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In this paper we define the Boolean Lifting Property (BLP) for residuated lattices to be the property that all Boolean elements can be lifted modulo every filter, and study residuated lattices with BLP. Boolean algebras, chains, local and…
One of the simplest matrix-valued function with a single variable matrix $X$ is given by $A + BXC$. In this this note, analytical formulas are established for calculating the maximal and minimal ranks of $A + BXC$ when the rank of the…
One matrix structure in the area of monotone Boolean functions is defined here. Some of its combinatorial, algebraic and algorithmic properties are derived. On the base of these properties, three algorithms are built. First of them…
A characteristic-dependent linear rank inequality is a linear inequality that holds by ranks of subspaces of a vector space over a finite field of determined characteristic, and does not in general hold over other characteristics. In this…
We develop a notion of {\em inner rank} as a tool for obtaining lower bounds on the rank of matrix multiplication tensors. We use it to give a short proof that the border rank (and therefore rank) of the tensor associated with $n\times n$…
Recent results suggest that the use of ensembles in Statistical Mechanics may not be necessary for isolated systems, since typically the states of the Hilbert space would have properties similar to the ones of the ensemble. Nevertheless, it…
This paper discusses and evaluates ideas of data balancing and data augmentation in the context of mathematical objects: an important topic for both the symbolic computation and satisfiability checking communities, when they are making use…
In this paper, we review the problem of matrix completion and expose its intimate relations with algebraic geometry, combinatorics and graph theory. We present the first necessary and sufficient combinatorial conditions for matrices of…
In many classification problems, we want a classifier that is robust to a range of non-semantic transformations. For example, a human can identify a dog in a picture regardless of the orientation and pose in which it appears. There is…
The matrix rank and its positive versions are robust for small approximations, i.e. they do not decrease under small perturbations. In contrast, the multipartite tensor rank can collapse for arbitrarily small errors, i.e. there may be a gap…
Let K be an infinite field such that its characteristic is not 2. We show that, for every $A\in\mathcal{M}_n(K)$ such that $\mathrm{rank}(A)\geq n/2$, there exists $B\in\mathcal{M}_n(K)$ such that $B$ is similar to $A$ and $A+B$ is…
We construct the augmentation representation. It is a representation of the fundamental group of the link complement associated to an augmentation of the framed cord algebra. This construction connects representations of two link invariants…
This paper presents a bijection between ascent sequences and upper triangular matrices whose non-negative entries are such that all rows and columns contain at least one non-zero entry. We show the equivalence of several natural statistics…
Given a tract $F$ in the sense of Baker and Bowler and a matrix $A$ with entries in $F$, we define several notions of rank for $A$. In this way, we are able to unify and find conceptually satisfying proofs for various results about ranks of…
Some general properties of abstract relations are closely examined. These include generalizations of linearity, and properties based on `pinning' an inequality by a pair of families of endomorphisms.To each property we try to associate a…
We consider the problem of exact low-rank matrix completion from a geometric viewpoint: given a partially filled matrix M, we keep the positions of specified and unspecified entries fixed, and study how the minimal completion rank depends…
We investigate whether eigenvectors, also known as critical rank-one approximations, of a symmetric tensor can be used to increase or decrease its Waring rank. First, we study the variety of degree-d rank-r forms which admit an eigenvector…
Given an integral domain A we consider the set of all integral elements over A that can occur as an eigenvalue of a symmetric matrix over A. We give a sufficient criterion for being such an element. In the case where A is the ring of…
The authors and Fischer recently proved that any hereditary property of two-dimensional matrices (where the row and column order is not ignored) over a finite alphabet is testable with a constant number of queries, by establishing the…
Let A and B be normal matrices with coefficients that are continuous complex-valued functions on a topological space X that has the homotopy type of a CW complex, and suppose these matrices have the same distinct eigenvalues at each point…