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18.786 Topics in Algebraic Number Theory (MIT) 18.786 Topics in Algebraic Number Theory (MIT)

Description

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

Description

In 8.323, Relativistic Quantum Field Theory I, concepts and basic techniques are developed through applications in elementary particle physics, and condensed matter physics.Topics include: Classical field theory, symmetries, and Noether's theorem. Quantization of scalar fields and spin 1/2 fields. Interacting fields and Feynman diagrams. In 8.323, Relativistic Quantum Field Theory I, concepts and basic techniques are developed through applications in elementary particle physics, and condensed matter physics.Topics include: Classical field theory, symmetries, and Noether's theorem. Quantization of scalar fields and spin 1/2 fields. Interacting fields and Feynman diagrams.

Subjects

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

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18.022 Calculus (MIT) 18.022 Calculus (MIT)

Description

This is an undergraduate course on calculus of several variables. It covers all of the topics covered in Calculus II (18.02), but presents them in greater depth. These topics are vector algebra in 3-space, determinants, matrices, vector-valued functions of one variable, space motion, scalar functions of several variables, partial differentiation, gradient, optimization techniques, double integrals, line integrals in the plane, exact differentials, conservative fields, Green's theorem, triple integrals, line and surface integrals in space, the divergence theorem, and Stokes' theorem. Additional topics covered in 18.022 are geometry, vector fields, and linear algebra. This is an undergraduate course on calculus of several variables. It covers all of the topics covered in Calculus II (18.02), but presents them in greater depth. These topics are vector algebra in 3-space, determinants, matrices, vector-valued functions of one variable, space motion, scalar functions of several variables, partial differentiation, gradient, optimization techniques, double integrals, line integrals in the plane, exact differentials, conservative fields, Green's theorem, triple integrals, line and surface integrals in space, the divergence theorem, and Stokes' theorem. Additional topics covered in 18.022 are geometry, vector fields, and linear algebra.

Subjects

vector algebra | vector algebra | determinant | determinant | matrix | matrix | matrices | matrices | vector-valued | vector-valued | functions | 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

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17.950 Understanding Modern Military Operations (MIT) 17.950 Understanding Modern Military Operations (MIT)

Description

A proper understanding of modern military operations requires a prior understanding of both the material side of war, including especially weapon, sensor, communication, and information processing technologies, and the human or organizational side of war, including especially military doctrine, which is an institutionalized vision within military organizations that predicts how the material tools of war will be wielded on future battlefields. Military doctrine makes assumptions about the nature of future battlefields, and determines what the division of labor on those battlefields will be between different military tools. Doctrine also therefore determines the organizational hierarchy among the various branches of the military which wield those tools. Thus, one way to think of the relation A proper understanding of modern military operations requires a prior understanding of both the material side of war, including especially weapon, sensor, communication, and information processing technologies, and the human or organizational side of war, including especially military doctrine, which is an institutionalized vision within military organizations that predicts how the material tools of war will be wielded on future battlefields. Military doctrine makes assumptions about the nature of future battlefields, and determines what the division of labor on those battlefields will be between different military tools. Doctrine also therefore determines the organizational hierarchy among the various branches of the military which wield those tools. Thus, one way to think of the relation

Subjects

Political science | Political science | military | military | modern | modern | operations | operations | material | material | war | war | weapon | weapon | sensor | sensor | communication | communication | information processing | information processing | technologies | technologies | human | human | organizational | organizational | doctrine | doctrine | future | future | battlefields | battlefields | organizational hierarchy | organizational hierarchy | branches. | branches. | branches | branches

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22.105 Electromagnetic Interactions (MIT) 22.105 Electromagnetic Interactions (MIT)

Description

Principles and applications of electromagnetism, starting from Maxwell's equations, with emphasis on phenomena important to nuclear engineering and radiation sciences. Solution methods for electrostatic and magnetostatic fields. Charged particle motion in those fields. Particle acceleration and focussing. Collisions with charged particles and with atoms. Electromagnetic waves, wave emission by accelerated particles, Bremsstrahlung. Compton scattering. Photoionization. Elementary applications to ranging, shielding, imaging, and radiation effects. Principles and applications of electromagnetism, starting from Maxwell's equations, with emphasis on phenomena important to nuclear engineering and radiation sciences. Solution methods for electrostatic and magnetostatic fields. Charged particle motion in those fields. Particle acceleration and focussing. Collisions with charged particles and with atoms. Electromagnetic waves, wave emission by accelerated particles, Bremsstrahlung. Compton scattering. Photoionization. Elementary applications to ranging, shielding, imaging, and radiation effects.

Subjects

electromagnetism | | electromagnetism | | Maxwell's equations | | Maxwell's equations | | electrostatic fields | | electrostatic fields | | magnetostatic fields | | magnetostatic fields | | Charged particle motion | | Charged particle motion | | Particle acceleration | | Particle acceleration | | Electromagnetic waves | | Electromagnetic waves | | Bremsstrahlung | | Bremsstrahlung | | Compton scattering | | Compton scattering | | Photoionization | Photoionization

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8.282J Introduction to Astronomy (MIT) 8.282J Introduction to Astronomy (MIT)

