Intro to Upper Exam 1

What are the roles of the experimental scientist, mathematical modeler, and computational scientist in the process of mathematical modeling?

(a) The experimental scientist collects experimental data. This involves designing experiments, making observations, taking measurements, etc. (b) The mathematical modeler writes the mathematical equation(s) of the model. This requires using the data to make informed decisions, and making simplifying modeling assumptions to translate the behavior of a real-life system into a set of equations. (c) The computational scientist solves the model. Most models are solved by designing an algorithm, implemented by a computer, to nd a solution.

What are the requirements that the axioms must satisfy to dene a valid axiomatic system?

(a) They must be consistent. An axiom cannot contradict any other axiom in the axiomatic system. (b) They must be complete. They must be sucient to prove all the theorems in the theory. (c) They must be independent. None of the axioms should be able to be proved by the others

What is a conditional statement? What is its contrapositive?

A conditional statement is the form If P , then Q hypothesis,conclusion • Its contrapositive is If not(Q), then not(P).

What is a conjecture?

A conjecture is a mathematical statement believed to be true, but for which no one has been able to provide a valid proof yet. Once someone produces a proof, the conjecture becomes a theorem.

How is progress made in applied and pure mathematics?

Applied mathematics partners with an experimental science (natural or social) to build a mathematical model that describes a real-life system, and use the model to study the system. It uses experimental data collected within the framework of the scientific method. • Pure mathematics expands a mathematical theory by proving a new theorem that generalizes and/or extends an existing theorem. It begins with a conjecture, which becomes a theorem when it is proved deductively using the axiomatic method.

Notable Fields Medalists:

Caucher Birkar, Alessio Figalli, Peter Scholze, and Akshay Venkatesh are the most recent Fields Medalists (2018). • Maryam Mirzakhani was the first woman ever to receive a Fields Medal, in 2014. She died in 2017. • Grigori Perelman was awarded a Fields Medal in 2006 for solving the Poincare Conjecture (one of the Millennium Problems).

Who was Maryam Mirzakhani and what was she known for?

Mirzakhani was the first woman to be awarded a Fields Medal in 2014. She died of cancer in 2017 at the age of 40 .

Who is Grigori Perelman and what is he known for?

Perelman is known for solving the Poincaré Conjecture, one of the seven Millennium Problems. He was awarded the Fields Medal and one million dollars in 2006, but refused to accept either.

What are the current fields of open research in pure and applied mathematics?

Pure math: Number Theory, Algebra, Topology/Geometry, Analysis. • Applied math: Differential Equations, Statistics, Operations Research, Numerical Analysis.

What are the Millennium Problems proposed by the Clay Institute in the year 2000?

The Millennium Problems are a set of 7 math problems proposed by the Clay Institute in the year 2000 as the most important unsolved problems in mathematics. The Clay Institute promises to award 1 million dollars to the first person who solves each of these problems. One of the problems (the "Poincaré Conjecture") was solved in 2006 by Grigori Perelman, so at this point in time there are only 6 unsolved Millennium Problems remaining.

The Fields Medal

awarded every 4 years to up to 4 young mathematicians (recipients must be less than 40 years old to be eligible). Fields Medals are awarded at the International Congress of Mathematicians, an event that takes place every 4 years.

The Abel Prize

awarded to one mathematician every year by the King of Norway. It is the equivalent of the Nobel Prize in other disciplines

Review research

consists in compiling, organizing, or comparing the original research of other people.

Expository research

consists in explaining the research of other people to a broader audience.

What is original research?

refers to working on a problem that has not be solved before, such as proving a new theorem or developing a new method. Original research expands the bounds of human knowledge, and must be published by the person(s) who did the research.

Analysis

studies infinity, infinitesimals, and the related concepts encountered in calculus (limits, series, derivatives, integrals).

Algebra

studies mathematical objects (numbers, vectors, matrices) and the properties of operations between them

Topology

studies the properties of geometrical objects that are preserved under a continuous deformation, that is, how their points are connected to each other regardless of their actual shape.

