MathWiki - User contributions [en]

Math 425: Mathematical Biology

2008-08-18T23:43:37Z

Rcb3:

Math 425

Title

Mathematical Biology.

(Credit Hours: Lecture Hours: Lab Hours)

(3:3:0)

Prerequisite

112

Description

How tools in mathematics can help biologists. How questions in biology can motivate new mathematics.

Desired Learning Outcomes

Students should gain a familiarity with how the disciplines of mathematics and biology can complement each other.

Prerequisites

A knowledge of calculus (and the mathematical maturity that having passed M112 entails) shoud suffice.

Minimal learning outcomes

Students should be familiar with the following discrete and continuous models of biological

phenomena. They should know the technical terms, and be able to implement the procedures
taught in the course to solve problems based on these models.

Basic notions concerning: Subcellular molecular systems. Cellular behavior. Physiological

problems. Population biology. Developmental biology. Mathematical techniques of phase

plane analysis, bifurcation theory, scientific computation, difference equations,

and stochastic processes.

Topics that will be covered within this program include

Signal transduction:

Menten Michaelis enzyme dynamics

Law of mass action

Dynamical systems

Bifurcation

Example systems:

Fitzhugh-Nagumo

Nerve and heart dynamics

Cell cycle model

cAMP

Population models:

Continuous predator-prey

Age structured models

Discrete dynamical systems

Time delayed differential equations

Stochastic models.

Additional Topics

These are at the discretion of the instructor as time allows.

Courses for which this course is prerequisite

None.

Discrete and continuous models of biological phenomena will be introduced including subcellular molecular systems, cellular behaviour, physiological problems

Math 425: Mathematical Biology

2008-08-18T23:42:15Z

Rcb3:

Math 425

Title

Mathematical Biology.

(Credit Hours: Lecture Hours: Lab Hours)

(3:3:0)

Prerequisite

112

Description

How tools in mathematics can help biologists. How questions in biology can motivate new mathematics.

Desired Learning Outcomes

Students should gain a familiarity with how the disciplines of mathematics and biology can complement each other.

Prerequisites

A knowledge of calculus (and the mathematical maturity that having passed M112 entails) shoud suffice.

Minimal learning outcomes

Students should be familiar with the following discrete and continuous models of biological
phenomena. They should know the technical terms, and be able to implement the procedures
taught in the course to solve problems based on these models.

Basic notions concerning: Subcellular molecular systems. Cellular behavior. Physiological
problems. Population biology. Developmental biology. Mathematical techniques of phase
plane analysis, bifurcation theory, scientific computation, difference equations,
and stochastic processes.

Topics that will be covered within this program include

Signal transduction:
Menten Michaelis enzyme dynamics
Law of mass action
Dynamical systems
Bifurcation

Example systems:
Fitzhugh-Nagumo
Nerve and heart dynamics
Cell cycle model
cAMP

Population models:
Continuous predator-prey
Age structured models
Discrete dynamical systems
Time delayed differential equations

Stochastic models.

Additional Topics

These are at the discretion of the instructor as time allows.

Courses for which this course is prerequisite

None.

Discrete and continuous models of biological phenomena will be introduced including subcellular molecular systems, cellular behaviour, physiological problems

Math 425: Mathematical Biology

2008-08-18T23:40:56Z

Rcb3: New page: Math 425 Title Mathematical Biology. (Credit Hours: Lecture Hours: Lab Hours) (3:3:0) Prerequisite 112 Description How tools in mathematics can help biologists. How questions...

Math 425

Title

Mathematical Biology.

(Credit Hours: Lecture Hours: Lab Hours)

(3:3:0)

Prerequisite

112

Description

How tools in mathematics can help biologists. How questions in biology can motivate new mathematics.

Desired Learning Outcomes

Students should gain a familiarity with how the disciplines of mathematics and biology can complement each other.

Prerequisites

A knowledge of calculus (and the mathematical maturity that having passed M112 entails) shoud suffice.

Minimal learning outcomes

Students should be familiar with the following discrete and continuous models of biological phenomena. They should know the technical terms, and be able to implement the procedures taught in the course to solve problems based on these models.

Basic notions concerning: Subcellular molecular systems. Cellular behavior. Physiological problems. Population biology. Developmental biology. Mathematical techniques of phase plane analysis, bifurcation theory, scientific computation, difference equations, and stochastic processes.

Topics that will be covered within this program include

Signal transduction:
Menten Michaelis enzyme dynamics
Law of mass action
Dynamical systems
Bifurcation

Example systems:
Fitzhugh-Nagumo
Nerve and heart dynamics
Cell cycle model
cAMP

Population models:
Continuous predator-prey
Age structured models
Discrete dynamical systems
Time delayed differential equations

Stochastic models.

Additional Topics

These are at the discretion of the instructor as time allows.

Courses for which this course is prerequisite

None.

Discrete and continuous models of biological phenomena will be introduced including subcellular molecular systems, cellular behaviour, physiological problems

Math 485: Mathematical Cryptography

2008-08-18T23:07:56Z

Rcb3: New page: Math 485 Title Introduction to cryptography. Prerequisite Math 371. Description A mathematical introduction to some of the high points of modern cryptography. Desired Learn...

Math 485

Title

Introduction to cryptography.

Prerequisite

Math 371.

Description

A mathematical introduction to some of the high points of modern cryptography.

Desired Learning Outcomes

This is a course in the mathematics and algorithms of modern cryptography. It complements, rather than being equivalent to, the current CS course on Information Security.

