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András Bátkai

Publications and source records attributed to András Bátkai.

17 recordsLinked to original sources
arXiv2026

Über Nichts

This paper investigates the concept of Nothing from both philosophical and mathematical perspectives, distinguishing between absolute non-being (nihil) and relational negation as a principle of difference. It explores how mathematics formalizes absence through two fundamental concepts: the number zero and the empty set. The text traces the historical evolution of zero from a positional marker in Babylonian systems to an independent algebraic entity in Indian mathematics, emphasizing its structural roles as a neutral and absorbing element. Parallel to this, the empty set is analyzed not as a void, but as an existing axiomatic object that serves as the foundation for constructing the number system in modern set theory. Furthermore, the operational utility of active nothing is demonstrated through its role in algebraic manipulation -- such as adding zero to reveal structure -- and in physical conservation laws. Finally, drawing on the philosophy of Nikolaus von Kues and the discovery of non-Euclidean geometry, the paper interprets mathematics as a free, creative act of the human mind emerging ex nihilo.

math.HO↗
arXiv2025

Abstract boundary delay systems and application to network flow

This paper investigates the well-posedness and positivity of solutions to a class of delayed transport equations on a network. The material flow is delayed at the vertices and along the edges. The problem is reformulated as an abstract boundary delay equation, and well-posedness is proved by using the Staffans-Weiss theory. We also establish spectral theory for the associated delay operators and provide conditions for the positivity of the semigroup.

math.AP↗
arXiv2024

Math goes to Hollywood: Stereotypen in Filmen und Serien dekodieren

Stereotype Vorstellungen von Mathematik und Mathematiker*innen beeinflussen das Interesse von Jugendlichen an MINT-Fächern. Daher plädiert dieser Beitrag dafür, populäre Filme und erfolgreiche Serien nicht nur im Fremdsprachenunterricht, sondern auch im Mathematikunterricht einzusetzen. Durch die Analyse audiovisueller Medien im Unterricht können verzerrte Darstellungen über Mathematiker*innen erkannt, die Gründe dafür benannt und alternative Sichtweisen entwickelt werden. Dies trägt dazu bei, das Bild der Mathematik zu entmystifizieren und den Spaß an Mathe und Englisch zu fördern. Die Einbindung sozialwissenschaftlicher Themen steigert zudem die Motivation der Schüler*innen.

math.HO↗
arXiv2023

Mathematik und Ethik: eine Überlegung für zukünftige Lehrpersonen

Mathematical modelling and ethics have more touching points than most of us would like to admit. Everyday decisions are often reasoned by mathematical arguments. Mathematics teachers belong to those mathematically literate, who must point out mistakes in public discussions and reflect on them in their teaching to educate students to become critical thinkers. Ethical questions arise in mathematical modelling problems from two possible contexts. The first question is what values are reflected in the modelling problem. The other question is about the ethical implications of applying mathematical models to real problems.

math.HO↗
arXiv2016

Operator splitting for dissipative delay equations

We investigate Lie-Trotter product formulae for abstract nonlinear evolution equations with delay. Using results from the theory of nonlinear contraction semigroups in Hilbert spaces, we explain the convergence of the splitting procedure. The order of convergence is also investigated in detail, and some numerical illustrations are presented.

math.FA↗
arXiv2014

PDE approximation of large systems of differential equations

A large system of ordinary differential equations is approximated by a parabolic partial differential equation with dynamic boundary condition and a different one with Robin boundary condition. Using the theory of differential operators with Wentzell boundary conditions and similar theories, we give estimates on the order of approximation. The theory is demonstrated on a voter model where the Fourier method applied to the PDE is of great advantage.

math.FA↗
arXiv2012

Semigroups and linear partial differential equations with delay

We prove the equivalence of the well-posedness of a partial differential equation with delay and an associated abstract Cauchy problem. This is used to derive sufficient conditions for well-posedness, exponential stability and norm continuity of the solutions. Applications to a reaction-diffusion equation with delay are given.

math.FA↗
arXiv2012

Stability and Convergence of Product Formulas for Operator Matrices

We present easy to verify conditions implying stability estimates for operator matrix splittings which ensure convergence of the associated Trotter, Strang and weighted product formulas. The results are applied to inhomogeneous abstract Cauchy problems and to boundary feedback systems.

math.FA↗
arXiv2012

Operator splitting for nonautonomous delay equations

We provide a general product formula for the solution of nonautonomous abstract delay equations. After having shown the convergence we obtain estimates on the order of convergence for differentiable history functions. Finally, the theoretical results are demonstrated on some typical numerical examples.

math.FA↗
arXiv2011

Decomposition of operator semigroups on W*-algebras

We consider semigroups of operators on a W$^*$-algebra and prove, under appropriate assumptions, the existence of a Jacobs-DeLeeuw-Glicksberg type decomposition. This decomposition splits the algebra into a "stable" and "reversible" part with respect to the semigroup and yields, among others, a structural approach to the Perron-Frobenius spectral theory for completely positive operators on W$^*$-algebras.

math.OA↗
arXiv2011

On shape preserving semigroups

Motivated by positivity-, monotonicity-, and convexity preserving differential equations, we introduce a definition of shape preserving operator semigroups and analyze their fundamental properties. In particular, we prove that the class of shape preserving semigroups is preserved by perturbations and taking limits. These results are applied to partial delay differential equations.

math.FA↗
arXiv2011

Differential equation approximations of stochastic network processes: an operator semigroup approach

The rigorous linking of exact stochastic models to mean-field approximations is studied. Starting from the differential equation point of view the stochastic model is identified by its Kolmogorov equations, which is a system of linear ODEs that depends on the state space size ($N$) and can be written as $\dot u_N=A_N u_N$. Our results rely on the convergence of the transition matrices $A_N$ to an operator $A$. This convergence also implies that the solutions $u_N$ converge to the solution $u$ of $\dot u=Au$. The limiting ODE can be easily used to derive simpler mean-field-type models such that the moments of the stochastic process will converge uniformly to the solution of appropriately chosen mean-field equations. A bi-product of this method is the proof that the rate of convergence is $\mathcal{O}(1/N)$. In addition, it turns out that the proof holds for cases that are slightly more general than the usual density dependent one. Moreover, for Markov chains where the transition rates satisfy some sign conditions, a new approach for proving convergence to the mean-field limit is proposed. The starting point in this case is the derivation of a countable system of ordinary differential equations for all the moments. This is followed by the proof of a perturbation theorem for this infinite system, which in turn leads to an estimate for the difference between the moments and the corresponding quantities derived from the solution of the mean-field ODE.

math.DS↗
arXiv2010

Operator splitting for non-autonomous evolution equations

We provide general product formulas for the solutions of non-autonomous abstract Cauchy problems. The main technical tool is the application of evolution semigroup methods, allowing the direct application of existing results on autonomous problems. The results are then illustrated by the example of a imaginary time Schrödinger equation with time dependent potential. We also obtain convergence rates for the Strang-splitting applied to this problem.

math.FA↗
arXiv2009

Operator splittings and spatial approximations for evolution equations

The convergence of various operator splitting procedures, such as the sequential, the Strang and the weighted splitting, is investigated in the presence of a spatial approximation. To this end a variant of Chernoff's product formula is proved. The methods are applied to abstract partial delay differential equations.

math.FA↗