https://math.byu.edu/wiki/api.php?action=feedcontributions&user=Cpg&feedformat=atomMathWiki - User contributions [en]2020-05-28T08:22:33ZUser contributionsMediaWiki 1.26.3https://math.byu.edu/wiki/index.php?title=Math_413_Advanced_Linear_Algebra&diff=3708Math 413 Advanced Linear Algebra2019-10-24T21:33:28Z<p>Cpg: </p>
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<div>== Catalog Information ==<br />
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=== Title ===<br />
Advanced Linear Algebra<br />
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=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
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=== Offered ===<br />
F and possibly W<br />
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=== Prerequisite ===<br />
[[Math 371]] recommended, but not required.<br />
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=== Description ===<br />
Theory and advanced topics of linear algebra.<br />
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== Desired Learning Outcomes ==<br />
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<strong>Prove:</strong> Students will be able to prove central linear-algebraic results, as well as other results with similar derivations.<br />
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<strong>Distinguish:</strong> Students will be able to distinguish between true and plausibly-sounding false propositions in the language of linear algebra.<br />
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<strong>Construct:</strong> Students will be able to construct examples and counterexamples illustrating relations between different linear-algebraic concepts.<br />
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<strong>Categorize:</strong> Students will be able to categorize linear-algebraic structures according to their properties.<br />
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<strong>Calculate:</strong> Students will be able to calculate precisely and efficiently, choosing appropriate methods.<br />
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=== Prerequisites ===<br />
[[Math 371]] recommended, but not required.<br />
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=== 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 />
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=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
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[[Category:Courses|413]]</div>Cpghttps://math.byu.edu/wiki/index.php?title=Garbage_2&diff=2411Garbage 22015-05-04T21:04:33Z<p>Cpg: moved Math 320: Algorithm Design and Optimization 1 to Garbage: Wish page would go away. Delete it if you can.</p>
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<div></div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_320:_Algorithm_Design_and_Optimization_1&diff=2412Math 320: Algorithm Design and Optimization 12015-05-04T21:04:33Z<p>Cpg: moved Math 320: Algorithm Design and Optimization 1 to Garbage: Wish page would go away. Delete it if you can.</p>
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<div>#REDIRECT [[Garbage]]</div>Cpghttps://math.byu.edu/wiki/index.php?title=Garbage_2&diff=2410Garbage 22015-05-04T20:52:07Z<p>Cpg: Blanked the page</p>
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<div></div>Cpghttps://math.byu.edu/wiki/index.php?title=Garbage_2&diff=2409Garbage 22015-05-04T20:36:57Z<p>Cpg: Created page with "== Catalog Information == === Title === Algorithms Design and Optimization 1 === (Credit Hours:Lecture Hours:Lab Hours) === (3:3:0) === Offered === F === Prerequisite === [[M..."</p>
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<div>== Catalog Information ==<br />
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=== Title ===<br />
Algorithms Design and Optimization 1<br />
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=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
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=== Offered ===<br />
F<br />
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=== Prerequisite ===<br />
[[Math 290]], [[Math 313]], [[Math 314]], [[Math 341]]; concurrent with [[Math 321]], [[Math 334]], [[Math 344]]<br />
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=== 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 />
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== Desired Learning Outcomes ==<br />
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=== Prerequisites ===<br />
[[Math 290]], [[Math 313]], [[Math 314]], [[Math 341]]; concurrent with [[Math 321]], [[Math 334]], [[Math 344]]<br />
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=== 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 />
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# 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 />
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=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
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=== Additional topics ===<br />
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=== Courses for which this course is prerequisite ===<br />
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[[Category:Courses|320]]</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_439:_Modeling_with_Dynamics_and_Control_2_Lab&diff=1876Math 439: Modeling with Dynamics and Control 2 Lab2012-06-06T18:29:46Z<p>Cpg: Created page with "== Catalog Information == === Title === === (Credit Hours:Lecture Hours:Lab Hours) === === Offered === === Prerequisite === === Description === == Desired Learning Outcomes..."</p>
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Possible textbooks for this course include (but are not limited to):<br />
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[[Category:Courses|439]]</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_438:_Modeling_with_Dynamics_and_Control_2&diff=1875Math 438: Modeling with Dynamics and Control 22012-06-06T18:29:23Z<p>Cpg: Created page with "== Catalog Information == === Title === === (Credit Hours:Lecture Hours:Lab Hours) === === Offered === === Prerequisite === === Description === == Desired Learning Outcomes..."</p>
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Possible textbooks for this course include (but are not limited to):<br />
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[[Category:Courses|438]]</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_437:_Modeling_with_Dynamics_and_Control_1_Lab&diff=1874Math 437: Modeling with Dynamics and Control 1 Lab2012-06-06T18:29:08Z<p>Cpg: Created page with "== Catalog Information == === Title === === (Credit Hours:Lecture Hours:Lab Hours) === === Offered === === Prerequisite === === Description === == Desired Learning Outcomes..."</p>
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=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
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[[Category:Courses|437]]</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_436:_Modeling_with_Dynamics_and_Control_1&diff=1873Math 436: Modeling with Dynamics and Control 12012-06-06T18:28:55Z<p>Cpg: Created page with "== Catalog Information == === Title === === (Credit Hours:Lecture Hours:Lab Hours) === === Offered === === Prerequisite === === Description === == Desired Learning Outcomes..."</p>
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=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
