https://math.byu.edu/wiki/api.php?action=feedcontributions&user=Ls5&feedformat=atomMathWiki - User contributions [en]2019-10-22T21:55:43ZUser contributionsMediaWiki 1.26.3https://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>Ls5https://math.byu.edu/wiki/index.php?title=Math_371:_Abstract_Algebra_1.&diff=3693Math 371: Abstract Algebra 1.2019-08-22T20:24:46Z<p>Ls5: /* Prerequisite */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Abstract Algebra 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 290]], [[Math 213 Elementary Linear Algebra|213]].<br />
<br />
=== Description ===<br />
Groups, group homomorphisms, rings, ideals, and polynomials.<br />
<br />
== Desired Learning Outcomes ==<br />
This course is aimed at undergraduate mathematics and mathematics education majors. It is a first course in abstract algebra. 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. Thus it is a core requirement for all mathematics majors. Outside of mathematics, algebra also has applications in cryptography, coding theory, quantum chemistry, and physics.<br />
<br />
=== Prerequisites ===<br />
The chief prerequisite for this course is [[Math 290]]. In [[Math 290]], students should learn basic logic, basic set theory, the division algorithm, Euclidean algorithm, and unique factorization theorem for integers, equivalence relations, functions, and mathematical induction. As these topics are of high importance in Math 371, it might be prudent for the instructor to review them at the beginning of the semester.<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 />
# Group Theory<br />
#* Basic Definitions<br />
#* Examples of groups<br />
#* Subgroups<br />
#* Lagrange’s Theorem<br />
#* Homomorphisms<br />
#* Normal Subgroups<br />
#* Quotient Groups<br />
#* Isomorphism Theorems<br />
#* Cauchy’s Theorem<br />
#* Direct Products<br />
#* The Symmetric Group<br />
#* Even and odd Permutations<br />
#* Cycle Decompositions<br />
# Ring Theory<br />
#* Basic Definitions<br />
#* Examples of rings (both commutative and noncommutative)<br />
#* Ideals<br />
#* Ring homomorphisms<br />
#* Quotient rings<br />
#* Prime and maximal ideals<br />
#* Polynomial rings<br />
#* Factorization in polynomial rings<br />
#* Field of fractions of a domain<br />
<br />
<br />
<br />
<br />
<br />
<br />
</div><br />
<br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Hungerford: Abstract Algebra: An Introduction<br />
* Herstein: Abstract Algebra<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): Sylow’s theorems, the fundamental theorem of finite abelian groups, the simplicity of ''A<sub>n</sub>'', Burnside’s Theorem, Polya counting, isometries of '''R'''<sup>3</sup> and regular solids, straight-edge and compass constructions, the 2- and 4-squares theorems, the RSA algorithm, wallpaper groups, coding theory, or latin squares. Instructors are free to use new approaches to the teaching of the material, as long as the minimal learning outcomes are achieved.<br />
=== Courses for which this course is prerequisite ===<br />
This course is required for [[Math 372]], [[Math 450]], [[Math 487]], [[Math 561]], [[Math 586]]. As a result it is essential that all required learning objectives be covered.<br />
<br />
[[Category:Courses|371]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_355:_Graph_Theory&diff=3692Math 355: Graph Theory2019-08-22T20:24:10Z<p>Ls5: /* Prerequisites */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Graph Theory.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
W<br />
<br />
=== Prerequisite ===<br />
[[Math 213]].<br />
<br />
=== Description ===<br />
Graphs and subgraphs, coloring problems, applications.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
<br />
[[Math 213]].<br />
<br />
=== Minimal learning outcomes ===<br />
Definition of a graph, examples (trees, paths, cycles, Petersen graph, etc.), First Theorem of Graph Theory, isomorphisms of graphs and graph invariants, chromatic number, connectivity, planar graphs and conditions for planarity, statements of Four Color Theorem and Kuratowski's theorem, Hamiltonian cycles.<br />
<br />
Additional topics from 1) proof of Kuratowski's theorem, 2) definition of the Euler characteristic and proof of Euler's formula for the genus, 3) proof of the Chvatal-Erdos theorem on the existence of Hamiltonian cycles.<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* ''Graph Theory'' by Rusell Merris (Chap. 1-5).<br />
<br />
A good additional resource is An Introduction to Graph Theory by Douglas B. West, but the book is probably too encyclopedic to use as a main text.<br />
<br />
=== Additional topics ===<br />
<br />
<br />
Possible other topics include spectral graph theory (networkings, expanders, Ramanujan graphs), characterization of Ramanujan graphs by the Riemann hypothesis for its zeta function. A good source for this topic is the paper by M. Ram Murty, Ramanujan Graphs, J. Ramanujan Math. Soc. 18(2003) 1-20.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|355]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_355:_Graph_Theory&diff=3691Math 355: Graph Theory2019-08-22T20:23:54Z<p>Ls5: /* Prerequisites */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Graph Theory.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
W<br />
<br />
=== Prerequisite ===<br />
[[Math 213]].<br />
<br />
=== Description ===<br />
Graphs and subgraphs, coloring problems, applications.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
<br />
Math 213.<br />
<br />
=== Minimal learning outcomes ===<br />
Definition of a graph, examples (trees, paths, cycles, Petersen graph, etc.), First Theorem of Graph Theory, isomorphisms of graphs and graph invariants, chromatic number, connectivity, planar graphs and conditions for planarity, statements of Four Color Theorem and Kuratowski's theorem, Hamiltonian cycles.<br />
<br />
Additional topics from 1) proof of Kuratowski's theorem, 2) definition of the Euler characteristic and proof of Euler's formula for the genus, 3) proof of the Chvatal-Erdos theorem on the existence of Hamiltonian cycles.<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* ''Graph Theory'' by Rusell Merris (Chap. 1-5).<br />
<br />
A good additional resource is An Introduction to Graph Theory by Douglas B. West, but the book is probably too encyclopedic to use as a main text.<br />
<br />
=== Additional topics ===<br />
<br />
<br />
Possible other topics include spectral graph theory (networkings, expanders, Ramanujan graphs), characterization of Ramanujan graphs by the Riemann hypothesis for its zeta function. A good source for this topic is the paper by M. Ram Murty, Ramanujan Graphs, J. Ramanujan Math. Soc. 18(2003) 1-20.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|355]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_355:_Graph_Theory&diff=3690Math 355: Graph Theory2019-08-22T20:23:37Z<p>Ls5: /* Prerequisite */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Graph Theory.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
W<br />
<br />
=== Prerequisite ===<br />
[[Math 213]].<br />
<br />
=== Description ===<br />
Graphs and subgraphs, coloring problems, applications.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
<br />
Math 313.<br />
<br />
=== Minimal learning outcomes ===<br />
Definition of a graph, examples (trees, paths, cycles, Petersen graph, etc.), First Theorem of Graph Theory, isomorphisms of graphs and graph invariants, chromatic number, connectivity, planar graphs and conditions for planarity, statements of Four Color Theorem and Kuratowski's theorem, Hamiltonian cycles.<br />
<br />
Additional topics from 1) proof of Kuratowski's theorem, 2) definition of the Euler characteristic and proof of Euler's formula for the genus, 3) proof of the Chvatal-Erdos theorem on the existence of Hamiltonian cycles.<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* ''Graph Theory'' by Rusell Merris (Chap. 1-5).<br />
<br />
A good additional resource is An Introduction to Graph Theory by Douglas B. West, but the book is probably too encyclopedic to use as a main text.<br />
<br />
=== Additional topics ===<br />
<br />
<br />
Possible other topics include spectral graph theory (networkings, expanders, Ramanujan graphs), characterization of Ramanujan graphs by the Riemann hypothesis for its zeta function. A good source for this topic is the paper by M. Ram Murty, Ramanujan Graphs, J. Ramanujan Math. Soc. 18(2003) 1-20.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|355]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_342:_Theory_of_Analysis_2&diff=3689Math 342: Theory of Analysis 22019-08-22T20:23:08Z<p>Ls5: /* Prerequisites */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Theory of Analysis 2.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F, W<br />
<br />
=== Prerequisite ===<br />
[[Math 213]], [[Math 341|341]].<br />
<br />
=== Description ===<br />
Rigorous treatment of calculus of several real variables; metric spaces, geometry and topology of Euclidean space, differentiation(,?) implicity(?) function theorem, integration on sets and manifolds.<br />
<br />
== Desired Learning Outcomes ==<br />
Math 342 is the multivariable sequel to [[Math 341]]. It provides a rigorous treatment of multivariable calculus. Unlike the case with [[Math 341]], Math 342 students are not required to have had an introductory course in the subject matter, so Math 342 needs to treat both computation and theory in some depth.<br />
<br />
=== Prerequisites ===<br />
[[Math 341]] provides the single-variable results on which many of the multivariable results of Math 342 are based. [[Math 213]] provides the tools for describing the spaces in which differentiation and integration is performed.<br />
<br />
=== Minimal learning outcomes ===<br />
Outlined below are topics that all successful Math 342 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, including calculations.<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
# Metric Spaces<br />
#* Open and closed sets<br />
#* Limit points and sequential limits<br />
#* Compactness (sequences and open covers)<br />
#* Connectedness<br />
#* Subspaces<br />
#* Boundedness<br />
#* Completeness<br />
#* Functional limits<br />
#* Continuity (including uniform and Lipschitz)<br />
#* The Contraction Mapping Theorem<br />
# Geometry and topology of '''R'''<sup>n</sup><br />
#* Dot product and Euclidean norm<br />
#* Components and convergence<br />
#* The Bolzano-Weierstrass Theorem<br />
#* The Heine-Borel Theorem<br />
#* Coordinate functions and continuity<br />
#* Algebraic continuity rules<br><br><br><br />
# Differentiation of ''f'': ''D'' ⊆ '''R'''<sup>m</sup> → '''R'''<sup>n</sup><br />
#* Directional, partial, and total derivatives<br />
#* Algebraic differentiation rules<br />
#* The Chain Rule<br />
#* The Mean Value Theorem<br />
#* Higher-order derivatives<br />
#* Taylor's Theorem<br />
#* The Second Derivative Test<br />
#* Lagrange multipliers<br />
#* Newton's method<br />
#* The Inverse Function Theorem<br />
#* The Implicit Function Theorem<br />
# Integration on ''n''-dimensional subsets of '''R'''<sup>n</sup><br />
#* Integrability of continuous functions<br />
#* Iterated integrals and Fubini's Theorem<br />
#* The Jacobian and change of variables<br />
#* Polar and spherical coordinates<br />
# Integration on manifolds<br />
#* Line and surface integrals<br />
#* Stokes' Theorem and special cases<br />
<br />
</div><br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
The theory of exterior algebra and differential forms is not a required topic, but instructors might find it useful to cover it. Instructors have the option of covering either the Riemann or (probably an abbreviated version of) the Lebesgue theory of integration.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
Math 342 is required for [[Math 547]] and [[Math 565]] and is recommended for some other courses.<br />
<br />
[[Category:Courses|342]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_342:_Theory_of_Analysis_2&diff=3688Math 342: Theory of Analysis 22019-08-22T20:22:48Z<p>Ls5: /* Prerequisite */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Theory of Analysis 2.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F, W<br />
<br />
=== Prerequisite ===<br />
[[Math 213]], [[Math 341|341]].<br />
<br />
=== Description ===<br />
Rigorous treatment of calculus of several real variables; metric spaces, geometry and topology of Euclidean space, differentiation(,?) implicity(?) function theorem, integration on sets and manifolds.<br />
<br />
== Desired Learning Outcomes ==<br />
