Communications in Contemporary Mathematics
Journal Prestige (SJR): 1.668 Citation Impact (citeScore): 1 Number of Followers: 0 Hybrid journal (It can contain Open Access articles) ISSN (Print) 02191997  ISSN (Online) 17936683 Published by World Scientific [120 journals] 
 Hspace and loop space structures for intermediate curvatures

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Authors: Mark Walsh, David J. Wraith
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
For dimensions [math] and [math], we show that the space of metrics of [math]positive Ricci curvature on the sphere [math] has the structure of an [math]space with a homotopy commutative, homotopy associative product operation. We further show, using the theory of operads and results of Boardman, Vogt and May, that the path component of this space containing the round metric is weakly homotopy equivalent to an [math]fold loop space.
Citation: Communications in Contemporary Mathematics
PubDate: 20220520T07:00:00Z
DOI: 10.1142/S0219199722500171

 Regularity and symmetry for semilinear elliptic equations in bounded
domains
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Authors: Louis Dupaigne, Alberto Farina
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
In this paper, we investigate the regularity and symmetry properties of weak solutions to semilinear elliptic equations which are stable or more generally of finite Morse index or even more generally locally stable.
Citation: Communications in Contemporary Mathematics
PubDate: 20220520T07:00:00Z
DOI: 10.1142/S0219199722500183

 Dirichlet problem for complex Monge–Ampère equation near an
isolated KLT singularity
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Authors: Xin Fu
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We solve the Dirichlet problem for complex Monge–Ampère equation near an isolate Klt singularity, which generalizes the result of Eyssidieux et al. [Singular Kähler–Einstein metrics, J. Amer. Math. Soc. 22 (2009) 607–639], where the Monge–Ampère equation is solved on singular varieties without boundary. As a corollary, we construct solutions to Monge–Ampère equation with isolated singularity on strongly pseudoconvex domain [math] contained in [math].
Citation: Communications in Contemporary Mathematics
PubDate: 20220520T07:00:00Z
DOI: 10.1142/S0219199722500195

 Symmetry and symmetry breaking for Hénontype problems involving the
1Laplacian operator
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Authors: Marcos T. O. Pimenta, Anderson dos Santos Gonzaga
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
In this work, we study a class of Hénontype equations which involve the [math]Laplacian operator in the unit ball. Under mild assumptions on the nonlinearity, the existence of radial solutions is proved and, for a parameter in a certain range, the existence of symmetry breaking is proved, through the presence of nonradial solutions. The approach is based on an approximation scheme, where a thorough analysis of the solutions of the associated [math]Laplacian problems is necessary.
Citation: Communications in Contemporary Mathematics
PubDate: 20220520T07:00:00Z
DOI: 10.1142/S0219199722500213

 Fourdimensional generalized Ricci flows with nilpotent symmetry

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Authors: Steven Gindi, Jeffrey Streets
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We study solutions to generalized Ricci flow on fourmanifolds with a nilpotent, codimension [math] symmetry. We show that all such flows are immortal, and satisfy type III curvature and diameter estimates. Using a new kind of monotone energy adapted to this setting, we show that blowdown limits lie in a canonical finitedimensional family of solutions. The results are new for Ricci flow.
Citation: Communications in Contemporary Mathematics
PubDate: 20220520T07:00:00Z
DOI: 10.1142/S0219199722500250

 Geometric theory of Weyl structures

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Authors: Andreas Čap, Thomas Mettler
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
Given a parabolic geometry on a smooth manifold [math], we study a natural affine bundle [math], whose smooth sections can be identified with Weyl structures for the geometry. We show that the initial parabolic geometry defines a reductive Cartan geometry on [math], which induces an almost biLagrangian structure on [math] and a compatible linear connection on [math]. We prove that the splitsignature metric given by the almost biLagrangian structure is Einstein with nonzero scalar curvature, provided that the parabolic geometry is torsionfree and [math]graded. We proceed to study Weyl structures via the submanifold geometry of the image of the corresponding section in [math]. For Weyl structures satisfying appropriate nondegeneracy conditions, we derive a universal formula for the second fundamental form of this image. We also show that for locally flat projective structures, this has close relations to solutions of a projectively invariant Monge–Ampere equation and thus to properly convex projective structures.
Citation: Communications in Contemporary Mathematics
PubDate: 20220520T07:00:00Z
DOI: 10.1142/S0219199722500262

 Flexibility of sections of nearly integrable Hamiltonian systems

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Authors: Dmitri Burago, Dong Chen, Sergei Ivanov
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
Given any symplectomorphism on [math] which is [math] close to the identity, and any completely integrable Hamiltonian system [math] in the proper dimension, we construct a [math] perturbation of [math] such that the resulting Hamiltonian flow contains a “local Poincaré section” that “realizes” the symplectomorphism. As a (motivating) application, we show that there are arbitrarily small perturbations of any completely integrable Hamiltonian system which are entropy nonexpansive (and, in particular, exhibit hyperbolic behavior on a set of positive measure). We use some results in Berger–Turaev [On Herman’s positive entropy conjecture, Adv. Math. 349 (2019) 1234—1288], though in higher dimensions we could simply apply a construction from [D. Burago and S. Ivanov, Boundary distance, lens maps and entropy of geodesic ows of Finsler metrics, Geom. & Topol. 20 (2016) 469–490].
Citation: Communications in Contemporary Mathematics
PubDate: 20220520T07:00:00Z
DOI: 10.1142/S0219199722500286

 A note on the compactness of Poincaré–Einstein manifolds

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Authors: Fang Wang, Huihuang Zhou
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
For a conformally compact Poincaré–Einstein manifold [math], we consider two types of compactifications for it. One is [math], where [math] is a fixed smooth defining function; the other is the adapted (including Fefferman–Graham) compactification [math] with a continuous parameter [math]. In this paper, we mainly prove that for a set of conformally compact Poincaré–Einstein manifolds [math] with conformal infinity of positive Yamabe type, [math] is compact in [math] topology if and only if [math] is compact in some [math] topology, provided that [math] and [math] has positive scalar curvature for each [math]. See Theorem 1.1 and Corollary 1.1 for the exact relation of [math] and [math].
Citation: Communications in Contemporary Mathematics
PubDate: 20220518T07:00:00Z
DOI: 10.1142/S0219199722500158

 On the dynamics of charged particles in an incompressible flow: From
kineticfluid to fluid–fluid models
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Authors: YoungPil Choi, Jinwook Jung
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
In this paper, we are interested in the dynamics of charged particles interacting with the incompressible viscous flow. More precisely, we consider the Vlasov–Poisson or Vlasov–Poisson–Fokker–Planck equation coupled with the incompressible Navier–Stokes system through the drag force. For the proposed kineticfluid model, we study the asymptotic regime corresponding to strong local alignment and diffusion forces. Under suitable assumptions on wellprepared initial data, we rigorously derive a coupled isothermal/pressureless Euler–Poisson system and incompressible Navier–Stokes system (EPNS system). For this hydrodynamic limit, we employ the modulated kinetic, internal, interaction energy estimates. We also construct a globalintime strong solvability for the isothermal/pressureless EPNS system. In particular, this globalintime solvability gives the estimates of hydrodynamic limit hold for all times.
Citation: Communications in Contemporary Mathematics
PubDate: 20220510T07:00:00Z
DOI: 10.1142/S0219199722500122

 The zero mass problem for Klein–Gordon equations

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Authors: Shijie Dong
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
In this paper, we are interested in the global existence result for a class of Klein–Gordon equations, particularly in the unified time decay result concerning a possibly vanishing mass parameter. We give for the first time a rigorous proof for this problem, which relies on both the flat foliation and the hyperboloidal foliation of the Minkowski spacetime. To take advantage of both foliations, an iteration procedure is used.
Citation: Communications in Contemporary Mathematics
PubDate: 20220510T07:00:00Z
DOI: 10.1142/S0219199722500298

 Approximation of random diffusion equations by nonlocal diffusion
equations in free boundary problems of one space dimension
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Authors: Yihong Du, Wenjie Ni
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We show how the Stefan type free boundary problem with random diffusion in one space dimension can be approximated by the corresponding free boundary problem with nonlocal diffusion. The approximation problem is a slightly modified version of the nonlocal diffusion problem with free boundaries considered in [J. Cao, Y. Du, F. Li and W.T. Li, The dynamics of a Fisher–KPP nonlocal diffusion model with free boundaries, J. Functional Anal. 277 (2019) 2772–2814; C. Cortazar, F. Quiros and N. Wolanski, A nonlocal diffusion problem with a sharp free boundary, Interfaces Free Bound. 21 (2019) 441–462]. The proof relies on the introduction of several auxiliary free boundary problems and constructions of delicate upper and lower solutions for these problems. As usual, the approximation is achieved by choosing the kernel function in the nonlocal diffusion term of the form [math] for small [math], where [math] has compact support. We also give an estimate of the error term of the approximation by some positive power of [math].
Citation: Communications in Contemporary Mathematics
PubDate: 20220418T07:00:00Z
DOI: 10.1142/S0219199722500043

 Uniqueness of inverse source problems for general evolution equations

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Authors: Yavar Kian, Yikan Liu, Masahiro Yamamoto
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
In this paper, we investigate inverse source problems for a wide range of PDEs of parabolic and hyperbolic types as well as timefractional evolution equations by partial interior observation. Restricting the source terms to the form of separated variables, we establish uniqueness results for simultaneously determining both temporal and spatial components without nonvanishing assumptions at [math], which seems novel to the best of our knowledge. Remarkably, mostly we allow a rather flexible choice of the observation time not necessarily starting from [math], which fits into various situations in practice. Our main approach is based on the combination of the Titchmarsh convolution theorem with unique continuation properties and timeanalyticity of the PDEs under consideration.
Citation: Communications in Contemporary Mathematics
PubDate: 20220402T07:00:00Z
DOI: 10.1142/S0219199722500092

 Chern degree functions

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Authors: Martí Lahoz, Andrés Rojas
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We introduce Chern degree functions for complexes of coherent sheaves on a polarized surface, which encode information given by stability conditions on the boundary of the [math]plane. We prove that these functions extend to continuous real valued functions and we study their differentiability in terms of stability. For abelian surfaces, Chern degree functions coincide with the cohomological rank functions defined by Jiang–Pareschi. We illustrate in some geometrical situations a general strategy to compute these functions.
Citation: Communications in Contemporary Mathematics
PubDate: 20220330T07:00:00Z
DOI: 10.1142/S0219199722500079

 Attractors of dissipative homeomorphisms of the infinite surface
homeomorphic to a punctured sphere
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Authors: Grzegorz Graff, Rafael Ortega, Alfonso RuizHerrera
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
A class of dissipative orientation preserving homeomorphisms of the infinite annulus, pairs of pants, or generally any infinite surface homeomorphic to a punctured sphere is considered. We prove that in some isotopy classes the local behavior of such homeomorphisms at a fixed point, namely the existence of socalled inverse saddle, impacts the topology of the attractor — it cannot be arcwise connected.
Citation: Communications in Contemporary Mathematics
PubDate: 20220330T07:00:00Z
DOI: 10.1142/S0219199722500109

