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Math 290: Fundamentals of Mathematics.

2019-07-26T17:51:37Z

Dmd54: Added the current textbook.

== Catalog Information ==

=== Title ===
Fundamentals of Mathematics.

=== (Credit Hours:Lecture Hours:Lab Hours) ===
(3:3:0)

=== Offered ===
F, W, Sp

=== Prerequisite ===
[[Math 112]] or concurrent enrollment with instructor's consent.

=== Description ===
Achieving maturity in mathematical communication. Introduction to mathematical proof; methods of proof; analysis of proof; induction; logical reasoning.

== Desired Learning Outcomes ==
This course is aimed at undergraduate mathematics and mathematics education majors. It is a first course in mathematical thinking. It is intended as an introduction to mathematical proof, and students who finish the course should achieve maturity in mathematical communication.

=== Prerequisites ===
This course has no prerequisites.

=== Minimal learning outcomes ===
Students should achieve mastery of the topics listed below. This means that they should know all relevant definitions, correct statements of the major theorems (including their hypotheses and limitations), and examples and non-examples of the various concepts. The students should be able to demonstrate their mastery by solving non-trivial problems related to these concepts, and by proving simple (but non-trivial) theorems about the below concepts, related to, but not identical to, statements proven by the text or instructor.

<div style="-moz-column-count:2; column-count:2;">
# Set Theory
#* Set builder notation
#* Venn diagrams
#* De Morgan’s Laws
# Logic
#* Truth Tables
#* Quantifiers
#* Negations of statements with quantifiers
#* Implications
#* Biconditionals
# Proof Techniques
#* Direct proof
#* Proof by contrapositive
#* Proof by contradiction
# Relations
#* Reflexive, irreflexive, symmetric, transitive relations
#* Equivalence relations
#* Equivalence classes
# Functions
#* One-to-one and onto
#* Function composition
#* Inverse functions
#* Bijective functions
#* Permutations
# Mathematical Induction
#* Well ordering principle
#* Mathematical induction
#* Strong induction
#* The method of descent
# Cardinal Numbers
#* Numerical equivalence
#* Countable and uncountable sets
#* Schröder-Bernstein theorem
# Number Theory
#* Division algorithm
#* Euclid’s Algorithm
#* Infinitude of primes
#* Unique factorization

</div>
In addition, on completion of the course, students should understand the basic mathematical language concerning logic, sets, the standard number systems, deductive and inductive reasoning, and the structure of proof. They should be able to translate a mathematical statement into logical form and discuss its negation and its implications. They should be able to translate a simple argument into logical form and detect logical validity and flaws. They should be able to read, write, listen and speak using standard mathematical terminology and reasoning.

=== Textbooks ===
Possible textbooks for this course include (but are not limited to):

* Darrin Doud and Pace P. Nielsen, <em>A Transition to Advanced Mathematics</em>, available at https://math.byu.edu/~doud/Transition/

* Gary Chartrand, Albert D. Polimeni, and Ping Zhang, ''Mathematical Proofs: A Transition to Advanced Mathematics (2nd Edition)'', Addison Wesley, 2007.

=== Additional topics ===
Beyond the minimal learning outcomes, instructors are free to cover additional topics. These may include (but are certainly not limited to): concepts of set theory, number theory, geometry, analysis, group theory, and ring theory. Instructors are free to use new approaches to the teaching of the material, as long as the core topics are adequately covered.
=== Courses for which this course is prerequisite ===
This course is required for almost all upper division course in the Mathematics department. There is a strong expectation that students who have taken this course will have a certain minimal preparation in mathematical thinking. As a result it is essential that all required learning outcomes be thoroughly covered.

[[Category:Courses|290]]

Math 290: Fundamentals of Mathematics.

2019-07-26T17:48:30Z

Dmd54:

== Catalog Information ==

=== Title ===
Fundamentals of Mathematics.

=== (Credit Hours:Lecture Hours:Lab Hours) ===
(3:3:0)

=== Offered ===
F, W, Sp

=== Prerequisite ===
[[Math 112]] or concurrent enrollment with instructor's consent.

