Related papers: A Cubic Whitney and Further Developments in Geomet…
Finite element spaces by Whitney $k$-forms on cubical meshes in $\mathbb{R}^n$ are presented. Based on the spaces, compatible discretizations to $H\Lambda^k$ problems are provided, and discrete de Rham complexes and commutative diagrams are…
On an oriented 4-manifold, we examine the geometry that arises when the curvature operator of a Riemannian or Lorentzian metric $g$ commutes, not with its own Hodge star operator, but rather with that of another semi-Riemannian metric $h$…
We develop a geometric framework for generalized Milnor classifying spaces in the setting of diffeological spaces and infinite-dimensional geometry. Starting from Milnor's construction, we introduce spherical and projective models endowed…
In this technical note we give a purely geometric understanding of discrete torsion, as an analogue of orbifold Wilson lines for two-form tensor field potentials. In order to introduce discrete torsion in this context, we describe gerbes…
Condensed mathematics, developed by Clausen and Scholze over the last few years, proposes a generalization of topology with better categorical properties. It replaces the concept of a topological space by that of a condensed set, which can…
This manuscript introduces novel approaches to three phenomena. First, we extend the algebraic formulation of kinetic theory within the contact framework by making explicit the gauge freedom, thereby obtaining a formulation in which the…
We introduce higher analytic geometry, a novel framework extending Lurie's derived complex analytic spaces. This theory generalizes classical complex analytic geometry, enabling the study of derived K\"ahler spaces with non-trivial higher…
We adapt the Gradient Discretisation Method (GDM), originally designed for elliptic and parabolic partial differential equations, to the case of a linear scalar hyperbolic equations. This enables the simultaneous design and convergence…
In various areas of modern physics and in particular in quantum gravity or foundational space-time physics it is of great importance to be in the possession of a systematic procedure by which a macroscopic or continuum limit can be…
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…
The earlier approach is used for description of qubits and geometric phase parameters, the things critical in the area of topological quantum computing. The used tool, Geometric (Clifford) Algebra is the most convenient formalism for that…
Cohomology operations (including the cohomology ring) of a geometric object are finer algebraic invariants than the homology of it. In the literature, there exist various algorithms for computing the homology groups of simplicial complexes…
We introduce the dual Koenigs lattices, which are the integrable discrete analogues of conjugate nets with equal tangential invariants, and we find the corresponding reduction of the fundamental transformation. We also introduce the notion…
We introduce a notion of ellipticity of complexes of linear pseudodifferential operators acting on sections of $A$-Hilbert bundles over smooth manifolds, $A$ being a $C^*$-algebra. We prove that the cohomology groups of an $A$-elliptic…
Variables adapted to the quantum dynamics of spherically symmetric models are introduced, which further simplify the spherically symmetric volume operator and allow an explicit computation of all matrix elements of the Euclidean and…
We present a new method for the quantization of totally constrained systems including general relativity. The method consists in constructing discretized theories that have a well defined and controlled continuum limit. The discrete…
The geodesic flow on a finite discrete q-manifold with or without boundary is defined as as a permutation of its ordered q-simplices. This allows to define geodesic sheets and a notion of sectional curvature.
In this paper, we present a unified approach using model category theory and an associative law to compare some classic variants of the geometric realization functor.
Solving the classical equations of motion in general relativity recursively, we consider the metric of a spatially localized and stationary source of matter. Having in mind a star of general composition, we characterize it by means of its…
A generalization of the triangle inequality is introduced by a mapping similar to a t-conorm mapping. This generalization leads us to a notion for which we use the $\star$-metric terminology. We are interested in the topological space…