E sub-n according to the Berry Conjecture. For the right half plane, it is given by the Dirichlet eta function, and it may be extended by the reflection functional equation to the entire complex plane, though it contains a singularity for the case s equals 1. FTP, name this special function defined as the natural analytic extension of the sum over every positive integer n of 1 over n to the s, whose non-trivial zeros may have real part equal to one half.
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The curtate variety of this figure contains a fixed point on its inside, which contrasts with its prolate variety. Huygens used its isochronous properties to design the first pendulum clock guaranteeing regular swing independent of pendulum height. This figure is represented parametrically by the equations x equals r times the quantity one minus sine of t, and y equals r times the quantity one minus cosine of t. It provides the solution to the problem of finding the curve of fastest descent, also known as the brachistochrone problem. For 10 points, name this figure traced out by a point on a rolling circle.
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The first English translations of this mathematician's work were done by G.B. Halsted in the early 20th century. Possibly the greatest student of Johann Bartels, this man preceded Swiss mathematician Carl Graeffe in finding a method for approximating the roots of algebraic equations. While working at the University of Kazan he did his most important work, similar to that being done by Hungarian Janos Bolyai. FTP, identify this Russian mathematician, best known for his development of a theory of non-Euclidan geometry.
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The first isogonic center of a triangle, which for an acute triangle is the interior point that minimizes the sum of the distances to the vertices, is known as his point. He proved that an odd prime can be uniquely expressed as the difference of two squares, and also that the sum of two squares cannot be equal to 3 modulo 4. The latter statement was a claim of Diophantus, which this man proved in preparing an edition of Diophantus' work. More famous are his claim that "a to the p minus 1 is equal to 1 mod p," known as his "little theorem." FTP, name this French mathematician whose most famous conjecture was proven by Andrew Wiles.
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The first polynomial-time test for this property was written by Shih and Hsu in 1999, but it is still outperformed by the 1974 algorithm of Hopcroft and Tarjans in many cases. A famous instructive example regarding this, the Petersen star, does not possess it because removing an outer vertex creates an object homeomorphic to K-three-three. Kuratowski's theorem given the necessary and sufficient conditions for it to hold for a graph and every graph with this property is four-colorable by the four-color theorem. FTP, name this property of graphs that may be unambiguously embedded on a surface, or drawn without crossing a line or lifting the drawing implement.
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The formulator of this concept hypothesized a connectionist gymnasium to rebut Paul and Patricia Churchland's objection to it. Based on Schank and Abelson's "Script Applier Mechanism," this argument can be derived from four axioms, including "Brains cause Minds." Pat Hayes defined cognitive science as an ongoing research project to refute this argument, which is critiqued in the Robot and Many Mansions replies. This argument compares the behavior of an English speaker who manipulates symbols rather than understanding them to a computer. For 10 points, identify this argument against Strong Artifical Intelligence, a thought experiment by John Searle named for a foreign language.
identify these objects, familiar as sets of points in n-dimensional Euclidean space a fixed distance from a given point.
The generalized Schoenflies Theorem concerns embedding one of these in another one dimension higher. For n=4 and higher, it is known that any compact n-manifold is homotopy equivalent to one of them if and only if it is homeomorphic to one, and that statement for n=3 is the original Poincaré conjecture. In high numbers of dimensions, most of their area is concentrated in a small band in coordinate space, and the numerical value of the volume of a unit one is largest when enclosing a five-space volume. For 10 points
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The iterated version of it was solved by Ernie Brickell in 1984 using a Cray-1, and a special case of it is used in the currently unbroken Chor-Rivest public key system. Its decision version was proved to be NP-complete by reducing the exact cover problem to it. One special case of it is the decision version with the constraint that the cost equals the value, and is equivalent to asking whether there exists a subset of a set of integers which has a given sum. For 10 points, name this combinatorial optimization problem, whose simplest formulation gives a set of items, each with a different weight and value, and asks you to maximize the value while being able to fit them in a certain bag.
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The mathematician Bhaksara solved the special case where d equals 61. Although William Brounker provided a first proof to Fermat's conjecture regarding this equation, Euler mistakenly misattributed it to another British mathematician. Euler solved other special cases, but Lagrange fully proved in 1768 that this equation always has nontrivial solutions for positive nonsquare integers d. FTP, identify this Diophantine equation of the form x squared minus d times y squared equals one.
name these additional aspects of spacetime that are searched for at current colliders?
The orbifold type of these can provide candidates for theories like the Standard Model of particle physics without any Higgs boson. The warped type, through the AdS-CFT correspondence, might provide a new way to study technicolor-like theories. Because Newton's law of gravity has only been tested to about a tenth of millimeter, flat ones could be about that size, and string theory predicts six of them. For 10 points
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The original author of this software package came in second to Miguel de Icaza's development of the Gnome desktop system at the 2nd Annual Free Software Foundation Awards. First developed at Stanford in 1984, it was extended in 1985 by Leslie Lamport to allow for a set of macros that would later serve as the template for SGML and HTML. FTP, name this text formatting language developed by Donald Knuth [KAH- nooth], the standard formatting language for nearly all scientific papers.
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The p-adic numbers satisfy a strong version of this statement, and a reversed form of it holds for non-positive definite metric spaces, such as a Minkowski space. A more general version of this statement was discovered by Bunyakovsky in its integral form. It is not obeyed by the L-0 norm, but it is a special case of the Cauchy-Schwartz inequality for the Euclidian norm. For 10 points, identify this statement, also known as subadditivity, which says that the length of any side of an eponymous shape must be less than or equal to the sum of the other two sides.
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The parameters of this name are four parameters that determine the state of polarization of a wave of monochromatic radiation from observations of the beam. The unit of this name is a cgs unit of kinematic viscosity equal to the ratio of the viscosity of a fluid in poises to its density in grams per cubic centimeter, while the law of this name is given in equation form as F equals 6 pi r eta v, and predicts the frictional force on a spherical ball moving through a viscous medium. FTP, what is this name most associated with the theorem extending Green's theorem to higher dimensions?
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The period of these numbers modulo m is equal to m if and only if 24 times 5 to the lambda where lambda is an integer greater than or equal to zero. Cassini's Identity is on these numbers. The Euclidean Algorithm's worst case occurs when finding the greatest common denominator of two of these numbers according to Lame's Theorem. The closed form solution is called Binet's formula, and as n goes to infinity, the nth one of these numbers divided by the previous one approaches the golden ratio. They were used by their namesake to solve a problem about the population of breeding rabbits. For 10 points, name this sequence of numbers that start with 1, 1, 2, 3, and 5.
