MathWiki - User contributions [en]

Math 341: Theory of Analysis 1

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== Catalog Information ==

=== Title ===
Theory of Analysis 1.

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

=== Offered ===
F, W

=== Prerequisite ===
[[Math 113]], [[Math 290|290]].

=== Description ===
Rigorous treatment of calculus of a single real variable: topology, order, completeness of real numbers; continuity, differentiability, integrability, and convergence of functions.

== Desired Learning Outcomes ==
The main purpose of this course is to provide students with an understanding of the real number system and of real-valued functions of a single real variable, with the focus being on the theoretical and logical foundations of single-variable calculus. A secondary purpose of this course is to reinforce students' prior training in discovering and writing valid mathematical proofs.

=== Prerequisites ===
The prerequisites for this course are Math [[Math 113|113]] and [[Math 290|290]]. The first is to ensure that the student has had a complete course in single-variable calculus at the introductory level, and the second is to ensure that the student knows how to read and write proofs and is familiar with the fundamental objects of advanced mathematics.

=== Minimal learning outcomes ===
Outlined below are topics that all successful Math 341 students should understand well. As evidence of that understanding, students should be able to demonstrate mastery of all relevant vocabulary, familiarity with common examples and counterexamples, knowledge of the content of the major theorems, understanding of the ideas in their proofs, and ability to make direct application of those results to related problems.

<div style="-moz-column-count:2; column-count:2;">

# Basic properties of '''R'''
#* Characterization as a complete, ordered field
#* Archimedean Property
#* Density of '''Q''' and '''R''' \ '''Q'''
#* Uncountability of each interval
# Convergence of real sequences and series
#* Convergence of bounded monotonic sequences
#* Algebraic and order rules for limits
#* Cauchy criterion for sequences and series
#* Common convergence tests for series
#* Convergence of rearranged series
# Basic topology of '''R'''
#* Open and closed sets
#* Limit points and limits
#* Characterizations of compactness
#** Sequential
#** Open coverings
#** The Heine-Borel Theorem
#* The Bolzano-Weierstrass Theorem
#* Connectedness
# Continuity of ''f'': ''D'' ⊆ '''R''' → '''R'''
#* Functional limits
#* Metric, sequential, and topological characterizations of continuity
#* Combinations of continuous functions
#* Continuity vs. uniform continuity
#* Preservation of compactness
#** The Extreme Value Theorem
#* Uniform continuity for compact domains
#* Preservation of connectedness
#** The Intermediate Value Theorem
# Differentiability of ''f'': ''D'' ⊆ '''R''' → '''R'''
#* Algebraic differential rules
#* Chain rule
#* Characterizing extrema
#* Rolle's Theorem
#* The Mean Value Theorem
#* The Generalized Mean Value Theorem
#* L'Hôpital's Rule
# Integrability of ''f'': [''a'',''b''] → '''R'''
#* The Darboux integral
#* The Riemann integral
#* Integrability of continuous functions
#* Integrability of monotonic functions
#* Rules for combining and comparing integrals
#* The Fundamental Theorem(s) of Calculus
# Convergence of sequences and series of functions
#* Pointwise vs. uniform convergence
#* Relation of uniform convergence to:
#** Continuity
#** Differentiation
#** Integration
#* The Weierstrass ''M''-Test
#* The Weierstrass Approximation Theorem
#* Power series
#** Continuity
#** Absolute and uniform convergence
#** Termwise differentiability
#** Taylor's Theorem
#** Analyticity vs. smoothness

</div>

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

* Stephen Abbott, ''Understanding Analysis, 2nd Edition'', Springer, 2015.

=== Additional topics ===
Among other options, instructors may want to discuss constructing the real numbers using
Dedekind cuts or Cauchy sequences.

=== Courses for which this course is prerequisite ===
As the foundational course in real analysis, Math 341 is a prerequisite for many advanced undergraduate and graduate courses in pure and applied analysis: Math [[Math 342|342]], [[Math 451|451]], [[Math 465|465]], [[Math 534|534]], [[Math 541|541]], and [[Math 634|634]].

