Related papers: Umbral Calculus, Difference Equations and the Disc…
Linear algebraic primitives are at the core of many modern algorithms in engineering, science, and machine learning. Hence, accelerating these primitives with novel computing hardware would have tremendous economic impact. Quantum computing…
In this paper we describe a quantum algorithm to solve sparse systems of nonlinear differential equations whose nonlinear terms are polynomials. The algorithm is nondeterministic and its expected resource requirements are polylogarithmic in…
We propose a numerical scheme to solve the time dependent linear Schr\"odinger equation. The discretization is carried out by combining a Runge-Kutta time-stepping scheme with a finite element discretization in space. Since the…
The purpose of this work is to present a method based on the factorizations used in one dimensional quantum mechanics in order to find the symmetries of quantum and classical superintegrable systems in higher dimensions. We apply this…
General molecular dynamic approach, making possible direct calculation of eigen values and eigen functions for a quantum-mechanical system of an arbitrary symmetry is proposed. The method is based on analogy between discrete representation…
Differential complexes such as the de Rham complex have recently come to play an important role in the design and analysis of numerical methods for partial differential equations. The design of stable discretizations of systems of partial…
We study variational problems for curves approximated by B-spline curves. We show that, one can obtain discrete Euler-Lagrange equations, for the data describing the approximated curves. Our main application is to the curve completion…
We discuss the left-covariant 3-dimensional differential calculus on the quantum sphere $SU_q (2)/U(1) $. The $SU_q (2)$-spinor harmonics are treated as coordinates of the quantum sphere. We consider the gauge theory for the quantum group…
Differential calculus on discrete spaces is studied in the manner of non-commutative geometry by representing the differential calculus by an operator algebra on a suitable Krein space. The discrete analogue of a (pseudo-)Riemannian metric…
This paper explores the explicit design of quantum circuits for quantum simulation of partial differential equations (PDEs) with physical boundary conditions. These equations and/or their discretized forms usually do not evolve via unitary…
In quantum theory, the so-called "spinless Salpeter equation," the relativistic generalization of the nonrelativistic Schroedinger equation, is used to describe both bound states of scalar particles and the spin-averaged spectra of bound…
Two quantum Hopf structures for the Schr\"odinger algebra as well as their corresponding differential-difference realizations are presented. For each case a (space or time) discretization of the Schr\"odinger equation is deduced and the…
We propose an approach to quantize discrete networks (graphs with discrete edges). We introduce a new exact solution of discrete Schrodinger equation that is used to write the solution for quantum graphs. Formulation of the problem and…
In this paper we present a new Eulerian finite element method for the discretization of scalar partial differential equations on evolving surfaces. In this method we use the restriction of standard space-time finite element spaces on a…
A procedure for obtaining a "minimal" discretization of a partial differential equation, preserving all of its Lie point symmetries is presented. "Minimal" in this case means that the differential equation is replaced by a partial…
We numerically benchmark methods for computing harmonic maps into the unit sphere, with particular focus on harmonic maps with singularities. For the discretization we compare two different approaches, both based on Lagrange finite…
A procedure allowing to construct rigorously discrete as well as continuum deterministic evolution equations from stochastic evolution equations is developed using a Dirac's bra and ket notation. This procedure is an extension of an…
This paper presents a systematic study of the calculus of interval-valued functions and its application to interval differential equations. To this end, first, we introduce new interval arithmetic operations. Under new operations, the space…
Marginal problems naturally arise in a variety of different fields: basically, the question is whether some marginal/partial information is compatible with a joint probability distribution. To this aim, the characterization of marginal sets…
We establish sharp-in-time kernel and dispersive estimates for the Schr\"odinger equation on non-compact Riemannian symmetric spaces of any rank. Due to the particular geometry at infinity and the Kunze-Stein phenomenon, these properties…