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[[Category:Courses|436]]</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_405:_Modeling_with_Uncertainty_and_Data_2_Lab&diff=1872Math 405: Modeling with Uncertainty and Data 2 Lab2012-06-06T18:28:31Z<p>Cpg: Created page with "== Catalog Information == === Title === === (Credit Hours:Lecture Hours:Lab Hours) === === Offered === === Prerequisite === === Description === == Desired Learning Outcomes..."</p>
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Possible textbooks for this course include (but are not limited to):<br />
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[[Category:Courses|405]]</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_404:_Modeling_with_Uncertainty_and_Data_2&diff=1871Math 404: Modeling with Uncertainty and Data 22012-06-06T18:28:15Z<p>Cpg: Created page with "== Catalog Information == === Title === === (Credit Hours:Lecture Hours:Lab Hours) === === Offered === === Prerequisite === === Description === == Desired Learning Outcomes..."</p>
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Possible textbooks for this course include (but are not limited to):<br />
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[[Category:Courses|404]]</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_403:_Modeling_with_Uncertainty_and_Data_1_Lab&diff=1870Math 403: Modeling with Uncertainty and Data 1 Lab2012-06-06T18:28:00Z<p>Cpg: Created page with "== Catalog Information == === Title === === (Credit Hours:Lecture Hours:Lab Hours) === === Offered === === Prerequisite === === Description === == Desired Learning Outcomes..."</p>
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Possible textbooks for this course include (but are not limited to):<br />
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[[Category:Courses|403]]</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_402:_Modeling_with_Uncertainty_and_Data_1&diff=1869Math 402: Modeling with Uncertainty and Data 12012-06-06T18:27:45Z<p>Cpg: Created page with "== Catalog Information == === Title === === (Credit Hours:Lecture Hours:Lab Hours) === === Offered === === Prerequisite === === Description === == Desired Learning Outcomes..."</p>
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[[Category:Courses|402]]</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_347:_Mathematical_Analysis_2_Lab&diff=1868Math 347: Mathematical Analysis 2 Lab2012-06-06T18:27:20Z<p>Cpg: Created page with "== Catalog Information == === Title === === (Credit Hours:Lecture Hours:Lab Hours) === === Offered === === Prerequisite === === Description === == Desired Learning Outcomes..."</p>
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[[Category:Courses|347]]</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_346:_Mathematical_Analysis_2&diff=1867Math 346: Mathematical Analysis 22012-06-06T18:27:06Z<p>Cpg: Created page with "== Catalog Information == === Title === === (Credit Hours:Lecture Hours:Lab Hours) === === Offered === === Prerequisite === === Description === == Desired Learning Outcomes..."</p>
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[[Category:Courses|346]]</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_345:_Mathematical_Analysis_1_Lab&diff=1866Math 345: Mathematical Analysis 1 Lab2012-06-06T18:26:50Z<p>Cpg: Created page with "== Catalog Information == === Title === === (Credit Hours:Lecture Hours:Lab Hours) === === Offered === === Prerequisite === === Description === == Desired Learning Outcomes..."</p>
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[[Category:Courses|345]]</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_344:_Mathematical_Analysis_1&diff=1865Math 344: Mathematical Analysis 12012-06-06T18:26:35Z<p>Cpg: Created page with "== Catalog Information == === Title === === (Credit Hours:Lecture Hours:Lab Hours) === === Offered === === Prerequisite === === Description === == Desired Learning Outcomes..."</p>
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[[Category:Courses|344]]</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_323:_Algorithm_Design_and_Optimization_2_Lab&diff=1864Math 323: Algorithm Design and Optimization 2 Lab2012-06-06T18:26:13Z<p>Cpg: Created page with "== Catalog Information == === Title === === (Credit Hours:Lecture Hours:Lab Hours) === === Offered === === Prerequisite === === Description === == Desired Learning Outcomes..."</p>
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Possible textbooks for this course include (but are not limited to):<br />
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[[Category:Courses|323]]</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_322:_Algorithm_Design_and_Optimization_2&diff=1863Math 322: Algorithm Design and Optimization 22012-06-06T18:25:55Z<p>Cpg: Created page with "== Catalog Information == === Title === === (Credit Hours:Lecture Hours:Lab Hours) === === Offered === === Prerequisite === === Description === == Desired Learning Outcomes..."</p>
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[[Category:Courses|322]]</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_321:_Algorithm_Design_and_Optimization_1_Lab&diff=1862Math 321: Algorithm Design and Optimization 1 Lab2012-06-06T18:25:35Z<p>Cpg: Created page with "== Catalog Information == === Title === === (Credit Hours:Lecture Hours:Lab Hours) === === Offered === === Prerequisite === === Description === == Desired Learning Outcomes..."</p>
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[[Category:Courses|321]]</div>Cpghttps://math.byu.edu/wiki/index.php?title=Garbage_3&diff=1861Garbage 32012-06-06T18:25:09Z<p>Cpg: Created page with "== Catalog Information == === Title === === (Credit Hours:Lecture Hours:Lab Hours) === === Offered === === Prerequisite === === Description === == Desired Learning Outcomes..."</p>
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[[Category:Courses|320]]</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_480&diff=1860Math 4802012-06-06T18:20:57Z<p>Cpg: Blanked the page</p>
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<div>== Catalog Information ==<br />
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=== Title ===<br />
Mathematical Models.<br />
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=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
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=== Offered ===<br />
On demand<br />
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=== Prerequisite ===<br />
[[Math 334]], [[Math 410|410]].<br />
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=== Description ===<br />
Construction, solution, and interpretation of discrete and continuous models applied to problems in the physical, natural, and social sciences.<br />
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== Desired Learning Outcomes ==<br />
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=== Prerequisites ===<br />
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Possible textbooks for this course include (but are not limited to):<br />
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=== Courses for which this course is prerequisite ===</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_97&diff=1841Math 972012-05-22T19:13:01Z<p>Cpg: </p>
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<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Intermediate Algebra<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(0:2:1)<br />
<br />
=== Offered ===<br />
F, W, Sp, Su<br />
<br />
=== Prerequisite ===<br />
High school algebra.<br />
<br />
=== Description ===<br />
