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vertices minus edges plus faces. This sixth decimal digit of pi is also the ones digit of the constant which is the base of the natural logarithm, Euler's number "e." For 10 points, name this smallest number of points in Euclidean space needed to determine a line, which is also the smallest prime number.

2

This thinker proved the central limit theorem using the Gaussian error function, and he included a state register in one mathematical model. Lambda calculus demonstrates a "completeness" named for this thinker. This thinker notably used the Banburismus process, which relied on comparing corresponding (*) letters in different sentences. That system was used by this thinker in collaboration with Joan Clarke to create bombes for decryption. A head that reads and writes bits on an infinitely long strip of tape are this scientist's namesake machines. For 10 points, name this British computer scientist who helped crack the Enigma code.

Alan (Mathing) Turing

Alonzo Church's lambda calculus satisfies this man's namesake "completeness", and along with Church, this man names a thesis that states that all computable functions can be computed using a namesake device. That hypothetical device named for this man manipulates symbols on an infinite strip of(*)

Alan Turing

This mathematician is the secondnamesake of a theorem that uses the undecidability of a finitely axiomatized definition of arithmetic to describe the various classes of computing functions. This man showed the undecidability of the halting problem, and he names a "Thesis" with Alonzo (*) Church. This man described a model of computation using an infinite strip of tape with input symbols called his namesake "machine," and a program that can pass as human is said to pass his namesake "Test." For 10 points, name this British computer scientist and mathematician known for helping to break the German enigma code.

Alan Turing

state is multiplied by a transition matrix to find probabilities of the next state. Random walks and other stochastic processes can be studied using—for 10 points—what Russian mathematician's namesake "chains"?

Andrey Markov

residue theorem and an integral formula. Sequences named for this mathematician satisfy the property that the differences between adjacent terms approach zero as n approaches infinity, and sequences of real numbers converge if and only if they satisfy that property. For 10 points, name this French mathematician who shares the name of a famous inequality with Hermann Schwarz and the name of a pair of differential equations with Bernhard Riemann.

Augustin-Louis Cauchy [koh-shee]

This was the first language in its lineage to use class objects, an introduction inspired by the language Simula. Unlike later languages in its lineage, this language requires manual garbage collection. This programming language supports four types of memory management (*) objects described by the adjectives standard, thread, automatic, and dynamic. A print function in this language is called by using the command c-out. This language allows for object-oriented programming, as well as other decomposition paradigms, like modular programming. For 10 points, name this object-oriented extension to C.

C++

Upon being noticed by G. H. Hardy, Ramanujan started publishing papers at this University. Isaac Newton, Stephen Hawking and James Clerk Maxwell all studied at this school.

Cambridge University

We won't actually be using these tools, but mathematicians have spent a lot of time thinking about them, including this German who worked out in 1796 which polygons can be constructed from them. He also proved the fundamental theorem of algebra and found out how to add the numbers from 1 to 100 when he was eight years old.

Carl Friedrich Gauss

Ramanujan was famous for his solving of these kinds of equations. These equations are polynomials, usually in at least two unknowns, for which only integer solutions are sought.

Diophantine equations

This algorithm can be used to find the gcd of two numbers by repeatedly subtracting one number from the other until one reaches zero. Its "extended" form can be used to find the coefficients that solve Bezout's identity.

Euclid's algorithm

The "left-pad" controversy referred to a package written in this language. Google developed the V8 engine for use with this language, and toolkits like Grunt and Gulp can be used along with transpilers to help "bundle" files written in this language. Microsoft wrote a "superscript" of this language to address its loose typing, and (*) Node is an adaptation of this language for server-side use, though it's usually executed client-side. Facebook's React framework and the JQuery library augment this language. For 10 points, name this programming language used with CSS and HTML to make websites interactive, not to be confused with the similarly named Java.

JavaScript

harmonic series minus the natural logarithm at the limit is named for this man and Mascheroni. This mathematician names the identity "e to the i-pi equals negative 1." For 10 points, name this 18th century Swiss mathematician, who named the base of the natural logarithm e.

