https://math.byu.edu/wiki/api.php?action=feedcontributions&user=Ls5&feedformat=atomMathWiki - User contributions [en]2019-02-21T05:56:49ZUser contributionsMediaWiki 1.26.3https://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>Ls5https://math.byu.edu/wiki/index.php?title=Math_465:_Intro_to_Differential_Geometry&diff=2537Math 465: Intro to Differential Geometry2016-02-05T20:48:50Z<p>Ls5: </p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Introduction to Differential Geometry.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F (odd years)<br />
<br />
=== Prerequisite ===<br />
[[Math 314]]; [[Math 341|341]] or equivalent.<br />
<br />
=== Recommended ===<br />
[[Math 342]] or equivalent.<br />
<br />
=== Description ===<br />
Geometry of smooth curves and surfaces. Topics include the first and second fundamental forms, the Gauss map, orientability of surfaces, Gaussian and mean curvature, geodesics, minimal surfaces and the Gauss-Bonnet Theorem.<br />
<br />
== Desired Learning Outcomes ==<br />
The main purpose of this course is to provide students with an understanding of the geometry of curves and surfaces, with the focus being on the theoretical and logical foundations of differential geometry. <br />
<br />
=== Prerequisites ===<br />
The prerequisite of [[Math 314]] is to ensure that students have some understanding of partial derivatives and differentiation for functions of more than one variable. [[Math 341|341]] is required so students have a rigorous understanding of the real number system and of real-valued functions. [[Math 342]] is recommended so students have a rigorous understanding of functions with more than one variable.<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
Outlined below are topics that all successful Math 465 students should understand well. As evidence of that understanding, students should be able to demonstrate mastery of all relevant vocabulary, familiarity with common examples and counterexamples, knowledge of the content of the major theorems, understanding of the ideas in their proofs, and ability to make direct application of those results to related problems.<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
# Basic Properties of Curves<br />
#* Parametrized curves<br />
#* Regular curves<br />
#* Arc length<br />
# Regular surfaces<br />
#* Regular surfaces as inverse images of regular values<br />
#* Change of parameters<br />
#* The tangent plane<br />
#* The first fundamental form<br />
#*Orientation of surfaces<br />
# The geometry of the Gauss map<br />
#* Definition of the Gauss map and fundamental properties<br />
#* The Gauss map in local coordinates<br />
#* Minimal surfaces<br />
#Intrinsic geometry of surfaces<br />
#* Isometries and conformal maps<br />
#* Geodesics and parallel transport<br />
#* The Gauss-Bonnet Theorem and applications<br />
#* The exponential map<br />
<br />
</div><br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*Manfredo P. Do Carmo, <i>Differential Geometry of Curves and Surfaces</i>, Prentice Hall, 1976.<br />
<br />
=== Additional topics ===<br />
<br />
Among other topics instructors may want to cover Jacobi fields and conjugate points, covering spaces, and the Hopf-Rinow Theorem.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
None<br />
<br />
[[Category:Courses|465]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_451:_Intro_to_Topology&diff=2536Math 451: Intro to Topology2016-02-05T20:47:50Z<p>Ls5: /* Offered */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Introduction to Topology.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
W (even years)<br />
<br />
=== Prerequisite ===<br />
[[Math 341]].<br />
<br />
=== Description ===<br />
Developing topological concepts, beginning from a linear setting. Developing proofs or counterexamples from axioms to a structured sequence of topological propositions using only notes provided.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
1. Students should demonstrate an ability to write and present mathematical proofs beyond that learned in Math 290.<br />
<br />
2. Students should demonstrate understanding of the following concepts by proving theorems about them: linear ordering, connectedness, continuity, open and closed sets, density, compactness, local connectedness, local compactness.<br />
<br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|451]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_450:_Combinatorics&diff=2535Math 450: Combinatorics2016-02-05T20:47:29Z<p>Ls5: /* Offered */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Combinatorics.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
Su<br />
<br />
=== Prerequisite ===<br />
[[Math 371]].<br />
<br />
=== Description ===<br />
Permutations, combinations, recurrence relations, applications. Students will learn the basics of combinatorics and its relation to the other areas of mathematics, including algebra and analysis.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
<br />
Math 371.<br />
<br />
=== Minimal learning outcomes ===<br />
Permutations and combinations basics, including the Pigeonhole Principle, binomial coefficients and the Binomial Theorem, Stirling's Approximation, Inclusion/exclusion, Generating functions and recurrence relations (rational functions), Groups, permutations and counting problems -- Polya's Theorem<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 />
* Chapters 5-9 of Alan Tucker's ''Applied Combinatorics'', John Wiley & Sons, Inc., 2002.<br />
* Chapters 1-4 of Russell Merris' ''Combinatorics'', 2nd edition, John Wiley & Sons, Inc., 2003.<br />
* Richard A. Brualdi, ''Introductory Combinatorics (5th Edition)'', Prentice Hall, 2010.<br />
<br />
=== Additional topics ===<br />
Possibly Ramsey theory<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|450]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_447:_Intro_to_Partial_Differential_Equations&diff=2534Math 447: Intro to Partial Differential Equations2016-02-05T20:47:06Z<p>Ls5: /* Offered */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Introduction to Partial Differential Equations.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F (even years)<br />
<br />
=== Prerequisite ===<br />
[[Math 303]]; or [[Math 314|314]] and [[Math 334|334]].<br />
<br />
=== Description ===<br />
Boundary value problems; transform methods; Fourier series; Bessel functions; Legendre polynomials.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
The main purpose of this course is to teach students how to solve the canonical linear second-order partial differential equations on simple domains. Secondarily, students should be introduced to the theory concerning the validity of such solutions.<br />
<br />
=== Prerequisites ===<br />
<br />
Current prerequisites ensure that students have had instruction in multivariable calculus and ordinary differential equations.<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
Primarily, students should be able to use the solution techniques described below. Students should gain a basic understanding of issues concerning solvability and convergence, but the current prerequisites don't guarantee that incoming students will have had any prior exposure to the theory of the convergence of sequences of functions, so expectations in that area are modest.<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
# Basic classification of PDEs<br />
#* Linearity<br />
#* Homogeneity <br />
#* Order<br />
#* Elliptic, parabolic, or hyperbolic<br />
# Basic Modeling<br />
#* Derivation of the heat equation<br />
#* Derivation of the wave equation<br />
# Basic principles, techniques, and theory<br />
#* Principle of superposition<br />
#* Method of separation of variables<br />
#* Definition of eigenvalues and eigenfunctions corresponding to two-point BVPs<br />