=== Description ===
Achieving maturity in mathematical communication. Introduction to mathematical proof; methods of proof; analysis of proof; induction; logical reasoning.

== Desired Learning Outcomes ==
This course is aimed at undergraduate mathematics and mathematics education majors. It is a first course in mathematical thinking. It is intended as an introduction to mathematical proof, and students who finish the course should achieve maturity in mathematical communication.

=== Prerequisites ===
This course has no prerequisites.

=== Minimal learning outcomes ===
Students should achieve mastery of the topics listed below. This means that they should know all relevant definitions, correct statements of the major theorems (including their hypotheses and limitations), and examples and non-examples of the various concepts. The students should be able to demonstrate their mastery by solving non-trivial problems related to these concepts, and by proving simple (but non-trivial) theorems about the below concepts, related to, but not identical to, statements proven by the text or instructor.

<div style="-moz-column-count:2; column-count:2;">
# Set Theory
#* Set builder notation
#* Venn diagrams
#* De Morgan’s Laws
# Logic
#* Truth Tables
#* Quantifiers
#* Negations of statements with quantifiers
#* Implications
#* Biconditionals
# Proof Techniques
#* Direct proof
#* Proof by contrapositive
#* Proof by contradiction
# Relations
#* Reflexive, irreflexive, symmetric, transitive relations
#* Equivalence relations
#* Equivalence classes
# Functions
#* One-to-one and onto
#* Function composition
#* Inverse functions
#* Bijective functions
#* Permutations
# Mathematical Induction
#* Well ordering principle
#* Mathematical induction
#* Strong induction
#* The method of descent
# Cardinal Numbers
#* Numerical equivalence
#* Countable and uncountable sets
#* Schröder-Bernstein theorem
# Number Theory
#* Division algorithm
#* Euclid’s Algorithm
#* Infinitude of primes
#* Unique factorization

</div>
In addition, on completion of the course, students should understand the basic mathematical language concerning logic, sets, the standard number systems, deductive and inductive reasoning, and the structure of proof. They should be able to translate a mathematical statement into logical form and discuss its negation and its implications. They should be able to translate a simple argument into logical form and detect logical validity and flaws. They should be able to read, write, listen and speak using standard mathematical terminology and reasoning.

=== Textbooks ===
Possible textbooks for this course include (but are not limited to):

* Darrin Doud and Pace P. Nielsen, <em><A HREF="https:math.byu.edu/~doud/Transition">A Transition to Advanced Mathematics</a></em>,

* Gary Chartrand, Albert D. Polimeni, and Ping Zhang, ''Mathematical Proofs: A Transition to Advanced Mathematics (2nd Edition)'', Addison Wesley, 2007.

=== Additional topics ===
Beyond the minimal learning outcomes, instructors are free to cover additional topics. These may include (but are certainly not limited to): concepts of set theory, number theory, geometry, analysis, group theory, and ring theory. Instructors are free to use new approaches to the teaching of the material, as long as the core topics are adequately covered.
=== Courses for which this course is prerequisite ===
This course is required for almost all upper division course in the Mathematics department. There is a strong expectation that students who have taken this course will have a certain minimal preparation in mathematical thinking. As a result it is essential that all required learning outcomes be thoroughly covered.

[[Category:Courses|290]]

Math 587: Introduction to Analytic Number Theory.

2019-07-26T17:42:18Z

Dmd54:

== Catalog Information ==

=== Title ===
Introduction to Analytic Number Theory.

=== Credit Hours ===
(3:3:0)

=== Offered ===
Contact Department

=== Prerequisite ===
([[Math 352]]) or equivalent; instructor's consent.

=== Description ===
Arithmetical functions; distribution of primes; Dirichlet characters; Dirichlet's theorem; Gauss sums; primitive roots; Dirichlet L-functions; Riemann zeta-function; prime number theorem; partitions.

== Desired Learning Outcomes ==
Students should gain a familiarity with the problems and tools of analytic number theory at beginning graduate level.

=== Prerequisites ===
A knowledge of complex analysis at the level of a first course such as [[Math 352]] should suffice.

