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Questions tagged [tensor]

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Vector-valued forms in Riemannian geometry

Suppose $(M,g)$ is a Riemannian manifold. I want to find a vector-valued $2-$form $T$ such that, for any vector fields $X,Y,Z$ on $M$, $$ g(T(X,Y),Z)=g(T(Z,X),Y)\,. $$ As a motivation, consider the ...
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76 views

DIstance on a Riemannian manifold [closed]

Given a Riemannian manifold $(M,g)$ is it possible to calculate the distance between two points on this manifold. Is it possible the inverse? That means: given a formula of the distance, for example: ...
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186 views

Generalization of Carleman coefficients to multivariable functions - Carleman tensor?

Recently I learned about a matrix called Carleman matrix. It is a matrix used to represent function iteration with matrix multiplying. Carleman linearization is a technique used to embed a finite ...
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1answer
135 views

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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50 views

Dual space of polynomial one-form

Recently I read a paper "Quasi-particles models for the representations of Lie algebras and geometry of flag manifold". In section 2, author gives a fact without proof. Now I rephrase this fact as ...
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1answer
189 views

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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75 views

Product of cotton-york tensor with ricci tensor

In the process of calculation of a problem in tensor analysis, I have encountered with an expression given by $$C^{ij}R_{ij}=0.$$ This is for a Riemannian manifold of dimension 3. Assuming the ...
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2answers
427 views

A kind of “Curvature tensor” for higher dimensional tensors

I begin my question with a multilinear question then I will consider two local smooth analogies: Assume that $\alpha$ is a real valued symmetric $k$-tensor, that is a $k$-linear map $\alpha:\...
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1answer
355 views

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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1answer
124 views

Strassen-like algorithm for Hadamard product of $2\times 2$ matrices

Let $A=\pmatrix{a_1& a_2\\a_3&a_4}, B=\pmatrix{b_1& b_2\\b_3&b_4}$ be two matrices. Let $C$ be the Hadamard product of $A$ and $B$. $$C=\pmatrix{c_1& c_2\\c_3&c_4}=\pmatrix{...
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1answer
83 views

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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Bounds on the tensor and border rank ratios of tensor unfoldings?

Consider $T_1 \in \mathbb{C}^{d_1 \times \cdots \times d_k}$ as an order-$k$ tensor of the format $(d_1, \cdots , d_k)$. Now, let $T_2$ be an unfolding of $T_1$ that is obtained by merging $\mathbb{C}^...
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Non Negative Tensor Tucker Decomposition Error Degradation

I have been working on iterative decomposition methods of tensors with non negativity constraints. I have noticed that $\textbf{N}$on negative $\textbf{T}$ensor $\textbf{F}$actorization "NTF" which is ...
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1answer
257 views

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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1answer
112 views

Multi-dimensional permanent

Is there a particularly natural / "correct" way of generalizing permanents to tensors? (I mean of course, 'square' tensors.) There seem to be very few resources on the matter. There needs to be a ...
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1answer
138 views

Vanishing of determinant of Cotton York tensor

suppose $\Omega \subset \mathbb{R}^3$ is a Riemannian manifold with $g= dx_1^2 + dx_2^2 + c(x_1,x_2,x_3)dx_3^2$. Is it true that $det(CY)=0$? Thanks
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1answer
207 views

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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Finding the particular and general solutions to Einstein Field Equations under generalized Vaidya Geometry

The problem I have is on finding the particular and general solutions to Einstein Field Equations under generalized Vaidya Geometry, which comes from the following paper : https://journals.aps.org/prd/...
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2answers
308 views

Tensor Field Decomposition in Space time

For vector fields in $R^3$ one knows that there exists a unique decomposition of vector fields in to solenoidal (divergence free) and potential parts. A generalization of this Theorem for symmetric ...
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1answer
141 views

Relation between curl and tensor divergence

The following seemingly-simple problem came up when working on a problem in the fluid theory of plasmas. Given a vector field $\mathbf{A}$, find a symmetric tensor $\mathbf{P}$ such that $\...
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2answers
326 views

Derivative of eigenvalues w.r.t. a tensor

$E$ is a real, positive-definitive 3x3 symmetric tensor (I am thinking about the strain tensor in solid mechanics). We perform eigendecomposition and get: $$E_p=\sum_{i=1}^{3}λ_iN_i⊗N_i$$ into its ...
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Doubt on the best low rank approximation of a symmetric tensor

I have a matrix $M\in\mathbb{R}^{n\times k}$, with $k<n$ whose columns $m_i$ are linearly independent. So we have $M := [m_1|..|m_k]$. From the columns of $M$ I can define the following matrix $$ ...
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1answer
188 views