#* Basic Sturm-Liouville theory<br />
# Special eigensystems<br />
#* Fourier<br />
#** Series representations<br />
#*** Effect of symmetry and modifications and combinations of functions<br />
#*** Theorems on pointwise, uniform, and ''L''<sup>2</sup> convergence<br />
#**** Bessel's Inequality and Parseval's Equation<br />
#** Integral representations<br />
#* Bessel<br />
#* Legendre<br />
# Representation of solutions to the canonical equations on simple domains<br />
#* Laplace's equation on rectangles, rectangular strips, quarter-planes, half-planes, disks, and balls<br />
#* Wave equation on bounded intervals, half-lines, lines, disks, and balls<br />
#* Heat equation on bounded intervals, half-lines, lines, rectangles, disks, and balls<br />
</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Richard Haberman, ''Applied Partial Differential Equations (4th Edition)'', Prentice Hall, 2003.<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
Students taking [[Math 511]] are supposed to have had either Math 447 or [[Math 303]]. It is proposed that Math 447 become a prerequisite (or at least recommended) for [[Math 547]], so that there will be less duplication of material in the PDE curriculum.<br />
<br />
[[Category:Courses|447]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_435:_Mathematical_Finance&diff=2533Math 435: Mathematical Finance2016-02-05T20:46:40Z<p>Ls5: /* Offered */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Mathematical Finance.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
W (odd years)<br />
<br />
=== Prerequisite ===<br />
[[One of Math 431, Stat 341, Stat 370]].<br />
<br />
=== Description ===<br />
The binomial asset pricing model (discrete probability). Martingales, pricing of derivative securities, random walk in financial models, random interest rates.<br />
<br />
== Desired Learning Outcomes ==<br />
The minimal expectation for this course is that students learn about mathematical finance ''in the context of discrete time and finite state-spaces.'' It is therefore not required that students be taught about Brownian motion, the Black-Scholes model, etc.<br />
<br />
=== Prerequisites ===<br />
Students should have had an introductory course in probability.<br />
<br />
=== Minimal learning outcomes ===<br />
Within the context mentioned above, students should be able to compute prices for derivative securities. They should be conversant with the standard terminology of mathematical finance and be able to use this terminology correctly in answering questions. At a minimum, students should understand the following concepts in the context of binomial decision trees:<br />
<div style="-moz-column-count:2; column-count:2;"><br />
# Martingales<br />
# Markov processes<br />
# Arbitrage<br />
# Risk neutrality<br />
# State prices<br><br><br />
# Options<br />
#* Call and put<br />
#* American and European<br />
# Stopping times<br />
# Simple random walks<br />
# Interest rate models<br />
</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Steven E. Shreve, ''Stochastic Calculus for Finance I: The Binomial Asset Pricing Model'', Springer, 2005.<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
None.<br />
<br />
[[Category:Courses|435]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_425:_Mathematical_Biology&diff=2532Math 425: Mathematical Biology2016-02-05T20:46:17Z<p>Ls5: /* Offered */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title === <br />
Mathematical Biology.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
W (even years)<br />
<br />
=== Prerequisite ===<br />
[[Math 334]].<br />
<br />
=== Description ===<br />
Using tools in mathematics to help biologists. Motivating new mathematics with questions in biology.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
Students should gain a familiarity with how the disciplines of mathematics and biology can complement each other.<br />
<br />
=== Prerequisites ===<br />
<br />
A knowledge of calculus (and the mathematical maturity that having passed [[Math 112]] entails) should suffice.<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
Students should become familiar with discrete and continuous models of biological phenomena. They should know the technical terms, and be able to implement the procedures taught in the course to solve problems based on these models. Possible topics include:<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
# Signal Transduction<br />
#* Menten Michaelis enzyme dynamics<br />
#* Law of mass action<br />
#* Dynamical systems<br />
#* Bifurcation<br />
# Example systems<br />
#* Fitzhugh-Nagumo<br />
#* Nerve and heart dynamics<br />
#* Cell cycle model<br />
#* cAMP<br />
# Population models<br />
#* Continuous predator-prey<br />
#* Age structured models<br />
#* Discrete dynamical systems<br />
#* Time delayed differential equations<br />
#* Stochastic models<br />
</div><br />
<br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* A Course in Mathematical Biology. Quantitative Modeling with Mathematical and Computational Methods. By Gerda de Vries, Thomas Hillen, Mark Lewis, Johannes Muller, Birgitt Schonfisch<br />
<br />
=== Additional Topics ===<br />
<br />
These are at the discretion of the instructor as time allows.<br />
<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
None.<br />
<br />
[[Category:Courses|425]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_431:_Probability_Theory&diff=2531Math 431: Probability Theory2016-02-05T20:45:53Z<p>Ls5: /* Offered */</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 313]].<br />
<br />
=== Description ===<br />
Axiomatic probability theory, conditional probability, discrete / continuous random variables, expectation, conditional expectation, moments, functions of random variables, multivariate distributions, laws of large numbers, central limit theorem.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
This course is a calculus-based first course in probability. It is cross-listed with EC En 370.<br />
<br />
=== Prerequisites ===<br />
<br />
The current prerequisite is linear algebra. Because of the need to work with joint distributions of continuous random variables in Math 431, the department should consider adding multivariable calculus as a prerequisite.<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
Primarily, students should be able to do basic computation of probabilistic quantities, including those involving applications. Students should be able to recall the most common types of discrete and continuous random variables and describe and compute their properties. Students should understand the theory of probability <i>in an elementary context</i>.<br />
<br />
<div style="column-count:2;-moz-column-count:2;-webkit-column-count:2"><br />
# Basic principles of counting<br />
#* Product sets<br />
#* Disjoint unions<br />
#* Combinations<br />
#* Permutations<br />
# Axiomatic probability<br />
#* Outcomes<br />
#* Events<br />
#* Probability measures<br />
#** Additivity<br />
#** Continuity<br />
# Discrete random variables<br />
#* Probability mass function<br />
#* Cumulative distribution function<br />
#* Moments<br />
#** Expectation<br />
#*** Of a function of a random variable<br />
#** Variance<br />
#* Common types<br />
#** Bernoulli<br />
#** Binomial<br />
#** Poisson <br><br><br><br><br />
# Continuous random variables<br />
#* Probability density function<br />
#* Cumulative distribution function<br />
#* Moments<br />
#** Expectation<br />
#*** Of a function of a random variable<br />
#** Variance<br />
#* Common types<br />
#** Uniform<br />
#** Exponential<br />
#** Normal<br />
# Conditional probability<br />
#* As a probability <br />
#* Bayes' Formula<br />
#* Independence<br />
#** Events<br />
#** Random variables<br />
# Joint distributions<br />
#* Covariance<br />