Math 342 is the multivariable sequel to [[Math 341]]. It provides a rigorous treatment of multivariable calculus. Unlike the case with [[Math 341]], Math 342 students are not required to have had an introductory course in the subject matter, so Math 342 needs to treat both computation and theory in some depth.<br />
<br />
=== Prerequisites ===<br />
[[Math 341]] provides the single-variable results on which many of the multivariable results of Math 342 are based. [[Math 313]] provides the tools for describing the spaces in which differentiation and integration is performed.<br />
<br />
=== Minimal learning outcomes ===<br />
Outlined below are topics that all successful Math 342 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, including calculations.<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
# Metric Spaces<br />
#* Open and closed sets<br />
#* Limit points and sequential limits<br />
#* Compactness (sequences and open covers)<br />
#* Connectedness<br />
#* Subspaces<br />
#* Boundedness<br />
#* Completeness<br />
#* Functional limits<br />
#* Continuity (including uniform and Lipschitz)<br />
#* The Contraction Mapping Theorem<br />
# Geometry and topology of '''R'''<sup>n</sup><br />
#* Dot product and Euclidean norm<br />
#* Components and convergence<br />
#* The Bolzano-Weierstrass Theorem<br />
#* The Heine-Borel Theorem<br />
#* Coordinate functions and continuity<br />
#* Algebraic continuity rules<br><br><br><br />
# Differentiation of ''f'': ''D'' ⊆ '''R'''<sup>m</sup> → '''R'''<sup>n</sup><br />
#* Directional, partial, and total derivatives<br />
#* Algebraic differentiation rules<br />
#* The Chain Rule<br />
#* The Mean Value Theorem<br />
#* Higher-order derivatives<br />
#* Taylor's Theorem<br />
#* The Second Derivative Test<br />
#* Lagrange multipliers<br />
#* Newton's method<br />
#* The Inverse Function Theorem<br />
#* The Implicit Function Theorem<br />
# Integration on ''n''-dimensional subsets of '''R'''<sup>n</sup><br />
#* Integrability of continuous functions<br />
#* Iterated integrals and Fubini's Theorem<br />
#* The Jacobian and change of variables<br />
#* Polar and spherical coordinates<br />
# Integration on manifolds<br />
#* Line and surface integrals<br />
#* Stokes' Theorem and special cases<br />
<br />
</div><br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
The theory of exterior algebra and differential forms is not a required topic, but instructors might find it useful to cover it. Instructors have the option of covering either the Riemann or (probably an abbreviated version of) the Lebesgue theory of integration.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
Math 342 is required for [[Math 547]] and [[Math 565]] and is recommended for some other courses.<br />
<br />
[[Category:Courses|342]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_334:_Ordinary_Differential_Equations&diff=3687Math 334: Ordinary Differential Equations2019-08-22T20:18:36Z<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|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 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=3686Math 334: Ordinary Differential Equations2019-08-22T20:18:15Z<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|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 313]] or be concurrently enrolled in [[Math 313]].<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_313:_Elementary_Linear_Algebra&diff=3682Math 313: Elementary Linear Algebra2019-07-25T18:04:47Z<p>Ls5: /* Title */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Elementary Linear Algebra. <strong>THIS CLASS WILL NO LONGER BE TAUGHT AFTER SUMMER TERM 2019.</strong><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 112]]. Students are recommended to take [[Math 290]] before taking Math 313<br />
<br />
=== Description ===<br />
Linear systems, matrices, vectors and vector spaces, linear transformations, determinants, inner product spaces, eigenvalues, and eigenvectors.<br />
<br />
This course is aimed at majors in mathematics, the physical sciences, engineering, and other students interested in applications of mathematics to their disciplines. Linear algebra is used more than any other form of advanced mathematics in industry and science. A key idea is the mathematical modeling of a problem via systems of linear equations.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
<br />
[[Math 112]]. [[Math 290]] is encouraged.<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
Upon completion of this course, the successful student will be able to:<br />
<br />
# Use Gaussian elimination to do all of the following: solve a linear system with reduced row echelon form, solve a linear system with row echelon form and backward substitution, find the inverse of a given matrix, and find the determinant of a given matrix.<br />
# Demonstrate proficiency at matrix algebra. For matrix multiplication demonstrate understanding of the associative law, the reverse order law for inverses and transposes, and the failure of the commutative law and the cancellation law.<br />
# Use Cramer's rule to solve a linear system.<br />
# Use cofactors to find the inverse of a given matrix and the determinant of a given matrix.<br />
# Determine whether a set with a given notion of addition and scalar multiplication is a vector space. Here, and in relevant numbers below, be familiar with both finite and infinite dimensional examples.<br />
# Determine whether a given subset of a vector space is a subspace.<br />
# Determine whether a given set of vectors is linearly independent, spans, or is a basis.<br />
# Determine the dimension of a given vector space or of a given subspace.<br />
# Find bases for the null space, row space, and column space of a given matrix, and determine its rank.<br />
# Demonstrate understanding of the Rank-Nullity Theorem and its applications.<br />
# Given a description of a linear transformation, find its matrix representation relative to given bases.<br />
# Demonstrate understanding of the relationship between similarity and change of basis.<br />
# Find the norm of a vector and the angle between two vectors in an inner product space.<br />
# Use the inner product to express a vector in an inner product space as a linear combination of an orthogonal set of vectors.<br />
# Find the orthogonal complement of a given subspace.<br />
# Demonstrate understanding of the relationship of the row space, column space, and nullspace of a matrix (and its transpose) via orthogonal complements.<br />
# Demonstrate understanding of the Cauchy-Schwartz inequality and its applications.<br />
# Determine whether a vector space with a (sesquilinear) form is an inner product space.<br />
# Use the Gram-Schmidt process to find an orthonormal basis of an inner product space. Be capable of doing this both in '''R'''<sup>n</sup> and in function spaces that are inner product spaces.<br />
# Use least squares to fit a line (''y'' = ''ax'' + ''b'') to a table of data, plot the line and data points, and explain the meaning of least squares in terms of orthogonal projection.<br />
# Use the idea of least squares to find orthogonal projections onto subspaces and for polynomial curve fitting.<br />
# Find (real and complex) eigenvalues and eigenvectors of 2 &times; 2 or 3 &times; 3 matrices.<br />
# Determine whether a given matrix is diagonalizable. If so, find a matrix that diagonalizes it via similarity.<br />
# Demonstrate understanding of the relationship between eigenvalues of a square matrix and its determinant, its trace, and its invertibility/singularity.<br />
# Identify symmetric matrices and orthogonal matrices.<br />
# Find a matrix that orthogonally diagonalizes a given symmetric matrix.<br />
# Know and be able to apply the spectral theorem for symmetric matrices.<br />
# Know and be able to apply the Singular Value Decomposition.<br />
# Correctly define terms and give examples relating to the above concepts.<br />
# Prove basic theorems about the above concepts.<br />
# Prove or disprove statements relating to the above concepts.<br />
# Be adept at hand computation for row reduction, matrix inversion and similar problems; also, use MATLAB or a similar program for linear algebra problems.<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 />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Math 334]], [[Math 342]], [[Math 355]], [[Math 371]], [[Math 431]], [[Math 485]], [[Math 570]]. It is clear from this list that it is imperative to cover all the minimal learning outcomes.<br />
<br />
[[Category:Courses|313]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_313:_Elementary_Linear_Algebra&diff=3681Math 313: Elementary Linear Algebra2019-07-25T18:04:19Z<p>Ls5: /* Title */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Elementary Linear Algebra. <strong>THIS CLASS WILL NO LONGER BE TAUGHT STARTING IN FALL SEMESTER 2019.</strong><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 112]]. Students are recommended to take [[Math 290]] before taking Math 313<br />
<br />
=== Description ===<br />
Linear systems, matrices, vectors and vector spaces, linear transformations, determinants, inner product spaces, eigenvalues, and eigenvectors.<br />
<br />
This course is aimed at majors in mathematics, the physical sciences, engineering, and other students interested in applications of mathematics to their disciplines. Linear algebra is used more than any other form of advanced mathematics in industry and science. A key idea is the mathematical modeling of a problem via systems of linear equations.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
<br />
[[Math 112]]. [[Math 290]] is encouraged.<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
Upon completion of this course, the successful student will be able to:<br />
<br />
# Use Gaussian elimination to do all of the following: solve a linear system with reduced row echelon form, solve a linear system with row echelon form and backward substitution, find the inverse of a given matrix, and find the determinant of a given matrix.<br />
# Demonstrate proficiency at matrix algebra. For matrix multiplication demonstrate understanding of the associative law, the reverse order law for inverses and transposes, and the failure of the commutative law and the cancellation law.<br />
# Use Cramer's rule to solve a linear system.<br />
# Use cofactors to find the inverse of a given matrix and the determinant of a given matrix.<br />
# Determine whether a set with a given notion of addition and scalar multiplication is a vector space. Here, and in relevant numbers below, be familiar with both finite and infinite dimensional examples.<br />
# Determine whether a given subset of a vector space is a subspace.<br />
# Determine whether a given set of vectors is linearly independent, spans, or is a basis.<br />
# Determine the dimension of a given vector space or of a given subspace.<br />
# Find bases for the null space, row space, and column space of a given matrix, and determine its rank.<br />
# Demonstrate understanding of the Rank-Nullity Theorem and its applications.<br />
# Given a description of a linear transformation, find its matrix representation relative to given bases.<br />
# Demonstrate understanding of the relationship between similarity and change of basis.<br />
# Find the norm of a vector and the angle between two vectors in an inner product space.<br />
# Use the inner product to express a vector in an inner product space as a linear combination of an orthogonal set of vectors.<br />
# Find the orthogonal complement of a given subspace.<br />
# Demonstrate understanding of the relationship of the row space, column space, and nullspace of a matrix (and its transpose) via orthogonal complements.<br />
# Demonstrate understanding of the Cauchy-Schwartz inequality and its applications.<br />
# Determine whether a vector space with a (sesquilinear) form is an inner product space.<br />
# Use the Gram-Schmidt process to find an orthonormal basis of an inner product space. Be capable of doing this both in '''R'''<sup>n</sup> and in function spaces that are inner product spaces.<br />
# Use least squares to fit a line (''y'' = ''ax'' + ''b'') to a table of data, plot the line and data points, and explain the meaning of least squares in terms of orthogonal projection.<br />
# Use the idea of least squares to find orthogonal projections onto subspaces and for polynomial curve fitting.<br />
# Find (real and complex) eigenvalues and eigenvectors of 2 &times; 2 or 3 &times; 3 matrices.<br />
# Determine whether a given matrix is diagonalizable. If so, find a matrix that diagonalizes it via similarity.<br />
# Demonstrate understanding of the relationship between eigenvalues of a square matrix and its determinant, its trace, and its invertibility/singularity.<br />
# Identify symmetric matrices and orthogonal matrices.<br />
# Find a matrix that orthogonally diagonalizes a given symmetric matrix.<br />
# Know and be able to apply the spectral theorem for symmetric matrices.<br />
# Know and be able to apply the Singular Value Decomposition.<br />
# Correctly define terms and give examples relating to the above concepts.<br />
# Prove basic theorems about the above concepts.<br />
# Prove or disprove statements relating to the above concepts.<br />
# Be adept at hand computation for row reduction, matrix inversion and similar problems; also, use MATLAB or a similar program for linear algebra problems.<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 />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Math 334]], [[Math 342]], [[Math 355]], [[Math 371]], [[Math 431]], [[Math 485]], [[Math 570]]. It is clear from this list that it is imperative to cover all the minimal learning outcomes.<br />
<br />
