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18.996 Random Matrix Theory and Its Applications (MIT) 18.996 Random Matrix Theory and Its Applications (MIT)

Description

This course is an introduction to the basics of random matrix theory, motivated by engineering and scientific applications. This course is an introduction to the basics of random matrix theory, motivated by engineering and scientific applications.

Subjects

Random matrix theory | Random matrix theory | Matrix Jacobians | Matrix Jacobians | Wishart Matrices | Wishart Matrices | Wigner's Semi-Circular laws | Wigner's Semi-Circular laws | Matrix beta ensembles | Matrix beta ensembles | free probability | free probability | spherical coordinates | spherical coordinates | wedging | wedging | Plucker coordinates | Plucker coordinates | matrix factorizations | matrix factorizations | householder transformations | householder transformations | Stiefel manifold | Stiefel manifold | Cauchey-Binet theorem | Cauchey-Binet theorem | Telatar's paper | Telatar's paper | level densities | level densities | orthogonal polynomials | orthogonal polynomials | matrix integrals | matrix integrals | hypergeometric functions | hypergeometric functions | wireless communictions | wireless communictions | eigenvalue density | eigenvalue density | sample covariance matrices | sample covariance matrices | Marcenko-Pastur theorem | Marcenko-Pastur theorem | wireless communications | wireless communications

License

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7.06 Cell Biology (MIT) 7.06 Cell Biology (MIT)

Description

This course deals with the biology of cells of higher organisms: The structure, function, and biosynthesis of cellular membranes and organelles; cell growth and oncogenic transformation; transport, receptors, and cell signaling; the cytoskeleton, the extracellular matrix, and cell movements; chromatin structure and RNA synthesis. This course deals with the biology of cells of higher organisms: The structure, function, and biosynthesis of cellular membranes and organelles; cell growth and oncogenic transformation; transport, receptors, and cell signaling; the cytoskeleton, the extracellular matrix, and cell movements; chromatin structure and RNA synthesis.

Subjects

Biology | Biology | cells | cells | organisms | organisms | biosynthesis | biosynthesis | cellular membranes | cellular membranes | organelles | organelles | cell growth | cell growth | oncogenic transformation | oncogenic transformation | transport | transport | receptors | receptors | cell signaling | cell signaling | cytoskeleton | cytoskeleton | extracellular matrix | extracellular matrix | matrix | matrix | cell movements | cell movements | chromatin | chromatin | RNA | RNA | RNA synthesis | RNA synthesis

License

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18.409 Algorithmic Aspects of Machine Learning (MIT) 18.409 Algorithmic Aspects of Machine Learning (MIT)

Description

This course is organized around algorithmic issues that arise in machine learning. Modern machine learning systems are often built on top of algorithms that do not have provable guarantees, and it is the subject of debate when and why they work. In this class, we focus on designing algorithms whose performance we can rigorously analyze for fundamental machine learning problems. This course is organized around algorithmic issues that arise in machine learning. Modern machine learning systems are often built on top of algorithms that do not have provable guarantees, and it is the subject of debate when and why they work. In this class, we focus on designing algorithms whose performance we can rigorously analyze for fundamental machine learning problems.

Subjects

Machine learning | Machine learning | nonnegative matrix factorization | nonnegative matrix factorization | tensor decomposition | tensor decomposition | tensor rank | tensor rank | border rank | border rank | sparse coding | sparse coding | sparse recovery | sparse recovery | learning mixture model | learning mixture model | matrix completion | matrix completion

License

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18.304 Undergraduate Seminar in Discrete Mathematics (MIT) 18.304 Undergraduate Seminar in Discrete Mathematics (MIT)

Description

This course is a student-presented seminar in combinatorics, graph theory, and discrete mathematics in general. Instruction and practice in written and oral communication is emphasized, with participants reading and presenting papers from recent mathematics literature and writing a final paper in a related topic. This course is a student-presented seminar in combinatorics, graph theory, and discrete mathematics in general. Instruction and practice in written and oral communication is emphasized, with participants reading and presenting papers from recent mathematics literature and writing a final paper in a related topic.