 Positive solutions for a Minkowskicurvature equation with indefinite
weight and superexponential nonlinearity
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Authors: Alberto Boscaggin, Guglielmo Feltrin, Fabio Zanolin
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We investigate the existence of positive solutions for a class of Minkowskicurvature equations with indefinite weight and nonlinear term having superlinear growth at zero and superexponential growth at infinity. As an example, for the equation ( u′ 1 − (u′ )2)′ + a(t)(eup − 1) = 0, where [math] and [math] is a signchanging function satisfying the meanvalue condition [math], we prove the existence of a positive solution for both periodic and Neumann boundary conditions. The proof relies on a topological degree technique.
Citation: Communications in Contemporary Mathematics
PubDate: 20220328T07:00:00Z
DOI: 10.1142/S0219199722500055

 Liouvilletype theorems for higherorder Lane–Emden system in
exterior domains
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Authors: Yuxia Guo, Shaolong Peng
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
In this paper, we are mainly concerned with the following system in an exterior domains: (−Δ)mu = vp,u ≥ 0 in ℝN∖B¯,(−Δ)mv = uq,v ≥ 0 in ℝN∖B¯,Δiu = 0, Δiv = 0,i = 0,…,m − 1on ∂B, where [math], [math] is an integer, [math], and [math] is the polyharmonic operator. We prove the nonexistence of positive solutions to the above system for [math] if [math], and [math] if [math]. The novelty of the paper is that we do not ask [math] satisfy any symmetry and asymptotic conditions at infinity. By proving the superharmonic properties of the solutions, we establish the equivalence between systems of partial differential equations (PDEs) and integral equations (IEs), then the method of scaling sphere in integral form can be applied to prove the nonexistence of the solutions.
Citation: Communications in Contemporary Mathematics
PubDate: 20220316T07:00:00Z
DOI: 10.1142/S0219199722500067

 24 rational curves on K3 surfaces

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Authors: Sławomir Rams, Matthias Schütt
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
Given [math], we prove that all smooth K3 surfaces (over any field of characteristic [math]) of degree greater than [math] contain at most 24 rational curves of degree at most [math]. In the exceptional characteristics, the same bounds hold for nonunirational K3 surfaces, and we develop analogous results in the unirational case. For [math], we also construct K3 surfaces of any degree greater than [math] with 24 rational curves of degree exactly [math], thus attaining the above bounds.
Citation: Communications in Contemporary Mathematics
PubDate: 20220310T08:00:00Z
DOI: 10.1142/S0219199722500080

 On the singular Weinstein conjecture and the existence of escape orbits
for [math]Beltrami fields
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Authors: Eva Miranda, Cédric Oms, Daniel PeraltaSalas
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
Motivated by Poincaré’s orbits going to infinity in the (restricted) threebody problem [see H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste, Vol. 3 (GauthierVillars, 1899) and A. Chenciner, Poincaré and the threebody problem, in Henri Poincaré, 1912–2012 (Birkhäuser, Basel, 2015), pp. 51–149], we investigate the generic existence of heterocliniclike orbits in a neighborhood of the critical set of a [math]contact form. This is done by using a singular counterpart [R. Cardona, E. Miranda and D. PeraltaSalas, Euler flows and singular geometric structures, Philos. Trans. R. Soc. A 377(2158) (2019) 20190034] of Etnyre–Ghrist’s contact/Beltrami correspondence [J. Etnyre and R. Ghrist, Contact topology and hydrodynamics: I. Beltrami fields and the Seifert conjecture, Nonlinearity 13(2) (2000) 441–458], and genericity results concerning eigenfunctions of the Laplacian established by Uhlenbeck [Generic properties of eigenfunctions, Amer. J. Math. 98(4) (1976) 1059–1078]. Specifically, we analyze the [math]Beltrami vector fields on [math]manifolds of dimension [math] and prove that for a generic asymptotically exact [math]metric they exhibit escape orbits. We also show that a generic asymptotically symmetric [math]Beltrami vector field on an asymptotically flat [math]manifold has a generalized singular periodic orbit and at least four escape orbits. Generalized singular periodic orbits are trajectories of the vector field whose [math] and [math]limit sets intersect the critical surface. These results are a first step towards proving the singular Weinstein conjecture.
Citation: Communications in Contemporary Mathematics
PubDate: 20220302T08:00:00Z
DOI: 10.1142/S0219199721500760

 Propagation phenomena in a diffusion system with the
Belousov–Zhabotinskii chemical reaction
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Authors: WeiJie Sheng, Mingxin Wang, ZhiCheng Wang
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
This paper is concerned with the propagation phenomena in a diffusion system with the Belousov–Zhabotinskii chemical reaction in [math] under the bistability assumption. We prove that there is a new type of entire solution originated from three moving planar traveling fronts, and evolved to a Vshaped traveling front as time changes, which means that the profile of this solution is not invariant at all. Here an entire solution is referred to a solution that is defined for all time [math] and in the whole space [math]. Furthermore, we show that not only the entire solution but also all transition fronts share the same global mean speed by constructing suitable radially symmetric expanding and retracting super and subsolutions.
Citation: Communications in Contemporary Mathematics
PubDate: 20220302T08:00:00Z
DOI: 10.1142/S0219199722500018

 Quantitative characterization of traces of Sobolev maps

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Authors: Katarzyna Mazowiecka, Jean Van Schaftingen
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We give a quantitative characterization of traces on the boundary of Sobolev maps in [math], where [math] and [math] are compact Riemannian manifolds, [math]: the Borelmeasurable maps [math] that are the trace of a map [math] are characterized as the maps for which there exists an extension energy density [math] that controls the Sobolev energy of extensions from [math]dimensional subsets of [math] to [math]dimensional subsets of [math].
Citation: Communications in Contemporary Mathematics
PubDate: 20220225T08:00:00Z
DOI: 10.1142/S0219199722500031

 Geometric inequalities for antiblocking bodies

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Authors: Shiri ArtsteinAvidan, Shay Sadovsky, Raman Sanyal
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We study the class of (locally) antiblocking bodies as well as some associated classes of convex bodies. For these bodies, we prove geometric inequalities regarding volumes and mixed volumes, including Godbersen’s conjecture, nearoptimal bounds on Mahler volumes, SaintRaymondtype inequalities on mixed volumes, and reverse Kleitman inequalities for mixed volumes. We apply our results to the combinatorics of posets and prove Sidorenkotype inequalities for linear extensions of pairs of [math]dimensional posets. The results rely on some elegant decompositions of differences of antiblocking bodies, which turn out to hold for antiblocking bodies with respect to general polyhedral cones.
Citation: Communications in Contemporary Mathematics
PubDate: 20220221T08:00:00Z
DOI: 10.1142/S0219199721501133

 Global wellposedness for volume–surface reaction–diffusion
systems
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Authors: Jeff Morgan, Bao Quoc Tang
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We study the global existence of classical solutions to volume–surface reaction–diffusion systems with control of mass. Such systems appear naturally from modeling evolution of concentrations or densities appearing both in a volume domain and its surface, and therefore have attracted considerable attention. Due to the characteristic volume–surface coupling, global existence of solutions to general systems is a challenging issue. In particular, the duality method, which is powerful in dealing with mass conserved systems in domains, is not applicable on its own. In this paper, we introduce a new family of [math]energy functions and combine them with a suitable duality method for volume–surface systems, to ultimately obtain global existence of classical solutions under a general assumption called the intermediate sum condition. For systems that conserve mass, but do not satisfy this condition, global solutions are shown under a quasiuniform condition, that is, under the assumption that the diffusion coefficients are close to each other. In the case of mass dissipation, we also show that the solution is bounded uniformly in time by studying the system on each timespace cylinder of unit size, and showing that the solution is supnorm bounded independently of the cylinder. Applications of our results include global existence and boundedness for systems arising from membrane protein clustering or activation of Cdc42 in cell polarization.
Citation: Communications in Contemporary Mathematics
PubDate: 20220221T08:00:00Z
DOI: 10.1142/S021919972250002X

 Small toric resolutions of toric varieties of string polytopes with small
indices
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Authors: Yunhyung Cho, Yoosik Kim, Eunjeong Lee, KyeongDong Park
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
Let [math] be a semisimple algebraic group over [math]. For a reduced word [math] of the longest element in the Weyl group of [math] and a dominant integral weight [math], one can construct the string polytope [math], whose lattice points encode the character of the irreducible representation [math]. The string polytope [math] is singular in general and combinatorics of string polytopes heavily depends on the choice of [math]. In this paper, we study combinatorics of string polytopes when [math], and present a sufficient condition on [math] such that the toric variety [math] of the string polytope [math] has a small toric resolution. Indeed, when [math] has small indices and [math] is regular, we explicitly construct a small toric resolution of the toric variety [math] using a Bott manifold. Our main theorem implies that a toric variety of any string polytope admits a small toric resolution when [math]. As a byproduct, we show that if [math] has small indices then [math] is integral for any dominant integral weight [math], which in particular implies that the anticanonical limit toric variety [math] of a partial flag variety [math] is Gorenstein Fano. Furthermore, we apply our result to symplectic topology of the full flag manifold [math] and obtain a formula of the disk potential of the Lagrangian torus fibration on [math] obtained from a flat toric degeneration of [math] to the toric variety [math].
Citation: Communications in Contemporary Mathematics
PubDate: 20220217T08:00:00Z
DOI: 10.1142/S0219199721501121

 Determinant of Friedrichs Dirichlet Laplacians on 2dimensional hyperbolic
cones
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Authors: Victor Kalvin
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We explicitly express the spectral determinant of Friedrichs Dirichlet Laplacians on the 2dimensional hyperbolic (Gaussian curvature [math]) cones in terms of the cone angle and the geodesic radius of the boundary.
Citation: Communications in Contemporary Mathematics
PubDate: 20220124T08:00:00Z
DOI: 10.1142/S0219199721501078

 Averaging of magnetic fields and applications

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Authors: Ayman Kachmar, Mohammad Wehbe
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
In this paper, we estimate the magnetic Laplacian energy norm in appropriate planar domains under a weak regularity hypothesis on the magnetic field. Our main contribution is an averaging estimate, valid in small cells, allowing us to pass from nonuniform to uniform magnetic fields. As a matter of application, we derive new upper and lower bounds of the lowest eigenvalue of the Dirichlet Laplacian which match in the regime of large magnetic field intensity. Furthermore, our averaging technique allows us to estimate the nonlinear Ginzburg–Landau energy, and as a byproduct, yields a nonGaussian trial state for the Dirichlet magnetic Laplacian.
Citation: Communications in Contemporary Mathematics
PubDate: 20220121T08:00:00Z
DOI: 10.1142/S021919972150108X