Description

Introduction to Astronomy provides a quantitative introduction to physics of the solar system, stars, interstellar medium, the galaxy, and universe, as determined from a variety of astronomical observations and models.Topics include: planets, planet formation; stars, the Sun, "normal" stars, star formation; stellar evolution, supernovae, compact objects (white dwarfs, neutron stars, and black holes), plusars, binary X-ray sources; star clusters, globular and open clusters; interstellar medium, gas, dust, magnetic fields, cosmic rays; distance ladder; galaxies, normal and active galaxies, jets; gravitational lensing; large scaling structure; Newtonian cosmology, dynamical expansion and thermal history of the Universe; cosmic microwave background radiation; big-bang nucleosynthesis Introduction to Astronomy provides a quantitative introduction to physics of the solar system, stars, interstellar medium, the galaxy, and universe, as determined from a variety of astronomical observations and models.Topics include: planets, planet formation; stars, the Sun, "normal" stars, star formation; stellar evolution, supernovae, compact objects (white dwarfs, neutron stars, and black holes), plusars, binary X-ray sources; star clusters, globular and open clusters; interstellar medium, gas, dust, magnetic fields, cosmic rays; distance ladder; galaxies, normal and active galaxies, jets; gravitational lensing; large scaling structure; Newtonian cosmology, dynamical expansion and thermal history of the Universe; cosmic microwave background radiation; big-bang nucleosynthesis

Subjects

solar system; stars; interstellar medium; the Galaxy; the Universe; planets; planet formation; star formation; stellar evolution; supernovae; compact objects; white dwarfs; neutron stars; black holes; plusars | binary X-ray sources; star clusters; globular and open clusters; interstellar medium | gas | dust | magnetic fields | cosmic rays; distance ladder; | solar system; stars; interstellar medium; the Galaxy; the Universe; planets; planet formation; star formation; stellar evolution; supernovae; compact objects; white dwarfs; neutron stars; black holes; plusars | binary X-ray sources; star clusters; globular and open clusters; interstellar medium | gas | dust | magnetic fields | cosmic rays; distance ladder; | solar system | solar system | stars | stars | interstellar medium | interstellar medium | the Galaxy | the Galaxy | the Universe | the Universe | planets | planets | planet formation | planet formation | star formation | star formation | stellar evolution | stellar evolution | supernovae | supernovae | compact objects | compact objects | white dwarfs | white dwarfs | neutron stars | neutron stars | black holes | black holes | plusars | binary X-ray sources | plusars | binary X-ray sources | star clusters | star clusters | globular and open clusters | globular and open clusters | interstellar medium | gas | dust | magnetic fields | cosmic rays | interstellar medium | gas | dust | magnetic fields | cosmic rays | distance ladder | distance ladder | galaxies | normal and active galaxies | jets | galaxies | normal and active galaxies | jets | gravitational lensing | gravitational lensing | large scaling structure | large scaling structure | Newtonian cosmology | dynamical expansion and thermal history of the Universe | Newtonian cosmology | dynamical expansion and thermal history of the Universe | cosmic microwave background radiation | cosmic microwave background radiation | big-bang nucleosynthesis | big-bang nucleosynthesis | pulsars | pulsars | binary X-ray sources | binary X-ray sources | gas | gas | dust | dust | magnetic fields | magnetic fields | cosmic rays | cosmic rays | galaxy | galaxy | universe | universe | astrophysics | astrophysics | Sun | Sun | supernova | supernova | globular clusters | globular clusters | open clusters | open clusters | jets | jets | Newtonian cosmology | Newtonian cosmology | dynamical expansion | dynamical expansion | thermal history | thermal history | normal galaxies | normal galaxies | active galaxies | active galaxies | Greek astronomy | Greek astronomy | physics | physics | Copernicus | Copernicus | Tycho | Tycho | Kepler | Kepler | Galileo | Galileo | classical mechanics | classical mechanics | circular orbits | circular orbits | full kepler orbit problem | full kepler orbit problem | electromagnetic radiation | electromagnetic radiation | matter | matter | telescopes | telescopes | detectors | detectors | 8.282 | 8.282 | 12.402 | 12.402

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6.881 Representation and Modeling for Image Analysis (MIT) 6.881 Representation and Modeling for Image Analysis (MIT)

Description

Most algorithms in computer vision and image analysis can be understood in terms of two important components: a representation and a modeling/estimation algorithm. The representation defines what information is important about the objects and is used to describe them. The modeling techniques extract the information from images to instantiate the representation for the particular objects present in the scene. In this seminar, we will discuss popular representations (such as contours, level sets, deformation fields) and useful methods that allow us to extract and manipulate image information, including manifold fitting, markov random fields, expectation maximization, clustering and others. For each concept -- a new representation or an estimation algorithm -- a lecture on the mathematical f Most algorithms in computer vision and image analysis can be understood in terms of two important components: a representation and a modeling/estimation algorithm. The representation defines what information is important about the objects and is used to describe them. The modeling techniques extract the information from images to instantiate the representation for the particular objects present in the scene. In this seminar, we will discuss popular representations (such as contours, level sets, deformation fields) and useful methods that allow us to extract and manipulate image information, including manifold fitting, markov random fields, expectation maximization, clustering and others. For each concept -- a new representation or an estimation algorithm -- a lecture on the mathematical f

Subjects

computer vision | computer vision | image analysis | image analysis | representation algorithm | representation algorithm | modeling | modeling | estimation algorithm | estimation algorithm | information | information | objects | objects | modeling techniques | modeling techniques | images | images | representations | representations | contours | contours | level sets | level sets | deformation fields | deformation fields | image information | image information | manifold fitting | manifold fitting | markov random fields | markov random fields | expectation maximization | expectation maximization | clustering | clustering | mathematical foundations | mathematical foundations | medical and biological imaging | medical and biological imaging

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18.702 Algebra II (MIT) 18.702 Algebra II (MIT)

Description

This undergraduate level course follows Algebra I. Topics include group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, and Galois theory. This undergraduate level course follows Algebra I. Topics include group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, and Galois theory.