[[Category:Courses|341]]

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Math 320: Computation & Optimization 1

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Math 320

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== Catalog Information ==

=== Title ===
Algorithms Design and Optimization 1

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

=== Offered ===
F

=== Prerequisite ===
[[Math 290]], [[Math 313]], [[Math 314]], [[Math 341]]; concurrent with [[Math 321]], [[Math 334]], [[Math 344]]

=== Description ===
A treatment of algorithms used to solve these problems. Specific topics include Complexity and Data, Approximation Theory, Recursive Algorithms, Linear Optimization, Unconstrained Optimization, Constrained Optimization, Global Optimization.

== Desired Learning Outcomes ==

=== Prerequisites ===
[[Math 290]], [[Math 313]], [[Math 314]], [[Math 341]]; concurrent with [[Math 321]], [[Math 334]], [[Math 344]]

=== Minimal learning outcomes ===
Students will have a solid understanding of the concepts listed below. They will be able to prove many of the theorems that are central to this material. They will understand the model specifications for the optimization algorithms, and be able to recognize whether they apply to a given application or not. They will be able to perform the relevant computations on small, simple problems. They will be able to describe the optimization and approximation algorithms well enough that they could program simple versions of them, and will have a basic knowledge of the computational strengths and weaknesses of the algorithms covered.

# Complexity and Data
#* Asymptotic Analysis
#* Combinatorics
#* Graphs and Trees
#* Complexity (P, NP, NP Complete)
# Approximation Theory
#* Interpolation and Splines
#* Stone-Weierstrass Theorem
#* Bezier Curves
#* B-Splines
# Recursive Algorithms
#* Difference Calculus, including Summation by Parts
#* Simple linear recurrences
#* General linear recurrences
#* Generating functions
# Linear Optimization
#* Problem Formulation
#* Simplex Method
#* Duality
#* Applications
# Unconstrained Optimization
#* Steepest Descent
#* Newton
#* Broyden
#* Conjugate Gradient
#* Applications
# Constrained Optimization
#* Equality Constrained, Lagrange Multipliers
#* Inequality Constrained, KKT Condition
#* Applications
# Global Optimization
#* Interior Point Methods
#* Genetic Algorithms
#* Simulated Annealing

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

*

=== Additional topics ===

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

[[Category:Courses|320]]

Math 320: Computation & Optimization 1

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Math 320: Algorithm Design and Optimization 1

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== Catalog Information ==

=== Title ===
Algorithm Design and Optimization 1

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

=== Offered ===
F

=== Prerequisite ===
[[Math 290]], [[Math 313]], [[Math 314]], [[Math 341]]; concurrent with [[Math 321]], [[Math 334]], [[Math 344]]

=== Description ===
A treatment of algorithms used to solve these problems. Specific topics include Complexity and Data, Approximation Theory, Recursive Algorithms, Linear Optimization, Unconstrained Optimization, Constrained Optimization, Global Optimization.

== Desired Learning Outcomes ==

=== Prerequisites ===
[[Math 290]], [[Math 313]], [[Math 314]], [[Math 341]]; concurrent with [[Math 321]], [[Math 334]], [[Math 344]]

=== Minimal learning outcomes ===
Students will have a solid understanding of the concepts listed below. They will be able to prove many of the theorems that are central to this material. They will understand the model specifications for the optimization algorithms, and be able to recognize whether they apply to a given application or not. They will be able to perform the relevant computations on small, simple problems. They will be able to describe the optimization and approximation algorithms well enough that they could program simple versions of them, and will have a basic knowledge of the computational strengths and weaknesses of the algorithms covered.