[[Category:Courses|313]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_313:_Elementary_Linear_Algebra&diff=3680Math 313: Elementary Linear Algebra2019-07-25T18:02:21Z<p>Ls5: /* Title */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Elementary Linear Algebra. THIS CLASS WILL NO LONGER BE TAUGHT STARTING IN FALL SEMESTER 2019.<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 112]]. Students are recommended to take [[Math 290]] before taking Math 313<br />
<br />
=== Description ===<br />
Linear systems, matrices, vectors and vector spaces, linear transformations, determinants, inner product spaces, eigenvalues, and eigenvectors.<br />
<br />
This course is aimed at majors in mathematics, the physical sciences, engineering, and other students interested in applications of mathematics to their disciplines. Linear algebra is used more than any other form of advanced mathematics in industry and science. A key idea is the mathematical modeling of a problem via systems of linear equations.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
<br />
[[Math 112]]. [[Math 290]] is encouraged.<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
Upon completion of this course, the successful student will be able to:<br />
<br />
# Use Gaussian elimination to do all of the following: solve a linear system with reduced row echelon form, solve a linear system with row echelon form and backward substitution, find the inverse of a given matrix, and find the determinant of a given matrix.<br />
# Demonstrate proficiency at matrix algebra. For matrix multiplication demonstrate understanding of the associative law, the reverse order law for inverses and transposes, and the failure of the commutative law and the cancellation law.<br />
# Use Cramer's rule to solve a linear system.<br />
# Use cofactors to find the inverse of a given matrix and the determinant of a given matrix.<br />
# Determine whether a set with a given notion of addition and scalar multiplication is a vector space. Here, and in relevant numbers below, be familiar with both finite and infinite dimensional examples.<br />
# Determine whether a given subset of a vector space is a subspace.<br />
# Determine whether a given set of vectors is linearly independent, spans, or is a basis.<br />
# Determine the dimension of a given vector space or of a given subspace.<br />
# Find bases for the null space, row space, and column space of a given matrix, and determine its rank.<br />
# Demonstrate understanding of the Rank-Nullity Theorem and its applications.<br />
# Given a description of a linear transformation, find its matrix representation relative to given bases.<br />
# Demonstrate understanding of the relationship between similarity and change of basis.<br />
# Find the norm of a vector and the angle between two vectors in an inner product space.<br />
# Use the inner product to express a vector in an inner product space as a linear combination of an orthogonal set of vectors.<br />
# Find the orthogonal complement of a given subspace.<br />
# Demonstrate understanding of the relationship of the row space, column space, and nullspace of a matrix (and its transpose) via orthogonal complements.<br />
# Demonstrate understanding of the Cauchy-Schwartz inequality and its applications.<br />
# Determine whether a vector space with a (sesquilinear) form is an inner product space.<br />
# Use the Gram-Schmidt process to find an orthonormal basis of an inner product space. Be capable of doing this both in '''R'''<sup>n</sup> and in function spaces that are inner product spaces.<br />
# Use least squares to fit a line (''y'' = ''ax'' + ''b'') to a table of data, plot the line and data points, and explain the meaning of least squares in terms of orthogonal projection.<br />
# Use the idea of least squares to find orthogonal projections onto subspaces and for polynomial curve fitting.<br />
# Find (real and complex) eigenvalues and eigenvectors of 2 &times; 2 or 3 &times; 3 matrices.<br />
# Determine whether a given matrix is diagonalizable. If so, find a matrix that diagonalizes it via similarity.<br />
# Demonstrate understanding of the relationship between eigenvalues of a square matrix and its determinant, its trace, and its invertibility/singularity.<br />
# Identify symmetric matrices and orthogonal matrices.<br />
# Find a matrix that orthogonally diagonalizes a given symmetric matrix.<br />
# Know and be able to apply the spectral theorem for symmetric matrices.<br />
# Know and be able to apply the Singular Value Decomposition.<br />
# Correctly define terms and give examples relating to the above concepts.<br />
# Prove basic theorems about the above concepts.<br />
# Prove or disprove statements relating to the above concepts.<br />
# Be adept at hand computation for row reduction, matrix inversion and similar problems; also, use MATLAB or a similar program for linear algebra problems.<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 />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Math 334]], [[Math 342]], [[Math 355]], [[Math 371]], [[Math 431]], [[Math 485]], [[Math 570]]. It is clear from this list that it is imperative to cover all the minimal learning outcomes.<br />
<br />
[[Category:Courses|313]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_215:_Computational_Linear_Algebra&diff=3679Math 215: Computational Linear Algebra2019-03-28T01:59:06Z<p>Ls5: /* Desired Learning Outcomes */</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 313]], [[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=3678Math 215: Computational Linear Algebra2019-03-28T01:58:53Z<p>Ls5: /* Desired Learning Outcomes */</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 313]], [[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 />
</div><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=3677Math 215: Computational Linear Algebra2019-03-28T01:58:35Z<p>Ls5: /* Desired Learning Outcomes */</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 313]], [[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 />
<div style="-moz-column-count:2; column-count:2;"><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 />
</div><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_213:_Elementary_Linear_Algebra&diff=3676Math 213: Elementary Linear Algebra2019-03-28T01:57:48Z<p>Ls5: /* Desired Learning Outcomes */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Elementary Linear Algebra.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(2:2:0)<br />
<br />
=== Offered ===<br />
F, W, Sp, Su<br />
<br />
=== Prerequisite ===<br />
[[Math 112]]. <br />
<br />
=== Description ===<br />
Concepts and applications of linear systems, matrices, vectors and vector spaces, linear transformations, determinants, inner product spaces, eigenvalues, and eigenvectors.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
Upon completion of this course, the successful student will be able to:<br />
<br />
# Vectors and vector operations<br />
# Lines and planes and their equations<br />
# Systems of linear equations<br />
# Spanning sets and linear independence<br />
# Matrix algebra<br />
# The inverse of a matrix<br />
# Subspaces, basis, dimension, and rank<br />
# Linear transformations<br />
# Eigenvalues and eigenvectors<br />
# Determinants<br />
# Similarity and diagonalization<br />
# Orthogonality, orthogonal complements and projections<br />
# Gram-Schmidt and QR factorization<br />
# Vector spaces and subspaces<br />
# Singular Value Decomposition<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 />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
<br />
<br />
[[Category:Courses|213]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_215:_Computational_Linear_Algebra&diff=3675Math 215: Computational Linear Algebra2019-03-28T01:57:17Z<p>Ls5: /* Catalog Information */</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 313]], [[Math 213]], or [[Math 302]] (recommended).<br />
<br />
=== Description ===<br />
Practical linear algebraic computations and applications.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
<br />
<br />
=== Minimal learning outcomes ===<br />
<div style="-moz-column-count:2; column-count:2;"><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 />
=== 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 />
[[Category:Courses|215]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_215:_Computational_Linear_Algebra&diff=3674Math 215: Computational Linear Algebra2019-03-28T01:54:43Z<p>Ls5: /* Desired Learning Outcomes */</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 313]], [[Math 213]], or [[Math 302]] (recommended).<br />
<br />
=== Description ===<br />
Practical linear algebraic computations and applications..<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
<br />
<br />
=== Minimal learning outcomes ===<br />
<div style="-moz-column-count:2; column-count:2;"><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 />
=== 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 />
[[Category:Courses|215]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_213:_Elementary_Linear_Algebra&diff=3673Math 213: Elementary Linear Algebra2019-03-28T01:54:04Z<p>Ls5: /* Desired Learning Outcomes */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Elementary Linear Algebra.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(2:2:0)<br />
<br />
=== Offered ===<br />
F, W, Sp, Su<br />
<br />
=== Prerequisite ===<br />
[[Math 112]]. <br />
<br />
=== Description ===<br />
Concepts and applications of linear systems, matrices, vectors and vector spaces, linear transformations, determinants, inner product spaces, eigenvalues, and eigenvectors.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
Upon completion of this course, the successful student will be able to:<br />
<br />
# Vectors and vector operations<br />
# Lines and planes and their equations<br />
# Systems of linear equations<br />
# Spanning sets and linear independence<br />
# Matrix algebra<br />
# The inverse of a matrix<br />
# Subspaces, basis, dimension, and rank<br />
# Linear transformations<br />
# Eigenvalues and eigenvectors<br />
# Determinants<br />
# Similarity and diagonalization<br />
# Orthogonality, orthogonal complements and projections<br />
# Gram-Schmidt and QR factorization<br />
# Vector spaces and subspaces<br />
# Singular Value Decomposition<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 />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
<br />
<br />
[[Category:Courses|113]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_215:_Computational_Linear_Algebra&diff=3672Math 215: Computational Linear Algebra2019-03-28T01:48:09Z<p>Ls5: Created page with "== Catalog Information == === Title === Computational Linear Algebra === (Credit Hours:Lecture Hours:Lab Hours) === (1:0:1) === Offered === F, W === Prerequisite === Mat..."</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 313]], [[Math 213]], or [[Math 302]] (recommended).<br />
<br />
=== Description ===<br />
Practical linear algebraic computations and applications..<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
<br />
<br />
=== Minimal learning outcomes ===<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 />
=== 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 />
[[Category:Courses|215]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_213:_Elementary_Linear_Algebra&diff=3671Math 213: Elementary Linear Algebra2019-03-28T01:42:18Z<p>Ls5: Created page with "== Catalog Information == === Title === Elementary Linear Algebra. === (Credit Hours:Lecture Hours:Lab Hours) === (2:2:0) === Offered === F, W, Sp, Su === Prerequisite ===..."</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Elementary Linear Algebra.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(2:2:0)<br />
