Related papers: The distance formula in algebraic spacetime theori…
In 1997, Erd\H{o}s asked whether for arbitrarily large $n$ there exists a set of $n$ points in $\mathbb{R}^2$ that determines $O(\frac{n}{\sqrt{\log n}})$ distinct distances while satisfying the local constraint that every 4-point subset…
In this article we introduce a notion of normalized angle for Lorentzian pre-length spaces. This concept allows us to prove some equivalences to the definition of timelike curvature bounds from below for Lorentzian pre-length spaces.…
The matching distance is a pseudometric on multi-parameter persistence modules, defined in terms of the weighted bottleneck distance on the restriction of the modules to affine lines. It is known that this distance is stable in a reasonable…
We motivate and construct a mathematical theory for the separation of space and time in general relativity. The formalism only requires a single observer and an optional choice of reference frame at each instant. As the splitting is done…
Motivated by an intention to apply partial wave analysis to systems containing photons, we construct eigenvectors of the distance of separation between two photons. A choice is made that makes the case of two photons most closely resemble…
This thesis is developed in the context of the spin-foam approach to quantum gravity; all results are concerned with the Lorentzian theory and with semiclassical methods. A correspondence is given between Majorana 2-spinors and time-like…
I show in this letter that it is possible to construct a Hamiltonian description for Lorentzian General Relativity in terms of two real $SO(3)$ connections. The constraints are simple polynomials in the basic variables. The present…
We define a one-parameter family of canonical volume measures on Lorentzian (pre-)length spaces. In the Lorentzian setting, this allows us to define a geometric dimension - akin to the Hausdorff dimension for metric spaces - that…
We deduce Tolman's formula of luminosity-distance in Cosmology from Poincare's definition of light-distance with Lorentz Transformation (LT).In Minkowskian metric, if distance is proper time (as it is often argued) then light-distance must…
We refine a recent distributional notion of d'Alembertian of a signed Lorentz distance function to an achronal set in a metric measure spacetime obeying the timelike measure contraction property. We show precise representation formulas and…
In this paper, we will give a new perspective to the Cameron-Martin-Maruyama-Girsanov formula by giving a totally algebraic proof to it. It is based on the exponentiation of the Malliavin-type differentiation and its adjointness.
We provide a new proof of the splitting theorems from Lorentzian geometry, in which simplicity is gained by sacrificing linearity of the d'Alembertian to recover ellipticity. We exploit a negative homogeneity (non-uniformly) elliptic…
Given two trace class operators A and B on a separable Hilbert space we provide an upper bound for the Hausdorff distance of their spectra involving only the distance of A and B in operator norm and the singular values of A and B. By…
We obtain the following version of Lidskii theorem. Let L, M, N be p-dimensional subspaces in R^n. Let \psi_j be the angles between L and M, let \phi_j be the angles between M and N, and let \theta_j be the angles between L and N. Consider…
Koplienko gave a trace formula for perturbations of self-adjoint operators by operators of Hilbert-Schmidt class $\mathcal{B}_2(\mathcal{H})$. Recently Gesztesy, Pushnitski and Simon gave an alternative proof of the trace formula when the…
The inner product provides a conceptually and algorithmically simple method for calculating the comoving distance between two cosmological objects given their redshifts, right ascension and declination, and arbitrary constant curvature. The…
We propose a new algorithm for computing the luminosity distance in the flat universe with a cosmological constant based on Shchigolev's homotopy perturbation method, where the optimization idea is applied to prevent the arbitrariness of…
An analogy with real Clifford algebras on even-dimensional vector spaces suggests to assign a couple of space and time dimensions modulo 8 to any algebra (represented over a complex Hilbert space) containing two self-adjoint involutions and…
In the present paper we calculate the Gromov-Hausdorff distance between an arbitrary simplex (a metric space all whose non-zero distances are the same) and a finite metric space whose non-zero distances take two distinct values (so-called…
The luminosity-angular distance relation for an expanding universe is given by $D_L=D_A(1+z)^2$. This relation is commonly proven by geometrical considerations from the source point of view assuming an expanding universe. In this note, the…