# Complexity and Data
#* Asymptotic Analysis
#* Combinatorics
#* Graphs and Trees
#* Complexity (P, NP, NP Complete)
# Approximation Theory
#* Interpolation and Splines
#* Stone-Weierstrass Theorem
#* Bezier Curves
#* B-Splines
# Recursive Algorithms
#* Difference Calculus, including Summation by Parts
#* Simple linear recurrences
#* General linear recurrences
#* Generating functions
# Linear Optimization
#* Problem Formulation
#* Simplex Method
#* Duality
#* Applications
# Unconstrained Optimization
#* Steepest Descent
#* Newton
#* Broyden
#* Conjugate Gradient
#* Applications
# Constrained Optimization
#* Equality Constrained, Lagrange Multipliers
#* Inequality Constrained, KKT Condition
#* Applications
# Global Optimization
#* Interior Point Methods
#* Genetic Algorithms
#* Simulated Annealing

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

*

=== Additional topics ===

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

[[Category:Courses|320]]

Math 320: Algorithm Design and Optimization 1

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Math 437: Modeling with Dynamics and Control 1 Lab

2015-05-04T22:44:02Z

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== Catalog Information ==

=== Title ===
Modeling with Dynamics and Control 1 Lab

=== (Credit Hours:Lecture Hours:Lab Hours) ===
(1:0:2)

=== Offered ===
F

=== Prerequisite ===
CS 142, [[Math 323]], [[Math 347]]; concurrent with [[Math 436]]

=== Description ===
Using and developing software to implement the content of [[Math 436]]. Developing models and applying results of computations to several application domains. Introduction to parallel computation.

== Desired Learning Outcomes ==

=== Prerequisites ===

=== Minimal learning outcomes ===
Students will be able to model many real-world applications effectively using the concepts covered in [[Math 436]]. They will be able to use commercial software to do the computations that are central to these concepts, and will be able to interpret and apply the results of those computations in the contexts in which the models originated.

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

*

=== Additional topics ===

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

[[Category:Courses|437]]

Math 436: Modeling with Dynamics and Control 1

2015-05-04T22:43:48Z

Dac47: /* Offered */

== Catalog Information ==

=== Title ===
Modeling with Dynamics and Control 1

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

=== Offered ===
F

=== Prerequisite ===
[[Math 322]], [[Math 341]], [[Math 346]]; concurrent with [[Math 437]]

=== Description ===
Theory and applicatiions of dynamic systems and partial differential equations. Topics include dynamic systems; bifurcation theory; control theory; hyperbolic, parabolic, and elliptic partial differential equations; commonly-used algorithms.

== Desired Learning Outcomes ==

=== Prerequisites ===
[[Math 322]], [[Math 341]], [[Math 346]]; concurrent with [[Math 437]]

=== Minimal learning outcomes ===
Students will have a solid understanding of the concepts listed below. They will be able to prove theorems that are central to this material, including theorems that they have not seen before. They will be able to recognize whether differential equation models apply in the context of a given application or not. They will be able to perform the relevant computations on small, simple problems.

# Dynamical Systems
#* Matrix Exponentiation
#* Linear Stability
#* Variation of Constants Formula
#* Lyapunov Matrices
#* Nonlinear Stability
#* Stable, Center, and Unstable Manifolds
#* Poincare-Bendixson Theorem
#* Chaotic Dynamics
# Bifurcation Theory
#* Saddle-Node Bifurcation
#* Trans-critical Bifurcation
#* Pitchfork Bifurcation
#* Hopf Bifurcation
# Control Theory
#* State-Space Models and Realizations
#* Observability and Controllability
#* Transfer Functions and Minimal Realizations
#* State Feedback
# Hyperbolic PDE
#* Method of Characteristics
#* Shock Waves and Rarefactions
#* Wave Equation
#* Fourier Transform
#* Separation of Variables
# Parabolic PDE
#* Heat Equation
#* Maximum Principle
#* Separation of Variables
#* Convolution and the Gaussian Kernel
# Elliptic PDE
#* Laplace equation
#* Separation of Variables
#* Green’s Function

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

*

=== Additional topics ===

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

[[Category:Courses|436]]

Math 403: Modeling with Uncertainty and Data 1 Lab

2015-05-04T22:43:25Z

Dac47: /* Offered */

== Catalog Information ==

=== Title ===
Modeling with Uncertainty and Data 1 Lab

=== (Credit Hours:Lecture Hours:Lab Hours) ===
(1:0:2)

=== Offered ===
F

=== Prerequisite ===
CS 142, [[Math 323]], [[Math 347]]; concurrent with [[Math 402]]

=== Description ===
Using and developing algorithms for the content of [[Math 402]]; developing I/O wrappers for numerical libraries. Applications presented. Developing models and applying results of computations.