<br />
=== Offered ===<br />
F, W, Sp, Su<br />
<br />
=== Prerequisite ===<br />
[[Math 112]]. <br />
<br />
=== Description ===<br />
Concepts and applications of linear systems, matrices, vectors and vector spaces, linear transformations, determinants, inner product spaces, eigenvalues, and eigenvectors.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
<br />
[[Math 112]]. [[Math 290]] is encouraged.<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
Upon completion of this course, the successful student will be able to:<br />
<br />
# Vectors and vector operations<br />
# Lines and planes and their equations<br />
# Systems of linear equations<br />
# Spanning sets and linear independence<br />
# Matrix algebra<br />
# The inverse of a matrix<br />
# Subspaces, basis, dimension, and rank<br />
# Linear transformations<br />
# Eigenvalues and eigenvectors<br />
# Determinants<br />
# Similarity and diagonalization<br />
# Orthogonality, orthogonal complements and projections<br />
# Gram-Schmidt and QR factorization<br />
# Vector spaces and subspaces<br />
# Singular Value Decomposition<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 />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
<br />
<br />
[[Category:Courses|113]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_303:_Math_for_Engineering_2&diff=3670Math 303: Math for Engineering 22018-08-30T21:31:09Z<p>Ls5: /* Desired Learning Outcomes */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Mathematics for Engineering 2.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(4:4:0)<br />
<br />
=== Offered ===<br />
F, W<br />
<br />
=== Prerequisite ===<br />
[[Math 302]] or [[Math 314]].<br />
<br />
=== Description ===<br />
ODEs, Laplace transforms, Fourier series, PDEs.<br />
<br />
== Desired Learning Outcomes ==<br />
This course is designed to give students from the College of Engineering the mathematics background necessary to succeed in their chosen field.<br />
<br />
=== Prerequisites ===<br />
Students are expected to have completed [[Math 302]] or [[Math 314]].<br />
<br />
=== Minimal learning outcomes ===<br />
Students should achieve mastery of the topics below.<br />
<div style="-moz-column-count:2; column-count:2;"><br />
# Some Basic Mathematical Models; Direction Fields<br />
#* Model physical processes using differential equations.<br />
#* Sketch the direction field (or slope field) of a differential equation using a computer graphing program.<br />
#* Describe the behavior of the solutions of a differential equation by analyzing its slope field. Identify any equilibrium solutions.<br />
# Solutions of Some Differential Equations; Classification of Differential Equations<br />
#* Solve basic initial value problems; obtain explicit solutions if possible.<br />
#* Characterize the solutions of a differential equation with respect to initial values.<br />
#* Use the solution of an initial value problem to answer questions about a physical system.<br />
#* Determine the order of an ordinary differential equation. Classify an ordinary differential equation as linear or nonlinear.<br />
#* Verify solutions to ordinary differential equations.<br />
#* Determine the order of a partial differential equation. Classify a partial differential equation as linear or nonlinear.<br />
#* Verify solutions to partial differential equations.<br />
# Linear First Order Equations with Variable Coefficients<br />
#* Identify and solve first order linear equations.<br />
#* Analyze the behavior of solutions.<br />
#* Solve initial value problems for first order linear equations.<br />
# Separable First Order Equations<br />
#* Identify and solve separable equations; obtain explicit solutions if possible.<br />
#* Solve initial value problems for separable equations, and analyze their solutions.<br />
#* Apply the transformation $y=xv(x)$ to obtain a separable equation, if possible.<br />
# Modeling with First Order Equations<br />
#* Construct models of tank problems using differential equations. Analyze the models to answer questions about the physical system modeled.<br />
#* Construct growth and decay problems using differential equations. Analyze the models to answer questions about the physical system modeled.<br />
#* Construct models of problems involving force and motion using differential equations. Analyze the models to answer questions about the physical system modeled.<br />
#Differences Between Linear and Nonlinear Equations<br />
#* Recall and apply the existence and uniqueness theorem for first order linear differential equations.<br />
#* Recall and apply the existence and uniqueness theorem for first order differential equations (both linear and nonlinear).<br />
#* Summarize the nice properties of linear equations. Contrast with nonlinear equations.<br />
# Autonomous Equations and Population Dynamics<br />
#* Determine and classify the equilibrium solutions of an autonomous equation as asymptotically stable, semistable or unstable by analyzing a graph of $\dfrac{dy}{dt}$ versus $y$. Sketch the phase line.<br />
#* Analyze solutions of the logistic equation and other autonomous equations.<br />
# Exact Equations and Integrating Factors<br />
#* Identify whether or not a differential equation is exact.<br />
#* Solve exact differential equations with or without initial conditions, and obtain explicit solutions if possible.<br />
#* Use integrating factors to convert a differential equation to an exact equation and then solve.<br />
#* Determine an integrating factor of the form $\mu(x)$ or $\mu(y)$ which will convert a non-exact differential equation to an exact equation, if possible.<br />
# Introduction to Second Order Equations<br />
#* Determine the characteristic equation of a second order linear differential equation with constant coefficients.<br />
#* Solve second order linear differential equations with constant coefficients that have a characteristic equation with real and distinct roots.<br />
#* Describe the behavior of solutions.<br />
#* Convert a second order differential equation to a first order differential equation in the following cases: i) y"=f(t,y'), ii) y"=f(y,y').<br />
# Fundamental Solutions of Linear Homogeneous Equations; the Wronskian<br />
#* Recall and apply the existence and uniqueness theorem for second order linear differential equations.<br />
#* Recall and verify the principal of superposition for solutions of second order linear differential equations.<br />
#* Evaluate the Wronskian of two functions.<br />
#* Determine whether or not a pair of solutions of a second order linear differential equations constitute a fundamental set of solutions.<br />
#* Recall and apply Abel's theorem.<br />
# Complex Roots of the Characteristic Equation<br />
#* Use Euler's formula to rewrite complex expressions in different forms.<br />
#* Solve second order linear differential equations with constant coefficients that have a characteristic equation with complex roots.<br />
#* Solve initial value problems and analyze the solutions.<br />
# Repeated Roots; Reduction of Order<br />
#* Solve second order linear differential equations with constant coefficients that have a characteristic equation with repeated roots.<br />
#* Solve initial value problems and analyze the solutions.<br />
#* Apply the method of reduction of order to find a second solution to a given differential equation.<br />
# Nonhomogeneous Equations; Method of Undetermined Coefficients<br />
#* For a nonhomogeneous second order linear differential equation, determine a suitable form of a particular solution that can be used in the method of undetermined coefficients.<br />
#* Apply the method of undetermined coefficients to solve nonhomogeneous second order linear differential equations.<br />
#* Solve initial value problems and analyze the solutions.<br />
# Variation of Parameters; Reduction of Order<br />
#* Apply the method of variation of parameters to solve nonhomogeneous second order linear differential equations with or without initial conditions.<br />
#* Apply the method of reduction of order to solve nonhomogeneous second order linear differential equations with or without initial conditions.<br />
# Mechanical Vibrations<br />
#* Model undamped mechanical vibrations with second order linear differential equations, and then solve. Analyze the solution. In particular, evaluate the frequency, period, amplitude, phase shift, and the position at a given time.<br />
#* Model damped mechanical vibrations with second order linear differential equations, and then solve. Analyze the solution. In particular, evaluate the quasi frequency, quasi period, and the behavior at infinity.<br />
#* Define critically damped and overdamped. Identify when these conditions exist in a system.<br />
# Forced Vibrations<br />
#* Model forced, undamped mechanical vibrations with second order linear differential equations, and then solve. Analyze the solution.<br />
#* Describe the phenomena of beats and resonance. Determine the frequency at which resonance occurs.<br />
#* Model forced, damped mechanical vibrations with second order linear differential equations, and then solve. Determine and analyze the solutions, including the steady state and transient parts.<br />
# General Theory of nth Order Linear Equations<br />
#* Recall and apply the existence and uniqueness theorem for higher order linear differential equations.<br />
#* Recall the definition of linear independence for a finite set of functions. Determine whether a set of functions is linearly independent or linearly dependent.<br />
#* Use the Wronskian to determine if a set of solutions form a fundamental set of solutions.<br />
#* Recall the relationship between Wronskian and linear independence for a set of functions, and for a set of solutions.<br />
#* Apply the method of reduction of order to solve higher order linear differential equations.<br />
# Homogeneous Equations with Constant Coefficients<br />
#* Apply Euler's formula to write a complex number in exponential form. Find the distinct complex roots of a number.<br />
#* Determine the characteristic equation of higher order linear differential equations.<br />
#* Solve higher order linear differential equations.<br />
#* Solve initial value problems.<br />
# The Method of Undetermined Coefficients<br />
#* For a nonhomogeneous higher order linear differential equation, determine a suitable form of a generalized particular solution that can be applied in the method of undetermined coefficients.<br />
#* Use the method of undetermined coefficients to solve nonhomogeneous higher order linear differential equations.<br />
#* Solve initial value problems.<br />
# The Method of Variation of Parameters<br />
#* Use the method of variation of parameters to solve nonhomogeneous higher order linear differential equations.<br />
#* Solve initial value problems.<br />
# Review of Power Series<br />
#* Determine the radius of convergence of a power series.<br />
#* Find the power series expansion of a function.<br />
#* Manipulate expressions involving summation notation. Change the index of summation.<br />
# Series Solutions near an Ordinary Point, Part I<br />
#* Find the general solution of a differential equation using power series.<br />
#* Solve initial value problems. Analyze the solution.<br />
# Series Solutions near an Ordinary Point, Part II<br />
#* Given an initial value problem, use the differential equation to inductively determine the terms in the power series of the solution, expanded about the initial value.<br />
#* Determine a lower bound for the radius of convergence of a series solution.<br />
# Euler Equations<br />
#* Find the general solution to an Euler equation in the cases of real distinct roots, equal roots, and complex roots.<br />
#* Solve initial value problems for Euler equations and analyze their solutions.<br />
# Definition of Laplace Transform<br />
#* Sketch a piecewise defined function. Determine if it is continuous, piecewise continuous or neither.<br />
#* Evaluate Laplace transforms from the definition.<br />
#* Determine whether an infinite integral converges or diverges.<br />
# Solution of Initial Value Problems<br />
#* Evaluate inverse Laplace transforms.<br />
#* Use Laplace transforms to solve initial value problems.<br />
#* Evaluate Laplace transforms using the derivative identity given in Problem 28 (p. 322) of the textbook.<br />
# Step Functions<br />
#* Sketch the graph of a function that is defined in terms of step functions.<br />
#* Convert piecewise defined functions to functions defined in terms of step functions and vice versa.<br />
#* Find the Laplace transform of a piecewise defined function.<br />