Subjects

discrete math; discrete mathematics; discrete; math; mathematics; seminar; presentations; student presentations; oral; communication; stable marriage; dych; emergency; response vehicles; ambulance; game theory; congruences; color theorem; four color; cake cutting; algorithm; RSA; encryption; numberical integration; sorting; post correspondence problem; PCP; ramsey; van der waals; fibonacci; recursion; domino; tiling; towers; hanoi; pigeonhole; principle; matrix; hamming; code; hat game; juggling; zero-knowledge; proof; repeated games; lewis carroll; determinants; infinitude of primes; bridges; konigsberg; koenigsberg; time series analysis; GARCH; rational; recurrence; relations; digital; image; compression; quantum computing | discrete math; discrete mathematics; discrete; math; mathematics; seminar; presentations; student presentations; oral; communication; stable marriage; dych; emergency; response vehicles; ambulance; game theory; congruences; color theorem; four color; cake cutting; algorithm; RSA; encryption; numberical integration; sorting; post correspondence problem; PCP; ramsey; van der waals; fibonacci; recursion; domino; tiling; towers; hanoi; pigeonhole; principle; matrix; hamming; code; hat game; juggling; zero-knowledge; proof; repeated games; lewis carroll; determinants; infinitude of primes; bridges; konigsberg; koenigsberg; time series analysis; GARCH; rational; recurrence; relations; digital; image; compression; quantum computing | discrete math | discrete math | discrete mathematics | discrete mathematics | discrete | discrete | math | math | mathematics | mathematics | seminar | seminar | presentations | presentations | student presentations | student presentations | oral | oral | communication | communication | stable marriage | stable marriage | dych | dych | emergency | emergency | response vehicles | response vehicles | ambulance | ambulance | game theory | game theory | congruences | congruences | color theorem | color theorem | four color | four color | cake cutting | cake cutting | algorithm | algorithm | RSA | RSA | encryption | encryption | numberical integration | numberical integration | sorting | sorting | post correspondence problem | post correspondence problem | PCP | PCP | ramsey | ramsey | van der waals | van der waals | fibonacci | fibonacci | recursion | recursion | domino | domino | tiling | tiling | towers | towers | hanoi | hanoi | pigeonhole | pigeonhole | principle | principle | matrix | matrix | hamming | hamming | code | code | hat game | hat game | juggling | juggling | zero-knowledge | zero-knowledge | proof | proof | repeated games | repeated games | lewis carroll | lewis carroll | determinants | determinants | infinitude of primes | infinitude of primes | bridges | bridges | konigsberg | konigsberg | koenigsberg | koenigsberg | time series analysis | time series analysis | GARCH | GARCH | rational | rational | recurrence | recurrence | relations | relations | digital | digital | image | image | compression | compression | quantum computing | quantum computing

License

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2.785J Cell-Matrix Mechanics (MIT) 2.785J Cell-Matrix Mechanics (MIT)

Description

Mechanical forces play a decisive role during development of tissues and organs, during remodeling following injury as well as in normal function. A stress field influences cell function primarily through deformation of the extracellular matrix to which cells are attached. Deformed cells express different biosynthetic activity relative to undeformed cells. The unit cell process paradigm combined with topics in connective tissue mechanics form the basis for discussions of several topics from cell biology, physiology, and medicine. Mechanical forces play a decisive role during development of tissues and organs, during remodeling following injury as well as in normal function. A stress field influences cell function primarily through deformation of the extracellular matrix to which cells are attached. Deformed cells express different biosynthetic activity relative to undeformed cells. The unit cell process paradigm combined with topics in connective tissue mechanics form the basis for discussions of several topics from cell biology, physiology, and medicine.

Subjects

2.785 | 2.785 | 3.97 | 3.97 | 20.411 | 20.411 | HST.523 | HST.523 | cell | cell | matrix | matrix | mechanics | mechanics | tissue | tissue | organ | organ | development | development | injury | injury | stress field | stress field | cell function | cell function | deformed cells | deformed cells | biosynthetic activity | biosynthetic activity | unit cell | unit cell | connective tissue | connective tissue | cell biology | cell biology | physiology | physiology | medicine | medicine | cytoplasm | cytoplasm | extracellular matrix | extracellular matrix | skeleton | skeleton | bone | bone

License

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1.050 Solid Mechanics (MIT) 1.050 Solid Mechanics (MIT)

Description

Includes audio/video content: AV faculty introductions. 1.050 is a sophomore-level engineering mechanics course, commonly labelled "Statics and Strength of Materials" or "Solid Mechanics I." This course introduces students to the fundamental principles and methods of structural mechanics. Topics covered include: static equilibrium, force resultants, support conditions, analysis of determinate planar structures (beams, trusses, frames), stresses and strains in structural elements, states of stress (shear, bending, torsion), statically indeterminate systems, displacements and deformations, introduction to matrix methods, elastic stability, and approximate methods. Design exercises are used to encourage creative student initiative and systems thinking. Includes audio/video content: AV faculty introductions. 1.050 is a sophomore-level engineering mechanics course, commonly labelled "Statics and Strength of Materials" or "Solid Mechanics I." This course introduces students to the fundamental principles and methods of structural mechanics. Topics covered include: static equilibrium, force resultants, support conditions, analysis of determinate planar structures (beams, trusses, frames), stresses and strains in structural elements, states of stress (shear, bending, torsion), statically indeterminate systems, displacements and deformations, introduction to matrix methods, elastic stability, and approximate methods. Design exercises are used to encourage creative student initiative and systems thinking.

