Tensors

Nov 17, 2014

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Tensors. A general transformation can be expressed as a matrix. Partial derivatives between two systems Jacobian N  N real matrix Element of the general linear group Gl ( N , r ) Cartesian coordinate transformations have an additional symmetry.

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vector
volume element
direct product
contravariant vector
vector direct product
special linear group sl

wayne-figueroa

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Tensors

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Tensors

Tensors. Transformation Rule. A Cartesian vector can be defined by its transformation rule. Another transformation matrix T transforms similarly. x 3. x 2. x 1. For a Cartesian coordinate system a tensor is defined by its transformation rule.

741 views • 8 slides

Tensors and Component Analysis

. Tensor: Generalization of an n-dimensional array. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vector: order-1 tensor. Matrix: order-2 tensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Order-3 tensor. . Re

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Physical interpretation of DC and non-DC components of moment tensors Václav Vavryčuk Institute of G eo physics, Pragu

Physical interpretation of DC and non-DC components of moment tensors Václav Vavryčuk Institute of G eo physics, Prague. MT for simple types of seismic sources. Explosive (implosive) source. Source process. Moment tensor. Force equivalent. explosion. three linear dipoles. implosion.

714 views • 45 slides

Appendix A: Tensors

Appendix A: Tensors. Instructor: Dr. Gleb V. Tcheslavski Contact: gleb@ee.lamar.edu Office Hours: TR Class web site: http://www.ee.lamar.edu/gleb/em/Index.htm. “tensor” by Kevin McCormick. The Euler’s formulas. (A.2.1). (A.2.2). (A.2.3). (A.2.4).

331 views • 8 slides

Differential Manifolds and Tensors

taken from wikipedia.org. Differential Manifolds and Tensors. A presentation of the topic as in: B. F. Schutz, Geometrical methods of mathematical Physics, chap. 2. What are they?. basic analytic tools. Metric spaces.

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Decomposition and Visualization of Fourth-Order Elastic-Plastic Tensors

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Multi-linear Systems and Invariant Theory in the Context of Computer Vision and Graphics Class 2: Homography Tensors CS

Multi-linear Systems and Invariant Theory in the Context of Computer Vision and Graphics Class 2: Homography Tensors CS329 Stanford University. Amnon Shashua. Material We Will Cover Today. 2D-2D mapping of a “ dynamic ” point configuration.

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Review (2 nd order tensors):

Review (2 nd order tensors):. Tensor – Linear mapping of a vector onto another vector. Tensor components in a Cartesian basis (3x3 matrix):. Basis change formula for tensor components. Dyadic vector product. General dyadic expansion. Routine tensor operations. Addition.

774 views • 26 slides

A Physicists’ Introduction to Tensors

Prof Geraint F. Lewis Sydney Institute for Astronomy The University of Sydney. A Physicists’ Introduction to Tensors. Tensors. The Goals: Understand what a Tensor is Understand what a Tensor does Tensor Algebra Tensor Contractions The Metric Tensor Coordinates and Invariance

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Mining Large Graphs: Spectral Methods, Tensors and Influence propagation

Mining Large Graphs: Spectral Methods, Tensors and Influence propagation. Christos Faloutsos CMU. Thanks. Alex Smola Jia Yu (Tim) Pan. Roadmap. Graph problems: G1: Fraud detection – BP G2: Botnet detection – spectral G3: Beyond graphs: tensors and ``NELL’’

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Earthquake source modelling by second degree moment tensors

Earthquake source modelling by second degree moment tensors . Petra Adamo vá Jan Šílený. Geophysical Institute, Academy of Sciences, Prague, Czech Republic e-mail: adamova@ig.cas.cz, fax: +420-272761549. Introduction, motivation. Finite source parameters from point source approximation.

390 views • 21 slides

TENSORS/ 3-D STRESS STATE

TENSORS/ 3-D STRESS STATE. Tensors Tensors are specified in the following manner: A zero-rank tensor is specified by a sole component, independent of the system of reference (e.g., mass, density).

410 views • 18 slides

EARTHQUAKES (2): WAVEFORM MODELING, MOMENT TENSORS, & SOURCE PARAMETERS

EARTHQUAKES (2): WAVEFORM MODELING, MOMENT TENSORS, & SOURCE PARAMETERS. Kikuchi and Kanamori, 1991. SOMETIMES FIRST MOTIONS DONâ€™T CONSTRAIN FOCAL MECHANISM Especially likely when Few nearby stations, as in the oceans, so arrivals are near center of focal sphere

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Extracting Structure from Matrices & Tensors by Random Sampling

Extracting Structure from Matrices & Tensors by Random Sampling. Michael W. Mahoney Yale University michael.mahoney@yale.edu (joint work with Petros Drineas and Ravi Kannan ) @ ARCC Workshop on Tensor Decompositions. Motivation.