Subjects

Sylow theorems | Sylow theorems | Group Representations | Group Representations | definitions | definitions | unitary representations | unitary representations | characters | characters | Schur's Lemma | Schur's Lemma | Rings: Basic Definitions | Rings: Basic Definitions | homomorphisms | homomorphisms | fractions | fractions | Factorization | Factorization | unique factorization | unique factorization | Gauss' Lemma | Gauss' Lemma | explicit factorization | explicit factorization | maximal ideals | maximal ideals | Quadratic Imaginary Integers | Quadratic Imaginary Integers | Gauss Primes | Gauss Primes | quadratic integers | quadratic integers | ideal factorization | ideal factorization | ideal classes | ideal classes | Linear Algebra over a Ring | Linear Algebra over a Ring | free modules | free modules | integer matrices | integer matrices | generators and relations | generators and relations | structure of abelian groups | structure of abelian groups | Rings: Abstract Constructions | Rings: Abstract Constructions | relations in a ring | relations in a ring | adjoining elements | adjoining elements | Fields: Field Extensions | Fields: Field Extensions | algebraic elements | algebraic elements | degree of field extension | degree of field extension | ruler and compass | ruler and compass | symbolic adjunction | symbolic adjunction | finite fields | finite fields | Fields: Galois Theory | Fields: Galois Theory | the main theorem | the main theorem | cubic equations | cubic equations | symmetric functions | symmetric functions | primitive elements | primitive elements | quartic equations | quartic equations | quintic equations | quintic equations

License

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18.702 Algebra II (MIT) 18.702 Algebra II (MIT)

Description

The course covers group theory and its representations, and focuses on the Sylow theorem, Schur's lemma, and proof of the orthogonality relations. It also analyzes the rings, the factorization processes, and the fields. Topics such as the formal construction of integers and polynomials, homomorphisms and ideals, the Gauss' lemma, quadratic imaginary integers, Gauss primes, and finite and function fields are discussed in detail. The course covers group theory and its representations, and focuses on the Sylow theorem, Schur's lemma, and proof of the orthogonality relations. It also analyzes the rings, the factorization processes, and the fields. Topics such as the formal construction of integers and polynomials, homomorphisms and ideals, the Gauss' lemma, quadratic imaginary integers, Gauss primes, and finite and function fields are discussed in detail.

Subjects

Sylow theorems | Sylow theorems | Group Representations | Group Representations | definitions | definitions | unitary representations | unitary representations | characters | characters | Schur's Lemma | Schur's Lemma | Rings: Basic Definitions | Rings: Basic Definitions | homomorphisms | homomorphisms | fractions | fractions | Factorization | Factorization | unique factorization | unique factorization | Gauss' Lemma | Gauss' Lemma | explicit factorization | explicit factorization | maximal ideals | maximal ideals | Quadratic Imaginary Integers | Quadratic Imaginary Integers | Gauss Primes | Gauss Primes | quadratic integers | quadratic integers | ideal factorization | ideal factorization | ideal classes | ideal classes | Linear Algebra over a Ring | Linear Algebra over a Ring | free modules | free modules | integer matrices | integer matrices | generators and relations | generators and relations | structure of abelian groups | structure of abelian groups | Rings: Abstract Constructions | Rings: Abstract Constructions | relations in a ring | relations in a ring | adjoining elements | adjoining elements | Fields: Field Extensions | Fields: Field Extensions | algebraic elements | algebraic elements | degree of field extension | degree of field extension | ruler and compass | ruler and compass | symbolic adjunction | symbolic adjunction | finite fields | finite fields | Fields: Galois Theory | Fields: Galois Theory | the main theorem | the main theorem | cubic equations | cubic equations | symmetric functions | symmetric functions | primitive elements | primitive elements | quartic equations | quartic equations | quintic equations | quintic equations

License

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

Description

The class will cover 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 3.012 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, fourier analysis and random walks.Technical RequirementsMathematica® software is required to run the .nb files found on this course site. The class will cover 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 3.012 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, fourier analysis and random walks.Technical RequirementsMathematica® software is required to run the .nb files found on this course site.

Subjects

energetics | energetics | materials structure and symmetry: applied fields | materials structure and symmetry: applied fields | mechanics and physics of solids and soft materials | mechanics and physics of solids and soft materials | linear algebra | linear algebra | orthonormal basis | orthonormal basis | eigenvalues | eigenvalues | eigenvectors | eigenvectors | quadratic forms | quadratic forms | tensor operations | tensor operations | symmetry operations | symmetry operations | calculus | calculus | complex analysis | complex analysis | differential equations | differential equations | theory of distributions | theory of distributions | fourier analysis | fourier analysis | random walks | random walks | mathematical technicques | mathematical technicques | materials science | materials science | materials engineering | materials engineering | materials structure | materials structure | symmetry | symmetry | applied fields | applied fields | materials response | materials response | solids mechanics | solids mechanics | solids physics | solids physics | soft materials | soft materials | multi-variable calculus | multi-variable calculus | ordinary differential equations | ordinary differential equations | partial differential equations | partial differential equations | applied mathematics | applied mathematics | mathematical techniques | mathematical techniques

License

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Magnetism with an Experimental Focus (MIT) Magnetism with an Experimental Focus (MIT)

Description

This course is an introduction to electromagnetism and electrostatics. Topics include: electric charge, Coulomb's law, electric structure of matter, conductors and dielectrics, concepts of electrostatic field and potential, electrostatic energy, electric currents, magnetic fields, Ampere's law, magnetic materials, time-varying fields, Faraday's law of induction, basic electric circuits, electromagnetic waves, and Maxwell's equations. The course has an experimental focus, and includes several experiments that are intended to illustrate the concepts being studied. Acknowledgements Prof. Roland wishes to acknowledge that the structure and content of this course owe much to the contributions of Prof. Ambrogio Fasoli. This course is an introduction to electromagnetism and electrostatics. Topics include: electric charge, Coulomb's law, electric structure of matter, conductors and dielectrics, concepts of electrostatic field and potential, electrostatic energy, electric currents, magnetic fields, Ampere's law, magnetic materials, time-varying fields, Faraday's law of induction, basic electric circuits, electromagnetic waves, and Maxwell's equations. The course has an experimental focus, and includes several experiments that are intended to illustrate the concepts being studied. Acknowledgements Prof. Roland wishes to acknowledge that the structure and content of this course owe much to the contributions of Prof. Ambrogio Fasoli.

