Related papers: On linear equations arising in Combinatorics (Part…
An algorithm is presented that generates sets of size equal to the degree of a given variety defined by a homogeneous ideal. This algorithm suggests a versatile framework to study various problems in combinatorial algebraic geometry and…
Inspired by the formulation of the Batalin-Vilkovisky method of quantization in terms of ``odd time'', we show that for a class of gauge theories which are first order in the derivatives, the kinetic term is bilinear in the fields, and the…
Symmetries and reductions of some algebraic equations are considered. Transformations that preserve the form of several algebraic equations, as well as transformations that reduce the degree of these equations, are described. Illustrative…
For any given sequence of integers there exists a quantum field theory whose Feynman rules produce that sequence. An example is illustrated for the Stirling numbers. The method employed here offers a new direction in combinatorics and graph…
This paper introduces Scavenger, the first theorem prover for pure first-order logic without equality based on the new conflict resolution calculus. Conflict resolution has a restricted resolution inference rule that resembles (a…
The main result is a generalization of Keller's recursion equation for finding a prime number given the previous primes. We also examine the convergence of the limit in Keller's equation and the convergence of the limit in the general…
A quantitative model of concurrent interaction is introduced. The basic objects are linear combinations of partial order relations, acted upon by a group of permutations that represents potential non-determinism in synchronisation. This…
We consider the homogenized linear feasibility problem, to find an $x$ on the unit sphere, satisfying $n$ line ar inequalities $a_i^Tx\ge 0$. To solve this problem we consider the centers of the insphere of spherical simpl ices, whose…
Linear first order systems of partial differential equations of the form $\nabla f = M\nabla g,$ where $M$ is a constant matrix, are studied on vector spaces over the fields of real and complex numbers, respectively. The Cauchy--Riemann…
We derive formulae for the calculation of Taylor coefficients of solutions to systems of Volterra integral equations, both linear and nonlinear, either without singularities or with singularities of Abel type and logarithmic type. We also…
This book is mainly an exposition of the author's works and his joint works with his former students on explicit representations of finite-dimensional simple Lie algebras, related partial differential equations, linear orthogonal algebraic…
This is the first installment of an exposition of an ACL2 formalization of elementary linear algebra, focusing on aspects of the subject that apply to matrices over an arbitrary commutative ring with identity, in anticipation of a future…
This is an introduction to algebraic combinatorics, written for a quarter-long graduate course. It starts with a rigorous introduction to formal power series with some combinatorial applications, then discusses integer partitions (proving…
In this work, we illustrate and explore the use of Taylor series as solutions of differential equations. For a large a number of classes of differential equations in the literature, there are plenty of sources where the well known Taylor…
Assuming that the Generalized Riemann Hypothesis (GRH) holds, we prove an explicit formula for the number of representations of an integer as a sum of $k\geq 5$ primes. Our error terms in such a formula improve by some logarithmic factors…
We introduce the notion of difference equation defined on a structured set. The symmetry group of the structure determines the set of difference operators. All main notions in the theory of difference equations are introduced as invariants…
Linear algebra's main concerns are sets of vectors, linear functions, subspaces, linear systems, matrices and concepts about those, such as whether the solution of linear system exists or is unique; a set of vectors is linearly independent…
A prime geodesic theorem is proven for singular geodesics in quotients of SL(4). This is a case where regularity assumptions of previous papers fail. As a consequence, the analysis becomes much more involved. For applications in number…
This paper studies a new class of integration schemes for the numerical solution of semi-explicit differential-algebraic equations of differentiation index 2 in Hessenberg form. Our schemes provide the flexibility to choose different…
In this article, I introduce a group-theoretical method to prove positivity of certain linear combinations (with coefficients generally lying in $\mathbb{C}$) of exponential functions under a set of semidefinite linear constraints. The…