Description

Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODE's) deal with functions of one variable, which can often be thought of as time. Topics include: Solution of first-order ODE's by analytical, graphical and numerical methods; Linear ODE's, especially second order with constant coefficients; Undetermined coefficients and variation of parameters; Sinusoidal and exponential signals: oscillations, damping, resonance; Complex numbers and exponentials; Fourier series, periodic solutions; Delta functions, convolution, and Laplace transform methods; Matrix and first order linear systems: eigenvalues and Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODE's) deal with functions of one variable, which can often be thought of as time. Topics include: Solution of first-order ODE's by analytical, graphical and numerical methods; Linear ODE's, especially second order with constant coefficients; Undetermined coefficients and variation of parameters; Sinusoidal and exponential signals: oscillations, damping, resonance; Complex numbers and exponentials; Fourier series, periodic solutions; Delta functions, convolution, and Laplace transform methods; Matrix and first order linear systems: eigenvalues and

Subjects

Ordinary Differential Equations | Ordinary Differential Equations | ODE | ODE | modeling physical systems | modeling physical systems | first-order ODE's | first-order ODE's | Linear ODE's | Linear ODE's | second order ODE's | second order ODE's | second order ODE's with constant coefficients | second order ODE's with constant coefficients | Undetermined coefficients | Undetermined coefficients | variation of parameters | variation of parameters | Sinusoidal signals | Sinusoidal signals | exponential signals | exponential signals | oscillations | oscillations | damping | damping | resonance | resonance | Complex numbers and exponentials | Complex numbers and exponentials | Fourier series | Fourier series | periodic solutions | periodic solutions | Delta functions | Delta functions | convolution | convolution | Laplace transform methods | Laplace transform methods | Matrix systems | Matrix systems | first order linear systems | first order linear systems | eigenvalues and eigenvectors | eigenvalues and eigenvectors | Non-linear autonomous systems | Non-linear autonomous systems | critical point analysis | critical point analysis | phase plane diagrams | phase plane diagrams | constant coefficients | constant coefficients | complex numbers | complex numbers | exponentials | exponentials | eigenvalues | eigenvalues | eigenvectors | eigenvectors

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Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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Dr Richard Earl of the Mathematical Institute, Oxford presents a talk about prime numbers. What they are and their role in internet security. Wales; http://creativecommons.org/licenses/by-nc-sa/2.0/uk/

Subjects

maths | numbers | prime numbers | Internet security | maths | numbers | prime numbers | Internet security | 2013-12-09

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http://creativecommons.org/licenses/by-nc-sa/2.0/uk/

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Dr Richard Earl of the Mathematical Institute, Oxford presents a talk about prime numbers. What they are and their role in internet security. Wales; http://creativecommons.org/licenses/by-nc-sa/2.0/uk/

Subjects

maths | numbers | prime numbers | Internet security | maths | numbers | prime numbers | Internet security | 2013-12-09

License

http://creativecommons.org/licenses/by-nc-sa/2.0/uk/

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You may have met complex numbers before, but not had experience in manipulating them. This unit gives an accessible introduction to complex numbers, which are very important in science and technology, as well as mathematics. The unit includes definitions, concepts and techniques which will be very helpful and interesting to a wide variety of people with a reasonable background in algebra and trigonometry.

Subjects

imaginary_numbers | real_numbers | mathematics and statistics | complex_numbers | Education | X000

License

Attribution-Noncommercial-Share Alike 2.0 UK: England & Wales Attribution-Noncommercial-Share Alike 2.0 UK: England & Wales http://creativecommons.org/licenses/by-nc-sa/2.0/uk/ http://creativecommons.org/licenses/by-nc-sa/2.0/uk/