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The rate at which the operation implied by this statement occurs is given by the Berry-Esseen theorem. Paul Levy proved a version of this statement using the technique of characteristic functions borrowed from Poincare, while various local proofs of it were given by Richard von Mises. A version of this statement is named after Lyapunov and requires the satisfaction of his namesake conditions, while a weaker condition than Lyapunov's is due to Lindeberg. The Edgeworth expansion was developed as an improvement of this statement, the proof of which is similar to the proof of the law of large numbers. For ten points, identify this theorem which holds that the distribution of the average of a set of arbitrarily distributed variables converges to the normal distribution.
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The second of this man's namesake functions is sometimes approximated as x minus the log of two pi. That the remainder when dividing prime numbers by 4 is more likely to be 3 than 1 is called this man's "bias". His namesake distance operation is sometimes called the chessboard distance, and his two namesake polynomial sequences, symbolized T sub n and U sub n, have recursive definitions, as they are solutions to his namesake differential equation. He names a relation which was actually formulated by Bienaymé and states that no more than one over k squared of the total values are more than k standard deviations away from the mean of any probability distribution. For 10 points, name this Russian mathematician best known for that namesake inequality.
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The work containing this man's most important mathematical result was discovered in 1896. One of his devices, a "magic horse," describes an automaton simulating a horse that drinks after being decapitated. Other devices listed in his Pneumatica include a fountain, and a double forcing-pump for a fire engine. FTP, identify this 1st century AD native of Alexandria, who, in his treatise Metrica, proves a formula for finding the area of a triangle in terms of its sides.
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The work of Vaughan Jones has connected these objects to the Ising model of statistical mechanics, while Dehn's theorem demonstrates that if one of these objects' namesake group is isomorphic to the integers, then the object itself is isomorphic to a closed loop. Dowker notation can be used to enumerate prime ones, examples of which are the hyperbolic, torus, and satellite types. The determinants of these structures are given by evaluating the Alexander polynmial at negative one, and other invariants related to these objects are the Jones and HOMFLY polynomials. Manipulated by Reidemesiter moves and generalizable in three dimensions to links, for ten points, identify these mathematical objects, examples of which include the bend, trefoil, and pretzel varieties.
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There exists a real number A such that the floor of the quantity A to the three to the n gives one of these numbers for every positive integer n according to Mills' Theorem. Two to the quantity p minus one is congruent to one modulo p squared is satisfied by the Weifrich type of these. The Lucas-Lehmer Test is used to determine if a number fits a specific category of these and every even number can be written as the sum of two of these according to Goldbach's conjecture. Euclid showed that there are infinitely many of, for 10 points, what type positive integers, with two being the smallest member, which are only divisible by one and themselves.
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These constructs are used in statistical sampling to randomly cover an area with a fixed set of elements. Gaston Terry's proof by exhaustion that there is no solution to the 36 officer problem proved that a more generalized form of them of size 6 by 6 does not exist. Circulent matrices are an example of them, and the Cayley table of a group is one of them. There are two orthogonal ones of size 4 by 4, and they also exist in a Greco- variety. For 10 points, identify this type of n by n array, exemplified by Sudoku, in which each row and column have permutations of the same entries, named by Euler for their traditional use with letters of the namesake alphabet.
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These numbers are the input to a Möbius transformation, and Gaussian integers are this sort of number. They can be represented by a two by two matrix with top row x negative y and bottom row y x. Holomorphic functions take these numbers as inputs. Phasors are another way to represent them, and they are extended by quaternions. For a given one of these numbers, flipping the sign of the second component in their standard representation yields its conjugate. If a quadratic polynomial has a negative discriminant, then its roots are of this type. For 10 points, name these numbers that have real and imaginary components.
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These structures are used in solving the cycle detection problem because the solving algorithm halts on the second detection of the smallest value of a sequence. A slotted channel is used in an algorithm employing these structures in collision resolution, and they are also used in LRU page-management. They are used in a method of memory allocation contrasted with heap-based allocation, and to implement reverse Polish notation calculators. The SP and SS registers on the Intel x86 architecture are used to implement this data structure first proposed by Friedrich Bauer. Containing two basic operations for adding and removing data, push and pop, FTP, identify these last-in-first-out data structures.
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They typically lack memory allocation and explicit variable assignments, and the absence in them of side effects ensures referential transparency. Pure ones have no variables at all and are therefore thread-safe, and the first one was known as IPL and was developed for the JOHNNIAC at the RAND Corporation. They possess two different types of evaluation, a strict way which evaluates all arguments first, and a lazy way, which evaluates arguments as they are needed and is exemplified by the Haskell language. Typically making heavy use of recursion, they had their foundation in the development by Alonzo Church of the lambda calculus. Typically contrasted with imperative languages, for ten points, identify this type of programming language, most famously exemplified by Scheme and LISP, and which models itself on the evaluation of certain mathematical relations.
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They typically reach conclusions by the method of confidences, and they can either reason from data to goals or from goals to data, processes known as backward and forward chaining. The first one, known as Dendral, was built in 1965 at Stanford in order to analyze chemical compounds, and the things they know are typically represented as "if-then" production rules. Gavin Riley developed one for NASA called CLIPS, which also includes an object-oriented language called COOL for writing more of them. Frequently incorporating heuristics and comprising a knowledge base and an inference engine, for ten points, identify these artificial intelligence programs which make decisions based on the analysis of information and take their name from their similarity to human specialists.
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This adjective describes a complete algebraic variety that is also an algebraic group, and the differential of this type is analytic on a compact Riemann surface. A field extension has this property if its associated Galois group is this. The Kronecker Decomposition Theorem states that any finite group of this type may be written as a direct product of prime-ordered cyclic groups. All subgroups of this type of group are normal, and all cyclic groups are of this type. FTP, identify this adjective derived from the name of a Norwegian mathematician, which is most famously used to describe a group in which every element commutes.
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This construct's namesake trick, often used in machine learning algorithms like support vector machines, uses Mercer's theorem to map a linear classification scheme into a non-linear one. In wavelet transforms, they define the nature and complexity of the transform pair, and in image processing, the convolution variety defines what sort of filtering is to be done. In operating systems, micro ones run most of their services in user space, while monolithic ones run them all in the namesake space. Their duties include inter-process communication, memory management, and executing programs on the CPU. FTP name this lowest abstraction layer and central component of most operating systems, including XNU for Macs and the freely available Linux one developed by Linus Torvalds, not Orville Redenbacher.