#* Conditional distributions<br />
# Conditional expectation<br />
# Limit theorems<br />
#* Weak Law of Large Numbers<br />
#* Strong Law of Large Numbers<br />
#* Central Limit Theorem<br />
</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Sheldon Ross, ''A First Course in Probability (8th edition)'', Prentice Hall, 2009.<br />
<br />
=== Additional topics ===<br />
<br />
If time permits, geometric, negative binomial, hypergeometric, gamma, Weibull, Cauchy, and/or beta random variables might be studied.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
Currently, Math 431 is only a prerequisite for [[Math 435]]. Consideration should perhaps be given to making it a prerequisite for [[Math 543]].<br />
<br />
[[Category:Courses|431]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_411:_Numerical_Methods&diff=2530Math 411: Numerical Methods2016-02-05T20:45:30Z<p>Ls5: /* Offered */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Numerical Methods.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
W (odd years)<br />
<br />
=== Prerequisite ===<br />
[[Math 334]], [[Math 410|410]].<br />
<br />
=== Description ===<br />
Iterative solvers for linear systems, eigenvalue, eigenvector approximations, numerical solutions to nonlinear systems, numerical techniques for initial and boundary value problems, elementary solvers for PDEs. [This official course description appears to differ with current standard practice, in that iterative solvers of linear systems are taught in [[Math 410]].]<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
<br />
The formal prerequisites reflect the fact that incoming students should have basic knowledge of ordinary differential equations and have had a first course in numerical methods. Indirectly, the prerequisites ensure that students have had multivariable calculus.<br />
<br />
=== Minimal learning outcomes ===<br />
Students should be able to describe, derive, and implement the numerical methods listed below. They should be able to explain the advantages and disadvantages of each method. They should understand error analysis and be able to make practical decisions based on the outcomes of that analysis.<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
# Numerical solution of initial-value problems<br />
#* Taylor methods<br />
#** Euler's method<br />
#* Runge-Kutta methods<br />
#** Runge-Kutta-Fehlberg method<br />
#* Multi-step methods<br />
#* Implicit methods<br />
#* Extrapolation methods<br />
#* Stability<br />
#* Stiff differential equations<br />
# Numerical solution of boundary-value problems<br />
#* Shooting methods<br />
#* Finite-difference methods<br />
#* Rayleigh-Ritz method<br />
# Numerical solution of nonlinear systems of equations<br />
#* Newton's method<br />
#* Quasi-Newton methods<br />
#* Steepest-descent methods<br />
# Approximation theory<br />
#* Least-squares approximation<br />
#* Orthogonal polynomials<br />
#** Chebyshev polynomials<br />
#* Rational function approximation<br />
#* Trigonometric polynomial approximation<br />
#* Fast Fourier transforms<br />
# Numerical computation of eigenvalues and eigenvectors<br />
#* Power Method<br />
# Partial differential equations<br />
#* Finite-difference methods<br />
#** For elliptic equations<br />
#** For parabolic equations<br />
#** For hyperbolic equations<br />
<br />
<br />
<br />
<br />
<br />
<br />
</div><br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Richard L. Burden and J. Douglas Faires, ''Numerical Analysis (9th Edition)'', Brooks Cole, 2010.<br />
<br />
=== Additional topics ===<br />
<br />
If time permits, students could be given an introduction to finite element methods.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|411]]<br />
<br />
None.</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_410:_Intro_to_Numerical_Methods&diff=2529Math 410: Intro to Numerical Methods2016-02-05T20:44:16Z<p>Ls5: /* Offered */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Introduction to Numerical Methods.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F (even years)<br />
<br />
=== Prerequisite ===<br />
[[Math 314]].<br />
<br />
=== Description ===<br />
Root finding, interpolation, curve fitting, numerical differentiation and integration, multiple integrals, direct solvers for linear systems, least squares, rational approximations, Fourier and other orthogonal methods. [This official course description appears to differ with current standard practice, in that iterative solvers of linear systems are taught in this course, while "Fourier and other orthogonal methods" are postponed until [[Math 411]].]<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
<br />
Students are required to have had multivariable calculus.<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
Students should be able to describe, derive, and implement the numerical methods listed below. They should be able to explain the advantages and disadvantages of each method. They should understand error analysis and be able to make practical decisions based on the outcomes of that analysis.<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
# Numerical solution of equations of one variable<br />
#* Bisection method<br />
#* Secant method<br />
#* Fixed-point iteration<br />
#** Newton's method<br />
#** Error analysis<br />
#* Polynomial equations<br />
# Interpolation<br />
#* Lagrange interpolation<br />
#* Divided-difference methods<br />
#* Hermite interpolation<br />
#* Cubic spline interpolation<br />
# Numerical differentiation<br />
#* Derivation of formulas<br />
#** Backward-difference<br />
#** Forward-difference<br />
#** Centered-difference<br />
#** Error analysis<br />
#* Richardson's extrapolation<br />
# Numerical integration<br />
#* Newton-Cotes formulas<br />
#* Composite integration<br />
#* Adaptive quadrature<br />
#* Gaussian quadrature<br />
#* Multiple integrals<br />
#* Error analysis<br />
# Numerical solution of linear systems<br />
#* Direct methods<br />
#** Gaussian elimination<br />
#*** Pivoting strategies<br />
#** Factorization methods<br />
#* Iterative methods<br />
#** Jacobi iteration<br />
#** Gauss-Seidel iteration<br />
#** Relaxation methods<br />
<br />
<br />
<br />
<br />
</div><br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Richard L. Burde and J. Douglas Faires, ''Numerical Analysis (9th Edition)'', Brooks Cole, 2010.<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
Math 410 is the introductory numerical analysis course and is a prerequisite for the other 3 numerical analysis courses: Math [[Math 411|411]], [[Math 510|510]], and [[Math 511|511]]. It is also a prerequisite for [[Math 480]].<br />
<br />
<br />
[[Category:Courses|410]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_362:_Survey_of_Geometry&diff=2528Math 362: Survey of Geometry2016-02-05T20:43:53Z<p>Ls5: /* Offered */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Survey of Geometry.<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]].<br />
<br />
=== Description ===<br />
This course studies the foundations of geometry going back more than two thousand years to Euclid and the ancient Greeks. This course is especially aimed at understanding the importance of Euclid’s parallel postulate and the alternative non-Euclidean geometries that arise from alternative axioms. This course places an emphasis on logical thinking and clear mathematical writing. Geometry software such as Geometer’s Sketchpad should be used throughout the course when appropriate. <br />
<br />
== Desired Learning Outcomes ==<br />
Students should gain familiarity with axioms of geometry, both Euclidean and non-Euclidean. Students should be able to prove the major theorems of geometry based on the axioms.<br />
=== Prerequisites ===<br />
A knowledge of calculus and a maturity developed in mathematical communication.<br />