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Public Seminar Series, Hilary term 2013. Seminar by dr Alice Edwards (UNHCR) recorded on 13 February 2013 at the Oxford Department of International Development, University of Oxford. The United Nations Refugee Convention recognises that the problem of refugees is inherently international and cannot be solved by a single State alone. Yet achieving international cooperation, or even achieving consensus on what this means, has had a long and chequered history. This lecture will examine the many ways in which a focus on asylum statistics has impacted on the international protection regime for refugees, and what needs to be done about it. Wales; http://creativecommons.org/licenses/by-nc-sa/2.0/uk/

Subjects

international burden-sharing | refugee numbers | international burden-sharing | refugee numbers

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http://creativecommons.org/licenses/by-nc-sa/2.0/uk/

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This course provides an elementary introduction to number theory with no algebraic prerequisites. Topics include primes, congruences, quadratic reciprocity, diophantine equations, irrational numbers, continued fractions and elliptic curves. This course provides an elementary introduction to number theory with no algebraic prerequisites. Topics include primes, congruences, quadratic reciprocity, diophantine equations, irrational numbers, continued fractions and elliptic curves.

Subjects

number theory with no algebraic prerequisites | number theory with no algebraic prerequisites | primes | congruences | primes | congruences | quadratic reciprocity | quadratic reciprocity | diophantine equations | diophantine equations | irrational numbers | irrational numbers | continued fractions | continued fractions | partitions | partitions

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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18.014, Calculus with Theory, covers the same material as 18.01 (Calculus), but at a deeper and more rigorous level. It emphasizes careful reasoning and understanding of proofs. The course assumes knowledge of elementary calculus. Topics: Axioms for the real numbers; the Riemann integral; limits, theorems on continuous functions; derivatives of functions of one variable; the fundamental theorems of calculus; Taylor's theorem; infinite series, power series, rigorous treatment of the elementary functions. Dr. Lachowska wishes to acknowledge Andrew Brooke-Taylor, Natasha Bershadsky, and Alex Retakh for their help with this course web site. 18.014, Calculus with Theory, covers the same material as 18.01 (Calculus), but at a deeper and more rigorous level. It emphasizes careful reasoning and understanding of proofs. The course assumes knowledge of elementary calculus. Topics: Axioms for the real numbers; the Riemann integral; limits, theorems on continuous functions; derivatives of functions of one variable; the fundamental theorems of calculus; Taylor's theorem; infinite series, power series, rigorous treatment of the elementary functions. Dr. Lachowska wishes to acknowledge Andrew Brooke-Taylor, Natasha Bershadsky, and Alex Retakh for their help with this course web site.

Subjects

axioms for the real numbers | axioms for the real numbers | the Riemann integral | the Riemann integral | limits | limits | theorems on continuous functions | theorems on continuous functions | derivatives of functions of one variablethe fundamental theorems of calculus | derivatives of functions of one variablethe fundamental theorems of calculus | Taylor's theorem | Taylor's theorem | infinite series | infinite series | power series | power series | rigorous treatment of the elementary functions | rigorous treatment of the elementary functions

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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This course introduces students to probability and random variables. Topics include distribution functions, binomial, geometric, hypergeometric, and Poisson distributions. The other topics covered are uniform, exponential, normal, gamma and beta distributions; conditional probability; Bayes theorem; joint distributions; Chebyshev inequality; law of large numbers; and central limit theorem. This course introduces students to probability and random variables. Topics include distribution functions, binomial, geometric, hypergeometric, and Poisson distributions. The other topics covered are uniform, exponential, normal, gamma and beta distributions; conditional probability; Bayes theorem; joint distributions; Chebyshev inequality; law of large numbers; and central limit theorem.

Subjects

Probability spaces | Probability spaces | random variables | random variables | distribution functions | distribution functions | Binomial | Binomial | geometric | geometric | hypergeometric | hypergeometric | Poisson distributions | Poisson distributions | Uniform | Uniform | exponential | exponential | normal | normal | gamma and beta distributions | gamma and beta distributions | Conditional probability | Conditional probability | Bayes theorem | Bayes theorem | joint distributions | joint distributions | Chebyshev inequality | Chebyshev inequality | law of large numbers | law of large numbers | and central limit theorem. | and central limit theorem.

