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`Umbral calculus' deals with representations of the canonical commutation relations. We present a short exposition of it and discuss how this calculus can be used to discretize continuum models and to construct representations of Lie…
The following work demonstrates the viability of Poincar\'e symmetry in a discrete universe. We develop the technology of the discrete principal Poincar\'e bundle to describe the pairing of (1) a hypercubic lattice `base manifold' labeled…
Geometric discretisation draws analogies between discrete objects and operations on a complex with continuum ones on a manifold. We generalise the theory to the cubic case and incorporate metric, by adding volume factors to our discrete…
We define discrete Hamiltonian systems in the framework of discrete embeddings. An explicit comparison with previous attempts is given. We then solve the discrete Helmholtz's inverse problem for the discrete calculus of variation in the…
Given a planar graph derived from a spherical, euclidean or hyperbolic tessellation, one can define a discrete curvature by combinatorial properties, which after embedding the graph in a compact 2d-manifold, becomes the Gaussian curvature.
We propose the difference discrete variational principle in discrete mechanics and symplectic algorithm with variable step-length of time in finite duration based upon a noncommutative differential calculus established in this paper. This…
We define in a global manner the notion of a connective structure for a gerbe on a space X. When the gerbe is endowed with trivializing data with respect to an open cover of X, we describe this connective structure in two separate ways,…
On contact manifolds we describe a notion of (contact) finite-type for linear partial differential operators satisfying a natural condition on their leading terms. A large class of linear differential operators are of finite-type in this…
We construct numerical basis function sets on a lattice, whose spatial extension is scalable from single lattice sites to the continuum limit. They allow us to compute small and large bound states with comparable, moderate effort. Adopting…
The hypothesis of a discrete fabric of the universe--the "Planck scale"--is always on stage, since it solves mathematical and conceptual problems in the infinitely small. However, it clashes with special relativity, which is designed for…
In this paper we relate the study of actions of discrete groups over connected manifolds to that of their orbit spaces seen as differentiable stacks. We show that the orbit stack of a discrete dynamical system on a simply connected manifold…
Curvature and torsion of linear transports along paths in, respectively, vector bundles and the tangent bundle to a differentiable manifold are defined and certain their properties are derived.
Recently, a set of tools has been developed with the purpose of the study of Quantum Gravity. Until now, there have been very few attempts to put these tools into a rigorous mathematical framework. This is the case, for example, of the so…
We summarize ideas and methods that apply to rigidity and connectivity percolation. These include: constraint counting concepts, exact calculations on diluted trees and numerical results on diluted lattices using matching algorithms. The…
We construct a connection and a curving on a bundle gerbe associated with lifting a structure group of a principal bundle to a central extension. The construction is based on certain structures on the bundle, i.e. connections and…
The connection between curvature and topology is a very well-studied theme in the subject of differential geometry. By suitably defining curvature on networks, the study of this theme has been extended into the domain of network analysis as…
Some particular examples of classical and quantum systems on the lattice are solved with the help of orthogonal polynomials and its connection to continuous models are explored.
The main purpose of this paper is to study the lattice structure of variable precision rough sets. The notion of variation in precision of rough sets have been further extended to variable precision rough set with variable classification…
We study the differential forms over the frame bundle of the based loop space. They are stochastics in the sense that we put over this frame bundle a probability measure. In order to understand the curvatures phenomena which appear when we…
Tangent categories provide an axiomatic approach to key structural aspects of differential geometry that exist not only in the classical category of smooth manifolds but also in algebraic geometry, homological algebra, computer science, and…