https://math.byu.edu/wiki/api.php?action=feedcontributions&user=Ls5&feedformat=atomMathWiki - User contributions [en]2020-06-02T21:36:28ZUser contributionsMediaWiki 1.26.3https://math.byu.edu/wiki/index.php?title=Math_215:_Computational_Linear_Algebra&diff=3744Math 215: Computational Linear Algebra2020-02-06T15:21:39Z<p>Ls5: /* Prerequisite */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Computational Linear Algebra<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(1:0:1)<br />
<br />
=== Offered ===<br />
F, W<br />
<br />
=== Prerequisite ===<br />
[[Math 112]]. Concurrent or previous enrollment in [[Math 213]] or [[Math 302]] (recommended).<br />
<br />
=== Description ===<br />
Practical linear algebraic computations and applications.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
Upon completion of this course, the successful student will be able to:<br />
<br />
# Solving large-scale linear algebraic problems.<br />
# Applying matrix and vectors to analyze scientific and technological systems.<br />
# Implementing linear algebraic techniques in suitable computing environments.<br />
<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
<br />
<br />
[[Category:Courses|215]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_215:_Computational_Linear_Algebra&diff=3743Math 215: Computational Linear Algebra2020-02-06T15:21:25Z<p>Ls5: /* Prerequisite */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Computational Linear Algebra<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(1:0:1)<br />
<br />
=== Offered ===<br />
F, W<br />
<br />
=== Prerequisite ===<br />
[[Math 112]]. Concurrent or previous enrollment in [[Math 213]], or [[Math 302]] (recommended).<br />
<br />
=== Description ===<br />
Practical linear algebraic computations and applications.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
Upon completion of this course, the successful student will be able to:<br />
<br />
# Solving large-scale linear algebraic problems.<br />
# Applying matrix and vectors to analyze scientific and technological systems.<br />
# Implementing linear algebraic techniques in suitable computing environments.<br />
<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
<br />
<br />
[[Category:Courses|215]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_525:_Network_Theory&diff=3742Math 525: Network Theory2020-01-02T21:52:42Z<p>Ls5: Created page with "== Catalog Information == === Title === Network Theory === (Credit Hours:Lecture Hours:Lab Hours) === (3:3:0) === Offered === Winter even years === Prerequisite === Math..."</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Network Theory<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
Winter even years<br />
<br />
=== Prerequisite ===<br />
[[Math 521]] is recommended<br />
<br />
=== Description ===<br />
Representing networks mathematically. Measures and metrics, computer algorithms, random graphs, and large-scale structures. Percolation and network resilience. Dynamical systems on networks.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
<br />
=== Prerequisites ===<br />
[[Math 521]] is recommended<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|525]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_673:_Algebra_3&diff=3741Math 673: Algebra 32019-11-14T17:48:43Z<p>Ls5: /* Catalog Information */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Algebra 3.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Offered ===<br />
F<br />
<br />
=== Prerequisite ===<br />
[[Math 572]] or equivalent.<br />
<br />
=== Description ===<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
[[Math 572]] or equivalent.<br />
<br />
=== Minimal learning outcomes ===<br />
Students should achieve an advanced mastery of the topics listed below. <br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
#Finite group theory: Sylow theory, Simple groups, solvable groups<br />
#Representations of finite groups:<br />
#Representations of associative algebras<br />
#Semisimple rings<br />
#Grobner bases<br />
<br />
<br />
</div><br />
=== Textbooks ===<br />
<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
<br />
<br />
=== Additional topics ===<br />
<br />
<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|673]]<br />
This course is a prerequisite for [[Math 674]].</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_656:_Algebraic_Topology&diff=3740Math 656: Algebraic Topology2019-11-14T17:42:33Z<p>Ls5: /* Catalog Information */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Algebraic Topology.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Offered ===<br />
W (odd years)<br />
<br />
=== Prerequisite ===<br />
[[Math 371]] (or equivalent) and [[Math 553]].<br />
<br />
=== Description ===<br />
A rigorous treatment of the fundamentals of algebraic topology, including homotopy (fundamental group and higher homotopy groups) and homology and cohomology of spaces.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
<br />
Topology of manifolds, tensors, and orientation is assumed from [[Math 655]]. A basic knowledge of fundamental groups and covering spaces, as in [[Math 554]], will also be required.<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
Outlined below are topics that all successful Math 656 students should understand well. Students should be able to demonstrate mastery of relevant vocabulary, and use the vocabulary fluently in their work. They should know common examples and counterexamples, and be able prove that these examples and counterexamples have properties as claimed. Additionally, students should know the content (and limitations) of major theorems and the ideas of the proofs, and apply results of these theorems to solve suitable problems, or use techniques of the proofs to prove additional related results, or to make calculations and computations.<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
# Fundamental group and homotopy<br />
#* Constructions<br />
#* Van Kampen Theorem<br />
#* Covering spaces and group actions<br />
#* Higher homotopy groups<br />
# Homology<br />
#* Simplicial, singular, cellular<br />
#* Exact sequences and excision<br />
#* Mayer-Vietoris sequences<br />
#* Homology with coefficients<br />
#* Homology and the fundamental group<br />
# Cohomology<br />
#* Universal coefficient theorem<br />
#* Cup product<br />
#* Poincare duality<br />
<br />
<br />
</div><br />
<br />
=== Additional topics ===<br />
<br />
At the discretion of the instructor as time allows. Possible topics include de Rham cohomology and the de Rham theorem, additional duality theorems, including Alexander and Lefschetz duality, additional homotopy theory, including Whitehead's theorem, the Hurewicz theorem, fiber bundles, fibrations, obstruction theory.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Math 751R]]<br />
<br />
[[Category:Courses|656]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_655:_Differential_Topology&diff=3739Math 655: Differential Topology2019-11-14T17:42:16Z<p>Ls5: /* Catalog Information */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Differential Topology<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Offered ===<br />
F (even years)<br />
<br />
=== Prerequisite ===<br />
[[Math 342|342]] or equivalent.<br />
<br />
=== Description ===<br />
An introduction to manifolds and smooth manifolds and their topology.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
<br />
Knowledge of basic point set topology from [[Math 553]], [[Math 554|554]] will be assumed. This includes topological spaces, basis and countability, metric spaces, quotient spaces, fundamental group, and covering maps. Basic knowledge of linear algebra ([[Math 313]]) and introductory analysis ([[Math 341]] and [[Math 342|342]]) will also be assumed.<br />
<br />
=== Minimal learning outcomes ===<br />
Outlined below are topics that all successful Math 655 students should understand well. Students should be able to demonstrate mastery of relevant vocabulary, and use the vocabulary fluently in their work. They should know common examples and counterexamples, and be able prove that these examples and counterexamples have properties as claimed. Additionally, students should know the content (and limitations) of major theorems and the ideas of the proofs, and apply results of these theorems to solve suitable problems, or use techniques of the proofs to prove additional related results, or to make calculations and computations.<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
# Manifolds<br />
#* Topological and smooth manifolds<br />
#* Manifolds with boundary<br />
#* Tangent vectors<br />
#* Tangent bundles<br />
#* Vector bundles and bundle maps<br />
#* Cotangent bundles<br />
# Submanifolds<br />
#* Submersions, immersions, embeddings<br />
#* Inverse and implicit function theorems<br />
#* Transversality<br />
#* Embedding and approximation theorems<br />
# Differential forms and tensors<br />
#* Wedge product<br />
#* Exterior derivative<br />
#* Orientations<br />
#* Stoke's Theorem<br />
<br />
</div><br />
<br />
=== Additional topics ===<br />
<br />
At the discretion of the instructor as time allows. Topics might include Lie groups and homogeneous spaces, Morse functions, de Rham cohomology and the de Rham theorem, Jordan curve theorem, Lefschetz fixed-point theory, degree, Gauss-Bonnet theorem, etc.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
[[Math 656]]<br />
<br />
[[Category:Courses|655]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_641:_Functions_of_a_Real_Variable&diff=3738Math 641: Functions of a Real Variable2019-11-14T17:36:16Z<p>Ls5: /* Catalog Information */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Functions of a Real Variable.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Offered ===<br />
W (odd years)<br />
<br />
=== Prerequisite ===<br />
[[Math 541]] or instructor's consent<br />
<br />
=== Description ===<br />
Abstract measure and integration theory; L(p)(?) spaces; measures on topological and Euclidean spaces.<br />
<br />
== Desired Learning Outcomes ==<br />
Math 641 is a course in abstract measure and integration theory. There will be some repetition of topics between [[Math 541]] and Math 641, but it is felt that the repetition will help solidify student understanding, and there will be a difference in approach, with the lower-level course taking a concrete approach restricted to Lebesgue measure.<br />
<br />
=== Prerequisites ===<br />
We strongly recommend that students take [[Math 541]] prior to Math 641.<br />
<br />
=== Minimal learning outcomes ===<br />
Outlined below are topics that all successful Math 641 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.<br />
<div style="-moz-column-count:2; column-count:2;"><br />
# Abstract measure theory<br />
#* σ-algebras<br />
#* Measures<br />
#** Positive measures<br />
#** Signed measures<br />
#** σ-finite measures<br />
#** Complete measures<br />
#* Measurable spaces<br />
#* Measure spaces<br />
# Abstract integration theory<br />
#* Abstract measurable mappings<br />
#* Measurable real- and extended-real-valued functions<br />
#* Integrating simple functions<br />
#* Integrating nonnegative functions<br />
#* Integrating L<sup>1</sup> functions<br />
#* Integration on a measurable set<br />
#* Measures defined through integration<br />
#* Absolute continuity of integration<br />
#* Linearity of integration<br />
#* Monotone Convergence Theorem<br />
#* Fatou's Lemma<br />
#* Dominated Convergence Theorem<br />
#* Effect of sets of measure zero<br />
# Operations on measures<br />
#* Absolutely continuous measures<br />
#* Mutually singular measures<br />
#* Lebesgue Decomposition Theorem<br />
#* Radon-Nikodym Theorem<br />
#* Hahn Decomposition Theorem<br />
#* Jordan Decomposition Theorem<br />
# L<sup>p</sup> spaces<br />
#* H&#246;lder's Inequality<br />
#* Minkowski's Inequality<br />
#* Completeness of L<sup>p</sup><br />
#* Density of C<sub>c</sub> in L<sup>p</sup><br />
#* Inclusion of L<sup>p</sup> spaces<br />
#* Duality of L<sup>p</sup> spaces<br />
# Convergence results<br />
#* Types of convergence<br />
#** Convergence in L<sup>p</sup>-norm<br />
#** Almost-everywhere convergence<br />
#** Almost-uniform convergence<br />
#** Convergence in measure<br />
#* Relationships between different types of convergence<br />
#** Egoroff's Theorem<br />
# Measures on abstract product spaces<br />
#* Existence of product measure<br />
#* Tonelli's Theorem<br />
#* Fubini's Theorem<br />
# Measures on topological spaces<br />
#* Borel σ-algebras<br />
#* Locally compact Hausdorff spaces<br />
#* Urysohn's Lemma<br />
#* Partitions of unity<br />
#* Borel measures<br />
#* Locally finite measures<br />
#* Regular measures<br />
#* Radon measures<br />
#* Riesz Representation Theorem (for positive linear functionals on C<sub>c</sub>)<br />
#* Lusin's Theorem<br />
# Lebesgue measure on <b>R</b><sup>n</sup><br />
#* Existence<br />
#* Composition with affine maps<br />
#* Change of variable formula for integration<br />
#* Differentiation and integration on <b>R</b><br />
#** Derivative of integral is the integrand a.e.<br />
#** Functions of bounded variation<br />
#** Absolutely continuous functions<br />
#** Integrating derivatives of absolutely continuous functions<br />
<br><br><br><br><br />
<br />
</div><br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
Math 641 is not a prerequisite for any course.<br />
<br />
[[Category:Courses|641]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_640:_Nonlinear_Analysis&diff=3737Math 640: Nonlinear Analysis2019-11-14T17:35:47Z<p>Ls5: /* Catalog Information */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Nonlinear Analysis.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
W<br />
<br />
=== Recommended(?) ===<br />
[[Math 540]].<br />
<br />
=== Description ===<br />
Differential calculus in normed spaces, fixed point theory, and abstract critical point theory.<br />
<br />
== Desired Learning Outcomes ==<br />
This course is intended as a natural nonlinear sequel to [[Math 540]]. Like its prequel, the focus would be on operators on abstract Banach spaces.<br />
<br />
=== Prerequisites ===<br />
Students need to have a good understanding of basic linear analysis, whether this comes from taking the [[Math 540]] or some other way.<br />
<br />
=== Minimal learning outcomes ===<br />
Students should obtain a thorough understanding of the topics listed below. In particular they should be able to define and use relevant terminology, compare and contrast closely-related concepts, and state (and, where feasible, prove) major theorems.<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
# Differential calculus on normed spaces<br />
#* Fréchet derivatives<br />
#* Gâteaux derivatives<br />
#* Inverse Function theorem<br />
#* Implicit Function theorem<br />
#* Lyapunov-Schmidt reduction<br />
# Fixed point theory<br />
#* Metric spaces<br />
#** Banach’s contraction mapping principle<br />
#** Parametrized contraction mapping principle<br />
#* Finite-dimensional spaces<br />
#** Brouwer fixed point theorem<br />
#* Normed spaces<br />
#** Schauder fixed point theorem<br />
#** Leray-Schauder alternative<br />
#* Ordered Banach spaces<br />
#** Monotone iterative method<br />
#* Monotone operators<br />
#** Browder-Minty theorem<br />
# Abstract critical point theory<br />
#* Functional properties<br />
#** Convexity<br />
#** Coercivity<br />
#** Lower semi-continuity<br />
#* Existence of global minimizers<br />
#* Existence of constrained minimizers<br />
#* Minimax results<br />
#** Ambrosetti-Rabinowitz mountain pass theorem<br />