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODE's) deal with functions of one variable, which can often be thought of as time. Topics include: Solution of first-order ODE's by analytical, graphical and numerical methods; Linear ODE's, especially second order with constant coefficients; Undetermined coefficients and variation of parameters; Sinusoidal and exponential signals: oscillations, damping, resonance; Complex numbers and exponentials; Fourier series, periodic solutions; Delta functions, convolution, and Laplace transform methods; Matrix and first order linear systems: eigenvalues and Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODE's) deal with functions of one variable, which can often be thought of as time. Topics include: Solution of first-order ODE's by analytical, graphical and numerical methods; Linear ODE's, especially second order with constant coefficients; Undetermined coefficients and variation of parameters; Sinusoidal and exponential signals: oscillations, damping, resonance; Complex numbers and exponentials; Fourier series, periodic solutions; Delta functions, convolution, and Laplace transform methods; Matrix and first order linear systems: eigenvalues and

Subjects

Ordinary Differential Equations | Ordinary Differential Equations | ODE | ODE | modeling physical systems | modeling physical systems | first-order ODE's | first-order ODE's | Linear ODE's | Linear ODE's | second order ODE's | second order ODE's | second order ODE's with constant coefficients | second order ODE's with constant coefficients | Undetermined coefficients | Undetermined coefficients | variation of parameters | variation of parameters | Sinusoidal signals | Sinusoidal signals | exponential signals | exponential signals | oscillations | oscillations | damping | damping | resonance | resonance | Complex numbers and exponentials | Complex numbers and exponentials | Fourier series | Fourier series | periodic solutions | periodic solutions | Delta functions | Delta functions | convolution | convolution | Laplace transform methods Matrix systems | Laplace transform methods Matrix systems | first order linear systems | first order linear systems | eigenvalues and eigenvectors | eigenvalues and eigenvectors | Non-linear autonomous systems | Non-linear autonomous systems | critical point analysis | critical point analysis | phase plane diagrams | phase plane diagrams

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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This course introduces students to probability and random variables. Topics include distribution functions, binomial, geometric, hypergeometric, and Poisson distributions. The other topics covered are uniform, exponential, normal, gamma and beta distributions; conditional probability; Bayes theorem; joint distributions; Chebyshev inequality; law of large numbers; and central limit theorem. This course introduces students to probability and random variables. Topics include distribution functions, binomial, geometric, hypergeometric, and Poisson distributions. The other topics covered are uniform, exponential, normal, gamma and beta distributions; conditional probability; Bayes theorem; joint distributions; Chebyshev inequality; law of large numbers; and central limit theorem.

Subjects

Probability spaces | Probability spaces | random variables | random variables | distribution functions | distribution functions | Binomial | Binomial | geometric | geometric | hypergeometric | hypergeometric | Poisson distributions | Poisson distributions | Uniform | Uniform | exponential | exponential | normal | normal | gamma and beta distributions | gamma and beta distributions | Conditional probability | Conditional probability | Bayes theorem | Bayes theorem | joint distributions | joint distributions | Chebyshev inequality | Chebyshev inequality | law of large numbers | law of large numbers | central limit theorem | central limit theorem

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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This course covers the laws of large numbers and central limit theorems for sums of independent random variables. It also analyzes topics such as the conditioning and martingales, the Brownian motion and the elements of diffusion theory. This course covers the laws of large numbers and central limit theorems for sums of independent random variables. It also analyzes topics such as the conditioning and martingales, the Brownian motion and the elements of diffusion theory.

