Related papers: Vector Partition Identities for $2$D, $3$D and $n$…

A novel Neural Network architecture is proposed using the mathematically and physically rich idea of vector fields as hidden layers to perform nonlinear transformations in the data. The data points are interpreted as particles moving along…

We investigate the structure of two-dimensional partial cubes, i.e., of isometric subgraphs of hypercubes whose vertex set defines a set family of VC-dimension at most 2. Equivalently, those are the partial cubes which are not contractible…

We study the set of visible lattice points in multidimensional hypercubes. The problems we investigate mix together geometric, probabilistic and number theoretic tones. For example, we prove that almost all self-visible triangles with…

We characterize sequences of positive integers $(a_1,a_2,\ldots,a_n)$ for which the $2\times2$ matrix $\left( \begin{array}{cc} a_n&-1 1&0 \end{array} \right) \left( \begin{array}{cc} a_{n-1}&-1 1&0 \end{array} \right) \cdots \left(…

In high-dimensional classification problems, a commonly used approach is to first project the high-dimensional features into a lower dimensional space, and base the classification on the resulting lower dimensional projections. In this…

This paper gives an exposition of well known results on vector partition functions. The exposition is based on works of M. Brion, A. Szenes and M. Vergne and is geared toward explicit computer realizations. In particular, the paper presents…

We generalize the well known Glaisher partition bijection result. For given positive integers n, d, both greater than 1, we provide a rich family of bijections between the set of partitions of n where at least one part is divisible by d,…

An internal or friendly partition of a vertex set $V(G)$ of a graph $G$ is a partition to two nonempty sets $A\cup B$ such that every vertex has at least as many neighbours in its own class as in the other one. Motivated by Diwan's…

Given a finite set of points in $\mathbb{R}^d$, Tverberg's theorem guarantees the existence of partitions of this set into parts whose convex hulls intersect. We introduce a graph structured on the family of Tverberg partitions of a given…

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…

We investigate the definability (reducts) lattice of the order of integers and describe a sublattice generated by relations 'between', 'cycle', 'separation', 'neighbor', '1-codirection', 'order' and equality'. Some open questions are…

We introduce and study the notion of orthosymmetric spaces over an Archimedean vector lattice as a generalization of finite-dimentional Euclidean inner spaces. A special attention has been paid to linear operators on these spaces.

We revisit the following problem (along with its higher dimensional variant): Given a set $S$ of $n$ points inside an axis-parallel rectangle $U$ in the plane, find a maximum-area axis-parallel sub-rectangle that is contained in $U$ but…

We investigate integrable 2-dimensional Hamiltonian systems with scalar and vector potentials, admitting second invariants which are linear or quadratic in the momenta. In the case of a linear second invariant, we provide some examples of…

We suggest a generalization of vector calculus for the case of non-integer dimensional space. The first and second orders operations such as gradient, divergence, the scalar and vector Laplace operators for non-integer dimensional space are…

The space of degree d single-variable monic and centered complex polynomial vector fields can be decomposed into loci in which the vector fields have the same topological structure. We analyze the geometric structure of these loci and…

We prove various results in infinite-dimensional differential calculus which relate differentiability properties of functions and associated operator-valued functions (e.g., differentials). The results are applied in two areas: 1. in the…

We introduce a collection of convex polytopes associated to a torus-equivariant vector bundle on a smooth complete toric variety. We show that the lattice points in these polytopes correspond to generators for the space of global sections…

Non-point spatial objects (e.g., polygons, linestrings, etc.) are ubiquitous. We study the problem of indexing non-point objects in memory for range queries and spatial intersection joins. We propose a secondary partitioning technique for…

We devise an algorithm which allows one to count the number of Killing vectors for a Lorentzian manifold of dimension 3. Our algorithm relies on the principal traces of powers of the Ricci tensor and branches intricately according to the…