Jürgen Jost - Academia.edu (original) (raw)

Papers by Jürgen Jost

$Research paper thumbnail of The differential equation <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>u</mi><mo>=</mo><mn>8</mn><mi>π</mi><mo>−</mo><mn>8</mn><mi>π</mi><mi>h</mi><mi>exp</mi><mo>⁡</mo><mi>u</mi></mrow><annotation encoding="application/x-tex">Δu = 8π- 8πh\exp {u}</annotation></semantics></math>Δu=8π−8πhexpu on a compact Riemann surface$

arXiv (Cornell University), 1997

Let MMM be a compact Riemann surface, h(x)h(x)h(x) a positive smooth function on MMM. In this paper, we... more Let MMM be a compact Riemann surface, h(x)h(x)h(x) a positive smooth function on MMM. In this paper, we consider the functional J(u)={1/2}\int \sb{M}|\bigtriangledown u|\sp 2 + 8\pi \int\sb{M}u -8\pi \log\int\sb{M}h\exp {u}$$. We give a sufficient condition under which JJJ achieves its minimum.

arXiv (Cornell University), May 26, 2020

At a mixed Nash equilibrium, the payoff of a player does not depend on her own action, as long as... more At a mixed Nash equilibrium, the payoff of a player does not depend on her own action, as long as her opponent sticks to his. In a periodic strategy, a concept developed in a previous paper [3], in contrast, the own payoff does not depend on the opponent's action. Here, we generalize this to multi-player simultaneous perfect information strategic form games. We show that also in this class of games, there always exists at least one periodic strategy, and we investigate the mathematical properties of such periodic strategies. In addition, we demonstrate that periodic strategies may exist in games with incomplete information; we shall focus on Bayesian games. Moreover we discuss the differences between the periodic strategies formalism and cooperative game theory. In fact, the periodic strategies are obtained in a purely non-cooperative way, and periodic strategies are as cooperative as the Nash equilibria are. Finally, we incorporate the periodic strategies in an epistemic game theory framework, and discuss several features of this approach.

Zenodo (CERN European Organization for Nuclear Research), Jan 11, 2022

We systematically analyze the conceptual structure of modern physics, i.e., the theory of relativ... more We systematically analyze the conceptual structure of modern physics, i.e., the theory of relativity and quantum mechanics and quantum field theory, from the perspective of Leibniz's three basic principles of identity, sufficient reason and continuity. We find that Leibniz's way of thinking can still provide insight into fundamental problems of physics, such as the nature of physical constants, the relation between physical invariances and covariant coordinate representations, the role of the Planck scale and quantum mechanical indeterminism and the measurement problem. Leibniz himself could not possess all those insights, simply because physics was not yet sufficiently developed then, but he had already forged some conceptual tools with which some fundamental problems of modern physics can be addressed. *** * Acknowledgement: I am grateful to Richard Arthur and Vincenzo De Risi for critical comments and extensive discussions which provided me with much insight.

Springer eBooks, 2014

Die ausere Geometrie einer Immersion $X:U\to \mathbb{E}$ beschreibt die Lage des Tangentialraum... more Die ausere Geometrie einer Immersion $X:U\to \mathbb{E}$ beschreibt die Lage des Tangentialraums T u und des Normalraums $ {N_u} = {({T_u})^ \bot } $ im umgebenden Raum $\mathbb{E}$. Wie die erste Fundamentalform g zur inneren Geometrie, so gehort die zweite Fundamentalform h zur auseren. Sie beschreibt, wie der Tangentialraum T u in Abhangigkeit von u variiert und ubernimmt damit die Aufgabe der Krummung im Fall von Kurven.

Peking University series in mathematics, Dec 28, 2017

Extending work of Caffarelli-Yang and Tarantello, we present a variational existence proof for tw... more Extending work of Caffarelli-Yang and Tarantello, we present a variational existence proof for two-vortex solutions of the periodic Chern-Simons Higgs model and analyze the asymptotic behavior of these solutions as the parameter coupling the gauge field with the scalar field tends to 0. * dg-ga/9710006

Minimalflachen lassen sich z.B. durch Seifenhaute realisieren, denn diese haben die physikalische... more Minimalflachen lassen sich z.B. durch Seifenhaute realisieren, denn diese haben die physikalische Eigenschaft, sich moglichst klein zusammenzuziehen. Taucht man daher eine Drahtschlinge in Seifenlauge, so hat der sich ausbildende Seifenfilm minimalen Flacheninhalt in folgendem Sinne: Bei allen kleinen Deformationen, die von derselben Drahtschlinge berandet werden, wird der Flacheninhalt groser. Man kann also durch Seifenfilme experimentell Minimalflachen herstellen. Ist insbesondere die berandende Drahtschlinge eine nicht-ebene Kurve, so bildet der Seifenfilm eine nicht-ebene Minimalflache aus. Diese Beobachtung wurde um 1849 durch die Seifenhaut-Experimente des belgischen Physikers Plateau verbreitet.