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This distribution appears in and is the reason for the name of the distribution solution to Poisson's equation. Generalizations of this to vector arguments are non-unique but can be determined by normalizing its all-space integral in light of any relevant symmetries. Reciprocity for Green's functions for self-adjoint operators results from the odd parity of this since Green's functions are the solutions to operator equations with this as the inhomogeneous term. Its convolution with a function gives the value of that function at the point where its argument vanishes by its sifting property. FTP, name this function defined by Dirac and named for a Greek letter.
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This language has two operators for inequality because its author couldn't decide which he liked better. Among its more powerful constructs are dynamic typing and binding as well as garbage collection, it is one of the few languages other than Java that can be converted into Java bytecode. Its most controversial feature is that it uses syntactically significant whitespace to delimit statement groups. Using features from Modula-3 and the ABC language, its creator developed it one Christmas while his Dutch lab was closed. First developed by Guido van Rossum, FTP, identify this object-oriented language with a reptilian name.
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This man collaborated with Richard B. Blackman and Hendrik Bode on a project entitled "Data Smoothing and Prediction in Fire-Control Systems," and one essay described an encoding process by which the length of English language texts could be reduced by 40%. In addition to writing "The Redundancy of English," this man designed devices like a roulette wheel to predict probabilistic outcomes of the stock market and used copper whiskers on a mechanical mouse, Theseus, which would remember paths through a maze he had completed. A theorem he collaborated on states that for a characteristic probability density function f vanishing outside of an symmetric open interval, f is uniquely determined by values one-half pi times f of n pi over lambda. For ten points, name this computer scientist who shares credit for a sampling theorem with Nyquist and who founded information theory.
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This man devoted much of his time working out the probability of various Biblical miracles, and calculated the chances of a man rising from the dead were 1 in 10^12. He wrote the 1832 work On the Economy of Machinery and Manufacturers, and earlier assisted John Herschel with astronomical calculations, which he thought could be better done by machines. FTP, name this English mathematician whose difference engine and analytical engine are the precursors of digital computers.
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This man formulated a mechanism that checks whether a new state is safe or not when granting requests in the Banker's Algorithm. That algorithm was subsequently used in an early operating system he developed, known as the "T-H-E multiprogramming system". He also formulated a method that uses a stack to convert standard syntax into Reverse Polish Notation, the Shunting Yard Algorithm. With C. A. R. Hoare this man lends his name to a 1972 text book called Structured Programming that advocated his position against the GOTO statement. His extensive correspondences are collectively known as the "EWD" series, and one of his eponymous creations falls apart when graphs have negative edge weights and has a heuristic modification known as A Star. FTP, identify this computer scientist best known for lending his name to a greedy algorithm that finds the single source shortest path from a node in a graph.
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This man is the namesake of a formula relating the Euler characteristics of two surfaces when one is a ramified covering of the other with Hurwitz. He names a theta function whose expansion begins with a t over 2 log of t over 2 pi term together with Siegel, with whom he also names a formula approximating his namesake function. With Christoffel, he names a curvature tensor which is central in his namesake geometry, where all tangent spaces feature smooth inner products. With Roch, he names a theorem giving the dimensions of certain spaces of meromorphic functions, and the fact that any open set can be holomorphically mapped to the unit disk is his namesake mapping theorem. Also naming the conjecture that a certain function has no zeroes with real part one half, for 10 points, identify this German mathematician with a namesake zeta function and hypothesis.
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This man names a line passing through a triangle's orthocenter, circumcenter, and centroid, while a cuboid with integer edges and face diagonals is known as his brick. A generalization of Kurrah's rule for finding amicable pairs is his rule, and his formula relates the number of vertices, edges, and faces in a convex polyhedron. His namesake method is used to solve initial value problems in ordinary differential equations, and his totient function is used in a generalization of Fermat's Little Theorem. For 10 points, this is what namesake of a constant along with Mascheroni, a Swiss mathematician who is also the namesake of the base of the natural logarithm?
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This man names the special case of the Archimedean spiral when n is set to two. Gauss showed that any number which can be expressed as a power of two times the product of numbers named for this man can be the number of sides in a constructible polygon. He is also the namesake of the statement that a to the p is equal to a modulo p if p is prime, and he proposed that light waves travel through the principle of least time. His most famous idea was proven by Andrew Wiles. For 10 points, name this namesake of a type of primes and a theorem that states that x^n plus y^n equals z^n has no solution for n greater than 2, his "last theorem."
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This man proposed a number system that used 2i as a base, and he wrote a paper about the evolution of refrains from rich ballads entitled "The Complexity of Songs." This primary developer of the MMIX (em-mix) architecture names a string-searching algorithm with Morris and Pratt, and his solution to the exact cover problem is known as "Algorithm X." He developed a method of displaying repeated exponentiation called his namesake "up-arrow notation," and he sends reward checks for errors found in his books. For 10 points, name this "father of algorithm analysis" and developer of the typesetting language TeX (tek), the author of The Art of Computer Programming.
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This man proposed the axiom of foundations and introduced the idea of a proper class to resolve Russell's Paradox. His work on Hilbert's Sixth Problem led him to axiomatize quantum mechanics by viewing physical quantities as operators in a Hilbert space, and with Bernays and Godel he gives his name to a set theory equivalent to Zermelo-Frankel. He proved the minimax theorem characterizing zero-sum games with perfect information, and the equivalence of cellular automata and Turing machines. In a report on the EDVAC, he introduced a namesake computer architecture. Responsible for introducing backwards induction in Theory of Games and Economic Behavior, FTP identify this Hungarian-born mathematician.
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This man proved that a function can be represented as a unique series of trigonometric functions. Along with Bernstein, this man names a theorem stating that two sets are equinumerous if each is equinumerous to a subset of the other. This man showed that there is a bijection between the points of the unit segment and all the points of an n-dimensional space. His namesake theorem states that the cardinality of a set is less than that of its power set. He introduced the idea that there are no sets with cardinalities between those of the naturals and the reals, the continuum hypothesis. For 10 points, name this German-born mathematician who used a diagonal argument to prove the uncountability of the reals.
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This man was the first person to sum a series with infinite number of terms when he calculated the area under a parabola, which may have been the first use of integral calculus. Although he didn't use the name pi, he proved that that number enters the formulas for the area and circumference of a circle, and gave a method of calculating the number to arbitrary accuracy. In physics he invented the field of statics, was the first to identify the center of gravity, and formulated the law of equilibrium of fluids and the law of the lever. FTP, identify this Greek mathematician and namesake of a spiral with equation r equals a theta, who was killed by a Roman soldier while presumably studying the squaring of the circle and is probably best known for his principle of buoyancy.
