Related papers: Tensor Complementarity Problem and Semi-positive T…

Classical matching theory can be defined in terms of matrices with nonnegative entries. The notion of Positive operator, central in Quantum Theory, is a natural generalization of matrices with nonnegative entries. Based on this point of…

We provide a sufficient condition for solvability of a system of real quadratic equations $p_i(x)=y_i$, $i=1, \ldots, m$, where $p_i: {\mathbb R}^n \longrightarrow {\mathbb R}$ are quadratic forms. By solving a positive semidefinite…

While every matrix admits a singular value decomposition, in which the terms are pairwise orthogonal in a strong sense, higher-order tensors typically do not admit such an orthogonal decomposition. Those that do have attracted attention…

Let $n \geq 3$ and $R_{abcd}$ be a $(4,0)$ sectionally positive curvature-type tensor (a tensor possessing all the local symmetries of the $(4,0)$ curvature tensor). Then there exists a metric tensor $g_{ab}$ such that $R_{abcd}\; g^{bd} =…

Let $A$ be a hereditary algebra over an algebraically closed field $k$ and $A^{(m)}$ be the $m$-replicated algebra of $A$. Given an $A^{(m)}$-module $T$, we denote by $\delta (T)$ the number of non isomorphic indecomposable summands of $T$.…

The covariance tensors in statistics{, elasticity tensor in solid mechanics, Riemann curvature tensor in relativity theory are all biquadratic tensors that are weakly symmetric, but not symmetric in general. Motivated by this, in this…

Let $P$ be a finitely generated commutative semiring. It was shown recently that if $P$ is a parasemifield (i.e. the multiplicative reduct of $P$ is a group) then $P$ cannot contain the positive rationals $\mathbb{Q}^+$ as its subsemiring.…

4x4x3 absolutely nonsingular tensors are characterized by their determinant polynomial. Non-quivalence among absolutely nonsingular tensors with respect to a class of linear transformations, which do not chage the tensor rank,is studied. It…

A symmetric matrix $C$ is completely positive (CP) if there exists an entrywise nonnegative matrix $B$ such that $C=BB^T$. The CP-completion problem is to study whether we can assign values to the missing entries of a partial matrix (i.e.,…

We prove that multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NP-hard. Our list here includes: determining the feasibility of a system of bilinear equations, deciding whether a 3-tensor…

This article explores a new type of nonlinear complementarity problem, namely the horizontal tensor complementarity problem (HTCP), which is a natural extension of the horizontal linear complementarity problem studied in [12]. We extend the…

We study finite-dimensional representations of quantum affine algebras using q-characters. We prove the conjectures from math.QA/9810055 and derive some of their corollaries. In particular, we prove that the tensor product of fundamental…

Let X be a (connected and reduced) complex space. A q-collar of X is a bounded domain whose boundary is a union of a strongly q-pseudoconvex, a strongly q-pseudoncave and two flat (i.e. locally zero sets of pluriharmonic functions)…

The quantum eigenvalue problem arises in the study of the geometric measure of the quantum entanglement. In this paper, we convert the quantum eigenvalue problem to the Z-eigenvalue problem of a real symmetric tensor. In this way, the…

In the noisy tensor completion problem we observe $m$ entries (whose location is chosen uniformly at random) from an unknown $n_1 \times n_2 \times n_3$ tensor $T$. We assume that $T$ is entry-wise close to being rank $r$. Our goal is to…

The geometry of the set of restrictions of rank-one tensors to some of their coordinates is studied. This gives insight into the problem of rank-one completion of partial tensors. Particular emphasis is put on the semialgebraic nature of…

Multilinear systems of equations arise in various applications, such as numerical partial differential equations, data mining, and tensor complementarity problems. In this paper, we propose a homotopy method for finding the unique positive…

An exact charged solution with axial symmetry is obtained in the teleparallel equivalent of general relativity (TEGR). The associated metric has the structure function $G(\xi)=1-{\xi}^2-2mA{\xi}^3-q^2A^2{\xi}^4$. The fourth order nature of…

Given proper cones $K_1$ and $K_2$ in $\mathbb{R}^n$ and $\mathbb{R}^m$, respectively, an $m \times n$ matrix $A$ with real entries is said to be semipositive if there exists a $x \in K_1^{\circ}$ such that $Ax \in K_2^{\circ}$, where…

We give an algorithm for completing an order-$m$ symmetric low-rank tensor from its multilinear entries in time roughly proportional to the number of tensor entries. We apply our tensor completion algorithm to the problem of learning…