</div><br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
In addition to the minimal learning outcomes above, instructors should give serious consideration to covering the following specific topics:<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
# Differential calculus on normed spaces<br />
#* Nash-Moser theorem<br />
# Fixed point theory<br />
#* Metric spaces<br />
#** Caristi fixed point theorem<br />
#* Hilbert spaces<br />
#** Browder-Göhde-Kirk theorem<br />
#* Ordered Banach spaces<br />
#** Krasnoselski’s fixed point theorem <br />
#** Krein-Rutman theorem<br />
#* Monotone operators<br />
#** Hartman-Stampacchia theorem<br />
# Abstract critical point theory<br />
#* Minimax results<br />
#** Ky Fan’s minimax inequality<br />
#** Ekeland’s variational principle<br />
#** Schechter’s bounded mountain pass theorem<br />
#** Rabinowitz saddle point theorem<br />
#** Rabinowitz linking theorem<br />
<br />
</div><br />
<br />
Furthermore, it is anticipated that instructors will want to motivate the abstract theory by considering appropriate concrete examples.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
It is proposed that this course be a prerequisite for [[Math 647]].<br />
<br />
[[Category:Courses|640]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_637:_Advanced_Probability_2&diff=3736Math 637: Advanced Probability 22019-11-14T17:35:11Z<p>Ls5: /* Catalog Information */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Advanced Probability 2.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Offered ===<br />
W (contact department)<br />
<br />
=== Prerequisite ===<br />
[[Math 636]].<br />
<br />
=== Recommended ===<br />
[[Math 341]], [[Math 342|342]], Stat 441(?); or equivalents.<br />
<br />
=== Description ===<br />
Advanced concepts in modern probability. Convergence theorems and laws of large numbers. Stationary processes and ergodic theorems. Martingales. Diffusion processes and stochastic integration.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
This course has [[Math 636]] as a prerequisite, so it can build on the work done in that class.<br />
<br />
=== Minimal learning outcomes ===<br />
Outlined below are topics that all successful Math 637 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.<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
#The Daniell-Kolmogorov Theorem<br />
#Stochastic processes and filtrations<br />
#Continuous-time Martingales<br />
#Brownian motion<br />
#Gaussian processes<br />
#Levy processes<br />
#Regular conditional probabilities<br />
#Markov processes<br />
#Stochastic integration<br />
# Ito’s formula<br><br><br><br><br><br><br />
</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Achim Klenke, ''Probability Theory: A Comprehensive Course'', Springer, 2008.<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|637]]<br />
None</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_636_Advanced_Probability_1&diff=3735Math 636 Advanced Probability 12019-11-14T17:34:34Z<p>Ls5: /* Catalog Information */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Advanced Probability 1.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Offered ===<br />
F (contact department)<br />
<br />
=== Prerequisite ===<br />
[[Math 314]] and [[Math 341]]; and [[Math 431]] or Stat 370; or equivalents.<br />
<br />
=== Description ===<br />
Foundations of the modern theory of probability with applications. Probability spaces, random variables, independence, conditioning, expectation, generating functions, and Markov chains.<br />
<br />
== Desired Learning Outcomes ==<br />
This should be an ''advanced'' course in probability and, therefore, clearly distinguishable from an introductory course like [[Math 431]]. Furthermore, it is supposed to be a course in the ''modern'' theory of probability, which suggests that it should be based on Kolmogorov's measure-theoretic approach or something equivalent.<br />
<br />
=== Prerequisites ===<br />
The official prerequisite is multivariable calculus. Other prior courses that will contribute to student success include:<br />
* an introductory course in probability;<br />
* a course in rigorous mathematical reasoning;<br />
* an introductory course in analysis.<br />
<br />
=== Minimal learning outcomes ===<br />
Outlined below are topics that all successful Math 543 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.<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
# Probability spaces<br />
#* Sigma-algebras and Borel sets<br />
#* Kolmogorov axioms<br />
#* Carathéodory's Extension Theorem<br />
#* Lebesgue-Stieltjes measure<br />
# Random variables<br />
#* Measurable maps<br />
#* Distributions and distribution functions<br />
# Independence<br />
#* Of events and classes of events<br />
#* Of random variables<br />
#* Borel-Cantelli Lemmas<br />
# Expectation<br />
#* Of arbitrary nonnegative random variables<br />
#* Of integrable real-valued random variables<br />
#* Of compositions<br />
#* Monotone Convergence Theorem<br />
#* Uniform integrability and dominated convergence<br />
# Conditioning<br />
#* Probability conditioned on a non-null set<br />
#* Expectation conditioned on a sigma-algebra<br />
#* Expectation conditioned on a random variable<br />
# Probability measures on product spaces<br />
# Strong Law of Large Numbers<br />
# Central Limit Theorem<br />
# Convergence of random variables<br />
#* Almost sure<br />
#* In probability<br />
#* L^p<br />
#* weak<br />
# Discrete-time Martingales<br><br><br><br><br><br><br><br><br />
</div><br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Achim Klenke, ''Probability Theory: A Comprehensive Course'', Springer, 2008.<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
[[Math 637]]<br />
<br />
[[Category:Courses|636]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_586:_Introduction_to_Algebraic_Number_Theory.&diff=3734Math 586: Introduction to Algebraic Number Theory.2019-11-14T17:32:58Z<p>Ls5: /* Catalog Information */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Introduction to Algebraic Number Theory.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Offered ===<br />
On demand (contact department)<br />
<br />
=== Prerequisite ===<br />
[[Math 372]] or equivalent; instructor's consent.<br />
<br />
=== Description ===<br />
Algebraic integers; different and discriminant; decomposition of primes; class group; Dirichlet unit theorem; Dedekind zeta function; cyclotomic fields; valuations; completions.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
[[Math 372]] is a prerequisite for this course. In particular, students should be familiar with the concepts of groups and rings, and they should understand constructions of quotient groups and quotient rings. By this point in their mathematical career, students should be skilled at proving theorems by themselves.<br />
<br />
=== Minimal learning outcomes ===<br />
Students should achieve an advanced mastery of the topics listed below. This means that they should know all relevant definitions, correct statements and proofs 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 difficult problems related to these concepts, and by proving theorems about the below concepts, even if the theorems go beyond the material in the text.<br />
<div style="-moz-column-count:2; column-count:2;"><br />
# Number Fields<br />
#*Algebraic Numbers<br />
#*Algebraic Integers<br />
#*Cyclotomic Fields<br />
#*Trace and Norm<br />
#*Discriminants<br />
#*Integral Bases<br />
#*Computing Integral Bases<br />
#Prime decomposition in rings of integers<br />
#*Ideal theory of Dedekind domains<br />
#*Splitting, ramification, inertia of primes<br />
#*Computing prime decompositions<br />
#*Decomposition and inertia groups<br />
#*Frobenius maps<br />
#*Functorial properties of the Frobenius<br />
#Ideal Class Group<br />
#*Finiteness of the class group<br />
#*Minkowski bounds<br />
#*Distribution of ideals in ideal classes<br />
#*Class group computations in quadratic fields<br />
#Dirichlet's unit theorem<br />
#*Computation of fundamental units in quadratic fields<br />
#Cebotarev Density Theorem (Statement)<br />
#Dedekind zeta function<br />
#* Class number formula<br />
<br />
</div><br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Daniel Marcus, Number Fields<br />
* Serge Lang, Algebraic Number Fields<br />
* Pierre Samuel, Algebraic Theory of Numbers<br />
* Gerald Janusz, Algebraic Number Fields<br />
<br />
=== Additional topics ===<br />
As time permits, additional topics that might be considered include Galois representations, class field theory, module theory over Dedekind domains, etc.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|586]]<br />
This course is not a prerequisite for any other courses.</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_565:_Differential_Geometry&diff=3733Math 565: Differential Geometry2019-11-14T17:08:12Z<p>Ls5: /* Catalog Information */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Differential Geometry<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Offered ===<br />
On demand (contact department)<br />
<br />
=== Prerequisite ===<br />
[[Math 342]] or equivlaent.<br />
<br />
=== Description ===<br />
<br />
A rigorous treatment of the theory of differential geometry.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
This course is aimed at graduate students in Mathematics as well as graduate students in Physics and Engineering. This course contributes to all the expected learning outcomes of the Mathematics M.S. and Ph.D. The topics include differential topology, Riemannian metrics, geodesics, curvature, and integration on manifolds. <br />
<br />
=== Prerequisites ===<br />
<br />
Students should have taken [[Math 342]] prior to taking this course. Math 342 provides a rigorous background of analysis that is needed to understand many of the proofs in this course. <br />
<br />
=== Minimal learning outcomes ===<br />
<br />
Students should achieve mastery of the topics<br />
below. This means that they should know all relevant definitions,<br />
full statements of the major theorems, and examples of the various<br />
concepts. Further, students should be able to solve non-trivial problems<br />
related to these concepts, and prove theorems in analogy to proofs<br />
given by the instructor.<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
# Differential topology<br />
#* Differentiable manifolds and smooth maps<br />
#* Tangent space, tangent bundle, derivative of a smooth map<br />
#* Immersions, submersions, and embeddings<br />
#* Orientation<br />
#* Vector fields, brackets<br />
# Riemannian metrics<br />
#* Definition of Riemannian metrics<br />
#* Affine connections<br />
#* Riemannian connections<br />
# Geodesics<br />
#* Definition of geodesics<br />
#* Geodesic flow<br />
#* Minimizing properties of geodesics<br />
#* Exponential map<br />
#* Convex neighborhoods<br />
# Curvature<br />
#* Definitions of curvature, curvature tensor<br />
#* Second fundamental form<br />
#* Sectional and Ricci curvature<br />
#* Jacobi fields<br />
# Integration on manifolds<br />
#* Tensor and vector bundles<br />
#* Exterior algebra<br />
#* Differential forms and exterior derivative<br />
#* Stokes Theorem<br><br><br><br><br><br><br><br />
</div><br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
<br />
These are at the instructor's discretion as time allows examples are: Hopf-Rinow theorem, spaces of constant curvature, De Rham cohomology, fixed points and intersection numbers, and Morse theory.<br />
<br />
=== Recommended Texts ===<br />
<br />
Riemannian geometry, by Manfredo P. Do Carmo; Differential Geometry and Topology, with a view to dynamical systems, by Keith Burns and Marian Gidea<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
None.<br />
<br />
[[Category:Courses|565]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_562:_Intro_to_Algebraic_Geometry_2&diff=3732Math 562: Intro to Algebraic Geometry 22019-11-14T17:07:54Z<p>Ls5: /* Catalog Information */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Introduction to Algebraic Geometry 2.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Offered ===<br />
On demand (contact department)<br />
<br />
=== Prerequisite ===<br />
[[Math 561]] or concurrent enrollment.<br />
<br />
=== Description ===<br />
Local properties of quasi-projective varieties. Divisors and differential forms.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
[[Math 561]]<br />
=== Minimal learning outcomes ===<br />
Students should achieve mastery of the topics listed below. This means 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 concepts below, related to, but not identical to, statements proven by the text or instructor.<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
#Local properties of algebraic varieties<br />
#* The local ring at a point<br />
#* Zariski tangent space<br />
#* Singular points<br />
#* The tangent space<br />
#Power series expansions<br />
#* Local parameters<br />
#* The completion of a local ring<br />
#Properties of nonsingular points<br><br />
#Birational maps<br />
#* Blowup in projective space<br />
#* Local blowup<br />
#* Behavior of a subvariety under a blowup<br />
#Normal varieties and normalization<br />
#Divisors<br />
#* Cartier divisors<br />
#* Weil divisors<br />
#Differential forms<br />
</div><br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
I. R. Shararevich, Basic Algebraic Geometry I, Varieties in Projective Space<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
As this is a terminal course, it may be possible to substitute other topics for the above, especially items 6 and 7. Some instructors may wish to give an overview on the moduli of curves and its relation to mathematical physics.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
None<br />
[[Category:Courses|562]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_561:_Intro_to_Algebraic_Geometry_1&diff=3731Math 561: Intro to Algebraic Geometry 12019-11-14T17:07:30Z<p>Ls5: /* Catalog Information */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Introduction to Algebraic Geometry 1.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Offered ===<br />
On demand (contact department)<br />
<br />
=== Prerequisite ===<br />
[[Math 571]] or concurrent enrollment.<br />
<br />
=== Description ===<br />
Basic definitions and theorems on affine, projective, and quasi-projective varieties.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
[[Math 571]] or concurrent enrollment.<br />
<br />
=== Minimal learning outcomes ===<br />
Students should achieve mastery of the topics listed below. This means 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 concepts below, related to, but not identical to, statements proven by the text or instructor.<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
#Algebraic plane curves<br />
#* Rational curves<br />
#* Relation with field theory<br />
#* Rational maps<br />
#* Singular and nonsingular points<br />
#* Projective spaces<br />
#Affine varieties<br />
#* Affine space and the Zariski topology<br />
#* Regular functions<br />
#* Regular maps<br />
#Rational functions and rational maps<br />
#Quasiprojective varieties<br />
#* The Zariski topology on projective space<br />
#* Regular and rational functions<br />
#* Examples<br />
#Products and maps of quasi-projective space<br />
#* Definition of products<br />
#* Properness of projective maps<br />
#* Finite maps<br />
#* Normalization<br />
#Dimension<br />
#* Definition of dimension<br />
#* Dimension of intersection with a hypersurface<br />
#* Dimension of fibres<br />