Subjects

Electromagnetism | Electromagnetism | electrostatics | electrostatics | electric charge | electric charge | Coulomb's law | Coulomb's law | electric structure of matter | electric structure of matter | conductors | conductors | dielectrics | dielectrics | electrostatic field | electrostatic field | electrostatic potential | electrostatic potential | electrostatic energy | electrostatic energy | electric current | electric current | magnetic field | magnetic field | Ampere's law | Ampere's law | magnetic | magnetic | electric | electric | time-varying fields | time-varying fields | Faraday's law | Faraday's law | induction | induction | circuits | circuits | electromagnetic waves | electromagnetic waves | Maxwell's equations | Maxwell's equations | 8.02 | 8.02

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8.282J Introduction to Astronomy (MIT) 8.282J Introduction to Astronomy (MIT)

Description

Introduction to Astronomy provides a quantitative introduction to the physics of the solar system, stars, the interstellar medium, the galaxy, and the universe, as determined from a variety of astronomical observations and models. Introduction to Astronomy provides a quantitative introduction to the physics of the solar system, stars, the interstellar medium, the galaxy, and the universe, as determined from a variety of astronomical observations and models.

Subjects

solar system; stars; interstellar medium; the Galaxy; the Universe; planets; planet formation; star formation; stellar evolution; supernovae; compact objects; white dwarfs; neutron stars; black holes; plusars | binary X-ray sources; star clusters; globular and open clusters; interstellar medium | gas | dust | magnetic fields | cosmic rays; distance ladder; | solar system; stars; interstellar medium; the Galaxy; the Universe; planets; planet formation; star formation; stellar evolution; supernovae; compact objects; white dwarfs; neutron stars; black holes; plusars | binary X-ray sources; star clusters; globular and open clusters; interstellar medium | gas | dust | magnetic fields | cosmic rays; distance ladder; | solar system | solar system | stars | stars | interstellar medium | interstellar medium | the Galaxy | the Galaxy | the Universe | the Universe | planets | planets | planet formation | planet formation | star formation | star formation | stellar evolution | stellar evolution | supernovae | supernovae | compact objects | compact objects | white dwarfs | white dwarfs | neutron stars | neutron stars | black holes | black holes | plusars | binary X-ray sources | plusars | binary X-ray sources | star clusters | star clusters | globular and open clusters | globular and open clusters | interstellar medium | gas | dust | magnetic fields | cosmic rays | interstellar medium | gas | dust | magnetic fields | cosmic rays | distance ladder | distance ladder | galaxies | normal and active galaxies | jets | galaxies | normal and active galaxies | jets | gravitational lensing | gravitational lensing | large scaling structure | large scaling structure | Newtonian cosmology | dynamical expansion and thermal history of the Universe | Newtonian cosmology | dynamical expansion and thermal history of the Universe | cosmic microwave background radiation | cosmic microwave background radiation | big-bang nucleosynthesis | big-bang nucleosynthesis | pulsars | pulsars | binary X-ray sources | binary X-ray sources | gas | gas | dust | dust | magnetic fields | magnetic fields | cosmic rays | cosmic rays | galaxy | galaxy | universe | universe | astrophysics | astrophysics | Sun | Sun | supernova | supernova | globular clusters | globular clusters | open clusters | open clusters | jets | jets | Newtonian cosmology | Newtonian cosmology | dynamical expansion | dynamical expansion | thermal history | thermal history | normal galaxies | normal galaxies | active galaxies | active galaxies | Greek astronomy | Greek astronomy | physics | physics | Copernicus | Copernicus | Tycho | Tycho | Kepler | Kepler | Galileo | Galileo | classical mechanics | classical mechanics | circular orbits | circular orbits | full kepler orbit problem | full kepler orbit problem | electromagnetic radiation | electromagnetic radiation | matter | matter | telescopes | telescopes | detectors | detectors | 8.282 | 8.282 | 12.402 | 12.402 | plusars | plusars | galaxies | galaxies | normal and active galaxies | normal and active galaxies | dynamical expansion and thermal history of the Universe | dynamical expansion and thermal history of the Universe

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8.321 Quantum Theory I (MIT) 8.321 Quantum Theory I (MIT)

Description

8.321 is the first semester of a two-semester subject on quantum theory, stressing principles. Topics covered include: Hilbert spaces, observables, uncertainty relations, eigenvalue problems and methods for solution thereof, time-evolution in the Schrodinger, Heisenberg, and interaction pictures, connections between classical and quantum mechanics, path integrals, quantum mechanics in EM fields, angular momentum, time-independent perturbation theory, density operators, and quantum measurement. 8.321 is the first semester of a two-semester subject on quantum theory, stressing principles. Topics covered include: Hilbert spaces, observables, uncertainty relations, eigenvalue problems and methods for solution thereof, time-evolution in the Schrodinger, Heisenberg, and interaction pictures, connections between classical and quantum mechanics, path integrals, quantum mechanics in EM fields, angular momentum, time-independent perturbation theory, density operators, and quantum measurement.