Subjects

solid mechanics | solid mechanics | engineering design | engineering design | open ended exercises | open ended exercises | matrix analysis of structures | matrix analysis of structures | structural mechanics | structural mechanics | static equilibrium | static equilibrium | force resultants | force resultants | support conditions | support conditions | determinate planar structures | determinate planar structures | beams | beams | trusses | trusses | frames | frames | stress | stress | strain | strain | shear | shear | bending | bending | torsion | torsion | matrix methods | matrix methods | elastic stability | elastic stability | design exercises | design exercises | interactive exercises | interactive exercises | systems thinking | systems thinking

License

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HST.523J Cell-Matrix Mechanics (MIT) HST.523J Cell-Matrix Mechanics (MIT)

Description

Mechanical forces play a decisive role during development of tissues and organs, during remodeling following injury as well as in normal function. A stress field influences cell function primarily through deformation of the extracellular matrix to which cells are attached. Deformed cells express different biosynthetic activity relative to undeformed cells. The unit cell process paradigm combined with topics in connective tissue mechanics form the basis for discussions of several topics from cell biology, physiology, and medicine. Mechanical forces play a decisive role during development of tissues and organs, during remodeling following injury as well as in normal function. A stress field influences cell function primarily through deformation of the extracellular matrix to which cells are attached. Deformed cells express different biosynthetic activity relative to undeformed cells. The unit cell process paradigm combined with topics in connective tissue mechanics form the basis for discussions of several topics from cell biology, physiology, and medicine.

Subjects

cell | cell | tissue | tissue | organ | organ | unit cell process | unit cell process | cell matrix | cell matrix | tissue structure | tissue structure | extracellular matrix | extracellular matrix | adhesion protein | adhesion protein | integrin | integrin | cell force | cell force | cell contraction | cell contraction | healing | healing | skin | skin | scar | scar | tendon | tendon | ligament | ligament | cartilage | cartilage | bone | bone | collagen | collagen | muscle | muscle | nerve | nerve | implant | implant | HST.523 | HST.523 | 2.785 | 2.785 | 3.97 | 3.97 | 20.411 | 20.411

License

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18.304 Undergraduate Seminar in Discrete Mathematics (MIT) 18.304 Undergraduate Seminar in Discrete Mathematics (MIT)

Description

This course is a student-presented seminar in combinatorics, graph theory, and discrete mathematics in general. Instruction and practice in written and oral communication is emphasized, with participants reading and presenting papers from recent mathematics literature and writing a final paper in a related topic. This course is a student-presented seminar in combinatorics, graph theory, and discrete mathematics in general. Instruction and practice in written and oral communication is emphasized, with participants reading and presenting papers from recent mathematics literature and writing a final paper in a related topic.

Subjects

discrete math; discrete mathematics; discrete; math; mathematics; seminar; presentations; student presentations; oral; communication; stable marriage; dych; emergency; response vehicles; ambulance; game theory; congruences; color theorem; four color; cake cutting; algorithm; RSA; encryption; numberical integration; sorting; post correspondence problem; PCP; ramsey; van der waals; fibonacci; recursion; domino; tiling; towers; hanoi; pigeonhole; principle; matrix; hamming; code; hat game; juggling; zero-knowledge; proof; repeated games; lewis carroll; determinants; infinitude of primes; bridges; konigsberg; koenigsberg; time series analysis; GARCH; rational; recurrence; relations; digital; image; compression; quantum computing | discrete math; discrete mathematics; discrete; math; mathematics; seminar; presentations; student presentations; oral; communication; stable marriage; dych; emergency; response vehicles; ambulance; game theory; congruences; color theorem; four color; cake cutting; algorithm; RSA; encryption; numberical integration; sorting; post correspondence problem; PCP; ramsey; van der waals; fibonacci; recursion; domino; tiling; towers; hanoi; pigeonhole; principle; matrix; hamming; code; hat game; juggling; zero-knowledge; proof; repeated games; lewis carroll; determinants; infinitude of primes; bridges; konigsberg; koenigsberg; time series analysis; GARCH; rational; recurrence; relations; digital; image; compression; quantum computing | discrete math | discrete math | discrete mathematics | discrete mathematics | discrete | discrete | math | math | mathematics | mathematics | seminar | seminar | presentations | presentations | student presentations | student presentations | oral | oral | communication | communication | stable marriage | stable marriage | dych | dych | emergency | emergency | response vehicles | response vehicles | ambulance | ambulance | game theory | game theory | congruences | congruences | color theorem | color theorem | four color | four color | cake cutting | cake cutting | algorithm | algorithm | RSA | RSA | encryption | encryption | numberical integration | numberical integration | sorting | sorting | post correspondence problem | post correspondence problem | PCP | PCP | ramsey | ramsey | van der waals | van der waals | fibonacci | fibonacci | recursion | recursion | domino | domino | tiling | tiling | towers | towers | hanoi | hanoi | pigeonhole | pigeonhole | principle | principle | matrix | matrix | hamming | hamming | code | code | hat game | hat game | juggling | juggling | zero-knowledge | zero-knowledge | proof | proof | repeated games | repeated games | lewis carroll | lewis carroll | determinants | determinants | infinitude of primes | infinitude of primes | bridges | bridges | konigsberg | konigsberg | koenigsberg | koenigsberg | time series analysis | time series analysis | GARCH | GARCH | rational | rational | recurrence | recurrence | relations | relations | digital | digital | image | image | compression | compression | quantum computing | quantum computing

License

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18.996 Random Matrix Theory and Its Applications (MIT)

Description

This course is an introduction to the basics of random matrix theory, motivated by engineering and scientific applications.