 Rapid exponential stabilization of a Boussinesq system of KdV–KdV
Type
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Authors: Roberto de A. Capistrano–Filho, Eduardo Cerpa, Fernando A. Gallego
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
This paper studies the exponential stabilization of a Boussinesq system describing the twoway propagation of small amplitude gravity waves on the surface of an ideal fluid, the socalled Boussinesq system of the Korteweg–de Vries type. We use a Gramianbased method introduced by Urquiza to design our feedback control. By means of spectral analysis and Fourier expansion, we show that the solutions of the linearized system decay uniformly to zero when the feedback control is applied. The decay rate can be chosen as large as we want. The main novelty of our work is that we can exponentially stabilize this system of two coupled equations using only one scalar input.
Citation: Communications in Contemporary Mathematics
PubDate: 20220120T08:00:00Z
DOI: 10.1142/S021919972150111X

 Primitive settheoretic solutions of the Yang–Baxter equation

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Authors: F. Cedó, E. Jespers, J. Okniński
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
To every involutive nondegenerate settheoretic solution [math] of the Yang–Baxter equation on a finite set [math] there is a naturally associated finite solvable permutation group [math] acting on [math]. We prove that every primitive permutation group of this type is of prime order [math]. Moreover, [math] is then a socalled permutation solution determined by a cycle of length [math]. This solves a problem recently asked by A. BallesterBolinches. The result opens a new perspective on a possible approach to the classification problem of all involutive nondegenerate settheoretic solutions.
Citation: Communications in Contemporary Mathematics
PubDate: 20220119T08:00:00Z
DOI: 10.1142/S0219199721501054

 Functional convergence of continuoustime random walks with continuous
paths
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Authors: Marcin Magdziarz, Piotr Zebrowski
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
Continuoustime random walks (CTRWs) are generic models of anomalous diffusion and fractional dynamics in statistical physics. They are typically defined in the way that their trajectories are discontinuous step functions. In this paper, we propose alternative definition of CTRWs with continuous trajectories. We also give the scaling limit theorem for sequence of such random walks. In general case this result requires the use of strong Skorohod [math] topology instead of Skorohod [math] topology, which is usually used in limit theorems for ordinary CTRW processes. We also give additional conditions under which convergence of sequence of considered random walks holds in the [math] topology.
Citation: Communications in Contemporary Mathematics
PubDate: 20220119T08:00:00Z
DOI: 10.1142/S0219199721501066

 Existence of solutions for critical [math]Laplacian equations in [math]

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Authors: Laura Baldelli, Roberta Filippucci
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
In this paper, we are mainly interested in existence properties for a class of nonlinear PDEs driven by the ([math])Laplace operator where the reaction combines a powertype nonlinearity at critical level with a subcritical term. In addition, nonnegative nontrivial weights and a positive parameter [math] are included in the nonlinearity. An important role in the analysis developed is played by the two potentials. Precisely, under suitable conditions on the exponents of the nonlinearity, first a detailed proof of the tight convergence of a sequence of measures is given, then the existence of a nontrivial weak solution is obtained provided that the parameter [math] is far from [math]. Our proofs use concentration compactness principles by Lions and Mountain Pass Theorem by Ambrosetti and Rabinowitz.
Citation: Communications in Contemporary Mathematics
PubDate: 20220119T08:00:00Z
DOI: 10.1142/S0219199721501091

 Asymptotic expansion and optimal symmetry of minimal gradient graph
equations in dimension 2
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Authors: Zixiao Liu, Jiguang Bao
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
In this paper, we study asymptotic expansion at infinity and symmetry of zero mean curvature equations of gradient graph in dimension 2, which include the Monge–Ampère equation, inverse harmonic Hessian equation and the special Lagrangian equation. This refines the research of asymptotic behavior, gives the precise gap between exterior minimal gradient graph and the entire case, and extends the classification results of Monge–Ampère equations.
Citation: Communications in Contemporary Mathematics
PubDate: 20220119T08:00:00Z
DOI: 10.1142/S0219199721501108

 Scalar positive immersions

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Authors: Luis A. Florit, Bernhard Hanke
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
As shown by Gromov–Lawson and Stolz the only obstruction to the existence of positive scalar curvature metrics on closed simply connected manifolds in dimensions at least five appears on spin manifolds and is given by the nonvanishing of the [math]genus of Hitchin. When unobstructed we shall realize a positive scalar curvature metric by an immersion into Euclidean space whose dimension is uniformly close to the classical Whitney upper bound for smooth immersions. Our main tool is an extrinsic counterpart of the wellknown Gromov–Lawson surgery procedure for constructing positive scalar curvature metrics.
Citation: Communications in Contemporary Mathematics
PubDate: 20211220T08:00:00Z
DOI: 10.1142/S0219199721500930

 The compactness of minimizing sequences for a nonlinear Schrödinger
system with potentials
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Authors: Norihisa Ikoma, Yasuhito Miyamoto
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
In this paper, we consider the following minimizing problem with two constraints: inf{E(u) u = (u1,u2),∥u1∥L22 = α 1,∥u2∥L22 = α 2}, where [math] and [math] is defined by E(u) := ∫RN 1 2∑i=12( ∇u i 2 + V i(x) ui 2) −∑ i=12 μi 2pi + 2 ui 2pi+2 − β p3 + 1 u1 p3+1 u 2 p3+1 dx. Here [math], [math] and [math] [math] are given functions. For [math], we consider two cases: (i) both of [math] and [math] are bounded, (ii) one of [math] and [math] is bounded. Under some assumptions on [math] and [math], we discuss the compactness of any minimizing sequence.
Citation: Communications in Contemporary Mathematics
PubDate: 20211217T08:00:00Z
DOI: 10.1142/S0219199721501030

 On the representation theory of the vertex algebra [math]

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Authors: Dražen Adamović, Ozren Perše, Ivana Vukorepa
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We study the representation theory of nonadmissible simple affine vertex algebra [math]. We determine an explicit formula for the singular vector of conformal weight four in the universal affine vertex algebra [math], and show that it generates the maximal ideal in [math]. We classify irreducible [math]modules in the category [math], and determine the fusion rules between irreducible modules in the category of ordinary modules [math]. It turns out that this fusion algebra is isomorphic to the fusion algebra of [math]. We also prove that [math] is a semisimple, rigid braided tensor category. In our proofs, we use the notion of collapsing level for the affine [math]algebra, and the properties of conformal embedding [math] at level [math] from D. Adamovic et al. [Finite vs infinite decompositions in conformal embeddings, Comm. Math. Phys. 348 (2016) 445–473.]. We show that [math] is a collapsing level with respect to the subregular nilpotent element [math], meaning that the simple quotient of the affine [math]algebra [math] is isomorphic to the Heisenberg vertex algebra [math]. We prove certain results on vanishing and nonvanishing of cohomology for the quantum Hamiltonian reduction functor [math]. It turns out that the properties of [math] are more subtle than in the case of minimal reduction.
Citation: Communications in Contemporary Mathematics
PubDate: 20211211T08:00:00Z
DOI: 10.1142/S0219199721501042

 A note on the concordance invariants Upsilon and phi

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Authors: Shida Wang
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
Dai, Hom, Stoffregen and Truong defined a family of concordance invariants [math]. The example of a knot with zero Upsilon invariant but nonzero epsilon invariant previously given by Hom also has nonzero phi invariant. We show there are infinitely many such knots that are linearly independent in the smooth concordance group. In the opposite direction, we build infinite families of linearly independent knots with zero phi invariant but nonzero Upsilon invariant. We also give a recursive formula for the phi invariant of torus knots.
Citation: Communications in Contemporary Mathematics
PubDate: 20211209T08:00:00Z
DOI: 10.1142/S021919972150098X

 The global solvability of the Hallmagnetohydrodynamics system in critical
Sobolev spaces
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Authors: Raphaël Danchin, Jin Tan
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We are concerned with the 3D incompressible Hallmagnetohydrodynamic system (HallMHD). Our first aim is to provide the reader with an elementary proof of a global wellposedness result for small data with critical Sobolev regularity, in the spirit of Fujita–Kato’s theorem [On the Navier–Stokes initial value problem I, Arch. Ration. Mech. Anal. 16 (1964) 269–315] for the Navier–Stokes equations. Next, we investigate the longtime asymptotics of global solutions of the HallMHD system that are in the Fujita–Kato regularity class. A weakstrong uniqueness statement is also proven. Finally, we consider the socalled 2[math]D flows for the HallMHD system (that is, 3D flows independent of the vertical variable), and establish the global existence of strong solutions, assuming only that the initial magnetic field is small. Our proofs strongly rely on the use of an extended formulation involving the socalled velocity of electron [math] and as regards [math]D flows, of the auxiliary vectorfield [math] that comes into play in the total magnetohelicity balance for the HallMHD system.
Citation: Communications in Contemporary Mathematics
PubDate: 20211209T08:00:00Z
DOI: 10.1142/S0219199721500991

 Normalized solutions to a Schrödinger–Bopp–Podolsky system under
Neumann boundary conditions
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Authors: Danilo G. Afonso, Gaetano Siciliano
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
In this paper, we study a Schrödinger–Bopp–Podolsky (SBP) system of partial differential equations in a bounded and smooth domain of [math] with a nonconstant coupling factor. Under a compatibility condition on the boundary data we deduce existence of solutions by means of the Ljusternik–Schnirelmann theory.
Citation: Communications in Contemporary Mathematics
PubDate: 20211209T08:00:00Z
DOI: 10.1142/S0219199721501005

 Chebyshev polynomials and inequalities for Kleinian groups

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Authors: Hala Alaqad, Jianhua Gong, Gaven Martin
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
The principal character of a representation of the free group of rank two into [math] is a triple of complex numbers that determines an irreducible representation uniquely up to conjugacy. It is a central problem in the geometry of discrete groups and low dimensional topology to determine when such a triple represents a discrete group which is not virtually abelian, that is, a Kleinian group. A classical necessary condition is Jørgensen’s inequality. Here, we use certain shifted Chebyshev polynomials and trace identities to determine new families of such inequalities, some of which are best possible. The use of these polynomials also shows how we can identify the principal character of some important subgroups from that of the group itself.
Citation: Communications in Contemporary Mathematics
PubDate: 20211204T08:00:00Z
DOI: 10.1142/S0219199721501029

