Questions tagged [tensor]
The tensor tag has no usage guidance.
48
questions with no upvoted or accepted answers
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answers
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views
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 ...
- 1,010
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votes
0
answers
98
views
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 ...
- 2,085
4
votes
0
answers
120
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Is there a fast way to do this tensor power/trace operation?
Well, I asked this question on Math SE, and didn't get any responses, so I'm trying it here.
Given an $M*N*P$ tensor $T$, is there a fast way of computing the following "eighth-power trace"?
...
- 525
4
votes
0
answers
217
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Pseudo-tensor- and tensor-densities: Sections of what bundle?
Let $\mathcal{M}$ be a smooth manifold. A tensor field is then usually defined to be a section of the tensor bundle
$$\bigotimes_{i=1}^{p}T\mathcal{M}\otimes\bigotimes_{i=1}^{q}T^{\ast}\mathcal{M}.$$
...
- 987
4
votes
0
answers
151
views
Given a specific tensor on $V^{\otimes d}$, which $GL(V)$ orbit does the tensor belong to?
Suppose we have a vector space $V$ over a field $K$, with basis vectors $\hat{\bf{e}}_k$, and suppose we define a tensor
$$
\Lambda = \lambda_{i_1, ..., i_d} (\hat{\bf{e}}_{i_1} \otimes ... \otimes \...
- 89
4
votes
0
answers
179
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Are mixed discriminants and hyper-determinants the same thing?
Premise
Hello, I'm trying to learn more about hyperdeterminants, but as I'm sifting through the literature I'm finding more and more papers on Mixed discriminant and I find their definitions extremely ...
- 143
4
votes
0
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159
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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 ...
- 17.5k
3
votes
0
answers
73
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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/...
- 31
3
votes
0
answers
362
views
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 ...
- 4,304
2
votes
0
answers
40
views
$1$-parameter family of metrics preserving the normal direction
Let $(M^n,g)$ be a compact Riemannian manifold with boundary, $n \geq 2$, and let $N$ be the unit outward normal to $\partial M$. Denote by $S^2(M)$ the symmetric covariant $2$-tensors, by $S^2_0(M)$ ...
- 2,085
2
votes
0
answers
83
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Is 'weak' Strassen Conjecture true?
$\newcommand{\rank}{\mathop{\mathrm{rank}}}$Strassen conjectured for two tensors $T_{1}$ and $T_{2}$, $\rank(T_{1}\oplus T_{2})=\rank(T_{1})+\rank(T_{2})$. This is not generally true according to ...
- 151
2
votes
0
answers
264
views
What are examples of "perfect tensors"?
A "perfect tensor" is defined on the nLab very abstractly as "its tensor/hom-adjuncts $V^{\otimes k} \to V^{\otimes n - k}$ for $k \le n/2$ are isometries". The only example I'm ...
- 451
2
votes
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answers
114
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The Ricci curvature is bounded below by scalar curvature
So I have more questions coming from Dr Hamilton's "Three-Manifolds with Positive Ricci Curvature" paper here. I'm working in section 9 on Preserving Positive Ricci Curvature. In theorem 9.4,...
- 263
2
votes
0
answers
85
views
Estimates for tensors using local coordinates
Suppose that we have a Kähler manifold $(M, \omega_0)$ and another $(1,1)$ form $\eta$ on $M$. Let $\varphi$ be a smooth function such that $\omega_{\varphi} = \omega_0 + i \partial \bar \partial \...
- 121
2
votes
0
answers
163
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Naive generalization of determinant from matrices to higher rank tensors
Recall that using the Levi-Cevita symbol the determinant can be written as
$$\operatorname{det} A=\frac{1}{n!}\epsilon_{i_1\dots i_n}\epsilon_{j_1\dots j_n}A_{i_1j_1}\dots A_{i_nj_n}$$
Some ...
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2
votes
0
answers
92
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For a manifold of positive curvature, can we lower bound the distance between unit normals?
