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47
votes
11answers
5k views

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 ...
17
votes
3answers
3k views

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 ...
11
votes
2answers
1k views

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?
10
votes
4answers
1k views

Is there anyway 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 anyway (or any algorithm) ...
9
votes
3answers
549 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 ...
8
votes
2answers
1k views

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
votes
1answer
2k views

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 ...
4
votes
1answer
373 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 ...
4
votes
2answers
554 views

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 ...
4
votes
1answer
398 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 ...
3
votes
2answers
1k views

Maxwell Stress Tensor and Equations in Mathematician's Language [closed]

In my language, a differential two-form on $\mathbb{R}^4$ (viewed as a differentiable manifold with coordinates $t,x,y,z$) is a differentiable choice at each point of an alternating bilinear function ...
3
votes
1answer
473 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..." ...
3
votes
1answer
820 views

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, ...
2
votes
1answer
697 views

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) ...
2
votes
1answer
195 views

How many flavors should a notational system offer for rank-1 tensors?

The notation for tensors is like the plumbing in a very old Vermont farmhouse. It may once have been intentionally designed, but after that it just evolved. As an example, it seems that depending on ...
1
vote
1answer
414 views

A property on the Green-St Venant strain tensor

Green-St Venant strain tensor is defined by $E(u)={1\over 2}[\nabla u+(\nabla u)^T+(\nabla u)^T\nabla u]$, where $\nabla u$ is the displacement gradient. Show that $u\in H^1(\Omega), E(u)\in ...
1
vote
1answer
113 views

Number of linear independent equations

Is there any general rule to find the number of linearly independent equations such that $$L_i(T_{\mu\nu},\partial_\eta T_{\mu\nu},\partial_\omega\partial_\eta T_{\mu\nu},...)=0$$ where $L_i$ is a ...
1
vote
1answer
250 views

Decomposition of order-3 tensors over the complex numbers

This is a question about decomposition of order-3 tensors. The survey Tensor Decompositions and Applications give a good account of recent developments in this area. Let $T$ be an order-3 tensor, ...
1
vote
0answers
235 views

How to find the tensor product of modules that we don't know a basis for them?

Hi I know how to calculate some easy tensor products like $\mathbb{Z}/m\mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Z}/n\mathbb{Z}\cong_{\mathbb{Z}} \mathbb{Z}/(m,n)\mathbb{Z} $ or $F[X] \otimes_{F} F[Y] ...
1
vote
0answers
253 views

tensor/hypermatrix analogues of $GL(n,\mathbb{C})$?

Please excuse me if this question turns out to be incredibly silly for one reason or another. Are there tensor/hypermatrix analogues of $GL(n,\mathbb{C})$ that are interesting? What I'm mainly ...
0
votes
1answer
81 views

How does a high order tensor irreducible decompose? [closed]

I know the two tensor case, but how about high order?
0
votes
2answers
504 views

Tensor algebra question [closed]

1)Why embedding of ( not necessarily finite-dimensional) vector spaces $V\rightarrow W$ produces embedding of tensor algebras $T(V)\rightarrow T(W)$. I can prove it using Hamel basis in $W$ but is ...
0
votes
1answer
256 views

Geometric interpretation of tracing [closed]

Let $(M,g)$ be a Riemannian manifold. Is it true that for any symmetric 2-tensor $\alpha$ we have: $Trace_g(\alpha)=1/\omega_n\int_{S^{n-1}}\alpha(V,V)dvol(V)$ where $w_n$ is the volume of $S^{n-1}$? ...
0
votes
0answers
15 views

Derivation of tensor (with einstein summation) [migrated]

Let $f$ be a scalar defined by: $$f = a_{iik}a_{kjj} (1 \leq i, j, k \leq 3)$$ in which $a$ is a third-rank tensor and the summation convention for repeated indices is employed. Now, we define $g = ...
0
votes
0answers
41 views

Questions about some special tensor transformation

Suppose tensor $U_{i\alpha\beta}$ with dimension $M*N*N$ satisfy following condition: $$U_{i\beta\alpha}=W^1_{\alpha\alpha'}W^2_{\beta\beta'}U_{i\alpha'\beta'}$$ where $W^1$ and $W^2$ are $N*N$ ...
0
votes
0answers
67 views

Bounding the absolute value of a polynomial involving a Diophantine equation

Let $\mathbf{z}\in\mathbb{C}^n$ with entries $z_1,z_2,\ldots,z_n$. I would like to bound the following quantity \begin{equation*} ...
0
votes
0answers
71 views

Null vector fields given Bondi metric

I'm trying to understand how to compute the null future-directed vector fields if I have a given (Bondi) metric $g=-e^{2\nu}du^{2}-2e^{\nu+\lambda}dudr+r^{2}d\Omega$ with $d\Omega$-standard metric ...