 [math]universal Hopf algebras (co)acting on [math]algebras

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Authors: A. L. Agore, A. S. Gordienko, J. Vercruysse
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We develop a theory which unifies the universal (co)acting bi/Hopf algebras as studied by Sweedler, Manin and Tambara with the recently introduced [A. L. Agore, A. S. Gordienko and J. Vercruysse, On equivalences of (co)module algebra structures over Hopf algebras, J. Noncommut. Geom., doi: 10.4171/JNCG/428.] bi/Hopfalgebras that are universal among all support equivalent (co)acting bi/Hopf algebras. Our approach uses vector spaces endowed with a family of linear maps between tensor powers of [math], called [math]algebras. This allows us to treat algebras, coalgebras, braided vector spaces and many other structures in a unified way. We study [math]universal measuring coalgebras and [math]universal comeasuring algebras between [math]algebras [math] and [math], relative to a fixed subspace [math] of [math]. By considering the case [math], we derive the notion of a [math]universal (co)acting bialgebra (and Hopf algebra) for a given algebra [math]. In particular, this leads to a refinement of the existence conditions for the Manin–Tambara universal coacting bi/Hopf algebras. We establish an isomorphism between the [math]universal acting bi/Hopf algebra and the finite dual of the [math]universal coacting bi/Hopf algebra under certain conditions on [math] in terms of the finite topology on [math].
Citation: Communications in Contemporary Mathematics
PubDate: 20211124T08:00:00Z
DOI: 10.1142/S0219199721500954

 Classical BV formalism for group actions

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Authors: Marco Benini, Pavel Safronov, Alexander Schenkel
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We study the derived critical locus of a function [math] on the quotient stack of a smooth affine scheme [math] by the action of a smooth affine group scheme [math]. It is shown that [math] is a derived quotient stack for a derived affine scheme [math], whose dgalgebra of functions is described explicitly. Our results generalize the classical BV formalism in finite dimensions from Lie algebra to group actions.
Citation: Communications in Contemporary Mathematics
PubDate: 20211118T08:00:00Z
DOI: 10.1142/S0219199721500942

 On a new functional for extremal metrics of the conformal Laplacian in
high dimensions
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Authors: Yannick Sire, Hang Xu
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
In this paper, we introduce a new functional for the conformal spectrum of the conformal Laplacian on a closed manifold [math] of dimension at least 3. For this new functional we provide a Korevaar type result. The main body of the paper deals with the case of the sphere but a section is devoted to more general closed manifolds. The functional introduced here has been investigated in the recent work [S. PerezAyala, Extremal eigenvalues of the conformal Laplacian under Sire–Xu normalization, preprint (2019); arXiv:2011.06018].
Citation: Communications in Contemporary Mathematics
PubDate: 20211117T08:00:00Z
DOI: 10.1142/S0219199721500966

 Power concavity for elliptic and parabolic boundary value problems on
rotationally symmetric domains
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Authors: Kazuhiro Ishige, Paolo Salani, Asuka Takatsu
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We study power concavity of rotationally symmetric solutions to elliptic and parabolic boundary value problems on rotationally symmetric domains in Riemannian manifolds. As applications of our results to the hyperbolic space [math], we have The first (positive) Dirichlet eigenfunction of the Laplacian on a ball in [math] raised to some power [math] is strictly concave; Let [math] be the heat kernel on [math]. Then [math] is strictly logconcave in [math] for [math] and [math].
Citation: Communications in Contemporary Mathematics
PubDate: 20211117T08:00:00Z
DOI: 10.1142/S0219199721500978

 Thermodynamic formalism for quantum channels: Entropy, pressure, Gibbs
channels and generic properties
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Authors: Jader E. Brasil, Josué Knorst, Artur O. Lopes
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
Denote [math] the set of complex [math] by [math] matrices. We will analyze here quantum channels [math] of the following kind: given a measurable function [math] and the measure [math] on [math] we define the linear operator [math], via the expression [math]. A recent paper by T. Benoist, M. Fraas, Y. Pautrat, and C. Pellegrini is our starting point. They considered the case where [math] was the identity. Under some mild assumptions on the quantum channel [math] we analyze the eigenvalue property for [math] and we define entropy for such channel. For a fixed [math] (the a priori measure) and for a given a Hamiltonian [math] we present a version of the Ruelle Theorem: a variational principle of pressure (associated to such [math]) related to an eigenvalue problem for the Ruelle operator. We introduce the concept of Gibbs channel. We also show that for a fixed [math] (with more than one point in the support) the set of [math] such that it is [math]Erg (also irreducible) for [math] is a generic set. We describe a related process [math], [math], taking values on the projective space [math] and analyze the question of the existence of invariant probabilities. We also consider an associated process [math], [math], with values on [math] ([math] is the set of density operators). Via the barycenter, we associate the invariant probability mentioned above with the density operator fixed for [math].
Citation: Communications in Contemporary Mathematics
PubDate: 20211027T07:00:00Z
DOI: 10.1142/S0219199721500905

 Boundary separated and clustered layer positive solutions for an elliptic
Neumann problem with large exponent
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Authors: Yibin Zhang
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
Given a smooth bounded domain [math] in [math] with [math], we study the existence and the profile of positive solutions for the following elliptic Neumann problem: − Δυ + υ = υp,υ> 0in ð’Ÿ,∂υ ∂ν = 0 on ∂ð’Ÿ, where [math] is a large exponent and [math] denotes the outer unit normal vector to the boundary [math]. For suitable domains [math], by a constructive way we prove that, for any nonnegative integers [math], [math] with [math], if [math] is large enough, such a problem has a family of positive solutions with [math] boundary layers and [math] interior layers which concentrate along [math] distinct [math]dimensional minimal submanifolds of [math], or collapse to the same [math]dimensional minimal submanifold of [math] as [math].
Citation: Communications in Contemporary Mathematics
PubDate: 20211025T07:00:00Z
DOI: 10.1142/S0219199721500887

 Multiplicative connections and their Lie theory

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Authors: Fabrizio Pugliese, Giovanni Sparano, Luca Vitagliano
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We define and study multiplicative connections in the tangent bundle of a Lie groupoid. Multiplicative connections are linear connections satisfying an appropriate compatibility with the groupoid structure. Our definition is natural in the sense that a linear connection on a Lie groupoid is multiplicative if and only if its torsion is a multiplicative tensor in the sense of Bursztyn–Drummond [Lie theory of multiplicative tensors, Mat. Ann. 375 (2019) 1489–1554, arXiv:1705.08579] and its geodesic spray is a multiplicative vector field. We identify the obstruction to the existence of a multiplicative connection. We also discuss the infinitesimal version of multiplicative connections in the tangent bundle, that we call infinitesimally multiplicative (IM) connections and we prove an integration theorem for IM connections. Finally, we present a few toy examples.
Citation: Communications in Contemporary Mathematics
PubDate: 20211025T07:00:00Z
DOI: 10.1142/S0219199721500929

 Mapping class group representations from nonsemisimple TQFTs

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Authors: Marco De Renzi, Azat M. Gainutdinov, Nathan Geer, Bertrand PatureauMirand, Ingo Runkel
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
In [M. De Renzi, A. Gainutdinov, N. Geer, B. PatureauMirand and I. Runkel, 3dimensional TQFTs from nonsemisimple modular categories, preprint (2019), arXiv:1912.02063[math.GT]], we constructed 3dimensional topological quantum field theories (TQFTs) using not necessarily semisimple modular categories. Here, we study projective representations of mapping class groups of surfaces defined by these TQFTs, and we express the action of a set of generators through the algebraic data of the underlying modular category [math]. This allows us to prove that the projective representations induced from the nonsemisimple TQFTs of the above reference are equivalent to those obtained by Lyubashenko via generators and relations in [V. Lyubashenko, Invariants of 3manifolds and projective representations of mapping class groups via quantum groups at roots of unity, Comm. Math. Phys. 172(3) (1995) 467–516, arXiv:hepth/9405167]. Finally, we show that, when [math] is the category of finitedimensional representations of the small quantum group of [math], the action of all Dehn twists for surfaces without marked points has infinite order.
Citation: Communications in Contemporary Mathematics
PubDate: 20211023T07:00:00Z
DOI: 10.1142/S0219199721500917

 Explicit values of the oscillation bounds for linear delay differential
equations with monotone argument
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Authors: Mihály Pituk, Ioannis P. Stavroulakis, John Ioannis Stavroulakis
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
The problem of finding the oscillation bounds for firstorder linear delay differential equations has been in the focus of the oscillation theory for a long time. Although numerous estimates for the oscillation bounds are available in the literature, their explicit values were not known. In this paper, we give the oscillation bounds explicitly in terms of the real branches of the Lambert [math] function.
Citation: Communications in Contemporary Mathematics
PubDate: 20211022T07:00:00Z
DOI: 10.1142/S0219199721500875

 Perturbation theory of the quadratic Lotka–Volterra double center

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Authors: Jean–Pierre Françoise, Lubomir Gavrilov
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We revisit the bifurcation theory of the Lotka–Volterra quadratic system X0 : ẋ = −y − x2 + y2,ẏ = x − 2xy with respect to arbitrary quadratic deformations. The system has a double center, which is moreover isochronous. We show that the deformed system can have at most two limit cycles on the finite plane, with possible distribution [math], where [math]. Our approach is based on the study of pairs of bifurcation functions associated to the centers, expressed in terms of iterated path integrals of length two.
Citation: Communications in Contemporary Mathematics
PubDate: 20211014T07:00:00Z
DOI: 10.1142/S0219199721500644

 The eigenvalue problem for Hessian type operator

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Authors: Xinqun Mei
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
In this paper, we establish a global [math] estimates for a Hessian type equation with homogeneous Dirichlet boundary. By the method of sub and sup solution, we get an existence and uniqueness result for the eigenvalue problem of a Hessian type operator.
Citation: Communications in Contemporary Mathematics
PubDate: 20211014T07:00:00Z
DOI: 10.1142/S0219199721500899

 The Varchenko matrix for topoplane arrangements

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Authors: Hery Randriamaro
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
A topoplane is a mild deformation of a linear hyperplane contained in a given smooth manifold that is homeomorphic to a Euclidean space. We consider solidly transsective topoplane arrangements. These collections generalize pseudohyperplane arrangements. Even though the topoplane arrangements locally look like hyperplane arrangements, the global coning procedure is absent here. The main aim of the paper is to introduce the Varchenko matrix in this context and show that the determinant has a similar factorization as in the case of hyperplane arrangements. We achieve this by suitably generalizing the strategy of Aguiar and Mahajan. We also study a system of linear equations introduced by them and describe its solution space in the context of topoplane arrangements.
Citation: Communications in Contemporary Mathematics
PubDate: 20211008T07:00:00Z
DOI: 10.1142/S0219199721500863

 Bernstein–Sato functional equations, [math]filtrations, and multiplier
ideals of direct summands
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Authors: Josep Àlvarez Montaner, Daniel J. Hernández, Jack Jeffries, Luis NúñezBetancourt, Pedro Teixeira, Emily E. Witt
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
This paper investigates the existence and properties of a Bernstein–Sato functional equation in nonregular settings. In particular, we construct [math]modules in which such formal equations can be studied. The existence of the Bernstein–Sato polynomial for a direct summand of a polynomial over a field is proved in this context. It is observed that this polynomial can have zero as a root, or even positive roots. Moreover, a theory of [math]filtrations is introduced for nonregular rings, and the existence of these objects is established for what we call differentially extensible summands. This family of rings includes toric, determinantal, and other invariant rings. This new theory is applied to the study of multiplier ideals and Hodge ideals of singular varieties. Finally, we extend known relations among the objects of interest in the smooth case to the setting of singular direct summands of polynomial rings.
Citation: Communications in Contemporary Mathematics
PubDate: 20211006T07:00:00Z
DOI: 10.1142/S0219199721500838

