Related papers: Generalized column distances
A first order inference system, called R-calculus, is defined to develop the specifications. It is used to eliminate the laws which is not consistent with the user's requirements. The R-calculus consists of the structural rules, an axiom, a…
We define two new families of polynomials that generalize permanents and prove upper and lower bounds on their determinantal complexities comparable to the known bounds for permanents. One of these families is obtained by replacing…
Significant research has been carried out in the past half-century on defining generalised determinants for transformations between (typically real) vector spaces of different dimensions. We review three different generalisations of the…
We study the differential properties of generalized arc schemes, and geometric versions of Kolchin's Irreducibility Theorem over arbitrary base fields. As an intermediate step, we prove an approximation result for arcs by algebraic curves.
Recent progress in generalised geometry and extended field theories suggests a deep connection between consistent truncations and dualities, which is not immediately obvious. A prime example is generalised Scherk-Schwarz reductions in…
There are various notions of dimension in fractal geometry to characterise (random and non-random) subsets of $\mathbb R^d$. In this expository text, we discuss their analogues for infinite subsets of $\mathbb Z^d$ and, more generally, for…
Intersection bodies represent a remarkable class of geometric objects associated with sections of star bodies and invoking Radon transforms, generalized cosine transforms, and the relevant Fourier analysis. The main focus of this article is…
Invariants of generalized tensor fields on a line are classified using special polynomials P_mk^(-1/lambda) introduced here for this purpose. For the case of positive characteristic, a new invariant of formal power series, a width, is…
Correlations may affect propagation processes on complex networks. To analyze their effect, it is useful to build ensembles of networks constrained to have a given value of a structural measure, such as the degree-degree correlation $r$,…
Numerous results on self-reciprocal polynomials over finite fields have been studied. In this paper we generalize some of these to a-self reciprocal polynomials defined in [4]. We consider some properties of the divisibility of a-reciprocal…
This paper introduces a generalization of Convolutional Neural Networks (CNNs) from low-dimensional grid data, such as images, to graph-structured data. We propose a novel spatial convolution utilizing a random walk to uncover the relations…
Fractal geometry proved to be an effective mathematical tool for exploring real geographical space based on digital maps and remote sensing images. Whether the fractal theory tool can be applied to abstract geographical space has not been…
A generalized definition of the determinant of matrices is given, which is compatible with the usual determinant for square matrices and keeps many important properties, such as being an alternating multilinear function, keeping…
The extended persistence diagram is an invariant of piecewise linear functions, which is known to be stable under perturbations of functions with respect to the bottleneck distance as introduced by Cohen-Steiner, Edelsbrunner, and Harer. We…
We characterize linear lexicographic $p$-ary codes. Using this characterization, when $p \ge 3$, we determine the dimensions of linear lexicographic codes obtained from several bases including the standard basis, except for those of certain…
An occurrence of a classical pattern p in a permutation \pi is a subsequence of \pi whose letters are in the same relative order (of size) as those in p. In an occurrence of a generalized pattern, some letters of that subsequence may be…
It is known that maximum distance separable and maximum distance profile convolutional codes exist over large enough finite fields of any characteristic for all parameters $(n,k,\delta)$. It has been conjectured that the same is true for…
This is a concise overview of the definitions and properties of the linking number and its higher-order generalization, Milnor invariants.
We develop a general framework for reasoning about distances between transition systems with quantitative information. Taking as starting point an arbitrary distance on system traces, we show how this leads to natural definitions of a…
In this article, we study the generalized Poincare problem from the opposite perspective, by establishing lower bounds on the degree of the vector field in terms of invariants of the variety.