Hausdorff Center for Mathematics

This paper provides a quantitative version of the recent result of Knüpfer and Muratov (Commun. Pure Appl. Math. 66 (2013), 1129–1162) concerning the solutions of an extension of the classical isoperimetric problem in which a non-local repulsive term involving Riesz potential is present. There it was shown that in two space dimensions the minimizer of the considered problem is either a ball or does not exist, depending on whether or not the volume constraint lies in an explicit interval around zero, provided that the Riesz kernel decays sufficiently slowly. Here we give an explicit estimate for the exponents of the Riesz kernel for which the result holds. 1

Data envelopment analysis (DEA) and free disposal hull (FDH) estimators are widely used to estimate efficiency of production. Practitioners use DEA estimators far more frequently than FDH estimators, implicitly assuming that production sets are convex. Moreover, use of the constant returns to scale (CRS) version of the DEA estimator requires an assumption of CRS. Although bootstrap methods have been developed for making inference about the efficiencies of individual units, until now no methods exist for making consistent inference about differences in mean efficiency across groups of producers or for testing hypotheses about model structure such as returns to scale or convexity of the production set. We use central limit theorem results from our previous work to develop additional theoretical results permitting consistent tests of model structure and provide Monte Carlo evidence on the performance of the tests in terms of size and power. In addition, the variable returns to scale version of the DEA estimator is proved to attain the faster convergence rate of the CRS-DEA estimator under CRS. Using a sample of U.S. commercial banks, we test and reject convexity of the production set, calling into question results from numerous banking studies that have imposed convexity assumptions. Supplementary materials for this article are available online.

Evolutionary game theory is the study of frequency-dependent selection. The success of an individual depends on the frequencies of strategies that are used in the population. We propose a new model for studying evolutionary dynamics in games with a continuous strategy space. The population size is finite. All members of the population use the same strategy. A mutant strategy is chosen from some distribution over the strategy space. The fixation probability of the mutant strategy in the resident population is calculated. The new mutant takes over the population with this probability. In this case, the mutant becomes the new resident. Otherwise, the existing resident remains. Then, another mutant is generated. These dynamics lead to a stationary distribution over the entire strategy space. Our new approach generalizes classical adaptive dynamics in three ways: (i) the population size is finite; (ii) mutants can be drawn non-locally and (iii) the dynamics are stochastic. We explore reactive strategies in the repeated Prisoner's Dilemma. We perform 'knock-out experiments' to study how various strategies affect the evolution of cooperation. We find that 'tit-for-tat' is a weak catalyst for the emergence of cooperation, while 'always cooperate' is a strong catalyst for the emergence of defection. Our analysis leads to a new understanding of the optimal level of forgiveness that is needed for the evolution of cooperation under direct reciprocity.

The classical Faber–Krahn inequality asserts that balls (uniquely) minimize the first eigenvalue of the Dirichlet Laplacian among sets with given volume. In this article we prove a sharp quantitative enhancement of this result, thus confirming a conjecture by Nadirashvili and by Bhattacharya and Weitsman. More generally, the result applies to every optimal Poincaré–Sobolev constant for the embeddings W 0 1 , 2 ( Ω ) ↪ L q ( Ω ) .

The Kardar–Parisi–Zhang (KPZ) equation is conjectured to universally describe the fluctuations of weakly asymmetric interface growth. Here we provide the first intrinsic well-posedness result for the stationary KPZ equation on the real line by showing that its <italic>energy solutions</italic> , as introduced by Gonçalves and Jara in 2010 and refined by Gubinelli and Jara, are unique. This is the first time that a singular stochastic PDE can be tackled using probabilistic methods, and the combination of the convergence results of the first work and many follow-up papers with our uniqueness proof establishes the weak KPZ universality conjecture for a wide class of models. Our proof builds on an observation of Funaki and Quastel from 2015, and a remarkable consequence is that the energy solution to the KPZ equation is not equal to the Cole–Hopf solution, but it involves an additional drift <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t slash 12"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>12</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">t/12</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .

On a star graph \mathcal{G} , we consider a nonlinear Schrödinger equation with focusing nonlinearity of power type and an attractive Dirac's delta potential located at the vertex. The equation can be formally written as i\partial _{t}\Psi (t) = −\mathrm{\Delta }\Psi (t)−|\Psi (t)|^{2\mu }\Psi (t) + \alpha \delta _{0}\Psi (t) , where the strength α of the vertex interaction is negative and the wave function Ψ is supposed to be continuous at the vertex. The values of the mass and energy functionals are conserved by the flow. We show that for 0 &lt; \mu ⩽2 the energy at fixed mass is bounded from below and that for every mass m below a critical mass m^{⁎} it attains its minimum value at a certain \hat \Psi _{m} \in H^{1}(\mathcal{G}) . Moreover, the set of minimizers has the structure \mathcal{M} = \{e^{i\theta }\hat \Psi _{m},\:\theta \in \mathbb{R}\} . Correspondingly, for every m &lt; m^{⁎} there exists a unique \omega = \omega (m) such that the standing wave \hat \Psi _{\omega }e^{i\omega t} is orbitally stable. To prove the above results we adapt the concentration-compactness method to the case of a star graph. This is nontrivial due to the lack of translational symmetry of the set supporting the dynamics, i.e. the graph. This affects in an essential way the proof and the statement of concentration-compactness lemma and its application to minimization of constrained energy. The existence of a mass threshold comes from the instability of the system in the free (or Kirchhoff's) case, that in our setting corresponds to \alpha = 0 .

We study the two-dimensional stochastic nonlinear wave equations (SNLW) with an additive space-time white noise forcing. In particular, we introduce a time-dependent renormalization and prove that SNLW is pathwise locally well-posed. As an application of the local well-posedness argument, we also establish a weak universality result for the renormalized SNLW.

In this paper, the authors propose a new framework under which a theory of generalized Besov-type and Triebel&#8211;Lizorkin-type function spaces is developed. Many function spaces appearing in harmonic analysis fall under the scope of this new framework.

We define the Schrödinger equation with focusing, cubic nonlinearity on one-vertex graphs. We prove global well-posedness in the energy domain and conservation laws for some self-adjoint boundary conditions at the vertex, i.e. Kirchhoff boundary condition and the so-called δ and δ′ boundary conditions. Moreover, in the same setting, we study the collision of a fast solitary wave with the vertex and we show that it splits in reflected and transmitted components. The outgoing waves preserve a soliton character over a time which depends on the logarithm of the velocity of the ingoing solitary wave. Over the same timescale, the reflection and transmission coefficients of the outgoing waves coincide with the corresponding coefficients of the linear problem. In the analysis of the problem, we follow ideas borrowed from the seminal paper [17] about scattering of fast solitons by a delta interaction on the line, by Holmer, Marzuola and Zworski. The present paper represents an extension of their work to the case of graphs and, as a byproduct, it shows how to extend the analysis of soliton scattering by other point interactions on the line, interpreted as a degenerate graph.

Abstract This paper is concerned with a study of the classical isoperimetric problem modified by an addition of a nonlocal repulsive term. We characterize existence, nonexistence, and radial symmetry of the minimizers as a function of mass in the situation where the nonlocal term is generated by a kernel given by an inverse power of the distance. We prove that minimizers of this problem exist for sufficiently small masses and are given by disks with prescribed mass below a certain threshold when the interfacial term in the energy is dominant. At the same time, we prove that minimizers fail to exist for sufficiently large masses due to the tendency of the low‐energy configuration to split into smaller pieces when the nonlocal term in the energy is dominant. In the latter regime, we also establish linear scaling of energy with mass, suggesting that for large masses low‐energy configurations consist of many roughly equal‐size pieces far apart. In the case of slowly decaying kernels, we give a complete characterization of the minimizers. © 2012 Wiley Periodicals, Inc.

Abstract. We provide information on a non trivial structure of phase space of the cubic NLS on a three-edge star graph. We prove that, contrarily to the case of the standard NLS on the line, the energy associated to the cubic focusing Schrödinger equation on the three-edge star graph with a free (Kirchhoff) vertex does not attain a minimum value on any sphere of constant L2-norm. We moreover show that the only stationary state with prescribed L2-norm is indeed a saddle point. 1.

We study optimal interfacial structures in multiferroic materials with a biquadratic coupling between two order parameters. We discover a new duality relation between the strong coupling and the weak coupling regime for the case of isotropic gradient terms. We analyze the phase diagram depending on the coupling constant and anisotropy of the gradient term, and show that in a certain regime the secondary order parameter becomes activated only in the interfacial region.

We prove weak-strong uniqueness in the class of admissible measure-valued solutions for the isentropic Euler equations in any space dimension and for the Savage-Hutter model of granular flows in one and two space dimensions. For the latter system, we also show the complete dissipation of momentum in finite time, thus rigorously justifying an assumption that has been made in the engineering and numerical literature.