This position paper looks into the formation of language and shows ties between structural proper... more This position paper looks into the formation of language and shows ties between structural properties of the words in the English language and their polysemy. Using Ollivier-Ricci curvature over a large graph of synonyms to estimate polysemy it shows empirically that the words that arguably are easier to pronounce also tend to have multiple meanings.

Journal of Zoological Systematics and Evolutionary Research, Aug 1, 2004

We propose a model that allows to deal with a group of populations constituting a species and to ... more We propose a model that allows to deal with a group of populations constituting a species and to define those terms that are crucial in a description of those factors that denote the various types of population-composition that may occur within a reproductive unit. Here, a population is understood in the sense of population biology, that is, a concrete set of organisms. The concept of a species thus acquires a meaning beyond the concept of a population that makes it relevant in the study of micro-evolutionary mechanisms. Zusammenfassung Wir betrachten eine Art als den dynamischen Ausgleich zwischen divergierenden (durch intrinsische Mutationen und differentiellen externen Umweltdruck hervorgerufenen) und konvergierenden (durch sexuelle Rekombination bewirkten) Tendenzen innerhalb von oder zwischen Populationen. Eine Population ist hierbei als eine konkrete Gruppe von Organismen im Sinne der Populationsbiologie verstanden. Der Artbegriff erlangt hierdurch eine Bedeutung, die weit uber den Populationsbegriff hinausreicht und fur die Untersuchung mikroevolutiver Mechanismen relevant ist.

arXiv (Cornell University), Jan 3, 2019

We investigate a parabolic-elliptic system for maps (u, v) from a compact Riemann surface M into ... more We investigate a parabolic-elliptic system for maps (u, v) from a compact Riemann surface M into a Lorentzian manifold N × R with a warped product metric. That system turns the harmonic map type equations into a parabolic system, but keeps the v-equation as a nonlinear second order constraint along the flow. We prove a global existence result of the parabolic-elliptic system by assuming either some geometric conditions on the target Lorentzian manifold or small energy of the initial maps. The result implies the existence of a Lorentzian harmonic map in a given homotopy class with fixed boundary data. Résumé: Nousétudions un système parabolique-elliptique pour les applications (u, v) d'une surface Riemannienne compacte M dans une varié té lorentzienne N ×R avec une mtrique de produit déformée. Nous transformons leséquations de type application harmonique en un systéme parabolique, mais conservons l'équation de v comme contrainte non linéaire du second ordre le long du flux. Nous démontrons un résultat global de l'existence du systéme parabolique-elliptique en supposant soit certaines conditions géométriques sur la varié té lorentzienne ou la petitesse de l'énergie des valeurs initiales. Le résultat implique l'existence d'une application harmonique lorentzienne dans une classe d'homotopie donnée avec des données a bord fixe. A critical point of the energy functional E in C 2 (M, N) is called a harmonic map. By Nash's embedding theorem [32], we can embed N isometrically into some Euclidian space R K and the corresponding Euler-Lagrange equation is ∆ h u = A(u)(∇u, ∇u),

arXiv (Cornell University), May 22, 2014

arXiv (Cornell University), Jan 15, 2013

We prove an analogue of Yau's Caccioppoli-type inequality for nonnegative subharmonic functions o... more We prove an analogue of Yau's Caccioppoli-type inequality for nonnegative subharmonic functions on graphs. We then obtain a Liouville theorem for harmonic or non-negative subharmonic functions of class L q , 1 ≤ q < ∞, on any graph, and a quantitative version for q > 1. Also, we provide counterexamples for Liouville theorems for 0 < q < 1.

Journal of Mathematical Biology, May 2, 2019

In this paper, we provide a complete mathematical construction for a stochastic leakyintegrate-an... more In this paper, we provide a complete mathematical construction for a stochastic leakyintegrate-and-fire model (LIF) mimicking the interspike interval (ISI) statistics of a stochastic FitzHugh-Nagumo neuron model (FHN) in the excitable regime, where the unique fixed point is stable. Under specific types of noises, we prove that there exists a global random attractor for the stochastic FHN system. The linearization method is then applied to estimate the firing time and to derive the associated radial equation representing a LIF equation. This result confirms the previous prediction in Ditlevsen and Greenwood (J Math Biol 67(2):239-259, 2013) for the Morris-Lecar neuron model in the bistability regime consisting of a stable fixed point and a stable limit cycle.

