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Published by: Massachusetts Institute of Technology  Language: English
Published by: Massachusetts Institute of Technology  Language: English
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This course offers an introduction to discrete and computational geometry. Emphasis is placed on teaching methods in combinatorial geometry. Many results presented are recent, and include open (as yet unsolved) problems.
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Tag(s):
 mathematics
 discrete geometry
 computational geometry
 convex partitions
 binary space partitions
 art gallery problems
 planar graphs
 pseudotriangulations
 encompassing graphs
 geometric graphs
 crossing numbers
 extremal graph theory
 gallaisylvester problems
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Published by: Massachusetts Institute of Technology  Language: English
Published by: Massachusetts Institute of Technology  Language: English
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László Tisza was Professor of Physics Emeritus at MIT, where he began teaching in 1941. This online publication is a reproduction the original lecture notes for the course "Applied Geometric Algebra" taught by Professor Tisza in the Spring of 1976.
Over the last 100 years, the mathematical tools employed by physicists have expanded considerably, from differential calculus, vector algebra and geometry, to advanced linear algebra, tensors, Hilbert space, spinors, Group theory and many others. These sophisticated tools provide powerful machinery for describing the physical world, however, their physical interpretation is often not intuitive. These course notes represent Prof. Tisza's attempt at bringing conceptual clarity and unity to the application and interpretation of these advanced mathematical tools. In particular, there is an emphasis on the unifying role that Group theory plays in classical, relativistic, and quantum physics. Prof. Tisza revisits many elementary problems with an advanced treatment in order to help develop the geometrical intuition for the algebraic machinery that may carry over to more advanced problems.
The lecture notes came to MIT OpenCourseWare by way of Samuel Gasster, '77 (Course 18), who had taken the course and kept a copy of the lecture notes for his own reference. He dedicated dozens of hours of his own time to convert the typewritten notes into LaTeX files and then publicationready PDFs. You can read about his motivation for wanting to see these notes published in his Preface. Professor Tisza kindly gave his permission to make these notes available on MIT OpenCourseWare.
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Published by: Massachusetts Institute of Technology  Language: English
Published by: Massachusetts Institute of Technology  Language: English
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This course covers a collection of geometric techniques that apply broadly in modern algorithm design.
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Tag(s):
 mathematics
 spectral graph theory
 iterative methods for linear algebra
 convex geometry
 lattices and basis reduction
 lps and sdps for approximating nphard problems
 graph laplacians
 cheeger inequalities
 fritz john?s theorem
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Published by: Massachusetts Institute of Technology  Language: English
Published by: Massachusetts Institute of Technology  Language: English
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This graduate level course focuses on nonlinear dynamics with applications. It takes an intuitive approach with emphasis on geometric thinking, computational and analytical methods and makes extensive use of demonstration software.
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Tag(s):
 mathematics
 phase plane
 limit cycles
 poincarebendixson theory
 timedependent systems
 floquet theory
 poincare maps
 averaging
 stability of equilibria
 nearequilibrium dynamics
 center manifolds
 elementary bifurcations
 normal forms
 chaos
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Published by: Massachusetts Institute of Technology  Language: English
Published by: Massachusetts Institute of Technology  Language: English
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This course will focus on fundamental subjects in (deterministic) optimization, connected through the themes of convexity, geometric multipliers, and duality. The aim is to develop the core analytical and computational issues of continuous optimization, duality, and saddle point theory using a handful of unifying principles that can be easi
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 electrical engineering and computer science
 convexity
 optimization
 geometric duality
 lagrangian duality
 fenchel duality
 cone programming
 semidefinite programming
 subgradients
 constrained optimization
 gradient projection
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Published by: Massachusetts Institute of Technology  Language: English
Published by: Massachusetts Institute of Technology  Language: English
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This is an advanced undergraduate course dealing with calculus in one complex variable with geometric emphasis. Since the course Analysis I (18.100B) is a prerequisite, topological notions like compactness, connectedness, and related properties of continuous functions are taken for granted.