 A functorial approach to monomorphism categories for species I

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Authors: Nan Gao, Julian Külshammer, Sondre Kvamme, Chrysostomos Psaroudakis
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We introduce a very general extension of the monomorphism category as studied by Ringel and Schmidmeier which in particular covers generalized species over locally bounded quivers. We prove that analogues of the kernel and cokernel functor send almost split sequences over the path algebra and the preprojective algebra to split or almost split sequences in the monomorphism category. We derive this from a general result on preservation of almost split morphisms under adjoint functors whose counit is a monomorphism. Despite of its generality, our monomorphism categories still allow for explicit computations as in the case of Ringel and Schmidmeier.
Citation: Communications in Contemporary Mathematics
PubDate: 20211005T07:00:00Z
DOI: 10.1142/S0219199721500693

 WKB analysis of the logarithmic nonlinear Schrödinger equation in an
analytic framework
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Authors: Guillaume Ferriere
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We are interested in a WKB analysis of the logarithmic nonlinear Schrödinger equation with “Riemannlike” variables in an analytic framework in semiclassical regime. We show that the Cauchy problem is locally well posed uniformly in the semiclassical constant and that the semiclassical limit can be performed. In particular, our framework is not only compatible with the Gross–Pitaevskii equation with logarithmic nonlinearity, but also allows initial data (and solutions) which can converge to [math] at infinity.
Citation: Communications in Contemporary Mathematics
PubDate: 20210929T07:00:00Z
DOI: 10.1142/S0219199721500826

 Knudsen type group for time in [math] and related Boltzmann type equations

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Authors: JörgUwe Löbus
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We consider certain Boltzmann type equations on a bounded physical and a bounded velocity space under the presence of both reflective as well as diffusive boundary conditions. We introduce conditions on the shape of the physical space and on the relation between the reflective and the diffusive part in the boundary conditions such that the associated Knudsen type semigroup can be extended to time [math]. Furthermore, we provide conditions under which there exists a unique global solution to a Boltzmann type equation for time [math] or for time [math] for some [math] which is independent of the initial value at time 0. Depending on the collision kernel, [math] can be arbitrarily small.
Citation: Communications in Contemporary Mathematics
PubDate: 20210925T07:00:00Z
DOI: 10.1142/S0219199721500723

 Reverse Alexandrov–Fenchel inequalities for zonoids

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Authors: Károly J. Böröczky, Daniel Hug
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
The Alexandrov–Fenchel inequality bounds from below the square of the mixed volume [math] of convex bodies [math] in [math] by the product of the mixed volumes [math] and [math]. As a consequence, for integers [math] with [math] the product [math] of suitable powers of the volumes [math] of the convex bodies [math], [math], is a lower bound for the mixed volume [math], where [math] is the multiplicity with which [math] appears in the mixed volume. It has been conjectured by Betke and Weil that there is a reverse inequality, that is, a sharp upper bound for the mixed volume [math] in terms of the product of the intrinsic volumes [math], for [math]. The case where [math], [math], [math] has recently been settled by the present authors (2020). The case where [math], [math], [math] has been treated by ArtsteinAvidan et al. under the assumption that [math] is a zonoid and [math] is the Euclidean unit ball. The case where [math], [math] is the unit ball and [math] are zonoids has been considered by Hug and Schneider. Here, we substantially generalize these previous contributions, in cases where most of the bodies are zonoids, and thus we provide further evidence supporting the conjectured reverse Alexandrov–Fenchel inequality. The equality cases in all considered inequalities are characterized. More generally, stronger stability results are established as well.
Citation: Communications in Contemporary Mathematics
PubDate: 20210925T07:00:00Z
DOI: 10.1142/S021919972150084X

 1Laplacian type problems with strongly singular nonlinearities and
gradient terms
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Authors: Daniela Giachetti, Francescantonio Oliva, Francesco Petitta
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We show optimal existence, nonexistence and regularity results for nonnegative solutions to Dirichlet problems as − Δ1u = g(u) Du + h(u)fin Ω,u = 0 on ∂Ω, where [math] is an open bounded subset of [math], [math] belongs to [math], and [math] and [math] are continuous functions that may blow up at zero. As a noteworthy fact we show how a nontrivial interaction mechanism between the two nonlinearities [math] and [math] produces remarkable regularizing effects on the solutions. The sharpness of our main results is discussed through the use of appropriate explicit examples.
Citation: Communications in Contemporary Mathematics
PubDate: 20210918T07:00:00Z
DOI: 10.1142/S0219199721500814

 Projective dynamics and an integrable Boltzmann billiard model

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Authors: Lei Zhao
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
The aim of this note is to explain the integrability of an integrable Boltzmann billiard model, previously established by Gallavotti and Jauslin [G. Gallavotti and I. Jauslin, A theorem on Ellipses, an integrable system and a theorem of Boltzmann, preprint (2020); arXiv:2008.01955], alternatively via the viewpoint of projective dynamics. We show that the energy of a corresponding spherical problem leads to an additional first integral of the system equivalent to Gallavotti–Jauslin’s first integral. The approach also leads to a family of integrable billiard models in the plane and on the sphere defined through the planar and spherical Kepler–Coulomb problems.
Citation: Communications in Contemporary Mathematics
PubDate: 20210917T07:00:00Z
DOI: 10.1142/S0219199721500851

 Universal Hardy–Sobolev inequalities on hypersurfaces of Euclidean
space
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Authors: Xavier Cabré, Pietro Miraglio
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
In this paper, we study Hardy–Sobolev inequalities on hypersurfaces of [math], all of them involving a mean curvature term and having universal constants independent of the hypersurface. We first consider the celebrated Sobolev inequality of Michael–Simon and Allard, in our codimension one framework. Using their ideas, but simplifying their presentations, we give a quick and easytoread proof of the inequality. Next, we establish two new Hardy inequalities on hypersurfaces. One of them originates from an application to the regularity theory of stable solutions to semilinear elliptic equations. The other one, which we prove by exploiting a “ground state” substitution, improves the Hardy inequality of Carron. With this same method, we also obtain an improved Hardy or Hardy–Poincaré inequality.
Citation: Communications in Contemporary Mathematics
PubDate: 20210916T07:00:00Z
DOI: 10.1142/S0219199721500632

 On the global existence and timedecay rates for a parabolic–hyperbolic
model arising from chemotaxis
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Authors: Fuyi Xu, Xinliang Li
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
In this paper, we are concerned with the study of the Cauchy problem for a parabolic–hyperbolic model arising from chemotaxis in any dimension [math]. We first prove the global existence of the model in [math] critical regularity framework with respect to the scaling of the associated equations. Furthermore, under a mild additional decay assumption involving only the low frequencies of the data, we also establish the timedecay rates for the constructed global solutions. Our analyses mainly rely on Fourier frequency localization technology and on a refined timeweighted energy inequalities in different frequency regimes.
Citation: Communications in Contemporary Mathematics
PubDate: 20210906T07:00:00Z
DOI: 10.1142/S0219199721500784

 Quasiclean rings and strongly quasiclean rings

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Authors: Gaohua Tang, Huadong Su, Pingzhi Yuan
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
An element [math] of a ring [math] is called a quasiidempotent if [math] for some central unit [math] of [math], or equivalently, [math], where [math] is a central unit and [math] is an idempotent of [math]. A ring [math] is called a quasiBoolean ring if every element of [math] is quasiidempotent. A ring [math] is called (strongly) quasiclean if each of its elements is a sum of a quasiidempotent and a unit (that commute). These rings are shown to be a natural generalization of the clean rings and strongly clean rings. An extensive study of (strongly) quasiclean rings is conducted. The abundant examples of (strongly) quasiclean rings state that the class of (strongly) quasiclean rings is very larger than the class of (strongly) clean rings. We prove that an indecomposable commutative semilocal ring is quasiclean if and only if it is local or [math] has no image isomorphic to [math]; For an indecomposable commutative semilocal ring [math] with at least two maximal ideals, [math]([math]) is strongly quasiclean if and only if [math] is quasiclean if and only if [math], [math] is a maximal ideal of [math]. For a prime [math] and a positive integer [math], [math] is strongly quasiclean if and only if [math]. Some open questions are also posed.
Citation: Communications in Contemporary Mathematics
PubDate: 20210903T07:00:00Z
DOI: 10.1142/S0219199721500796

 Global Hölder continuity of solutions to quasilinear equations with
Morrey data
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Authors: SunSig Byun, Dian K. Palagachev, Pilsoo Shin
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We deal with general quasilinear divergenceform coercive operators whose prototype is the [math]Laplacean operator. The nonlinear terms are given by Carathéodory functions and satisfy controlled growth structure conditions with data belonging to suitable Morrey spaces. The fairly nonregular boundary of the underlying domain is supposed to satisfy a capacity density condition which allows domains with exterior corkscrew property. We prove global boundedness and Hölder continuity up to the boundary for the weak solutions of such equations, generalizing this way the classical [math]result of Ladyzhenskaya and Ural’tseva to the settings of the Morrey spaces.
Citation: Communications in Contemporary Mathematics
PubDate: 20210827T07:00:00Z
DOI: 10.1142/S0219199721500620

 A microlocal approach to renormalization in stochastic PDEs

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Authors: Claudio Dappiaggi, Nicolò Drago, Paolo Rinaldi, Lorenzo Zambotti
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We present a novel framework for the study of a large class of nonlinear stochastic partial differential equations (PDEs), which is inspired by the algebraic approach to quantum field theory. The main merit is that, by realizing random fields within a suitable algebra of functionalvalued distributions, we are able to use techniques proper of microlocal analysis which allow us to discuss renormalization and its associated freedom without resorting to any regularization scheme and to the subtraction of infinities. As an example of the effectiveness of the approach we apply it to the perturbative analysis of the stochastic [math] model.
Citation: Communications in Contemporary Mathematics
PubDate: 20210827T07:00:00Z
DOI: 10.1142/S0219199721500759

 Nonwellordered lower and upper solutions for semilinear systems of PDEs

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Authors: Alessandro Fonda, Giuliano Klun, Andrea Sfecci
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We prove existence results for systems of boundary value problems involving elliptic secondorder differential operators. The assumptions involve lower and upper solutions, which may be either wellordered, or not at all. The results are stated in an abstract framework, and can be translated also for systems of parabolic type.
Citation: Communications in Contemporary Mathematics
PubDate: 20210827T07:00:00Z
DOI: 10.1142/S0219199721500802