Suppose $M \subset \mathbb R^d$ is a $C^2$ $(d-1)$-manifold. In particular I am interested in when $M$ is the set $\{x \in \mathbb R^d: f(x) \le 1\}$ for some $C^2$ function $f:\mathbb R^d \to \...
- 1,761
2
votes
0
answers
77
views
Rank-1 decomposability of symmetric tensors
My question is about rank-1 decomposability of symmetric tensors over the reals.
Let $v_1,\dots,v_n\in\mathbb{R}^d$ be vectors. Construct the object:
$$
V=\sum_{j=1}^n \underbrace{v_j\otimes v_j\...
- 1,096
2
votes
0
answers
139
views
Example Petrov Classification
I would like to calculate the Petrov type for a specific spacetime, unfortunately I am not able to find a step by step algorithm or example for the process, either using null-tetrad, nor calculating ...
- 649
2
votes
0
answers
439
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
...
- 123
2
votes
0
answers
168
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 ...
2
votes
0
answers
69
views
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
$$
...
2
votes
0
answers
208
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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2
votes
0
answers
637
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{...
- 219
2
votes
0
answers
43
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 ...
- 121
1
vote
0
answers
305
views
About "residual" scalar curvature in Einstein warped product manifold
I have an Einstein warped product manifold $M=B \times_f F$ with warped metric $g=g^B+f^2g^F$, where $F$ is Ricci flat, then its scalar curvature is $S^F=0$.
It is well known that the scalar curvature ...
- 242
1
vote
0
answers
121
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Some kind of product of two 2d tensors to create a 3d tensor?
I recently need to apply the following concept of product of two 2d tensors to create a 3d tensor (tensors understood as generalized arrays):
given two 2d tensors $A_{m\times n}$ and $B_{n\times p}$, ...
- 461
1
vote
0
answers
77
views
tensor dimension/reshaping group
Consider an $N$ dimensional tensor $T$ using the strided view representation used by PyTorch, i.e. we have a storage vector $S$ projected into $N$ dimensions using a size tuple $s$ and a stride tuple $...
- 11
1
vote
0
answers
63
views
Tucker decompositions over arbitrary fields
Given an $n$-mode tensor $\mathcal{T}\in\mathbb{R}^{d_1\times\dotsb\times d_n}$, there exists a Tucker decomposition of $\mathcal T$ of the form
$$\mathcal{T} = \mathcal{X}\times_1 W_1\times_2\dotsb\...
- 228
1
vote
0
answers
124
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Adjacency matrix/tensor operations for graph sequences?
Consider a graph $G=(V,E)$. Its adjacency matrix $A$ is defined by $A_{u,v} = 1$ if $(u,v)\in E$, $0$ otherwise.
Consider a vector $x$ that associates a value $x_v$ to each vertex of $G$. Consider the ...
- 1,424
1
vote
0
answers
98
views
References Request: Bach tensor
Recently I want to study about Bach tensor in detail. From the fundemental definition and properties to conformal invariant. Is there any references to me about Bach tensor? If there are some origins ...
- 11
1
vote
0
answers
75
views
Spectral theorem for symmetric real tensors
Is there a definition of eigenvalues that allows to use a spectral theorem?
Let $\mathbf{T}$ be a real fully symmetric tensor of order $3$ and size $N$. Its components can be represented as $T_{ijk}\...
- 97
1
vote
1
answer
272
views
Best approximation with tensors of rank $\ge2$
Let $k\in\mathbb N$, $H_i$ be a (finite-dimensional, if necessary) $\mathbb R$-Hilbert space for $i\in I:=\{1,\ldots,k\}$, $H:=\bigotimes_{i\in I}H_i$ denote the tensor product$^1$ of $(H_i)_{i\in I}$ ...
- 161
1
vote
0
answers
157
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Compute Frobenius inner product of two tensor-trains in terms of tensor contractions
Let $p\in\mathbb N$, $n\in\mathbb N^p$ and identify the Hilbert space tensor $\bigotimes_{k=1}^p\mathbb R^{n_k}$ with $\mathbb R^{n_1\times\cdots\times n_p}$ (equipped with the Euclidean inner product)...