#* Application to lines on surfaces (optional)<br><br><br><br><br><br><br />
</div><br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*I. R. Shafarevich, Basic Algebraic Geometry 1, Varieties in Projective Space<br />
<br />
=== Additional topics ===<br />
If time permits, additional topics may be covered. Possibilities include the 27 lines on a cubic surface, or an introduction to elliptic curves.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
[[Math 562]]<br />
[[Category:Courses|561]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_554:_Foundations_of_Topology_2&diff=3730Math 554: Foundations of Topology 22019-11-14T17:07:06Z<p>Ls5: /* Catalog Information */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Foundations of Topology 2.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
W<br />
<br />
=== Prerequisite ===<br />
[[Math 553]] or instructor's consent.<br />
<br />
=== Description ===<br />
Fundamental group, retractions and fixed points, homotopy types, separation theorems, classification of surfaces, Seifert-van Kampen Theorem,<br />
classification of covering spaces, and applications to group theory.<br />
<br />
== Desired Learning Outcomes ==<br />
Students should gain a familiarity with surfaces, fundamental group, and covering spaces.<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
# The Fundamenatal Group<br />
# The topology of the plane<br />
#* Jordan Curve Theorem<br />
# Seifert-van Kampen Theorem<br />
# Classification of Surfaces<br />
# Classification of Covering Spaces<br />
# Group Theory<br />
#* Free groups<br />
#* Free abelian groups<br />
#* Presentations of groups<br />
#* Subgroups of free groups<br />
<br />
</div><br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|554]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_547:_Partial_Differential_Equations_1&diff=3729Math 547: Partial Differential Equations 12019-11-14T17:06:12Z<p>Ls5: /* Catalog Information */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Partial Differential Equations 1. [Recommended change: Applied Partial Differential Equations]<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Offered ===<br />
On demand (contact department)<br />
<br />
=== Prerequisite ===<br />
[[Math 334]], [[Math 342|342]]; or equivalents.<br />
<br />
=== Recommended ===<br />
[[Math 352]] or equivalent.<br />
<br />
=== Description ===<br />
Methods of analysis for hyperbolic, elliptic, and parabolic equations, including characteristic manifolds, distributions, Green's functions, maximum principles and Fourier analysis.<br />
<br />
== Desired Learning Outcomes ==<br />
It is proposed that the focus of this course be '''applied''' partial differential equations. Thus, the students ought to be less concerned with knowing existence/uniqueness proofs than with understanding the properties and representation of solutions and with using PDEs to model important phenomena.<br />
<br />
=== Prerequisites ===<br />
Since ODEs appear in certain approaches to PDEs (in particular, the method of characteristics), [[Math 334]] is a prerequisite. [[Math 342]] is a prerequisite to ensure that students are reasonably comfortable with analysis in several dimensions.<br />
<br />
=== Minimal learning outcomes ===<br />
Outlined below are topics that all successful Math 547 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, and ability to make direct application of those results to related problems, including calculations. <br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
# General Cauchy problem<br />
#* Cauchy-Kowalevski Theorem<br />
#* Lewy Example<br />
# Method of characteristics for first-order equations<br />
#* Semilinear case<br />
#* Quasilinear case<br />
#* General case<br />
# Quasilinear systems of conservation laws on a line<br />
#* Riemann problem<br />
#* Rankine-Hugoniot jump condition<br />
#* Entropy condition<br />
#* Shocks<br />
#* Rarefaction waves<br />
# Classification of general second-order equations<br />
# Canonical forms for semilinear second-order equations<br><br><br><br><br />
# Hyperbolic equations<br />
#* The wave equation<br />
#* Cauchy problem<br />
#* Problems with boundary data<br />
#* Huygens' principle<br />
#* Applications<br />
# Elliptic equations<br />
#* Laplace's equation<br />
#* Poisson's equation<br />
#* Green's functions<br />
#* Maximum principles<br />
#* Applications<br />
# Parabolic equations<br />
#* The heat equation<br />
#* Green's functions<br />
#* The heat kernel<br />
#* Maximum principles<br />
#* Applications<br />
</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* John Ockendon, Sam Howison, Andrew Lacey, and Alexander Movchan, ''Applied Partial Differential Equations (Revised Edition)'', Oxford University Press, 1999.<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
Currently this course is listed as a prerequisite for [[Math 647]]. It is recommended that this connection be dropped, because the intended clientele of the two courses are quite different.<br />
<br />
[[Category:Courses|547]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_541:_Real_Analysis&diff=3728Math 541: Real Analysis2019-11-14T17:05:41Z<p>Ls5: /* Catalog Information */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Real Analysis.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Offered ===<br />
F<br />
<br />
=== Prerequisite ===<br />
[[Math 341]]; [[Math 314|314]] or [[Math 342|342]]; or equivalents.<br />
<br />
=== Recommended ===<br />
[[Math 352]] or equivalent.<br />
<br />
=== Description ===<br />
Rigorous treatment of differentiation and integration theory; Lebesque<br />
measure; Banach spaces.<br />
<br />
== Desired Learning Outcomes ==<br />
Math 541 is a one-semester course specifically on Lebesgue integration in Euclidean space and Fourier analysis.<br />
<br />
=== Prerequisites ===<br />
Currently, Math 541 requires a semester of single-variable real analysis and a semester of multi-variable Calculus. Replacing these prerequisites by [[Math 342]] would imply that the new version of Math 541 could presuppose that students had been exposed to the geometry of <b>R</b><sup>n</sup> and to metric spaces, which would make it easier to cover the core topics listed below.<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
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.<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
# Lebesgue measure on <b>R</b><sup>n</sup><br />
#* Inner and outer measures<br />
#* Construction of Lebesgue measure<br />
#* Properties of Lebesgue measure<br />
#** Effect of basic set operations<br />
#** Limiting properties<br />
#** Its domain<br />
#** Approximation properties<br />
#** Sets of outer measure zero<br />
#** Invariance w.r.t. isometries<br />
#** Effect of dilations<br />
#* Existence of nonmeasurable sets<br />
# Lebesgue integration on <b>R</b><sup>n</sup><br />
#* Measurable functions<br />
#* Simple functions<br />
#* Approximation of measurable functions with simple functions<br />
#* The extended reals<br />
#* Integrating nonnegative functions<br />
#* Integrating absolutely-integrable functions<br />
#* Integrating on measurable sets<br />
#* Basic properties of the Lebesgue integral<br />
#** Linearity<br />
#** Monotonicity<br />
#** Effects of sets of measure zero<br />
#** Absolute continuty of integration<br />
#** Fatou's Lemma<br />
#** Monotone Convergence Theorem<br />
#** Dominated Convergence Theorem<br />
#** Differentiation w.r.t. a parameter<br />
#** Linear changes of variable<br />
#** Compatibility with Riemann integration<br />
# Fubini's Theorem for <b>R</b><sup>n</sup><br />
# L<sup>1</sup>, L<sup>2</sup>, and L<sup>&#8734;</sup><br />
#* Completeness<br />
#* Approximation by smooth functions<br />
#* Continuity of translation<br />
# Fourier transform on <b>R</b><sup>n</sup><br />
#* Convolutions<br />
#* Basic properties of Fourier transforms<br />
#** Composition with translation, dilation, inversion, differentiation, convolution, etc.<br />
#** Regularity of transformed functions<br />
#** Riemann-Lebesgue Lemma<br />
#* Inversion Theorem for L<sup>1</sup><br />
#* Schwartz class<br />
#* Fourier-Plancherel Transform on L<sup>2</sup><br />
#** Its inversion<br />
#** Isomorphism<br />
#* Fourier series<br />
#** Dirichlet and Fej&#233;r kernels<br />
#** L<sup>2</sup> convergence<br />
#** Pointwise convergence<br />
#** Convergence of Ces&#224;ro means<br />
</div><br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
<br />
Extra time could be used to go into Fourier analysis in more depth.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
Math 541 is a prerequisite for [[Math 641]], [[Math 644|644]], and [[Math 647|647]].<br />
<br />
[[Category:Courses|541]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_540:_Linear_Analysis&diff=3727Math 540: Linear Analysis2019-11-14T17:05:14Z<p>Ls5: /* Catalog Information */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Linear Analysis.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
W<br />
<br />
=== Recommended ===<br />
[[Math 342]] or equivalent. [[Math 352]] or equivalent.<br />
<br />
=== Description ===<br />
Normed vector spaces and linear maps between them.<br />
<br />
== Desired Learning Outcomes ==<br />
The course is designed to cover elementary abstract linear functional analysis. "Elementary" means that methods dependent on complex analysis or measure-theoretic integration are not core topics. "Abstract" means that applications to specific function spaces are not core topics.<br />
<br />
=== Prerequisites ===<br />
The official prerequisite is [[Math 342]]. What's important is that incoming students be familiar with linear algebra and metric spaces and be mathematically mature.<br />
<br />
=== Minimal learning outcomes ===<br />
Students should obtain a thorough understanding of the topics listed below. In particular they should be able to define and use relevant terminology, compare and contrast closely-related concepts, and state (and, where feasible, prove) major theorems.<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
# Normed spaces<br />
#* Basics<br />
#** Banach spaces<br />
#** Special linear operators<br />
#*** Continuous/bounded<br />
#*** Compact<br />
#*** Finite rank<br />
#* Duality<br />
#** Dual spaces<br />
#*** Their completeness<br />
#** Adjoints of bounded linear operators<br />
#** Second duals<br />
#*** Reflexivity<br />
#** Weak and weak-star topologies<br />
#*** Banach-Alaoglu theorem<br />
#* Structure<br />
#** Hamel and Schauder bases<br />
#** Biorthogonal systems<br />
#** Separability<br />
#** Direct sums<br />
#** Quotient spaces<br />
#* Finite-dimensional spaces<br />
#** Equivalence of all norms<br />
#** Completeness<br />
#** Continuity of all linear operators<br />
#** Characterization: unit ball is compact<br />
#* Fundamental theorems<br />
#** Baire category theorem<br />
#** Hahn-Banach extension theorem<br />
#** Banach-Steinhaus theorem<br />
#** Open mapping theorem<br />
#** Closed graph theorem<br />
#** Bounded inverse theorem<br />
# Inner product spaces<br />
#* Basics<br />
#** Hilbert spaces<br />
#** Special linear operators<br />
#*** Self-adjoint<br />
#*** Unitary<br />
#*** Normal<br />
#*** Orthogonal projections<br />
#*** Hilbert-Schmidt operators<br />
#* Structure<br />
#** Orthogonality<br />
#*** Complements and direct sums<br />
#*** Bases<br />
#** Representation theorems<br />
#*** Riesz-Frechet theorem<br />
#*** Lax-Milgram theorem<br />
#** Abstract Fourier theory<br />
#*** Riesz-Fischer theorem<br />
#*** Bessel’s inequality<br />
#*** Parseval’s identities<br />
# Spectral theory<br />
#* Banach algebras<br />
#* Bounded operators on Banach spaces<br />
#** Gelfand’s spectral-radius formula<br />
#* Compact operators on Banach spaces<br />
#** Riesz-Schauder theory including Fredholm Alternative<br />
#* Compact normal operators on Hilbert spaces<br />
#* Compact self-adjoint operators on Hilbert spaces<br><br><br><br><br><br><br />
<br />
</div><br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* David Promislow, <i>A First Course in Functional Analysis</i>, Wiley, 2008.<br />
<br />
=== Additional topics ===<br />
While the focus of the course is on abstract theory, this theory should probably be motivated and illustrated with appropriate concrete examples.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
This course is recommended for [[Math 640]]. Indirectly (through the [[Math 640]]), this course will possibly become be a prerequisite for [[Math 647]].<br />
<br />
[[Category:Courses|540]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_534:_Intro_to_Dynamical_Systems_1&diff=3726Math 534: Intro to Dynamical Systems 12019-11-14T17:04:37Z<p>Ls5: /* Offered */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Introduction to Dynamical Systems 1(?).<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Offered ===<br />
F W On demand (contact department)<br />
<br />
=== Prerequisite ===<br />
[[Math 341]] or equivalent(?).<br />
<br />
=== Recommended ===<br />
[[Math 352]] or equivalent.<br />
<br />
=== Description ===<br />
Discrete dynamical systems; iterations of maps on the line and the plane; bifurcation theory; chaos, Julia sets, and fractals. Computational experimentation.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
</div><br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|534]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_534:_Intro_to_Dynamical_Systems_1&diff=3725Math 534: Intro to Dynamical Systems 12019-11-14T17:04:00Z<p>Ls5: /* Catalog Information */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Introduction to Dynamical Systems 1(?).<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Offered ===<br />
On demand (contact department)<br />
<br />
=== Prerequisite ===<br />
[[Math 341]] or equivalent(?).<br />
<br />
=== Recommended ===<br />
[[Math 352]] or equivalent.<br />
<br />
=== Description ===<br />
Discrete dynamical systems; iterations of maps on the line and the plane; bifurcation theory; chaos, Julia sets, and fractals. Computational experimentation.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
</div><br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|534]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_532:_Complex_Analysis&diff=3724Math 532: Complex Analysis2019-11-14T17:03:36Z<p>Ls5: /* Catalog Information */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Complex Analysis.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Offered ===<br />
On demand (contact department)<br />
<br />
=== Prerequisite ===<br />
[[Math 352]] or instructor`s consent.<br />
<br />
=== Description ===<br />
A second course in complex analysis including the theory of infinite products, gamma and zeta functions, elliptic functions and the Riemann mapping theorem.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
A knowledge of complex analysis at the level of a first course such as [[Math 352]].<br />
<br />
=== Minimal learning outcomes ===<br />
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.<br />
#'''Essential results from a first course.''' Review of power series, integration along curves, Goursat theorem, Cauchy's theorem in a disc, Taylor series, Morera's theorem, singularities, residue calculus, Laurent series, argument principle, harmonic functions, maximum modulus principle.<br />
#'''Entire functions.''' Jensen's formula, functions of finite order, Weierstrass infinite products, Hadamard factorization theorem.<br />
#'''The gamma and zeta functions.''' Analytic continuation of gamma function, further properties of &Gamma;, functional equation and analytic continuation of zeta function.<br />
