Questions tagged [tensor]
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157 questions
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Why are matrices ubiquitous but hypermatrices rare?
I am puzzled by the amazing utility and therefore ubiquity of
two-dimensional matrices in comparison to the relative
paucity of multidimensional arrays of numbers, hypermatrices.
Of course ...
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Is there any way to rewrite a partial differential equation using language of differential forms, tensors, etc?
My question is: usually, a partial differential equation, for example, those coming from physics, is written in a language of vector calculus in a local coordinate. Is there any way (or any algorithm) ...
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Geometrical meaning of the Ricci Tensor and its Symmetry
Let $M$ be a smooth, pseudo-Riemannian manifold with $\dim(M) \ge 2.$ Let $\nabla$ be any affine connection on $M$. No reason for it to be the Levi-Civita connection. All we assume is that it has zero ...
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Hessian as a tensor, multi-dimensional taylor series, and generalizations
The Hessian matrix $\{\partial_i \partial_j f \}$ of a function $f:\mathbb{R}^n \to \mathbb{R}$ depends on the coordinate system you choose. If $x_1,\cdots,x_n$ and $y_1,\cdots,y_n$ are two sets of ...
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Best known bounds on tensor rank of matrix multiplication of 3×3 matrices
Years ago I attended a conference where they taught us that matrix multiplication can be represented by a tensor. The rank of the tensor is important, because putting it into minimal rank form ...
- 3,359
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Equations satisfied by the Riemann curvature tensor
It is well known that the Riemann curvature tensor of a metric satisfies
\begin{eqnarray}
R_{jikl}=-R_{ijkl}=R_{ijlk},(1)\\
R_{klij}=R_{ijkl},(2)\\
R_{i[jkl]}=0 \mbox{(1st Bianchi identity)}.(3)
\end{...
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Exponentiation of vector spaces?
This question occurred to me while thinking on another one here, Name for an operation on matrices?
Can one define in an invariant way a binary operation on finite-dimensional vector spaces - let us ...
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What is the largest tensor rank of $n \times n \times n$ tensor?
The tensor rank of a three dimensional array $M[i,j,k], i,j,k\in [1,\ldots,n]$ is the minimal number of vectors $x_i,y_i,z_i$, such that $M=\sum_{i=1}^d x_i\otimes y_i\otimes z_i$.
From dimension ...
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Who coined the name tensor and why?
Who coined the name "tensor" and why? What does the word "tensor" really mean, not the mathematical definition?
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Derivative of eigenvectors of a matrix with respect to its components
Suppose that $B$ is a real, positive-definitive symmetric ($3\times3$) matrix (more accurately, $B$ is a tensor) with distinct eigenvalues, and that we can write it as
$$
B= \sum_{i=1}^3 \lambda_{i}(...
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Does the hyperdeterminant calculate a quantity akin to the volume of a parallelepiped?
If $M$ is an $n \times n$ matrix, $|\det(M)|$ is the volume of the $n$-dimensional
parallelepiped spanned by the column vectors of $M$.
...
- 151k
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votes
1
answer
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Geometric Interpretation of Multiplication in Pure Cubic Number Fields and Beyond
I got interested in the question of possible geometric interpretations of the multiplication in algebraic number fields of degree $>2$ (with application to multiplication of units in the ring of ...
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A generalization of Boolean matrix multiplication for order-3 tensors
The Boolean matrix product of two 0-1 $n \times n$ matrices $A$ and $B$ is the matrix $C$ defined as
$$C[i,j] = \vee_{k=1}^n (A[i,k] \wedge B[k,j]).$$ If $A = B$ and the matrix is an adjacency matrix ...
- 4,729
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Recovering a smooth manifold from its tensor fields
1) Consider the algebra $\mathcal T (M) = \bigoplus \limits _{p, q \ge 0} \mathcal T ^{p, q} (M)$, where $\mathcal T ^{p, q} (M)$ is the space of tensor fields of type $(p,q)$. I should endow it with ...
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Question on Nash's paper on $C^1$ isometric immersions: Why approximating the error tensor $\delta$?
I am trying to go through the classical paper by Nash on the existance of $C^1$ isometric immersion of a Riemannian manifold $(M,g)$ (here is the Jstor link: https://www.jstor.org/stable/1969840?seq=1#...
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Generalization of a standard algebraic group theory result for a tensor problem
$\DeclareMathOperator\GL{GL}$Let $X$, $Y$, $Z$ be $\mathbb{C}$-vector spaces, and let $A\subseteq X$ and $B\subseteq Y$ and $C\subseteq Z$ be linear subspaces. Let $V=X \otimes Y \otimes Z$, acted on ...
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Non-tensor-representable ultrafilters on $\omega$
If ${\cal U}$ and ${\cal V}$ are ultrafilters on non-empty sets $A$ and $B$ respectively, then the tensor product ${\cal U}\otimes{\cal V}$ is the following ultrafilter on $A\times B$:
$$\big\{X\...