Subjects

eigenstates | eigenstates | uncertainty relation | uncertainty relation | observables | observables | eigenvalues | eigenvalues | probabilities of the results of measurement | probabilities of the results of measurement | transformation theory | transformation theory | equations of motion | equations of motion | constants of motion | constants of motion | Symmetry in quantum mechanics | Symmetry in quantum mechanics | representations of symmetry groups | representations of symmetry groups | Variational and perturbation approximations | Variational and perturbation approximations | Systems of identical particles and applications | Systems of identical particles and applications | Time-dependent perturbation theory | Time-dependent perturbation theory | Scattering theory: phase shifts | Scattering theory: phase shifts | Born approximation | Born approximation | The quantum theory of radiation | The quantum theory of radiation | Second quantization and many-body theory | Second quantization and many-body theory | Relativistic quantum mechanics of one electron | Relativistic quantum mechanics of one electron | probability | probability | measurement | measurement | motion equations | motion equations | motion constants | motion constants | symmetry groups | symmetry groups | quantum mechanics | quantum mechanics | variational approximations | variational approximations | perturbation approximations | perturbation approximations | identical particles | identical particles | time-dependent perturbation theory | time-dependent perturbation theory | scattering theory | scattering theory | phase shifts | phase shifts | quantum theory of radiation | quantum theory of radiation | second quantization | second quantization | many-body theory | many-body theory | relativistic quantum mechanics | relativistic quantum mechanics | one electron | one electron | Hilbert spaces | Hilbert spaces | time evolution | time evolution | Schrodinger picture | Schrodinger picture | Heisenberg picture | Heisenberg picture | interaction picture | interaction picture | classical mechanics | classical mechanics | path integrals | path integrals | EM fields | EM fields | electromagnetic fields | electromagnetic fields | angular momentum | angular momentum | density operators | density operators | quantum measurement | quantum measurement | quantum statistics | quantum statistics | quantum dynamics | quantum dynamics

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18.702 Algebra II (MIT) 18.702 Algebra II (MIT)

Description

This undergraduate level course follows Algebra I. Topics include group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, and Galois theory. This undergraduate level course follows Algebra I. Topics include group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, and Galois theory.

Subjects

Sylow theorems | Sylow theorems | Group Representations | Group Representations | definitions | definitions | unitary representations | unitary representations | characters | characters | Schur's Lemma | Schur's Lemma | Rings: Basic Definitions | Rings: Basic Definitions | homomorphisms | homomorphisms | fractions | fractions | Factorization | Factorization | unique factorization | unique factorization | Gauss' Lemma | Gauss' Lemma | explicit factorization | explicit factorization | maximal ideals | maximal ideals | Quadratic Imaginary Integers | Quadratic Imaginary Integers | Gauss Primes | Gauss Primes | quadratic integers | quadratic integers | ideal factorization | ideal factorization | ideal classes | ideal classes | Linear Algebra over a Ring | Linear Algebra over a Ring | free modules | free modules | integer matrices | integer matrices | generators and relations | generators and relations | structure of abelian groups | structure of abelian groups | Rings: Abstract Constructions | Rings: Abstract Constructions | relations in a ring | relations in a ring | adjoining elements | adjoining elements | Fields: Field Extensions | Fields: Field Extensions | algebraic elements | algebraic elements | degree of field extension | degree of field extension | ruler and compass | ruler and compass | symbolic adjunction | symbolic adjunction | finite fields | finite fields | Fields: Galois Theory | Fields: Galois Theory | the main theorem | the main theorem | cubic equations | cubic equations | symmetric functions | symmetric functions | primitive elements | primitive elements | quartic equations | quartic equations | quintic equations | quintic equations

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18.786 Topics in Algebraic Number Theory (MIT) 18.786 Topics in Algebraic Number Theory (MIT)

Description

This course provides an introduction to algebraic number theory. Topics covered include dedekind domains, unique factorization of prime ideals, number fields, splitting of primes, class group, lattice methods, finiteness of the class number, Dirichlet's units theorem, local fields, ramification, discriminants. This course provides an introduction to algebraic number theory. Topics covered include dedekind domains, unique factorization of prime ideals, number fields, splitting of primes, class group, lattice methods, finiteness of the class number, Dirichlet's units theorem, local fields, ramification, discriminants.

Subjects

number fields | number fields | dedekind domain | dedekind domain | prime ideal | prime ideal | class group | class group | lattice method | lattice method

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18.786 Topics in Algebraic Number Theory (MIT)

Description

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

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8.323 Relativistic Quantum Field Theory I (MIT)

Description

In 8.323, Relativistic Quantum Field Theory I, concepts and basic techniques are developed through applications in elementary particle physics, and condensed matter physics.Topics include: Classical field theory, symmetries, and Noether's theorem. Quantization of scalar fields and spin 1/2 fields. Interacting fields and Feynman diagrams.

Subjects

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

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18.024 Calculus with Theory II (MIT) 18.024 Calculus with Theory II (MIT)

Description

This course is a continuation of 18.014. It covers the same material as 18.02 (Calculus), but at a deeper level, emphasizing careful reasoning and understanding of proofs. There is considerable emphasis on linear algebra and vector integral calculus.Topics include: Calculus of several variables. Vector algebra in 3-space, determinants, matrices. Vector-valued functions of one variable, space motion. Scalar functions of several variables: partial differentiation, gradient, optimization techniques. Double integrals and line integrals in the plane; exact differentials and conservative fields; Green's theorem and applications, triple integrals, line and surface integrals in space, Divergence theorem, Stokes' theorem; applications. Dr. Lachowska wishes to acknowledge Andrew Brooke-Taylor This course is a continuation of 18.014. It covers the same material as 18.02 (Calculus), but at a deeper level, emphasizing careful reasoning and understanding of proofs. There is considerable emphasis on linear algebra and vector integral calculus.Topics include: Calculus of several variables. Vector algebra in 3-space, determinants, matrices. Vector-valued functions of one variable, space motion. Scalar functions of several variables: partial differentiation, gradient, optimization techniques. Double integrals and line integrals in the plane; exact differentials and conservative fields; Green's theorem and applications, triple integrals, line and surface integrals in space, Divergence theorem, Stokes' theorem; applications. Dr. Lachowska wishes to acknowledge Andrew Brooke-Taylor