Subjects

Random matrix theory | Matrix Jacobians | Wishart Matrices | Wigner's Semi-Circular laws | Matrix beta ensembles | free probability | spherical coordinates | wedging | Plucker coordinates | matrix factorizations | householder transformations | Stiefel manifold | Cauchey-Binet theorem | Telatar's paper | level densities | orthogonal polynomials | matrix integrals | hypergeometric functions | wireless communictions | eigenvalue density | sample covariance matrices | Marcenko-Pastur theorem | wireless communications

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 https://ocw.mit.edu/terms/index.htm

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20.441 Biomaterials-Tissue Interactions (BE.441) (MIT) 20.441 Biomaterials-Tissue Interactions (BE.441) (MIT)

Description

This course is an introduction to principles of materials science and cell biology underlying the design of medical implants, artificial organs, and matrices for tissue engineering. Topics include methods for biomaterials surface characterization and analysis of protein adsorption on biomaterials. Molecular and cellular interactions with biomaterials are analyzed in terms of unit cell processes, such as matrix synthesis, degradation, and contraction. It also covers mechanisms underlying wound healing and tissue remodeling following implantation in various organs. Other areas include tissue and organ regeneration; design of implants and prostheses based on control of biomaterials-tissue interactions; comparative analysis of intact, biodegradable, and bioreplaceable implants by reference to This course is an introduction to principles of materials science and cell biology underlying the design of medical implants, artificial organs, and matrices for tissue engineering. Topics include methods for biomaterials surface characterization and analysis of protein adsorption on biomaterials. Molecular and cellular interactions with biomaterials are analyzed in terms of unit cell processes, such as matrix synthesis, degradation, and contraction. It also covers mechanisms underlying wound healing and tissue remodeling following implantation in various organs. Other areas include tissue and organ regeneration; design of implants and prostheses based on control of biomaterials-tissue interactions; comparative analysis of intact, biodegradable, and bioreplaceable implants by reference to

Subjects

medical implants | medical implants | artificial organs | artificial organs | tissue engineering | tissue engineering | matrix | matrix | biomaterials | biomaterials | protein adsorption | protein adsorption | unit cell process | unit cell process | wound healing | wound healing | tissue remodeling | tissue remodeling | tissue regeneration | tissue regeneration | organ regeneration | organ regeneration | prosthesis | prosthesis | biodegradable | biodegradable | bioreplaceable implants | bioreplaceable implants | BE.441 | BE.441 | 2.79 | 2.79 | 3.96 | 3.96 | HST.522 | HST.522

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18.06 Linear Algebra (MIT) 18.06 Linear Algebra (MIT)

Description

Basic subject on matrix theory and linear algebra, emphasizing topics useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, similarity, and positive definite matrices. Applications to least-squares approximations, stability of differential equations, networks, Fourier transforms, and Markov processes. Uses MATLAB®. Compared with 18.700 [also Linear Algebra], more emphasis on matrix algorithms and many applications. MATLAB® is a trademark of The MathWorks, Inc. Basic subject on matrix theory and linear algebra, emphasizing topics useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, similarity, and positive definite matrices. Applications to least-squares approximations, stability of differential equations, networks, Fourier transforms, and Markov processes. Uses MATLAB®. Compared with 18.700 [also Linear Algebra], more emphasis on matrix algorithms and many applications. MATLAB® is a trademark of The MathWorks, Inc.

Subjects

Generalized spaces | Generalized spaces | Linear algebra | Linear algebra | Algebra | Universal | Algebra | Universal | Mathematical analysis | Mathematical analysis | Calculus of operations | Calculus of operations | Line geometry | Line geometry | Topology | Topology | matrix theory | matrix theory | systems of equations | systems of equations | vector spaces | vector spaces | systems determinants | systems determinants | eigen values | eigen values | positive definite matrices | positive definite matrices | Markov processes | Markov processes | Fourier transforms | Fourier transforms | differential equations | differential equations

License

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18.409 Algorithmic Aspects of Machine Learning (MIT)

Description

This course is organized around algorithmic issues that arise in machine learning. Modern machine learning systems are often built on top of algorithms that do not have provable guarantees, and it is the subject of debate when and why they work. In this class, we focus on designing algorithms whose performance we can rigorously analyze for fundamental machine learning problems.

Subjects

Machine learning | nonnegative matrix factorization | tensor decomposition | tensor rank | border rank | sparse coding | sparse recovery | learning mixture model | matrix completion

License

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ESD.36J System and Project Management (MIT) ESD.36J System and Project Management (MIT)