Elementary logic, real number system, equations and inequalities (linear, polynomial, rational, and radical expressions), graphing, function notation, inverse function, exponential functions, systems of equations, variations.<br />
<br />
== Desired Learning Outcomes ==<br />
???<br />
<br />
=== Prerequisites ===<br />
???<br />
<br />
=== Minimal learning outcomes ===<br />
???<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
???<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|097]]</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_97&diff=1840Math 972012-05-22T19:12:33Z<p>Cpg: Created page with "== Catalog Information == === Title === Intermediate Algebra === (Credit Hours:Lecture Hours:Lab Hours) === (0:2:1) === Offered === F, W, Sp, Su === Prerequisite === High sch..."</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Intermediate Algebra<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(0:2:1)<br />
<br />
=== Offered ===<br />
F, W, Sp, Su<br />
<br />
=== Prerequisite ===<br />
High school algebra.<br />
<br />
=== Description ===<br />
Elementary logic, real number system, equations and inequalities (linear, polynomial, rational, and radical expressions), graphing, function notation, inverse function, exponential functions, systems of equations, variations.<br />
<br />
== Desired Learning Outcomes ==<br />
???<br />
<br />
=== Prerequisites ===<br />
???<br />
<br />
=== Minimal learning outcomes ===<br />
???<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
???<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|97]]</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_431:_Probability_Theory&diff=1839Math 431: Probability Theory2012-05-16T15:32:59Z<p>Cpg: </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<br />
<br />
=== Prerequisite ===<br />
[[Math 313]].<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>Cpghttps://math.byu.edu/wiki/index.php?title=Math_540:_Linear_Analysis&diff=1817Math 540: Linear Analysis2011-11-14T18:34:25Z<p>Cpg: /* Textbooks */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Linear Analysis.<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Recommended ===<br />
[[Math 342]] 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>Cpghttps://math.byu.edu/wiki/index.php?title=Math_640:_Nonlinear_Analysis&diff=1798Math 640: Nonlinear Analysis2011-08-03T21:48:35Z<p>Cpg: </p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Nonlinear Analysis.<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<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>Cpghttps://math.byu.edu/wiki/index.php?title=Math_648:_Theory_of_Partial_Differential_Equations_2&diff=1796Math 648: Theory of Partial Differential Equations 22011-05-31T19:12:25Z<p>Cpg: /* Desired Learning Outcomes */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Theory of Partial Differential Equations 2.<br />
<br />
=== 3Credit Hours ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F<br />
<br />
=== Prerequisite ===<br />
[[Math 641]], [[Math 540]], recommended [[Math 640]], [[Math 647]]. Suggestion: Since the standard textbook does its own functional analysis, it's not clear that functional analysis prerequisites are appropriate.<br />
<br />
=== Description ===<br />
Advanced theory of partial differential equations. Functional-analytic techniques.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
Students need a thorough understanding of real analysis.<br />
<br />
=== Minimal learning outcomes ===<br />
Outlined below are topics that all successful Math 648 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 />
# Second-order elliptic equations<br />
#* Classification<br />
#* Weak solutions<br />
#** Lax-Milgram theorem<br />
#** Energy estimates<br />
#** Fredholm alternative<br />
#* Regularity<br />
#** Interior<br />
#** Boundary<br />
#* Maximum principles<br />
#** Weak<br />
#** Strong<br />
#** Harnack's inequality<br />
#* Eigenpairs of elliptic operators<br />
#** Symmetric<br />
#** Nonsymmetric<br />
# Linear Evolution Equations<br />
#* Second-order parabolic equations<br />
#** Weak solutions<br />
#** Regularity<br />
#** Maximum principles<br />
#* Second-order hyperbolic equations<br />
#** Weak solutions<br />
#** Regularity<br />
# Calculus of Variations<br />
#* Euler-Lagrange equation<br />
#* Coercivity<br />
#* Convexity<br />
#* Semicontinuity<br />
#* Weak Solutions<br />
#* Regularity<br />
#* Constraints<br />
#* Critical points<br />
#** Mountain pass theorem<br />
# Hamilton-Jacobi equations<br />
#* Viscosity solutions<br />
</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Lawrence C. Evans, ''Partial Differential Equations (Second Edition)'', American Mathematical Society, 2010.<br />
<br />
=== Additional topics ===<br />
If time permits, topics that could be discussed include hyperbolic systems, semigroup theory, systems of convservation laws, and nonvariational techniques for nonlinear equations.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|648]]<br />
None</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_648:_Theory_of_Partial_Differential_Equations_2&diff=1795Math 648: Theory of Partial Differential Equations 22011-05-31T19:12:08Z<p>Cpg: /* Prerequisites */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Theory of Partial Differential Equations 2.<br />
<br />
=== 3Credit Hours ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F<br />
<br />
=== Prerequisite ===<br />
[[Math 641]], [[Math 540]], recommended [[Math 640]], [[Math 647]]. Suggestion: Since the standard textbook does its own functional analysis, it's not clear that functional analysis prerequisites are appropriate.<br />
<br />
=== Description ===<br />
Advanced theory of partial differential equations. Functional-analytic techniques.<br />
<br />
== Desired Learning Outcomes ==<br />
Students should gain a familiarity with abstract methods for studying boundary value and initial boundary value problems for<br />
partial differential equations including a working familiarity with the function spaces which are most often used in these methods.<br />
<br />
=== Prerequisites ===<br />
Students need a thorough understanding of real analysis.<br />
<br />
=== Minimal learning outcomes ===<br />
Outlined below are topics that all successful Math 648 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 />
# Second-order elliptic equations<br />
#* Classification<br />
#* Weak solutions<br />
#** Lax-Milgram theorem<br />
#** Energy estimates<br />
#** Fredholm alternative<br />
#* Regularity<br />
#** Interior<br />
#** Boundary<br />
#* Maximum principles<br />
#** Weak<br />
#** Strong<br />
#** Harnack's inequality<br />
#* Eigenpairs of elliptic operators<br />
#** Symmetric<br />
#** Nonsymmetric<br />
# Linear Evolution Equations<br />
#* Second-order parabolic equations<br />
#** Weak solutions<br />
#** Regularity<br />
#** Maximum principles<br />
#* Second-order hyperbolic equations<br />
#** Weak solutions<br />
#** Regularity<br />
# Calculus of Variations<br />
#* Euler-Lagrange equation<br />
#* Coercivity<br />
#* Convexity<br />
#* Semicontinuity<br />
#* Weak Solutions<br />
#* Regularity<br />
#* Constraints<br />
#* Critical points<br />
#** Mountain pass theorem<br />
# Hamilton-Jacobi equations<br />
#* Viscosity solutions<br />
</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Lawrence C. Evans, ''Partial Differential Equations (Second Edition)'', American Mathematical Society, 2010.<br />
<br />
=== Additional topics ===<br />