<br />
=== Minimal learning outcomes ===<br />
<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 many of the theorems.<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
# Euclid’s Elements<br />
#* Understand the historical importance of Euclid’s Elements.<br />
#* Understand and interpret Euclid’s definitions, axioms, and common notions.<br />
#* Identify the logical gaps in Euclid’s proofs.<br />
# Axiomatic Systems and Incidence Geometry<br />
#* Give examples of axiomatic systems and understand the axioms for Incidence Geometry.<br />
#* Give examples of systems that satisfy the axioms of Incidence Geometry.<br />
#* State theorems about Incidence Geometry.<br />
#* Write direct and indirect proofs of theorems of Incidence Geometry.<br />
# Set Theory and Real Numbers<br />
#* Understand basic facts from set theory including sets, elements, intersections, unions, and distinguish between “element of” and “subset of.”<br />
#* Write correct mathematical statements using the vocabulary of set theory.<br />
#* Understand the properties of the real numbers including trichotomy, density of rational numbers, the Archimedean property, and the least upper bound property. <br />
# Axioms for Plane Geometry<br />
#* Identify undefined terms for neutral geometry.<br />
#* Understand existence and incidence postulates and vocabulary for neutral geometry in terms of basic facts from set theory.<br />
#* Give examples of different metrics to show an understanding of the Ruler Postulate and coordinate functions on lines.<br />
#* Understand the need for and use the Plane Separation Postulate, and use it to define the interior of polygon.<br />
#* Define angles, and angle measure using the Protractor Postulate.<br />
#* Define and prove theorems about betweenness for points and rays.<br />
#* Prove theorems about vertical angles, linear pairs, and perpendicular bisectors.<br />
#* Define congruence of triangles and the need for the side-angle-side Postulate.<br />
#* Understand the Euclidean, elliptic, and hyperbolic parallel postulates and examples of geometries that satisfy each of the parallel postulates.<br />
# Theorems in Neutral Geometry<br />
#* Prove theorems about isosceles triangles and perpendicular lines.<br />
#* Prove and understand the Exterior Angle Theorem.<br />
#* Prove and understand congruence theorems for triangles.<br />
#* Prove and understand triangle inequality theorems relating sides and angles (Scalene Inequality, Triangle Inequality, Hinge Theorem).<br />
#* Prove and understand theorems about parallel lines cut by a transversal.<br />
#* Prove and understand the Saccheri-Legendre Theorem about the sum of the angles of a triangle.<br />
#* Prove and understand theorems about quadrilaterals including Saccheri and Lambert quadrilaterals.<br />
#* Be able list and prove that statements are equivalent to the parallel postulate.<br />
# Basic Theorems of Euclidean Geometry<br />
#* Prove and understand the Euclidean theorems about parallel lines cut by a transversal.<br />
#* Prove and understand the angle sum theorem for triangles.<br />
#* Prove and understand theorems about quadrilaterals including squares, rectangles, parallelograms, and trapezoids.<br />
#* Prove and understand theorems about the ratio of lengths of segments of transversals to three parallel lines.<br />
#* Prove and understand theorems about similar triangles.<br />
#* Prove and understand the Pythagorean Theorem.<br />
#* Work with triangles including altitudes, medians, angle bisectors and perpendicular bisectors. Prove and understand the concurrency theorems and the Euler Line Theorem.<br />
# Hyperbolic Geometry<br />
#* Prove and understand the angle sum theorem for triangles and define the angle defect of a triangle.<br />
#* Prove and understand theorems about quadrilaterals including Saccheri and Lambert quadrilaterals.<br />
#* Prove and understand theorems about parallel lines and transversals.<br />
#* Prove and understand that triangles with congruent angles are congruent.<br />
#* Describe and understand limiting parallel rays and asymptotically parallel lines.<br />
#* Model the hyperbolic plane with the Poincaré disk or the upper half plane.<br />
# Transformations<br />
#* Contrast the transformational perspective with the Euclidean perspective of Euclid.<br />
#* Define an isometry and prove that isometries for a geometry from a group.<br />
#* Show that translations, rotations, and reflections are isometries.<br />
#* Prove that the group of isometries for the plane is generated by reflections about a line.<br />
#* Develop plane geometry by replacing the side-angle-side postulate by the reflection postulate and defining congruence in terms of isometries. <br />
# Van Hiele Levels<br />
#* Be familiar with the van Hiele Model of the development of geometric thought.<br><br><br><br><br><br><br><br><br><br><br><br />
</div><br />
<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; paper models of the hyperbolic plane and area in Euclidean and hyperbolic geometry would be useful.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
[[Mathed 562]]<br />
<br />
[[Category:Courses|362]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_352:_Introduction_to_Complex_Analysis&diff=2527Math 352: Introduction to Complex Analysis2016-02-05T20:43:26Z<p>Ls5: /* Offered */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Introduction to Complex Analysis.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F, W, Su<br />
<br />
=== Prerequisite ===<br />
[[Math 290]], and either [[Math 341|Math 341]] or concurrent enrollment.<br />
<br />
=== Description ===<br />
Complex algebra, analytic functions, integration in the complex plane, infinite series, theory of residues, conformal mapping.<br />
<br />
== Desired Learning Outcomes ==<br />
This course is aimed at undergraduates majoring in mathematical and physical sciences and engineering. In addition to being an important branch of mathematics in its own right, complex analysis is an important tool for differential equations (ordinary and partial), algebraic geometry and number theory. Thus it is a core requirement for all mathematics majors. It 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 and mastered [[Math 290]], and to have taken or to have concurrent enrollment in [[Math 341]] (Theory of Analysis) to provide the necessary understanding of the modes of thought of mathematical analysis.<br />
<br />
=== Minimal learning outcomes ===<br />
Students should achieve mastery of the topics listed below. This means that they should know all relevant definitions, the 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 />
# Complex numbers, moduli, exponential form, arguments of products and quotients, roots of complex numbers, regions in the complex plane.<br />
# Limits, including those involving the point at infinity. Open, closed and connected sets. Continuity, derivatives.<br />
# Analytic functions, Cauchy-Riemann equations, harmonic functions, finding the harmonic conjugate.<br />
# Elementary functions in the complex plane: exponential and log functions, complex exponents, trigonometric and hyperbolic functions and their inverses.<br />
# Contour integrals, upper bounds for moduli, primitives, Cauchy-Goursat theorem, Cauchy integral formulae, Liouville theorem, maximum modulus theorem.<br />
# Taylor series, Laurent series, integration and differentiation of power series, uniqueness of series representation, multiplication and division of power series.<br />
# Isolated singularities, behavior near a singularity. Residue theorem, its application to improper integrals, Jordan's lemma. Argument principle, Rouche's theorem.<br />
#Conformal mappings. Moebius transformations.<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. Possibilities include applications of complex analysis in physics.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