Number Theory

the study of integers

What are the 5 axioms of Euclidean geometry?

• (1) Each pair of points denes a line. • (2) A finite segment can be extended indenitely in a straight line. • (3) Given two distinct points O and A, there is a circle centered at O with radius OA. • (4) All right angles are equal. • (5) If L1 is a line and P is a point not on L1, there is a unique line L2 through P parallel to L1.

. In what year (approximately) did the Western (European) civilizations begin pursuing mathematics as a scholarly activity? What events prompted this?

• 1200 AD. Several things happened at that time: (a) European merchants started importing mathematical concepts from their travels to the Middle East and India, such basic concepts of algebra. (b) Europeans started using the Hindu-Arabic base-10 number system, which is more amenable to developing mathematics than Roman numerals. (c) The Christian Church established the rst universities, which required all students to study various forms of math (such as logic, geometry, arithmetic)

What characterizes a mathematical denition? How is it dierent than a descriptive denition used in everyday speech?

• A mathematical denition is a mathematical statement that assigns a word to an object or property. Its purpose is not to describe, clarify, or explain a term, but rather to provide an equivalent mathematical statement (which must be decidable and either true or false). On the other hand, a descriptive denition (as used in a dictionary, for example) may be vague or subjective. The same word may have both a mathematical denition and a (very dierent) descriptive denition used in everyday English (for example continuous, similar, group, or integral).

What is a mathematical model? What does mathematical modeling mean?

• A mathematical model is a mathematical equation whose solution describes a natural phenomenon. Mathematical modeling means using a mathematical model within the context of the scientific method to help the progress of science by discovering patterns and mechanisms that govern how the physical world works.

What characterizes a mathematical statement? What are the rules of noncontradiction and excluded middle?

• A mathematical statement is a declarative statement that must be either true or false. It must be decidable, and satisfy the rule of noncontradiction (it cannot be both true and false) and excluded middle (it cannot be neither true nor false).

What is a non-Euclidean geometry?

• A non-Euclidean geometry is axiomatic system that replaces the 5th axiom of Euclidean geometry by an alternate one, for example: • (5) If L1 is a line and P is a point not on L1, there are no lines through P parallel to L1. • (5) If L1 is a line and P is a point not on L1, there are more than one line through P parallel to L1.

What characterizes a peer-reviewed journal?

• A peer-reviewed journal enlists a team of outside experts (the "peer-reviewers") in the publication process. Peer-reviewers are not employed by the journal directly and do not make the final determination on whether or not a manuscript gets published. But their comments are taken very seriously by the journal editors who do make the final decision.

6. Who can access articles in a peer-reviewed journal?

• A reader who has a subscription to the journal can access all the articles published in that journal. • A reader who does not have a subscription can access a specific article by paying its pay-per-view (PPV) price. • The author of an article can pay an open access fee (OA) to the journal, in order to make the article accessible to anyone for free

What are the differences between a scientific theory and a mathematical theory?

• A scientic theory is established by the scientic method, when an experiment (or series of experiments) conrms or rejects a hypothesis about a system's behavior. A scientic theory is constantly subject to revision and correction as new experimental results become available. • A mathematical theory is established by the axiomatic method. It consists of a set of theorems which are proved on the basis of a set of axioms (called an axiomatic system). All theorems in a theory are proved by logical deduction based on previously-proved theorems and axioms. The theorems in a theory are only true within the context of the axiomatic system, and may fall apart if the set of axioms is changed.

In addition to the axioms and theorems, what must be provided to properly dene a theory?

• A theory must also provide precise mathematical denitions for the terms it uses. Denitions are built on previously-dened terms in sequence, and can all be traced back to a set of primitive terms that are assumed to be clear enough as to not require a denition (such as point or line).

What is an axiom, an axiomatic system, and a mathematical theory?

• An axiom is a statement that is accepted as being true by denition. • An axiomatic system is a set of axioms used as a foundation to prove theorems. • A mathematical theory is the set of all theorems build on a common set of axioms

Who is Andrew Wiles and what is he known for?