Subjects

linear algebra | linear algebra | vector integral calculus | vector integral calculus | Calculus of several variables | Calculus of several variables | Vector algebra in 3-space | Vector algebra in 3-space | determinants | determinants | matrices | matrices | Vector-valued functions of one variable | Vector-valued functions of one variable | space motion | space motion | Scalar functions of several variables: partial differentiation | Scalar functions of several variables: partial differentiation | gradient | gradient | optimization techniques | optimization techniques | Double integrals and line integrals in the plane | Double integrals and line integrals in the plane | exact differentials and conservative fields | exact differentials and conservative fields | Green's theorem and applications | Green's theorem and applications | triple integrals | triple integrals | line and surface integrals in space | line and surface integrals in space | Divergence theorem | Divergence theorem | Stokes' theorem | Stokes' theorem | applications | applications

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6.453 Quantum Optical Communication (MIT) 6.453 Quantum Optical Communication (MIT)

Description

This course is offered to graduate students and covers topics in five major areas of quantum optical communication: quantum optics, single-mode and two-mode quantum systems, multi-mode quantum systems, nonlinear optics, and quantum systems theory. Specific topics include the following.  Quantum optics: Dirac notation quantum mechanics; harmonic oscillator quantization; number states, coherent states, and squeezed states; radiation field quantization and quantum field propagation; P-representation and classical fields.  Linear loss and linear amplification: commutator preservation and the Uncertainty Principle; beam splitters; phase-insensitive and phase-sensitive amplifiers. Quantum photodetection: direct detection, heterodyne detection, and homodyne detection.&a This course is offered to graduate students and covers topics in five major areas of quantum optical communication: quantum optics, single-mode and two-mode quantum systems, multi-mode quantum systems, nonlinear optics, and quantum systems theory. Specific topics include the following.  Quantum optics: Dirac notation quantum mechanics; harmonic oscillator quantization; number states, coherent states, and squeezed states; radiation field quantization and quantum field propagation; P-representation and classical fields.  Linear loss and linear amplification: commutator preservation and the Uncertainty Principle; beam splitters; phase-insensitive and phase-sensitive amplifiers. Quantum photodetection: direct detection, heterodyne detection, and homodyne detection.&a

Subjects

Quantum optics: Dirac notation quantum mechanics | Quantum optics: Dirac notation quantum mechanics | harmonic oscillator quantization | harmonic oscillator quantization | number states | coherent states | and squeezed states | number states | coherent states | and squeezed states | radiation field quantization and quantum field propagation | radiation field quantization and quantum field propagation | P-representation and classical fields | P-representation and classical fields | Linear loss and linear amplification: commutator preservation and the Uncertainty Principle | Linear loss and linear amplification: commutator preservation and the Uncertainty Principle | beam splitters | beam splitters | phase-insensitive and phase-sensitive amplifiers | phase-insensitive and phase-sensitive amplifiers | Quantum photodetection: direct detection | heterodyne detection | and homodyne detection | Quantum photodetection: direct detection | heterodyne detection | and homodyne detection | Second-order nonlinear optics: phasematched interactions | Second-order nonlinear optics: phasematched interactions | optical parametric amplifiers | optical parametric amplifiers | generation of squeezed states | photon-twin beams | non-classical fourth-order interference | and polarization entanglement | generation of squeezed states | photon-twin beams | non-classical fourth-order interference | and polarization entanglement | Quantum systems theory: optimum binary detection | Quantum systems theory: optimum binary detection | quantum precision measurements | quantum precision measurements | quantum cryptography | quantum cryptography | quantum teleportation | quantum teleportation

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22.00J Introduction to Modeling and Simulation (MIT) 22.00J Introduction to Modeling and Simulation (MIT)

Description

This course surveys the basic concepts of computer modeling in science and engineering using discrete particle systems and continuum fields. It covers techniques and software for statistical sampling, simulation, data analysis and visualization, and uses statistical, quantum chemical, molecular dynamics, Monte Carlo, mesoscale and continuum methods to study fundamental physical phenomena encountered in the fields of computational physics, chemistry, mechanics, materials science, biology, and applied mathematics. Applications are drawn from a range of disciplines to build a broad-based understanding of complex structures and interactions in problems where simulation is on equal footing with theory and experiment. A term project allows development of individual interests. Students are mentor This course surveys the basic concepts of computer modeling in science and engineering using discrete particle systems and continuum fields. It covers techniques and software for statistical sampling, simulation, data analysis and visualization, and uses statistical, quantum chemical, molecular dynamics, Monte Carlo, mesoscale and continuum methods to study fundamental physical phenomena encountered in the fields of computational physics, chemistry, mechanics, materials science, biology, and applied mathematics. Applications are drawn from a range of disciplines to build a broad-based understanding of complex structures and interactions in problems where simulation is on equal footing with theory and experiment. A term project allows development of individual interests. Students are mentor

Subjects

computer modeling | computer modeling | discrete particle system | discrete particle system | continuum | continuum | continuum field | continuum field | statistical sampling | statistical sampling | data analysis | data analysis | visualization | visualization | quantum | quantum | quantum method | quantum method | chemical | chemical | molecular dynamics | molecular dynamics | Monte Carlo | Monte Carlo | mesoscale | mesoscale | continuum method | continuum method | computational physics | computational physics | chemistry | chemistry | mechanics | mechanics | materials science | materials science | biology; applied mathematics | biology; applied mathematics | fluid dynamics | fluid dynamics | heat | heat | fractal | fractal | evolution | evolution | melting | melting | gas | gas | structural mechanics | structural mechanics | FEM | FEM | finite element | finite element | biology | biology | applied mathematics | applied mathematics | 1.021 | 1.021 | 2.030 | 2.030 | 3.021 | 3.021 | 10.333 | 10.333 | 18.361 | 18.361 | HST.588 | HST.588 | 22.00 | 22.00