This course is required for [[Math 532]] and [[Math 587|587]]. It is needed by anyone proceeding to graduate studies in mathematics. As a result it is essential that ALL required learning objectives be covered.<br />
<br />
[[Category:Courses|352]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_290:_Fundamentals_of_Mathematics.&diff=2526Math 290: Fundamentals of Mathematics.2016-02-05T20:43:09Z<p>Ls5: /* Offered */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Fundamentals of Mathematics.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:0)<br />
<br />
=== Offered ===<br />
F, W, Sp<br />
<br />
=== Prerequisite ===<br />
[[Math 112]] or concurrent enrollment with instructor's consent.<br />
<br />
=== Description ===<br />
Achieving maturity in mathematical communication. Introduction to mathematical proof; methods of proof; analysis of proof; induction; logical reasoning.<br />
<br />
== Desired Learning Outcomes ==<br />
This course is aimed at undergraduate mathematics and mathematics education majors. It is a first course in mathematical thinking. It is intended as an introduction to mathematical proof, and students who finish the course should achieve maturity in mathematical communication.<br />
<br />
=== Prerequisites ===<br />
This course has no prerequisites.<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 />
# Set Theory<br />
#* Set builder notation<br />
#* Venn diagrams<br />
#* De Morgan’s Laws<br />
# Logic<br />
#* Truth Tables<br />
#* Quantifiers<br />
#* Negations of statements with quantifiers<br />
#* Implications<br />
#* Biconditionals<br />
# Proof Techniques<br />
#* Direct proof<br />
#* Proof by contrapositive<br />
#* Proof by contradiction<br />
# Relations<br />
#* Reflexive, irreflexive, symmetric, transitive relations<br />
#* Equivalence relations<br />
#* Equivalence classes<br />
# Functions<br />
#* One-to-one and onto<br />
#* Function composition<br />
#* Inverse functions<br />
#* Bijective functions<br />
#* Permutations<br />
# Mathematical Induction<br />
#* Well ordering principle<br />
#* Mathematical induction<br />
#* Strong induction<br />
#* The method of descent<br />
# Cardinal Numbers<br />
#* Numerical equivalence<br />
#* Countable and uncountable sets<br />
#* Schröder-Bernstein theorem<br />
# Number Theory<br />
#* Division algorithm<br />
#* Euclid’s Algorithm<br />
#* Infinitude of primes<br />
#* Unique factorization<br />
<br />
</div><br />
In addition, on completion of the course, students should understand the basic mathematical language concerning logic, sets, the standard number systems, deductive and inductive reasoning, and the structure of proof. They should be able to translate a mathematical statement into logical form and discuss its negation and its implications. They should be able to translate a simple argument into logical form and detect logical validity and flaws. They should be able to read, write, listen and speak using standard mathematical terminology and reasoning.<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Gary Chartrand, Albert D. Polimeni, and Ping Zhang, ''Mathematical Proofs: A Transition to Advanced Mathematics (2nd Edition)'', Addison Wesley, 2007.<br />
<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): concepts of set theory, number theory, geometry, analysis, group theory, and ring theory. Instructors are free to use new approaches to the teaching of the material, as long as the core topics are adequately covered.<br />
=== Courses for which this course is prerequisite ===<br />
This course is required for almost all upper division course in the Mathematics department. There is a strong expectation that students who have taken this course will have a certain minimal preparation in mathematical thinking. As a result it is essential that all required learning outcomes be thoroughly covered.<br />
<br />
[[Category:Courses|290]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_111:_Trigonometry&diff=2525Math 111: Trigonometry2016-02-05T20:42:22Z<p>Ls5: /* Offered */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Trigonometry.<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(2:2:0)<br />
<br />
=== Offered ===<br />
F, W, Sp<br />
<br />
=== Prerequisite ===<br />
[[Math 110]] or equivalent.<br />
<br />
=== Description ===<br />
Circular functions, triangle relationships, identities, inverse trig functions, trigonometric equations, complex numbers, DeMoivre's theorem.<br />
<br />
== Desired Learning Outcomes ==<br />
Students should gain familiarity and proficiency with the basic theorems of trigonometry.<br />
=== Prerequisites ===<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
# Trigonometric Functions<br />
#*Include angles and their measure, the six trigonometric functions via the unit circle, properties of trigonometric functions (including domain, range, period, fundamental identities, etc.), and graphs of trigonometric functions.<br />
# Analytic Trigonometry<br />
#*Include inverse trigonometric functions, trigonometric identities (including sum and difference formulas,, double-angle and half-angle formulas), and solving trigonometric equations.<br><br><br />
# Applications of Trigonometric Functions<br />
#*Include the Law of Sines, the Law of Cosines, and finding the area of a triangle (including Heron's Formula).<br />
# Polar Coordinates<br />
#*Include polar coordinates, graphs in polar coordinates, the complex plane, and De Moivre's Theorem.<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 />
Vectors.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Math 112]]<br />
[[Category:Courses|111]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_102:_Quantitative_Reasoning&diff=2524Math 102: Quantitative Reasoning2016-02-05T20:41:50Z<p>Ls5: /* Offered */</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, Sp<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>Ls5https://math.byu.edu/wiki/index.php?title=Math_561:_Intro_to_Algebraic_Geometry_1&diff=2371Math 561: Intro to Algebraic Geometry 12015-02-13T18:18:19Z<p>Ls5: /* Prerequisites */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Introduction to Algebraic Geometry 1.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Prerequisite ===<br />
[[Math 571]] or concurrent enrollment.<br />
<br />
=== Description ===<br />
Basic definitions and theorems on affine, projective, and quasi-projective varieties.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
[[Math 571]] or concurrent enrollment.<br />
<br />
=== Minimal learning outcomes ===<br />
Students should achieve mastery of the topics listed below. This means they should know all relevant definitions, correct statements of the major theorems (including their hypotheses and limitations), and examples and non-examples of the various concepts. The students should be able to demonstrate their mastery by solving non-trivial problems related to these concepts, and by proving simple (but non-trivial) theorems about the concepts below, related to, but not identical to, statements proven by the text or instructor.<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
#Algebraic plane curves<br />
#* Rational curves<br />
#* Relation with field theory<br />
#* Rational maps<br />
#* Singular and nonsingular points<br />
#* Projective spaces<br />
#Affine varieties<br />
#* Affine space and the Zariski topology<br />
#* Regular functions<br />
#* Regular maps<br />
#Rational functions and rational maps<br />
#Quasiprojective varieties<br />
#* The Zariski topology on projective space<br />
#* Regular and rational functions<br />
#* Examples<br />
#Products and maps of quasi-projective space<br />
#* Definition of products<br />
#* Properness of projective maps<br />
#* Finite maps<br />