#* Apply the shifting theorems (Theorems 6.3.1 and 6.3.2) to evaluate Laplace transforms and inverse Laplace transforms.<br />
# Differential Equations with Discontinuous Forcing Functions<br />
#* Use Laplace transforms to solve differential equations with discontinuous forcing functions.<br />
#* Analyze the solutions of differential equations with discontinuous forcing functions.<br />
# Impulse Functions<br />
#* Define an idealized unit impulse function.<br />
#* Use Laplace transforms to solve differential equations that involve impulse functions.<br />
#* Analyze the solutions of differential equations that involve impulse functions.<br />
# The Convolution Integral<br />
#* Evaluate the convolution of two functions from the definition.<br />
#* Prove and disprove properties of the convolution operator.<br />
#* Evaluate the Laplace transform of a convolution of functions.<br />
#* Use the convolution theorem to evaluate inverse Laplace transforms.<br />
#* Solve initial value problems using convolution.<br />
# Introduction to Systems of First Order Equations<br />
#* Transform a higher order linear differential equation into a system of first order linear equations.<br />
#* Transform a system of first order linear equations to a single higher order linear equation.<br />
#* Model physical systems that involve more than one unknown function with a system of differential equations.<br />
#* Recall and apply methods of linear algebra.<br />
# Basic Theory of Systems of First Order Linear Equations<br />
#* Recall and verify the superposition principle for first order linear systems.<br />
#* Relate the Wronskian to linear independence and a fundamental set of solutions.<br />
# Homogeneous Linear Systems with Constant Coefficients<br />
#* Sketch a direction field and a phase portrait for a homogeneous linear system with constant coefficients.<br />
#* Find the general solution of a homogeneous linear system with constant coefficients in the case of real, distinct eigenvalues.<br />
#* Determine if the origin is a saddle point or a node for a homogeneous linear system. Classify a node as asymptotically stable or unstable.<br />
#* Find general solutions, solve initial value problems, and analyze their solutions.<br />
# Complex Eigenvalues<br />
#* Sketch a direction field and a phase portrait for a homogeneous linear system with constant coefficients.<br />
#* Find the general solution of a homogeneous linear system with constant coefficients in the case of complex eigenvalues.<br />
#* Classify the origin as a saddle point, a node, a spiral point or a center.<br />
#* Solve and analyze physical problems modeled by systems of differential equations.<br />
# Fundamental Matrices<br />
#* Determine a fundamental matrix for a system of equations.<br />
#* Solve initial value problems using a fundamental matrix.<br />
#* Prove identities using fundamental matrices.<br />
# Repeated Eigenvalues<br />
#* Sketch a direction field and a phase portrait for a homogeneous linear system with constant coefficients.<br />
#* Find the general solution of a homogeneous linear system with constant coefficients in the case of repeated eigenvalues.<br />
#* Identify improper nodes. Classify them as asymptotically stable or unstable.<br />
#* Solve initial value problems.<br />
# Nonhomogeneous Linear Systems<br />
#* Use diagonalization to solve nonhomogeneous linear systems.<br />
#* Use the method of undetermined coefficients to solve nonhomogeneous linear systems.<br />
#* Use the method of variation of parameters to solve nonhomogeneous linear systems.<br />
#* Solve initial value problems.<br />
# Two-Point Boundary Value Problems<br />
#* Solve boundary value problems involving linear differential equations.<br />
#* Find the eigenvalues and the corresponding eigenfunctions of a boundary value problem.<br />
# Fourier Series<br />
#* Identify functions that are periodic. Determine their periods.<br />
#* Find the Fourier series for a function defined on a closed interval.<br />
#* Determine the $m$th partial sum of the Fourier series of a function. Compare to the function.<br />
# The Fourier Convergence Theorem<br />
#* Find the Fourier series for a periodic function.<br />
#* Recall and apply the convergence theorem for Fourier series.<br />
# Even and Odd Functions<br />
#* Determine whether a given function is even, odd or neither.<br />
#* Sketch the even and odd extensions of a function defined on the interval [0,L].<br />
#* Find the Fourier sine and cosine series for the function defined on [0,L].<br />
#* Establish identities involving infinite sums from Fourier series.<br />
# Separation of Variables; Heat Conduction in a Rod<br />
#* Apply the method of separation of variables to solve partial differential equations, if possible.<br />
#* Find the solutions of heat conduction problems in a rod using separation of variables.<br />
# Other Heat Conduction Problems<br />
#* Solve steady state heat conduction problems in a rod with various boundary conditions.<br />
#* Analyze the solutions.<br />
# The Wave Equation; Vibrations of an Elastic String<br />
#* Solve the wave equation that models the vibration of a string with fixed ends.<br />
#* Describe the motion of a vibrating string.<br />
# Laplace's Equation<br />
#* Solve Laplace's equation over a rectangular region for various boundary conditions.<br />
#* Solve Laplace's equation over a circular region for various boundary conditions.<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 />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|303]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_302:_Math_for_Engineering_1&diff=3669Math 302: Math for Engineering 12018-08-30T21:30:19Z<p>Ls5: /* Desired Learning Outcomes */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Mathematics for Engineering 1.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(4:4:0)<br />
<br />
=== Offered ===<br />
F, W<br />
<br />
=== Prerequisite ===<br />
[[Math 113]] and passing grade on required preparatory exam taken during first week of class. (Practice exams available on class website.)<br />
<br />
=== Description ===<br />
Multivariable calculus, linear algebra, and numerical methods.<br />
<br />
== Desired Learning Outcomes ==<br />
This course is designed to give students from the College of Engineering the mathematics background necessary to succeed in their chosen field.<br />
<br />
=== Prerequisites ===<br />
<br />
Students are expected to have completed [[Math 113]].<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
# Rectangular Space Coordinates; Vectors in Three-Dimensional Space<br />
#* Define the following:<br />
#** Cartesian coordinates of a point<br />
#** sphere<br />
#** symmetry about a point, a line, and a plane<br />
#** vector<br />
#** components of a vector<br />
#** vector addition<br />
#** scalar multiplication<br />
#** zero vector<br />
#** vector subtraction<br />
#** vector norm (magnitude, length)<br />
#** unit vector<br />
#** coordinate unit vectors i, j, k<br />
#** linear combination of unit vectors<br />
#* Plot points in three-dimensional space.<br />
#* Calculate the distance between two points in two-dimensional space and 3-dimensional space<br />
#* Write the equation of a sphere centered about a given point with a given radius. Determine the center and radius of a sphere, given its equation.<br />
#* Write the component equations of a line that passes through two given points.<br />
#* Write the component equations of a line segment with given endpoints.<br />
#* Find the midpoint of a given line segment.<br />
#* Find the points of symmetry about a point, line, or plane.<br />
#* Represent a vector by each of the following:<br />
#** components<br />
#** a linear combination of coordinate unit vectors<br />
#* Carry out the vector operations:<br />
#** addition<br />
#** scalar multiplication<br />
#** subtraction<br />
#* Represent the operations of vector addition, scalar multiplication and norm geometrically.<br />
#* Find the norm (magnitude, length) of a vector. Determine whether two vectors are parallel.<br />
#* Recall, apply and verify the basic properties of vector addition, scalar multiplication and norm.<br />
#* Model and solve application problems using vectors.<br />
# The Dot Product<br />
#* Define the following:<br />
#** dot product.<br />
#** perpendicular vectors.<br />
#** unit vector in the direction of a vector a, denoted u_a.<br />
#** the projection of a on b, denoted proj_b a.<br />
#** the b-component of a, denoted comp_b a.<br />
#** the direction cosines of a vector.<br />
#** the direction angles of a vector.<br />
#** the Schwarz Inequality.<br />
#** the work done by a constant force on an object.<br />
#** the dot product test for perpendicular vectors.<br />
#** the dot product test for parallel vectors.<br />
#** geometric interpretation of the dot product<br />
#* Evaluate a dot product from the coordinate formula or the angle formula.<br />
#* Interpret the dot product geometrically.<br />
#* Evaluate the following using the dot product:<br />
#** the length of a vector.<br />
#** the angle between two vectors.<br />
#** u_a, the unit vector in the direction of a vector a.<br />
#** proj_b a, the projection of a on b.<br />
#** comp_b a, the b-component of a.<br />
#** the direction cosines of a vector.<br />
#** the direction angles of a vector.<br />
#** the work done by a constant force on an object.<br />
#* Prove and verify the Schwarz Inequality.<br />
#* Prove and apply the dot product tests for perpendicular and parallel vectors.<br />
#* Recall and apply the properties of the dot product.<br />
#* Prove identities involving the dot product.<br />
#* Solve application problems involving the dot product.<br />
#* Extend the vector operations and related identities for addition, scalar multiplication, and dot product to higher dimensions.<br />
# The Cross Product<br />
#* Define the following:<br />
#** the cross product of two vectors<br />
#** scalar triple product<br />
#* Evaluate a cross product from the the coordinate formula or angle formula.<br />
#* Interpret the cross product geometrically.<br />
#* Evaluate the following using the cross product:<br />
#** a vector perpendicular to two given vectors.<br />
#** the area of a parallelogram.<br />
#** the area or a triangle.<br />
#** moment of force or moment of torque.<br />
#* Evaluate scalar triple products.<br />
#* Use the scalar triple product to determine the following:<br />
#** volume of a parallelepiped.<br />
#** whether or not three vectors are coplanar.<br />
#* Recall and apply the properties of the cross product and scalar triple product.<br />
#* Prove identities involving the cross product and the scalar triple product.<br />
#* Solve application problems involving the cross product and scalar triple product.<br />
# Lines<br />
#*Define the following:<br />
#** direction vector for a line<br />
#** vector equation of a line<br />
#** scalar parametric equations of a line<br />
#** Cartesian equations or symmetric form of a line<br />
#* Represent a line in 3-space by:<br />
#** a vector equation<br />
#** scalar parametric equations<br />
#** Cartesian equations<br />
#* Find the equation(s) representing a line given information about<br />
#** a point of the line and the direction of the line or<br />
#** two points contained in the line.<br />
#** a point and a parallel line.<br />
#** a point and perpendicular to a plane.<br />
#** two planes intersecting in the line.<br />
#* Find the distance from a point to a line.<br />
#* Solve application problems involving lines.<br />
# Planes<br />
#* Define the following:<br />
#** normal vector to a plane<br />
#** cartesian equation of a plane<br />
#** parametric equation of a plane<br />
#* Find the equation of a plane in 3-space given a point and a normal vector, three points, or a geometric description of the plane.<br />
#* Determine a normal vector and the intercepts of a given plane.<br />
#* Represent a plane by parametric equations.<br />
#* Find the distance from a point to a plane.<br />
#* Find the angle between a line and a plane.<br />
#* Determine a point of intersection between a line and a surface.<br />
#* Sketch planes given their equations.<br />
#* Solve application problems involving planes.<br />
# Systems of Linear Equations<br />
#* Define the following:<br />
#** linear system of m equations in n unknowns<br />
#** consistent and inconsistent<br />
#** solution set<br />
#** coefficient matrix<br />
#** elementary row operations<br />
#* Identify linear systems.<br />
#* Represent a system of linear equations as an augmented matrix and vice versa.<br />