#'''Conformal mappings.''' Conformal equivalence, Schwarz lemma, Montel's theorem, Riemann mapping theorem.<br />
#'''Elliptic Functions.''' Liouville's Theorems, Poles and zeros of elliptic functions, Weierstrass elliptic functions.<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
</div><br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
These are at the discretion of the instructor as time allows. Some possible additional topics include: <br />
<br />
'''Modular Functions and Theta Functions.''' The modular group, Eisenstein series, product formula for Jacobi theta function, transformation laws of theta functions, application to sums of squares.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|532]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_522:_Methods_of_Applied_Math_2&diff=3723Math 522: Methods of Applied Math 22019-11-14T17:02:39Z<p>Ls5: /* Catalog Information */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Methods of Applied Mathematics 2.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Offered ===<br />
On demand (contact department)<br />
<br />
=== Prerequisite ===<br />
[[Math 334]] or equivalent.<br />
<br />
=== Recommended ===<br />
[[Math 352]] or equivalent.<br />
<br />
=== Description ===<br />
Possible topics include variational, integral, and partial differential equations; spectral and transform methods; nonlinear waves; Green's functions; scaling and asymptotic analysis; perturbation theory; continuum<br />
mechanics.<br />
<br />
== Desired Learning Outcomes ==<br />
The object of this course is to familiarize students with classical techniques in applied mathematics and demonstrate their application to specific problems. <br />
=== Prerequisites ===<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
</div><br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|522]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_521:_Methods_of_Applied_Math_1&diff=3722Math 521: Methods of Applied Math 12019-11-14T17:02:14Z<p>Ls5: /* Catalog Information */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Methods of Applied Mathematics 1.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Offered ===<br />
On demand (contact department)<br />
<br />
=== Prerequisite ===<br />
[[Math 334]] or equivalents. Students are recommended to take [[Math 352]] or equivalent before taking Math 521.<br />
<br />
=== Description ===<br />
Possible topics include variational, integral, and partial differential equations; spectral and transform methods; nonlinear waves; Green's functions; scaling and asymptotic analysis; perturbation theory; continuum<br />
mechanics.<br />
<br />
== Desired Learning Outcomes ==<br />
The object of this course is to familiarize students with classical techniques in applied mathematics and demonstrate their application to specific problems. The list above gives examples of possible techniques and problems. <br />
=== Prerequisites ===<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
</div><br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|521]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_513R:_Advanced_Topics_in_Applied_Math&diff=3721Math 513R: Advanced Topics in Applied Math2019-11-14T17:01:47Z<p>Ls5: /* Catalog Information */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Advanced Topics in Applied Mathematics.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Offered ===<br />
On demand (contact department)<br />
<br />
=== Prerequisite ===<br />
instructor's consent<br />
<br />
=== Description ===<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
</div><br />
<br />
Students should gain a familiarity with a particular area of applied mathematics selected by the instructor. In general, a significant, coherent set of readings will be defined, and the student's base of mathematical knowledge and expertise will increase in notable, measurable ways. The student is expected to demonstrate mastery of the material in a manner that is acceptable to the professor.<br />
<br />
=== Textbooks ===<br />
<br />
Textbooks will be chosen by the professor as appropriate.<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|513]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_406R:_Topics_in_Mathematics&diff=3720Math 406R: Topics in Mathematics2019-11-14T17:01:16Z<p>Ls5: /* Offered */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Topics in Mathematics<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:1)<br />
<br />
=== Offered ===<br />
On demand (contact department)<br />
<br />
=== Prerequisite ===<br />
Instructor's consent<br />
<br />
=== Description ===<br />
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.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
<br />
<br />
=== Prerequisites ===<br />
<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|406R]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_393:_Survey_of_Advanced_Mathematics&diff=3719Math 393: Survey of Advanced Mathematics2019-11-14T17:00:33Z<p>Ls5: /* Offered */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Survey of Advanced Mathematics<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(0.5:1:0)<br />
<br />
=== Offered ===<br />
On demand (contact department)<br />
<br />
=== Prerequisite ===<br />
Majors only<br />
<br />
=== Description ===<br />
A series of lectures designed to give students an overview of active research areas in advanced mathematics.<br />
<br />
== Desired Learning Outcomes ==<br />
Students should be familiar with the areas of mathematical research described (naturally in a highly schematic way) by the lecturers in the course. The areas of research will be essentially those that are currently being pursued by BYU faculty in the mathematics department. Students will be able to make an informed choice if they wish to begin a program of undergraduate research with one of the faculty.<br />
<br />
=== Prerequisites ===<br />
<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|393]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_511:_Numerical_Methods_for_PDEs&diff=3718Math 511: Numerical Methods for PDEs2019-11-14T16:59:18Z<p>Ls5: /* Catalog Information */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Numerical Methods for Partial Differential Equations.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Offered ===<br />
W<br />
<br />
=== Prerequisite ===<br />
[[Math 303]] or [[Math 347|347]]; [[Math 410|410]]; or equivalents.<br />
<br />
=== Description ===<br />
Finite difference and finite volume methods for partial differential equations. Stability, consistency, and convergence theory.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
This course is designed to prepare students to solve mathematical<br />
models represented by initial or boundary value problems involving<br />
partial differential equations that cannot be solved directly<br />
using standard mathematical techniques but are amenable to a<br />
computational approach. Numerical solution of partial differential equations has important applications in many application areas. Students are introduced to the<br />
discretization methodologies, with particular emphasis on the<br />
finite difference method, that allows the construction of<br />
accurate and stable numerical schemes. In depth discussion of<br />
theoretical aspects such as stability analysis and convergence<br />
will be used to enhance the students' understanding of the<br />
numerical methods. Students will also be required to perform some<br />
programming and computation so as to gain experience in<br />
implementing the schemes and to be able to observe the numerical<br />
performance of the various numerical methods.<br />
<br />
The course addresses the University goal of developing the skills<br />
of sound thinking, effective communication and quantitative<br />
reasoning. The course also allow students, especially<br />
undergraduate students, to develop some depth and consequently<br />
competence in an important area of applied mathematics.<br />
<br />
This course requires knowledge of higher level courses in<br />
mathematics and serves as an introductory graduate<br />
level course to prepare the students to apply the methods learned<br />
in their research projects.<br />
<br />
=== Prerequisites ===<br />
<br />
Understanding of basic theory and properties of solutions of partial differential equations;<br />
<br />
Basic programming skill in matlab;<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
</div><br />
Students are expected to acquire the following knowledge and skills:<br />
<br />
Derive finite difference schemes using Taylor series.<br />
<br />
Derive finite volume schemes using flux balance.<br />
<br />
Understand how finite volume scheme and finite difference scheme are related.<br />
<br />
Determine the consistency of a difference scheme.<br />
<br />
Explain the proper function spaces and discrete norms for<br />
grid functions for use in analysis of stability.<br />
<br />
Establish the stability of a difference scheme using <br />
(1) Heuristic approach <br />
(2) Energy method <br />
(3) von Neumann method <br />
(4) Matrix method.<br />
<br />
Recall the CFL condition its relation with stability.<br />
<br />
Explain the convergence of the finite difference<br />
approximations and its relation with consistency and stability via<br />
Lax theorem;<br />
<br />
Determine the order of accuracy of a finite difference<br />
scheme.<br />
<br />
Implement finite difference schemes on computers and perform<br />
numerical studies of the stability and convergence properties of<br />
the schemes.<br />
<br />
Explain the role and the control of numerical diffusion and<br />
dispersion in computation ; to determine how numerical phase speed<br />
and group velocity may deviate from the theoretical phase speed<br />
and group velocity and the numerical techniques to handle such<br />
issues.<br />
<br />
Recall numerical methods that efficiently handle a<br />
multidimensional problem.<br />
<br />
Recall alternating direction methods that reduce higher<br />
dimensional problems into a sequence of one dimensional problems.<br />
<br />
Recall the maximum principles for numerical schemes for<br />
Laplace equations.<br />
<br />
Recall iterative techniques for solving the linear systems<br />
resulting from finite difference or finite element discretization.<br />
<br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
John Strikwerda, Finite Difference Schemes and Partial Differential Equations, 2nd Ed., SIAM, 2007;<br />
ISBN: 089871639X, 978-0898716399<br />
<br />
Randall Leveque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems, SIAM 2007;<br />
ISBN: 0898716292, 978-0898716290<br />
<br />
K. W. Morton and D. F. Mayers, Numerical Solution of Partial Differential Equations: An Introduction, 2nd Ed., Cambridge University Press, 2005;<br />
ISBN: 0521607930, 978-0521607933<br />
<br />
Arieh Iserles, A First Course in the Numerical Analysis of Differential Equations, 2nd Ed, Cambridge University Press, 2008;<br />
ISBN: 0521734908, 978-0521734905<br />
<br />
Claes Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Dover, 2009;<br />
ISBN: 048646900X, 978-0486469003<br />
<br />
J.W. Thomas, Numerical Partial Differential Equations: Finite Difference Methods, 2nd Ed., Springer, 2010;<br />
ISBN-10: 1441931058, 978-1441931054<br />
<br />
=== Additional topics ===<br />
Finite element method; Method of lines; Parallel computing<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|511]]<br />
Math 303 or 347; 410; or equivalents.</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_510:_Numerical_Methods_for_Linear_Algebra&diff=3717Math 510: Numerical Methods for Linear Algebra2019-11-14T16:58:51Z<p>Ls5: /* Catalog Information */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Numerical Methods for Linear Algebra.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Offered ===<br />
F<br />
<br />
=== Prerequisite ===<br />
[[Math 314]], CS 142; or equivalent.<br />
<br />
=== Description ===<br />
Numerical matrix algebra, orthogonalization and least squares methods, unsymmetric and symmetric eigenvalue problems, iterative<br />
methods, advanced solvers for partial differential equations.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
This course is designed to prepare students to solve linear algebra<br />
problems arising from many applications such as mathematical models of physical or engineering processes.<br />
Students are introduced to modern concepts and methodologies in numerical linear algebra, with particular emphasis on t<br />
methods that can be used to solve very large scale problems. In depth discussion of<br />
theoretical aspects such as stability and convergence<br />
will be used to enhance the students' understanding of the<br />
numerical methods. Students will also be required to perform some<br />
programming and computation so as to gain experience in<br />
implementing and observing the numerical<br />
performance of the various numerical methods.<br />
<br />
The course addresses the University goal of developing the skills<br />
of sound thinking, effective communication and quantitative<br />
reasoning. The course also allow students, especially<br />
undergraduate students, to develop some depth and consequently<br />
competence in an important area of applied mathematics.<br />
<br />
This course requires knowledge of higher level courses in<br />
mathematics. The course also serves as an introductory graduate<br />
level course to prepare the students to apply the methods learned<br />
in their research projects.<br />
<br />
=== Prerequisites ===<br />
Mastery of materials in an undergraduate course in linear algebra. Knowledge of matlab.<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
</div><br />
(Must know)<br />
<br />
Know properties of unitary matrices<br />
<br />
Know general and practical definitions and properties of norms<br />
<br />
Know definition and properties of SVD <br />
<br />
Know definitions of and properties of projectors and orthogonal projectors<br />
<br />
Be able to state and apply classical Gram-Schmidt and modified Gram-Schmidt algorithms<br />
<br />
Know definition and properties of a Householder reflector<br />
<br />
Know how to construct Householder QR factorization <br />
<br />
Know how least squares problems arise from a polynomial fitting problem<br />
<br />
Know how to solve least square problems using <br />
(1) normal equations/pseudoinverse, <br />
(2) QR factorization and <br />
(3) SVD<br />
<br />
Be able to define the condition of a problem and related condition number<br />
Know how to calculate the condition number of a matrix <br />
<br />
Understand the concepts of well-conditioned and ill-conditioned problems<br />
<br />
Know how to derive the conditioning bounds<br />
<br />
Know the precise definition of stability and backward stability<br />
<br />
Be able to apply the fundamental axiom of floating point arithmetic to determine stability <br />
<br />
Know the difference between stability and conditioning<br />
<br />
Know the four condition numbers of a least squares problem <br />
<br />
Know how to construct LU and PLU factorizations<br />
<br />
Know how PLU is related to Gaussian elimination<br />
<br />
Understand Cholesky decomposition <br />
<br />
Know properties of eigenvalues and eigenvectors under similarity transformation and shift<br />
<br />
Know various matrix decomposition related to eigenvalue calculation: <br />
(1) spectral decomposition<br />
(2) unitary diagonaliation<br />
(2) Schur decomposition<br />
