Related papers: Laplacians on discrete and quantum geometries
The geometric formulation of quantum mechanics is a very interesting field of research which has many applications in the emerging field of quantum computation and quantum information, such as schemes for optimal quantum computers. In this…
Computational complexity is a new quantum information concept that may play an important role in holography and in understanding the physics of the black hole interior. We consider quantum computational complexity for $n$ qubits using…
I introduce a family of closeness functions between causal Lorentzian geometries of finite volume and arbitrary underlying topology. When points are randomly scattered in a Lorentzian manifold, with uniform density according to the volume…
Coordination geometries describe how the neighbours of a central particle are arranged around it. Such geometries can be thought to lie in an abstract topological space; a model of this space could provide a mathematical basis for…
This paper is a contribution to the development of a framework, to be used in the context of semiclassical canonical quantum gravity, in which to frame questions about the correspondence between discrete spacetime structures at "quantum…
Starting from the working hypothesis that both physics and the corresponding mathematics have to be described by means of discrete concepts on the Planck-scale, one of the many problems one has to face is to find the discrete protoforms of…
Inspired by the work of Z. Lu and G. Tian \cite{lutian}, in this paper we address the problem of studying those \K\ manifolds satisfying the $\Delta$-property, i.e. such that on a neighborhood of each of its points the $k$-th power of the…
Suppose a Lagrangian is constructed from its fields and their derivatives. When the field configuration is a distribution, it is unambiguously defined as the limit of a sequence of smooth fields. The Lagrangian may or may not be a…
We present an axiomatic approach to finite- and infinite-dimensional differential calculus over arbitrary infinite fields (and, more generally, suitable rings). The corresponding basic theory of manifolds and Lie groups is developed.…
For real hyperbolic spaces, the dynamics of individual isometries and the geometry of the limit set of nonelementary discrete isometry groups have been studied in great detail. Most of the results were generalised to discrete isometry…
Cluster varieties are geometric objects that have recently found applications in several areas of mathematics and mathematical physics. This thesis studies the geometry of a large class of cluster varieties associated to compact oriented…
We construct some intrinsically defined discrete model of the magnetic Laplacian. The existence and uniqueness of solutions of the Dirichlet problem for the difference Poisson type equation are proved. We study in detail properties of the…
Using Legendrian immersions and, in particular, Legendre curves in odd dimensional spheres and anti De Sitter spaces, we provide a method of construction of new examples of Hamiltonian-minimal Lagrangian submanifolds in complex projective…
In a series of publications we developed "differential geometry" on discrete sets based on concepts of noncommutative geometry. In particular, it turned out that first order differential calculi (over the algebra of functions) on a discrete…
A super-Laplacian is a set of differential operators in superspace whose highest-dimensional component is given by the spacetime Laplacian. Symmetries of super-Laplacians are given by linear differential operators of arbitrary finite degree…
We informally review the construction of spacetime geometries with multifractal and, more generally, multiscale properties. Based on fractional calculus, these continuous spacetimes have their dimension changing with the scale; they display…
Given a complete non-compact surface embedded in R^3, we consider the Dirichlet Laplacian in a layer of constant width about the surface. Using an intrinsic approach to the layer geometry, we generalise the spectral results of an original…
In this paper, we introduce and study a noncommutative extension of the Gross Laplacian, called quantum Gross Laplacian. Then, applying the quantum Gross Laplacian to the particular case where the operator is the multiplication operator, we…
In $\mathbb{R}^3$, the primal and dual constructions yield completely different discrete Laplacians for tetrahedral meshes.In this article, we prove that the discrete Laplacian satisfies the Euler-Lagrange equation of the Dirichlet energy…
We characterize harmonic spaces in terms of the dimensions of various spaces of radial eigen-spaces of the Laplacian $\Delta^0$ on functions and the Laplacian $\Delta^1$ on 1-forms. We examine the nature of the singularity as the geodesic…