268 views • 13 slides

Part B Tensors

Part B Tensors. Or, inverting. The graph above represents a transformation of coordinates when the system is rotated at an angle  CCW. 2B1 Tensors.

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Fault systems and Paleo-stress tensors in the Indus Suture Zone (NW Pakistan)

Fault systems and Paleo-stress tensors in the Indus Suture Zone (NW Pakistan). GEOLOGICAL INSTITUTE. ZURICH. Gerold Zeilinger, Jean- Pierre Burg, Nawaz Chaudhry, Hamid Dawood & Shahid Hussain. Outline. Overview Field examples Data Method Results Interpretation.

361 views • 16 slides

Theory of Large Deformation and Alternate Stress Tensors

Theory of Large Deformation and Alternate Stress Tensors. Brian Sweetman Advisor : Dr . Andreas A. Linninger Wednesday, April 29, 200 9 Laboratory for Product and Process Design, Department of Bioengineering, University of Illinois, Chicago, IL. Some questions raised last week.

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LECTURE 6: SEISMIC MOMENT TENSORS

LECTURE 6: SEISMIC MOMENT TENSORS Represent other types of seismic sources as well as slip on a fault Give additional insight into the rupture process Simplify inverting (rather than forward modeling ) seismograms to estimate source parameters

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SPARSE TENSORS DECOMPOSITION SOFTWARE

SPARSE TENSORS DECOMPOSITION SOFTWARE. Papa S. Diaw, Master’s Candidate Dr. Michael W. Berry, Major Professor. Introduction. Large data sets Nonnegative Matrix Factorization (NMF) Insights on the hidden relationships Arrange multi-way data into a matrix Computation memory and higher CPU

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Fast and Simple Calculus on Tensors in the Log-Euclidean Framework

8th International Conference on M edical I mage C omputing and C omputer A ssisted I ntervention, Oct 26 to 30, 2005. Fast and Simple Calculus on Tensors in the Log-Euclidean Framework. Vincent Arsigny, Pierre Fillard, Xavier Pennec, Nicholas Ayache.

489 views • 36 slides

2. Differentiable Manifolds And Tensors

2. Differentiable Manifolds And Tensors. 2.1 Definition Of A Manifold 2.2 The Sphere As A Manifold 2.3 Other Examples Of Manifolds 2.4 Global Considerations 2.5 Curves 2.6 Functions On M 2.7 Vectors And Vector Fields 2.8 Basis Vectors And Basis Vector Fields 2.9 Fiber Bundles

596 views • 54 slides

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Tensors

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Presentation Transcript

Tensors

Tensors and Component Analysis

Physical interpretation of DC and non-DC components of moment tensors Václav Vavryčuk Institute of G eo physics, Pragu

Appendix A: Tensors

Differential Manifolds and Tensors

Decomposition and Visualization of Fourth-Order Elastic-Plastic Tensors

Multi-linear Systems and Invariant Theory in the Context of Computer Vision and Graphics Class 2: Homography Tensors CS

Review (2 nd order tensors):

A Physicists’ Introduction to Tensors

Mining Large Graphs: Spectral Methods, Tensors and Influence propagation

Earthquake source modelling by second degree moment tensors

TENSORS/ 3-D STRESS STATE

EARTHQUAKES (2): WAVEFORM MODELING, MOMENT TENSORS, &amp; SOURCE PARAMETERS

Extracting Structure from Matrices &amp; Tensors by Random Sampling

Part B Tensors

Fault systems and Paleo-stress tensors in the Indus Suture Zone (NW Pakistan)

Theory of Large Deformation and Alternate Stress Tensors

LECTURE 6: SEISMIC MOMENT TENSORS

SPARSE TENSORS DECOMPOSITION SOFTWARE

Fast and Simple Calculus on Tensors in the Log-Euclidean Framework

2. Differentiable Manifolds And Tensors

EARTHQUAKES (2): WAVEFORM MODELING, MOMENT TENSORS, & SOURCE PARAMETERS

Extracting Structure from Matrices & Tensors by Random Sampling