• Andrew Wiles is known for solving proving Fermat's Last Theorem in 1995, a problem that had been open for over 350 years. • Fermat's Last Theorem says that there are no positive integers a, b, and c that satisfy the equation a^n + b^n = c^n for any integer n > 2. It was proposed as a conjecture by Fermat in 1637, and proved by Andrew Wiles in 1995.

What is applied math, and how is it different than pure math?

• Applied math is the branch of mathematics connected to science, that is, to the empirical study (using one's senses) of how the physical world works. Applied math is connected to the scientific method, as it requires using experimental data collected by observing the physical world. Its primary activity is writing and working with mathematical models, which describe the laws of nature and other equations that govern how the world works. • Pure math often arises when the tools developed for applied mathematics are generalized and abstracted without regard to applications or connections to the physical world. Thus, pure math is primarily non-empirical and does not use the scientific method, but rather the axiomatic method. Unlike applied math, research in pure mathematics does not have to be practical, or even be be consistent with physical reality.

What is contained in the introduction of an article of original research in applied math?

• Background information about the topic and field of research. • Scope, description, goal, and relevance of the research project. • Review of previous published works that are used as starting points for the current research project. • Explanation of the way in which the current research claims to be original

What are the advantages and disadvantages of the peer-reviewed process?

• Because peer-reviewed articles have been through several rounds of improvement (as guided by the peer-reviewers), they are of very high quality. Also they are (usually) free of errors and guaranteed to be truly original. • The peer-review process is slow and expensive.

What does the Mathematization of the Sciences refer to?

• Before the 20th century, applied mathematics was mostly restricted to applications in physics. The Mathematization of the Sciences refers to the gradual increase of mathematical applications to all other fields of natural and social science during 2nd half of the 20th century.

What are the two main types of mathematical models? What are their similarities and differences?

• Both require experimental data (a) A mechanistic model uses differential equations to discover the laws of nature or first principles that describe the mechanisms that drive the patterns observed in experimental data. Common in the natural sciences (b) An empirical model uses statistics to discover and describe patterns in the data, and infer cause-and-effect relationships. Common in the social sciences

What are some of the advantages of incorporating mathematical modeling into the scientific method?

• Compared to the cost of experimental science, mathematical models can be built and solved very fast and at very little cost. • Mathematical models can reveal patterns in experimental data that are invisible to the naked eye, and thus would go undiscovered by experimental science alone. • Mathematical models can simulate experiments quickly and cheaply to help predict future experimental results, and decide how to best invest money, time, and eort towards future experiments.

What is Euclid's Elements? Why is it significant?

• Euclid compiled all known mathematics of his day (300 BC) into a comprehensive work of 13 books called the Elements. The Elements is the perfect example of an axiomatic system. It begins with a set of primitive terms (23) and axioms (5 common notions and 5 postulates), and then builds over 465 theorems in logical sequence, where each proof is justified only by previously-proved theorems and axioms. -The Axiomatic Method exemplified in Euclid's Elements has become a template for organizing ideas and concepts in all fields of mathematics to this day. It is also used in many other fields outside of mathematics (philosophy, theology, law, and political science, among others).

Name one 17th century natural philosopher who used experiments and mathematics to study the world.

• Galileo studied gravity • Kepler studied planetary motion. • Newton studied light diffraction and motion

How do the words hypothesis and theory mean in the axiomatic vs. scientic method?

• In the axiomatic method, a hypothesis is the rst part of a conditional statement (If P,...). In the scientic method, a hypothesis is an educated guess about the behavior of a system, waiting to be supported or rejected by an experiment. • In the axiomatic system, a theory is the set of all theorems based on the same set of axioms. In the scientic method, a theory is an explanation of a physical system that has been supported by repeated experiments

Where does the base-10 number system that we use today come from?

• It comes from India (400 AD); the Arab Empire adopted it around 800 AD, and Europe adopted it around 1200 AD.

. Notable Abel Prize winners:

• Karen Uhlenbeck was the first women ever to receive the Abel Prize, in 2019. • Andrew Wiles was awarded the Abel Prize in 2016 for solving Fermat's Last Theorem (in 1995). • John Nash was awarded the Abel Prize in 2015 for inventing Game Theory and for his research in partial differential equations. He died shortly thereafter.