== Desired Learning Outcomes ==

=== Prerequisites ===

=== Minimal learning outcomes ===
Students will be able to model specific applications using the concepts covered in [[Math 402]], and to critique the models in the context from which the application was taken. They will be able to use software to do the computations that are central to these concepts, and will be able to interpret and apply the results of those computations in the contexts in which the models originated.

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

*

=== Additional topics ===

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

[[Category:Courses|403]]

Math 402: Modeling with Uncertainty and Data 1

2015-05-04T22:42:54Z

Dac47: /* Offered */

== Catalog Information ==

=== Title ===
Modeling with Uncertainty and Data 1

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

=== Offered ===
F

=== Prerequisite ===
[[Math 322]], [[Math 346]]; concurrent with [[Math 403]]

=== Description ===
Theory of probability and stochastic processes, emphasizing topics used in applications. Random spaces and variables, probability distributions, limit theorems, martingales, diffusion, Markov, Poisson and queuing processes, renewal theory, and information theory.

== Desired Learning Outcomes ==

=== Prerequisites ===
[[Math 322]], [[Math 346]]; concurrent with [[Math 403]]

=== Minimal learning outcomes ===
Students will have a solid understanding of the concepts listed below. They will be able to prove theorems that are central to this material, including theorems that they have not seen before. They will understand connections between the concepts taught, and will be able to perform the related computations on small, simple problems. They will understand the model specifications for martingales and for diffusion, Markov, Poisson, queuing and renewal theoretic processes, and be able to recognize whether they apply in the context of a given application or not. They will be able to perform the relevant computations on small, simple problems.

# Random Spaces and Variables
#* Probability Spaces (including σ-algebras)
#* Random Variables (including Measurable Functions)
#* Expectation (including Lebesgue Integration)
#* Independence
#* Conditional Expectation
#* Law of Large Numbers
# Distributions
#* Generating Functions and Characteristic Functions
#* Moments
#* Commonly Used Distributions
#* Joint and Conditional Distributions
# Limit Theorems
#* Weak Convergence
#* Central Limit Theorem
#* Applications
# Martingales and Diffusion
#* Stochastic Processes, Filtrations, Stopping Times
#* Martingales
#* Doob's Decomposition Theorem
#* Doob's Inequality and Convergence Theorems
# Markov Processes
#* The Markov Property
#* Finite Markov Chains
#* Asymptotic Behavior
#* Absorbing Markov Chains
#* Continuous-Time Markov Chains
# Poisson, Queuing, and Renewal Theory
#* Counting Integrals
#* Kolomogorov's Forward System
#* Poisson Processes
#* Queues
#* Renewal Processes
# Information Theory
#* Entropy
#* Conditional and Joint Entropy
#* Kullback-Lieber Distance
#* Channel Capacity

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

*

=== Additional topics ===

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

[[Category:Courses|402]]

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Math 406R: Topics in Mathematics

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== Catalog Information ==

=== Title ===
Topics in Mathematics

=== (Credit Hours:Lecture Hours:Lab Hours) ===
(3:3:1)

=== Offered ===

=== Prerequisite ===
Instructor's consent

=== Description ===
Topics selected from various aspects of mathematics. Possibilities include, but are not limited to: combinatorial design theory; factorization and primality testing; game theory; harmonic analysis; hyperbolic geometry; linear programming; Lie groups; p-adic numbers; set theory and mathematical logic; stochastic processes; supply chain management; voting theory.

== Desired Learning Outcomes ==

=== Prerequisites ===

=== Minimal learning outcomes ===

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

*

=== Additional topics ===

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

[[Category:Courses|406R]]

Math 406R - Topics in Mathematics

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Math 402

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Talk:Math 303

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Math 300: History and Philosophy of Mathematics

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== Catalog Information ==

=== Title ===
History and Philosophy of Mathematics.

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

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

=== Prerequisite ===
[[Math 113]].