- 48.9k
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Is $\max_{\|x\|_p=\|y\|_p=1} |\langle x, Ay\rangle|$ equivalent to $\max_{\|x\|_p=|} |\langle x, Ax\rangle|$ for symmetric $A$ & $p\geq 2$?
Let $A\in \mathbb{R}^{n\times n}$ be a symmetric matrix, and consider the $l_p$ norm ($p\geq 2$).
Can we prove that the following problems are equivalent:
$$\max_{\|x\|_p=\|y\|_p=1} \left| \langle x, ...
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votes
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How to count the number of tensors over a finite field of tensor rank $r$?
For simplicity, work over $\mathbb F_2$ and only consider order-$3$ equilateral tensors. For $r\in\mathbb Z_{>0}$, how many tensors $\mathfrak{T}\in\mathsf{Ten}_{n}^{\otimes 3}(\mathbb F_2)$ are ...
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Is there any sort of higher-order SVD (quadratic and above) for dimensionality reduction?
(Posted this on math.stackexchange and cross.correlated over more than a week ago, but didn't get an answer, and this is a question in my research so this seems like it might have been the better ...
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Does every ‘curvature’ tensor induce a metric? [duplicate]
So we know that given a Riemannian manifold $(M,g)$ we can calculate its Riemannian curvature $R$. This covariant $4$-tensor field then satisfies some important symmetries
\begin{gather*}
R_{ijkl} = - ...
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Waring rank of monomials, and how it depends on the ground field
The Waring rank of a degree-$d$ homogeneous polynomial $p$ is the least integer $r$ such that you can write $p$ as a linear combination of $r$ $d$-th powers of linear forms $\{\ell_k\}$:
$$
p = \sum_{...
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Is there a relation between the Hessian matrix and the structure Tensor?
for a 2 dimensional image, i am interested to find a relation between the Hessian
$H = \begin{pmatrix} \frac{\partial^2 I}{\partial x^2}&\frac{\partial^2 I}{\partial x \partial y}\\ \frac{\partial^...
- 61
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Tensor matricizations and their decompositions
Suppose we have a 4-index tensor $t_{ijkl}$ (all 4 dimensions are equal size). We can make a matrix out of it by taking first and last two indexes as new indexes: $t_{ijkl} \rightarrow M_{ij, kl}$. ...
- 71
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Local diagonalisation of a degenerated 2d metric tensor
Consider a smooth 2d-manifold $M$ and let $g$ be a smooth $(0,2)$-tensor satisfying $rk(g)\geq1$ everywhere. Obviously if $rk(g)=2$ at a point $p\in M$ then $g$ is locally diagonalisable (i.e. there ...
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Example of a curvature with no associated metric
Is there a concrete example of a $4$ tensor $R_{ijkl}$ with the same symmetries as the Riemannian curvature tensor, i.e.
\begin{gather*}
R_{ijkl} = - R_{ijlk},\quad R_{ijkl} = R_{jikl},\quad R_{ijkl} =...
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2
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Bounds for metric in normal coordinate
Let $M$ be a Riemannian $n$-manifold and $x \in M$. For the metric tensor $g_{ij}$ of geodesic normal coordinates at $x$, there is a formula $g_{ij}(y) = \delta_{ij} + \frac13 R_{kijl} y^k y^l + O(\|y\...
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What is expected (border) rank of the knonecker product of 3-tensors
Given two three order tensors $T$ and $S$ in $F^{m\times n\times p}$ and $F^{a\times b\times c}$. Clearly $\operatorname{rk}(T\otimes S)\le \operatorname{rk}(T)\operatorname{rk}(S)$. Does the equality ...
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Is a flattening rank a lower bound for the border rank?
Suppose $T \in V_1 \otimes \cdots \otimes V_k$ is a tensor, where each $V_i$ is a finite dimensional complex vector space. A $1$-flattening (or a flattening) is a realization of $T$ as a matrix in the ...
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Is the asymptotic rank of a tensor bounded by (naive) border rank?
Write $R(T)$ for the rank of an order-$3$ tensor $T \in \mathbb C^{m \times n \times p}$ over the complex numbers. If $T' \in \mathbb {C}^{m' \times n' \times p'}$ is another such tensor then let $T \...
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higher order analogues of sylvester's law of inertia?
Sylvester's law of inertia (here I quote wikipedia)
If A is the symmetric matrix that defines the quadratic form, and S is any invertible matrix such that D = SAS^{T} is diagonal, then the number ...
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Levi-Civita symbol
Is the Levi-Civita symbol a tensor?
R. A. Sharipov afirm (In "Quick Introduction to Tensor Analysis", page 30) that "...the Levi-Civita symbol is NOT a tensor..."