Understand why and how matrices can be reduced to Hessenberg form <br />
<br />
Know various form of power method and what Rayleigh quotient iteration and their properties and convergence rates<br />
<br />
Know the QR algorithm with shifts <br />
<br />
Understand simultaneous iteration and QR algorithm are mathematically equivalent <br />
<br />
Understand the Arnoldi algorithm and its properties<br />
<br />
Know the polynomial approximation problem associated with Arnoldi method<br />
<br />
Be able to state the GMRES algorithm<br />
<br />
Be able to state three term recurrence of Lanczos iteration for real symmetric matrices<br />
<br />
Understand the CG algorithm and its properties<br />
<br />
Know the v-cycle multigrid algorithm and the full multigrid algorithm<br />
<br />
Know how to construct preconditioners: (block) diagonal, incomplete LU and incomplete Cholesky preconditioners<br />
<br />
Know CGN and BCG and other Krylov space methods<br />
<br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
Lloyd N. Trefethen and David Bau III, Numerical Linear Algebra, SIAM; 1997; <br />
ISBN: 0898713617, 978-0898713619<br />
<br />
James W. Demmel, Applied Numerical Linear Algebra, SIAM; 1997; <br />
ISBN: 0898713897, 978-0898713893<br />
<br />
Gene H. Golub and Charles F. Van Loan, Matrix Computations, 3rd Ed., Johns Hopkins University Press, 1996;<br />
ISBN: 0801854148, 978-0801854149<br />
<br />
William L. Briggs, Van Emden Henson and Steve F. McCormick, A Multigrid Tutorial, 2nd Ed., SIAM, 2000;<br />
ISBN: 0898714621, 978-0898714623<br />
<br />
Barry Smith, Petter Bjorstad, William Gropp, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press, 2004;<br />
ISBN: 0521602866, 978-0521602860<br />
<br />
=== Additional topics ===<br />
Multigrid method;<br />
Domain decomposition method;<br />
Freely available linear algebra software;<br />
Fast multipole method for linear systems; Parallel processing<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|510]]<br />
Math 343, 410; or equivalents.</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_487:_Intro_to_Number_Theory&diff=3716Math 487: Intro to Number Theory2019-11-14T16:57:18Z<p>Ls5: /* Offered */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Number Theory.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
W<br />
<br />
=== Prerequisite ===<br />
[[Math 371]].<br />
<br />
=== Description ===<br />
Foundations; congruences; quadratic reciprocity; unique factorization, prime distribution or Diophantine equations.<br />
<br />
== Desired Learning Outcomes ==<br />
This course is aimed at undergraduate mathematics majors. It is a first course in number theory, and is intended to introduce students to number theoretic problems and to different areas of number theory. Number theory has a very long history compared to some other areas of mathematics, and has many applications, especially to coding theory and cryptography.<br />
<br />
=== Prerequisites ===<br />
[[Math 371]] is a prerequisite for this course. In particular, students should be familiar with the concepts of groups and rings, and they should understand constructions of quotient groups and quotient rings. By this point in their mathematical career, students should be comfortable proving theorems by themselves.<br />
<br />
=== Minimal learning outcomes ===<br />
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.<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
# Divisibility in the integers<br />
#* Prime numbers<br />
#* Unique factorization<br />
#* Euclid’s algorithm<br />
#* GCD and LCM<br />
# Congruence arithmetic<br />
#* Complete and reduced residue systems<br />
#* Linear congruences<br />
#* Chinese remainder theorem<br />
#* Polynomial congruences<br />
#* Hensel’s lemma<br />
#* Quadratic residues<br />
#* Legendre and Jacobi symbols<br />
#* Quadratic reciprocity<br />
# Primitive roots<br />
#* Existence of primitive roots<br />
#* Structure of units modulo nonprimes<br />
# Number Theoretic Functions<br />
#* Moebius Function<br />
#* Euler phi function<br />
#* Sum of divisors function<br />
#* Big Oh notation<br />
#* Little Oh notation<br />
# Distribution of Primes<br />
#* Definition of Pi(''x'')<br />
#* Estimates of Pi(''x'')<br />
#* Primes in arithmetic progressions<br />
#* Bertrand’s Hypothesis<br />
# Sums of Squares<br />
#* Representations of numbers as sums of two and four squares<br />
#* Statement of Waring’s Problem<br />
<br />
</div><br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Ivan Niven, Herbert S. Zuckerman, and Hugh L. Montgomery, <i>An Introduction to the Theory of Numbers (5th edition)</i>, Wiley, 1991.<br />
* George Andrews, <i>Number Theory</i>, Dover, 1994.<br />
* William LeVeque, <i>Fundamentals of Number Theory</i>, Dover, 1996.<br />
* Benjamin Fine and Gerhard Rosenberger, Number Theory, An Introduction via the Distribution of Primes, (available as a free Springer e-book at springerlink.com).<br />
<br />
=== Additional topics ===<br />
Beyond the minimal learning outcomes, instructors are free to cover additional topics. These may include (but are certainly not limited to): diophantine approximation, continued fractions, elliptic curves, cryptography, partition theory, Abel and Euler-Maclaurin summation formulae. It is expected that some topics beyond the minimal learning objectives will typically be discussed.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
This course is not a prerequisite for any other courses in the regular curriculum. Hence, this course may be the only opportunity for students to learn the topics listed here. As a result, it is important that all learning objectives be completed.<br />
<br />
[[Category:Courses|487]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_473:_Group_Representation_Theory.&diff=3715Math 473: Group Representation Theory.2019-11-14T16:56:32Z<p>Ls5: /* Offered */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Group Representation Theory.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F<br />
<br />
=== Prerequisite ===<br />
[[Math 371]].<br />
<br />
=== Description ===<br />
FG-modules; Maschke's theorem; Shur's lemma; characters of groups; orthogonality relations of characters; induced, lifted, and restricted characters; construction of character tables; Burnside's theorem.<br />
<br />
== Desired Learning Outcomes ==<br />
This course is aimed at undergraduate mathematics majors. It is a second course in abstract algebra, and covers the representation theory of finite groups. Representation theory is an important topic in mathematics, as well as having applications in physics and chemistry.<br />
<br />
=== Prerequisites ===<br />
[[Math 371]] is a prerequisite for this course. In particular, students should be familiar with the concepts of groups, rings, and fields, and they should understand quotient groups, quotient rings, and homomorphisms. By this point in their mathematical career, students should be comfortable proving theorems by themselves.<br />
<br />
=== Minimal learning outcomes ===<br />
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.<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
# FG-modules<br />
#* Groups and homomorphisms<br />
#* Vector spaces and linear transformations<br />
#* Group representations<br />
#* Group Algebras<br />
#* Definition of FG-modules<br />
#* FG-Homomorphisms<br />
# Reduciblity of modules<br />
#* Maschke’s Theorem<br />
#* Schur’s Lemma<br />
#* Irreducible modules<br />
# Character Theory<br />
#* Definition of Characters<br />
#* Inner products of characters<br />
#* Conjugacy classes<br />
#* The number of irreducible characters<br />
#* Orthogonality relations<br />
#* Character Tables<br />
# Operations on Characters<br />
#* Normal subgroups and lifted characters<br />
#* Tensor products<br />
#* Restriction to a subgroup<br />
#* Induced modules and characters<br />
# Properties of Characters<br />
#* Real representations<br />
#* Divisibility properties<br />
#* Properties of character tables<br />
# Applications of characters to group theory<br />
#* Characters of groups of order pq<br />
#* Characters of some p-groups<br />
#* Burnside’s pq theorem<br />
# Representations of symmetric groups<br />
#* Young tableaux<br />
#* Frobenius formula<br />
<br />
<br />
</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*Gordon James and Martin Liebeck, <i>Representations and Characters of Groups (2nd edition)</i>, Cambridge, 2001.<br />
*M. Issacs, <i>Character theory of finite groups</i>, AMS, 2006.<br />
<br />
=== Additional topics ===<br />
Beyond the minimal learning outcomes, instructors are free to cover additional topics. These may include (but are certainly not limited to): modular representation theory or applications of representation theory to physics and chemistry.<br />
=== Courses for which this course is prerequisite ===<br />
This course is not a prerequisite for any other courses in the regular curriculum. Hence, this course may be the only opportunity for students to learn the topics listed here. As a result, it is important that all learning objectives be completed.<br />
<br />
[[Category:Courses|473]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_451:_Intro_to_Topology&diff=3714Math 451: Intro to Topology2019-11-14T16:55:58Z<p>Ls5: /* Offered */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Introduction to Topology.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
W<br />
<br />
=== Prerequisite ===<br />
[[Math 341]].<br />
<br />
=== Description ===<br />
Developing topological concepts, beginning from a linear setting. Developing proofs or counterexamples from axioms to a structured sequence of topological propositions using only notes provided.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
1. Students should demonstrate an ability to write and present mathematical proofs beyond that learned in Math 290.<br />
<br />
2. Students should demonstrate understanding of the following concepts by proving theorems about them: linear ordering, connectedness, continuity, open and closed sets, density, compactness, local connectedness, local compactness.<br />
<br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|451]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_447:_Intro_to_Partial_Differential_Equations&diff=3713Math 447: Intro to Partial Differential Equations2019-11-14T16:55:20Z<p>Ls5: /* Offered */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Introduction to Partial Differential Equations.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
W (even years)<br />
<br />
=== Prerequisite ===<br />
[[Math 303]]; or [[Math 314|314]] and [[Math 334|334]].<br />
<br />
=== Description ===<br />
Boundary value problems; transform methods; Fourier series; Bessel functions; Legendre polynomials.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
The main purpose of this course is to teach students how to solve the canonical linear second-order partial differential equations on simple domains. Secondarily, students should be introduced to the theory concerning the validity of such solutions.<br />
<br />
=== Prerequisites ===<br />
<br />
Current prerequisites ensure that students have had instruction in multivariable calculus and ordinary differential equations.<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
Primarily, students should be able to use the solution techniques described below. Students should gain a basic understanding of issues concerning solvability and convergence, but the current prerequisites don't guarantee that incoming students will have had any prior exposure to the theory of the convergence of sequences of functions, so expectations in that area are modest.<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
# Basic classification of PDEs<br />
#* Linearity<br />
#* Homogeneity <br />
#* Order<br />
#* Elliptic, parabolic, or hyperbolic<br />
# Basic Modeling<br />
#* Derivation of the heat equation<br />
#* Derivation of the wave equation<br />
# Basic principles, techniques, and theory<br />
#* Principle of superposition<br />
#* Method of separation of variables<br />
#* Definition of eigenvalues and eigenfunctions corresponding to two-point BVPs<br />
#* Basic Sturm-Liouville theory<br />
# Special eigensystems<br />
#* Fourier<br />
#** Series representations<br />
#*** Effect of symmetry and modifications and combinations of functions<br />
#*** Theorems on pointwise, uniform, and ''L''<sup>2</sup> convergence<br />
#**** Bessel's Inequality and Parseval's Equation<br />
#** Integral representations<br />
#* Bessel<br />
#* Legendre<br />
# Representation of solutions to the canonical equations on simple domains<br />
#* Laplace's equation on rectangles, rectangular strips, quarter-planes, half-planes, disks, and balls<br />
#* Wave equation on bounded intervals, half-lines, lines, disks, and balls<br />
#* Heat equation on bounded intervals, half-lines, lines, rectangles, disks, and balls<br />
</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Richard Haberman, ''Applied Partial Differential Equations (4th Edition)'', Prentice Hall, 2003.<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
Students taking [[Math 511]] are supposed to have had either Math 447 or [[Math 303]]. It is proposed that Math 447 become a prerequisite (or at least recommended) for [[Math 547]], so that there will be less duplication of material in the PDE curriculum.<br />
<br />
[[Category:Courses|447]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_435:_Mathematical_Finance&diff=3712Math 435: Mathematical Finance2019-11-14T16:52:39Z<p>Ls5: /* Offered */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Mathematical Finance.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
W (even years)<br />
<br />
=== Prerequisite ===<br />
[[One of Math 431, Stat 341, Stat 370]].<br />
<br />
=== Description ===<br />
The binomial asset pricing model (discrete probability). Martingales, pricing of derivative securities, random walk in financial models, random interest rates.<br />
<br />
== Desired Learning Outcomes ==<br />
The minimal expectation for this course is that students learn about mathematical finance ''in the context of discrete time and finite state-spaces.'' It is therefore not required that students be taught about Brownian motion, the Black-Scholes model, etc.<br />
<br />
=== Prerequisites ===<br />
Students should have had an introductory course in probability.<br />
<br />
=== Minimal learning outcomes ===<br />
Within the context mentioned above, students should be able to compute prices for derivative securities. They should be conversant with the standard terminology of mathematical finance and be able to use this terminology correctly in answering questions. At a minimum, students should understand the following concepts in the context of binomial decision trees:<br />
<div style="-moz-column-count:2; column-count:2;"><br />
# Martingales<br />
# Markov processes<br />
# Arbitrage<br />
# Risk neutrality<br />
# State prices<br><br><br />
# Options<br />
#* Call and put<br />
#* American and European<br />
# Stopping times<br />
# Simple random walks<br />
# Interest rate models<br />
</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Steven E. Shreve, ''Stochastic Calculus for Finance I: The Binomial Asset Pricing Model'', Springer, 2005.<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
None.<br />
<br />
[[Category:Courses|435]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_425:_Mathematical_Biology&diff=3711Math 425: Mathematical Biology2019-11-14T16:51:47Z<p>Ls5: /* Offered */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title === <br />