Description

The course is designed for students in the System Design and Management (SDM) program and therefore assumes that you already have a basic knowledge of project management. The objective is to introduce advanced methods and tools of project management in a realistic context such that they can be taken back to the workplace to improve management of development projects. In contrast to traditional courses on the subject we will emphasize scenarios that cannot be fully predicted such as task iterations, unplanned rework, perceived versus actual progress and misalignments between tasks, product architectures and organizations. This class was also offered in Course 13 (Department of Ocean Engineering) as 13.615J. In 2005, ocean engineering subjects became part of Course 2 (Department of Mechanica The course is designed for students in the System Design and Management (SDM) program and therefore assumes that you already have a basic knowledge of project management. The objective is to introduce advanced methods and tools of project management in a realistic context such that they can be taken back to the workplace to improve management of development projects. In contrast to traditional courses on the subject we will emphasize scenarios that cannot be fully predicted such as task iterations, unplanned rework, perceived versus actual progress and misalignments between tasks, product architectures and organizations. This class was also offered in Course 13 (Department of Ocean Engineering) as 13.615J. In 2005, ocean engineering subjects became part of Course 2 (Department of Mechanica

Subjects

system and project management | system and project management | product development | product development | PERT | PERT | CPM | CPM | design structure matrix | design structure matrix | DSM | DSM | system dynamics | system dynamics | SD | SD | SPM | SPM | product development process | product development process | PDP | PDP | concurrent engineering | concurrent engineering | project monitoring | project monitoring | resource consumption | resource consumption | critical paths | critical paths | project progress | project progress | corrective action | corrective action | system dynamics models | system dynamics models | ESD.36 | ESD.36 | 1.432 | 1.432

License

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5.80 Small-Molecule Spectroscopy and Dynamics (MIT) 5.80 Small-Molecule Spectroscopy and Dynamics (MIT)

Description

The goal of this course is to illustrate the spectroscopy of small molecules in the gas phase: quantum mechanical effective Hamiltonian models for rotational, vibrational, and electronic structure; transition selection rules and relative intensities; diagnostic patterns and experimental methods for the assignment of non-textbook spectra; breakdown of the Born-Oppenheimer approximation (spectroscopic perturbations); the stationary phase approximation; nondegenerate and quasidegenerate perturbation theory (van Vleck transformation); qualitative molecular orbital theory (Walsh diagrams); the notation of atomic and molecular spectroscopy. The goal of this course is to illustrate the spectroscopy of small molecules in the gas phase: quantum mechanical effective Hamiltonian models for rotational, vibrational, and electronic structure; transition selection rules and relative intensities; diagnostic patterns and experimental methods for the assignment of non-textbook spectra; breakdown of the Born-Oppenheimer approximation (spectroscopic perturbations); the stationary phase approximation; nondegenerate and quasidegenerate perturbation theory (van Vleck transformation); qualitative molecular orbital theory (Walsh diagrams); the notation of atomic and molecular spectroscopy.

Subjects

spectroscopy | spectroscopy | harmonic oscillators | harmonic oscillators | matrix | matrix | hamiltonian | hamiltonian | heisenberg | heisenberg | vibrating rotor | vibrating rotor | Born-Oppenheimer | Born-Oppenheimer | diatomics | diatomics | laser schemes | laser schemes | angular momentum | angular momentum | hund's cases | hund's cases | energy levels | energy levels | second-order effects | second-order effects | perturbations | perturbations | Wigner-Eckart | Wigner-Eckart | Rydberg-Klein-Rees | Rydberg-Klein-Rees | rigid rotor | rigid rotor | asymmetric rotor | asymmetric rotor | vibronic coupling | vibronic coupling | wavepackets | wavepackets

License

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18.022 Calculus of Several Variables (MIT) 18.022 Calculus of Several Variables (MIT)

Description

This is a variation on 18.02 Multivariable Calculus. It covers the same topics as in 18.02, but with more focus on mathematical concepts. This is a variation on 18.02 Multivariable Calculus. It covers the same topics as in 18.02, but with more focus on mathematical concepts.

Subjects

vector algebra | vector algebra | determinant | determinant | matrix | matrix | matrices | matrices | vector-valued functions | vector-valued functions | space motion | space motion | scalar functions | scalar functions | partial differentiation | partial differentiation | gradient | gradient | optimization techniques | optimization techniques | double integrals | double integrals | line integrals | line integrals | exact differentials | exact differentials | conservative fields | conservative fields | Green's theorem | Green's theorem | triple integrals | triple integrals | surface integrals | surface integrals | divergence theorem | divergence theorem | Stokes' theorem | Stokes' theorem | geometry | geometry | vector fields | vector fields | linear algebra | linear algebra

License

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3.016 Mathematics for Materials Scientists and Engineers (MIT) 3.016 Mathematics for Materials Scientists and Engineers (MIT)

Description

This course covers the mathematical techniques necessary for understanding of materials science and engineering topics such as energetics, materials structure and symmetry, materials response to applied fields, mechanics and physics of solids and soft materials. The class uses examples from the materials science and engineering core courses (3.012 and 3.014) to introduce mathematical concepts and materials-related problem solving skills. Topics include linear algebra and orthonormal basis, eigenvalues and eigenvectors, quadratic forms, tensor operations, symmetry operations, calculus of several variables, introduction to complex analysis, ordinary and partial differential equations, theory of distributions, and fourier analysis. Users may find additional or updated materials at Professor C This course covers the mathematical techniques necessary for understanding of materials science and engineering topics such as energetics, materials structure and symmetry, materials response to applied fields, mechanics and physics of solids and soft materials. The class uses examples from the materials science and engineering core courses (3.012 and 3.014) to introduce mathematical concepts and materials-related problem solving skills. Topics include linear algebra and orthonormal basis, eigenvalues and eigenvectors, quadratic forms, tensor operations, symmetry operations, calculus of several variables, introduction to complex analysis, ordinary and partial differential equations, theory of distributions, and fourier analysis. Users may find additional or updated materials at Professor C