 Critical Gagliardo–Nirenberg, Trudinger, Brezis–Gallouet–Wainger
inequalities on graded groups and ground states
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Authors: Michael Ruzhansky, Nurgissa Yessirkegenov
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
In this paper, we investigate critical Gagliardo–Nirenberg, Trudingertype and Brezis–Gallouet–Wainger inequalities associated with the positive Rockland operators on graded Lie groups, which include the cases of [math], Heisenberg, and general stratified Lie groups. As an application, using the critical Gagliardo–Nirenberg inequality, the existence of least energy solutions of nonlinear Schrödinger type equations is obtained. We also express the best constant in the critical Gagliardo–Nirenberg and Trudinger inequalities in the variational form as well as in terms of the ground state solutions of the corresponding nonlinear subelliptic equations. The obtained results are already new in the setting of general stratified Lie groups (homogeneous Carnot groups). Among new technical methods, we also extend Folland’s analysis of Hölder spaces from stratified Lie groups to general homogeneous Lie groups.
Citation: Communications in Contemporary Mathematics
PubDate: 20210824T07:00:00Z
DOI: 10.1142/S0219199721500619

 Complexity of virtual multistrings

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Authors: David Freund
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
A virtual[math]string [math] consists of a closed, oriented surface [math] and a collection [math] of [math] oriented, closed curves immersed in [math]. We consider virtual [math]strings up to virtual homotopy, i.e. stabilizations, destabilizations, stable homeomorphism, and homotopy. Recently, Cahn proved that any virtual 1string can be virtually homotoped to a minimally filling and crossingminimal representative by monotonically decreasing both genus and the number of selfintersections. We generalize her result to the case of nonparallel virtual [math]strings. Cahn also proved that any two crossingirreducible representatives of a virtual 1string are related by isotopy, Type 3 moves, stabilizations, destabilizations, and stable homeomorphism. Kadokami claimed that this held for virtual [math]strings in general, but Gibson found a counterexample for 5strings. We show that Kadokami’s statement holds for nonparallel [math]strings and exhibit a counterexample for general virtual 3strings.
Citation: Communications in Contemporary Mathematics
PubDate: 20210824T07:00:00Z
DOI: 10.1142/S0219199721500668

 On the positivity of the first Chern class of an Ulrich vector bundle

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Authors: Angelo Felice Lopez
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We study the positivity of the first Chern class of a rank [math] Ulrich vector bundle [math] on a smooth [math]dimensional variety [math]. We prove that [math] is very positive on every subvariety not contained in the union of lines in [math]. In particular, if [math] is not covered by lines we have that [math] is big and [math]. Moreover we classify rank [math] Ulrich vector bundles [math] with [math] on surfaces and with [math] or [math] on threefolds (with some exceptions).
Citation: Communications in Contemporary Mathematics
PubDate: 20210824T07:00:00Z
DOI: 10.1142/S0219199721500711

 Estimates of the topological degree of a class of piecewise linear maps
with applications
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Authors: Laura Poggiolini, Marco Spadini
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We provide some new estimates for the topological degree of a class of continuous and piecewise linear maps based on a classical integral computation formula. We provide applications to some nonlinear problems that exhibit a local [math] structure.
Citation: Communications in Contemporary Mathematics
PubDate: 20210824T07:00:00Z
DOI: 10.1142/S0219199721500735

 On the classification of Smale–Barden manifolds with Sasakian
structures
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Authors: Vicente Muñoz, Aleksy Tralle
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
Smale–Barden manifolds [math] are classified by their second homology [math] and the Barden invariant [math]. It is an important and difficult question to decide when [math] admits a Sasakian structure in terms of these data. In this work, we show methods of doing this. In particular, we realize all [math] with [math] and [math] provided that [math], [math], [math] are pairwise coprime. We give a complete solution to the problem of the existence of Sasakian structures on rational homology spheres in the class of semiregular Sasakian structures. Our method allows us to completely solve the following problem of Boyer and Galicki in the class of semiregular Sasakian structures: determine which simply connected rational homology 5spheres admit negative Sasakian structures.
Citation: Communications in Contemporary Mathematics
PubDate: 20210824T07:00:00Z
DOI: 10.1142/S0219199721500772

 Closed geodesics on surfaces without conjugate points

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Authors: Vaughn Climenhaga, Gerhard Knieper, Khadim War
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We obtain Margulistype asymptotic estimates for the number of free homotopy classes of closed geodesics on certain manifolds without conjugate points. Our results cover all compact surfaces of genus at least 2 without conjugate points.
Citation: Communications in Contemporary Mathematics
PubDate: 20210821T07:00:00Z
DOI: 10.1142/S021919972150067X

 Torsors for difference algebraic groups

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Authors: Annette Bachmayr, Michael Wibmer
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We introduce a cohomology set for groups defined by algebraic difference equations and show that it classifies torsors under the group action. This allows us to compute all torsors for large classes of groups. We also present an application to the Galois theory of differential equations depending on a discrete parameter.
Citation: Communications in Contemporary Mathematics
PubDate: 20210821T07:00:00Z
DOI: 10.1142/S0219199721500681

 Singular fibers of very general Lagrangian fibrations

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Authors: Justin Sawon
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
Let [math] be a (holomorphic) Lagrangian fibration that is very general in the moduli space of Lagrangian fibrations. We conjecture that the singular fibers in codimension one must be semistable degenerations of abelian varieties. We prove a partial result towards this conjecture, and describe an example that provides further evidence.
Citation: Communications in Contemporary Mathematics
PubDate: 20210821T07:00:00Z
DOI: 10.1142/S021919972150070X

 Multiple positive solutions for a [math]Laplace Benci–Cerami type
problem ([math]), via Morse theory
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Authors: Giuseppina Vannella
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
Let us consider the quasilinear problem (Pðœ€) −ðœ€pΔ pu + up−1 = f(u)in Ω,u> 0 in Ω,u = 0 on ∂Ω, where [math] is a bounded domain in [math] with smooth boundary, [math], [math], [math] is a parameter and [math] is a continuous function with [math], having a subcritical growth. We prove that there exists [math] such that, for every [math], [math] has at least [math] solutions, possibly counted with their multiplicities, where [math] is the Poincaré polynomial of [math]. Using Morse techniques, we furnish an interpretation of the multiplicity of a solution, in terms of positive distinct solutions of a quasilinear equation on [math], approximating [math].
Citation: Communications in Contemporary Mathematics
PubDate: 20210819T07:00:00Z
DOI: 10.1142/S0219199721500656

 Wellposedness in weighted spaces for the generalized Hartree equation
with [math]
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Authors: Anudeep K. Arora, Oscar Riaño, Svetlana Roudenko
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We investigate the wellposedness in the generalized Hartree equation [math], [math], [math], for low powers of nonlinearity, [math]. We establish the local wellposedness for a class of data in weighted Sobolev spaces, following ideas of Cazenave and Naumkin, Local existence, global existence, and scattering for the nonlinear Schrödinger equation, Comm. Contemp. Math. 19(2) (2017) 1650038. This crucially relies on the boundedness of the Riesz transform in weighted Lebesgue spaces. As a consequence, we obtain a class of data that exists globally, moreover, scatters in positive time. Furthermore, in the focusing case in the [math]supercritical setting we obtain a subset of locally wellposed data with positive energy, which blows up in finite time.
Citation: Communications in Contemporary Mathematics
PubDate: 20210819T07:00:00Z
DOI: 10.1142/S0219199721500747

 Invariant Hermitian forms on vertex algebras

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Authors: Victor G. Kac, Pierluigi Möseneder Frajria, Paolo Papi
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We study invariant Hermitian forms on a conformal vertex algebra and on their (twisted) modules. We establish existence of a nonzero invariant Hermitian form on an arbitrary [math]algebra. We show that for a minimal simple [math]algebra [math] this form can be unitary only when its [math]grading is compatible with parity, unless [math] “collapses” to its affine subalgebra.
Citation: Communications in Contemporary Mathematics
PubDate: 20210811T07:00:00Z
DOI: 10.1142/S0219199721500590

 A Liouvilletype theorem for fully nonlinear CR invariant equations on the
Heisenberg group
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Authors: Bo Wang
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We obtain a Liouvilletype theorem for cylindrical viscosity solutions of fully nonlinear CR invariant equations on the Heisenberg group. As a byproduct, we also prove a comparison principle with finite singularities for viscosity solutions to more general fully nonlinear operators on the Heisenberg group.
Citation: Communications in Contemporary Mathematics
PubDate: 20210811T07:00:00Z
DOI: 10.1142/S0219199721500607

 Vectorvalued Maclaurin inequalities

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Authors: Silouanos Brazitikos, Finlay McIntyre
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We investigate a Maclaurin inequality for vectors and its connection to an Aleksandrovtype inequality for parallelepipeds.
Citation: Communications in Contemporary Mathematics
PubDate: 20210726T07:00:00Z
DOI: 10.1142/S0219199721500449

 [math]metrics and conformal metrics with [math]bounded scalar curvature

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Authors: Conghan Dong, Yuxiang Li, Ke Xu
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
A [math]metric on an [math]dimensional closed Riemannian manifold naturally induces a distance function, provided [math] is sufficiently close to [math]. If a sequence of metrics [math] converges in [math] to a limit metric [math], then the corresponding distance functions [math] subconverge to a limit distance function [math], which satisfies [math]. As an application, we show that the above convergence result applies to a sequence of conformal metrics with [math]bounded scalar curvatures, under certain geometric assumptions. In particular, in this special setting, the limit distance function [math] actually coincides with [math].
Citation: Communications in Contemporary Mathematics
PubDate: 20210624T07:00:00Z
DOI: 10.1142/S0219199721500474

 Boundary singularities of semilinear elliptic equations with LerayHardy
potential
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Authors: Huyuan Chen, Laurent Véron
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We study existence and uniqueness of solutions of ([math]) [math] in [math], [math] on [math], where [math] is a bounded smooth domain such that [math], [math] is a constant, [math] a continuous nondecreasing function satisfying some integral growth condition and [math] and [math] two Radon measures, respectively, in [math] and on [math]. We show that the situation differs considerably according the measure is concentrated at [math] or not. When [math] is a power we introduce a capacity framework which provides necessary and sufficient conditions for the solvability of problem ([math]).
Citation: Communications in Contemporary Mathematics
PubDate: 20210621T07:00:00Z
DOI: 10.1142/S0219199721500516

 Relaxed highestweight modules II: Classifications for affine vertex
algebras
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Authors: Kazuya Kawasetsu, David Ridout
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
This is the second of a series of papers devoted to the study of relaxed highestweight modules over affine vertex algebras and Walgebras. The first [K. Kawasetsu and D. Ridout, Relaxed highestweight modules I: Rank [math] cases, Commun. Math. Phys. 368 (2019) 627–663, arXiv:1803.01989 [math.RT]] studied the simple “rank[math]” affine vertex superalgebras [math] and [math], with the main results including the first complete proofs of certain conjectured character formulae (as well as some entirely new ones). Here, we turn to the question of classifying relaxed highestweight modules for simple affine vertex algebras of arbitrary rank. The key point is that this can be reduced to the classification of highestweight modules by generalizing Olivier Mathieu’s coherent families [O. Mathieu, Classification of irreducible weight modules, Ann. Inst. Fourier[math]Grenoble[math] 50 (2000) 537–592]. We formulate this algorithmically and illustrate its practical implementation with several detailed examples. We also show how to use coherent family technology to establish the nonsemisimplicity of category [math] in one of these examples.
Citation: Communications in Contemporary Mathematics
PubDate: 20210614T07:00:00Z
DOI: 10.1142/S0219199721500371