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8.033 Relativity (MIT) 8.033 Relativity (MIT)

Description

Relativity is normally taken by physics majors in their sophomore year. Topics include: Einstein's postulates; consequences for simultaneity, time dilation, length contraction, clock synchronization; Lorentz transformation; relativistic effects and paradoxes; Minkowski diagrams; invariants and four-vectors; momentum, energy and mass; and particle collisions. Also covered is: Relativity and electricity; Coulomb's law; and magnetic fields. Brief introduction to Newtonian cosmology. There is also an introduction to some concepts of General Relativity; principle of equivalence; the Schwarzchild metric; gravitational red shift, particle and light trajectories, geodesics, and Shapiro delay. Relativity is normally taken by physics majors in their sophomore year. Topics include: Einstein's postulates; consequences for simultaneity, time dilation, length contraction, clock synchronization; Lorentz transformation; relativistic effects and paradoxes; Minkowski diagrams; invariants and four-vectors; momentum, energy and mass; and particle collisions. Also covered is: Relativity and electricity; Coulomb's law; and magnetic fields. Brief introduction to Newtonian cosmology. There is also an introduction to some concepts of General Relativity; principle of equivalence; the Schwarzchild metric; gravitational red shift, particle and light trajectories, geodesics, and Shapiro delay.

Subjects

Einstein's postulates | Einstein's postulates | consequences for simultaneity | time dilation | length contraction | clock synchronization | consequences for simultaneity | time dilation | length contraction | clock synchronization | Lorentz transformation | Lorentz transformation | relativistic effects and paradoxes | relativistic effects and paradoxes | Minkowski diagrams | Minkowski diagrams | invariants and four-vectors | invariants and four-vectors | momentum | energy and mass | momentum | energy and mass | particle collisions | particle collisions | Relativity and electricity | Relativity and electricity | Coulomb's law | Coulomb's law | magnetic fields | magnetic fields | Newtonian cosmology | Newtonian cosmology | General Relativity | General Relativity | principle of equivalence | principle of equivalence | the Schwarzchild metric | the Schwarzchild metric | gravitational red shift | particle and light trajectories | geodesics | Shapiro delay | gravitational red shift | particle and light trajectories | geodesics | Shapiro delay | gravitational red shift | gravitational red shift | particle trajectories | particle trajectories | light trajectories | light trajectories | invariants | invariants | four-vectors | four-vectors | momentum | momentum | energy | energy | mass | mass | relativistic effects | relativistic effects | paradoxes | paradoxes | electricity | electricity | time dilation | time dilation | length contraction | length contraction | clock synchronization | clock synchronization | Schwarzchild metric | Schwarzchild metric | geodesics | geodesics | Shaprio delay | Shaprio delay | relativistic kinematics | relativistic kinematics | relativistic dynamics | relativistic dynamics | electromagnetism | electromagnetism | hubble expansion | hubble expansion | universe | universe | equivalence principle | equivalence principle | curved space time | curved space time | Ether Theory | Ether Theory | constants | constants | speed of light | speed of light | c | c | graph | graph | pythagorem theorem | pythagorem theorem | triangle | triangle | arrows | arrows

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BE.430J Fields, Forces, and Flows in Biological Systems (MIT) BE.430J Fields, Forces, and Flows in Biological Systems (MIT)

Description

This course covers the following topics: conduction, diffusion, convection in electrolytes; fields in heterogeneous media; electrical double layers; Maxwell stress tensor and electrical forces in physiological systems; and fluid and solid continua: equations of motion useful for porous, hydrated biological tissues. Case studies considered include membrane transport; electrode interfaces; electrical, mechanical, and chemical transduction in tissues; electrophoretic and electroosmotic flows; diffusion/reaction; and ECG. The course also examines electromechanical and physicochemical interactions in biomaterials and cells; orthopaedic, cardiovascular, and other clinical examples. This course covers the following topics: conduction, diffusion, convection in electrolytes; fields in heterogeneous media; electrical double layers; Maxwell stress tensor and electrical forces in physiological systems; and fluid and solid continua: equations of motion useful for porous, hydrated biological tissues. Case studies considered include membrane transport; electrode interfaces; electrical, mechanical, and chemical transduction in tissues; electrophoretic and electroosmotic flows; diffusion/reaction; and ECG. The course also examines electromechanical and physicochemical interactions in biomaterials and cells; orthopaedic, cardiovascular, and other clinical examples.

Subjects

biomaterials | biomaterials | conduction | conduction | diffusion | diffusion | convection in electrolytes | convection in electrolytes | fields in heterogeneous media | fields in heterogeneous media | electrical double layers | electrical double layers | Maxwell stress tensor | Maxwell stress tensor | fluid and solid continua | fluid and solid continua | biological tissues | biological tissues | membrane transport | membrane transport | electrode | electrode | transduction | transduction | electrophoretic flow | electrophoretic flow | electroosmotic flow | electroosmotic flow | diffusion reaction | diffusion reaction | ECG | ECG | orthopaedic | cardiovascular | orthopaedic | cardiovascular | 2.795J | 2.795J | 2.795 | 2.795 | 6.561J | 6.561J | 6.561 | 6.561 | 10.539J | 10.539J | 10.539 | 10.539 | HST.544J | HST.544J | HST.544 | HST.544