Understanding complex systems, 2017

Springer Complexity is an interdisciplinary program publishing the best research and academic-lev... more Springer Complexity is an interdisciplinary program publishing the best research and academic-level teaching on both fundamental and applied aspects of complex systemscutting across all traditional disciplines of the natural and life sciences, engineering, economics, medicine, neuroscience, social and computer science. Complex Systems are systems that comprise many interacting parts with the ability to generate a new quality of macroscopic collective behavior the manifestations of which are the spontaneous formation of distinctive temporal, spatial or functional structures. Models of such systems can be successfully mapped onto quite diverse "real-life" situations like the climate, the coherent emission of light from lasers, chemical reaction-diffusion systems, biological cellular networks, the dynamics of stock markets and of the internet, earthquake statistics and prediction, freeway traffic, the human brain, or the formation of opinions in social systems, to name just some of the popular applications. Although their scope and methodologies overlap somewhat, one can distinguish the following main concepts and tools: self-organization, nonlinear dynamics, synergetics, turbulence, dynamical systems, catastrophes, instabilities, stochastic processes, chaos, graphs and networks, cellular automata, adaptive systems, genetic algorithms and computational intelligence. The three major book publication platforms of the Springer Complexity program are the monograph series "Understanding Complex Systems" focusing on the various applications of complexity, the "Springer Series in Synergetics", which is devoted to the quantitative theoretical and methodological foundations, and the "SpringerBriefs in Complexity" which are concise and topical working reports, case-studies, surveys, essays and lecture notes of relevance to the field. In addition to the books in these two core series, the program also incorporates individual titles ranging from textbooks to major reference works.

Theory in Biosciences, Jun 16, 2006

We define a gestalt as the invariants of a collection of patterns that can mutually be transforme... more We define a gestalt as the invariants of a collection of patterns that can mutually be transformed into each other through a class of transformations encoded by, or conversely, determining that gestalt. The class of these transformations needs to satisfy structural regularities like the ones of the mathematical structure of a group. This makes an analysis of a gestalt possible in terms of relations between its representing patterns. While the gestalt concept has its origins in cognitive psychology, it has also important implications for morphology.

Der Grundgedanke der Differenzenverfahren besteht darin, eine gegebene Differentialgleichung durc... more Der Grundgedanke der Differenzenverfahren besteht darin, eine gegebene Differentialgleichung durch eine Differenzengleichung mit einer Schrittweitehzu ersetzen und zu versuchen, nachzuweisen, daß für h →0 die Lösungen der Differenzengleichung gegen eine Lösung der Differentialgleichung konvergieren. Es handelt sich hierbei um ein konstruktives Verfahren, welches insbesondere auch häufig zur numerischen (approximativen) Berechnung von Lösungen von Differentialgleichungen eingesetzt wird.

Wahrscheinlich sind die wesentlichen Aspekte der Molekularbiologie inzwischen allgemeines Bildung... more Wahrscheinlich sind die wesentlichen Aspekte der Molekularbiologie inzwischen allgemeines Bildungsgut geworden und daher auch den meisten mathematisch Ausgebildeten bekannt. Da sie die Grundlage der nachstehenden Erorterungen bilden, sollen sie trotzdem an dieser Stelle kurz vorgestellt werden, wobei ich aber unter Weglassung vieler biochemischer Einzelheiten eine moglichst formale Darstellung wahlen werde.

arXiv (Cornell University), Sep 19, 2018

Let {u n } be a sequence of maps from a compact Riemann surface M with smooth boundary to a gener... more Let {u n } be a sequence of maps from a compact Riemann surface M with smooth boundary to a general compact Riemannian manifold N with free boundary on a smooth submanifold K ⊂ N satisfying sup n ∇u n L 2 (M) + τ(u n) L 2 (M) ≤ Λ, where τ(u n) is the tension field of the map u n. We show that the energy identity and the no neck property hold during a blow-up process. The assumptions are such that this result also applies to the harmonic map heat flow with free boundary, to prove the energy identity at finite singular time as well as at infinity time. Also, the no neck property holds at infinity time. 1. introduction Let (M, g) be a compact Riemannian manifold with smooth boundary and (N, h) be a compact Riemannian manifold of dimension n. Let K ⊂ N be a k−dimensional closed submanifold where 1 ≤ k ≤ n. For a mapping u ∈ C 2 (M, N), the energy density of u is defined by e(u) = 1 2 |∇u| 2 = Trace g u * h, where u * h is the pull-back of the metric tensor h. The energy of the mapping u is defined as E(u) = M e(u)dvol g. Define C(K) = u ∈ C 2 (M, N); u(∂M) ⊂ K. A critical point of the energy E over C(K) is a harmonic map with free boundary u(∂M) on K. The problem of the existence, uniqueness and regularity of such harmonic maps with a free boundary was first systematically investigated in [8]. By Nash's embedding theorem, (N, h) can be isometrically embedded into some Euclidean space R N. Then we can get the Euler-Lagrange equation ∆ g u = A(u)(∇u, ∇u),

Klassische Texte der Wissenschaft, 2023