arXiv subjects

Ferenc Bartha

Publications and source records attributed to Ferenc Bartha.

3 recordsLinked to original sources

arXiv2020

Central diagonal sections of the $n$-cube

We prove that the volume of central hyperplane sections of a unit cube in $\mathbb{R}^n$ orthogonal to a diameter of the cube is a strictly monotonically increasing function of the dimension for $n\geq 3$. Our argument uses an integral formula that goes back to P\'olya \cite{P} (see also \cite{H} and \cite{B86}) for the volume of central sections of the cube, and Laplace's method to estimate the asymptotic behaviour of the integral. First we show that monotonicity holds starting from some specific $n_0$. Then, using interval arithmetic (IA) and automatic differentiation (AD), we compute an explicit bound for $n_0$, and check the remaining cases between $3$ and $n_0$ by direct computation.

math.MG↗

arXiv2017

Compile-Time Extensions to Hybrid ODEs

Reachability analysis for hybrid systems is an active area of development and has resulted in many promising prototype tools. Most of these tools allow users to express hybrid system as automata with a set of ordinary differential equations (ODEs) associated with each state, as well as rules for transitions between states. Significant effort goes into developing and verifying and correctly implementing those tools. As such, it is desirable to expand the scope of applicability tools of such as far as possible. With this goal, we show how compile-time transformations can be used to extend the basic hybrid ODE formalism traditionally supported in hybrid reachability tools such as SpaceEx or Flow*. The extension supports certain types of partial derivatives and equational constraints. These extensions allow users to express, among other things, the Euler-Lagrangian equation, and to capture practically relevant constraints that arise naturally in mechanical systems. Achieving this level of expressiveness requires using a binding time-analysis (BTA), program differentiation, symbolic Gaussian elimination, and abstract interpretation using interval analysis. Except for BTA, the other components are either readily available or can be easily added to most reachability tools. The paper therefore focuses on presenting both the declarative and algorithmic specifications for the BTA phase, and establishes the soundness of the algorithmic specifications with respect to the declarative one.

cs.PL↗

arXiv2002

Time evolution in the Morse potential using supersymmetry: dissociation of the NO molecule

We present an algebraic method for treating molecular vibrations in the Morse potential perturbed by an external laser field. By the help of a complete and normalizable basis we transform the Schrödinger equation into a system of coupled ordinary differential equations. We apply our method to calculate the dissociation probability of the NO molecule excited by chirped laser pulses. The dependence of the molecular dipole-moment on the interatomic separation is determined by a quantum-chemical method, and the corresponding transition dipole moments are given by approximate analytic expressions. These turn out to be very small between neighboring stationary states around the vibrational quantum number $m=42$, therefore we propose to use additional pulses in order to skip this trapping state, and to obtain a reasonable dissociation probability.

quant-ph↗