Related papers: Arc Spaces and Rogers-Ramanujan Identities
This paper describes a method to find a connection between combinatorial identities and hypergeometric series with a number of examples. Combinatorial identities can often be written as hypergeometric series with unit argument. In a number…
We give a combinatorial proof of the first Rogers-Ramanujan identity by using two symmetries of a new generalization of Dyson's rank. These symmetries are established by direct bijections.
Following the method of combinatorial telescoping for alternating sums given by Chen, Hou and Mu, we present a combinatorial telescoping approach to partition identities on sums of positive terms. By giving a classification of the…
An identity by Ramanujan is expressed using polar coordinates, so that its proof reduces to the verification of an elementary trigonometric identity. This approach produces a few variations on Ramanujan's original identity.
We usually define an algebraic structure by a set, some operations defined on this set and some propositions that the algebraic structure must validate. In some cases, we can replace these propositions by an algorithm on terms constructed…
This article has been written for an educational magazine whose target audience consists of students and teachers of mathematics in universities, colleges and schools. It concerns a notion of duality between rectangles. A proof is given…
Consider a rational map from a projective space to a product of projective spaces, induced by a collection of linear projections. Motivated by the the theory of limit linear series and Abel-Jacobi maps, we study the basic properties of the…
This paper considers a higher-dimensional generalization of the notion of Ramanujan graphs, defined by Lubotzky, Phillips, and Sarnak. Specifically the Ramanujan property is studied for cubical complexes which are uniformized by an ordered…
This paper is concerned with the construction of a small, but non-trivial, example of a polynomial identity algebra, which we call the \emph{Jackson algebra}, that will be used in sequels to this paper to study non-commutative arithmetic…
Hecke symmetries generalize the usual tensor symmetry of vector spaces $v\otimes w\arrow w\otimes v$ as well as the symmetry of vector superspaces. To a Hecke symmetry $R$ there associates a quadratic algebra which can be interpreted as the…
We study non-associative twisted group algebras over $(\Z_2)^n$ with cubic twisting functions. We construct a series of algebras that extend the classical algebra of octonions in the same way as the Clifford algebras extend the algebra of…
We focus on writing closed forms of generating functions for the number of partitions with gap conditions as double sums starting from a combinatorial construction. Some examples of the sets of partitions with gap conditions to be discussed…
A new algorithm is proposed to accelerate RANSAC model quality calculations. The method is based on partitioning the joint correspondence space, e.g., 2D-2D point correspondences, into a pair of regular grids. The grid cells are mapped by…
We set up a combinatorial framework for inclusion-exclusion on the partitions into distinct parts to obtain an alternative generating function of partitions into distinct and non-consecutive parts. In connection with Rogers-Ramanujan…
We extend a holomorphic projection argument of our earlier work to prove a novel divisibility result for non-holomorphic congruences of Hurwitz class numbers. This result allows us to establish Ramanujan-type congruences for Hurwitz class…
Compositionality is a key property for dealing with complexity, which has been studied from many points of view in diverse fields. Particularly, the composition of individual computations (or programs) has been widely studied almost since…
A finite-dimensional unital and associative algebra over $\mathbb{R}$, or what we shall call simply "an algebra" in this paper for short, generalities the construction by which we derive the complex numbers by "adjoining an element $i$" to…
Results concerning recurrence and ergodicity are proved in an abstract Hilbert space setting based on the proof of Khintchine's recurrence theorem for sets, and on the Hilbert space characterization of ergodicity. These results are carried…
We consider the scator space - a hypercomplex, non-distributive hyperbolic algebra introduced by Fern\'andez-Guasti and Zald\'ivar. We discuss isometries of the scator space and find consequent method for treating them algebraically, along…
This article analyzes the geometric properties of an idempotent, non-associative algebraic structure that extends the Max-Times semiring. This algebraic structure is useful for studying systems of Max-Times and Max-Plus equations, employing…