If time permits, topics that could be discussed include hyperbolic systems, semigroup theory, systems of convservation laws, and nonvariational techniques for nonlinear equations.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|648]]<br />
None</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_648:_Theory_of_Partial_Differential_Equations_2&diff=1794Math 648: Theory of Partial Differential Equations 22011-05-31T19:11:28Z<p>Cpg: /* Additional topics */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Theory of Partial Differential Equations 2.<br />
<br />
=== 3Credit Hours ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F<br />
<br />
=== Prerequisite ===<br />
[[Math 641]], [[Math 540]], recommended [[Math 640]], [[Math 647]]. Suggestion: Since the standard textbook does its own functional analysis, it's not clear that functional analysis prerequisites are appropriate.<br />
<br />
=== Description ===<br />
Advanced theory of partial differential equations. Functional-analytic techniques.<br />
<br />
== Desired Learning Outcomes ==<br />
Students should gain a familiarity with abstract methods for studying boundary value and initial boundary value problems for<br />
partial differential equations including a working familiarity with the function spaces which are most often used in these methods.<br />
<br />
=== Prerequisites ===<br />
A thorough knowledge of all the principle theorems of the Lebesgue integral is essential, especially the Riesz representation theorems for positive linear functionals and for the dual spaces for the ''L''<sup>''p''</sup> spaces and the space ''C''<sub>0</sub>. Understanding of the Radon Nikodym theorem is also essential. In addition, knowledge of the basic theorems of functional analysis is essential. The classical theory of partial differential equations is helpful but not essential.<br />
<br />
=== Minimal learning outcomes ===<br />
Outlined below are topics that all successful Math 648 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 />
# Second-order elliptic equations<br />
#* Classification<br />
#* Weak solutions<br />
#** Lax-Milgram theorem<br />
#** Energy estimates<br />
#** Fredholm alternative<br />
#* Regularity<br />
#** Interior<br />
#** Boundary<br />
#* Maximum principles<br />
#** Weak<br />
#** Strong<br />
#** Harnack's inequality<br />
#* Eigenpairs of elliptic operators<br />
#** Symmetric<br />
#** Nonsymmetric<br />
# Linear Evolution Equations<br />
#* Second-order parabolic equations<br />
#** Weak solutions<br />
#** Regularity<br />
#** Maximum principles<br />
#* Second-order hyperbolic equations<br />
#** Weak solutions<br />
#** Regularity<br />
# Calculus of Variations<br />
#* Euler-Lagrange equation<br />
#* Coercivity<br />
#* Convexity<br />
#* Semicontinuity<br />
#* Weak Solutions<br />
#* Regularity<br />
#* Constraints<br />
#* Critical points<br />
#** Mountain pass theorem<br />
# Hamilton-Jacobi equations<br />
#* Viscosity solutions<br />
</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Lawrence C. Evans, ''Partial Differential Equations (Second Edition)'', American Mathematical Society, 2010.<br />
<br />
=== Additional topics ===<br />
If time permits, topics that could be discussed include hyperbolic systems, semigroup theory, systems of convservation laws, and nonvariational techniques for nonlinear equations.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|648]]<br />
None</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_648:_Theory_of_Partial_Differential_Equations_2&diff=1793Math 648: Theory of Partial Differential Equations 22011-05-31T19:09:17Z<p>Cpg: /* Minimal learning outcomes */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Theory of Partial Differential Equations 2.<br />
<br />
=== 3Credit Hours ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F<br />
<br />
=== Prerequisite ===<br />
[[Math 641]], [[Math 540]], recommended [[Math 640]], [[Math 647]]. Suggestion: Since the standard textbook does its own functional analysis, it's not clear that functional analysis prerequisites are appropriate.<br />
<br />
=== Description ===<br />
Advanced theory of partial differential equations. Functional-analytic techniques.<br />
<br />
== Desired Learning Outcomes ==<br />
Students should gain a familiarity with abstract methods for studying boundary value and initial boundary value problems for<br />
partial differential equations including a working familiarity with the function spaces which are most often used in these methods.<br />
<br />
=== Prerequisites ===<br />
A thorough knowledge of all the principle theorems of the Lebesgue integral is essential, especially the Riesz representation theorems for positive linear functionals and for the dual spaces for the ''L''<sup>''p''</sup> spaces and the space ''C''<sub>0</sub>. Understanding of the Radon Nikodym theorem is also essential. In addition, knowledge of the basic theorems of functional analysis is essential. The classical theory of partial differential equations is helpful but not essential.<br />
<br />
=== Minimal learning outcomes ===<br />
Outlined below are topics that all successful Math 648 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 />
# Second-order elliptic equations<br />
#* Classification<br />
#* Weak solutions<br />
#** Lax-Milgram theorem<br />
#** Energy estimates<br />
#** Fredholm alternative<br />
#* Regularity<br />
#** Interior<br />
#** Boundary<br />
#* Maximum principles<br />
#** Weak<br />
#** Strong<br />
#** Harnack's inequality<br />
#* Eigenpairs of elliptic operators<br />
#** Symmetric<br />
#** Nonsymmetric<br />
# Linear Evolution Equations<br />
#* Second-order parabolic equations<br />
#** Weak solutions<br />
#** Regularity<br />
#** Maximum principles<br />
#* Second-order hyperbolic equations<br />
#** Weak solutions<br />
#** Regularity<br />
# Calculus of Variations<br />
#* Euler-Lagrange equation<br />
#* Coercivity<br />
#* Convexity<br />
#* Semicontinuity<br />
#* Weak Solutions<br />
#* Regularity<br />
#* Constraints<br />
#* Critical points<br />
#** Mountain pass theorem<br />
# Hamilton-Jacobi equations<br />
#* Viscosity solutions<br />
</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Lawrence C. Evans, ''Partial Differential Equations (Second Edition)'', American Mathematical Society, 2010.<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|648]]<br />
None</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_648:_Theory_of_Partial_Differential_Equations_2&diff=1792Math 648: Theory of Partial Differential Equations 22011-05-31T19:02:33Z<p>Cpg: /* Additional topics */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Theory of Partial Differential Equations 2.<br />
<br />
=== 3Credit Hours ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F<br />
<br />
=== Prerequisite ===<br />
[[Math 641]], [[Math 540]], recommended [[Math 640]], [[Math 647]]. Suggestion: Since the standard textbook does its own functional analysis, it's not clear that functional analysis prerequisites are appropriate.<br />
<br />
=== Description ===<br />
Advanced theory of partial differential equations. Functional-analytic techniques.<br />
<br />
== Desired Learning Outcomes ==<br />
Students should gain a familiarity with abstract methods for studying boundary value and initial boundary value problems for<br />
partial differential equations including a working familiarity with the function spaces which are most often used in these methods.<br />
<br />
=== Prerequisites ===<br />