#* Normalization<br />
#Dimension<br />
#* Definition of dimension<br />
#* Dimension of intersection with a hypersurface<br />
#* Dimension of fibres<br />
#* Application to lines on surfaces (optional)<br><br><br><br><br><br><br />
</div><br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*I. R. Shafarevich, Basic Algebraic Geometry 1, Varieties in Projective Space<br />
<br />
=== Additional topics ===<br />
If time permits, additional topics may be covered. Possibilities include the 27 lines on a cubic surface, or an introduction to elliptic curves.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
[[Math 562]]<br />
[[Category:Courses|561]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_691R_-_Graduate_Math_Colloquium&diff=2361Math 691R - Graduate Math Colloquium2014-12-01T21:55:16Z<p>Ls5: Created page with "== Catalog Information == === Title === Graduate Math Colloquium === (Credit Hours:Lecture Hours:Lab Hours) === (1:1:0) === Offered === F W Sp Su === Prerequisite === === D..."</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Graduate Math Colloquium<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(1:1:0)<br />
<br />
=== Offered ===<br />
F W Sp Su<br />
<br />
=== Prerequisite ===<br />
<br />
<br />
=== Description ===<br />
Attend a diverse set of talks at the graduate level. Students will broaden their knowledge of recent and current research in mathematics.<br />
<br />
Speakers will be faculty, visitors and students reporting on thesis work.<br />
<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
<br />
<br />
=== Prerequisites ===<br />
<br />
[[Math 371]] or equivalent<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|691R]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_406R:_Topics_in_Mathematics&diff=2360Math 406R: Topics in Mathematics2014-12-01T21:42:20Z<p>Ls5: /* Desired Learning Outcomes */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Topics in Mathematics<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:1)<br />
<br />
=== Offered ===<br />
<br />
<br />
=== Prerequisite ===<br />
Instructor's consent<br />
<br />
=== Description ===<br />
Topics selected from various aspects of mathematics. Possibilities include, but are not limited to: combinatorial design theory; factorization and primality testing; game theory; harmonic analysis; hyperbolic geometry; linear programming; Lie groups; p-adic numbers; set theory and mathematical logic; stochastic processes; supply chain management; voting theory.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
<br />
<br />
=== Prerequisites ===<br />
<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|406R]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_406R:_Topics_in_Mathematics&diff=2359Math 406R: Topics in Mathematics2014-12-01T21:41:44Z<p>Ls5: Created page with "== Catalog Information == === Title === Topics in Mathematics === (Credit Hours:Lecture Hours:Lab Hours) === (3:3:1) === Offered === === Prerequisite === Instructor's consen..."</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Topics in Mathematics<br />
<br />
=== (Credit Hours:Lecture Hours:Lab Hours) ===<br />
(3:3:1)<br />
<br />
=== Offered ===<br />
<br />
<br />
=== Prerequisite ===<br />
Instructor's consent<br />
<br />
=== Description ===<br />
Topics selected from various aspects of mathematics. Possibilities include, but are not limited to: combinatorial design theory; factorization and primality testing; game theory; harmonic analysis; hyperbolic geometry; linear programming; Lie groups; p-adic numbers; set theory and mathematical logic; stochastic processes; supply chain management; voting theory.<br />
<br />
== Desired Learning Outcomes ==<br />
Students should be familiar with the areas of mathematical research described (naturally in a highly schematic way) by the lecturers in the course. The areas of research will be essentially those that are currently being pursued by BYU faculty in the mathematics department. Students will be able to make an informed choice if they wish to begin a program of undergraduate research with one of the faculty.<br />
<br />
=== Prerequisites ===<br />
<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
=== Textbooks ===<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|406R]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_674:_Lie_Groups_and_Algebras&diff=2347Math 674: Lie Groups and Algebras2014-10-29T21:10:04Z<p>Ls5: </p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Lie Groups and Algebras<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Offered ===<br />
<br />
<br />
=== Prerequisite ===<br />
<br />
<br />
=== Description ===<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
[[Math 673]] or equivalent.<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<br />
<br />
=== Textbooks ===<br />
<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
<br />
<br />
=== Additional topics ===<br />
<br />
<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|674]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_674:_Lie_Groups_and_Algebras&diff=2345Math 674: Lie Groups and Algebras2014-10-29T21:08:16Z<p>Ls5: moved Math 674 to Math 674: Lie Groups and Algebras</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Lie Groups and Algebras<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Offered ===<br />
<br />
<br />
=== Prerequisite ===<br />
<br />
<br />
=== Description ===<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
[[Math 673]] or equivalent.<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<br />
<br />
=== Textbooks ===<br />
<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
<br />
<br />
=== Additional topics ===<br />
<br />
<br />
<br />
=== Courses for which this course is prerequisite ===</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_674&diff=2346Math 6742014-10-29T21:08:16Z<p>Ls5: moved Math 674 to Math 674: Lie Groups and Algebras</p>
<hr />
<div>#REDIRECT [[Math 674: Lie Groups and Algebras]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_674:_Lie_Groups_and_Algebras&diff=2344Math 674: Lie Groups and Algebras2014-10-29T21:07:33Z<p>Ls5: Created page with "== Catalog Information == === Title === Lie Groups and Algebras === Credit Hours === 3 === Offered === === Prerequisite === === Description === == Desired Learning Outcom..."</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Lie Groups and Algebras<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Offered ===<br />
<br />
<br />
=== Prerequisite ===<br />
<br />
<br />
=== Description ===<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
[[Math 673]] or equivalent.<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<br />
<br />
=== Textbooks ===<br />
<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
<br />
<br />
=== Additional topics ===<br />
<br />
<br />
<br />
=== Courses for which this course is prerequisite ===</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_676:_Commutative_Algebra.&diff=2339Math 676: Commutative Algebra.2014-09-29T21:22:20Z<p>Ls5: /* Prerequisite */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Commutative Algebra.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Prerequisite ===<br />
[[Math 572]]<br />
<br />
=== Description ===<br />
Commutative rings, modules, tensor products, localization, primary decomposition, Noetherian and Artinian rings, application to algebraic geometry and algebraic number theory.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
[[Math 672]]<br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
</div><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 concepts below, related to, but not identical to, statements proven by the text or instructor.<br />
<br />
*Commutative rings and ideals<br />
*Modules <br />
*Tensor products<br />
*Localization<br />
*Primary decomposition<br />
