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Topic: Rank of a tensor


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  Tensor - Wikipedia, the free encyclopedia
Tensors may be written down in terms of coordinate systems, as arrays of scalars, but are defined so as to be independent of any chosen frame of reference.
In the field of diffusion tensor imaging, for instance, a tensor quantity that expresses the differential permeability of organs to water in varying directions is used to produce scans of the brain.
Perhaps the most important engineering examples are the stress tensor and strain tensor, which are both 2nd rank tensors, and are related in a general linear material by a fourth rank elasticity tensor.
en.wikipedia.org /wiki/Tensor   (1888 words)

  
 Tensor - LearnThis.Info Enclyclopedia   (Site not responding. Last check: 2007-10-20)
In mathematics, a tensor is a certain kind of geometrical entity which generalizes the concepts of scalar, vector (spatial) and linear operator in a way that is independent of any chosen frame of reference.
The point of having a tensor theory is to explain the further implication of saying that a quantity is a tensor, beyond that specifying it requires a number of indexed components.
Note that the word "tensor" is often used as a shorthand for tensor field, which is a tensor value defined at every point in a manifold.
encyclopedia.learnthis.info /t/te/tensor.html   (1174 words)

  
 Tensor -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-10-20)
Tensors may be written down in terms of (A system that uses coordinates to establish position) coordinate systems, as arrays of scalars, but are defined so as to be independent of any chosen (A system of assumptions and standards that sanction behavior and give it meaning) frame of reference.
Tensors are of importance in (The science of matter and energy and their interactions) physics and (The discipline dealing with the art or science of applying scientific knowledge to practical problems) engineering.
indices at all), and vectors are rank one tensors.
www.absoluteastronomy.com /encyclopedia/t/te/tensor.htm   (2033 words)

  
 PlanetMath: tensor
A tensor is the mathematical idealization of a geometric or physical quantity that may be represented relative to a given frame of reference as an array of numbers
Actually, the tensor notion is quite general, and applies to all of the above examples; scalars and vectors are special kinds of tensors.
Tensor field concepts, which typically involved derivatives of some kind, are discussed elsewhere.
planetmath.org /encyclopedia/Tensor.html   (852 words)

  
 PlanetMath: simple tensor
In general it is not trivial to find the simplest way of expressing a tensor as a sum of simple tensors, so there is a name for the length of the shortest such sum.
There is an entirely different concept which is also called `the rank of a tensor', namely the number of components (factors) in the tensor product forming the space in which the tensor lives.
One area where the distinction between simple and non-simple tensors is particularly important is in Quantum Mechanics, because the state space of a pair of quantum systems is in general the tensor product of the state spaces of the component systems.
www.planetmath.org /encyclopedia/Entangled.html   (568 words)

  
 Tensor   (Site not responding. Last check: 2007-10-20)
In mathematics, a tensor is a certain kind of geometricalentity which generalizes the concepts of scalar, vector (spatial) and linear operator ina way that is independent of any chosen frame of reference.Tensors are of importance in physics and engineering.
But tensors are used also within other fields such as continuum mechanics, for example the straintensor, (see linear elasticity).
Note that the word "tensor" is often used as a shorthand for tensorfield, which is a tensor value defined at every point in a manifold.
www.therfcc.org /tensor-41664.html   (1114 words)

  
 Tensor : Tensors
In mathematics, a tensor is a certain kind of geometrical entity which generalizes the concepts of scalar, vector and linear operator in a way that is independent of any chosen frame of reference.
Tensors are of importance in differential geometry, physics and engineering.
Note that the word "tensor" is often used as a shorthand for tensor field, a concept which defines a tensor value at every point in a manifold.
www.fastload.org /te/Tensors.html   (903 words)

  
 Tensor (intrinsic definition) - Wikipedia, the free encyclopedia
In mathematics, the modern component-free approach to the theory of tensors views tensors initially as abstract objects, expressing some definite type of multi-linear concept.
The notion of tensor product can be generalised to vector spaces without a chosen basis, and even further; between modules.
If you are baffled by this, try reading the main tensor article and the classical treatment first.
www.wikipedia.org /wiki/Rank_of_a_tensor   (959 words)