Where did mathematics originate?

• Mathematics originated independently in every civilization to develop quantitative tools required to respond to to practical or utilitarian needs of everyday life in an urban society, including timekeeping, trade, commerce, accounting, taxing, surveying, construction, engineering, public works, astronomy, and navigation. • Some notable examples of early civilizations that developed advanced mathematics include: Mesopotamia (5000 BC) Egypt (4000 BC) China (2000 BC) Mesoamerica (2000 BC) Greece (600 BC) India (400 AD) Arab Empire (800 AD)

What is the rhetorical style of writing mathematics? When did symbolic notation become widespread in mathematics?

• Mathematics written in rhetorical style is written out using only words and sentences. Most mathematics was written in rhetorical style until symbolic notation and the use of symbols (such as +, −, =, √) became widespread in the 14th century. It took another 300 years for many symbols to become standardized to what we use today.

What is natural philosophy and how is it connected to applied math?

• Natural philosophy refers to to the study of how the physical world and universe work. In the 17th century, natural philosophers first started using the scientific method and experiments to study the physical world, and using mathematical tools to describe patterns in the experimental data.

. What is an article of primary literature? What is secondary literature?

• Primary literature refers to articles of original research written by the person(s) who did the research. • Secondary literature includes review literature (works that compile, compare, summarize, or organize the research of other people), and expository literature (works that explain or clarify the research of other people for a broader audience

What is notable about Greek mathematics?

• The Greek mathematicians developed geometry into an abstract system used to organize true statements into a coherent structure based on axioms, which holds together by chains of deductive reasoning. This method (called the Axiomatic method) is the framework used for all fields of mathematics to this day. true statements based on axioms held together by deductive reasoning

How is truth established by the axiomatic method? How is truth discerned in experimental science?

• The axiomatic method establishes the truth of a statement by a proof, which connects the statement back via a sequence of logical steps to a set of axioms which are, by denition, true. Therefore, true only means consistent with the axioms. It does not necessarily mean consistent with physical reality because the axiomatic system need not be consistent with physical reality. • In experimental science, the scientic method establishes something as true if it's consistent with physical reality and can be veried by repeated experiments.

What is the impact factor of a journal?

• The impact factor of a journal is a number that represents the average number of times that an article published in that journal is cited in other works. The number is seen as an indicator for the overall quality, prestige and importance of the journal.

Describe the peer-review process and what it means to be double-blind.

• The peer-review process involves (a) the author(s), (b) the journal editors, and (c) the peerreviewers. The peer-reviewers are outside experts enlisted by the journal to carefully evaluate a manuscript that the journal's editors have judged as potentially worthy of publication. Peerreviewers provide detailed feedback on how to extend, clarify, or correct the content. It is a double-blind system, meaning that the journal's editors transmit all back-and-forth communication between peer-reviewers and authors without revealing names. A manuscript often goes through multiple rounds of revision and resubmission, which can last many months or years, before the peer-reviewers are satisfied enough with the content to recommend it for publication.

What is the proof of a theorem?

• The proof of a theorem provides a sequence of equivalent statements that are true because they follow a valid rule of inference that connects them to a previously-proved theorem, or an axiom.

What is the role of the peer-reviewers in the peer-review process?

• They carefully read a submitted manuscript and check it for correctness and originality, • They make recommendations to the author(s) on how to extend, complete, improve, clarify, or correct the content, and • They make a recommendation to the journal editors about whether the manuscript deserves to be published.

Why are there so few women in the history of mathematics?

• Women were heavily discriminated against in Western academia until well into the 20th century. Women were not allowed to study mathematics, or even audit math classes at a university. Only a very small number of independently wealthy women had access to mathematics by hiring private tutors to teach them.

Besides peer-reviewed journals, what are other sources of published mathematical research?

• conference proceedings, • dissertations, • books, • working papers or manuscripts that have not yet completed the peer-review process.