Questions tagged [tensor-calculus]

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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_{...
6
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0answers
173 views

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 ...
1
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2answers
128 views

Of all probability matrix $P$ having stationary distribution $\pi$, find the one having smallest diagonal

Hello MathOverflow community, 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 ...
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15 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}...
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1answer
56 views

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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0answers
109 views

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}\...
3
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2answers
123 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 ...
3
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2answers
460 views

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 $\...