Subjects

Earth | Earth | Solar System | Solar System | Geophysics | Geophysics | Gravitational Field | Gravitational Field | Magnetic Field | Magnetic Field | Seismology | Seismology | Geodynamics | Geodynamics | Laws of large numbers | Laws of large numbers | central limit theorems for sums of independent random variables | central limit theorems for sums of independent random variables | conditioning and martingales | conditioning and martingales | Brownian motion and elements of diffusion theory | Brownian motion and elements of diffusion theory | functional limit theorems | functional limit theorems

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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This course covers the laws of large numbers and central limit theorems for sums of independent random variables. It also analyzes topics such as the conditioning and martingales, the Brownian motion and the elements of diffusion theory. This course covers the laws of large numbers and central limit theorems for sums of independent random variables. It also analyzes topics such as the conditioning and martingales, the Brownian motion and the elements of diffusion theory.

Subjects

Earth | Earth | Solar System | Solar System | Geophysics | Geophysics | Gravitational Field | Gravitational Field | Magnetic Field | Magnetic Field | Seismology | Seismology | Geodynamics | Geodynamics | Laws of large numbers | Laws of large numbers | central limit theorems for sums of independent random variables | central limit theorems for sums of independent random variables | conditioning and martingales | conditioning and martingales | Brownian motion and elements of diffusion theory | Brownian motion and elements of diffusion theory | functional limit theorems | functional limit theorems

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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This course is an introduction to numerical methods: interpolation, differentiation, integration, and systems of linear equations. It covers the solution of differential equations by numerical integration, as well as partial differential equations of inviscid hydrodynamics: finite difference methods, boundary integral equation panel methods. Also addressed are introductory numerical lifting surface computations, fast Fourier transforms, the numerical representation of deterministic and random sea waves, as well as integral boundary layer equations and numerical solutions.Technical RequirementMATLAB® software is required to run the .m files found on this course site. The .FIN and .OUT are simply data offest tables. They can be viewed with any text reader. RealOne™ This course is an introduction to numerical methods: interpolation, differentiation, integration, and systems of linear equations. It covers the solution of differential equations by numerical integration, as well as partial differential equations of inviscid hydrodynamics: finite difference methods, boundary integral equation panel methods. Also addressed are introductory numerical lifting surface computations, fast Fourier transforms, the numerical representation of deterministic and random sea waves, as well as integral boundary layer equations and numerical solutions.Technical RequirementMATLAB® software is required to run the .m files found on this course site. The .FIN and .OUT are simply data offest tables. They can be viewed with any text reader. RealOne™

Subjects

numerical methods | numerical methods | interpolation | interpolation | differentiation | differentiation | integration | integration | systems of linear equations | systems of linear equations | differential equations | differential equations | numerical integration | numerical integration | partial differential | partial differential | boundary integral equation panel methods | boundary integral equation panel methods | deterministic and random sea waves | deterministic and random sea waves | Fast Fourier Transforms | Fast Fourier Transforms | finite difference methods | finite difference methods | Integral boundary layer equations | Integral boundary layer equations | numerical lifting surface computations | numerical lifting surface computations | Numerical representation | Numerical representation | numerical solutions | numerical solutions | partial differential equations of inviscid hydrodynamics | partial differential equations of inviscid hydrodynamics | incompressible fluid mechanics | incompressible fluid mechanics | calculus | calculus | complex numbers | complex numbers | root finding | root finding | curve fitting | curve fitting | numerical differentiation | numerical differentiation | numerical errors | numerical errors | panel methods | panel methods | oscillating rigid objects | oscillating rigid objects | 2.29 | 2.29

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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Includes audio/video content: AV lectures. The laws of nature are expressed as differential equations. Scientists and engineers must know how to model the world in terms of differential equations, and how to solve those equations and interpret the solutions. This course focuses on the equations and techniques most useful in science and engineering. Includes audio/video content: AV lectures. The laws of nature are expressed as differential equations. Scientists and engineers must know how to model the world in terms of differential equations, and how to solve those equations and interpret the solutions. This course focuses on the equations and techniques most useful in science and engineering.