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This man's "resolvent" is used to solve cubic equations, and his "spectrum," which ends at Freiman's constant, consists of his "numbers." One theory named for him states that if p is a prime number and f a polynomial of degree n, then the equation f of x = 0 mod p has at most n solutions between 0 and p. Another namesake theorem states that the size of a subgroup must divide that of any group containing it. The maximum and minimum values of a function subject to a geometric constraint can be found with his multipliers. For 10 points, identify this man who found the five spots in a two body system where a third body of negligible mass will be stationary, known as his "points."
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This man's namesake duality is one of the most important theorems in homology theory and the 10-parameter group named for him represents the set of length-preserving transformations in Minkowski space. He contributed a dynamical fixed-point theorem with Birkhoff, and his section allows chaos theorists to look at systems stroboscopically. In addition, this polymath founded modern topology and is the namesake of a formula generalizing the Euler polyhedron theorem to objects with higher genus. FTP name the French mathematician who, in 1889, ironically won a well-publicized prize an incorrect proof of the stability of the solar system.
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This man's namesake inequality, a form of the Bonferroni inequality, states that for any finite or countable set of events, the probability of at least one of them occurring is no greater than the sum of the probabilities of the individual events. His namesake algebras satisfy the properties of a lattice, and his name is given to rings in which every element is idempotent. In 1938 Claude Shannon applied his theories to electrical switching circuits, showing that, with two variables, the operators "meet" and "join" could be used to simplify electromechanical relays, laying the basis for computer chips. For 10 points, name this mathematician whose work is associated with truth tables and binary mathematics.
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This man's namesake tetrahedra are two tetrahedra that mutually circumscribe and inscribe each other, and his namesake "mu function" gives a measure of the number of distinct primes in square-free integers. A variation of this man's most famous mathematical contribution provides an example of a non-trivial vector bundle over the circle. In 1818 he introduced his barycentric calculus, and later influenced projective geometry with his namesake net. Formulator of the "four-color" problem, FTP, who was this German mathematician best known for his one-sided strip?
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This mathematical object can be approximated with the Lanczos formula and represented in the Weierstrass form. The Legendre duplication formula gives the relationship between this function for an argument and for twice that argument, and its product with itself is given by the Gauss multiplication formula. It can be defined through a power series whose kth coefficient is proportional to the Riemann zeta function evaluated at k. Its values for half-integer arguments are proportional to the square root of pi and, as a function of z, it is typically given as the integral from 0 to infinity of t to the z minus 1, times e to the minus t, dt. FTP, identify this mathematical function; a generalization of the factorial to the complex numbers.
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This mathematician is the source of the famous quote that 'God made the integers; all the rest is the work of man." A student of Dirichlet and Kummer, he came from a wealthy family and was able to live as a private scholar at the Berlin Academy of Sciences before becoming professor there in 1883. He was involved in a controversy with Weierstrass and Cantor of the use of the infinite in mathematics, and published a systems of axioms in 1870 that were shown to govern finite Abelian groups. FTP, identify this Polish mathematician, whose namesake "delta" is useful in the evaluation of determinants.
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This mathematician was the first to use decimal points to separate the whole and fractional parts of numbers, and he also made advances in scientific farming, especially by his use of salt as a fertilizer. In 1617's Rabdologiae he described a mechanical method for claculating products and quotients using labelled rods usually made with ivory or bone, and which consequently became known as his "bones", but he is better known for developing a way to simplify scientific calculations. FTP, who was this English mathematician, the inventor of logarithms?
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This number is closely related to solutions of the equation "derivative of y with respect to x equals y" and is equal to the limit as x goes to 0 of the quantity 1 plus x all raised to the power one over x. Also equal to the sum as n goes from zero to infinity of 1 over n factorial is, for ten points, what transcendental number, the base of natural logarithms?
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This programming language was designed to contain mechanisms for concurrent programming, to ensure that it would be easy to verify the correctness of programs, and to support information hiding through modules. Program units are made up of subprograms, packages, and tasks, which are the core of the language's parallel processing ability. Originally intended for embedded computer systems, identify this language commissioned by the Department of Defense in the early 1970s, and, FTP, which is named for Babbage's collaborator on the Analytical Engine.
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This protocol recognizes five types of protocol data units, or PDUs, that can be sent between a management console and a monitored device. Four of them, get-request, get-next-request, set-request, and get-response, allow the manager to query the device's Management Information Base and get information on the device's status. The other, trap, allows the device to send unsolicited information to the manager when a major event occurs. FTPE, identify this system administration protocol, abbreviated SNMP.
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This protocol was first defined surprisingly late in RFC 821. Servers run on port 25. Without some provision for data transparency, the character sequence <CRLF>.<CRLF> ends its text. Its command keywords are four letters long; sessions begin with either the keyword EHLO or the keyword HELO, depending on whether extended features such as those used for sending large MIME objects are to be used. Usually returning the command RCPT for successfully sent data, the most notorious server for it is sendmail. FTP, identify this "simple" protocol used to send mail on the Internet.
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This quantity can be defined on a finite CW complex as an alternating sum, and an extension of this quantity to higher dimensions can be obtained through Betti numbers. Examining the trace of the matrix for a map homotopic to the identity shows that this quantity for such maps is equal to the Lefschetz number. It can be determined by taking the number of maxima and minima and subtracting saddle points or by subtracting twice the genus from two, and Poincare's formula for polyhedra gives it by subtracting the number of edges and faces from the number of vertices. For 10 points, what is this property of a compacted, closed surface that is zero for the Klein bottle and Möbius strip and positive 2 for a sphere, and which was first introduced by its namesake in his solution to the "bridges of Konigsberg" problem.
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This standard is named after a group formed in a cooperative effors of the ISO, ITU, and IEC. The method starts with a discrete cosine transform, which results in several 8x8 matrices of spatial frequencies. During the quantization phase, the matrices are modified by another matrix, usually chosen to compress well with minimal loss. The final encoding phase linearizes the two-dimensional data and then compresses it. Although it does result in information loss, this is usually tolerable with image data. FTP, indetify this standard for the compression of still images.