Mathematical Biology.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
W (odd years)<br />
<br />
=== Prerequisite ===<br />
[[Math 334]].<br />
<br />
=== Description ===<br />
Using tools in mathematics to help biologists. Motivating new mathematics with questions in biology.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
Students should gain a familiarity with how the disciplines of mathematics and biology can complement each other.<br />
<br />
=== Prerequisites ===<br />
<br />
A knowledge of calculus (and the mathematical maturity that having passed [[Math 112]] entails) should suffice.<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
Students should become familiar with 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. Possible topics include:<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
# Signal Transduction<br />
#* Menten Michaelis enzyme dynamics<br />
#* Law of mass action<br />
#* Dynamical systems<br />
#* Bifurcation<br />
# Example systems<br />
#* Fitzhugh-Nagumo<br />
#* Nerve and heart dynamics<br />
#* Cell cycle model<br />
#* cAMP<br />
# Population models<br />
#* Continuous predator-prey<br />
#* Age structured models<br />
#* Discrete dynamical systems<br />
#* Time delayed differential equations<br />
#* Stochastic models<br />
</div><br />
<br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* A Course in Mathematical Biology. Quantitative Modeling with Mathematical and Computational Methods. By Gerda de Vries, Thomas Hillen, Mark Lewis, Johannes Muller, Birgitt Schonfisch<br />
<br />
=== Additional Topics ===<br />
<br />
These are at the discretion of the instructor as time allows.<br />
<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
None.<br />
<br />
[[Category:Courses|425]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_372:_Abstract_Algebra_2.&diff=3710Math 372: Abstract Algebra 2.2019-11-14T16:48:22Z<p>Ls5: /* Offered */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Abstract Algebra 2.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
W<br />
<br />
=== Prerequisite ===<br />
[[Math 371]].<br />
<br />
=== Description ===<br />
Fields, Galois theory, solvability of polynomials by radicals.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
<br />
Students need to have mastered [[Math 313]] (linear algebra) and [[Math 371]] (group theory and ring theory) prior to enrolling in Math 372.<br />
<br />
This is a second course in abstract algebra focusing on field theory. The course is aimed at undergraduate mathematics majors, and it is strongly recommended for students intending to complete a graduate degree in mathematics. In addition to being an important branch of mathematics in its own right, abstract algebra is now an essential tool in number theory, geometry, topology, and, to a lesser extent, analysis. Outside of mathematics, algebra also has applications in cryptography, coding theory, quantum chemistry, and physics.<br />
<br />
=== Minimal learning outcomes ===<br />
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.<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
# Ring Theory<br />
#* Ideals and ring homomorphisms<br />
#* Quotient rings<br />
#* Prime and maximal ideals<br />
#* Polynomial rings over fields<br />
#* Factorization in polynomial rings<br />
#* Irreducible polynomials<br />
#* Polynomial division algorithm<br />
# Field Theory<br />
#* Extensions of fields<br />
#* Field extensions via quotients in polynomial rings<br />
#* Automorphisms of fields<br />
#* Finite fields<br />
#* Fields of characteristic 0 and prime characteristic<br />
#* Splitting fields<br />
#* Galois extensions and Galois groups<br />
#* The Galois correspondence<br />
#* Fundamental Theorem of Galois Theory<br />
#* Fundamental Theorem of Algebra<br />
#* Roots of unity<br />
#* Solvability by radicals<br />
#* Ruler and compass constructions<br />
#* Insolvability of the quintic<br />
<br />
</div><br />
<br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*Joseph Rotman, ''Galois Theory (Second Edition)'', Springer, 1998.<br />
<br />
*David Dummit and Richard Foote, ''Abstract Algebra (Third Edition)'', Wiley, 2003. (The chapters on fields and Galois theory, and some of the material on rings.)<br />
<br />
*Ian Stewart, ''Galois Theory (Third Edition)'', Chapman Hall, 2004.<br />
<br />
*David Cox, ''Galois Theory (Second Edition)'', Wiley, 2012.<br />
<br />
=== Additional topics ===<br />
<br />
The instructor may cover additional topics beyond the minimal requirements. Possible topics include (but are not limited to): introduction to algebraic numbers, applications of field extensions to cryptography, applications of field extensions to diophantine analysis, relations of field theory to algebraic geometry, construction of algebraically closed fields using Zorn's lemma, introduction to computer calculations in abstract algebra.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|372]]<br />
[[Math 586]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_341:_Theory_of_Analysis_1&diff=3709Math 341: Theory of Analysis 12019-11-14T16:46:31Z<p>Ls5: /* Offered */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Theory of Analysis 1.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F, W, Sp<br />
<br />
=== Prerequisite ===<br />
[[Math 113]], [[Math 290|290]].<br />
<br />
=== Description ===<br />
Rigorous treatment of calculus of a single real variable: topology, order, completeness of real numbers; continuity, differentiability, integrability, and convergence of functions.<br />
<br />
== Desired Learning Outcomes ==<br />
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.<br />
<br />
=== Prerequisites ===<br />
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.<br />
<br />
=== Minimal learning outcomes ===<br />
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.<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
# Basic properties of '''R'''<br />
#* Characterization as a complete, ordered field<br />
#* Archimedean Property<br />
#* Density of '''Q''' and '''R''' \ '''Q'''<br />
#* Uncountability of each interval<br />
# Convergence of real sequences and series<br />
#* Convergence of bounded monotonic sequences<br />
#* Algebraic and order rules for limits<br />
#* Cauchy criterion for sequences and series<br />
#* Common convergence tests for series<br />
#* Convergence of rearranged series<br />
# Basic topology of '''R'''<br />
#* Open and closed sets<br />
#* Limit points and limits<br />
#* Characterizations of compactness<br />
#** Sequential<br />
#** Open coverings<br />
#** The Heine-Borel Theorem<br />
#* The Bolzano-Weierstrass Theorem<br />
#* Connectedness<br />
# Continuity of ''f'': ''D'' ⊆ '''R''' → '''R'''<br />
#* Functional limits<br />
#* Metric, sequential, and topological characterizations of continuity<br />
#* Combinations of continuous functions<br />
#* Continuity vs. uniform continuity<br />
#* Preservation of compactness<br />
#** The Extreme Value Theorem<br />
#* Uniform continuity for compact domains<br />
#* Preservation of connectedness<br />
#** The Intermediate Value Theorem<br />
# Differentiability of ''f'': ''D'' ⊆ '''R''' → '''R'''<br />
#* Algebraic differential rules<br />
#* Chain rule<br />
#* Characterizing extrema<br />
#* Rolle's Theorem<br />
#* The Mean Value Theorem<br />
#* The Generalized Mean Value Theorem<br />
#* L'Hôpital's Rule<br />
# Integrability of ''f'': [''a'',''b''] → '''R'''<br />
#* The Darboux integral<br />
#* The Riemann integral<br />
#* Integrability of continuous functions<br />
#* Integrability of monotonic functions<br />
#* Rules for combining and comparing integrals<br />
#* The Fundamental Theorem(s) of Calculus<br />
# Convergence of sequences and series of functions<br />
#* Pointwise vs. uniform convergence<br />
#* Relation of uniform convergence to:<br />
#** Continuity<br />
#** Differentiation<br />
#** Integration<br />
#* The Weierstrass ''M''-Test<br />
#* The Weierstrass Approximation Theorem<br />
#* Power series<br />
#** Continuity<br />
#** Absolute and uniform convergence<br />
#** Termwise differentiability<br />
#** Taylor's Theorem<br />
#** Analyticity vs. smoothness<br />
<br />
<br />
</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Stephen Abbott, ''Understanding Analysis, 2nd Edition'', Springer, 2015.<br />
<br />
=== Additional topics ===<br />
Among other options, instructors may want to discuss constructing the real numbers using<br />
Dedekind cuts or Cauchy sequences.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
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]].<br />
<br />
[[Category:Courses|341]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_413_Advanced_Linear_Algebra&diff=3707Math 413 Advanced Linear Algebra2019-10-24T21:11:59Z<p>Ls5: /* Desired Learning Outcomes */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Advanced Linear Algebra<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F and possibly W<br />
<br />
=== Prerequisite ===<br />
[[Math 371]] recommended, but not required.<br />
<br />
=== Description ===<br />
Theory and advanced topics of linear algebra.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
<strong>Prove:</strong> Students will be able to prove central linear-algebraic results, as well as other results with similar derivations.<br />
<br />
<strong>Distinguish:</strong> Students will be able to distinguish between true and plausibly-sounding false propositions in the language of linear algebra.<br />
<br />
<strong>Construct:</strong> Students will be able to construct examples and counterexamples illustrating relations between different linear-algebraic concepts.<br />
<br />
<strong>Categorize:</strong> Students will be able to categorize linear-algebraic structures according to their properties.<br />
<br />
<strong>Calculate:</strong> Students will be able to calculate precisely and efficiently, choosing appropriate methods.<br />
<br />
=== Prerequisites ===<br />
[[Math 371]] recommended, but not required.<br />
<br />
=== Minimal learning outcomes ===<br />
# Linear equations (row operations, matrix multiplication, and invertibility)<br />
# Vector spaces<br />
# Linear transformations (algebra of linear transformations, isomorphisms, linear functionals, duality)<br />
# Polynomials and determinants (algebra of polynomials, polynomial ideals, determinant functions, permutations and uniqueness of determinants)<br />
# Jordan canonical form and elementary canonical forms (invariant subspaces, simultaneous diagonalization and triangulation, direct-sum decompositions, rational forms, Jordan form)<br />
# Inner product spaces (inner product spaces, linear functionals and adjoints, unitary operators, normal operators)<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_413_Advanced_Linear_Algebra&diff=3706Math 413 Advanced Linear Algebra2019-10-24T21:11:30Z<p>Ls5: Created page with "== Catalog Information == === Title === Advanced Linear Algebra === (Credit Hours:Lecture Hours:Lab Hours) === (3:3:0) === Offered === F and possibly W === Prerequisite ==..."</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Advanced Linear Algebra<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F and possibly W<br />
<br />
=== Prerequisite ===<br />
[[Math 371]] recommended, but not required.<br />
<br />
=== Description ===<br />
Theory and advanced topics of linear algebra.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
<strong>Prove:</strong> Students will be able to prove central linear-algebraic results, as well as other results with similar derivations.<br />
<br />
<strong>Distinguish:</strong> Students will be able to distinguish between true and plausibly-sounding false propositions in the language of linear algebra.<br />
<br />
<strong>Construct:</strong> Students will be able to construct examples and counterexamples illustrating relations between different linear-algebraic concepts.<br />
<br />
<strong>Categorize:</strong> Students will be able to categorize linear-algebraic structures according to their properties.<br />
<br />
<strong>Calculate:</strong> Students will be able to calculate precisely and efficiently, choosing appropriate methods.<br />
<br />
<br />
=== Prerequisites ===<br />
[[Math 371]] recommended, but not required.<br />
<br />
=== Minimal learning outcomes ===<br />
# Linear equations (row operations, matrix multiplication, and invertibility)<br />
# Vector spaces<br />
# Linear transformations (algebra of linear transformations, isomorphisms, linear functionals, duality)<br />
# Polynomials and determinants (algebra of polynomials, polynomial ideals, determinant functions, permutations and uniqueness of determinants)<br />
# Jordan canonical form and elementary canonical forms (invariant subspaces, simultaneous diagonalization and triangulation, direct-sum decompositions, rational forms, Jordan form)<br />
# Inner product spaces (inner product spaces, linear functionals and adjoints, unitary operators, normal operators)<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_344:_Mathematical_Analysis_1&diff=3705Math 344: Mathematical Analysis 12019-08-22T20:34:32Z<p>Ls5: /* Prerequisites */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Mathematical Analysis 1<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F<br />
<br />
=== Prerequisite ===<br />
[[Math 290]], [[Math 213]] and [[Math 215]] (or [[Math 313]]), [[Math 314]]; concurrent with [[Math 334]], [[Math 345]]<br />
<br />
=== Description ===<br />
Development of the theory of vector spaces, linear maps, inner product spaces, spectral theory, metric space topology, differentiation, contraction mappings and convex analysis.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
[[Math 290]], [[Math 213]] and [[Math 215]] (or [[Math 313]]), [[Math 314]]; concurrent with [[Math 334]], [[Math 345]]<br />
<br />
=== Minimal learning outcomes ===<br />
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 be able to relate them to other mathematical material that they have studied. They will be able to perform the related computations on small, simple problems.<br />
<br />
# Abstract Vector Spaces<br />
#* Vector Algebra<br />
#* Subspaces and Spans<br />
#* Linearly Independent Sets<br />
#* Products, Sums, and Complements<br />
# Linear Transformations and Linear Systems<br />
#* Definitions and Examples<br />
#* Rank, Range, and Nullity<br />
#* Matrix Representations<br />
#* Change of Basis and Similarity<br />
#* Invariant Subspaces<br />
#* Linear Systems (Elementary Matrices, Row Echelon Form, Inverses, Determinants)<br />
# Inner Product Spaces<br />
#* Definitions and Examples<br />
#* Orthonormal Sets<br />
#* Gram Schmidt Orthogonalization<br />
#* Normed Linear Spaces<br />
#* The Adjoint of a Linear Transformation<br />
#* Fundamental Subspaces<br />
#* Projectors<br />
#* Least Squares<br />
# Spectral Theory<br />
#* Eigenvalues<br />
#* Diagonalization<br />
#* Special Matrices<br />
#* Positive Definite Matrices<br />
#* The Singular Value Decomposition<br />
#* Generalized Eigenvectors<br />
# Metric Space Topology<br />
#* Metric Spaces<br />
#* Properties of Open and Closed Sets<br />
#* Continuous and Uniformly Continuous Mappings<br />
#* Cauchy Sequences and Completeness<br />
#* Compactness<br />
#* Connectedness<br />
# Differentiation<br />
#* Directional Derivatives<br />
#* The Derivative<br />
#* The Chain Rule<br />
#* Higher-Order Derivatives<br />
#* The Mean-Value Theorem<br />
#* Taylor’s Theorem<br />
#* Analytic Function Theory (Cauchy Reimann Theorem) <br />
# Contraction Mappings and Applications<br />
#* Contraction Mapping Principle<br />