This course offers biweekly problem sets with sol
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Tag(s):
 mathematics
 functions of one complex variable
 cauchy's theorem
 holomorphic functions
 meromorphic functions
 residues
 contour integrals
 conformal mapping
 infinite series and products
 the gamma function
 the mittagleffler theorem
 harmonic functions
 dirichlet's problem
 the riemann mapping theorem
 the riemann zeta function
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Published by: Massachusetts Institute of Technology  Language: English
Published by: Massachusetts Institute of Technology  Language: English
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This course will focus on various aspects of mirror symmetry. It is aimed at students who already have some basic knowledge in symplectic and complex geometry (18.966, or equivalent). The geometric concepts needed to formulate various mathematical versions of mirror symmetry will be introduced along the way, in variable levels of detail and
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Tag(s):
 mathematics
 mirror symmetry
 deformation
 hodge theory
 pseudoholomorphic
 gromovwitten
 cohomology
 yukawa
 monodromy
 picardfuchs
 lagrangian floer theory
 homology
 syz conjecture
 submanifolds
 k3 surfaces
 matrices
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Published by: Massachusetts Institute of Technology  Language: English
Published by: Massachusetts Institute of Technology  Language: English
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This course provides an introduction to the mechanics of solids with applications to science and engineering. We emphasize the three essential features of all mechanics analyses, namely: (a) the geometry of the motion and/or deformation of the structure, and conditions of geometric fit, (b) the forces on and within structures and assemblage
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Tag(s):
 mechanical engineering
 statics
 pressure
 deformation
 deformable solid
 equilibrium
 geometric compatibility
 material behavior
 stress
 strain
 shear
 elasticity
 thermal expansion
 failure modes
 biomechanics
 natural materials
 motion
 structure
 force
 moment
 member
 truss
 friction
 torsion
 bending
 displacement
 beam
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Published by: Massachusetts Institute of Technology  Language: English
Published by: Massachusetts Institute of Technology  Language: English
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This course explores the following topics: derivation of elastic and plastic stressstrain relations for plate and shell elements; the bending and buckling of rectangular plates; nonlinear geometric effects; postbuckling and ultimate strength of cold formed sections and typical stiffened panels used in naval architecture; the general theor
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Tag(s):
 mechanical engineering
 plates
 shells
 engineering strain
 strain measure
 bending moment
 structural plasticity
 membrane energy
 greenlagrangian strain
 bending theory of plates
 buckling theory of plates
 raleighritz quotient
 local buckling
 plastic buckling
 cylindrical shells
 axial load
 lateral pressure
 hydrostatic pressure
 torsion
 bending boundary c
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Published by: Massachusetts Institute of Technology  Language: English
Published by: Massachusetts Institute of Technology  Language: English
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This is a graduatelevel course in combinatorial theory. The content varies year to year, according to the interests of the instructor and the students. The topic of this course is hyperplane arrangements, including background material from the theory of posets and matroids.
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Tag(s):
 mathematics
 combinatorial theory
 hyperplane arrangements
 intersection poset
 matroids
 geometric lattices
 broken circuits
 modular elements
 supersolvability
 finite fields
 hyperplane
 arrangements
 intersection poset
 geometric lattices
 broken circuits
 modular elements
 supersolvability
 finite fields
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Published by: Massachusetts Institute of Technology  Language: English
Published by: Massachusetts Institute of Technology  Language: English
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Data structures play a central role in modern computer science. You interact with data structures much more often than with algorithms (think of Google, your mail server, and even your network routers). In addition, data structures are essential building blocks in obtaining efficient algorithms. This course will cover major results and curr
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Tag(s):
 electrical engineering and computer science
 dynamic optimality
 geometric data structures
 strings
 integers
 dictionaries
 dynamic graphs
 temporal data structures
 external memory
 cacheoblivious
 succinct data structures
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Published by: Massachusetts Institute of Technology  Language: English
Published by: Massachusetts Institute of Technology  Language: English
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This course introduces students to probability and random variables. Topics include distribution functions, binomial, geometric, hypergeometric, and Poisson distributions. The other topics covered are uniform, exponential, normal, gamma and beta distributions; conditional probability; Bayes theorem; joint distributions; Chebyshev inequality
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Tag(s):
 mathematics
 probability spaces
 random variables
 distribution functions
 binomial
 geometric
 hypergeometric
 poisson distributions
 uniform
 exponential
 normal
 gamma and beta distributions
 conditional probability
 bayes theorem
 joint distributions
 chebyshev inequality
 law of large numbers
 and central limit theorem
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Published by: Massachusetts Institute of Technology  Language: English
Published by: Massachusetts Institute of Technology  Language: English
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This course examines how randomization can be used to make algorithms simpler and more efficient via random sampling, random selection of witnesses, symmetry breaking, and Markov chains. Topics covered include: randomized computation; data structures (hash tables, skip lists); graph algorithms (minimum spanning trees, shortest paths, minimu
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Tag(s):
 electrical engineering and computer science
 randomized algorithms
 algorithms
 efficient in time and space
 randomization
 computational problems
 data structures
 graph algorithms
 optimization
 geometry
 markov chains
 sampling
 estimation
 geometric algorithms
 parallel and distributed algorithms
 parallel and ditributed algorithm
 parallel and distributed algorithm
 random sampli
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Published by: Massachusetts Institute of Technology  Language: English
Published by: Massachusetts Institute of Technology  Language: English
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In this undergraduate level seminar series, topics vary from year to year. Students present and discuss the subject matter, and are provided with instruction and practice in written and oral communication. Some experience with proofs required. The topic for fall 2008: Computational algebra and algebraic geometry.
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Tag(s):
 mathematics
 computational algebra
 algebraic geometry
 geometry
 algebra
 algorithms
 groebner bases
 elimination theory
 algebrageometry dictionary
 polynomial functions
 rational functions
 geometric theorem proving
 invariant theory of finite groups
 projective algebraic geometry
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