Subjects

Ordinary Differential Equations | Ordinary Differential Equations | ODE | ODE | modeling physical systems | modeling physical systems | first-order ODE's | first-order ODE's | Linear ODE's | Linear ODE's | second order ODE's | second order ODE's | second order ODE's with constant coefficients | second order ODE's with constant coefficients | Undetermined coefficients | Undetermined coefficients | variation of parameters | variation of parameters | Sinusoidal signals | Sinusoidal signals | exponential signals | exponential signals | oscillations | oscillations | damping | damping | resonance | resonance | Complex numbers and exponentials | Complex numbers and exponentials | Fourier series | Fourier series | periodic solutions | periodic solutions | Delta functions | Delta functions | convolution | convolution | Laplace transform methods | Laplace transform methods | Matrix systems | Matrix systems | first order linear systems | first order linear systems | eigenvalues and eigenvectors | eigenvalues and eigenvectors | Non-linear autonomous systems | Non-linear autonomous systems | critical point analysis | critical point analysis | phase plane diagrams | phase plane diagrams

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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Includes audio/video content: AV lectures. Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODE's) deal with functions of one variable, which can often be thought of as time. Includes audio/video content: AV lectures. Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODE's) deal with functions of one variable, which can often be thought of as time.

Subjects

Ordinary Differential Equations | Ordinary Differential Equations | ODE | ODE | modeling physical systems | modeling physical systems | first-order ODE's | first-order ODE's | Linear ODE's | Linear ODE's | second order ODE's | second order ODE's | second order ODE's with constant coefficients | second order ODE's with constant coefficients | Undetermined coefficients | Undetermined coefficients | variation of parameters | variation of parameters | Sinusoidal signals | Sinusoidal signals | exponential signals | exponential signals | oscillations | oscillations | damping | damping | resonance | resonance | Complex numbers and exponentials | Complex numbers and exponentials | Fourier series | Fourier series | periodic solutions | periodic solutions | Delta functions | Delta functions | convolution | convolution | Laplace transform methods | Laplace transform methods | Matrix systems | Matrix systems | first order linear systems | first order linear systems | eigenvalues and eigenvectors | eigenvalues and eigenvectors | Non-linear autonomous systems | Non-linear autonomous systems | critical point analysis | critical point analysis | phase plane diagrams | phase plane diagrams

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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Includes audio/video content: AV faculty introductions. This course is an introduction to numerical methods: interpolation, differentiation, integration, and systems of linear equations. It covers the solution of differential equations by numerical integration, as well as partial differential equations of inviscid hydrodynamics: finite difference methods, boundary integral equation panel methods. Also addressed are introductory numerical lifting surface computations, fast Fourier transforms, the numerical representation of deterministic and random sea waves, as well as integral boundary layer equations and numerical solutions. This course was originally offered in Course 13 (Department of Ocean Engineering) as 13.024. In 2005, ocean engineering subjects became part of Course 2 (Department Includes audio/video content: AV faculty introductions. This course is an introduction to numerical methods: interpolation, differentiation, integration, and systems of linear equations. It covers the solution of differential equations by numerical integration, as well as partial differential equations of inviscid hydrodynamics: finite difference methods, boundary integral equation panel methods. Also addressed are introductory numerical lifting surface computations, fast Fourier transforms, the numerical representation of deterministic and random sea waves, as well as integral boundary layer equations and numerical solutions. This course was originally offered in Course 13 (Department of Ocean Engineering) as 13.024. In 2005, ocean engineering subjects became part of Course 2 (Department

Subjects

numerical methods | numerical methods | interpolation | interpolation | differentiation | differentiation | integration | integration | systems of linear equations | systems of linear equations | differential equations | differential equations | numerical integration | numerical integration | partial differential | partial differential | boundary integral equation panel methods | boundary integral equation panel methods | deterministic and random sea waves | deterministic and random sea waves | Fast Fourier Transforms | Fast Fourier Transforms | finite difference methods | finite difference methods | Integral boundary layer equations | Integral boundary layer equations | numerical lifting surface computations | numerical lifting surface computations | Numerical representation | Numerical representation | numerical solutions | numerical solutions | partial differential equations of inviscid hydrodynamics | partial differential equations of inviscid hydrodynamics | incompressible fluid mechanics | incompressible fluid mechanics | calculus | calculus | complex numbers | complex numbers | root finding | root finding | curve fitting | curve fitting | numerical differentiation | numerical differentiation | numerical errors | numerical errors | panel methods | panel methods | oscillating rigid objects | oscillating rigid objects