#* Newton’s Method<br />
#* Implicit Function Theorem<br />
#* Inverse Function Theorem<br />
# Convex Analysis<br />
#* Convex Sets<br />
#* Convex Combinations<br />
#* Examples (Hyperplanes, Halfspaces, Cones) <br />
#* Geometry of Convex Sets (separation theorem) <br />
#* Convex Functions<br />
#* Graphs and Epigraphs<br />
#* Inequalities<br />
#* The Convex Conjugate<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|344]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_344:_Mathematical_Analysis_1&diff=3704Math 344: Mathematical Analysis 12019-08-22T20:34:18Z<p>Ls5: /* Prerequisite */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Mathematical Analysis 1<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F<br />
<br />
=== Prerequisite ===<br />
[[Math 290]], [[Math 213]] and [[Math 215]] (or [[Math 313]]), [[Math 314]]; concurrent with [[Math 334]], [[Math 345]]<br />
<br />
=== Description ===<br />
Development of the theory of vector spaces, linear maps, inner product spaces, spectral theory, metric space topology, differentiation, contraction mappings and convex analysis.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
[[Math 290]], [[Math 313]], [[Math 314]]; concurrent with [[Math 334]], [[Math 345]]<br />
<br />
=== Minimal learning outcomes ===<br />
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 be able to relate them to other mathematical material that they have studied. They will be able to perform the related computations on small, simple problems.<br />
<br />
# Abstract Vector Spaces<br />
#* Vector Algebra<br />
#* Subspaces and Spans<br />
#* Linearly Independent Sets<br />
#* Products, Sums, and Complements<br />
# Linear Transformations and Linear Systems<br />
#* Definitions and Examples<br />
#* Rank, Range, and Nullity<br />
#* Matrix Representations<br />
#* Change of Basis and Similarity<br />
#* Invariant Subspaces<br />
#* Linear Systems (Elementary Matrices, Row Echelon Form, Inverses, Determinants)<br />
# Inner Product Spaces<br />
#* Definitions and Examples<br />
#* Orthonormal Sets<br />
#* Gram Schmidt Orthogonalization<br />
#* Normed Linear Spaces<br />
#* The Adjoint of a Linear Transformation<br />
#* Fundamental Subspaces<br />
#* Projectors<br />
#* Least Squares<br />
# Spectral Theory<br />
#* Eigenvalues<br />
#* Diagonalization<br />
#* Special Matrices<br />
#* Positive Definite Matrices<br />
#* The Singular Value Decomposition<br />
#* Generalized Eigenvectors<br />
# Metric Space Topology<br />
#* Metric Spaces<br />
#* Properties of Open and Closed Sets<br />
#* Continuous and Uniformly Continuous Mappings<br />
#* Cauchy Sequences and Completeness<br />
#* Compactness<br />
#* Connectedness<br />
# Differentiation<br />
#* Directional Derivatives<br />
#* The Derivative<br />
#* The Chain Rule<br />
#* Higher-Order Derivatives<br />
#* The Mean-Value Theorem<br />
#* Taylor’s Theorem<br />
#* Analytic Function Theory (Cauchy Reimann Theorem) <br />
# Contraction Mappings and Applications<br />
#* Contraction Mapping Principle<br />
#* Newton’s Method<br />
#* Implicit Function Theorem<br />
#* Inverse Function Theorem<br />
# Convex Analysis<br />
#* Convex Sets<br />
#* Convex Combinations<br />
#* Examples (Hyperplanes, Halfspaces, Cones) <br />
#* Geometry of Convex Sets (separation theorem) <br />
#* Convex Functions<br />
#* Graphs and Epigraphs<br />
#* Inequalities<br />
#* The Convex Conjugate<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|344]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_320:_Algorithm_Design_and_Optimization_1&diff=3703Math 320: Algorithm Design and Optimization 12019-08-22T20:33:37Z<p>Ls5: /* Prerequisites */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Algorithm Design and Optimization 1<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F<br />
<br />
=== Prerequisite ===<br />
[[Math 290]], [[Math 213]] and [[Math 215]] (or [[Math 313]]), [[Math 314]], [[Math 341]]; concurrent with [[Math 321]], [[Math 334]], [[Math 344]]<br />
<br />
=== Description ===<br />
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.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
[[Math 290]], [[Math 213]] and [[Math 215]] (or [[Math 313]]), [[Math 314]], [[Math 341]]; concurrent with [[Math 321]], [[Math 334]], [[Math 344]]<br />
<br />
=== Minimal learning outcomes ===<br />
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.<br />
<br />
# Complexity and Data<br />
#* Asymptotic Analysis<br />
#* Combinatorics<br />
#* Graphs and Trees<br />
#* Complexity (P, NP, NP Complete)<br />
# Approximation Theory<br />
#* Interpolation and Splines<br />
#* Stone-Weierstrass Theorem<br />
#* Bezier Curves<br />
#* B-Splines<br />
# Recursive Algorithms<br />
#* Difference Calculus, including Summation by Parts<br />
#* Simple linear recurrences<br />
#* General linear recurrences<br />
#* Generating functions<br />
# Linear Optimization<br />
#* Problem Formulation<br />
#* Simplex Method<br />
#* Duality<br />
#* Applications<br />
# Unconstrained Optimization<br />
#* Steepest Descent<br />
#* Newton<br />
#* Broyden<br />
#* Conjugate Gradient<br />
#* Applications<br />
# Constrained Optimization<br />
#* Equality Constrained, Lagrange Multipliers<br />
#* Inequality Constrained, KKT Condition<br />
#* Applications<br />
# Global Optimization<br />
#* Interior Point Methods<br />
#* Genetic Algorithms<br />
#* Simulated Annealing<br />
<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|320]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_320:_Algorithm_Design_and_Optimization_1&diff=3702Math 320: Algorithm Design and Optimization 12019-08-22T20:33:10Z<p>Ls5: /* Prerequisite */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Algorithm Design and Optimization 1<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F<br />
<br />
=== Prerequisite ===<br />
[[Math 290]], [[Math 213]] and [[Math 215]] (or [[Math 313]]), [[Math 314]], [[Math 341]]; concurrent with [[Math 321]], [[Math 334]], [[Math 344]]<br />
<br />
=== Description ===<br />
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.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
[[Math 290]], [[Math 313]], [[Math 314]], [[Math 341]]; concurrent with [[Math 321]], [[Math 334]], [[Math 344]]<br />
<br />
=== Minimal learning outcomes ===<br />
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.<br />
<br />
# Complexity and Data<br />
#* Asymptotic Analysis<br />
#* Combinatorics<br />
#* Graphs and Trees<br />
#* Complexity (P, NP, NP Complete)<br />
# Approximation Theory<br />
#* Interpolation and Splines<br />
#* Stone-Weierstrass Theorem<br />
#* Bezier Curves<br />
#* B-Splines<br />
# Recursive Algorithms<br />
#* Difference Calculus, including Summation by Parts<br />
#* Simple linear recurrences<br />
#* General linear recurrences<br />
#* Generating functions<br />
# Linear Optimization<br />
#* Problem Formulation<br />
#* Simplex Method<br />
#* Duality<br />
#* Applications<br />
# Unconstrained Optimization<br />
#* Steepest Descent<br />
#* Newton<br />
#* Broyden<br />
#* Conjugate Gradient<br />
#* Applications<br />
# Constrained Optimization<br />
#* Equality Constrained, Lagrange Multipliers<br />
#* Inequality Constrained, KKT Condition<br />
#* Applications<br />
# Global Optimization<br />
#* Interior Point Methods<br />
#* Genetic Algorithms<br />
#* Simulated Annealing<br />
<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|320]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_334:_Ordinary_Differential_Equations&diff=3701Math 334: Ordinary Differential Equations2019-08-22T20:29:33Z<p>Ls5: /* Prerequisite */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Ordinary Differential Equations.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F, W, Sp, Su<br />
<br />
=== Prerequisite ===<br />
[[Math 113]], [[Math 213]] or [[Math 313]] or be concurrently enrolled in [[Math 213]].<br />
<br />
=== Description ===<br />
Methods and theory of ordinary differential equations.<br />
<br />
== Desired Learning Outcomes ==<br />
This course is aimed at students majoring in mathematical and physical sciences and mathematical education. The main purpose of the course is to introduce students to the theory and methods of ordinary differential equations. The course content contributes to all the expected learning outcomes of the Mathematics BS (see [http://learningoutcomes.byu.edu]).<br />
<br />
=== Prerequisites ===<br />
Students are expected to have completed [[Math 113]], and [[Math 213]] (or [[Math 313]]) or be concurrently enrolled in [[Math 213]].<br />
<br />
=== Minimal learning outcomes ===<br />
Students should achieve mastery of the topics below. This means that they should know all relevant definitions, full statements of the major theorems, and examples of the various concepts. Further, students should be able to solve non-trivial problems related to these concepts, and prove simple theorems in analogy to proofs given by the instructor.<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
# First order equations<br />
#* Linear, separable, and exact equations<br />
#* Existence and uniqueness of solutions<br />
#* Linear versus nonlinear equations<br />
#* Autonomous equations<br />
#* Models and Applications<br />
# Higher order equations<br />
#* Theory of linear equations<br />
#* Linear independence and the Wronskian<br />
#* Homogeneous linear equations with constant coefficients<br />
#* Nonhomogeneous linear equations, method of undetermined coefficients and variation of parameters<br />
#* Mechanical and electrical vibrations<br />
#* Power series solutions<br />
#* The Laplace transform – definitions and applications <br />
# Systems of equations<br />
#* General theory<br />
#* Eigenvalue-eigenvector method for systems with constant coefficients<br />
#* Homogeneous linear systems with constant coefficients<br />
#* Fundamental matrices<br />
#* Nonhomogeneous linear systems, method of undetermined coefficients and variation of parameters<br />
#* Stability, instability, asymptotic stability, and phase plane analysis<br />
#* Models and applications<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
These are at the instructor's discretion as time allows; applications to physical problems are particularly helpful.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
This course is required for [[Math 447]], [[Math 480]], [[Math 521]], [[Math 534]], [[Math 547]], and [[Math 634]].<br />
<br />
[[Category:Courses|334]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_334:_Ordinary_Differential_Equations&diff=3700Math 334: Ordinary Differential Equations2019-08-22T20:28:42Z<p>Ls5: /* Prerequisites */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Ordinary Differential Equations.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F, W, Sp, Su<br />
<br />
=== Prerequisite ===<br />
[[Math 113]], [[Math 213]] or [[Math 313]]<br />
<br />
=== Description ===<br />
Methods and theory of ordinary differential equations.<br />
<br />
== Desired Learning Outcomes ==<br />
This course is aimed at students majoring in mathematical and physical sciences and mathematical education. The main purpose of the course is to introduce students to the theory and methods of ordinary differential equations. The course content contributes to all the expected learning outcomes of the Mathematics BS (see [http://learningoutcomes.byu.edu]).<br />
<br />
=== Prerequisites ===<br />
Students are expected to have completed [[Math 113]], and [[Math 213]] (or [[Math 313]]) or be concurrently enrolled in [[Math 213]].<br />
<br />
=== Minimal learning outcomes ===<br />
Students should achieve mastery of the topics below. This means that they should know all relevant definitions, full statements of the major theorems, and examples of the various concepts. Further, students should be able to solve non-trivial problems related to these concepts, and prove simple theorems in analogy to proofs given by the instructor.<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
# First order equations<br />
#* Linear, separable, and exact equations<br />
#* Existence and uniqueness of solutions<br />
#* Linear versus nonlinear equations<br />
#* Autonomous equations<br />
#* Models and Applications<br />
# Higher order equations<br />
#* Theory of linear equations<br />
#* Linear independence and the Wronskian<br />
#* Homogeneous linear equations with constant coefficients<br />
#* Nonhomogeneous linear equations, method of undetermined coefficients and variation of parameters<br />
#* Mechanical and electrical vibrations<br />
#* Power series solutions<br />
#* The Laplace transform – definitions and applications <br />
# Systems of equations<br />
#* General theory<br />
#* Eigenvalue-eigenvector method for systems with constant coefficients<br />
#* Homogeneous linear systems with constant coefficients<br />
#* Fundamental matrices<br />
#* Nonhomogeneous linear systems, method of undetermined coefficients and variation of parameters<br />
#* Stability, instability, asymptotic stability, and phase plane analysis<br />
#* Models and applications<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
These are at the instructor's discretion as time allows; applications to physical problems are particularly helpful.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
This course is required for [[Math 447]], [[Math 480]], [[Math 521]], [[Math 534]], [[Math 547]], and [[Math 634]].<br />
<br />
[[Category:Courses|334]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_334:_Ordinary_Differential_Equations&diff=3699Math 334: Ordinary Differential Equations2019-08-22T20:28:11Z<p>Ls5: /* Prerequisite */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Ordinary Differential Equations.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F, W, Sp, Su<br />
<br />
=== Prerequisite ===<br />
[[Math 113]], [[Math 213]] or [[Math 313]]<br />
<br />
=== Description ===<br />
Methods and theory of ordinary differential equations.<br />
<br />
== Desired Learning Outcomes ==<br />
This course is aimed at students majoring in mathematical and physical sciences and mathematical education. The main purpose of the course is to introduce students to the theory and methods of ordinary differential equations. The course content contributes to all the expected learning outcomes of the Mathematics BS (see [http://learningoutcomes.byu.edu]).<br />
<br />
=== Prerequisites ===<br />
Students are expected to have completed [[Math 113]], and [[Math 213]] or be concurrently enrolled in [[Math 213]].<br />
<br />
=== Minimal learning outcomes ===<br />
Students should achieve mastery of the topics below. This means that they should know all relevant definitions, full statements of the major theorems, and examples of the various concepts. Further, students should be able to solve non-trivial problems related to these concepts, and prove simple theorems in analogy to proofs given by the instructor.<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
# First order equations<br />
#* Linear, separable, and exact equations<br />
#* Existence and uniqueness of solutions<br />
#* Linear versus nonlinear equations<br />
#* Autonomous equations<br />
#* Models and Applications<br />
# Higher order equations<br />
#* Theory of linear equations<br />
#* Linear independence and the Wronskian<br />
#* Homogeneous linear equations with constant coefficients<br />
#* Nonhomogeneous linear equations, method of undetermined coefficients and variation of parameters<br />
#* Mechanical and electrical vibrations<br />
#* Power series solutions<br />
#* The Laplace transform – definitions and applications <br />