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This structure was invented in 1972 by Rudolph Bayer, and its height is no more than 2 lg (n+1) ["twice the base 2 log of n-plus-one"] if it contains n nodes. A similar structure created by Adelson-Velsky and Landis is a type of this and shares the property that it may require up to two rotations per insertion and can insert or delete in O (lg n) ["big-O of log n"] time. In it, no leaf may be more than twice as far from the root as any other, and every simple path from a node to a descendent contains the same number of nodes of the second namesake type. FTP, name this type of nearly-balanced tree, of which the AVL tree is a type, and each of whose nodes contain a bit indicating which of two "colors" the node is.
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This system arose out of the need for a distributed database to replace the hosts.txt (hosts dot text) file maintained at Stanford until the early 1980s. Root servers at the top level can answer queries, while at the lower end, multiple zones are usually delegated for clarity. Resource records are stored as text files beginning with a Start of Authority section, and a typical machine connected to the internet will have a few nameservers listed for resolving addresses. FTP, identify this service, often implemented on UNIX by BIND (pronounced "bind"), which allows clients to map a hostname to an ip (pronounced "I P") address.
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This term refers to a knot which is defined in cylindrical polar coordinates by the equations "r equals 2 plus cosine of p theta over q", and "z equals sin of p theta over q", where p and q are relatively prime integers. The fundamental group of one of these objects is equal to the cartesian product of the integers with themselves. The Kolmogorov-Arnold-Moser theorem concerns quasiperiodic orbits described by trajectories moving on the surface of one of these objects, which can be viewed as the cartesian product of two circles. By the Heawood conjecture, 7 colors are needed to color this genus one surface, which can be obtained by gluing opposite edges of a rectangle together with no twists. For 10 points, identify this kind of three-dimensional surface with a single hole, whose name describes the shape of a donut.
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This theorem can be used to prove Taylor's Theorem, of which it is also a generalization of, while a generalized version of this theorem can be used to prove L'hopital's rule. That form states that the quotient of the derivatives of two sections of a function at a certain point is equal to the quotient of the differences of the endpoint values, and is named for Cauchy. A special case is of it states that if a continuous function has the same value at the end points of an interval, its derivative must equal zero on that interval, and is named for Rolle. For 10 points, name this theorem which states that there is a least one point on an interval at which the slope is equal to the average derivative.
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This theorem was first explicitly stated by Claude de Bachet in 1621, although it was probably known in ancient times. Like his last theorem, Fermat claimed to have a proof, but died without recording it. It can be used to find explicit values for g(k) [G of K] for k equals 2, 3, or 4, and built on Euler's earlier work in number theory. FTP, identify this assertion that every positive integer is expressible as the sum of at most four squares, named after an 18th century French mathematician.
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This type of algorithm is useful when the problem exhibits optimal substructure and when locally optimal choices lead to globally optimal solutions. Huffman coding, Dijkstra's (Dike-strahs) algorithm, and the fractional knapsack problem use it, although dynamic programming is a better solution to the "zero-one" knapsack problem. For ten points, what algorithm always makes the choice that looks best at the moment.
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This value can be calculated by Dodgson's condensation. The volume of a parallelepiped is equal to the absolute value of this for an object of which the shape's bounding vectors are the components. One can find this value by applying the Laplace expansion, which also called expansion by minors. Taking a ratio of them can be used to solve a linear system via Cramer's rule. For an identity matrix, the value for this is one, and if this zero, then the matrix is singular. For 10 points, name this value which for a two by two matrix can be found by taking the difference of the two diagonals.
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This was solved by the novel method of discharging, which forms the subtitle of the second part of the paper in which its solution was originally published. It does not generalize to surfaces of non-zero genus, though the Heawood conjecture is the analogous statement in those cases, with the exception of the Klein bottle. First noted in 1840, it was brought to public attention by the work of de Morgan, who found it infuriating. FTP, name this problem of graph theory that remained open until a 1977 "computer proof" by Appel and Haken showed that the vertices of "Every Planar Graph" can be uniquely labeled in the namesake way.
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Together with Copeland, this man gives his name to a normal number obtained by concatenating the primes, while the Diophantine equation "sum from j equals one to m minus one of j to the n equals m to the n" is named for him and Moser. He named and derived a lower bound for the happy end problem, and with Selberg he demonstrated a number-theoretic proof of the prime number theorem. In 1958, he linked graph theory with Ramsey theory and combinatorics. He referred to beautiful theorems as coming "straight from the Book." FTP, identify this prolific Hungarian-born mathematician, whose namesake number gives the collaborative "distance" from another mathematician to him.
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Topological spaces are termed Lindelof if every cover with this property has a countable subcover, while the existence of a finite subcover for every cover of this type commonly defines a compact space. A set possesses this property if every point in the set has an epsilon-neighborhood in the set. This property is not necessarily possessed by the intersection of an infinite number of sets of possessing it because that intersection can form a single point, which notably lacks this property. Often denoted for intervals by parentheses at the endpoints rather than brackets, for 10 points, identify this property, which, for a set, can be formed from the complement of a closed set.
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Tschirnhaus' cubic curve is sometimes called this man's eponymous cubic. He was one of the five original solvers of the brachistochrone problem, yet his solution wasn't published for nearly 300 years. He famously acknowledged Leibniz and both Jacob and his teacher Johann Bernoulli in his text, the first ever on differential calculus. FTP, name this French mathematician whose eponymous rule is used to find limits of indeterminate forms.
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Two answers needed. The Poncelet-Steiner theorem states that one is unnecessary if you can use the other one, and the Mohr-Mascheroni theorem says one of them is unnecessary. One can be used, along with dividers, to create the Pythagorean extensions of the rationals. Considered in algebraic terms, they can only produce constructible numbers, and as a result one cannot construct every regular polygon with them, nor can one double the volume of a cube with them. One also cannot use them to find a square with the area of a given circle, but with a quadratrix, one can trisect a curve with them. For 10 points, identify these two mechanical devices used in classical Greek geometry.
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Two generalizations of its continuous formulation are related by a Weitzenböck identity, while the discrete version of this operation evaluated on a graph vertex sums the difference between the function value at that vertex and all of its nearest neighbors. Under conditions of chemical equilibrium, it vanishes when applied to the density of a quantity like concentration, and a generalization of it is found in a relativistic version of the Schrödinger equation, the Klein-Gordon equation. Earnshaw's theorem invokes it to show that an electrical force derived from a potential function must have zero divergence, and when it is evaluated on Minkowski space, it is called the d'Alembertian. For 10 points, identify this elliptic operator which vanishes for harmonic functions, the divergence of the gradient, named for a Frenchman.