- 161
1
vote
0
answers
27
views
Show that a tensor-train is contained in a recursive sequence of subspaces
Let
$p\in\mathbb N$;
$n_k\in\mathbb N$ and $\left(e^{(k)}_1,\ldots,e^{(k)}_{n_k}\right)$ denote the standard basis of $\mathbb R^{n_k}$ for $k\in\{1,\ldots,p\}$;
$u\in\bigotimes_{k=1}^p\mathbb R^{n_k}...
- 161
1
vote
0
answers
227
views
Images and Kernels of tensor products of homomorphisms of modules
Let $\mathcal{H}$ be a Hopf algebra and let $M_1,M_2,N_1,N_2$ modules over $\mathcal{H}$. If $f:M_1\rightarrow N_1$ and $g:M_2\rightarrow N_2$ are homomorphisms of modules, then are the following ...
- 343
1
vote
0
answers
50
views
On symmetric tensors with same rank, different orders
Let $A,B$ be two symmetric tensors of same rank $m$; and orders $k$ and $\ell$, respectively. In particular, assume that $A,B$ admits the following structure: There exists $v_1,\dots,v_m\in\mathbb{R}^...
- 1,096
1
vote
0
answers
62
views
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}^...
- 490
1
vote
0
answers
31
views
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 ...
- 111
1
vote
0
answers
55
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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 ...
- 315
1
vote
0
answers
403
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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 ...
1
vote
0
answers
683
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] \...
- 11
1
vote
0
answers
302
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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 ...
- 1,075
0
votes
0
answers
122
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A question from Richard Hamilton's paper "A matrix Harnack estimate for the heat equation"
Richard Hamilton "A matrix Harnack estimate for the heat equation. Communications in Analysis and Geometry. 1(1993), 113-126."
On page 125, at the end of the proof of Theorem 4.3, I abstract ...
0
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0
answers
179
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Is the kernel $\vert d_X - d_Y \vert^p$ conditionally negative definite?
Given two finite metric spaces $(X,d_X)$ and $(Y,d_Y)$, for $p > 0$, define the kernel ($4$-D tensor) $K$ on $(X \times Y)^2$ by:
$$K\big( (x_i, y_k), (x_j, y_l) \big) = \vert d_X(x_i, x_j) - d_Y(...
- 101
0
votes
0
answers
52
views
Solving nonlinear differential multi-variable equation with block-matrices
Here is the problem:
Given a formula $f:\mathbb{R}^{n+k}\rightarrow\mathbb{R}^n$, written as $f(x_1(t),x_2(t),...,x_n(t);a1,...ak)$ with $k$ real unknown parameters $a_1,...,a_k$. For any $(a1,...ak)$,...
0
votes
0
answers
124
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Is it possible to reduce eigenvalues of tensors to an matrix eigenvalue problem?
Can we construct a larger matrix $M$ such that its eigenvalues are the same as the eigenvalues of a tensor $T$ of order 3?
Let $\mathbf{T}$ be a fully symmetric tensor of order $3$ and size $N$. Its ...
- 97
0
votes
0
answers
349
views
What is the physical meaning of torsion
The torsion tensor in 4 dimensions $S_{ab}^{\hphantom0\hphantom0 c}$ has 24 components and it can be split into a vector part $\hphantom0^{V}S_{ab}^{\hphantom0\hphantom0 c}=\frac{1}{3}(S_a\delta^c_b-...
- 1
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votes
0
answers
42
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How are the vector-valued trace and the unique linearization of $\mathfrak L(X,Y)\:\hat\otimes_π\:X→Y$ of $\mathfrak L(X,Y)×X→Y,\;(L,x)↦Lx$ related?
Let $X$ be a $\mathbb R$-Banach space and $X'\:\hat\otimes_\pi\:X$ denote the completion of the tensor product of $X'$ and $X$ with respect to the projective norm. The trace functional $\operatorname{...
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