A thorough knowledge of all the principle theorems of the Lebesgue integral is essential, especially the Riesz representation theorems for positive linear functionals and for the dual spaces for the ''L''<sup>''p''</sup> spaces and the space ''C''<sub>0</sub>. Understanding of the Radon Nikodym theorem is also essential. In addition, knowledge of the basic theorems of functional analysis is essential. The classical theory of partial differential equations is helpful but not essential.<br />
<br />
=== Minimal learning outcomes ===<br />
#The Bochner Integral<br />
#*The Pettis theorem<br />
#*The spaces ''L''<sup>''p''</sup>(&Omega;; ''X'')<br />
#*Vector measures and Radon Nikodym property in Banach space<br />
#*Riesz representation theorem for the duals of ''L''<sup>''p''</sup>(&Omega;;''X'')<br />
#*Embedding results of Lions and Simon<br />
#Surjectivity of nonlinear set valued operators.<br />
#Lion's method of elliptic regularization and evolution equations of mixed type.<br />
#Weak Derivatives<br />
#*Morrey's inequality and Rademacher's theorem<br />
#*Area formula<br />
#*Integration on manifolds<br />
#Sobolev spaces<br />
#*Embedding theorems for ''W''<sup>''m'', ''p''</sup>('''R'''<sup>''n''</sup>)<br />
#*Extension theorems for Lipschitz domains<br />
#*General embedding theorems<br />
#Korn's Inequality on bounded Lipschitz domains<br />
#Elliptic regularity and Nirenberg differences<br />
#The trace spaces of Lions<br />
#*Traces of Sobolev spaces and fractional order spaces<br />
#*The half space<br />
#*A right inverse for the trace for a half space<br />
#*Intrinsic norms<br />
#*Fractional order Sobolev spaces<br />
#*Reflexivity of fractional order Sobolev spaces<br />
#Sobolev spaces on manifolds<br />
#*Basic definitions<br />
#*The trace on the boundary of an open set<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
</div><br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Lawrence C. Evans, ''Partial Differential Equations (Second Edition)'', American Mathematical Society, 2010.<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|648]]<br />
None</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_648:_Theory_of_Partial_Differential_Equations_2&diff=1791Math 648: Theory of Partial Differential Equations 22011-05-31T19:01:53Z<p>Cpg: /* Description */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Theory of Partial Differential Equations 2.<br />
<br />
=== 3Credit Hours ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F<br />
<br />
=== Prerequisite ===<br />
[[Math 641]], [[Math 540]], recommended [[Math 640]], [[Math 647]]. Suggestion: Since the standard textbook does its own functional analysis, it's not clear that functional analysis prerequisites are appropriate.<br />
<br />
=== Description ===<br />
Advanced theory of partial differential equations. Functional-analytic techniques.<br />
<br />
== Desired Learning Outcomes ==<br />
Students should gain a familiarity with abstract methods for studying boundary value and initial boundary value problems for<br />
partial differential equations including a working familiarity with the function spaces which are most often used in these methods.<br />
<br />
=== Prerequisites ===<br />
A thorough knowledge of all the principle theorems of the Lebesgue integral is essential, especially the Riesz representation theorems for positive linear functionals and for the dual spaces for the ''L''<sup>''p''</sup> spaces and the space ''C''<sub>0</sub>. Understanding of the Radon Nikodym theorem is also essential. In addition, knowledge of the basic theorems of functional analysis is essential. The classical theory of partial differential equations is helpful but not essential.<br />
<br />
=== Minimal learning outcomes ===<br />
#The Bochner Integral<br />
#*The Pettis theorem<br />
#*The spaces ''L''<sup>''p''</sup>(&Omega;; ''X'')<br />
#*Vector measures and Radon Nikodym property in Banach space<br />
#*Riesz representation theorem for the duals of ''L''<sup>''p''</sup>(&Omega;;''X'')<br />
#*Embedding results of Lions and Simon<br />
#Surjectivity of nonlinear set valued operators.<br />
#Lion's method of elliptic regularization and evolution equations of mixed type.<br />
#Weak Derivatives<br />
#*Morrey's inequality and Rademacher's theorem<br />
#*Area formula<br />
#*Integration on manifolds<br />
#Sobolev spaces<br />
#*Embedding theorems for ''W''<sup>''m'', ''p''</sup>('''R'''<sup>''n''</sup>)<br />
#*Extension theorems for Lipschitz domains<br />
#*General embedding theorems<br />
#Korn's Inequality on bounded Lipschitz domains<br />
#Elliptic regularity and Nirenberg differences<br />
#The trace spaces of Lions<br />
#*Traces of Sobolev spaces and fractional order spaces<br />
#*The half space<br />
#*A right inverse for the trace for a half space<br />
#*Intrinsic norms<br />
#*Fractional order Sobolev spaces<br />
#*Reflexivity of fractional order Sobolev spaces<br />
#Sobolev spaces on manifolds<br />
#*Basic definitions<br />
#*The trace on the boundary of an open set<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
</div><br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Lawrence C. Evans, ''Partial Differential Equations (Second Edition)'', American Mathematical Society, 2010.<br />
<br />
=== Additional topics ===<br />
The above might be all there is time for, but if there is time for more, it would be nice to consider Mihlin's theorem and the $L^{p}$ theory of elliptic regularity. Other topics could include methods of interpolation in Banach space.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|648]]<br />
None</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_648:_Theory_of_Partial_Differential_Equations_2&diff=1790Math 648: Theory of Partial Differential Equations 22011-05-31T19:01:03Z<p>Cpg: /* Prerequisite */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Theory of Partial Differential Equations 2.<br />
<br />
=== 3Credit Hours ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F<br />
<br />
=== Prerequisite ===<br />
[[Math 641]], [[Math 540]], recommended [[Math 640]], [[Math 647]]. Suggestion: Since the standard textbook does its own functional analysis, it's not clear that functional analysis prerequisites are appropriate.<br />
<br />
=== Description ===<br />
This course develops abstract methods for studying partial differential equations and inclusions.<br />
<br />
== Desired Learning Outcomes ==<br />
Students should gain a familiarity with abstract methods for studying boundary value and initial boundary value problems for<br />
partial differential equations including a working familiarity with the function spaces which are most often used in these methods.<br />
<br />
=== Prerequisites ===<br />
A thorough knowledge of all the principle theorems of the Lebesgue integral is essential, especially the Riesz representation theorems for positive linear functionals and for the dual spaces for the ''L''<sup>''p''</sup> spaces and the space ''C''<sub>0</sub>. Understanding of the Radon Nikodym theorem is also essential. In addition, knowledge of the basic theorems of functional analysis is essential. The classical theory of partial differential equations is helpful but not essential.<br />
<br />
=== Minimal learning outcomes ===<br />
#The Bochner Integral<br />
#*The Pettis theorem<br />
#*The spaces ''L''<sup>''p''</sup>(&Omega;; ''X'')<br />
#*Vector measures and Radon Nikodym property in Banach space<br />
#*Riesz representation theorem for the duals of ''L''<sup>''p''</sup>(&Omega;;''X'')<br />
#*Embedding results of Lions and Simon<br />
#Surjectivity of nonlinear set valued operators.<br />