$\epsilon_{jkq}=\epsilon^{jkq}=\...
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1
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Bochner Laplacian in coordinates
Sorry if this is a too basic question, but I didn't find an answer anywhere:
The connection Laplacian, or Bochner Laplacian, is the differential operator acting on $k$-tensor fields $T\in\Gamma^{\...
- 1,171
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1
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Tell me something about these "component tensor" TQFT's
I noticed that there is a class of TQFT's that exists for every dimension $n\geq1$. It's probably well-known because it's quite simple, but I'm looking for a standard name or a better way to think ...
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Is there a generalization of eigenvalues and eigenvectors to tensors?
Two perhaps ill-posed or just silly questions:
Let $n>0$, $T$ be an $(n+2)$-tensor, and $\otimes$ denote the Kronecker product of tensors. Is there a tensor generalization for the fundamental ...
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Looking for a tractable algorithm or formula for the determinant of a tensor
It is possible to define the determinant of a tensor.
We think of a tensor as a collection of numbers but this collection easily extends to a proper multilinear map.
If $T:\{1,....,n\}^m\to \mathbb C$ ...
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How is this product of tensors defined?
I am reading the paper “ The first eigenvalue of a small geodesic ball in a riemannian manifold”, by Karp and Pinsky, from where I took the following:
Here, $\Delta_{-2}$ denotes the usual Laplacian ...
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Question on Lorentzian geometry
I apologize in advance if this is a too basic question.
Let $(M,g)$ be a Lorentzian manifold with signature convention $(-,+,\dots,+)$. Now, lets suppose $X\in\Gamma(TM)$ defines a global time-...
- 1,171
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Simple way to calculate the eigenvalues of a $2 \times 2 \times 2$ tensor
I am working with hypergraphs. The various matrices associated with hypergraphs are hypermatrix or tensors. I am interested in spectral aspects. In particular, I want to find all the eigenvalues ...
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Are constant connection coefficients uniquely determined by the (1,3) curvature coefficients?
Suppose that on a certain coordinate system the coefficients $\Gamma^i_{jk}$, $i,j,k=1,\cdots, n$, of a linear connection are constant. We do not require compatibility with a metric, however I am ...
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Symmetric tensor decomposition
Let $T$ be an order-$k$, rank-$m$ symmetric tensor, that is, $T=\sum_{j=1}^m v_j\otimes v_j \otimes \cdots \otimes v_j$, where the Segre outer product is taken $k$ times, with $v_j\in\mathbb{R}^d$ for ...
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Taylor's series for Lie groups
Let $G_1$ and $G_2$ be two (matrix) Lie groups, with $L(G_1)$ and $L(G_2)$ their respective Lie algebras.
I am interested to know if there is a well developed theory to approximate a (sufficiently) ...
4
votes
1
answer
747
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The Laplacian of an expression involving the Ricci tensor
While doing some computations on a compact Riemannian manifold I have reached the following expression:
$$ \Delta_y \big( Ric_y (\exp_y ^{-1} x, \exp_y ^{-1} x) \big) (x)$$
where $\Delta_y$ is the ...
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Indexed tensor manipulation CAS
hello.
I am looking for tensor manipulation software that would allow me:
declare indices
declare results of contraction (or simplification rules)
allow algebraic simplifications and expansion
index ...
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Singular value decomposition for tensor
I am looking at (the limitation of) the extension of the singular value decomposition to tensors. I would like to show that there is a tensor $A_{i,j,k}$ that cannot be decomposed in the following ...
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What is the current status of representation theory of $n$-ary groups in terms of hypermatrices?
An $n$-ary group is a generalization of the usual concept of a group where the binary operation (2-argument operation) is instead an $n$-ary ($n$-argument) operation. More info here on Wikipedia.
I ...
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Representation theory of (anti)self-dual tensors
I am using usual physics notations and I guess the physics motivations of this question are obvious.
Let a basis of the $SO(n,m)$ Lie algebra be denoted by $S^{\mu \nu}$ and the Lie algebra be, $[S^{...
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Construct a 4th-order tensor with matricization ranks $r$ that is not rank $r$
I ask you for this possibly not so simple task:
Explicitly construct a 4th-order tensor $A \in \mathbb{C}^{n_1 \times \ldots \times n_4}$ that does not have (border) rank $r$, but for which each ...
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Characterization of all-orthogonal tensors
In the paper [1], it is proven in Theorem 2 that any $n$-tensor $\mathcal{A}\in\mathbb{R}^{d_1\times...\times d_n}$ can be decomposed as
$$
\mathcal{A}=\mathcal{S} \times_1 U_1 ...\times_n U_n
$$
...
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Mathematica package for supergravity and string theory
I am looking for a Mathematica package that can manipulate tensors for supergravity, string theory or M-theory. I am particularly looking for a package that can do spinor and Clifford algebra ...
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