=== Minimal learning outcomes ===
Students should be familiar with the following concepts. They should know the technical terms, and be able to implement the methods taught in the course to work associated problems, including proving simple results.

<div style="-moz-column-count:2; column-count:2;">
#'''Arithmetic functions'''
#*convolution of arithmetic functions
#*Möbius inversion
#*multiplication function
#*Euler's summation formula, Abel's identity
#*average order of τ(''n''), σ(''n'') and φ(''n'')
#'''Elementary theorems on distribution of prime numbers'''
#*Chebyshev's inequalities
#*Mertens' theorem
#'''Finite abelian groups and their characters'''
#*characters of finite abelian groups
#*the character group
#*Dirichlet characters
#*nonvanishing of ''L''(1, χ) for real nonprincipal χ
#*Gauss sums associated with Dirichlet characters
#*Polya's inequality
#*Dirichlet's theorem on primes in arithmetic progression
#'''Dirichlet series and Euler product'''
#*Half plane of absolute convergence
#*multiplication of Dirichlet series
#*Euler products
#*Perron's formula for partial sums
#'''The zeta function and the Dirichlet L-functions'''
#*Properties of the gamma function
#*Hurwitz zeta function
#*Analytic continuation of ζ(''s'') and ''L''(''s'', χ)
#*functional equation for ζ(''s'') and ''L''(''s'', χ)
#*nonvanishing of ζ(''s'') on σ = 1
#'''Partitions'''
#*Generating functions for partitions
#*Euler's pentagonal-number theorem
#*Jacobi's triple product identity and its consequences.<br><br><br>
</div>
=== Textbooks ===

Possible textbooks for this course include (but are not limited to):

* Tom Apostol, <em>Introduction to Analytic Number Theory</em>

=== Additional topics ===
These are at the discretion of the instructor as time allows.

=== Courses for which this course is prerequisite ===
None

[[Category:Courses|587]]

Math 686R: Topics in Algebraic Number Theory.

2016-07-11T22:20:42Z

Dmd54:

== Catalog Information ==

=== Title ===
Topics in Algebraic Number Theory.

=== Credit Hours ===
3

=== Prerequisite ===
Math 372 and permission of the Instructor. In general, the prerequisites will depend on the material covered.

=== Description ===
Current topics of research interest.

== Desired Learning Outcomes ==
Gaining mastery over an advanced area of algebraic number theory of interest in research.

=== Prerequisites ===
Math 372 and permission of the Instructor. In general, the prerequisites will depend on the material covered.

=== Minimal learning outcomes ===
These cannot be specified uniquely for a topics course. Past topics have included "Primes of the form <i>x<sup>2</sup>+Ny<sup>2</sup></i>," "The Proof of Fermat's Last Theorem," and "Computational Algebraic Number Theory." The following example gives a clear indication of the level of difficulty appropriate to the course. Students should know the technical terms, and be able to implement the methods taught in the course to work associated problems, including proving results of suitable accessibility.

<div style="-moz-column-count:2; column-count:2;">
# Primes of the form <i>x<sup>2</sup>+Ny<sup>2</sup></i>
#* Classical Topics
#** Equivalence of binary quadratic forms
#** Composition of Binary quadratic forms
#** Relation between binary quadratic forms and ideal class groups of quadratic fields
#** Representations of integers by binary quadratic forms
#* Class field theory
#** Artin Reciprocity and the Artin Map
#** The Existence and Conductor Theorems
#** Cebotarev Density Theorem
#** Identifying Ray Class Fields and Ring Class Fields
#* Complex Analytic Techniques
#** Elliptic Functions
#** Properties of the Weierstrass p-function
#** Invariants of Lattices
#** Modular forms--definitions and properties
#** Singular values of modular forms
#** The modular equation
#** Computing generators for ring class fields
</div>

=== Textbooks ===

Possible textbooks for this course include, (but are not limited to):
* D. Cox, Primes of the form x^2+Ny^2
* H. Cohen, A course in computational number theory

=== Additional topics ===
None

=== Courses for which this course is prerequisite ===

None

[[Category:Courses|686]]