 Morse inequalities at infinity for a resonant mean field equation

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Authors: Mohameden Ahmedou, Mohamed Ben Ayed
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
In this paper, we study the following mean field type equation: (MF) − Δgu = ϱ Keu ∫ΣKeudVg − 1inΣ, where [math] is a closed oriented surface of unit volume [math] = 1, [math] positive smooth function and [math], [math]. Building on the critical points at infinity approach initiated in [M. Ahmedou, M. Ben Ayed and M. Lucia, On a resonant mean field type equation: A “critical point at infinity” approach, Discrete Contin. Dyn. Syst. 37(4) (2017) 1789–1818] we develop, under generic condition on the function [math] and the metric [math], a full Morse theory by proving Morse inequalities relating the Morse indices of the critical points, the indices of the critical points at infinity, and the Betti numbers of the space of formal barycenters [math]. We derive from these Morse inequalities at infinity various new existence as well as multiplicity results of the mean field equation in the resonant case, i.e. [math].
Citation: Communications in Contemporary Mathematics
PubDate: 20210614T07:00:00Z
DOI: 10.1142/S0219199721500541

 Delta invariant of curves on rational surfaces I. An analytic approach

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Authors: José Ignacio CogolludoAgustín, Tamás László, Jorge MartínMorales, András Némethi
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We prove that if [math] is a reduced curve germ on a rational surface singularity [math] then its delta invariant can be recovered by a concrete expression associated with the embedded topological type of the pair [math]. Furthermore, we also identify it with another (a priori) embedded analytic invariant, which is motivated by the theory of adjoint ideals. Finally, we connect our formulae with the local correction term at singular points of the global Riemann–Roch formula, valid for projective normal surfaces, introduced by Blache.
Citation: Communications in Contemporary Mathematics
PubDate: 20210609T07:00:00Z
DOI: 10.1142/S0219199721500528

 Isospectral finiteness on convex cocompact hyperbolic 3manifolds

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Authors: Gilles Courtois, Inkang Kim
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
In this paper, we show that given a set of lengths of closed geodesics, there are only finitely many convex cocompact hyperbolic 3manifolds with that specified length spectrum with multiplicity, homotopy equivalent to a given 3manifold without a handlebody factor, up to orientation preserving isometries.
Citation: Communications in Contemporary Mathematics
PubDate: 20210609T07:00:00Z
DOI: 10.1142/S0219199721500589

 On mixed Hodge–Riemann relations for translationinvariant valuations
and Aleksandrov–Fenchel inequalities
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Authors: Jan Kotrbatý, Thomas Wannerer
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
A version of the Hodge–Riemann relations for valuations was recently conjectured and proved in several special cases by [J. Kotrbatý, On Hodge–Riemann relations for translationinvariant valuations, preprint (2020), arXiv:2009.00310]. The Lefschetz operator considered there arises as either the product or the convolution with the mixed volume of several Euclidean balls. Here we prove that in (co)degree one, the Hodge–Riemann relations persist if the balls are replaced by several different (centrally symmetric) convex bodies with smooth boundary with positive Gauss curvature. While these mixed Hodge–Riemann relations for the convolution directly imply the Aleksandrov–Fenchel inequality, they yield for the dual operation of the product a new inequality. This new inequality strengthens classical consequences of the Aleksandrov–Fenchel inequality for lowerdimensional convex bodies and generalizes some of the geometric inequalities recently discovered by [S. Alesker, Kotrbatý’s theorem on valuations and geometric inequalities for convex bodies, preprint (2020), arXiv:2010.01859].
Citation: Communications in Contemporary Mathematics
PubDate: 20210607T07:00:00Z
DOI: 10.1142/S0219199721500498

 Integrability of close encounters in the spatial restricted threebody
problem
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Authors: Franco Cardin, Massimiliano Guzzo
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We extend to the spatial case a technique of integration of the close encounters formulated by Tullio LeviCivita for the planar restricted threebody problem. We consider the Hamiltonian introduced in the Kustaanheimo–Stiefel regularization and construct a complete integral of the related Hamilton–Jacobi equation by means of a series convergent in a neighborhood of the collisions with the primary or secondary body.
Citation: Communications in Contemporary Mathematics
PubDate: 20210604T07:00:00Z
DOI: 10.1142/S0219199721500401

 Margulis lemma and Hurewicz fibration theorem on Alexandrov spaces

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Authors: Shicheng Xu, Xuchao Yao
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
1 We prove the generalized Margulis lemma with a uniform index bound on an Alexandrov [math]space [math] with curvature bounded below, i.e. small loops at [math] generate a subgroup of the fundamental group of the unit ball [math] that contains a nilpotent subgroup of index [math], where [math] is a constant depending only on the dimension [math]. The proof is based on the main ideas of V. Kapovitch, A. Petrunin and W. Tuschmann, and the following results: (1) We prove that any regular almost Lipschitz submersion constructed by Yamaguchi on a collapsed Alexandrov space with curvature bounded below is a Hurewicz fibration. We also prove that such fibration is uniquely determined up to a homotopy equivalence. (2) We give a detailed proof on the gradient push, improving the universal pushing time bound given by V. Kapovitch, A. Petrunin and W. Tuschmann, and justifying in a specific way that the gradient push between regular points can always keep away from extremal subsets.
Citation: Communications in Contemporary Mathematics
PubDate: 20210604T07:00:00Z
DOI: 10.1142/S0219199721500486

 Asymptotic behavior of solutions to differential equations with
[math]Laplacian
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Authors: Zuzana Došlá, Kōdai Fujimoto
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
This paper deals with the secondorder nonlinear differential equation [math] involving [math]Laplacian. The existence and the uniqueness of nonoscillatory solutions of this equation in certain classes, which are related with integral conditions, are studied. Moreover, a minimal set for solutions of this equation is introduced as an extension of the concept of principal solutions for linear equations. Obtained results extend the results for equations with [math]Laplacian.
Citation: Communications in Contemporary Mathematics
PubDate: 20210525T07:00:00Z
DOI: 10.1142/S0219199721500462

 The fibers of the ramified Prym map

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Authors: Paola Frediani, Juan Carlos Naranjo, Irene Spelta
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We study the ramified Prym map [math] which assigns to a ramified double cover of a smooth irreducible curve of genus [math] ramified in [math] points the Prym variety of the covering. We focus on the six cases where the dimension of the source is strictly greater than the dimension of the target giving a geometric description of the generic fiber. We also give an explicit example of a totally geodesic curve which is an irreducible component of a fiber of the Prym map [math].
Citation: Communications in Contemporary Mathematics
PubDate: 20210521T07:00:00Z
DOI: 10.1142/S0219199721500309

 Thermodynamic formalism for invariant measures in iterated function
systems with overlaps
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Authors: Eugen Mihailescu
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We study images of equilibrium (Gibbs) states for a class of noninvertible transformations associated to conformal iterated function systems (IFSs) with overlaps [math]. We prove exact dimensionality for these image measures, and find a dimension formula using their overlap numbers. In particular, we obtain a geometric formula for the dimension of selfconformal measures for IFSs with overlaps, in terms of the overlap numbers. This implies a necessary and sufficient condition for dimension drop. If [math] is a selfconformal measure, then [math] if and only if the overlap number [math]. Examples are also discussed.
Citation: Communications in Contemporary Mathematics
PubDate: 20210521T07:00:00Z
DOI: 10.1142/S0219199721500413

 Existence of embedded minimal disks

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Authors: Baris Coskunuzer
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We give a generalization of Meeks–Yau’s celebrated embeddedness result for the solutions of the Plateau problem for extreme curves.
Citation: Communications in Contemporary Mathematics
PubDate: 20210521T07:00:00Z
DOI: 10.1142/S0219199721500504

 A special Calabi–Yau degeneration with trivial monodromy

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Authors: Sławomir Cynk, Duco van Straten
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
A wellknown theorem of Kulikov, Persson and Pinkham states that a degeneration of a family of K3surfaces with trivial monodromy can be completed to a smooth family. We give a simple example that an analogous statement does not hold for Calabi–Yau threefolds.
Citation: Communications in Contemporary Mathematics
PubDate: 20210521T07:00:00Z
DOI: 10.1142/S0219199721500553

 Optimal nonhomogeneous improvements for the series expansion of
Hardy’s inequality
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Authors: Konstantinos T. Gkikas, Georgios Psaradakis
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We consider the series expansion of the [math]Hardy inequality of [G. Barbatis, S. Filippas and A. Tertikas, Series expansion for [math] Hardy inequalities, Indiana Univ. Math. J. 52 (2003) 171–190], in the particular case where the distance is taken from an interior point of a bounded domain in [math] and [math]. For [math] we improve it by adding as a remainder term an optimally weighted critical Sobolev norm, generalizing the [math] result of [S. Filippas and A. Tertikas, Optimizing improved Hardy inequalities, J. Funct. Anal. 192 (2002) 186–233] and settling the open question raised in [G. Barbatis, S. Filippas and A. Tertikas, A unified approach to improved [math] Hardy inequalities with best constants, Trans. Amer. Math. Soc. 356 (2004) 2169–2196]. For [math] we improve it by adding as a remainder term the optimally weighted Hölder seminorm, extending the Hardy–Morrey inequality of [G. Psaradakis, An optimal Hardy–Morrey inequality, Calc. Var. Partial Differential Equations 45 (2012) 421–441] to the series case.
Citation: Communications in Contemporary Mathematics
PubDate: 20210517T07:00:00Z
DOI: 10.1142/S0219199721500310

 Blowing up solutions for supercritical Yamabe problems on manifolds with
nonumbilic boundary
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Authors: Marco G. Ghimenti, Anna Maria Micheletti
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We build blowingup solutions for a supercritical perturbation of the Yamabe problem on manifolds with boundary, provided the dimension of the manifold is [math] and the tracefree part of the second fundamental form is nonzero everywhere on the boundary.
Citation: Communications in Contemporary Mathematics
PubDate: 20210517T07:00:00Z
DOI: 10.1142/S0219199721500358

 Compactness of Sobolevtype embeddings with measures

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Authors: Paola Cavaliere, Zdeněk Mihula
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We study compactness of embeddings of Sobolevtype spaces of arbitrary integer order into function spaces on domains in [math] with respect to upper Ahlfors regular measures [math], that is, Borel measures whose decay on balls is dominated by a power of their radius. Sobolevtype spaces as well as target spaces considered in this paper are built upon general rearrangementinvariant function norms. Several sufficient conditions for compactness are provided and these conditions are shown to be often also necessary, yielding sharp compactness results. It is noteworthy that the only connection between the measure [math] and the compactness criteria is how fast the measure decays on balls. Applications to Sobolevtype spaces built upon Lorentz–Zygmund norms are also presented.
Citation: Communications in Contemporary Mathematics
PubDate: 20210517T07:00:00Z
DOI: 10.1142/S021919972150036X