Leonhard Euler

The golden ratio is also the limit of the ratio of consecutive numbers in a sequence named for the author of this work. That author, Fibonacci, did not mention the golden ratio in this work, instead spending most of the time rocking the Western world with the Hindu-Arabic numeral system and making up stories about rabbits.

Liber Abaci

radial coordinate "r," since it directly gives rise to the Euclidean distance formula. Sets of positive integers associated with this theorem are known as its namesake "triples." For 10 points, name this theorem that describes the relationship between side lengths of a right triangle.

Pythagorean theorem

One type of this entity operates using tunnel magnetoresistance. The first example of this component was the Williams tube, which was implemented in the Manchester SSEM in 1947. Its magnetic-core type relies on magnetized rings, and (*) Corsair and Kingston have the largest market share for this computer component. It comes in static, dynamic, and non-volatile types, all of which only maintain data when power is applied. DDR3 and DDR4 are types of it, and four to eight gigabytes of this entity are included in most computers. For 10 points, name this type of storage that holds data currently in use.

RAM [Random Access Memory]

The best-known Millennium Prize problem is to prove this hypothesis about the zeros of a complex-valued function, which has massive implications for the distribution of prime numbers.

Riemann hypothesis

email systems to detect spam via a "naive" application of his namesake theorem. This man names an interpretation of probability that considers reasonable expectation, in contrast with frequentist probability. For 10 points, name this mathematician whose namesake theorem computes probability based on prior knowledge of relevant conditions.

Thomas Bayes

halting problem is undecidable for these devices, meaning that this device cannot determine whether a program will terminate or run infinitely. These devices have a head for reading and writing on an infinitely long tape, which can be used to run any algorithm and computation as determined in a thesis co-named for Alonzo Church. For 10 points, name these theoretical computational devices named for a British computer scientist.

Turing machines

Name this function that returns the product of all integers from 1 to its argument

factorial

Many approximations for the factorial use this function, since this function of the factorial is slowly varying. This function's Taylor series starts with "x minus x squared over two plus x cubed over three."

natural logarithm

Give this word used to denote a vector or line that points out from a given plane or curve. The choice of the negative or positive one of these vectors reflects an object's orientation.

normal vector

The hyperbolic sine is this type of function, which can be represented mathematically by the relation sinh(-x) = -sinh(x) ["sinch of negative x equals negative sinch of x"].

odd

Surfaces such as the saddle, which have a hyperbolic geometry, do not satisfy this postulate from Euclid's Elements, since two lines on such a surface that locally have the namesake property will diverge.

parallel postulate

This time, however, you notice that the coefficients are variable with ordinary singular points, so you decide that the solution is one of these entities, an infinite series in increasing exponents of x. Taylor and Maclaurin series are types of these series.

power series

Eratosthenes was able to find these numbers via his namesake sieve. According to the fundamental theorem of arithmetic, all natural numbers greater than one can be expressed as a unique product of these numbers, known as these numbers' namesake factorization. For 10 points, name these numbers, including 2, 3, 5, and 7, which are divisible only by one and themselves.

prime numbers

This task is impossible to carry out with just a straightedge and compass because the square root of pi is a transcendental number. This task involves constructing two certain shapes so that they have equal areas.

squaring the circle

ratio associated with this man is a root of the quadratic x squared minus x minus 1; taking the limit of ratios of consecutive numbers named after this man yields that golden ratio. This man devised a recursive sequence to model the growth of a population of rabbits. For 10 points, name this mathematician whose namesake sequence begins 1, 1, 2, 3, 5.

Fibonacci (accept Leonardo of Pisa; prompt on Leonardo)

hypothesis." This mathematician also lends his name to a branch of geometry and to a type of surface used to visualize functions of complex variables. For 10 points, name this mathematician with a namesake integral, sum, and sphere.

Georg Friedrich Bernhard Riemann

TensorFlow. In March 2016, Lee Sedol was defeated by this company's game-playing AI called AlphaGo in a widely broadcast five-game match. For 10 points, name this company whose PageRank algorithm sorts results from their search engine.

Google Inc.