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6.763 Applied Superconductivity (MIT) 6.763 Applied Superconductivity (MIT)

Description

This course provides a phenomenological approach to superconductivity, with emphasis on superconducting electronics. Topics include: electrodynamics of superconductors, London's model, flux quantization, Josephson Junctions, superconducting quantum devices, equivalent circuits, high-speed superconducting electronics, and quantized circuits for quantum computing. The course also provides an overview of type II superconductors, critical magnetic fields, pinning, the critical state model, superconducting materials, and microscopic theory of superconductivity.Technical RequirementsMATLAB® software is required to run the .m files found on this course site.MATLAB® is a trademark of The MathWorks, Inc. This course provides a phenomenological approach to superconductivity, with emphasis on superconducting electronics. Topics include: electrodynamics of superconductors, London's model, flux quantization, Josephson Junctions, superconducting quantum devices, equivalent circuits, high-speed superconducting electronics, and quantized circuits for quantum computing. The course also provides an overview of type II superconductors, critical magnetic fields, pinning, the critical state model, superconducting materials, and microscopic theory of superconductivity.Technical RequirementsMATLAB® software is required to run the .m files found on this course site.MATLAB® is a trademark of The MathWorks, Inc.

Subjects

applied superconductivity | applied superconductivity | superconducting electronics | superconducting electronics | electrodynamics of superconductors | electrodynamics of superconductors | London's model | London's model | flux quantization | flux quantization | Josephson Junctions | Josephson Junctions | superconducting quantum devices | superconducting quantum devices | equivalent circuits | equivalent circuits | high-speed superconducting electronics | high-speed superconducting electronics | quantized circuits | quantized circuits | quantum computing | quantum computing | type II superconductors | type II superconductors | critical magnetic fields | critical magnetic fields | pinning | pinning | the critical state model | the critical state model | superconducting materials | superconducting materials | microscopic theory of superconductivity | microscopic theory of superconductivity | Electric conductivity | Electric conductivity

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6.451 Principles of Digital Communication II (MIT) 6.451 Principles of Digital Communication II (MIT)

Description

This course is the second of a two-term sequence with 6.450. The focus is on coding techniques for approaching the Shannon limit of additive white Gaussian noise (AWGN) channels, their performance analysis, and design principles. After a review of 6.450 and the Shannon limit for AWGN channels, the course begins by discussing small signal constellations, performance analysis and coding gain, and hard-decision and soft-decision decoding. It continues with binary linear block codes, Reed-Muller codes, finite fields, Reed-Solomon and BCH codes, binary linear convolutional codes, and the Viterbi algorithm.More advanced topics include trellis representations of binary linear block codes and trellis-based decoding; codes on graphs; the sum-product and min-sum algorithms This course is the second of a two-term sequence with 6.450. The focus is on coding techniques for approaching the Shannon limit of additive white Gaussian noise (AWGN) channels, their performance analysis, and design principles. After a review of 6.450 and the Shannon limit for AWGN channels, the course begins by discussing small signal constellations, performance analysis and coding gain, and hard-decision and soft-decision decoding. It continues with binary linear block codes, Reed-Muller codes, finite fields, Reed-Solomon and BCH codes, binary linear convolutional codes, and the Viterbi algorithm.More advanced topics include trellis representations of binary linear block codes and trellis-based decoding; codes on graphs; the sum-product and min-sum algorithms

Subjects

coding techniques | coding techniques | the Shannon limit of additive white Gaussian noise channels | the Shannon limit of additive white Gaussian noise channels | performance analysis | performance analysis | Small signal constellations | Small signal constellations | coding gain | coding gain | Hard-decision and soft-decision decoding | Hard-decision and soft-decision decoding | Introduction to binary linear block codes | Introduction to binary linear block codes | Reed-Muller codes | Reed-Muller codes | finite fields | finite fields | Reed-Solomon and BCH codes | Reed-Solomon and BCH codes | binary linear convolutional codes | binary linear convolutional codes | Viterbi and BCJR algorithms | Viterbi and BCJR algorithms | Trellis representations of binary linear block codes | Trellis representations of binary linear block codes | trellis-based ML decoding | trellis-based ML decoding | Codes on graphs | Codes on graphs | sum-product | sum-product | max-product | max-product | decoding algorithms | decoding algorithms | Turbo codes | Turbo codes | LDPC codes and RA codes | LDPC codes and RA codes | Coding for the bandwidth-limited regime | Coding for the bandwidth-limited regime | Lattice codes | Lattice codes | Trellis-coded modulation | Trellis-coded modulation | Multilevel coding | Multilevel coding | Shaping | Shaping

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8.07 Electromagnetism II (MIT) 8.07 Electromagnetism II (MIT)

Description

Survey of basic electromagnetic phenomena: electrostatics, magnetostatics, electromagnetic properties of matter. Time-dependent electromagnetic fields and Maxwell's equations. Electromagnetic waves, emission, absorption, and scattering of radiation. Relativistic electrodynamics and mechanics. Survey of basic electromagnetic phenomena: electrostatics, magnetostatics, electromagnetic properties of matter. Time-dependent electromagnetic fields and Maxwell's equations. Electromagnetic waves, emission, absorption, and scattering of radiation. Relativistic electrodynamics and mechanics.

Subjects

electromagnetic phenomena | electromagnetic phenomena | electrostatics | electrostatics | magnetostatics | magnetostatics | electromagnetic properties of matter | electromagnetic properties of matter | Time-dependent electromagnetic fields and Maxwell's equations | Time-dependent electromagnetic fields and Maxwell's equations | Electromagnetic waves | Electromagnetic waves | emission | emission | absorption | absorption | scattering of radiation | scattering of radiation | Relativistic electrodynamics | Relativistic electrodynamics | mechanics | mechanics

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