We consider a generalized nonlinear Schrödinger equation (NLS) with a power nonlinearity |ψ| 2μψ of focusing type describing propagation on the ramified structure given by N edges connected at a vertex (a star graph). To model the interaction at the junction, it is there imposed a boundary condition analogous to the δ potential of strength α on the line, including as a special case (α=0) the free propagation. We show that nonlinear stationary states describing solitons sitting at the vertex exist both for attractive (α<0, representing a potential well) and repulsive (α>0, a potential barrier) interaction. In the case of sufficiently strong attractive interaction at the vertex and power nonlinearity μ<2, including the standard cubic case, we characterize the ground state as minimizer of a constrained action and we discuss its orbital stability. Finally we show that in the free case, for even N only, the stationary states can be used to construct traveling waves on the graph

We generalize theorems of Kesten and Deuschel-Pisztora about the connectedness of the exterior boundary of a connected subset of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z Superscript d"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {Z}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where “connectedness” and “boundary” are understood with respect to various graphs on the vertices of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z Superscript d"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {Z}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . These theorems are widely used in statistical physics and related areas of probability. We provide simple and elementary proofs of their results. It turns out that the proper way of viewing these questions is graph theory instead of topology.

Abstract When two massive objects (black holes, neutron stars or stars) in our universe fly past each other, their gravitational interactions deflect their trajectories 1,2 . The gravitational waves emitted in the related bound-orbit system—the binary inspiral—are now routinely detected by gravitational-wave observatories 3 . Theoretical physics needs to provide high-precision templates to make use of unprecedented sensitivity and precision of the data from upcoming gravitational-wave observatories 4 . Motivated by this challenge, several analytical and numerical techniques have been developed to approximately solve this gravitational two-body problem. Although numerical relativity is accurate 5–7 , it is too time-consuming to rapidly produce large numbers of gravitational-wave templates. For this, approximate analytical results are also required 8–15 . Here we report on a new, highest-precision analytical result for the scattering angle, radiated energy and recoil of a black hole or neutron star scattering encounter at the fifth order in Newton’s gravitational coupling G , assuming a hierarchy in the two masses. This is achieved by modifying state-of-the-art techniques for the scattering of elementary particles in colliders to this classical physics problem in our universe. Our results show that mathematical functions related to Calabi–Yau (CY) manifolds, 2 n -dimensional generalizations of tori, appear in the solution to the radiated energy in these scatterings. We anticipate that our analytical results will allow the development of a new generation of gravitational-wave models, for which the transition to the bound-state problem through analytic continuation and strong-field resummation will need to be performed.

We argue that ℓ-loop Yangian-invariant fishnet integrals in two dimensions are connected to a family of Calabi-Yau ℓ folds. The value of the integral can be computed from the periods of the Calabi-Yau, while the Yangian generators provide its Picard-Fuchs differential ideal. Using mirror symmetry, we can identify the value of the integral as the quantum volume of the mirror Calabi-Yau. We find that, similar to what happens in string theory, for ℓ=1 and 2 the value of the integral agrees with the classical volume of the mirror, but starting from ℓ=3, the classical volume gets corrected by instanton contributions. We illustrate these claims on several examples, and we use them to provide for the first time results for 2- and 3-loop Yangian-invariant train track integrals in two dimensions for arbitrary external kinematics.

Abstract Sigma‐delta modulation is a popular method for analog‐to‐digital conversion of bandlimited signals that employs coarse quantization coupled with oversampling. The standard mathematical model for the error analysis of the method measures the performance of a given scheme by the rate at which the associated reconstruction error decays as a function of the oversampling ratio λ. It was recently shown that exponential accuracy of the form O (2 − r λ ) can be achieved by appropriate one‐bit sigma‐delta modulation schemes. By general information‐entropy arguments, r must be less than 1. The current best‐known value for r is approximately 0:088. The schemes that were designed to achieve this accuracy employ the “greedy” quantization rule coupled with feedback filters that fall into a class we call “minimally supported.” In this paper, we study the discrete minimization problem that corresponds to optimizing the error decay rate for this class of feedback filters. We solve a relaxed version of this problem exactly and provide explicit asymptotics of the solutions. From these relaxed solutions, we find asymptotically optimal solutions of the original problem, which improve the best‐known exponential error decay rate to r ≈ 0.102. Our method draws from the theory of orthogonal polynomials; in particular, it relates the optimal filters to the zero sets of Chebyshev polynomials of the second kind. © 2011 Wiley Periodicals, Inc.

We investigate the rate of convergence of interpolating splines with respect to sparse grids for Besov spaces of dominating mixed smoothness (tensor product Besov spaces). Main emphasis is given to the approximation by piecewise linear functions.

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