*Integral dependence<br />
*Noetherian and Artinian rings<br />
*Dedekind domains and discrete valuation rings<br />
*Applications to algebraic geometry<br />
<br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*Atiyah & Macdonald, Introduction to Commutative Algebra<br />
<br />
=== Additional topics ===<br />
Possible additional topics are: <br />
*Completions<br />
*Dimension theory<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|676]]<br />
[[Math 663]]<br />
[[Math 664]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_561:_Intro_to_Algebraic_Geometry_1&diff=2338Math 561: Intro to Algebraic Geometry 12014-09-29T21:21:24Z<p>Ls5: /* Prerequisite */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Introduction to Algebraic Geometry 1.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Prerequisite ===<br />
[[Math 571]] or concurrent enrollment.<br />
<br />
=== Description ===<br />
Basic definitions and theorems on affine, projective, and quasi-projective varieties.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
[[Math 671]] or concurrent enrollment.<br />
<br />
=== Minimal learning outcomes ===<br />
Students should achieve mastery of the topics listed below. This means they should know all relevant definitions, correct statements of the major theorems (including their hypotheses and limitations), and examples and non-examples of the various concepts. The students should be able to demonstrate their mastery by solving non-trivial problems related to these concepts, and by proving simple (but non-trivial) theorems about the concepts below, related to, but not identical to, statements proven by the text or instructor.<br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
#Algebraic plane curves<br />
#* Rational curves<br />
#* Relation with field theory<br />
#* Rational maps<br />
#* Singular and nonsingular points<br />
#* Projective spaces<br />
#Affine varieties<br />
#* Affine space and the Zariski topology<br />
#* Regular functions<br />
#* Regular maps<br />
#Rational functions and rational maps<br />
#Quasiprojective varieties<br />
#* The Zariski topology on projective space<br />
#* Regular and rational functions<br />
#* Examples<br />
#Products and maps of quasi-projective space<br />
#* Definition of products<br />
#* Properness of projective maps<br />
#* Finite maps<br />
#* Normalization<br />
#Dimension<br />
#* Definition of dimension<br />
#* Dimension of intersection with a hypersurface<br />
#* Dimension of fibres<br />
#* Application to lines on surfaces (optional)<br><br><br><br><br><br><br />
</div><br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*I. R. Shafarevich, Basic Algebraic Geometry 1, Varieties in Projective Space<br />
<br />
=== Additional topics ===<br />
If time permits, additional topics may be covered. Possibilities include the 27 lines on a cubic surface, or an introduction to elliptic curves.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
[[Math 562]]<br />
[[Category:Courses|561]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_677:_Homological_Algebra.&diff=2337Math 677: Homological Algebra.2014-09-29T21:20:44Z<p>Ls5: /* Prerequisite */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Homological Algebra.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Prerequisite ===<br />
[[Math 572]].<br />
<br />
=== Description ===<br />
Chain complexes, derived functors, cohomology of groups, ext and tor, spectral sequences, etc. Application to algebraic geometry and algebraic number theory.<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
<br />
=== Minimal learning outcomes ===<br />
Outlined below are topics that all successful Math 677 students should understand well. As evidence of that understanding, students should be able to demonstrate mastery of all relevant vocabulary, familiarity with common examples and counterexamples, knowledge of the content of the major theorems, understanding of the ideas in their proofs, and ability to make direct application of those results to related problems.<br />
<div style="-moz-column-count:2; column-count:2;"><br />
# Category Theory<br />
#* Definitions and Examples<br />
#** Modules<br />
#*** Snake Lemma<br />
#** Opposite Category<br />
#* Morphism properties<br />
#** Monic/Epic morphism<br />
#** Initial/Terminal object<br />
#*** Zero object<br />
#** Kernel/Cokernel<br />
#* Product/Coproduct<br />
#* Functors/Cofunctors<br />
#* Limits/Colimits<br />
#** Direct Limits<br />
#* Natural transformation<br />
#** Functor Categories<br />
#* Ab-categories<br />
#** Biproducts<br />
#** Additive Functors<br />
#* Additive Categories<br />
# Abelian Categories<br />
#* Left/Right Exact functors<br />
#* Freyd-Mitchell Embedding Theorem<br />
#* Category of Chain Complexes<br />
#** Homology Functor<br />
#* Projective/Injective Objects<br />
#* Projective/Injective Resolutions<br />
#** Comparison Theorem<br />
#** Horseshoe Lemma<br />
#* Derived Functors<br />
#** Ext/Tor<br />
#**Derived Functors of the Inverse Limit<br />
#** Acyclic Objects/Resolutions<br />
#** Univesal delta-functors<br />
#* Adjoint Functors<br />
# Spectral Sequences<br />
#* Convergence<br />
#*Spectral Sequence of a Filtration<br />
#** The Classical Convergence Theorem<br />
#** Complete Convergence Theorem<br />
#* Spectral Sequence of a Double Complex<br />
#** Balancing Ext and Tor<br />
#** Kuenneth Spectral Sequence<br />
#**Universal Coefficient Theorems<br />
#* Hyperhomology<br />
#**Cartan-Eilenberg Resolutions<br />
#*Grothendieck Spectral Sequence<br />
#* Spectral Sequence of an Exact Couple<br />
# Calculating Ext and Tor<br />
#* Tor for Abelian Groups<br />
#*Tor and Flatness<br />
#*Ext for Nice Rings<br />
#* Ext and Extensions<br />
# Dimension<br />
#* Projective/Injective Dimension<br />
#*Flat Dimension<br />
#* Change of Rings Theorems<br />
# Group Homology/Cohomology<br />
#*Shapiro's Lemma<br />
#* The Bar Resolution<br />
<br><br><br><br><br />
<br />
</div><br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
* Weibel: An introduction to homological algebra<br />
<br />
=== Additional topics ===<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|677]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_675R:_Special_Topics_in_Algebra.&diff=2336Math 675R: Special Topics in Algebra.2014-09-29T21:20:15Z<p>Ls5: /* Prerequisite */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Special Topics in Algebra.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Prerequisite ===<br />
[[Math 572]].<br />
<br />
=== Description ===<br />
<br />
Advanced topics in algebra drawn from pure and applied mathematics. Possible topics include: representation theory, Lie groups and Lie algebras, geometric group theory, Galois theory, algebraic number theory, computational algebra, Category theory, Grobner bases, algebraic geometry, algebraic combinatorics, cryptanalysis, finite group theory, modular forms, commutative algebra, homological algebra, group cohomology, character theory of finite groups, mathematical physics, ring theory.<br />
<br />
== Desired Learning Outcomes ==<br />
Students should gain a familiarity with a particular area of algebra selected by the instructor. <br />
<br />
=== Prerequisites ===<br />
Students are expected to have completed the graduate algebra sequence [[Math 671]] and [[Math 672]].<br />
<br />
=== Minimal learning outcomes ===<br />
Minimal learning outcomes cannot be specified for a course in which topics will vary from year to year. However, regardless of the topic, successful students will know terminology, statements and approaches to problems undergoing active research, and major results in the area and techniques used to prove them. Students will demonstrate this knowledge by working suitable problems and developing their own proofs, by presenting and writing work inside and outside of class, and participating in other activities expected of more advanced graduate students. <br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