The recursively-defined inductive dimension will always be less than or equal to this dimension of a fractal named for a German mathematician. For the Koch snowflake, this quantity is log-base-three of four, or about 1.26.

Hausdorff dimension

One mathematician from this country discovered a generalized formula for the area of cyclic quadrilaterals. For 10 points each: [10] Name this country, the birthplace of Brahmagupta and Srinivasa Ramanujan.

India

This was originally developed in 1991 with five unique design principles in mind, one of which was that it be "robust and secure." After the fourth edition, the developers of this decided to switch edition numbering from a decimal number scale to a whole number scale. Developed by James Gosling for Sun Microsystems, the publisher for this platform is now the Oracle (*) Corporation. While the Android operating system is largely written in C, apps for the Android OS are written in this strictly typed computer programming language. For 10 points, name this object-oriented computer programming language, one of the most popular currently in use, which was named for a type of coffee.

Java (programming language)

Because it is the root of the polynomial x2-x-1 [x-squared minus x minus one], the golden ratio is this type of number, which differs from transcendental numbers like pi in that it can be expressed as the root of a polynomial with non-zero, rational coefficients.

algebraic integer

These objects are the result of the function "explode" in php. The distance in memory between entries in these objects is known as its "stride." A lack of bound checking of these objects can cause overwriting of memory addresses. These objects can be dynamically resized to allow (*) retrieval time to remain the same. Maps and dictionaries are considered associative forms of these objects, and one-dimensional versions of these objects can be considered linear. In most languages, these objects' indices start at zero. For 10 points, name these objects whose two-dimensional forms are known as matrices.

arrays

In certain languages, this symbol can be used to spread an array into a function's parameters and is called the splat operator. A version of this symbol named for Kleene["KLAY-nee"] can be used in regular expressions to denote one or more of the preceding expression, and pointer variables in C can be (*) dereferenced with this symbol. This symbol frequently serves as a wildcard symbol, and in many languages, a forward slash followed by this symbol indicates a block comment. On QWERTY-keyboards, this symbol comes above the eight. For 10 points, name this symbol that typically also represents multiplication.

asterisk (or star)

A normal matrix must satisfy this property with respect to its Hermitian adjoint. Matrix multiplication, unlike multiplication of numbers, generally does not have this property.

commutative property

Name this set of numbers denoted by a capital C, whose members can be expressed as a sum of real and imaginary numbers.

complex numbers

dot product of two vectors is proportional to this function of the angle between them. A law named for this function is a generalized form of the Pythagorean Theorem, and one over this function yields the secant function. For 10 points, what trigonometric function is given by dividing the adjacent side of a right triangle by its hypotenuse, and is thus the cofunction of sine?

cosine

System F refers to an extension of the lambda calculus that utilizes these features. Overloading refers to writing a function that takes arguments that differ in this feature, and functional languages get their classification from the use of functions as atomic examples of these features. These features can be converted between each other using (*) casting, and Java is static because a user must explicitly declare this feature for every variable. Some examples of these features include enums and bignums, and "true" and "false" are typically the only values for the bool example. For 10 points, name these categories of data such as int, float, and char.

data types

Another method to solve second order differential equations finds a solution linearly independent to a given solution and is called the reduction of this quantity. This number is the highest derivative in the differential equation.

equation order

Mamma Mia! andOklahoma!, and in chess, two of these things denote an excellent move. The International Phonetic Alphabet uses this symbol to represent a postalveolar click sound, and in computing, this thing is called a "bang" and denotes a logical negation. For 10 points, name this symbol which mathematically denotes a factorial and which in English is used as a punctuation mark after an interjection.

exclamation point

computer in proving a theorem named partially for this number. This is the highest degree for which there exists a general solution for solving polynomial equations, which was shown by Abel and Ruffini. This is also the number of faces on the simplest Platonic solid. For 10 points, name this smallest composite integer.

four

The Koch snowflake is a type of this thing, other types of which are named for Sierpinski and Mandelbrot.