=== Description ===
Historical development of important mathematical ideas and philosophies; implications for the mathematical curriculum.

== Desired Learning Outcomes ==
The main purpose of this course is to learn mathematics through its history, to clarify and reinforce fundamental mathematical ideas, such as (but not limited to) not just who were the first to compute π to whatever degree of accuracy, but more importantly, why is π well-defined, ''i.e.'', why is the ratio circumference/diameter the same for all circles, or how the ancient civilizations figured out the exact formula for the area of circle without using integral calculus, as done in calculus courses.

=== Prerequisites ===
Knowledge of the topics whose history is being discussed, ''e.g.'', basic algebra, arithmetic, geometry and calculus.

=== Minimal learning outcomes ===
The students should be able to demonstrate a historical perspective of mathematics. They should be able to exhibit with clarity both the ideas and their evolution over the period of their history. The students are expected to know the following periods of mathematical history:

<div style="-moz-column-count:2; column-count:2;">
# '''Mathematics of ancient civilizations''', especially of Babylon and Egypt with special reference to:
#* '''Arithmetic'''. Representation of (natural) numbers in Egyptian, Chinese, Mayan and Roman numerals. The evolution of our own (Indo-Arabic) number system. Ability to do arithmetic in different number systems and conversion from one number system to another.
#* '''Geometry'''. Basic facts known at the time about triangles, circles, Pythagorean Theorem, Pythagorean triplets, Plimpton 322, Rhind Papyrus, proof of proportionality of sides of similar triangles, universality of π, and the formula for the area of a circle (without using integral calculus).
# '''Greek Mathematics'''. Evolution of Greek civilization with special reference to mathematics, its indebtedness to Babylonian and Egyptian math, life and contribution to math of Thales, Pythagoras, Zeno, Euclid, Apollonius, Eudoxus, Archimedes, Diophantus, Pappus and others. Discussion of Euclid's ''Elements'', a sketch of his self-contained proof of Pythagorean Theorem.
# '''Mathematics of Arabia'''. Demise of the Greek math, Dark Ages in Europe, migration of Greek mathematicians to Arabia, translation of Greek texts into Arabic at the House of Wisdom at Baghdad. Arab mathematicians, in particular al-Khwarizmi, their contribution to mathematics, including transmission of Indian mathematics to Europe.
# '''Mathematics in China''' with special reference to the Chinese Remainder Theorem.
# '''Mathematics of Medieval India'''. Indian mathematicians, esp., Āryabhaṭa, Brahmagupta and Bhāskara, their solutions of the quadratic equation and the Pell equation, using the Pell equation to approximate square roots.
# '''Mathematics of Medieval Europe'''. Life and work of Fibonacci, especially his rabbit problem and congruent numbers (numbers that are areas of right triangles with all sides rational).
# '''Mathematics of Renaissance'''. Famous story behind Cardano's formula for solving cubic equations, discussion of the cube roots of unity, Cardano's formula and its proof.
# '''Emergence of Modern Mathematics'''. History of modern mathematical symbols (+, -, ×, etc.), introduction of letters for variables and constants by Viète and Descartes. The beginnings of modern number theory with Fermat. Invention of:
#* Calculus, by Newton and Leibniz,
#* Algebraic geometry, by Descartes and Fermat.
# '''Bernoullis and Euler'''. Work of Euler and his standardization of current mathematical symbols and terminology.
# '''French and German schools of Mathematics'''. Gauss, Riemann, Cantor, Dirichlet, Weierstrass, . . ., and Cauchy, Lagrange, Laplace, Fourier, . . . .
# '''Hilbert and his 23 Problems''', and their impact on 20th century math.

</div>

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

*

=== Additional topics ===

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

[[Category:Courses|300]]

Math 300: History & Philosophy of Mathematics

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Math 410

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Math 303

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Math 439

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Math 437

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Math 436

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Math 436: Differential and Integral Equations 1

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Math 405

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Math 404

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Math 405: Probability and Statistics 2 Lab

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Math 403

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Math 402

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Math 322

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Math 352

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Math 347

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Math 347: Linear and Nonlinear Analysis 2 Lab

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