# Systems of equations<br />
#* General theory<br />
#* Eigenvalue-eigenvector method for systems with constant coefficients<br />
#* Homogeneous linear systems with constant coefficients<br />
#* Fundamental matrices<br />
#* Nonhomogeneous linear systems, method of undetermined coefficients and variation of parameters<br />
#* Stability, instability, asymptotic stability, and phase plane analysis<br />
#* Models and applications<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
These are at the instructor's discretion as time allows; applications to physical problems are particularly helpful.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
This course is required for [[Math 447]], [[Math 480]], [[Math 521]], [[Math 534]], [[Math 547]], and [[Math 634]].<br />
<br />
[[Category:Courses|334]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_334:_Ordinary_Differential_Equations&diff=3698Math 334: Ordinary Differential Equations2019-08-22T20:28:00Z<p>Ls5: /* Prerequisite */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Ordinary Differential Equations.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F, W, Sp, Su<br />
<br />
=== Prerequisite ===<br />
[[Math 113]], [[Math 213]] or [[Math313]]<br />
<br />
=== Description ===<br />
Methods and theory of ordinary differential equations.<br />
<br />
== Desired Learning Outcomes ==<br />
This course is aimed at students majoring in mathematical and physical sciences and mathematical education. The main purpose of the course is to introduce students to the theory and methods of ordinary differential equations. The course content contributes to all the expected learning outcomes of the Mathematics BS (see [http://learningoutcomes.byu.edu]).<br />
<br />
=== Prerequisites ===<br />
Students are expected to have completed [[Math 113]], and [[Math 213]] or be concurrently enrolled in [[Math 213]].<br />
<br />
=== Minimal learning outcomes ===<br />
Students should achieve mastery of the topics below. This means that they should know all relevant definitions, full statements of the major theorems, and examples of the various concepts. Further, students should be able to solve non-trivial problems related to these concepts, and prove simple theorems in analogy to proofs given by the instructor.<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
# First order equations<br />
#* Linear, separable, and exact equations<br />
#* Existence and uniqueness of solutions<br />
#* Linear versus nonlinear equations<br />
#* Autonomous equations<br />
#* Models and Applications<br />
# Higher order equations<br />
#* Theory of linear equations<br />
#* Linear independence and the Wronskian<br />
#* Homogeneous linear equations with constant coefficients<br />
#* Nonhomogeneous linear equations, method of undetermined coefficients and variation of parameters<br />
#* Mechanical and electrical vibrations<br />
#* Power series solutions<br />
#* The Laplace transform – definitions and applications <br />
# Systems of equations<br />
#* General theory<br />
#* Eigenvalue-eigenvector method for systems with constant coefficients<br />
#* Homogeneous linear systems with constant coefficients<br />
#* Fundamental matrices<br />
#* Nonhomogeneous linear systems, method of undetermined coefficients and variation of parameters<br />
#* Stability, instability, asymptotic stability, and phase plane analysis<br />
#* Models and applications<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
These are at the instructor's discretion as time allows; applications to physical problems are particularly helpful.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
This course is required for [[Math 447]], [[Math 480]], [[Math 521]], [[Math 534]], [[Math 547]], and [[Math 634]].<br />
<br />
[[Category:Courses|334]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_334:_Ordinary_Differential_Equations&diff=3697Math 334: Ordinary Differential Equations2019-08-22T20:27:45Z<p>Ls5: /* Prerequisite */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Ordinary Differential Equations.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F, W, Sp, Su<br />
<br />
=== Prerequisite ===<br />
[[Math 113]], [[Math 213]] (or [[Math313]])<br />
<br />
=== Description ===<br />
Methods and theory of ordinary differential equations.<br />
<br />
== Desired Learning Outcomes ==<br />
This course is aimed at students majoring in mathematical and physical sciences and mathematical education. The main purpose of the course is to introduce students to the theory and methods of ordinary differential equations. The course content contributes to all the expected learning outcomes of the Mathematics BS (see [http://learningoutcomes.byu.edu]).<br />
<br />
=== Prerequisites ===<br />
Students are expected to have completed [[Math 113]], and [[Math 213]] or be concurrently enrolled in [[Math 213]].<br />
<br />
=== Minimal learning outcomes ===<br />
Students should achieve mastery of the topics below. This means that they should know all relevant definitions, full statements of the major theorems, and examples of the various concepts. Further, students should be able to solve non-trivial problems related to these concepts, and prove simple theorems in analogy to proofs given by the instructor.<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
# First order equations<br />
#* Linear, separable, and exact equations<br />
#* Existence and uniqueness of solutions<br />
#* Linear versus nonlinear equations<br />
#* Autonomous equations<br />
#* Models and Applications<br />
# Higher order equations<br />
#* Theory of linear equations<br />
#* Linear independence and the Wronskian<br />
#* Homogeneous linear equations with constant coefficients<br />
#* Nonhomogeneous linear equations, method of undetermined coefficients and variation of parameters<br />
#* Mechanical and electrical vibrations<br />
#* Power series solutions<br />
#* The Laplace transform – definitions and applications <br />
# Systems of equations<br />
#* General theory<br />
#* Eigenvalue-eigenvector method for systems with constant coefficients<br />
#* Homogeneous linear systems with constant coefficients<br />
#* Fundamental matrices<br />
#* Nonhomogeneous linear systems, method of undetermined coefficients and variation of parameters<br />
#* Stability, instability, asymptotic stability, and phase plane analysis<br />
#* Models and applications<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
These are at the instructor's discretion as time allows; applications to physical problems are particularly helpful.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
This course is required for [[Math 447]], [[Math 480]], [[Math 521]], [[Math 534]], [[Math 547]], and [[Math 634]].<br />
<br />
[[Category:Courses|334]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_485:_Mathematical_Cryptography&diff=3696Math 485: Mathematical Cryptography2019-08-22T20:26:37Z<p>Ls5: /* Prerequisites */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Mathematical Cryptography.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F<br />
<br />
<!-- Inaccurate. Recently offered Fall 2009; scheduled to be offered Fall 2010. --><br />
<br />
=== Prerequisite ===<br />
[[Math 213]].<br />
<br />
=== Recommended ===<br />
[[Math 371]].<br />
<br />
=== Description ===<br />
A mathematical introduction to some of the high points of modern cryptography.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
This is a course in the mathematics and algorithms of modern cryptography. It complements, rather than being equivalent to, the current CS course on Computer Security (CS 465) and the current Information Technology course on Encryption and Compression (IT 531).<br />
<br />
=== Prerequisites ===<br />
<br />
The requirement for [[Math 213]] ensures both an appropriate level of mathematical maturity and a basic knowledge of linear algebra. The recommendation for [[Math 371]] encourages students to be familiar with the concepts of groups, rings, and fields.<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
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, or another appropriate computing language.<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
# Classical systems, including:<br />
#* Substitution theory<br />
#* Block ciphers<br />
#* Enigma<br />
# Elementary number theory as follows:<br />
#* Euclid's algorithm<br />
#* Modular arithmetic and the algorithm for modular exponentiation<br />
#* Chinese Remainder Theorem<br />
#* Fermat and Euler Theorems<br />
#* Primitive roots<br />
#* Legendre and Jacobi symbols<br />
#* Elementary continued fractions<br />
#* Simple discussion of finite fields<br />
# The DES and AES encryption standards<br />
# RSA and its strengths and weaknesses; attacks on RSA<br />
#* Wiener's continued fraction attack on low decryption exponent<br />
# Primality testing algorithms<br />
# Factorization techniques<br />
#* The Quadratic Sieve<br />
# Discrete logarithms<br />
#* Diffie-Hellman key exchange<br />
#* ElGamal<br />
<br />
</div><br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
If time allows, additional topics may include, but are not limited to: Elliptic curve cryptography, birthday attacks and probability, quantum cryptography (key distribution, Shor's algorithm), hash functions, digital signatures, and lattices and lattice algorithms (LLL algorithm, NTRU system, lattice attacks on RSA).<br />
<br />
=== Courses for which this course is prerequisite ===<br />
None.<br />
<br />
[[Category:Courses|485]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_485:_Mathematical_Cryptography&diff=3695Math 485: Mathematical Cryptography2019-08-22T20:25:38Z<p>Ls5: /* Prerequisite */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Mathematical Cryptography.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F<br />
<br />
<!-- Inaccurate. Recently offered Fall 2009; scheduled to be offered Fall 2010. --><br />
<br />
=== Prerequisite ===<br />
[[Math 213]].<br />
<br />
=== Recommended ===<br />
[[Math 371]].<br />
<br />
=== Description ===<br />
A mathematical introduction to some of the high points of modern cryptography.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
This is a course in the mathematics and algorithms of modern cryptography. It complements, rather than being equivalent to, the current CS course on Computer Security (CS 465) and the current Information Technology course on Encryption and Compression (IT 531).<br />
<br />
=== Prerequisites ===<br />
<br />
The requirement for [[Math 313]] ensures both an appropriate level of mathematical maturity and a basic knowledge of linear algebra. The recommendation for [[Math 371]] encourages students to be familiar with the concepts of groups, rings, and fields.<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
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, or another appropriate computing language.<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
# Classical systems, including:<br />
#* Substitution theory<br />
#* Block ciphers<br />
#* Enigma<br />
# Elementary number theory as follows:<br />
#* Euclid's algorithm<br />
#* Modular arithmetic and the algorithm for modular exponentiation<br />
#* Chinese Remainder Theorem<br />
#* Fermat and Euler Theorems<br />
#* Primitive roots<br />
#* Legendre and Jacobi symbols<br />
#* Elementary continued fractions<br />
#* Simple discussion of finite fields<br />
# The DES and AES encryption standards<br />
# RSA and its strengths and weaknesses; attacks on RSA<br />
#* Wiener's continued fraction attack on low decryption exponent<br />
# Primality testing algorithms<br />
# Factorization techniques<br />
#* The Quadratic Sieve<br />
# Discrete logarithms<br />
#* Diffie-Hellman key exchange<br />
#* ElGamal<br />
<br />
</div><br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
If time allows, additional topics may include, but are not limited to: Elliptic curve cryptography, birthday attacks and probability, quantum cryptography (key distribution, Shor's algorithm), hash functions, digital signatures, and lattices and lattice algorithms (LLL algorithm, NTRU system, lattice attacks on RSA).<br />
<br />
=== Courses for which this course is prerequisite ===<br />
None.<br />
<br />
[[Category:Courses|485]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_431:_Probability_Theory&diff=3694Math 431: Probability Theory2019-08-22T20:25:16Z<p>Ls5: /* Prerequisite */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Probability Theory.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F (odd years)<br />
<br />
=== Prerequisite ===<br />
[[Math 213]].<br />
<br />
=== Description ===<br />
Axiomatic probability theory, conditional probability, discrete / continuous random variables, expectation, conditional expectation, moments, functions of random variables, multivariate distributions, laws of large numbers, central limit theorem.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
This course is a calculus-based first course in probability. It is cross-listed with EC En 370.<br />
<br />
=== Prerequisites ===<br />
<br />
The current prerequisite is linear algebra. Because of the need to work with joint distributions of continuous random variables in Math 431, the department should consider adding multivariable calculus as a prerequisite.<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
Primarily, students should be able to do basic computation of probabilistic quantities, including those involving applications. Students should be able to recall the most common types of discrete and continuous random variables and describe and compute their properties. Students should understand the theory of probability <i>in an elementary context</i>.<br />
<br />
<div style="column-count:2;-moz-column-count:2;-webkit-column-count:2"><br />
# Basic principles of counting<br />
#* Product sets<br />
#* Disjoint unions<br />
#* Combinations<br />
#* Permutations<br />
# Axiomatic probability<br />
#* Outcomes<br />
#* Events<br />
#* Probability measures<br />
#** Additivity<br />
#** Continuity<br />
# Discrete random variables<br />
#* Probability mass function<br />
#* Cumulative distribution function<br />
#* Moments<br />
#** Expectation<br />
#*** Of a function of a random variable<br />
#** Variance<br />
#* Common types<br />
#** Bernoulli<br />
#** Binomial<br />
#** Poisson <br><br><br><br><br />
# Continuous random variables<br />
#* Probability density function<br />
#* Cumulative distribution function<br />
#* Moments<br />
#** Expectation<br />
#*** Of a function of a random variable<br />
#** Variance<br />
#* Common types<br />
#** Uniform<br />
#** Exponential<br />
#** Normal<br />
# Conditional probability<br />
#* As a probability <br />
#* Bayes' Formula<br />
#* Independence<br />
#** Events<br />
#** Random variables<br />
# Joint distributions<br />
#* Covariance<br />
#* Conditional distributions<br />
# Conditional expectation<br />
# Limit theorems<br />
#* Weak Law of Large Numbers<br />
#* Strong Law of Large Numbers<br />
#* Central Limit Theorem<br />
</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Sheldon Ross, ''A First Course in Probability (8th edition)'', Prentice Hall, 2009.<br />
<br />
=== Additional topics ===<br />
<br />
If time permits, geometric, negative binomial, hypergeometric, gamma, Weibull, Cauchy, and/or beta random variables might be studied.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
Currently, Math 431 is only a prerequisite for [[Math 435]]. Consideration should perhaps be given to making it a prerequisite for [[Math 543]].<br />
<br />
[[Category:Courses|431]]</div>Ls5