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In the Billions and Exponential Modeling

In the Billions and Exponential Modeling

This problem provides an opportunity to experiment with modeling real data. Populations ... (more)

This problem provides an opportunity to experiment with modeling real data. Populations are often modeled with exponential functions and in this particular case we see that, over the last 200 years, the rate of population growth accelerated rapidly, reaching a peak a little after the middle of the 20th century and now it is slowing down. (less)

Subject:
Mathematics and Statistics
Material Type:
Activities and Labs
Collection:
Illustrative Mathematics
Provider:
Institute for Mathematics and Education
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Introduction to Partial Differential Equations

Introduction to Partial Differential Equations

Partial differential equations (PDEs) describe the relationships among the derivatives of an ... (more)

Partial differential equations (PDEs) describe the relationships among the derivatives of an unknown function with respect to different independent variables, such as time and position. Experiment and observation provide information about the connections between rates of change of an important quantity, such as heat, with respect to different variables. Upon successful completion of this course, the student will be able to: State the heat, wave, Laplace, and Poisson equations and explain their physical origins; Define harmonic functions; State and justify the maximum principle for harmonic functions; State the mean value property for harmonic functions; Define linear operators and identify linear operations; Identify and classify linear PDEs; Identify homogeneous PDEs and evolution equations; Relate solving homogeneous linear PDEs to finding kernels of linear operators; Define boundary value problem and identify boundary conditions as periodic, Dirichlet, Neumann, or Robin (mixed); Explain physical significance of boundary conditions; Show uniqueness of solutions to the heat, wave, Laplace and Poisson equations with various boundary conditions; Define well-posedness; Define, characterize, and use inner products; Define the space of L2 functions, state its key properties, and identify L2 functions; Define orthogonality and orthonormal basis and show the orthogonality of certain trigonometric functions; Distinguish between pointwise, uniform, and L2 convergence and show convergence of Fourier series; Define Fourier series on [0,pi] and [0,L] and identify sufficient conditions for their convergence and uniqueness; Compute Fourier coefficients and construct Fourier series; Use the method of characteristics to solve linear and nonlinear first-order wave equations; Solve the one-dimensional wave equation using d'Alembert's formula; Use similarity methods to solve PDEs; Solve the heat, wave, Laplace, and Poisson equations using separation of variables and apply boundary conditions; Define the delta function and apply ideas from calculus and Fourier series to generalized functions; Derive Green's representation formula; Use Green's functions to solve the Poisson equation on the unit disk; Define the Fourier transform; Derive basic properties of the Fourier transform of a function, such as its relationship to the Fourier transform of the derivative; Show that the inverse Fourier transform of a product is a convolution; Compute Fourier transforms of functions; Use the Fourier transform to solve the heat and wave equations on unbounded domains. (Mathematics 222) (less)

Subject:
Mathematics and Statistics
Material Type:
Assessments
Full Course
Homework and Assignments
Readings
Syllabi
Textbooks
Collection:
Saylor Foundation
Provider:
The Saylor Foundation
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Linear Algebra II

Linear Algebra II

Linear Algebra is both rich in theory and full of interesting applications; ... (more)

Linear Algebra is both rich in theory and full of interesting applications; in this course the student will try to balance both. This course includes a review of topics learned in Linear Algebra I. Upon successful completion of this course, the student will be able to: Solve systems of linear equations; Define the abstract notions of vector space and inner product space; State examples of vector spaces; Diagonalize a matrix; Formulate what a system of linear equations is in terms of matrices; Give an example of a space that has the Archimedian property; Use the Euclidean algorithm to find the greatest common divisor; Understand polar form and geometric interpretation of the complex numbers; Explain what the fundamental theorem of algebra states; Determine when two matrices are row equivalent; State the Fredholm alternative; Identify matrices that are in row reduced echelon form; Find a LU factorization for a given matrix; Find a PLU factorization for a given matrix; Find a QR factorization for a given matrix; Use the simplex algorithm; Compute eigenvalues and eigenvectors; State Shur's Theorem; Define normal matrices; Explain the composition and the inversion of permutations; Define and compute the determinant; Explain when eigenvalues exist for a given operator; Normal form of a nilpotent operator; Understand the idea of Jordan blocks, Jordan matrices, and the Jordan form of a matrix; Define quadratic forms; State the second derivative test; Define eigenvectors and eigenvalues; Define a vector space and state its properties; State the notions of linear span, linear independence, and the basis of a vector space; Understand the ideas of linear independence, spanning set, basis, and dimension; Define a linear transformation; State the properties of linear transformations; Define the characteristic polynomial of a matrix; Define a Markov matrix; State what it means to have the property of being a stochastic matrix; Define a normed vector space; Apply the Cauchy Schwarz inequality; State the Riesz representation theorem; State what it means for a nxn matrix to be diagonalizable; Define Hermitian operators; Define a Hilbert space; Prove the Cayley Hamilton theorem; Define the adjoint of an operator; Define normal operators; State the spectral theorem; Understand how to find the singular-value decomposition of an operator; Define the notion of length for abstract vectors in abstract vector spaces; Define orthogonal vectors; Define orthogonal and orthonormal subsets of R^n; Use the Gram-Schmidt process; Find the eigenvalues and the eigenvectors of a given matrix numerically; Provide an explicit description of the Power Method. (Mathematics 212) (less)

Subject:
Mathematics and Statistics
Material Type:
Assessments
Full Course
Homework and Assignments
Readings
Syllabi
Textbooks
Collection:
Saylor Foundation
Provider:
The Saylor Foundation
Read the Fine Print
Linear Functions

Linear Functions

This task requires students to use the fact that on the graph ... (more)

This task requires students to use the fact that on the graph of the linear function h(x)=ax+b, the y-coordinate increases by a when x increases by one. Specific values for a and b were left out intentionally to encourage students to use the above fact as opposed to computing the point of intersection, (p,q), and then computing respective function values to answer the question. (less)

Subject:
Mathematics and Statistics
Material Type:
Activities and Labs
Collection:
Illustrative Mathematics
Provider:
Institute for Mathematics and Education
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Linear Or Exponential?

