Introduction To Modern Number Theory: Fundament...

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\"Introduction to Modern Number Theory\" surveys from a unified point of view both the modern state and the trends of continuing development of various branches of number theory. Motivated by elementary problems, the central ideas of modern theories are exposed. Some topics covered include non-Abelian generalizations of class field theory, recursive computability and Diophantine equations, zeta- and L-functions.

Description: An introduction to mathematical reasoning, construction of proofs, and careful mathematical writing in the context of continuous mathematics and calculus. Topics may include the real number system, limits and continuity, the derivative, integration, and compactness in terms of the real number system.

A set is pure if all of its members are sets, all members of its members are sets, and so on. For example, the set containing only the empty set is a nonempty pure set. In modern set theory, it is common to restrict attention to the von Neumann universe of pure sets, and many systems of axiomatic set theory are designed to axiomatize the pure sets only. There are many technical advantages to this restriction, and little generality is lost, because essentially all mathematical concepts can be modeled by pure sets. Sets in the von Neumann universe are organized into a cumulative hierarchy, based on how deeply their members, members of members, etc. are nested. Each set in this hierarchy is assigned (by transfinite recursion) an ordinal number α {\\displaystyle \\alpha } , known as its rank. The rank of a pure set X {\\displaystyle X} is defined to be the least ordinal that is strictly greater than the rank of any of its elements. For example, the empty set is assigned rank 0, while the set {{}} containing only the empty set is assigned rank 1. For each ordinal α {\\displaystyle \\alpha } , the set V α {\\displaystyle V_{\\alpha }} is defined to consist of all pure sets with rank less than α {\\displaystyle \\alpha } . The entire von Neumann universe is denoted V {\\displaystyle V} .

Solution of systems of nonlinear equations, solution of initial value problems, matrix norms and the analysis of iterative solutions, numerical solution of boundary value problems and partial differential equations, and introduction to the finite element method. NOTE: Offered in even-numbered years only.

This course combines traditional material with a modern systems approach. It presents a thorough introduction to differential equations, tempering a classic \"pure math\" approach with more practical applied aspects. The course covers key topics such as first order equations, matrix algebra, systems, and phase plane portraits. The focus is on interpreting and solving problems through the use of software support and technology projects. Using software tools graphics will be used to display the ideas in ODEs; modeling and applications; and projects. An objective is to provide students with the opportunity to bring together much of what they have learned, including analytical, computational, and interpretative skills.

This is the first semester in a year-long modern algebra sequence MATH 3098 - MATH 3101. It is a thorough introduction to the theory of groups and rings. NOTE: Students who have had limited exposure to proofs should consider taking MATH 2111 first.

The book is intended as a sort of survey of all of modern number theory, while at the same time serving as the introduction to the other number theory volumes in this series. The style is typical of broad surveys: big theorems are quoted carefully but not proved, but simple arguments and proofs are often given when they don't require a lot of space. Embedded in the text are a lot of interesting ideas, insights, and clues to how the authors think about the subject.

An introduction to mathematical logic and proof within the context of discrete structures. Topics include basic mathematical logic, elementary number theory, basic set theory, functions, and relations. Writing proficiency is required for a passing grade in this course. A student who does not write with the skill normally required of an upper-division student will not earn a passing grade, no matter how well the student performs in other areas of the course.

This course will give an overview of geometry from a modern point of view. Axiomatic, analytic, transformational, and algebraic approaches to geometry will be used. The relationship between Euclidean geometry, the geometry of complex numbers, and trigonometry will be emphasized.

This course is an introduction to nonlinear programming. Topics will include necessary and sufficient conditions for optimality, as well as basic theory and numerical algorithms for several traditional optimization methods, e.g., basic descent methods, conjugate direction methods, quasi-Newton methods, penalty and barrier methods, Lagrange multiplier methods. A brief introduction to selected modern topics may be added if time permits.

This course is an introduction to proofs and abstract mathematical thinking, serving as a bridge from introductory courses such as calculus to more advanced proof-based courses. The principal goal of this course is to help students develop skills for both reading and writing mathematical proofs. Topics covered may include fundamentals of logic, quantifiers, proof techniques, mathematical induction, elementary set theory, equivalence relations, functions, and the notions of countability and uncountability. Some topics in analysis will also be surveyed, such as open and closed sets in the real line, sequences of real numbers, and limits of functions. Additional topics may vary from year to year.

An introduction to the theory of rational integers; divisibility, the unique factorization theorem; congruences, quadratic residues. Selections from the following topics: cryptology; Diophantine equations; asymptotic prime number estimates; continued fractions; algebraic integers. Four class hours per week. Offered in alternate years.

The p-adic numbers were first introduced by Kurt Hensel near the end of the nineteenth century. Since their introduction they have become a central object in modern number theory, algebraic geometry, and algebraic topology. These numbers give a new set of fields (one for each prime p) that contain the rational numbers and behave in some ways like the real numbers, but they also possess some unique and unexpected properties. The main objective of this course will be to rigorously construct the p-adic numbers and explore some of the algebraic, analytic, and topological properties that make them interesting. Four class meetings per week.

An introduction to Lebesgue measure and integration; topology of the real numbers; inner and outer measures and measurable set; the approximation of continuous and measurable functions; the Lebesgue integral and associated convergence theorems; the Fundamental Theorem of Calculus. Four class hours per week. 59ce067264

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