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Let $p \geq 5$ be a prime, $N$ be an integer not divisible by $p$, $\bar\rho_0$ be a reducible, odd and semi-simple representation of $G_{\mathbb{Q},Np}$ of dimension $2$ and $\{\ell_1,\cdots,\ell_r\}$ be a set of primes not dividing $Np$.…
We study the decomposability and the subdifferential of the tensor nuclear norm. Both concepts are well understood and widely applied in matrices but remain unclear for higher-order tensors. We show that the tensor nuclear norm admits a…
Matrices of rank at most k are defined by the vanishing of polynomials of degree k + 1 in their entries (namely, their (k + 1)-times-(k + 1)-subdeterminants), regardless of the size of the matrix. We prove a qualitative analogue of this…
We show that in a finite tensor category, the tensor product property holds for support varieties if and only if it holds between indecomposable periodic objects. We apply this to certain Hopf algebras in the form of skew group algebras. In…
We show how well known tools of algebraic geometry for the study of finite sets can be fruitfully applied to the study of Waring decompositions of symmetric tensors (forms). We mainly focus on the uniqueness of a given decomposition (the…
In this article, we study a class of non-selfadjoint Schr{\"o}dinger operators H which are perturbation of some model operator H 0 satisfying a weighted coercive assumption. For the model operator H 0 , we prove that the derivatives of the…
A tensor $T$, in a given tensor space, is said to be $h$-identifiable if it admits a unique decomposition as a sum of $h$ rank one tensors. A criterion for $h$-identifiability is called effective if it is satisfied in a dense, open subset…
In a recent study of Engel Lie rings, Serena Cicalo` and Willem de Graaf have given a practical set of conditions for an additively finitely generated Lie ring L to satisfy an Engel condition. We present a simpler and more direct proof of…
A class of self-inversive polynomials includes all the self-reciprocal polynomials. Let A denote the set of all self-reciprocal polynomials with n+1 coefficients. Let B denote the set of certain self-inversive and non self-reciprocal…
In 1907, Oskar Perron showed that a positive square matrix has a unique largest positive eigenvalue with a positive eigenvector. This result was extended to irreducible nonnegative matrices by Geog Frobenius in 1912, and to irreducible…
The problem of simplifying tensor expressions is addressed in two parts. The first part presents an algorithm designed to put tensor expressions into a canonical form, taking into account the symmetries with respect to index permutations…
Latent variable models with hidden binary units appear in various applications. Learning such models, in particular in the presence of noise, is a challenging computational problem. In this paper we propose a novel spectral approach to this…
Observables in random tensor theory are polynomials in the entries of a tensor of rank $d$ which are invariant under $U(N)^d$. It is notoriously difficult to evaluate the expectations of such polynomials, even in the Gaussian distribution.…
Let $q=p^f$ be a prime power, $H \leq \mathrm{GL}_d(q)$ a subgroup containing a genuine Singer cycle $s$ of order $q^d-1$, and $W$ an $\mathbb{F}_q H$-module whose scalar extension restricts to an untwisted polynomial tensor representation…
Tensors are multidimensional analogs of matrices. In this paper, based on degree-theoretic ideas, we study homogeneous nonlinear complementarity problems induced by tensors. By specializing this to $Z$-tensors (which are tensors with…
In this paper, we present a partial survey of the tools borrowed from tensor algebra, which have been utilized recently in Statistics and Signal Processing. It is shown why the decompositions well known in linear algebra can hardly be…
We introduce the concept of mode-k generalized eigenvalues and eigenvectors of a tensor and prove some properties of such eigenpairs. In particular, we derive an upper bound for the number of equivalence classes of generalized tensor…
We prove inclusion theorems for both spectra and essential spectra as well as two-sided bounds for isolated eigenvalues for Klein-Gordon type Hamiltonian operators. We first study operators of the form $JG$, where $J$, $G$ are selfadjoint…
Third-order tensors are widely used as a mathematical tool for modeling physical properties of media in solid state physics. In most cases, they arise as constitutive tensors of proportionality between basic physics quantities. The…
Based on some previous results, one gives a general formula for introducing electromagnetic multipole expansions in terms of symmetric and traceless cartesian tensors.