Linear Or Exponential?

This task gives a variet of real-life contexts which could be modeled ... (more)

This task gives a variet of real-life contexts which could be modeled by a linear or exponential function. The key distinguishing feature between the two is whether the change by equal factors over equal intervals (exponential functions), or by a constant increase per unit interval (linear functions). (less)

Subject:
Mathematics and Statistics
Material Type:
Activities and Labs
Collection:
Illustrative Mathematics
Provider:
Institute for Mathematics and Education
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Linear Pair of Angles

Linear Pair of Angles

An interactive applet and associated web page that demonstrate a linear pair ... (more)

An interactive applet and associated web page that demonstrate a linear pair of angles. A pair of angles are shown and the user can drag the common side. It demonstrates that they a supplementary (add to 180 degrees) and have a common side. The angle measures can be turned off for class discussion. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com. (less)

Subject:
Mathematics and Statistics
Material Type:
Readings
Simulations
Collection:
Math Open Reference
Provider:
Math Open Reference
Author:
John Page
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Linear Vector Spaces

Linear Vector Spaces

Introduces tools and formulas to use when dealing with Linear Vector Spaces. ... (more)

Introduces tools and formulas to use when dealing with Linear Vector Spaces. Topics covered include: linear vector spaces, inner product spaces, norm, Schwarz inequality, and distance between two vectors (less)

Subject:
Mathematics and Statistics
Science and Technology
Material Type:
Readings
Syllabi
Collection:
Connexions
Provider:
Rice University
Author:
Don Johnson
No Strings Attached
Linear and Time-Invariant Systems

Linear and Time-Invariant Systems

Linear time-invariant (LTI) systems are the most important class of systems in ... (more)

Linear time-invariant (LTI) systems are the most important class of systems in communications. They ensure that a system acting on a signal can be modeled by the system acting individually on the component parts of the signal and summed. (less)

Subject:
Science and Technology
Material Type:
Readings
Syllabi
Collection:
Connexions
Provider:
Rice University
Author:
Don Johnson
No Strings Attached
Linear function graph display

Linear function graph display

An interactive applet that allows the user to graphically explore the properties ... (more)

An interactive applet that allows the user to graphically explore the properties of a linear functions. Specifically, it is designed to foster an intuitive understanding of the effects of changing the two coefficients in the function y=ax+b. The applet shows a large graph of a quadratic (ax + b) and has two slider controls, one each for the coefficients a and b. As the sliders are moved, the graph is redrawn in real time illustrating the effects of these variations. Applet can be enlarged to full screen size for use with a classroom projector. This resource is a component of the Math Open Reference Interactive Geometry textbook project at http://www.mathopenref.com. (less)

Subject:
Mathematics and Statistics
Material Type:
Readings
Simulations
Collection:
Math Open Reference
Provider:
Math Open Reference
Author:
John Page
Read the Fine Print
Modeling Dynamics and Control I, Spring 2005

Modeling Dynamics and Control I, Spring 2005

First of two-term sequence on modeling, analysis and control of dynamic systems. ... (more)

First of two-term sequence on modeling, analysis and control of dynamic systems. Mechanical translation, uniaxial rotation, electrical circuits and their coupling via levers, gears and electro-mechanical devices. Analytical and computational solution of linear differential equations and state-determined systems. Laplace transforms, transfer functions. Frequency response, Bode plots. Vibrations, modal analysis. Open- and closed-loop control, instability. Time-domain controller design, introduction to frequency-domain control design techniques. Case studies of engineering applications. (less)

Subject:
Science and Technology
Material Type:
Activities and Labs
Assessments
Case Study
Full Course
Homework and Assignments
Syllabi
Other
Collection:
MIT OpenCourseWare
Provider:
M.I.T.
Author:
Dubowsky, Steven
Trumper, David L.
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Numerical Methods for Partial Differential Equations (SMA 5212), Spring 2003

Numerical Methods for Partial Differential Equations (SMA 5212), Spring 2003

Covers the fundamentals of modern numerical techniques for a wide range of ... (more)

Covers the fundamentals of modern numerical techniques for a wide range of linear and nonlinear elliptic, parabolic, and hyperbolic partial differential and integral equations. Topics include: mathematical formulations; finite difference, finite volume, finite element, and boundary element discretization methods; and direct and iterative solution techniques. The methodologies described form the foundation for computational approaches to engineering systems involving heat transfer, solid mechanics, fluid dynamics, and electromagnetics; and engineering systems such as finance, traffic flow, and performance analysis of stochastic systems. Computer assignments requiring programming. (less)

Subject:
Science and Technology
Material Type:
Full Course
Homework and Assignments
Lecture Notes
Syllabi
Collection:
MIT OpenCourseWare
Provider:
M.I.T.
Author:
Peraire, Jaime
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