Subjects

energetics | energetics | visualization | visualization | graph | graph | plot | plot | chart | chart | materials science | materials science | DMSE | DMSE | structure | structure | symmetry | symmetry | mechanics | mechanics | physicss | physicss | solids and soft materials | solids and soft materials | linear algebra | linear algebra | orthonormal basis | orthonormal basis | eigenvalue | eigenvalue | eigenvector | eigenvector | quadratic form | quadratic form | tensor operation | tensor operation | symmetry operation | symmetry operation | calculus | calculus | complex analysis | complex analysis | differential equations | differential equations | ODE | ODE | solution | solution | vector | vector | matrix | matrix | determinant | determinant | theory of distributions | theory of distributions | fourier analysis | fourier analysis | random walk | random walk | Mathematica | Mathematica | simulation | simulation

License

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2.782J Design of Medical Devices and Implants (MIT) 2.782J Design of Medical Devices and Implants (MIT)

Description

Solution of clinical problems by use of implants and other medical devices. Systematic use of cell-matrix control volumes. The role of stress analysis in the design process. Anatomic fit: shape and size of implants. Selection of biomaterials. Instrumentation for surgical implantation procedures. Preclinical testing for safety and efficacy: risk/benefit ratio assessment. Evaluation of clinical performance: design of clinical trials. Project materials drawn from orthopedic devices, soft tissue implants, artificial organs, and dental implants. Solution of clinical problems by use of implants and other medical devices. Systematic use of cell-matrix control volumes. The role of stress analysis in the design process. Anatomic fit: shape and size of implants. Selection of biomaterials. Instrumentation for surgical implantation procedures. Preclinical testing for safety and efficacy: risk/benefit ratio assessment. Evaluation of clinical performance: design of clinical trials. Project materials drawn from orthopedic devices, soft tissue implants, artificial organs, and dental implants.

Subjects

clinical problems | clinical problems | implants | implants | medical devices | medical devices | cell-matrix control volumes | cell-matrix control volumes | stress analysis | stress analysis | Anatomic fit | Anatomic fit | biomaterials | biomaterials | surgical implantation procedures | surgical implantation procedures | Preclinical testing | Preclinical testing | risk/benefit ratio assessment | risk/benefit ratio assessment | clinical performance | clinical performance | clinical trials | clinical trials | orthopedic devices | orthopedic devices | soft tissue implants | soft tissue implants | artificial organs | artificial organs | dental implants | dental implants | BE.451J | BE.451J | 2.782 | 2.782 | 3.961 | 3.961 | BE.451 | BE.451 | HST.524 | HST.524 | 20.451 | 20.451

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18.02 Multivariable Calculus (MIT) 18.02 Multivariable Calculus (MIT)

Description

Includes audio/video content: AV lectures. This course covers vector and multi-variable calculus. It is the second semester in the freshman calculus sequence. Topics include vectors and matrices, partial derivatives, double and triple integrals, and vector calculus in 2 and 3-space. MIT OpenCourseWare offers another version of 18.02, from the Spring 2006 term. Both versions cover the same material, although they are taught by different faculty and rely on different textbooks. Multivariable Calculus (18.02) is taught during the Fall and Spring terms at MIT, and is a required subject for all MIT undergraduates. Includes audio/video content: AV lectures. This course covers vector and multi-variable calculus. It is the second semester in the freshman calculus sequence. Topics include vectors and matrices, partial derivatives, double and triple integrals, and vector calculus in 2 and 3-space. MIT OpenCourseWare offers another version of 18.02, from the Spring 2006 term. Both versions cover the same material, although they are taught by different faculty and rely on different textbooks. Multivariable Calculus (18.02) is taught during the Fall and Spring terms at MIT, and is a required subject for all MIT undergraduates.

Subjects

calculus | calculus | calculus of several variables | calculus of several variables | vector algebra | vector algebra | determinants | determinants | matrix | matrix | matrices | matrices | vector-valued function | vector-valued function | space motion | space motion | scalar function | scalar function | partial differentiation | partial differentiation | gradient | gradient | optimization techniques | optimization techniques | double integrals | double integrals | line integrals | line integrals | exact differential | exact differential | conservative fields | conservative fields | Green's theorem | Green's theorem | triple integrals | triple integrals | surface integrals | surface integrals | divergence theorem Stokes' theorem | divergence theorem Stokes' theorem | applications | applications

License

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18.310 Principles of Discrete Applied Mathematics (MIT) 18.310 Principles of Discrete Applied Mathematics (MIT)

Description

This course is an introduction to discrete applied mathematics. Topics include probability, counting, linear programming, number-theoretic algorithms, sorting, data compression, and error-correcting codes. This is a Communication Intensive in the Major (CI-M) course, and thus includes a writing component. This course is an introduction to discrete applied mathematics. Topics include probability, counting, linear programming, number-theoretic algorithms, sorting, data compression, and error-correcting codes. This is a Communication Intensive in the Major (CI-M) course, and thus includes a writing component.

