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Given a minimum measurable length underlying spacetime, the latter may be effectively regarded as discrete, at scales of order the Planck length. A systematic discretization of continuum physics may be effected most efficiently through the…
In this paper we survey a recent progress on continuous frames inspired by the solution of the Kadison-Singer problem by Marcus, Spielman, and Srivastava. We present an extension of Lyapunov's theorem for discrete frames due to Akemann and…
In this paper we consider on the notion of continuous frame of subspace and define a new concept of continuous frame, entitled {\it continuous atomic resolution of identity}, for arbitrary Hilbert space $\h$ which has a countable…
We apply the consistent discretization scheme to general relativity particularized to the Gowdy space-times. This is the first time the framework has been applied in detail in a non-linear generally-covariant gravitational situation with…
We generalize the sparsification algorithm of Batson, Spielman and Srivastava, making one part of the result dimension-independent. In particular, we recover discretization inequalities in $L_2$- and sup-norms on general finite-dimensional…
I point out that the authors' interpretation of their calculations differs from the standard interpretation, described in Sect. 84 of Landau-Lifshitz book. This casts doubt on the authors' claim that Sagnac effect "arises also in…
We define the notion of {\em rational presentation of a complete metric space} in order to study metric spaces from the algorithmic complexity point of view. In this setting, we study some presentations of the space $\czu$ of uniformly…
A time-changed discretization for the Dirac equation is proposed. More precisely, we consider a Dirac equation with discrete space and continuous time perturbed by a time-dependent diffusion term $\sigma^2Ht^{2H-1}$ that seamlessly…
The partial Hamiltonian analysis of the actions presented in the paper by M. Chaichian, M. Oksanen, A. Tureanu (Eur. Phys. J. C 71, 1657 (2011)) is incorrect; the true algebra of constraints differs from what they claim for their choice of…
A discretization of a continuum theory with constraints or conserved quantities is called mimetic if it mirrors the conserved laws or constraints of the continuum theory at the discrete level. Such discretizations have been found useful in…
This paper concerns the study and resolution of wave equations in the space of Schwartz distributions. Wave phenomena are widespread in many branches of physics and chemistry, such as optics, gravitation, quantum mechanics, chemical waves…
This paper is concerned with minimization of a fourth-order linearized Canham-Helfrich energy subject to Dirichlet boundary conditions on curves inside the domain. Such problems arise in the modeling of the mechanical interaction of…
A high-accuracy time discretization is discussed to numerically solve the nonlinear fractional diffusion equation forced by a space-time white noise. The main purpose of this paper is to improve the temporal convergence rate by modifying…
In this thesis, we study the Teichm\"uller geodesic flow on the space of translation surfaces by introducing two related discrete-time dynamical systems. First, we discuss the Rauzy-Veech induction, highlighting its connections to interval…
Our aim is to study the backward problem, i.e. recover the initial data from the terminal observation, of the subdiffusion with time dependent coefficients. First of all, by using the smoothing property of solution operators and a…
The present work aims at the application of finite element discretizations to a class of equilibrium problems involving moving constraints. Therefore, a Moreau--Yosida based regularization technique, controlled by a parameter, is discussed…
$R$-limited functions are multivariate generalization of band-limited functions whose Fourier transforms are supported within a compact region $R\subset\mathbb{R}^{n}$. In this work, we generalize sampling and interpolation theorems for…
We analyze the method for calculation of properties of non-relativistic quantum systems based on exact diagonalization of space-discretized short-time evolution operators. In this paper we present a detailed analysis of the errors…
This article reports on the confluence of two streams of research, one emanating from the fields of numerical analysis and scientific computation, the other from topology and geometry. In it we consider the numerical discretization of…
The paper studies the sampling discretization problem for integral norms on subspaces of $L^p(\mu)$. Several close to optimal results are obtained on subspaces for which certain Nikolskii-type inequality is valid. The problem of norms…