#Lion's method of elliptic regularization and evolution equations of mixed type.<br />
#Weak Derivatives<br />
#*Morrey's inequality and Rademacher's theorem<br />
#*Area formula<br />
#*Integration on manifolds<br />
#Sobolev spaces<br />
#*Embedding theorems for ''W''<sup>''m'', ''p''</sup>('''R'''<sup>''n''</sup>)<br />
#*Extension theorems for Lipschitz domains<br />
#*General embedding theorems<br />
#Korn's Inequality on bounded Lipschitz domains<br />
#Elliptic regularity and Nirenberg differences<br />
#The trace spaces of Lions<br />
#*Traces of Sobolev spaces and fractional order spaces<br />
#*The half space<br />
#*A right inverse for the trace for a half space<br />
#*Intrinsic norms<br />
#*Fractional order Sobolev spaces<br />
#*Reflexivity of fractional order Sobolev spaces<br />
#Sobolev spaces on manifolds<br />
#*Basic definitions<br />
#*The trace on the boundary of an open set<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
</div><br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Lawrence C. Evans, ''Partial Differential Equations (Second Edition)'', American Mathematical Society, 2010.<br />
<br />
=== Additional topics ===<br />
The above might be all there is time for, but if there is time for more, it would be nice to consider Mihlin's theorem and the $L^{p}$ theory of elliptic regularity. Other topics could include methods of interpolation in Banach space.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|648]]<br />
None</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_647:_Theory_of_Partial_Differential_Equations_1&diff=1789Math 647: Theory of Partial Differential Equations 12011-05-31T18:59:39Z<p>Cpg: /* Minimal learning outcomes */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Theory of Partial Differential Equations 1.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Prerequisite ===<br />
[[Math 541]], [[Math 547|547]]. It is proposed that [[Math 547]] be dropped as a prerequisite, as these courses have always operated independently of each other.<br />
<br />
=== Description ===<br />
Proposed: Classical theory of canonical linear PDEs. Introduction to Sobolev spaces.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
Students should understand analysis at the first-year graduate level.<br />
<br />
=== Minimal learning outcomes ===<br />
Outlined below are topics that all successful Math 647 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 />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
# Classical theory for canonical linear PDEs<br />
#* Transport equation<br />
#* Laplace's equation<br />
#** Fundamental solution<br />
#** Mean-value and maximum principles<br />
#** Energy methods<br />
#* Heat equation<br />
#** Fundamental solution<br />
#** Mean-value and maximum principles<br />
#** Energy methods<br />
#* Wave equation<br />
#** Spherical means<br />
#** Energy methods<br />
# Method of characteristics<br />
# Sobolev spaces<br />
#* Traces<br />
#* Sobolev inequalities<br />
#* Compactness<br />
</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Lawrence C. Evans, ''Partial Differential Equations (Second Edition)'', American Mathematical Society, 2010.<br />
<br />
=== Additional topics ===<br />
<br />
If time permits, Hamilton-Jacobi equations and/or conservation laws could be introduced.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
[[Math 648]]<br />
<br />
[[Category:Courses|647]]</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_647:_Theory_of_Partial_Differential_Equations_1&diff=1788Math 647: Theory of Partial Differential Equations 12011-05-31T18:59:10Z<p>Cpg: /* Description */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Theory of Partial Differential Equations 1.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Prerequisite ===<br />
[[Math 541]], [[Math 547|547]]. It is proposed that [[Math 547]] be dropped as a prerequisite, as these courses have always operated independently of each other.<br />
<br />
=== Description ===<br />
Proposed: Classical theory of canonical linear PDEs. Introduction to Sobolev spaces.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
Students should understand analysis at the first-year graduate level.<br />
<br />
=== Minimal learning outcomes ===<br />
Outlined below are topics that all successful Math 647 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 />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
# Foundational theory for canonical linear PDEs<br />
#* Transport equation<br />
#* Laplace's equation<br />
#** Fundamental solution<br />
#** Mean-value and maximum principles<br />
#** Energy methods<br />
#* Heat equation<br />
#** Fundamental solution<br />
#** Mean-value and maximum principles<br />
#** Energy methods<br />
#* Wave equation<br />
#** Spherical means<br />
#** Energy methods<br />
# Method of characteristics<br />
# Sobolev spaces<br />
#* Traces<br />
#* Sobolev inequalities<br />
#* Compactness<br />
</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Lawrence C. Evans, ''Partial Differential Equations (Second Edition)'', American Mathematical Society, 2010.<br />
<br />
=== Additional topics ===<br />
<br />
If time permits, Hamilton-Jacobi equations and/or conservation laws could be introduced.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
[[Math 648]]<br />
<br />
[[Category:Courses|647]]</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_647:_Theory_of_Partial_Differential_Equations_1&diff=1787Math 647: Theory of Partial Differential Equations 12011-05-31T18:57:57Z<p>Cpg: /* Courses for which this course is prerequisite */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Theory of Partial Differential Equations 1.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Prerequisite ===<br />
[[Math 541]], [[Math 547|547]]. It is proposed that [[Math 547]] be dropped as a prerequisite, as these courses have always operated independently of each other.<br />
<br />
=== Description ===<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
Students should understand analysis at the first-year graduate level.<br />
<br />
=== Minimal learning outcomes ===<br />
Outlined below are topics that all successful Math 647 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 />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
# Foundational theory for canonical linear PDEs<br />
#* Transport equation<br />
#* Laplace's equation<br />
#** Fundamental solution<br />
#** Mean-value and maximum principles<br />
#** Energy methods<br />
#* Heat equation<br />
#** Fundamental solution<br />
#** Mean-value and maximum principles<br />
#** Energy methods<br />
#* Wave equation<br />
#** Spherical means<br />
#** Energy methods<br />
# Method of characteristics<br />
# Sobolev spaces<br />
#* Traces<br />
#* Sobolev inequalities<br />
#* Compactness<br />
</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Lawrence C. Evans, ''Partial Differential Equations (Second Edition)'', American Mathematical Society, 2010.<br />
<br />
=== Additional topics ===<br />
<br />
If time permits, Hamilton-Jacobi equations and/or conservation laws could be introduced.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
[[Math 648]]<br />
<br />
[[Category:Courses|647]]</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_647:_Theory_of_Partial_Differential_Equations_1&diff=1786Math 647: Theory of Partial Differential Equations 12011-05-31T18:57:41Z<p>Cpg: /* Prerequisites */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Theory of Partial Differential Equations 1.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Prerequisite ===<br />