Subjects

probability | probability | probability theory counting | probability theory counting | pigeonhole principle | pigeonhole principle | Van der Waerden's theorem | Van der Waerden's theorem | Chernoff bounds | Chernoff bounds | counting | counting | coding | coding | sampling | sampling | random sampling | random sampling | Catalan families | Catalan families | generating functions | generating functions | chord diagrams | chord diagrams | linear programming | linear programming | simplex method | simplex method | Zero-Sum matrix | Zero-Sum matrix | network flows | network flows | maximum flow problem | maximum flow problem | sorting algorithms | sorting algorithms | QUICKSORT | QUICKSORT | median finding | median finding | sorting networks | sorting networks | Batcher's algorithm | Batcher's algorithm | Euclid's algorithm | Euclid's algorithm | Chinese Remainder Theorem | Chinese Remainder Theorem | cryptography | cryptography | RSA code | RSA code | primaility testing | primaility testing | FFT | FFT | Fast Fourier Transform | Fast Fourier Transform | Shannon's coding theorems | Shannon's coding theorems | Lempel-Ziv codes | Lempel-Ziv codes | linear codes | linear codes | hamming code | hamming code

License

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8.323 Relativistic Quantum Field Theory I (MIT) 8.323 Relativistic Quantum Field Theory I (MIT)

Description

8.323, Relativistic Quantum Field Theory I, is a one-term self-contained subject in quantum field theory. Concepts and basic techniques are developed through applications in elementary particle physics, and condensed matter physics. 8.323, Relativistic Quantum Field Theory I, is a one-term self-contained subject in quantum field theory. Concepts and basic techniques are developed through applications in elementary particle physics, and condensed matter physics.

Subjects

Classical field theory | Classical field theory | symmetries | symmetries | and Noether's theorem. Quantization of scalar fields | and Noether's theorem. Quantization of scalar fields | spin fields | spin fields | and Gauge bosons. Feynman graphs | and Gauge bosons. Feynman graphs | analytic properties of amplitudes and unitarity of the S-matrix. Calculations in quantum electrodynamics (QED). Introduction to renormalization. | analytic properties of amplitudes and unitarity of the S-matrix. Calculations in quantum electrodynamics (QED). Introduction to renormalization.

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8.325 Relativistic Quantum Field Theory III (MIT) 8.325 Relativistic Quantum Field Theory III (MIT)

Description

This course is the third and last term of the quantum field theory sequence. Its aim is the proper theoretical discussion of the physics of the standard model. Topics include: quantum chromodynamics; the Higgs phenomenon and a description of the standard model; deep-inelastic scattering and structure functions; basics of lattice gauge theory; operator products and effective theories; detailed structure of the standard model; spontaneously broken gauge theory and its quantization; instantons and theta-vacua; topological defects; introduction to supersymmetry. This course is the third and last term of the quantum field theory sequence. Its aim is the proper theoretical discussion of the physics of the standard model. Topics include: quantum chromodynamics; the Higgs phenomenon and a description of the standard model; deep-inelastic scattering and structure functions; basics of lattice gauge theory; operator products and effective theories; detailed structure of the standard model; spontaneously broken gauge theory and its quantization; instantons and theta-vacua; topological defects; introduction to supersymmetry.

Subjects

gauge symmetry | gauge symmetry | confinement | confinement | renormalization | renormalization | asymptotic freedom | asymptotic freedom | anomalies | anomalies | instantons | instantons | zero modes | zero modes | gauge boson and Higgs spectrum | gauge boson and Higgs spectrum | fermion multiplets | fermion multiplets | CKM matrix | CKM matrix | unification in SU(5) and SO(10) | unification in SU(5) and SO(10) | phenomenology of Higgs sector | phenomenology of Higgs sector | lepton and baryon number violation | lepton and baryon number violation | nonperturbative (lattice) formulation | nonperturbative (lattice) formulation

License

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20.442 Molecular Structure of Biological Materials (BE.442) (MIT) 20.442 Molecular Structure of Biological Materials (BE.442) (MIT)

Description

This course, intended for both graduate and upper level undergraduate students, will focus on understanding of the basic molecular structural principles of biological materials. It will address the molecular structures of various materials of biological origin, such as several types of collagen, silk, spider silk, wool, hair, bones, shells, protein adhesives, GFP, and self-assembling peptides. It will also address molecular design of new biological materials applying the molecular structural principles. The long-term goal of this course is to teach molecular design of new biological materials for a broad range of applications. A brief history of biological materials and its future perspective as well as its impact to the society will also be discussed. Several experts will be invited to gi This course, intended for both graduate and upper level undergraduate students, will focus on understanding of the basic molecular structural principles of biological materials. It will address the molecular structures of various materials of biological origin, such as several types of collagen, silk, spider silk, wool, hair, bones, shells, protein adhesives, GFP, and self-assembling peptides. It will also address molecular design of new biological materials applying the molecular structural principles. The long-term goal of this course is to teach molecular design of new biological materials for a broad range of applications. A brief history of biological materials and its future perspective as well as its impact to the society will also be discussed. Several experts will be invited to gi