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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This course will provide students with an introduction to numerical methods and MATLAB®. Topics covered throughout the course will include: errors, condition numbers and roots of equations; Navier-Stokes; direct and iterative methods for linear systems; finite differences for elliptic, parabolic and hyperbolic equations; Fourier decomposition, error analysis, and stability; high-order and compact finite-differences; finite volume methods; time marching methods; Navier-Stokes solvers; grid generation; finite volumes on complex geometries; finite element methods; spectral methods; boundary element and panel methods; turbulent flows; boundary layers; Lagrangian Coherent Structures. Subject includes a final research project. This course will provide students with an introduction to numerical methods and MATLAB®. Topics covered throughout the course will include: errors, condition numbers and roots of equations; Navier-Stokes; direct and iterative methods for linear systems; finite differences for elliptic, parabolic and hyperbolic equations; Fourier decomposition, error analysis, and stability; high-order and compact finite-differences; finite volume methods; time marching methods; Navier-Stokes solvers; grid generation; finite volumes on complex geometries; finite element methods; spectral methods; boundary element and panel methods; turbulent flows; boundary layers; Lagrangian Coherent Structures. Subject includes a final research project.

Subjects

errors | errors | condition numbers and roots of equations | condition numbers and roots of equations | Navier-Stokes | Navier-Stokes | direct and iterative methods for linear systems | direct and iterative methods for linear systems | finite differences for elliptic | finite differences for elliptic | parabolic and hyperbolic equations | parabolic and hyperbolic equations | Fourier decomposition | error analysis | and stability | Fourier decomposition | error analysis | and stability | high-order and compact finite-differences | high-order and compact finite-differences | finite volume methods | finite volume methods | time marching methods | time marching methods | Navier-Stokes solvers | Navier-Stokes solvers | grid generation | grid generation | finite volumes on complex geometries | finite volumes on complex geometries | finite element methods | finite element methods | spectral methods | spectral methods | boundary element and panel methods | boundary element and panel methods | turbulent flows | turbulent flows | boundary layers | boundary layers | Lagrangian Coherent Structures | Lagrangian Coherent Structures

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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This course will provide a solid foundation in probability and statistics for economists and other social scientists. We will emphasize topics needed for further study of econometrics and provide basic preparation for 14.32. Topics include elements of probability theory, sampling theory, statistical estimation, and hypothesis testing. This course will provide a solid foundation in probability and statistics for economists and other social scientists. We will emphasize topics needed for further study of econometrics and provide basic preparation for 14.32. Topics include elements of probability theory, sampling theory, statistical estimation, and hypothesis testing.

Subjects

statistics | statistics | economic applications | economic applications | probability theory | probability theory | sampling theory | sampling theory | statistical estimation | statistical estimation | regression analysis | regression analysis | hypothesis testing | hypothesis testing | Elementary econometrics | Elementary econometrics | statistical tools | statistical tools | economic data | economic data | economics | economics | statistical | statistical | probability distribution function | probability distribution function | cumulative distribution function | cumulative distribution function | normal | normal | Student's t | Student's t | chi-squared | chi-squared | central limit theorem | central limit theorem | law of large numbers | law of large numbers | Bayes theorem | Bayes theorem

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Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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18.014, Calculus with Theory, covers the same material as 18.01 (Single Variable Calculus), but at a deeper and more rigorous level. It emphasizes careful reasoning and understanding of proofs. The course assumes knowledge of elementary calculus. 18.014, Calculus with Theory, covers the same material as 18.01 (Single Variable Calculus), but at a deeper and more rigorous level. It emphasizes careful reasoning and understanding of proofs. The course assumes knowledge of elementary calculus.