#* Relate the following types of solution sets of a system of two or three variables to the intersections of lines in a plane or the intersection of planes in three space:<br />
#** a unique solution.<br />
#** infinitely many solutions.<br />
#** no solution.<br />
# Gaussian elimination<br />
#* Define the following:<br />
#** reduced row echelon form<br />
#** leading variables or pivots<br />
#** free variables<br />
#** row echelon form<br />
#** back substitution<br />
#** Gaussian elimination<br />
#** Gauss-Jordan elimination<br />
#** homogeneous<br />
#** trivial solution<br />
#** nontrivial solutions<br />
#* Identify matrices that are in row echelon form and reduced row echelon form.<br />
#* Determine whether a linear system is consistent or inconsistent from its reduced row echelon form. If the system is consistent, write the solution.<br />
#* Identify the lead variables and free variables of a system represented by an augmented matrix in reduced row echelon form.<br />
#* Solve systems of linear equations using Gaussian elimination and back substitution.<br />
#* Solve systems of linear equations using Gauss-Jordan elimination.<br />
#* Model and solve application problems using linear systems.<br />
# Matrices and Matrix Operations<br />
#* Define the following:<br />
#** vector, row vector, and column vector<br />
#** equal matrices<br />
#** scalar multiplication<br />
#** sum of matrices<br />
#** zero matrix<br />
#** scalar product<br />
#** linear combination<br />
#** matrix multiplication<br />
#** transpose<br />
#** trace<br />
#** identity matrix<br />
#* Perform the operations of matrix addition, scalar multiplication, transposition, trace, and matrix multiplication.<br />
#* Represent matrices in terms of double subscript notation.<br />
# Inverses; Rules of Matrix Arithmetic<br />
#* Define the following:<br />
#** commutative property<br />
#** singular<br />
#** nonsingular or invertible<br />
#** multiplicative inverse<br />
#* Recall, demonstrate, and apply algebraic properties for matrices.<br />
#* Recall that matrix multiplication is not commutative in general. Determine conditions under which matrices do commute.<br />
#* Recall and prove properties and identities involving the transpose operator.<br />
#* Recall and prove properties and identities involving matrix inverses.<br />
#* Recall and prove properties and identities involving matrix powers.<br />
#* Recall, demonstrate, and apply that the cancelation laws for scalar multiplication do not hold for matrix multiplication.<br />
#* Recall and apply the formula for the inverse of 2x2 matrices.<br />
# Elementary Matrices<br />
#* Define the following:<br />
#** elementary matrix<br />
#** row equivalent matrices<br />
#* Identify elementary matrices and find their inverses or show that their inverse does not exist.<br />
#* Relate elementary matrices to row operations.<br />
#* Factor matrices using elementary matrices.<br />
#* Find the inverse of a matrix, if possible, using elementary matrices.<br />
#* Prove theorems about matrix products and matrix inverses.<br />
#* Solve a linear equation using matrix inverses.<br />
# Further Results on Systems of Equations and Invertibility<br />
#* Solve matrix equations using matrix algebra.<br />
#* Recall and prove properties and identities involving matrix inverses.<br />
#* Recall equivalent conditions for invertibility.<br />
# Further Results on Systems of Equations and Invertibility<br />
#* Define the following:<br />
#** diagonal matrix<br />
#** upper and lower triangular matrices<br />
#** symmetric matrix<br />
#** skew-symmetric matrix<br />
#* Determine powers of diagonal matrices.<br />
#* Recall and prove properties and identities involving the transpose operator.<br />
#* Prove basic facts involving symmetric and skew-symmetric matrices.<br />
# Determinants<br />
#* Define the following:<br />
#** minor<br />
#** cofactor<br />
#** cofactor expansion<br />
#** determinant<br />
#** adjoint<br />
#** Cramer's Rule<br />
#* Apply cofactor expansion to evaluate determinants of nxn matrices.<br />
#* Recall and apply the properties of determinants to evaluate determinants.<br />
#* Evaluate the adjoint of a matrix.<br />
#* Determine whether or not a matrix has an inverse based on its determinant.<br />
#* Evaluate the inverse of a matrix using the adjoint method.<br />
#* Use Cramer's rule to solve a linear system.<br />
# Properties of Determinants<br />
#* Recall the effects that row operations have on the determinants of matrices. Relate to the determinants of elementary matrices.<br />
#* Recall, apply and verify the properties of determinants to evaluate determinants, including:<br />
#** det(AB) = det(A) det(B)<br />
#** det(kA) = k^n det(A)<br />
#** det(A^-1)= 1/det(A)<br />
#** det(A^T) = det(A)<br />
#** det(A) = 0 if and only if A is singular<br />
#* Evaluate the determinant of a matrix using row operations.<br />
#* Apply determinants to determine invertibility of matrix products.<br />
# Linear Transformations: Definitions and Examples<br />
#* Define the following:<br />
#** linear transformation<br />
#** image<br />
#** range<br />
#* Describe geometrically the effects of a linear operator.<br />
#* Determine whether or not a given transformation is linear.<br />
#* Prove theorems and solve application problems involving linear transformations.<br />
# Matrix Representations of Linear Transformations<br />
#* Define the following:<br />
#** standard matrix representation<br />
#** eigenvalues and eigenvectors<br />
#* Determine the matrix that represents a given linear transformation of vectors given an algebraic description.<br />
#* Determine the matrix that represents a given linear transformation of vectors given a geometric description.<br />
#* Prove theorems and solve application problems involving linear transformations.<br />
# Vector Spaces: Definitions and Examples<br />
#* Define the following:<br />
#** vector space<br />
#** vector space axioms<br />
#** vector space R^n<br />
#** vector space R^(mxn)<br />
#** vector space of real-valued functions<br />
#** additional properties of vector spaces<br />
#* Prove or disprove that a given set of vectors together with an addition and a scalar multiplication is a vector space.<br />
#* Prove and verify properties of a vector space.<br />
# Subspaces<br />
#* Define the following:<br />
#** subspace<br />
#** closure under addition<br />
#** closure under scalar multiplication<br />
#** zero subspace<br />
#** linear combination<br />
#** span (or subspace spanned by a set of vectors)<br />
#** spanning set<br />
#* Prove or disprove that a set of vectors forms a subspace.<br />
#* Prove or disprove a set of vectors is a spanning set for R^n.<br />
#* Prove or disprove a given vector is in the span of a set of vectors. Determine the span of a set of vectors.<br />
#* Prove theorems about vector spaces and spans.<br />
# Linear Independence<br />
#* Define the following:<br />
#** linearly independent<br />
#** linearly dependent<br />
#** Wronskian<br />
#* Determine whether a set of vectors is linearly dependent or linearly independent.<br />
#* Geometrically describe the span of a set of vectors. For sets that are linearly dependent, determine a dependence relation.<br />
#* Prove theorems about linear independence.<br />
# Basis and Dimension<br />
#* Define the following:<br />
#** basis<br />
#** dimension<br />
#** finite and infinite dimensional<br />
#** standard basis<br />
#* Prove or disprove a set of vectors forms a basis.<br />
#* Find a basis for a vector space.<br />
#* Determine the dimension of a vector space.<br />
#* Geometrically interpret the ideas of span, linear dependance, basis, and dimension.<br />
# Row Space, Column Space, and Null Space<br />
#* Define the following:<br />
#** row space<br />
#** column space<br />
#** null space<br />
#** particular solution<br />
#** general solution<br />
#* Express a product Ax as a linear combination of column vectors.<br />
#* Find a basis for a the column space, the row space, and the null space of a matrix.<br />
#* Find the basis for a span of vectors.<br />
# Rank and Nullity<br />
#* Define the following:<br />
#** rank<br />
#** nullity<br />
#** The Consistency Theorem<br />
#** equivalent statements of invertibility<br />
#* Find the rank and nullity of a matrix.<br />
#* Recall and prove identities involving rank and nullity<br />
#* Recall and apply the Consistency Theorm<br />
#* Recall and apply the equivalent statements of invertibility.<br />
# Eigenvalues and Eigenvectors<br />
#* Define the following:<br />
#** eigenvalue or characteristic value<br />
#** eigenvector or characteristic vector<br />
#** characteristic polynomial or characteristic polynomial<br />
#** equivalent statements of invertibility<br />
#* Find the eigenvalues and eigenvectors of an nxn matrix.<br />
#* Prove theorems and solve application problems involving eigenvalues and eigenvectors.<br />
# Diagonalization<br />
#* Define the following:<br />
#** diagonalizable<br />
#** algebraic multiplicity<br />
#** geometric multiplicity<br />
#* Determine whether or not a matrix is diagonalizable.<br />
#* Find the diagonalization of a matrix, if possible.<br />
#* Find powers of a matrix using the diagonalization of a matrix.<br />
#* Prove theorems and solve application problems involving the diagonalization of matrices.<br />
# Limit, Continuity, Vector Derivative; The Rules of Differentiation<br />
#* Define the following:<br />
#** scalar functions<br />
#** vector functions<br />
#** components of a vector function<br />
#** plane curve or space curve<br />
#** parametrization of a curve<br />
#** limit of a vector function<br />
#** a vector function continuous at a point<br />
#** derivative of a vector function<br />
#** a differentiable vector function<br />
#** integral of a vector function<br />
#* Graph a parametric curve.<br />
#* Identify a curve given its parametrization.<br />
#* Determine combinations of vector functions such as sums, vector products and scalar products.<br />
#* Evaluate limits, derivatives, and integrals of vector functions.<br />
#* Recall, derive and apply rules to combinations of vector functions for the following:<br />
#** limits<br />
#** differentiation<br />
#** integration<br />
#* Determine continuity of a vector-valued function.<br />
#* Prove theorems involving limits and derivatives of vector-valued functions.<br />
#* Solve application problems involving vector-valued functions.<br />
# Curves; Vector Calculus in Mechanics<br />
#* Define the following:<br />
#** directed path<br />
#** differentiable parameterized curve<br />
#** tangent vector<br />
#** tangent line<br />
#** unit tangent vector<br />
#** principal normal vector<br />
#** normal line<br />
#** osculation plane<br />
#** force vector<br />
#** momentum vector<br />
#** angular momentum vector<br />
#** torque<br />
#* Find the tangent vector and tangent line to a curve at a given point.<br />
#* Find the principle normal and normal line to a curve at a given point.<br />
#* Determine the osculating plane for a space curve at a given point.<br />
#* Reverse the direction of a curve.<br />
#* Solve application problems involving curves.<br />
#* Solve application problems involving force, momentum, angular momentum, and torque.<br />
# Arc Length<br />
#* Define the following:<br />
#** arc length<br />
#** arc length parametrization<br />
#* Evaluate the arc length of a curve.<br />
#* Determine whether a curve is arc length parameterized.<br />
#* Find the arc length parametrization of a curve.<br />
# Curvilinear Motion; Curvature<br />
#* Define the following:<br />
#** velocity vector function<br />
#** speed<br />
#** acceleration vector function<br />
#** uniform circular motion<br />
#** curvature<br />
#** tangential component of acceleration<br />
#** normal component of acceleration<br />
#* Given the position vector function of a moving object, calculate the velocity vector function, speed, and acceleration vector function, and vice versa.<br />
#* Calculate the curvature of a space curve.<br />
#* Recall the formulas for the curvature of a parameterized planar curve or a planar curve that is the graph of a function. Apply these formulas to calculate the curvature of a planar curve.<br />
#* Determine the tangential and normal components of acceleration for a given parameterized curve.<br />
#* Solve application problems involving curvilinear motion and curvature.<br />
# Functions of Several Variables; A Brief Catalogue of the Quadric Surfaces; Projections<br />
#* Define the following:<br />
#** real-valued function of several variables<br />