Subjects

protein | protein | hydration | hydration | amino acid | amino acid | ECM | ECM | extracellular matrix | extracellular matrix | peptide | peptide | helix | helix | DNA | DNA | RNA | RNA | biomaterial | biomaterial | biotech | biotech | biotechnology | biotechnology | nanomaterial | nanomaterial | beta-sheet | beta-sheet | beta sheet | beta sheet | molecular structure | molecular structure | bioengineering | bioengineering | silk | silk | biomimetic | biomimetic | self-assembly | self-assembly | keratin | keratin | collagen | collagen | adhesive | adhesive | GFP | GFP | fluorescent | fluorescent | polymer | polymer | lipid | lipid

License

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12.141 Electron Microprobe Analysis (MIT) 12.141 Electron Microprobe Analysis (MIT)

Description

Introduction to the theory of x-ray microanalysis through the electron microprobe including ZAF matrix corrections. Techniques to be discussed are wavelength and energy dispersive spectrometry, scanning backscattered electron, secondary electron, cathodoluminescence, and X-ray imaging. Lab sessions involve hands-on use of the electron microprobe.Offered for undergraduate credit, but persons interested in an in-depth discussion of quantitative x-ray analysis are invited to participate. Students will be required to complete lab exercises to obtain credit. Introduction to the theory of x-ray microanalysis through the electron microprobe including ZAF matrix corrections. Techniques to be discussed are wavelength and energy dispersive spectrometry, scanning backscattered electron, secondary electron, cathodoluminescence, and X-ray imaging. Lab sessions involve hands-on use of the electron microprobe.Offered for undergraduate credit, but persons interested in an in-depth discussion of quantitative x-ray analysis are invited to participate. Students will be required to complete lab exercises to obtain credit.

Subjects

x-ray microanalysis | x-ray microanalysis | electron microprobe | electron microprobe | ZAF matrix corrections | ZAF matrix corrections | wavelength and energy dispersive spectrometry | wavelength and energy dispersive spectrometry | scanning backscattered electron | scanning backscattered electron | secondary electron | secondary electron | cathodoluminescence | cathodoluminescence | X-ray imaging | X-ray imaging

License

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18.06 Linear Algebra (MIT) 18.06 Linear Algebra (MIT)

Description

This is a basic subject on matrix theory and linear algebra. Emphasis is given to topics that will be useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, similarity, and positive definite matrices. This is a basic subject on matrix theory and linear algebra. Emphasis is given to topics that will be useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, similarity, and positive definite matrices.

Subjects

Generalized spaces | Generalized spaces | Linear algebra | Linear algebra | Algebra | Universal | Algebra | Universal | Mathematical analysis | Mathematical analysis | Calculus of operations | Calculus of operations | Line geometry | Line geometry | Topology | Topology | matrix theory | matrix theory | systems of equations | systems of equations | vector spaces | vector spaces | systems determinants | systems determinants | eigen values | eigen values | positive definite matrices | positive definite matrices | Markov processes | Markov processes | Fourier transforms | Fourier transforms | differential equations | differential equations | linear algebra | linear algebra | determinants | determinants | eigenvalues | eigenvalues | similarity | similarity | least-squares approximations | least-squares approximations | stability of differential equations | stability of differential equations | networks | networks

License

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18.S096 Topics in Mathematics of Data Science (MIT) 18.S096 Topics in Mathematics of Data Science (MIT)

Description

This is a mostly self-contained research-oriented course designed for undergraduate students (but also extremely welcoming to graduate students) with an interest in doing research in theoretical aspects of algorithms that aim to extract information from data. These often lie in overlaps of two or more of the following: Mathematics, Applied Mathematics, Computer Science, Electrical Engineering, Statistics, and / or Operations Research. This is a mostly self-contained research-oriented course designed for undergraduate students (but also extremely welcoming to graduate students) with an interest in doing research in theoretical aspects of algorithms that aim to extract information from data. These often lie in overlaps of two or more of the following: Mathematics, Applied Mathematics, Computer Science, Electrical Engineering, Statistics, and / or Operations Research.

Subjects

Principal Component Analysis (PCA) | Principal Component Analysis (PCA) | random matrix theory | random matrix theory | spike model | spike model | manifold learning | manifold learning | Diffusion Maps | Diffusion Maps | Sobolev Embedding Theorem | Sobolev Embedding Theorem | Spectral Clustering | Spectral Clustering | Cheeger’s inequality | Cheeger’s inequality | Mesh Theorem | Mesh Theorem | Number Theory | Number Theory | Approximation algorithms | Approximation algorithms | Max-Cut problem | Max-Cut problem | Stochastic Block Model | Stochastic Block Model | Synchronization | Synchronization | inverse problems | inverse problems

License

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