fractal

Some of the "strategies" for evaluating these entities can include "CBN" or "CBV," and if they are termed "pure" then they must lack "side-effects." Alonzo Church defined his lambda calculus using nested examples of these entities, and they lend their name to a (*) paradigm of programming exemplified by Lisp and OCaml and contrasted with iterative programming. In Java, your primary program is declared using the "public static void main" example of these entities. These entities can be recursive, and they often return a value. For 10 points, name these procedures you can define in one place and call and run from another.

functions

The infinite continued fraction "one plus one over quantity one plus one over quantity one, and so forth, yields this number. For 10 points each: [10] Identify this constant, symbolized lowercase phi, which is equal to its reciprocal plus one.

golden ratio

In general, for integers x and y, Bézout's identity states that there exists a linear combination which equals this value for x and y. This value is the largest integer which divides each of a set of integers

greatest common factor

When this adjective is used with "cosine," it denotes the function $\frac{1}{2}(e^{x} + e^{- x})$ ["one-half quantity e to the x plus e to the minus x"]. For 10 points each: [10] Give this adjective that describes things related to the conic section exemplified by the function f$(x) = \frac{1}{x}$ ["f of x equals one over x"].

hyperbolic

zero vector, and modifying this property is the basis of non-Euclidean geometry. Euclid's fifth postulate, which is named for this property, implies that, since rectangles have sides that exhibit this property, all their angles are right angles. For 10 points, name this property possessed by two lines in the same plane that do not intersect.

parallel

Gram-Schmidt process. A plane is completely defined by a point and a vector with this property with respect to the plane. Two vectors have this property if their dot product is zero, and two lines have this property if their slopes are negative reciprocals of each other. For 10 points, give this adjective that describes two lines intersecting at right angles.

perpendicular

For a line segment connecting points A and B, by drawing circles of equal radii centered at points A and B, one can construct this type of line that passes through the circles' two intersections. In a triangle, three of these things meet at the circumcenter.

perpendicular bisector

The "tail" form of this thing results in a functional language not adding that call to the stack. For 10 points each: [10] Name this process whereby a function is applied within its own definition, useful in creating things like the Koch snowflake.

recursion

The Y Combinator is a lambda expression that allows for this type of process, and memoization can be a faster alternative to it. The complexity of functions that have this property can be analyzed by the Master Theorem. Divide-and-conquer algorithms like quicksort rely on this process, the (*) "tail" variety of which allows for machine optimization. This concept is usually taught to students with the Tower of Hanoi, and it typically features a "base case" at which the function ceases and returns up the stack. For 10 points, describe this type of computation in which a function calls itself.

recursion

A "beam" algorithm for this task is memory optimized and greedy. The Bellman-Ford algorithm for this task can notably handle negative weights, and pruning can be used to reduce the space involved in this task. The use of a heuristic differentiates the A-star algorithm for this task from a similar algorithm developed by (*) Edsger Dijkstra ["DIKE-struh"] for it. Algorithms that perform this task by adding visited nodes to a queue or stack include breadth- and depth-first, and splitting a list in half recursively is an example of its binary type. For 10 points, give the term for this task of locating a specific item on structures like a graph or list.

search

counting kind of this type of algorithm. The bubble variety of algorithm for this task has a quadratic runtime for its average and worst cases, and is less efficient than the selection and insertion kinds. For 10 points, name this kind of algorithm that places elements in order.

sorting algorithms

Riemann [REE-mahn]. Transforming from Cartesian coordinates to these objects' namesake coordinates involves the Jacobian "r-squared sine-theta," where theta is the polar angle. They're mapped to the plane by stereographic projection and they have surface area "4-pi r-squared." For 10 points, name these 3D analogues of circles.

spheres

In the normal distribution, 68% of the area lies less than this distance from the mean. This quantity is the square root of the variance, and it measures the spread of a set of data.

standard deviation

Within these objects in Python 3,enclosure in curly braces signals latermodification. These objects are operated onwith a "rolling" hash function in the Rabin-Karp algorithm. In one programming tool, thestart and end of these objects are signified by acarat and a dollar sign. The "main" method inJava always takes an array of these objects,traditionally assigned the variable name (*)

strings