[[Math 541]], [[Math 547|547]]. It is proposed that [[Math 547]] be dropped as a prerequisite, as these courses have always operated independently of each other.<br />
<br />
=== Description ===<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
Students should understand analysis at the first-year graduate level.<br />
<br />
=== Minimal learning outcomes ===<br />
Outlined below are topics that all successful Math 647 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 />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
# Foundational theory for canonical linear PDEs<br />
#* Transport equation<br />
#* Laplace's equation<br />
#** Fundamental solution<br />
#** Mean-value and maximum principles<br />
#** Energy methods<br />
#* Heat equation<br />
#** Fundamental solution<br />
#** Mean-value and maximum principles<br />
#** Energy methods<br />
#* Wave equation<br />
#** Spherical means<br />
#** Energy methods<br />
# Method of characteristics<br />
# Sobolev spaces<br />
#* Traces<br />
#* Sobolev inequalities<br />
#* Compactness<br />
</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Lawrence C. Evans, ''Partial Differential Equations (Second Edition)'', American Mathematical Society, 2010.<br />
<br />
=== Additional topics ===<br />
<br />
If time permits, Hamilton-Jacobi equations and/or conservation laws could be introduced.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|647]]</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_647:_Theory_of_Partial_Differential_Equations_1&diff=1785Math 647: Theory of Partial Differential Equations 12011-05-31T18:56:58Z<p>Cpg: /* Minimal learning outcomes */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Theory of Partial Differential Equations 1.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Prerequisite ===<br />
[[Math 541]], [[Math 547|547]]. It is proposed that [[Math 547]] be dropped as a prerequisite, as these courses have always operated independently of each other.<br />
<br />
=== Description ===<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
<br />
=== Minimal learning outcomes ===<br />
Outlined below are topics that all successful Math 647 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 />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
# Foundational theory for canonical linear PDEs<br />
#* Transport equation<br />
#* Laplace's equation<br />
#** Fundamental solution<br />
#** Mean-value and maximum principles<br />
#** Energy methods<br />
#* Heat equation<br />
#** Fundamental solution<br />
#** Mean-value and maximum principles<br />
#** Energy methods<br />
#* Wave equation<br />
#** Spherical means<br />
#** Energy methods<br />
# Method of characteristics<br />
# Sobolev spaces<br />
#* Traces<br />
#* Sobolev inequalities<br />
#* Compactness<br />
</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Lawrence C. Evans, ''Partial Differential Equations (Second Edition)'', American Mathematical Society, 2010.<br />
<br />
=== Additional topics ===<br />
<br />
If time permits, Hamilton-Jacobi equations and/or conservation laws could be introduced.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|647]]</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_647:_Theory_of_Partial_Differential_Equations_1&diff=1784Math 647: Theory of Partial Differential Equations 12011-05-31T18:56:02Z<p>Cpg: /* Additional topics */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Theory of Partial Differential Equations 1.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Prerequisite ===<br />
[[Math 541]], [[Math 547|547]]. It is proposed that [[Math 547]] be dropped as a prerequisite, as these courses have always operated independently of each other.<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 />
# Foundational theory for canonical linear PDEs<br />
#* Transport equation<br />
#* Laplace's equation<br />
#** Fundamental solution<br />
#** Mean-value and maximum principles<br />
#** Energy methods<br />
#* Heat equation<br />
#** Fundamental solution<br />
#** Mean-value and maximum principles<br />
#** Energy methods<br />
#* Wave equation<br />
#** Spherical means<br />
#** Energy methods<br />
# Method of characteristics<br />
# Sobolev spaces<br />
#* Traces<br />
#* Sobolev inequalities<br />
#* Compactness<br />
</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Lawrence C. Evans, ''Partial Differential Equations (Second Edition)'', American Mathematical Society, 2010.<br />
<br />
=== Additional topics ===<br />
<br />
If time permits, Hamilton-Jacobi equations and/or conservation laws could be introduced.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|647]]</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_647:_Theory_of_Partial_Differential_Equations_1&diff=1783Math 647: Theory of Partial Differential Equations 12011-05-31T18:54:59Z<p>Cpg: /* Minimal learning outcomes */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Theory of Partial Differential Equations 1.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Prerequisite ===<br />
[[Math 541]], [[Math 547|547]]. It is proposed that [[Math 547]] be dropped as a prerequisite, as these courses have always operated independently of each other.<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 />
# Foundational theory for canonical linear PDEs<br />
#* Transport equation<br />
#* Laplace's equation<br />
#** Fundamental solution<br />
#** Mean-value and maximum principles<br />
#** Energy methods<br />
#* Heat equation<br />
#** Fundamental solution<br />
#** Mean-value and maximum principles<br />
#** Energy methods<br />
#* Wave equation<br />
#** Spherical means<br />
#** Energy methods<br />
# Method of characteristics<br />
# Sobolev spaces<br />
#* Traces<br />
#* Sobolev inequalities<br />
#* Compactness<br />
</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Lawrence C. Evans, ''Partial Differential Equations (Second Edition)'', American Mathematical Society, 2010.<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|647]]</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_647:_Theory_of_Partial_Differential_Equations_1&diff=1782Math 647: Theory of Partial Differential Equations 12011-05-31T18:52:34Z<p>Cpg: /* Minimal learning outcomes */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Theory of Partial Differential Equations 1.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Prerequisite ===<br />
[[Math 541]], [[Math 547|547]]. It is proposed that [[Math 547]] be dropped as a prerequisite, as these courses have always operated independently of each other.<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 />
# Foundational theory for canonical linear PDEs<br />
#* Transport equation<br />
#* Laplace's equation<br />
#** Fundamental solution<br />
#** Mean-value and maximum principles<br />
#** Energy methods<br />
#* Heat equation<br />
#** Fundamental solution<br />
#** Mean-value and maximum principles<br />
#** Energy methods<br />
#* Wave equation<br />
#** Spherical means<br />
#** Energy methods<br />
</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Lawrence C. Evans, ''Partial Differential Equations (Second Edition)'', American Mathematical Society, 2010.<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|647]]</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_647:_Theory_of_Partial_Differential_Equations_1&diff=1781Math 647: Theory of Partial Differential Equations 12011-05-31T18:49:08Z<p>Cpg: /* Minimal learning outcomes */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Theory of Partial Differential Equations 1.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Prerequisite ===<br />