Subjects

axioms for the real numbers | axioms for the real numbers | the Riemann integral | the Riemann integral | limits | limits | theorems on continuous functions | theorems on continuous functions | derivatives of functions of one variable | derivatives of functions of one variable | the fundamental theorems of calculus | the fundamental theorems of calculus | Taylor's theorem | Taylor's theorem | infinite series | infinite series | power series | power series | rigorous treatment of the elementary functions | rigorous treatment of the elementary functions

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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This course is an introduction to analytic number theory, including the use of zeta functions and L-functions to prove distribution results concerning prime numbers (e.g., the prime number theorem in arithmetic progressions). This course is an introduction to analytic number theory, including the use of zeta functions and L-functions to prove distribution results concerning prime numbers (e.g., the prime number theorem in arithmetic progressions).

Subjects

analytic number theory | analytic number theory | Riemann zeta function | Riemann zeta function | L-functions | L-functions | prime number theorem | prime number theorem | Dirichlet's theorem | Dirichlet's theorem | Riemann Hypothesis | Riemann Hypothesis | Sieving methods | Sieving methods | Linnik | Linnik | Linnik's large sieve | Linnik's large sieve | Selberg | Selberg | Selberg's sieve | Selberg's sieve | distribution of prime numbers | distribution of prime numbers

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18.104 is an undergraduate level seminar for mathematics majors. Students present and discuss subject matter taken from current journals or books. Instruction and practice in written and oral communication is provided. The topics vary from year to year. The topic for this term is Applications to Number Theory. 18.104 is an undergraduate level seminar for mathematics majors. Students present and discuss subject matter taken from current journals or books. Instruction and practice in written and oral communication is provided. The topics vary from year to year. The topic for this term is Applications to Number Theory.

Subjects

Infinitude of the primes | Infinitude of the primes | Summing powers of integers | Summing powers of integers | Bernoulli polynomials | Bernoulli polynomials | sine product formula | sine product formula | $\zeta(2n)$ | $\zeta(2n)$ | Fermat's Little Theorem | Fermat's Little Theorem | Fermat's Great Theorem | Fermat's Great Theorem | Averages of arithmetic functions | Averages of arithmetic functions | arithmetic-geometric mean | arithmetic-geometric mean | Gauss' theorem | Gauss' theorem | Wallis's formula | Wallis's formula | Stirling's formula | Stirling's formula | prime number theorem | prime number theorem | Riemann's hypothesis | Riemann's hypothesis | Euler's proof of infinitude of primes | Euler's proof of infinitude of primes | Density of prime numbers | Density of prime numbers | Euclidean algorithm | Euclidean algorithm | Golden Ratio | Golden Ratio

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Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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This course is a first course in algebraic number theory. Topics to be covered include number fields, class numbers, Dirichlet's units theorem, cyclotomic fields, local fields, valuations, decomposition and inertia groups, ramification, basic analytic methods, and basic class field theory. An additional theme running throughout the course will be the use of computer algebra to investigate number-theoretic questions; this theme will appear primarily in the problem sets. This course is a first course in algebraic number theory. Topics to be covered include number fields, class numbers, Dirichlet's units theorem, cyclotomic fields, local fields, valuations, decomposition and inertia groups, ramification, basic analytic methods, and basic class field theory. An additional theme running throughout the course will be the use of computer algebra to investigate number-theoretic questions; this theme will appear primarily in the problem sets.

