Questions tagged [tensor-calculus]
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13
questions
7
votes
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Generalized differential geometry based on Penrose's abstract tensor systems?
Penrose graphical notation has been an important precursor of string diagrams for monoidal categories. It was introduced in Penrose's paper Applications of negative dimensional tensors with intended ...
4
votes
1
answer
138
views
An introductory reference for tensor networks
I found a good reference on Tensor Networks: https://arxiv.org/abs/1912.10049. But I need an introductory reference with detailed proofs on Tensor Networks. Do you know another reference?
3
votes
2
answers
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Curl as a divergence... Is it possible? [closed]
I want to know if it is possible to express the operation
$$
\nabla \phi \times (\nabla \times \mathbf A)
$$
as the divergence of second order tensor field $T$. Here $ \phi$ is a scalar field and $\...
3
votes
2
answers
139
views
Rank of order-3 tensor with all slices being rank-1
If some tensor $T=(t_{ijk})$ has that all of its (2 dimensional) slices (along all 3 axes) are of rank-1, does it follow that the tensor is also rank-1? That is, can be written as
$$ t_{ijk}=a_i b_j ...
2
votes
1
answer
95
views
Mass of the push forward of a k-current with fixed orientation
$\DeclareMathOperator{\Mass}{Mass}$Let $f: \mathbb{R}^n \to \mathbb{R}^n$ be a smoth map. Given a $2$-vector (in general a $k$-vector but let's stick to $2$) $v_1 \wedge v_2 \in \Lambda_2 (\mathbb{R}^...
2
votes
2
answers
303
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Of all probability matrix $P$ having stationary distribution $\pi$, find the one having smallest diagonal
I am requesting your help today trying to solve a somewhat odd problem. Is there a way to find through some numerical algorithm such as Newton's method the stochastic matrix $\boldsymbol{P}$ having ...
1
vote
1
answer
130
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Diagonalizability, orthogonal diagonalizability of higher order tensors and their being or not being dense in some suitable topology
For our discussion, we'll assume that we're working with $\mathbb{R}^m$ only, but much or all of the following discussion should be carried over immediately to any finite dimensional inner product ...
1
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0
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Show that a tensor-train is contained in a recursive sequence of subspaces
Let
$p\in\mathbb N$;
$n_k\in\mathbb N$ and $\left(e^{(k)}_1,\ldots,e^{(k)}_{n_k}\right)$ denote the standard basis of $\mathbb R^{n_k}$ for $k\in\{1,\ldots,p\}$;
$u\in\bigotimes_{k=1}^p\mathbb R^{n_k}...
0
votes
1
answer
259
views
Curvature collineation and the Killing identity
The Lie derivative of a general covariant $4$-tensor is given by
$$\mathcal{L}_{K}R_{abcd} = X^{e}\nabla_{e}R_{abcd} + R_{ebcd}\nabla_{a}X^{e} + R_{aecd}\nabla_{b}X^{e} + R_{abed}\nabla_{c}X^{e} + R_{...
0
votes
1
answer
129
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Cores in the tensor-train decomposition
Let $d_i\in\mathbb N$, $I_i:=\{1,\ldots,d_i\}$ and $u\in\mathbb R^{d_1}\otimes\mathbb R^{d_2}\otimes\mathbb R^{d_3}$. It's somehow clear to me that we may regard $u$ as a three-dimensional array (see ...
0
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66
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Is $\left|\frac{\det(A_{\mu\nu})}{\det(B_{\mu\nu})}\right|$ an invariant for two tensors $A_{\mu\nu}$ and $B_{\mu\nu}$ in a manifold?
I was doing some math around the determinant of 2nd-order covariant tensors. In a general $n$-dimensional manifold, I deduced that the determinant of a tensor $A_{\mu\nu}$ can be defined as
$$
\det(A_{...
0
votes
1
answer
62
views
Tensor nuclear norm for a binary 3rd-order tensor
I am interested in the low-rank approximation of a binary(01) third-order tensor. Does anyone know how to define its tensor nuclear norm based on whatever tensor decomposition methods? Could anyone ...
0
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Formula with indices
Does anyone recognize this as something familiar?
$\epsilon^{acm}\partial_m\Theta^{kb}+\frac12\epsilon^{abm}\partial_m\Theta^{kc}$
OR
$(\epsilon^{acm}\partial_m\Theta^{kb}
+\frac12\epsilon^{abm}\...