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Two major types of these can be converted between by the so-called "subset method." The trajectories of some of these are governed by the Cohen-Kung theorem; these include Langdon's Ant. Formally type-0 mininal automata, these exhaust the space of computable algorithms according to a thesis due to Church and their namesake. They were originally used to demonstrate the undecidability of the Halting Problem and consist of an action table, register, read/write head, and infinite tape. FTP, name these theoretical computing constructs; ideal computers named for a pioneering British computer scientist.
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Two nodes are conditionally independent if there exists a d-seperation between them for the current givens. Top nodes contain only raw probabilities, while all other nodes contain a table of conditional probabilities based upon possible combinations of parent values. FTP, name this type of belief network, widely used for probabalistic inference in AI, drawing its name from a central rule of conditional probability.
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Two of these objects are said to be Morita-equivalent if there is an equivalence between their module categories. Every local example of one of these objects has a finite Krull dimension and modules are defined over these objects. Examples of these objects satisfying the descending chain condition are known as Artinian ones. The ascending chain condition, the requirement that all ideals be finitely generated and that every ideal contain a maximal element are criteria that are satisfied by Noetherian ones of these. . Commutative examples of these objects have no zero divisors and are called integral domains, of which fields are a subset. For 10 points, identify these algebraic structures exemplified by the integers which unlike groups consist of a set and two binary operations, usually addition and multiplication.
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When Cardano applied it to the formula, "x cubed equals 15x plus 4," it gave an answer involving the square root of "negative 121," but he knew the equation had "x equals 4" as a solution, so he showed that it could work with quantities more general than real numbers. A false proof it was given by Leibniz when he asserted that "x to the fourth plus t to the fourth" could not be written as a product of two real quadratic factors, but he didn't notice that the square root of "i"could be written in the form "a plus (b times i)." It simply states that every polynomial equation of degree "n" with complex coefficients has "n" roots in the complex numbers. FTP, name this basic theorem of mathematics.
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When the parameter is set to one, it can be expanded as a series whose coefficients are Stieltjes constants, and it can be evaluated using Parseval's theorem. A generalization of it is called the Hurwitz function, and it resembles a p-series. When evaluated at 2, it gives the solution to the Basel problem, and Ramanujan showed that it is equal to minus one twelfths when evaluated at negative one. 1.5 million roots have been shown to lie on the critical strip, but it is not known whether all roots have a real part 1/2, and this function is closely associated with the prime number theorem. For 10 points, identify this function named for a German mathematician who has a namesake hypothesis
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When the result of this procedure has a unique value, then that value is a function of every sufficient statistic. The covariance matrices of quantities generated by this method are commonly approximated by the inverse of the Fisher information matrix. Under appropriate regularity conditions, this method gives results that are equivariant, consistent, asymptotically normal and asymptotically efficient. This method's namesake function is the data's joint density treated as a function of the parameter being estimated, and that namesake function appears in Bayes' theorem as the unnormalized ratio of the posterior and prior probability densities. In carrying out this procedure it is often more convenient to replace that namesake function by its logarithm. For 10 points, name this method of parameter estimation which gives the parameter values that make the observed data most probable.
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When using it on a larger scale, the expansion by minors method is extremely helpful but can get tedious. Given a hypothetical system of n simultaneous linear equations in n unknowns, it could give a solution of "x sub I" equals "B sub I" divided by "A," where "I" equals one, two, three, etc. However, if A equals zero, then there is no unique solution under this technique in which "A" and "B" are determinants. FTP, name this algebraic method for solving two-variable linear equations by means of determinants and named for a British mathematician.
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With Skolem, this person names a theorem about automorphisms of simple rings. This mathematician's namesake bound states that the ring of invariants in the finite basis problem is generated by homogeneous invariants of degree less than or equal to the finite group's order. This person's namesake problem is a specific case of the rationality problem, and, in topology, this person's namesake spaces have closed sets which satisfy the descending chain condition. In algebra, the rings that are named for this person have ideals which obey the ascending chain condition. For 10 points, name this mathematician whose theorem proving that differentiable symmetries of the actions of physical systems have linked conservation laws is central to the calculus of variations, a German Jewish woman.
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Zhan and Noon tested algorithms for solving this problem, finding approximate or double bucket modifications to one algorithm and Pallattino's algorithm with two queues were fastest on actual data. Johnson's algorithm solves this problem in the sparse case faster than the cubic Floyd's algorithm, both of which solve this for all pairs. Another approach uses dynamic programming, and checks all edges n times; that algorithm also detects negative cost cycles, and is named for Bellman and Ford. A Fibonacci heap is used to speed up a greedy approach, which works only for positive-weight edges, while if an admissible heuristic is available, the A* algorithm can be used. For 10 points, identify this problem in graph theory, most famously solved by Dijkstra's algorithm, which asks for a fast route between two nodes.
Bessel functions
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Bridges of Konnigsberg
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Cauchy-Goursat Theorem or Cauchy's Integral Theorem (do not accept Cauchy's Integral Formula)
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Colin Maclaurin
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Dirac delta function
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Fibonacci sequence or numbers
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Francis Galton
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Fundamental Theorem of Algebra
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Guillaume Francois Antoine de L'Hôpital, or the Marquis de St. Mesme
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Hyperbola
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Isaac Barrow
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Joseph Louis, Count Lagrange
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Kepler conjecture
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Lissajous figures
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Parity Bit
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Pierre de Fermat
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Quaternions
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Quicksort
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Riemann zeta function
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Roger Penrose
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Srinivasa Ramanujan
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The Fields Medal
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The Golden Ratio or the Golden Mean
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The Minimax Theorem
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Turing machine
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Von Neumann architecture (or Von Neumann computer)
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axiom of choice (accept Zorn's Lemma and the well-ordering principle before they are mentioned in the question)
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catenary
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comment markers or comments
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parallel (or fifth) axiom
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pi
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stack (prompt on "LIFO" or "last-in, first-out" on early buzz)
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Although Gauss solved one version, it was still part of Hilbert's 18th unsolved problem. While it's thought that Gauss's result of pi over the square root of 18 applies to non-lattice packings, only recently has Thomas Hales claimed he's shown the face-centered cubic lattice to be the most efficient arrangement to stack spheres. FTP, name this conjecture created in 1611 by a famous astronomer.