#** domain<br />
#** range<br />
#** bounded functions<br />
#** quadric surface<br />
#** intercepts<br />
#** traces<br />
#** sections<br />
#** center<br />
#** symmetry<br />
#** boundedness<br />
#** cylinder<br />
#** ellipsiod<br />
#** elliptic cone<br />
#** elliptic paraboloid<br />
#** hyperboloid of one sheet<br />
#** hyperboloid of two sheets<br />
#** hyperbolic paraboloid<br />
#** parabolic cylinder<br />
#** elliptic cylinder<br />
#** projection of a curve onto a coordinate plane<br />
#* Describe the domain and range of a function of several variables.<br />
#* Write a function of several variables given a description.<br />
#* Identify standard quadratic surfaces given their functions or graphs.<br />
#* Sketch the graph of a quadratic surface by sketching intercepts, traces, sections, centers, symmetry, boundedness.<br />
#* Find the projection of a curve, that is the intersection of two surfaces, to a coordinate plane.<br />
# Graphs; Level Curves and Level Surfaces<br />
#* Define the following:<br />
#** level curve<br />
#** level surface<br />
#* Describe the level sets of a function of several variables.<br />
#* Graphically represent a function of two variables by level curves or a function of three variables by level surfaces.<br />
#* Identify the characteristics of a function from its graph or from a graph of its level curves (or level surfaces).<br />
#* Solve application problems involving level sets. functions.<br />
# Partial Derivatives<br />
#* Define the following:<br />
#** partial derivative of a function of several variables<br />
#** second partial derivative<br />
#** mixed partial derivative<br />
#* Interpret the definition of a partial derivative of a function of two variables graphically.<br />
#* Evaluate the partial derivatives of a function of several variables.<br />
#* Evaluate the higher order partial derivatives of a function of several variables.<br />
#* Verify equations involving partial derivatives.<br />
#* Apply partial derivatives to solve application problems.<br />
# Open and Closed Sets; Limits and Continuity; Equity of Mixed Partials<br />
#* Define the following:<br />
#** neighborhood of a point<br />
#** deleted neighborhood of a point<br />
#** interior of a set<br />
#** boundary of a set<br />
#** open set<br />
#** closed set<br />
#** limit of a function of several variables at a point<br />
#** continuity of a function of several variables at a point<br />
#* Determine the boundary and interior of a set.<br />
#* Determine whether a set is open, closed, neither, or both.<br />
#* Evaluate the limit of a function of several variables or show that it does not exists.<br />
#* Determine whether or not a function is continuous at a given point.<br />
#* Recall and apply the conditions under which mixed partial derivatives are equal.<br />
# Differentiability and Gradient<br />
#* Define the following:<br />
#** differentiable multivariable function<br />
#** gradient of a multivariable function<br />
#* Evaluate the gradient of a function.<br />
#* Find a function with a given gradient.<br />
# Gradient and Directional Derivative<br />
#* Define the following:<br />
#** directional derivative<br />
#** isothermals<br />
#* Recall and prove identities involving gradients.<br />
#* Give a graphical interpretation of the gradient.<br />
#* Evaluate the directional derivative of a function.<br />
#* Give a graphical interpretation of directional derivative.<br />
#* Recall, prove, and apply the theorem that states that a differential function f increases most rapidly in the direction of the gradient (the rate of change is then ||f(x)||) and it decreases most rapidly in the opposite direction (the rate of change is then -||f(x)||).<br />
#* Find the path of a heat seeking or a heat repelling particle.<br />
#* Solve application problems involving gradient and directional derivatives.<br />
# The Mean-Value Theorem; The Chain Rule<br />
#* Define the following:<br />
#** the Mean Value Theorem for functions of several variables<br />
#** normal line<br />
#** chain rules for functions of several variables<br />
#** implicit differentiation<br />
#* Recall and apply the Mean Value Theorem for functions of several variables and its corollaries.<br />
#* Apply an appropriate chain rule to evaluate a rate of change.<br />
#* Apply implicit differentiation to evaluate rates of change.<br />
#* Solve application problems involving chain rules and implicit differentiation.<br />
# The Gradient as a Normal; Tangent Lines and Tangent Planes<br />
#* Define the following:<br />
#** normal vector<br />
#** tangent vector<br />
#** tangent line<br />
#** tangent plane<br />
#** normal line<br />
#* Use gradients to find the normal vector and normal line to a smooth planar curve at a given point.<br />
#* Use gradients to find the tangent vector and tangent line to a smooth planar curve at a given point.<br />
#* Use gradients to find the normal vector to a smooth surface at a given point.<br />
#* Use gradients to find the tangent plane to a smooth surface at a given point.<br />
#* Use gradients to find the normal line to a smooth surface at a given point.<br />
#* Solve application problems involving normals and tangents to curves and surfaces.<br />
# Local Extreme Values<br />
#* Define the following:<br />
#** local minimum and local maximum<br />
#** critical points<br />
#** stationary points<br />
#** saddle points<br />
#** discriminant<br />
#** Second Derivative Test<br />
#* Find the critical points of a function of two variables.<br />
#* Apply the Second-Partials Test to determine whether each critical point is a local minimum, a local maximum, or a saddle point.<br />
#* Solve word problems involving local extreme values.<br />
# Absolute Extreme Values<br />
#* Define the following:<br />
#** absolute minimum and absolute maximum<br />
#** bounded subset of a plane or three-space<br />
#** the Extreme Value Theorem<br />
#* Determine absolute extreme values of a function defined on a closed and bounded set.<br />
#* Apply the Extreme Value Theorem to justify the method for finding extreme values of functions defined on certain sets.<br />
#* Solve word problems involving absolute extreme values.<br />
# Maxima and Minima with Side Conditions<br />
#* Define the following:<br />
#** side conditions or constraints<br />
#** method of Lagrange<br />
#** Lagrange multipliers<br />
#** cross-product equation of the Lagrange condition<br />
#* Graphically interpret the method of Lagrange.<br />
#* Determine the extreme values of a function subject to a side conditions by applying the method of Lagrange.<br />
#* Apply the cross-product equation of the Lagrange condition to solve extreme value problems subject to side conditions.<br />
#* Apply the method of Lagrange to solve word problems.<br />
# Differentials; Reconstructing a Function from its Gradient<br />
#* Define the following:<br />
#** differential<br />
#** general solution<br />
#** particular solution<br />
#** connected open set<br />
#** open region<br />
#** simple closed curve<br />
#** simply connected open region<br />
#** partial derivative gradient test<br />
#* Determine the differential for a given function of several variables.<br />
#* Determine whether or not a vector function is a gradient.<br />
#* Given a vector function that is a gradient, find the functions with that gradient.<br />
# Multiple-Sigma Notation; The Double Integral over a Rectangle R; The Evaluation of Double Integrals by Repeated Integrals<br />
#* Define the following:<br />
#** double sigma notation<br />
#** triple sigma notation<br />
#** upper sum<br />
#** lower sum<br />
#** double integral<br />
#** integral formula for the volume of a solid bounded between a region Omega in the xy-plane and the graph of a non-negative function z = f(x,y) defined on Omega.<br />
#** integral formula for the area of region in a plane<br />
#** integral formula for the average of a function defined on a region Omega.<br />
#** projection of a region onto a coordinate axis<br />
#** Type I and Type II regions<br />
#** reduction formulas for double integrals<br />
#** the geometric interpretation of the reduction formulas for double integrals<br />
#* Evaluate double and triple sums given their sigma notation.<br />
#* Recall and apply summation identities.<br />
#* Approximate the integral of a function by a lower sum and an upper sum.<br />
#* Evaluate the integral of a function using the definition.<br />
#* Evaluate double integrals over a rectangle using the reduction formulas.<br />
#* Sketch planar regions and determine if they are Type I, Type II, or both.<br />
#* Evaluate double integrals over Type I and Type II regions.<br />
#* Change the order of integration of an integral.<br />
#* Apply double integrals to calculate volumes, areas, and averages.<br />
# The Double Integral as the Limit of Riemann Sums; Polar Coordinates<br />
#* Define the following:<br />
#** diameter of a set<br />
#** Riemann sum<br />
#** double integral as a limit of Riemann sums<br />
#** polar coordinates (r; theta)<br />
#** transformation formulas between Cartesian and polar coordinates<br />
#** double integral conversion formula between Cartesian and polar coordinates<br />
#* Represent a region in both Cartesian and polar coordinates.<br />
#* Evaluate double integrals in terms of polar coordinates.<br />
#* Evaluate areas and volumes using polar coordinates.<br />
#* Convert a double integral in Cartesian coordinates to a double integral in polar coordinates and then evaluate.<br />
# Further Applications of the Double Integral<br />
#* Define the following:<br />
#** integral formula for the mass of a plate<br />
#** integral formulas for the center of mass of a plate<br />
#** integral formulas for the centroid of a plate<br />
#** integral formulas for the moment of an inertia of a plate<br />
#** radius of gyration<br />
#** the Parallel Axis Theorem<br />
#* Evaluate the mass and center or mass of a plate<br />
#* Evaluate the centroid of a plate.<br />
#* Evaluate the moments of inertia of a plate.<br />
#* Calculate the radius of gyration of a plate.<br />
#* Recall and apply the parallel axis theorem.<br />
# Triple Integrals; Reduction to Repeated Integrals<br />
#* Define the following:<br />
#** triple integral<br />
#** integral formula for the volume of a solid<br />
#** integral formula for the mass of a solid<br />
#** integral formulas for the center of mass of a solid<br />
#* Evaluate physical quantities using triple integrals such as volume, mass, center of mass, and moments of intertia.<br />
#* Recall and apply the properties of triple integrals, including: linearity, order, additivity, and the mean-value condition.<br />
#* Sketch the domain of integration of an iterated integral.<br />
#* Change the order of integration of a triple integral.<br />
# Cylindrical Coordinates<br />
#* Define the following:<br />
#** cylindrical coordinates of a point<br />
#** coordinate transformations between Cartesian and cylindrical coordinates<br />
#** cylindrical element of volume<br />
#* Convert between Cartesian and cylindrical coordinates.<br />
#* Describe regions in cylindrical coordinates.<br />
#* Evaluate triple integrals using cylindrical coordinates.<br />
# Spherical Coordinates<br />
#* Define the following:<br />
#** spherical coordinates of a point<br />
#** coordinate transformations between Cartesian and spherical coordinates<br />
#** spherical element of volume<br />
#* Convert between Cartesian and spherical coordinates.<br />
#* Describe regions in spherical coordinates.<br />
#* Evaluate triple integrals using spherical coordinates.<br />
# Jacobians; Changing Variables in Multiple Integration<br />
#* Define the following:<br />
#** Jacobian<br />
#** change of variable formula for double integration<br />
#** change of variable formula for triple integration<br />
#* Find the Jacobian of a coordinate transformation.<br />
#* Use a coordinate transformation to evaluate double and triple integrals.<br />
# Line Integrals<br />
#* Define the following:<br />
#** work along a curved path<br />
#** smooth parametric curve<br />
#** directed or oriented curve<br />
#** path dependence<br />
#** closed curve<br />
#* Evaluate the work done by a varying force over a curved path.<br />
#* Evaluate line integrals in general including line integrals with respect to arc length.<br />
#* Evaluate the physical characteristics of a wire such as centroid, mass, and center of mass using line integrals.<br />
#* Determine whether or not a vector field is a gradient.<br />
#* Determine whether or not a differential form is exact.<br />