[[Math 541]], [[Math 547|547]]. It is proposed that [[Math 547]] be dropped as a prerequisite, as these courses have always operated independently of each other.<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 />
# Foundational theory for canonical linear PDEs<br />
#* Transport equation<br />
#* Laplace's equation<br />
#* Heat equation<br />
#* Wave equation<br />
</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Lawrence C. Evans, ''Partial Differential Equations (Second Edition)'', American Mathematical Society, 2010.<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|647]]</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_648:_Theory_of_Partial_Differential_Equations_2&diff=1780Math 648: Theory of Partial Differential Equations 22011-05-31T18:24:53Z<p>Cpg: /* Textbooks */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Theory of Partial Differential Equations 2.<br />
<br />
=== 3Credit Hours ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F<br />
<br />
=== Prerequisite ===<br />
[[Math 641]], [[Math 540]], recommended [[Math 640]], [[Math 647]]<br />
<br />
=== Description ===<br />
This course develops abstract methods for studying partial differential equations and inclusions.<br />
<br />
== Desired Learning Outcomes ==<br />
Students should gain a familiarity with abstract methods for studying boundary value and initial boundary value problems for<br />
partial differential equations including a working familiarity with the function spaces which are most often used in these methods.<br />
<br />
=== Prerequisites ===<br />
A thorough knowledge of all the principle theorems of the Lebesgue integral is essential, especially the Riesz representation theorems for positive linear functionals and for the dual spaces for the ''L''<sup>''p''</sup> spaces and the space ''C''<sub>0</sub>. Understanding of the Radon Nikodym theorem is also essential. In addition, knowledge of the basic theorems of functional analysis is essential. The classical theory of partial differential equations is helpful but not essential.<br />
<br />
=== Minimal learning outcomes ===<br />
#The Bochner Integral<br />
#*The Pettis theorem<br />
#*The spaces ''L''<sup>''p''</sup>(&Omega;; ''X'')<br />
#*Vector measures and Radon Nikodym property in Banach space<br />
#*Riesz representation theorem for the duals of ''L''<sup>''p''</sup>(&Omega;;''X'')<br />
#*Embedding results of Lions and Simon<br />
#Surjectivity of nonlinear set valued operators.<br />
#Lion's method of elliptic regularization and evolution equations of mixed type.<br />
#Weak Derivatives<br />
#*Morrey's inequality and Rademacher's theorem<br />
#*Area formula<br />
#*Integration on manifolds<br />
#Sobolev spaces<br />
#*Embedding theorems for ''W''<sup>''m'', ''p''</sup>('''R'''<sup>''n''</sup>)<br />
#*Extension theorems for Lipschitz domains<br />
#*General embedding theorems<br />
#Korn's Inequality on bounded Lipschitz domains<br />
#Elliptic regularity and Nirenberg differences<br />
#The trace spaces of Lions<br />
#*Traces of Sobolev spaces and fractional order spaces<br />
#*The half space<br />
#*A right inverse for the trace for a half space<br />
#*Intrinsic norms<br />
#*Fractional order Sobolev spaces<br />
#*Reflexivity of fractional order Sobolev spaces<br />
#Sobolev spaces on manifolds<br />
#*Basic definitions<br />
#*The trace on the boundary of an open set<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
</div><br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Lawrence C. Evans, ''Partial Differential Equations (Second Edition)'', American Mathematical Society, 2010.<br />
<br />
=== Additional topics ===<br />
The above might be all there is time for, but if there is time for more, it would be nice to consider Mihlin's theorem and the $L^{p}$ theory of elliptic regularity. Other topics could include methods of interpolation in Banach space.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|648]]<br />
None</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_647:_Theory_of_Partial_Differential_Equations_1&diff=1779Math 647: Theory of Partial Differential Equations 12011-05-31T18:24:35Z<p>Cpg: /* Textbooks */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Theory of Partial Differential Equations 1.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Prerequisite ===<br />
[[Math 541]], [[Math 547|547]]. It is proposed that [[Math 547]] be dropped as a prerequisite, as these courses have always operated independently of each other.<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 />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Lawrence C. Evans, ''Partial Differential Equations (Second Edition)'', American Mathematical Society, 2010.<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|647]]</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_647:_Theory_of_Partial_Differential_Equations_1&diff=1778Math 647: Theory of Partial Differential Equations 12011-05-31T16:55:59Z<p>Cpg: /* Prerequisite */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Theory of Partial Differential Equations 1.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Prerequisite ===<br />
[[Math 541]], [[Math 547|547]]. It is proposed that [[Math 547]] be dropped as a prerequisite, as these courses have always operated independently of each other.<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 />
=== 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|647]]</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_647:_Theory_of_Partial_Differential_Equations_1&diff=1777Math 647: Theory of Partial Differential Equations 12011-05-31T16:55:03Z<p>Cpg: /* Chris Grant's Proposed Core Topics for Math 647/648 */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Theory of Partial Differential Equations 1.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Prerequisite ===<br />
[[Math 541]], [[Math 547|547]].<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 />
=== 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|647]]</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_548:_Partial_Differential_Equations_2&diff=1776Math 548: Partial Differential Equations 22011-05-31T16:54:34Z<p>Cpg: </p>
<hr />
<div>It is proposed that this course be dropped. Core topics from applied PDE will be treated in [[Math 547]], and additional applied PDE topics can be treated in [[Math 513R]] and other courses.<br />
<br />
<br />
== Catalog Information ==<br />
<br />
=== Title ===<br />
Partial Differential Equations 2.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Prerequisite ===<br />
[[Math 547]].<br />
<br />
=== Recommended(?) ===<br />
[[Math 341]], [[Math 342|342]]; or equivalents.<br />
<br />
=== Description ===<br />
Tools for PDEs and special topics: spherical means, method of descent, subharmonic functions, Hamilton-Jacobi equations, Riemann invariants, conservation laws for linear and non-linear waves.<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|548]]</div>Cpghttps://math.byu.edu/wiki/index.php?title=Math_547:_Partial_Differential_Equations_1&diff=1775Math 547: Partial Differential Equations 12011-05-31T16:52:23Z<p>Cpg: /* Courses for which this course is prerequisite */</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 />
=== Prerequisite ===<br />
[[Math 334]], [[Math 342|342]]; or equivalents.<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>Cpg