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At the age of 9, this woman published a Latin discourse in defense of higher education for women. More famous as a mathematician, she has a crater on the face of Venus named in her honor. In 1750, she became the first woman to hold the mathematics chair at the University of Bologna and is most noted for her work in differential calculus. FTP, name the woman whose most famous invention is just a cubic curve, despite its mistranslation as "witch."
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Bailey, Borwein, and Plouffe found an algorithm for extracting any of its hexdecimal digits. Ramanujan provided a quartically-convergent algorithm for computing it. Buffon linked it to probability in his needle problem and von Lindemann proved it was transcendental. For ten points, what number did Archimedes note lay between 3 and one-seventh and 3 and ten over seventy-one.
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Both the generator and checker networks are logic circuits consisting of exclusive-OR gates and in one instance an exclusive-NOR gate. The checker can detect any odd number of errors; however, it cannot detect an even number of errors. In this error detection system, a binary one bit is added to each piece of information transmitted over a data network so that the total number of one bits in that piece of data is a number with agreed upon division properties. FTP, name this method of error detection, the setting on your modem that can be even or odd.
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Closely related to the Lucas numbers, it can be found in the sums of the shallow diagonals of Pascal's triangle. One-half of the sum of one and the square root of five, also called the golden mean, is the limiting ratio of terms of, for ten points, what sequence first derived in 13th century Italy by considering the reproduction of immortal rabbits.
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Comets that have high velocity and large mass cannot be captured by the sun's gravitational field move in an orbit shaped like this conic section. The navigational device Loran uses these to pinpoint position. An equilateral one has perpendicular asymptotes. FTP, identify this conic section which has two non-intersecting branches.
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Developed in 1970 by Alain Colmerauer of the University of Marseille, it operates from a pre-programmed domain, or set of facts, that is typically oriented toward a single scientific or technical field. By applying the formal rules of symbolic logic, a computer using this language can "learn" additional facts beyond its original data set and test for inconsistencies, but it has difficulty in areas that require fuzzy logic or uncertainty. For 10 points identify this artificial intelligence programming language, an early rival to LISP.
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Euler proved that the sum of the reciprocals of the prime numbers diverges by using his discovery that this function may be written as the product over all prime p of 1 over 1 minus p to the negative x. It was not until 1859 that its namesake extended its domain to the whole complex plane, and since then much attention has been paid to the values it takes in the so-called "critical strip" of complex numbers whose real parts lie between zero and one. FTP, name this function, defined as the sum as n goes from 1 to infinity of 1 over n to the z, which is the subject of the celebrated Riemann hypothesis.
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First demonstrated in 1799, one proof uses the result that entire, bounded, complex functions are constant. The original proof is a topological one using polynomials in the complex plane and was Gauss's doctoral thesis. For ten points, what result states that complex polynomials of degree "n" can be factored into "n" linear, complex factors.
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First given at Oslo in 1936, this prize is restricted to those under forty years of age, in order to reward promise as well as achievement, by the wishes of its endower. Winners have included Richard Borcherds, William Gowers, and Curtis McMullen, though not Gerald Lambreau, despite what you saw in Good Will Hunting. Andrew Wiles did not win, but was given a silver plaque for his proof of Fermat's Last Theorem. FTP, identify this prize, awarded every four years by the International Congress of Mathematicians.
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He extended the differential geometry of 2D surfaces, developed by his teacher Gauss, generalizing the concept of metrics and geodesics. Thus, in 1854, he almost stumbled on general relativity. He gave his name to the extension of Cauchy's equations in complex analysis and a curvature tensor which contracts to the Ricci tensor. For ten points, what German discovered the zeta function.
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He organized the defences of Edinburgh against the Jacobites in 1745, the year before his death. On the strength of 1720's Geometria organica, Newton recommended him for Edinburgh's chair of mathematics, and he went on to give a systematic geometric account of the calculus in his Treatise on Fluxions. FTP, name this mathematician, best known for a special case of an infinite power series of derivatives in which the variable is set to zero, a variant of the Taylor series.
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He was a young French marquis who, while learning Leibnizian mathematics from Jean Bernoulli, got his teacher to sign an agreement making over to him all his mathematical discoveries in exchange for a salary. Among them was a principle that now bears his name, since he published it in his Treatise on Conic Sections. FTP, name this man whose rule states that the limit of an indeterminate quotient is the limit of the quotient of the derivatives of the two functions.
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His career in academia has seen him teaching economics at Harvard, engineering at Yale, physiology at the Albert Einstein College of Medicine, and mathematics at Paris and Geneva. Although his groundbreaking work in the field did not begin until 1977, he was actually introduced to it in 1945 by his uncle, who gave him a copy of Julia's 1917 paper. By formulating the idea that objects can have dimensional values that are not whole numbers, he developed fractal geometry as a separate field of mathematics. FTP, name this Polish-born mathematician, creator of the concept of a "set" that has a border that is infinitely detailed with dimension between one and two.
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His time spent at Trinity College was miserable, thanks to tuberculosis and malnutrition caused by his vegetarianism. In 1976, George Andrews of Penn State discovered this man's "Lost Notebook," containing 600 theorems on loose sheets of paper. A failed student at Madras University, he was invited to Cambridge by G.H. Hardy, on the basis of a letter containing some hundred theorems. FTP, name this Indian who taught himself mathematics from an elementary English textbook.
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In 1741, Frederick the Great made him a member of the Berlin Academy, where he produced a steady stream of mathematical works, despite being blind in one eye. The author of the Konigsberg Bridge problem, the equation "e to the i times theta equals cosine theta plus i times sine theta" is known as his identity, and he was the first to introduce the capital sigma as a symbol for sum, as well as pi for that transcendental number. FTP, name this Swiss mathematician.
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In 1838 George Airy worked out the proof for the explanation given by Christian Huygens in 1678 that this curve was the catacaustic of a circle when the light source is infinity. The evolute of Cayley's sextic, this curve is essentially a two-cusped epicycloid formed by a circle of radius A rolling externally on a fixed circle of radius 2A. FTP name this curve, given its modern name in 1878 by R.A. Proctor, who thought that its shape was reminiscent of a kidney.
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In 1923, he gave the first rigorous mathematical treatment of Albert Einstein's model for Brownian motion, so that the motion is sometimes known by his name. He later applied that treatment to show that quantum theory is consistent with other sciences. His work on gunfire control led to the use of statistical methods in control and communication engineering. For ten points, name this mathematician who established the science of cybernetics.