# The Fundamental Theorem for Line Integrals; Work-Energy Formula; Conservation of Mechanical Energy<br />
#* Define the following:<br />
#** path-independent line integrals<br />
#** closed vector field<br />
#** simply connected<br />
#* Recall, apply, and verify the Fundamental Theorem for Line Integrals (Theorem 2 in Section 15.3).<br />
#* Determine whether or not a force field is closed on a given region, and if so, find its potential function.<br />
#* Solve application problems involving work done by a conservative vector field<br />
# Vector Fields<br />
#* Define the following:<br />
#** vector field<br />
#** open<br />
#** path connected<br />
#** region<br />
#** integral curve (field lines, flow lines, or streamlines)<br />
#** gradient vector field (or conservative vector field)<br />
#** potential function<br />
#** continuously differentiable vector field<br />
#* Sketch a vector field.<br />
#* Write the formula for a vector field from a description.<br />
#* Write the gradient vector field associated with a given scalar-valued function.<br />
#* Recover a function from its gradient or show it is not possible.<br />
#* Find the integral curves of a vector field.<br />
# Green's Theorem<br />
#* Define the following:<br />
#** Jordan curve<br />
#** Jordan region<br />
#** Green's Theorem<br />
#* Recall and verify Green's Theorem.<br />
#* Apply Green's Theorem to evaluate line integrals.<br />
#* Apply Green's Theorem to find the area of a region.<br />
#* Derive identities involving Green's Theorem<br />
# Parameterized Surfaces; Surface Area<br />
#* Define the following:<br />
#** parameterized surface<br />
#** fundamental vector product<br />
#** element of surface area for a parameterized surface<br />
#** surface integral<br />
#** integral formula for the surface area of a parameterized surface<br />
#** integral formula for the surface area of a surface z = f(x; y)<br />
#** upward unit normal<br />
#* parameterize a surface.<br />
#* evaluate the fundamental vector product for a parameterized surface.<br />
#* Calculate the surface area of a parameterized surface.<br />
#* Calculate the surface area of a surface z = f(x; y).<br />
# Surface Integrals<br />
#* Define the following:<br />
#** surface integral<br />
#** integral formulas for the surface area and centroid of a parameterized surface<br />
#** integral formulas for the mass and center of mass of a parameterized surface<br />
#** integral formulas for the moments of inertia of a parameterized surface<br />
#** integral formula for flux through a surface<br />
#* Calculate the surface area and centroid of a parameterized surface.<br />
#* Calculate the mass and center of mass of a parameterized surface.<br />
#* Calculate the moments of inertia of a parameterized surface.<br />
#* Evaluate the flux of a vector field through a surface.<br />
#* Solve application problems involving surface integrals.<br />
# The Vector Differential Operator Del<br />
#* Define the following:<br />
#** the vector differential operator Del<br />
#** divergence<br />
#** curl<br />
#** Laplacian<br />
#* Evaluate the divergence of a vector field.<br />
#* Evaluate the curl of a vector field<br />
#* Evaluate the Laplacian of a function.<br />
#* Recall, derive and apply formulas involving divergence, gradient and Laplacian.<br />
#* Interpret that divergence and curl of a vector fields physically.<br />
# The Divergence Theorem<br />
#* Define the following:<br />
#** outward unit normal<br />
#** the divergence theorem<br />
#** sink and source<br />
#** solenoidal<br />
#* Recall and verify the Divergence Theorem.<br />
#* Apply the Divergence Theorem to evaluate the flux through a surface.<br />
#* Solve application problems using the Divergence Theorem.<br />
# Stokes' Theorem<br />
#* Define the following:<br />
#** oriented surface<br />
#** outward, upward, and downward unit normal<br />
#** the positive sense around the boundary of a surface<br />
#** circulation<br />
#** component of curl in the normal direction<br />
#** irrotational<br />
#** Stokes' theorem<br />
#* Recall and verify Stoke's theorem.<br />
#* Use Stokes' Theorem to calculate the flux of a curl vector field through a surface by a line integral.<br />
#* Apply Stokes' theorem to calculate the work (or circulation) of a vector field around a simple closed curve.<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 />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
[[Math 303]]<br />
<br />
[[Category:Courses|302]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_636_Advanced_Probability_1&diff=3667Math 636 Advanced Probability 12018-03-29T22:45:44Z<p>Ls5: Ls5 moved page Math 636 to Math 636 Advanced Probability 1</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Advanced Probability 1.<br />
<br />
=== Credit Hours ===<br />
3<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_636&diff=3668Math 6362018-03-29T22:45:44Z<p>Ls5: Ls5 moved page Math 636 to Math 636 Advanced Probability 1</p>
<hr />
<div>#REDIRECT [[Math 636 Advanced Probability 1]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_636_Advanced_Probability_1&diff=3666Math 636 Advanced Probability 12018-03-29T22:43:46Z<p>Ls5: </p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Advanced Probability 1.<br />
<br />
=== Credit Hours ===<br />
3<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_636_Advanced_Probability_1&diff=3665Math 636 Advanced Probability 12018-03-29T22:37:19Z<p>Ls5: Replaced content with "{{db-g7}}"</p>
<hr />
<div>{{db-g7}}</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_637:_Advanced_Probability_2&diff=3664Math 637: Advanced Probability 22018-03-29T22:31:22Z<p>Ls5: </p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Advanced Probability 2.<br />
<br />
=== Credit Hours ===<br />
3<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_637:_Advanced_Probability_2&diff=3663Math 637: Advanced Probability 22018-03-29T22:31:06Z<p>Ls5: </p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Advanced Probability 2.<br />
<br />
=== Credit Hours ===<br />
3<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_544:_Advanced_Probability_2&diff=3662Math 544: Advanced Probability 22018-03-29T22:28:40Z<p>Ls5: </p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Advanced Probability 2.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Prerequisite ===<br />
[[Math 543]].<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 543]] 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 544 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 />
# Convergence of random variables<br />
#* Almost sure<br />
#* In probability<br />
#* In mean<br />
#* In distribution<br />
# Laws of Large Numbers<br />
#* Weak Law<br />
#* Strong Law<br />
# Stochastic processes<br />
#* Gaussian<br />
#* Stationary<br />
#* Stationary increments<br />
#* Independent increments<br />
#* Filtrations<br />
#** Adapted processes<br />
#** Predictable processes<br />
#** Stopping times<br />
# Ergodic theory<br />
#* Birkhoff's Ergodic Theorem<br />
#* Mixing<br />
# Martingales<br />
#* Submartingales and supermartingales<br />
#* Doob Decomposition Theorem<br />
#* Optional Stopping Theorem<br />
#* Optional Sampling Theorem<br />
#* Martingale Convergence Theorem<br />
#* Convergence of backwards martingales<br />
# Brownian motion<br />
#* Definition<br />
#* Existence<br />
#* Path properties<br />
# Itô integral<br />
#* With respect to Brownian motion<br />
#* With respect to diffusion processes<br />
#* Itô 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 />
None</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_636_Advanced_Probability_1&diff=3661Math 636 Advanced Probability 12018-03-29T22:26:34Z<p>Ls5: </p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Advanced Probability 1.<br />
<br />
=== Credit Hours ===<br />
3<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]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_636:_Advanced_Probability_1&diff=3660Math 636: Advanced Probability 12018-03-29T22:26:05Z<p>Ls5: Created page with "== Catalog Information == === Title === Advanced Probability 1. === Credit Hours === 3 === Prerequisite === Math 314 and Math 341; and Math 431 or Stat 370; or..."</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Advanced Probability 1.<br />
<br />
=== Credit Hours ===<br />
3<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_637:_Advanced_Probability_2&diff=3659Math 637: Advanced Probability 22018-03-29T22:24:33Z<p>Ls5: Created page with "== Catalog Information == === Title === Advanced Probability 2. === Credit Hours === 3 === Prerequisite === Math 636. === Recommended === Math 341, [[Math 342|342]..."</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Advanced Probability 2.<br />
<br />
=== Credit Hours ===<br />
3<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_543:_Advanced_Probability_1&diff=3658Math 543: Advanced Probability 12018-03-29T22:15:22Z<p>Ls5: </p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Advanced Probability 1.<br />
<br />
=== Credit Hours ===<br />
3<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 />
#* Bayes' Formula<br />
#* Regular conditional distributions<br />
# Probability measures on product spaces<br />
#* Product measures<br />
#* Kolmogorov Extension Theorem<br />
# Generating functions<br />
# Discrete Markov chains<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 544]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_637&diff=3657Math 6372018-03-29T19:02:03Z<p>Ls5: </p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Advanced Probability 2.<br />
<br />
=== Credit Hours ===<br />
3<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 646]] 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 544 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 />
# Itô'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 />
None</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_637&diff=3656Math 6372018-03-29T19:01:29Z<p>Ls5: Created page with "== Catalog Information == === Title === Advanced Probability 2. === Credit Hours === 3 === Prerequisite === Math 636. === Recommended === Math 341, [[Math 342|342]..."</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Advanced Probability 2.<br />
<br />
=== Credit Hours ===<br />
3<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 543]] 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 544 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 />
# Itô'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 />
None</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_636_Advanced_Probability_1&diff=3655Math 636 Advanced Probability 12018-03-29T18:54:57Z<p>Ls5: </p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Advanced Probability 1.<br />
<br />
=== Credit Hours ===<br />
3<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_636_Advanced_Probability_1&diff=3654Math 636 Advanced Probability 12018-03-29T18:52:55Z<p>Ls5: Created page with "== Catalog Information == === Title === Advanced Probability 1. === Credit Hours === 3 === Prerequisite === Math 314 and Math 341; and Math 431 or Stat 370; or..."</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Advanced Probability 1.<br />
<br />
=== Credit Hours ===<br />
3<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_102:_Quantitative_Reasoning&diff=3632Math 102: Quantitative Reasoning2017-10-13T21:39:11Z<p>Ls5: /* Title */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Quantitative Reasoning<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F, W<br />
<br />
=== Description ===<br />
Practicing and applying quantitative reasoning: personal finance, consumer statistics, etc.<br />
<br />
=== Note ===<br />
For students who do not need developmental algebra for subsequent courses.<br />
<br />
<br />
<br />
=== Prerequisites ===<br />
<br />
There are no prerequisites for this course.<br />
<br />
=== Minimal learning outcomes ===<br />
1. Solve problems using dimensional analysis.<br />
<br />
2. Identify the uses and abuses of percentages in real life applications.<br />
<br />
3. Calculate interest, payments, and earnings on mortgages, lines of credit, annuities, and other interest bearing investments and debts.<br />
<br />
4. Analyze statistical studies and judge their validity.<br />
<br />
5. Calculate probabilities associated with normally distributed data, and identify data that is likely to be normally distributed.<br />
<br />
6. Calculate probabilities of simple events and properly combine probabilities of independent events.<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Bennett and Brigs: Using and Understanding Mathematics<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|102]]</div>Ls5