Subjects

algebraic number theory | algebraic number theory | number fields | number fields | class numbers | class numbers | Dirichlet's units theorem | Dirichlet's units theorem | cyclotomic fields | cyclotomic fields | local fields | local fields | valuations | valuations | decomposition and inertia groups | decomposition and inertia groups | ramification | ramification | basic analytic methods | basic analytic methods | basic class field theory | basic class field theory

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Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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This course offers an introduction to discrete and computational geometry. Emphasis is placed on teaching methods in combinatorial geometry. Many results presented are recent, and include open (as yet unsolved) problems. This course offers an introduction to discrete and computational geometry. Emphasis is placed on teaching methods in combinatorial geometry. Many results presented are recent, and include open (as yet unsolved) problems.

Subjects

discrete geometry | discrete geometry | computational geometry | computational geometry | convex partitions | convex partitions | binary space partitions | binary space partitions | art gallery problems | art gallery problems | Planar graphs | Planar graphs | pseudo-triangulations | pseudo-triangulations | encompassing graphs | encompassing graphs | geometric graphs | geometric graphs | crossing numbers | crossing numbers | extremal graph theory | extremal graph theory | Gallai-Sylvester problems | Gallai-Sylvester problems

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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This course is a study of Behavior of Algorithms and covers an area of current interest in theoretical computer science. The topics vary from term to term. During this term, we discuss rigorous approaches to explaining the typical performance of algorithms with a focus on the following approaches: smoothed analysis, condition numbers/parametric analysis, and subclassing inputs. This course is a study of Behavior of Algorithms and covers an area of current interest in theoretical computer science. The topics vary from term to term. During this term, we discuss rigorous approaches to explaining the typical performance of algorithms with a focus on the following approaches: smoothed analysis, condition numbers/parametric analysis, and subclassing inputs.

Subjects

Condition number | Condition number | largest singluar value of a matrix | largest singluar value of a matrix | Smoothed analysis | Smoothed analysis | Gaussian elimination | Gaussian elimination | Growth factors of partial and complete pivoting | Growth factors of partial and complete pivoting | GE of graphs with low bandwidth or small separators | GE of graphs with low bandwidth or small separators | Spectral Partitioning of planar graphs | Spectral Partitioning of planar graphs | spectral paritioning of well-shaped meshes | spectral paritioning of well-shaped meshes | spectral paritioning of nearest neighbor graphs | spectral paritioning of nearest neighbor graphs | Turner's theorem | Turner's theorem | bandwidth of semi-random graphs. | bandwidth of semi-random graphs. | McSherry's spectral bisection algorithm | McSherry's spectral bisection algorithm | Linear Programming | Linear Programming | von Neumann's algorithm | von Neumann's algorithm | primal and dual simplex methods | and duality Strong duality theorem | primal and dual simplex methods | and duality Strong duality theorem | Renegar's condition numbers | Renegar's condition numbers

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The subject of enumerative combinatorics deals with counting the number of elements of a finite set. For instance, the number of ways to write a positive integer n as a sum of positive integers, taking order into account, is 2n-1. We will be concerned primarily with bijective proofs, i.e., showing that two sets have the same number of elements by exhibiting a bijection (one-to-one correspondence) between them. This is a subject which requires little mathematical background to reach the frontiers of current research. Students will therefore have the opportunity to do original research. It might be necessary to limit enrollment. The subject of enumerative combinatorics deals with counting the number of elements of a finite set. For instance, the number of ways to write a positive integer n as a sum of positive integers, taking order into account, is 2n-1. We will be concerned primarily with bijective proofs, i.e., showing that two sets have the same number of elements by exhibiting a bijection (one-to-one correspondence) between them. This is a subject which requires little mathematical background to reach the frontiers of current research. Students will therefore have the opportunity to do original research. It might be necessary to limit enrollment.

Subjects

enumerative combinatorics | enumerative combinatorics | finite set | finite set | sum of positive integers | sum of positive integers | bijective proofs | bijective proofs | bijection (one-to-one correspondence) | bijection (one-to-one correspondence) | permutations | permutations | partitions | partitions | Catalan numbers | Catalan numbers | Young tableaux | Young tableaux | lattice paths and tilings | lattice paths and tilings

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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