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In a zero-sum game, a rational player will hedge his bets by choosing a strategy that gives his opponent the worst best-possible outcome. The other player, knowing that the first player is acting rationally, will expect him to play the strategy just described and so will choose the one responding strategy that gives him the best outcome under that scenario. As a result, every finite, two-person, zero-sum game has a single outcome that can be predicted with certainty if both players are rational. For 10 points identify this fundamental theorem of game theory, developed by John von Newmann.
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In addition to proving several of Fermat's theorems, he solved the problem of finding an integer x such that the quantity n times x squared plus one is a square if n is another non-square integer. At the age of 19, he became a professor at the Royal Artillery School in Turin, and in 1766 he succeeded Euler as Director of the Berlin Academy of Sciences. FTP, name this French mathematician, who used the calculus of variations and of four-dimensional space to treat mechanics in his best known work, Analytical Mechanics.
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It is equivalent to the statement that the cartesian product of an infinite family of sets is non-empty; to Zorn's Lemma; and to the well-ordering principle. It is an assumption independent of the other axioms in Zermelo-Fraenkel set theory. FTP, what name is given to the assertion that from any infinite family of sets a new set may be created containing exactly one element from each set in the family?
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Its average time complexity is of the order n log(n). To execute it, arbitrarily choose an element x in the set. Split all other elements in the set into those greater than or equal to x and those less than x. Apply the same function recursively to the newly generated subsets until each subset has just one member. You have just performed, for 10 points, what sorting algorithm?
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Its first published proof used the calculus of variations, although a letter to Bessel shows that Gauss was aware of the result 14 years earlier. In an 1883 article, a later mathematician offered a proof that removed the unstated hypothesis that the function's derivative must be continuous; however, the existence and continuity of all derivatives follow from an important corollary. FTP, name this fundamental result of complex analysis, which states that the integral of an analytic function around a closed loop is 0.
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Leonard Euler discovered that this curve, the evolute of a tractrix, forms the only minimal surface of revolution when it is revolved about its asymptote. It can be defined as the locus of the mid-point of the vertical line segment between the curves "e to the x" and "e to the negative x." With the general equation of "y equals a times the hyperbolic cosine of the quantity x over a," this curve can be generated as the locus of the focus of a parabola rolling along a straight line. FTP, identify this shape that is formed by a chain suspended by its endpoints and only acted on by gravity.
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Originally nominated for a Greek professorship, his Lectiones Geometricas was based on lectures delivered during the Great Plague. When he became chaplain to Charles II in 1669, he recommended one of his students to replace him in his previous post. FTP, name the man who discovered and proved the Fundamental Theorem of Calculus, the first holder of Cambridge's Lucasian Chair of mathematics, who was replaced by Isaac Newton.
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Plato, in his Timaeus, considered it the most binding of all mathematical relations and makes it the key to the physics of the cosmos. During the Renaissance, it served as the "hermetic" structure on which some of the great masterpieces were composed. Phidias closely studied it and used it in his work, and consequently it was given the name phi. FTP, name this constant, which accurate to three decimal places is 1.618.
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Shown by Lord Rayleigh to arise from a cylindrical formulation of LaPlace's equation, forms of these functions had been studied by Daniel Bernoulli and Leonhard Euler, but they were not derived in full until 1817. Commonly combined to form Hankel and Neumann functions, they were used to study the three-body problem and later shown to appear in the mathematical descriptions of the diffraction of light, the motions of fluids, and many other physical phenomena. FTP, identify these functions designated J sub n, which are solutions to the equation x squared times y double prime plus x times y prime plus y times the quantity x squared minus n squared equals 0, named for a German astronomer.
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The Gauss-Markov theorem describes the ordinary method of this as BLUE. In the case of perfect multicollinearity the variances and standard errors of the betas are infinite, and in the case of heteroskedasticity (het-er-o-ske-das-tis-it-ee) the ordinary method is no longer perfectly efficient. FTP, name this method used by economists and statisticians to achieve the best linear unbiased estimates of the average values of a predicted variable on the basis of known values of other variables, which involves minimizing a certain function of the residual differences.
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The Pregel River, formed by the confluence of its two branches, runs through the town and flows on either side of the island of Kneiphof. The townspeople had wondered whether it was possible to go for a walk and cross each of the seven bridges spanning the river and its branches once and once only before returning home. Since there are four vertices of odd order, Euler showed that this was impossible. FTP, name this problem associated with Kaliningrad that became one of the bases of graph theory.
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The extended version has three groups of combinations: One for numbers and punctuation marks, one for uppercase letters and a few other punctuation marks, and a final group used for lowercase letters. For ten points identify the most common data-transmission code used in personal computers to represent textual information and non-input device commands.
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The first tables of these for integers were calculated by Burgi, at the time employed as the court watch-maker for Rudolph II. A much more powerful continuous form was calculated later in the late sixteenth century and given to Tycho Brahe and it was the use of these functions by Kepler that demonstrated their use to the scientific community in general. FTP, name this concept, developed separately by Stifil and Napier, which allows for the transformation of multiplication into addition or subtraction.
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The polynomial "x cubed minus 2" is irreducible over the rational numbers and so its roots have degree three, as does a simple extension over the rationals by the real roots. However, any simple extension by a constructible point over the rationals must be a power of two, thus, for ten points, proving the impossibility of what straightedge-and-compass construction which baffled the ancient Greeks along with angle trisection and quadrature of the circle?
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Their discovery marked a break from the commutative property of multiplication. Their discoverer cut into the Brougham Bridge over Dublin's Royal Canal their fundamental formulas, those being I squared equals k squared equals j squared equals I times j times k equals negative one. They provide an axiomatic geometry for complex numbers in three-dimensional space. FTP, name these ordered quadruplets discovered by Sir William Rowan Hamilton.
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This man proposed that every positive integer is a sum of at most three triangular numbers, four square numbers, and, in general, n n-polygonal numbers. He also noted that the area of a rational right triangle cannot be a square number. His primes are of the form 2-to-the-2-to-the-n plus 1 for integer n, and his "little theorem" states that if p is a prime number and q is a natural number, q-to-the-p is q mod p. FTP identify the French mathematician best known for a marginal notation in his copy of Diophantus' Arithmetica.
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This man's intersection theorem states that in a complete metric space, any nested sequence of closed sets whose diameters approach zero contains a unique intersection point. His "paradox" arises from the assumption that there is an all-inclusive infinite set, and his ternary set is created by removing the open middle third of the interval [0,1], and then recursively removing the middle third of the remaining intervals. FTP, identify this German mathematician, whose diagonal theorem can be used to show that the real numbers are uncountable.