  
 Tensor   (Site not responding. Last check: 2007-10-20)
In mathematics a tensor is a certain kind of geometrical which generalizes the concepts of scalar vector (spatial) and linear operator in a way that is independent any chosen frame of reference.
Note that the word "tensor" is often as a shorthand for tensor field which is a tensor value defined every point in a manifold.
Such a relationship is described by tensor of type (1 1) (that is say it transforms a vector into another The tensor can be represented as a matrix which when multiplied by a vector in another vector.
www.freeglossary.com /Tensors   (1517 words)

  
 deal.II base library: Tensor< rank_, dim > Class Template Reference
The Tensor class provides an indexing operator and a bit of infrastructure, but most functionality is recursively handed down to tensors of rank 1 or put into external templated functions, e.g.
Using this tensor class for objects of rank 2 has advantages over matrices in many cases since the dimension is known to the compiler as well as the location of the data.
Contract a tensor of rank 3 with a tensor of rank 3.
www.dealii.org /developer/doxygen/base/classTensor.html   (2046 words)

  
 Rank of tensor - Physics Help and Math Help - Physics Forums   (Site not responding. Last check: 2007-10-20)
The rank of a tensor is just the total number of (free) indices that it has.
This is a definition of the rank of a tensor in terms of its components.
Note that in the tensor e, all indices must be different for the tensor to be nonzero.
www.physicsforums.com /showthread.php?p=761692#post761692   (1160 words)

  
 Method and apparatus for second-rank tensor generation - Patent 5005954
A real-time optical second-rank tensor generator as defined in claim 1 wherein said first and second vector beams and said plane wave pumping beam are generated at the same frequency.
A tensor is an element of an abstract system used to denote position determined within the context of more than one coordinate system, a special case of which is a vector that is determined in a single coordinate system.
In accordance with the present invention, a real-time tensor generator utilizes means for generating first and second amplitude modulated coherent vector beams orthogonally disposed in space, and incident in exact opposition on parallel sides of a nonlinear refractive crystal.
www.freepatentsonline.com /5005954.html   (1917 words)

  
 Wolfram Research, Inc.
that specify a particular element in the tensor correspond to the coordinates in the cuboid.
In physics, the tensors that occur typically have indices which run over the possible directions in space or spacetime.
The rank of a tensor is equal to the number of indices needed to specify each element.
documents.wolfram.com /v3/MainBook/3.7.11.html   (644 words)

  
 Tensor Products   (Site not responding. Last check: 2007-10-20)
As depicted in the following diagram, the two irreducible tensors combine to form a reducible tensor with rank equal to the sum of the ranks of two tensor which formed it.
To demonstrate the mathematics we shall first consider the buildup of a rank 2 spatial tensor from the product of two irreducible rank 1 spatial tensors whose components are the normalized spherical harmonics.
Keep in mind that this results from the specific build up of a rank 2 spatial tensor from two spherical irreducible rank1 spatial tensors whose components were the normalized spherical harmonics.
gamma.magnet.fsu.edu /html/modules/level1/spacet7.htm   (960 words)

  
 Tensor (Orbital)
The rank is the number of dimensions needed as indices for the components of this tensor.
Since every implementation is allowed to choose the tensor's internal data representation freely, you should not rely on such behaviour of mutable arithmetic objects to provoke inner state changes.
The reason is that one-dimensional indices are meaningless for tensors of rank r>1 and that adding a single component to a tensor is not allowed as it would destroy its rectangular form.
www.functologic.com /orbital/Orbital-doc/api/orbital/math/Tensor.html   (820 words)

  
 Rank 1 Tensors   (Site not responding. Last check: 2007-10-20)
Given the Cartesian components, the function A1 returns a rank 1 spatial tensor or a specified irreducible spherical component of that tensor.
For a spatial rank 1 tensor these components are explicitly given by equation (9-5).
For a spatial rank 1 tensor these components are given by A1,1, A1,0, and A1,-1, the three values taken from the function arguments of SphA1.
gamma.magnet.fsu.edu /html/modules/level1/spacet2.htm   (824 words)

  
 Tensor   (Site not responding. Last check: 2007-10-20)
Energy-momentum tensor for the electromagnetic flux in an...
The energy and momentum content of an electromagnetic field can be expressed entirely in terms of the fields through the energy-momentum tensor with no mention...
NEW YORK Tensor Group Inc., Woodridge, Ill., added equipment to press lines at Pioneer Press, Northfield, Ill., to meet demand for increased four-color...
www.wikiverse.org /tensor   (1294 words)