Prerequisite

The requirement for Math 371 ensures both an appropriate level of mathematical maturity and a basic knowledge of linear algebra.

Minimal Learning Outcomes

The student should gain a understanding of the following topics. In particular this includes knowing the definitions, being familiar with standard examples, and being able to solve mathematical and algorithmic problems by directly using the material taught in the course. This includes appropriate use of Maple, Mathematica and Matlab.

1. Classical systems, including substitution theory, block ciphers and Enigma.

2. Elementary number theory as follows: Euclid's algorithm. Modular arithmetic and the algorithm for modular exponentiation. Chinese Remainder Theorem. Fermat and Euler Theorems. Primitive roots. Elementary continued fractions. Simple discussion of finite fields.

3. The DES and AES encryption standards.

4.RSA and its weaknesses. Primality testing and factorization. The Quadratic Sieve. Wiener's continued fraction attack on low decryption exponent.

5. Discrete logarithms. Diffie-Hellman key exchange. ElGamal.

6. Lattices and Lattice Algorithms. The LLL algorithm. The NTRU system. Lattice attacks on RSA.

Additional Topics

If time allows: Elliptic curve cryptography.

Courses for which this course is prerequisite

None.

Garbage 4

2008-08-15T22:25:08Z

Rcb3: Proposed Linear Analysis Course moved to M540: Integrate with other courses approaching final dept. approval.

#REDIRECT [[M540]]

Math 541: Real Analysis

2008-08-15T22:21:15Z

Rcb3: Math 540 moved to Math 541 over redirect: Previous move was an error

== Catalog Information ==

=== Title ===
Real Analysis.

=== Credit Hours ===
3

=== Prerequisite ===
[[Math 315]], [[Math 343|343]]; [[Math 214|214]] or [[Math 316|316]].

=== Description ===
Rigorous treatment of differentiation and integration theory; Lebesque
measure; Banach spaces.

== Desired Learning Outcomes ==

Math 541 is currently the first half of a two-semester sequence on Lebesgue integration in Euclidean space and several related topics, but it is proposed that it become a one-semester course specifically on Lebesgue integration in Euclidean space and Fourier analysis.

=== Prerequisites ===

Currently, Math 541 requires a semester of single-variable real analysis and a semester of multi-variable Calculus. Replacing these prerequisites by [[Math 316]] would imply that the new version of Math 541 could presuppose that students had been exposed to the geometry of Rn and to metric spaces, which would make it easier to cover the core topics listed below.

=== Minimal learning outcomes ===

Outlined below are topics that all successful Math 541 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;">
# Lebesgue measure on Rn
#* Inner and outer measures
#* Construction of Lebesgue measure
#* Properties of Lebesgue measure
#** Effect of basic set operations
#** Limiting properties
#** Its domain
#** Approximation properties
#** Sets of outer measure zero
#** Invariance w.r.t. isometries
#** Effect of dilations
#* Existence of nonmeasurable sets
# Lebesgue integration on Rn
#* Measurable functions
#* Simple functions
#* Approximation of measurable functions with simple functions
#* The extended reals
#* Integrating nonnegative functions
#* Integrating absolutely-integrable functions
#* Integrating on measurable sets
#* Basic properties of the Lebesgue integral
#** Linearity
#** Monotonicity
#** Effects of sets of measure zero
#** Absolute continuty of integration
#** Fatou's Lemma
#** Monotone Convergence Theorem
#** Dominated Convergence Theorem
#** Differentiation w.r.t. a parameter
#** Linear changes of variable
#** Compatibility with Riemann integration
# Fubini's Theorem for Rn
# L1, L2, and L∞
#* Completeness
#* Approximation by smooth functions
#* Continuity of translation
# Fourier transform on Rn
#* Convolutions
#* Basic properties of Fourier transforms
#** Composition with translation, dilation, inversion, differentiation, convolution, etc.
#** Regularity of transformed functions
#** Riemann-Lebesgue Lemma
#* Inversion Theorem for L1
#* Schwartz class
#* Fourier-Plancherel Transform on L2
#** Its inversion
#** Isomorphism
#* Fourier series
#** Dirichlet and Fejér kernels
#** L2 convergence
#** Pointwise convergence
#** Convergence of Cesàro means
</div>

=== Additional topics ===

Extra time could be used to go into Fourier analysis in more depth.

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

Currently, it is not a formal prerequisite for any class. Under the proposed changes, it will probably be recommended for those taking [[Math 641]].

[[Category:Courses|541]]

Talk:Math 541: Real Analysis

2008-08-15T22:21:15Z

Rcb3: Talk:Math 540 moved to Talk:Math 541 over redirect: Previous move was an error

== Proposed Catalog Changes ==
* Rename as "Real Analysis 1".--[[User:Cpg|Cpg]] 13:01, 8 May 2008 (MDT)
* Delete Math 343 as a prerequisite. (It's redundant.)--[[User:Cpg|Cpg]] 13:01, 8 May 2008 (MDT)

Talk:Math 540: Linear Analysis

2008-08-15T22:21:15Z

Rcb3: Talk:Math 540 moved to Talk:Math 541 over redirect: Previous move was an error

#REDIRECT [[Talk:Math 541]]

Talk:Math 541: Real Analysis

2008-08-15T22:18:58Z

Rcb3: Talk:Math 541 moved to Talk:Math 540: New curriculum content; requires a new number

Math 541: Real Analysis

2008-08-15T22:18:58Z

Rcb3: Math 541 moved to Math 540: New curriculum content; requires a new number