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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182 views

Smoothness of a (given) global section of a vector bundle over G(2,6)

Let $G=Gr(2,6)$ the Grassmannian of two planes in $V=\mathbb C^6$, and let $\mathcal Q(1)$ the rank four quotient bundle on it twisted with $\mathcal O_G(1) \cong $ det$(S^*)$, $S$ being the ...
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1answer
114 views

Proof about affine connections

I'm reading Nomizu & Sasaki's "Affine Differential Geometry: Geometry of Affine Immersions" and I'm having some trouble with Proposition 1.4. I have an immersed surface in $M \hookrightarrow \...
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1answer
182 views

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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1answer
732 views

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^...
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0answers
167 views

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: $t_{ijkl} \rightarrow M_{ij, kl}$. ...
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0answers
4k views

Divergence of a second order tensor [closed]

I think that I have found 2 seemingly conflicting sources relating to the divergence of a second order tensor. I am not sure which is correct. Suppose you would like to compute the components of a ...
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2answers
1k views

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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1answer
459 views

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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0answers
125 views

Functional composition of Hadamard product

Let $\Bbb K$ be a ring. Are there universal functions $$f,h:\Bbb K^{n\times n}\times\Bbb K^{n\times n}\times\Bbb K^{n\times n}\times\Bbb K^{n\times n}\rightarrow\Bbb K^{n\times n}$$ $$g:\Bbb K^{n\...
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330 views

Mixed tensor index position significance

What is the significance of tensor index position? For example the fourth order Riemann curvature tensor \begin{align} R^m_{ijk} \end{align} or \begin{align} R^{\phantom{i}m}_{i\phantom{m}jk}. \end{...
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Obtaining the metric from the mixed Ricci tensor $R^i{}_j$

In chapter 5 of the book "Einstein Manifolds", Arthur Besse discusses the possibility to find the metric $g$ when knowing the Ricci curvature tensor $Ric(g)$ ($=R_{ij}$). But what do we know about ...
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38 views

Tensor of Relative Bases

Suppose $U,V,W$ are the cyclotomic fields $\mathbb{Q}(\zeta_{m_3})$, $\mathbb{Q}_(\zeta_{m_2})$, and $\mathbb{Q}(\zeta_{m_1})$ respectively, where $m_1 \mid m_2 \mid m_3$ so that $U/V/W$ is a tower of ...
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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 $M$.                  ...
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125 views

Cannot multivectors be classified more easily than general tensors?

This is sort of a spinoff of Is there a useful generalization of the Schmidt decomposition to the tensoring together of 3 or more vector spaces? - seems to be almost hopeless, but maybe some partial ...
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2answers
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Calculating the Riemann Christoffel tensor for a diagonal metric

I am trying to calculate the entries of the Riemann curvature tensor $R^m_{\phantom{m}ijk}$ for the metric $g_{ij}$. The Riemann-Christoffel tensor is given as \begin{align} R^m_{\phantom{m}ijk} = \...
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1answer
688 views

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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0answers
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Possibility Of Curvature and/or Mellin based approach to (Non-linear) system Identification?

I have some experience in non-linear system identification (from an experimental point of view) using higher oder spectral analysis. I see this is the most popular way of identifying non-linearities ...
2
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1answer
286 views

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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2answers
735 views

Derivative of an eigenvector with respect to his own 3x3 real symmetric matrix

$\mathbf{C}$ is a real, positive-definitive 3x3 symmetric matrix (I am thinking about the right Cauchy-Green tensor in solid mechanics). We perform eigendecomposition and get: $$\mathbf{C} = \sum_{i=...
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1answer
231 views

Petrov classification/Weyl scalars

There is one calculation in Chandrasekhar's "Mathematical Theory of Black Holes" that I cannot understand. Here is the setup: We want to show that Petrov type D (i.e. two principal null directions) ...
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4answers
2k views

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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1answer
799 views

Commutation of tensor products with inverse limits in a specific case

For $X,Y$ sets, let's denote $Y^X$ the set of all mappings $X\rightarrow Y$. If $Y(=R)$ is a ring, $R^X$ is a $R$-module (well, a bi-module but my question is - at first - concerning commutative rings)...
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1answer
186 views

Global geometry measures for Riemannian manifolds

I'm working on a stochastic algorithm and considering it to apply in case of any curved space (manifolds). But in order to make the algorithm as efficient as possible I want to include in it some ...
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1answer
447 views

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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1answer
763 views

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 ...
4
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1answer
3k views

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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3answers
1k views

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 ...