  
 Geometry-Algebra-Singularities-Combinatorics Seminar Talk   (Site not responding. Last check: 2007-10-20)
In the case where n=2 tensor rank is easily described and the maximum tensor rank of a tensor is the same as the "generic" tensor rank of a tensor.
The two numbers "maximal tensor rank" and "generic tensor rank" are no longer equal and there are only conjectures about these numbers.
In my talk I will explain why there are really TWO distinct problems concerning tensor rank (when n > 2) that should be considered (in analogy with the Waring Problem for Forms) and also give some results about these problems as well as a counterexample to a published conjecture on these topics.
www.math.neu.edu /GASC/GAS/geramita.html   (204 words)

  
 Rank 2 Interaction Parameters
This section describes how an ASCII file may be constructed that is self readable by a rank 2 interaction.
The angle theta which relates the rank 2 interactions orientation down from the z-axis of its PAS may be set.
The angle phi which relates the rank 2 interactions orientation over from the x-axis of its PAS may be set.
www.gamma.ethz.ch /html/modules/rank2/intrana8.htm   (306 words)

  
 [No title]   (Site not responding. Last check: 2007-10-20)
An appropriate set of topics is tensor description of crystal properties, derivation of the restrictions that symmetry imposes on tensor elements, and the nature of the resulting anisotropy in physical properties.
The viewing perspective may be varied to convey appreciation of both the shape of the surface and that it conforms to the point group of the crystal.
The magnitude of any tensor element in the polynomial may be "zoomed" up or down to show its contribution to the shape of the surface.
www.za.iucr.org /iucr-top/cong/17/iucr/abstracts/abstracts/E1341.html   (271 words)

  
 Talk:Tensor product - InformationBlast
The 'rank' of a tensor is actually not defined by the columns, rows, ect., but by an equation such as:
Nothing is wrong if the rank is the rank_(tensor), but it is linked to Rank of a matrix, which is (loosely speaking) the degree of linear dependency of a matrix and was what I talked about in my remark.
You could say that the whole point of tensor theory is to understand what this operation does for you.
www.informationblast.com /Talk:Tensor_product.html   (1245 words)

  
 James Tauber : Tensors
Now, a tensor of rank (1,0) is a linear mapping from one-forms to the real numbers.
In other words, a tensor of rank (1,0) can be thought of as a constant mapping to a vector.
A tensor of rank (1,1) can be viewed as a mapping from a one-form and vector to a real number or as a mapping from a vector to a vector or as a mapping from a one-form to a one-form.
www.jtauber.com /tensors   (391 words)

  
 General-purpose support functions   (Site not responding. Last check: 2007-10-20)
In some cases, heuristic ordering of tensor contractions so that the number of operations to be performed is minimized fails to produce the best solution.
Compute the n-antisymmetrizing delta symbol, a rank-2n tensor which is identical to its "square" and antisymmetric in the first and last n indices.
For higher :sum-rank, this corresponds to multiplication of rank-2n tensors, where the last n indices of a tensor are contracted with the first n indices of its successor.
www.cip.physik.uni-muenchen.de /~tf/lambdatensor/LambdaTensor1.1.4/doc/sp-array.html   (3078 words)

  
 Tensors and tensor algebra   (Site not responding. Last check: 2007-10-20)
transforming it according to the rules for a second rank tensor it is the same in all rotated frames of reference.
Clearly the sum or difference of two tensors of the same rank is also a tensor, and similarly if one multiplies all elements of a tensor by a scalar it is still a tensor.
Setting two indices of a tensor equal and summing reduces the the rank of the tensor by two.
astron.berkeley.edu /~jrg/ay202/node185.html   (790 words)

  
 PlanetMath: scalar
A scalar is a quantity that is invariant under coordinate transformation, also known as a tensor of rank 0.
As such, a scalar can be an element of a field over which a vector space is defined.
Cross-references: vector space, field, Variable, rank, tensor, transformation, coordinate, invariant
planetmath.org /encyclopedia/Scalar.html   (102 words)

  
 The Stress Tensor of the Electromagnetic Field
A tensor is a bookkeeping device designed to keep together elements that transform in a similar way.
Together, the energy density(W), Poynting's vector (Sa) and the Maxwell stress tensor (m_ab) are all the components of the stress tensor of the electromagnetic field.
The rest of that tensor is generated by the second line.
world.std.com /~sweetser/quaternions/EandM/tensor/tensor.html   (614 words)

  
 ipedia.com: Tensor Article   (Site not responding. Last check: 2007-10-20)
For more technical Wiki articles on tensors, see the section later in this article.
In mathematics, a tensor is a certain kind of geometrical entity which generalizes the concepts of scalar, vector an...
The "components" of the tensor are the indices of the array.
www.ipedia.com /tensor.html   (1234 words)

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