Related papers: Enumeration of Splitting Subspaces over Finite Fie…

We develop the theory of frames and Parseval frames for finite-dimensional vector spaces over the binary numbers. This includes characterizations which are similar to frames and Parseval frames for real or complex Hilbert spaces, and the…

In this paper we extend our findings in [3] and answer further questions regarding continuity and discontinuity of seminorms on infinite-dimensional vector spaces.

Diagram semigroups are interesting algebraic and combinatorial objects, several types of them originating from questions in computer science and in physics. Here we describe diagram semigroups in a general framework and extend our…

A theorem of Wiegerinck asserts that the Bergman space of an open subset of the complex numbers is either infinite-dimensional or trivial. Recently, this has been generalized to holomorphic vector bundles over the projective line by the…

In this survey we discuss the results on the finitistic dimension of various stratified algebras. We describe what is already known, present some recent estimates, and list some open problems.

Saturating sets are combinatorial objects in projective spaces over finite fields that have been intensively investigated in the last three decades. They are related to the so-called covering problem of codes in the Hamming metric. In this…

We propose a simple, powerful, and flexible machine learning framework for (i) reducing the search space of computationally difficult enumeration variants of subset problems and (ii) augmenting existing state-of-the-art solvers with…

We solve an elementary number theory problem on sums of fractional parts, using methods from group theory. We apply our result to deduce the finiteness of certain monodromy representations.

Let $f_{1}, \ldots, f_{k}$ be polynomials defining an algebraic set in affine $n$-space over a finite field. Suppose $k>n$. We prove that there exists a system of polynomials $g_{1}, \ldots, g_{n}$, each being a linear combination with…

We show that a wide class of geometrically defined overdetermined semilinear partial differential equations may be explicitly prolonged to obtain closed systems. As a consequence, in the case of linear equations we extract sharp bounds on…

We discuss the concept of invariant subspaces for unbounded linear operators, point out some shortcomings of known definitions, and propose our own.

We study the question how many subgroups, cosets or subspaces are needed to cover a finite Abelian group or a vector space if we have some natural restrictions on the structure of the covering system. For example we determine, how many…

When a problem has more than one solution, it is often important, depending on the underlying context, to enumerate (i.e., to list) them all. Even when the enumeration can be done in polynomial delay, that is, spending no more than…

We investigate the finite subgroups that occur in the Hamiltonian quaternion algebra over the real subfield of cyclotomic fields. When possible, we investigate their distribution among the maximal orders.

Let A be a k-vector space of dimension a. A subvector space M of End(A) is said to be of rank r if every non-zero f in M has rank r. The problem considered in this paper is to determine l(r;a) the maximal dimension of a rank r subspace of…

Recently, the second and third author showed that complete geometric graphs on $2n$ vertices in general cannot be partitioned into $n$ plane spanning trees. Building up on this work, in this paper, we initiate the study of partitioning into…

We discuss the notions of indices of vector fields and 1-forms and their generalizations to singular varieties and varieties with actions of finite groups, as well as indices of collections of vector fields and 1-forms.

Although a good portion of elementary linear algebra concerns itself with matrices over a field such as $\mathbb{R}$ or $\mathbb{C}$, many combinatorial problems naturally surface when we instead work with matrices over a finite field. As…

We give a combinatorial model for the bounded derived category of graded modules over the dual numbers in terms of arcs on the integer line with a point at infinity. Using this model we describe the lattice of thick subcategories of the…

We give necessary and sufficient conditions for the sum of n subspaces of a Hilbert space to be closed. We also present various properties of n-tuples of subspaces with closed sum.