<br />
</div><br />
<br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
<br />
Past topics chosen include: <br />
<br />
# Representations of algebras and finite groups, using the book by Curtis and Reiner. <br />
# Modular forms using a book by Kilford "Modular forms: a classical and computational introduction." <br />
# Lie algebras, using a book by J Humphreys.<br />
# Lie groups, using the book "Lie groups, Lie algberas and representations", by Brian C. Hall.<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|675]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_571:_Algebra_1&diff=2335Math 571: Algebra 12014-09-29T20:22:46Z<p>Ls5: /* Minimal learning outcomes */</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Algebra 1 (formerly Math 671)<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Prerequisite ===<br />
[[Math 372]] or equivalent.<br />
<br />
=== Description ===<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
[[Math 372]] is a prerequisite for this course. In particular, students should be familiar with the concepts of groups and rings, and they should understand constructions of quotient groups and quotient rings. By this point in their mathematical career, students should be skilled at proving theorems by themselves.<br />
<br />
=== Minimal learning outcomes ===<br />
Students should achieve an advanced mastery of the topics listed below. This means that they should know all relevant definitions, correct statements and proofs of the major theorems (including their hypotheses and limitations), and examples and non-examples of the various concepts The students should be able to demonstrate their mastery by solving difficult problems related to these concepts, and by proving theorems about the below concepts, even if the theorems go beyond the material in the text.<br />
<div style="-moz-column-count:2; column-count:2;"><br />
#Group Theory<br />
#* Axioms and Examples<br />
#* Homomorphisms and Isomorphisms<br />
#* Subgroups<br />
#* Centralizers and Normalizers<br />
#* Cyclic groups and subgroups<br />
#* Quotient Groups<br />
#* Lagrange's Theorem<br />
#* Isomorphism theorems<br />
#* Group Actions<br />
#* Permutation Representations<br />
#* Cayley's Theorem<br />
#* The class equation<br />
#* Sylow theorems<br />
#* Direct and semidirect products<br />
#* Solvable and Nilpotent groups<br />
#Ring Theory<br />
#* Definitions and Examples<br />
#* Homomorphisms and quotient rings<br />
#* Ideals<br />
#* Rings of fractions<br />
#* Chinese remainder theorem<br />
#* Euclidean Domains, PID's and UFD's<br />
#* Polynomial Rings<br />
#Module Theory<br />
#* Definitions and Examples<br />
#* Quotient modules and homomorphisms<br />
#* Direct sums<br />
#* Free Modules<br />
#* Tensor Products<br />
#* Exact Sequences<br />
#* Projectives, Injectives, Flats<br />
<br />
</div><br />
<br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
Most additional topics in this course will be covered in the second course of the sequence, [[Math 572: Algebra 2]]. Depending on time constraints, some or all of the module theory listed here may be moved to the beginning of [[Math 572: Algebra 2]]<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|571: Algebra 1]]<br />
This course is a prerequisite for [[Math 572: Algebra 2]], which is a prerequisite for [[Math 663]], [[Math 675R]], [[Math 676]], and [[Math 677]],</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_673:_Algebra_3&diff=2334Math 673: Algebra 32014-09-29T20:21:01Z<p>Ls5: </p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Algebra 3.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Prerequisite ===<br />
[[Math 572]] or equivalent.<br />
<br />
=== Description ===<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
[[Math 572]] or equivalent.<br />
<br />
=== Minimal learning outcomes ===<br />
Students should achieve an advanced mastery of the topics listed below. <br />
<br />
<div style="-moz-column-count:2; column-count:2;"><br />
#Localization<br />
#Associate primes and primary decompositions<br />
#Integral dependence<br />
#Nullstellensatz<br />
#Filtrations<br />
#Dimension<br />
#Tensor products<br />
<br />
</div><br />
=== Textbooks ===<br />
<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
<br />
<br />
=== Additional topics ===<br />
<br />
<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|673]]<br />
This course is a prerequisite for [[Math 674]].</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_673:_Algebra_3&diff=2333Math 673: Algebra 32014-09-29T20:18:16Z<p>Ls5: </p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Algebra 3.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Prerequisite ===<br />
[[Math 572]] or equivalent.<br />
<br />
=== Description ===<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
[[Math 572]] is a prerequisite for this course. <br />
<br />
=== Minimal learning outcomes ===<br />
<br />
Localization<br />
Associate primes and primary decompositions<br />
Integral dependence<br />
Nullstellensatz<br />
Filtrations<br />
Dimension<br />
Tensor products<br />
<br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
<br />
<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|673]]<br />
This course is a prerequisite for [[Math 674]].</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_673:_Algebra_3&diff=2331Math 673: Algebra 32014-09-29T20:05:48Z<p>Ls5: moved Math 673 to Math 673: Algebra 3</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Algebra 3.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Prerequisite ===<br />
[[Math 572]] or equivalent.<br />
<br />
=== Description ===<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
[[Math 572]] is a prerequisite for this course. <br />
<br />
=== Minimal learning outcomes ===<br />
<br />
<br />
<br />
</div><br />
=== Textbooks ===<br />
<br />
Possible textbooks for this course include (but are not limited to):<br />
<br />
*<br />
<br />
=== Additional topics ===<br />
<br />
<br />
<br />
=== Courses for which this course is prerequisite ===<br />
<br />
[[Category:Courses|673]]<br />
This course is a prerequisite for [[Math 674]].</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_673&diff=2332Math 6732014-09-29T20:05:48Z<p>Ls5: moved Math 673 to Math 673: Algebra 3</p>
<hr />
<div>#REDIRECT [[Math 673: Algebra 3]]</div>Ls5https://math.byu.edu/wiki/index.php?title=Math_673:_Algebra_3&diff=2330Math 673: Algebra 32014-09-29T20:04:57Z<p>Ls5: Created page with "== Catalog Information == === Title === Algebra 3. === Credit Hours === 3 === Prerequisite === Math 572 or equivalent. === Description === == Desired Learning Outcomes =..."</p>
<hr />
<div>== Catalog Information ==<br />
<br />
=== Title ===<br />
Algebra 3.<br />
<br />
=== Credit Hours ===<br />
3<br />
<br />
=== Prerequisite ===<br />
[[Math 572]] or equivalent.<br />
<br />
=== Description ===<br />
<br />
== Desired Learning Outcomes ==<br />
<br />
=== Prerequisites ===<br />
[[Math 572]] is a prerequisite for this course. <br />
<br />
=== Minimal learning outcomes ===<br />
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=== Textbooks ===<br />
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Possible textbooks for this course include (but are not limited to):<br />
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*<br />
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=== Additional topics ===<br />
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=== Courses for which this course is prerequisite ===<br />
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[[Category:Courses|673]]<br />
This course is a prerequisite for [[Math 674]].</div>Ls5