 Multiplicity of negativeenergy solutions for singularsuperlinear
Schrödinger equations with indefinitesign potential
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Authors: Ricardo Lima Alves, Carlos Alberto Santos, Kaye Silva
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We are concerned with the multiplicity of positive solutions for the singular superlinear and subcritical Schrödinger equation −Δu + V (x)u = λa(x)u−γ + b(x)upin ℝN, beyond the Nehari extremal value, as defined in [Y. Il’yasov, On extreme values of Nehari manifold via nonlinear Rayleigh’s quotient, Topol. Methods Nonlinear Anal. 49 (2017) 683–714], when the potential [math] may change its sign, [math], [math] is a positive continuous function, [math] and [math] is a real parameter. The main difficulties come from the nondifferentiability of the energy functional and the fact that the intersection of the boundaries of the connected components of the Nehari set is nonempty. We overcome these difficulties by exploring topological structures of that boundary to build nonempty sets whose boundaries have empty intersection and minimizing over them by controlling the energy level.
Citation: Communications in Contemporary Mathematics
PubDate: 20210517T07:00:00Z
DOI: 10.1142/S0219199721500425

 Small codimension components of the Hodge locus containing the Fermat
variety
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Authors: R. Villaflor Loyola
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We characterize the smallest codimension components of the Hodge locus of smooth degree [math] hypersurfaces of the projective space [math] of even dimension [math], passing through the Fermat variety (with [math]). They correspond to the locus of hypersurfaces containing a linear algebraic cycle of dimension [math]. Furthermore, we prove that among all the local Hodge loci associated to a nonlinear cycle passing through Fermat, the ones associated to a complete intersection cycle of type [math] attain the minimal possible codimension of their Zariski tangent spaces. This answers a conjecture of Movasati, and generalizes a result of Voisin about the first gap between the codimension of the components of the Noether–Lefschetz locus to arbitrary dimension, provided that they contain the Fermat variety.
Citation: Communications in Contemporary Mathematics
PubDate: 20210517T07:00:00Z
DOI: 10.1142/S021919972150053X

 Symplectic induction, prequantum induction, and prequantum multiplicities

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Authors: Tudor S. Ratiu, François Ziegler
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
Frobenius reciprocity asserts that induction from a subgroup and restriction to it are adjoint functors in categories of unitary [math]modules. In the 1980s, Guillemin and Sternberg established a parallel property of Hamiltonian [math]spaces, which (as we show) unfortunately fails to mirror the situation where more than one [math]module “quantizes” a given Hamiltonian [math]space. This paper offers evidence that the situation is remedied by working in the category of prequantum [math]spaces, where this ambiguity disappears; there, we define induction and multiplicity spaces and establish Frobenius reciprocity as well as the “induction in stages” property.
Citation: Communications in Contemporary Mathematics
PubDate: 20210517T07:00:00Z
DOI: 10.1142/S0219199721500577

 Local derivative estimates for the heat equation coupled to the Ricci flow

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Authors: Hong Huang
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
In this paper, we obtain local derivative estimates of Shitype for the heat equation coupled to the Ricci flow. As applications, in part combining with Kuang’s work, we extend some results of Zhang and Bamler–Zhang including distance distortion estimates and a backward pseudolocality theorem for Ricci flow on compact manifolds to the noncompact case.
Citation: Communications in Contemporary Mathematics
PubDate: 20210511T07:00:00Z
DOI: 10.1142/S0219199721500437

 Integrability of positive solutions of the integral system involving the
Riesz potentials
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Authors: Xiaoqian Liu, Yutian Lei
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
In this paper, we are concerned with the following integral system ui(x) =∫ℝnui+1pi+1(y) x − y n−α dy,i = 1, 2,…,m − 1, um(x) =∫ℝn u1p1(y) x − y n−αdy, m ≥ 1,n ≥ 1, where [math], [math], and [math] ([math]). When [math], such an integral system is associated with the best constants of the Hardy–Littlewood–Sobolev inequality. Chen, Li and their cooperators obtained optimal integrability intervals of the finite energy solutions by an argument of contraction and shrinking operators. This result is helpful to well understand the classification of the extremal functions of the Hardy–Littlewood–Sobolev inequality. The critical condition plays a key role in their work. In this paper, we study optimal integrability intervals when the positive solutions have some initial integrability. Now, the critical condition is not necessary, and we apply a weaker condition, the Serrintype condition, to establish some important relations of exponents which come into play to lift the regularity. In addition, we also generalize this result to the case of [math].
Citation: Communications in Contemporary Mathematics
PubDate: 20210510T07:00:00Z
DOI: 10.1142/S0219199721500322

 Nonexistence of dead cores in fully nonlinear elliptic models

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Authors: João Vitor da Silva, Disson dos Prazeres, Humberto Ramos Quoirin
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We investigate nonexistence of nonnegative dead core solutions for the problem Du γF(x,D2u) + a(x)uq = 0in Ω,u = 0on ∂Ω. Here, [math] is a bounded smooth domain, [math] is a fully nonlinear elliptic operator, [math] is a signchanging weight, [math], and [math]. We show that this problem has no nontrivial dead core solutions if either [math] is close enough to [math] or the negative part of [math] is sufficiently small. In addition, we obtain the existence and uniqueness of a positive solution under these conditions on [math] and [math]. Our results extend previous ones established in the semilinear case, and are new even for the simple model [math], where [math] is a uniformly elliptic and nonnegative matrix.
Citation: Communications in Contemporary Mathematics
PubDate: 20210510T07:00:00Z
DOI: 10.1142/S0219199721500395

 Exotic periodic points

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Authors: DucViet Vu
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We introduce the notion of exotic periodic points of a meromorphic selfmap. We then establish the expected asymptotic for the number of isolated or exotic periodic points for holomorphic selfmaps with a simple action on the cohomology groups on a compact Kähler manifold.
Citation: Communications in Contemporary Mathematics
PubDate: 20210505T07:00:00Z
DOI: 10.1142/S0219199721500292

 Spacelike translating solitons in Lorentzian product spaces: Nonexistence,
Calabi–Bernstein type results and examples
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Authors: Márcio Batista, Henrique F. de Lima
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We establish nonexistence results for complete spacelike translating solitons immersed in a Lorentzian product space [math], under suitable curvature constraints on the curvatures of the Riemannian base [math]. In particular, we obtain Calabi–Bernstein type results for entire translating graphs constructed over [math]. For this, we prove a version of the Omori–Yau’s maximum principle for complete spacelike translating solitons. Besides, we also use other two analytical tools related to an appropriate drift Laplacian: a parabolicity criterion and certain integrability properties. Furthermore, under the assumption that the base [math] is nonpositively curved, we close our paper constructing new examples of rotationally symmetric spacelike translating solitons embedded into [math].
Citation: Communications in Contemporary Mathematics
PubDate: 20210505T07:00:00Z
DOI: 10.1142/S0219199721500346

 An asymptotic expansion for the fractional [math]Laplacian and for
gradientdependent nonlocal operators
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Authors: Claudia Bucur, Marco Squassina
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
Mean value formulas are of great importance in the theory of partial differential equations: many very useful results are drawn, for instance, from the wellknown equivalence between harmonic functions and mean value properties. In the nonlocal setting of fractional harmonic functions, such an equivalence still holds, and many applications are nowadays available. The nonlinear case, corresponding to the [math]Laplace operator, has also been recently investigated, whereas the validity of a nonlocal, nonlinear, counterpart remains an open problem. In this paper, we propose a formula for the nonlocal, nonlinear mean value kernel, by means of which we obtain an asymptotic representation formula for harmonic functions in the viscosity sense, with respect to the fractional (variational) [math]Laplacian (for [math]) and to other gradientdependent nonlocal operators.
Citation: Communications in Contemporary Mathematics
PubDate: 20210326T07:00:00Z
DOI: 10.1142/S0219199721500218

 A singular periodic Ambrosetti–Prodi problem of Rayleigh equations
without coercivity conditions
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Authors: Xingchen Yu, Shiping Lu
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
In this paper, we use the Leray–Schauder degree theory to study the following singular periodic problems: [math], [math], where [math] is a continuous function with [math], function [math] is continuous with an attractive singularity at the origin, and [math] is a constant. We consider the case where the friction term [math] satisfies a local superlinear growth condition but not necessarily of the Nagumo type, and function [math] does not need to satisfy coercivity conditions. An Ambrosetti–Prodi type result is obtained.
Citation: Communications in Contemporary Mathematics
PubDate: 20210319T07:00:00Z
DOI: 10.1142/S0219199721500127

 [math]actions of Lie algebroids

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Authors: Olivier Brahic, Marco Zambon
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We consider homotopy actions of a Lie algebroid on a graded manifold, defined as suitable [math]algebra morphisms. On the “semidirect product” we construct a homological vector field that projects to the Lie algebroid. Our main theorem states that this construction is a bijection. Since several classical geometric structures can be described by homological vector fields as above, we can display many explicit examples, involving Lie algebroids (including extensions, representations up to homotopy and their cocycles) as well as transitive Courant algebroids.
Citation: Communications in Contemporary Mathematics
PubDate: 20210318T07:00:00Z
DOI: 10.1142/S0219199721500139

 Symmetric solutions for a 2D critical Dirac equation

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Authors: William Borrelli
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
In this paper, we show the existence of infinitely many symmetric solutions for a cubic Dirac equation in two dimensions, which appears as effective model in systems related to honeycomb structures. Such equation is critical for the Sobolev embedding and solutions are found by variational methods. Moreover, we also prove smoothness and exponential decay at infinity.
Citation: Communications in Contemporary Mathematics
PubDate: 20210316T07:00:00Z
DOI: 10.1142/S021919972150019X

 Serrin’s type problems in warped product manifolds

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Authors: Alberto Farina, Alberto Roncoroni
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
In this paper, we consider Serrin’s overdetermined problems in warped product manifolds and we prove Serrin’s type rigidity results by using the [math]function approach introduced by Weinberger.
Citation: Communications in Contemporary Mathematics
PubDate: 20210316T07:00:00Z
DOI: 10.1142/S0219199721500206

 Stationary measures on infinite graphs

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Authors: Alexandre Baraviera, Pedro Duarte, Maria Joana Torres
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We extend the theory of isospectral reductions of Bunimovich and Webb to infinite graphs, and describe an application of this extension to the problems of existence and approximation of stationary measures on infinite graphs.
Citation: Communications in Contemporary Mathematics
PubDate: 20210316T07:00:00Z
DOI: 10.1142/S0219199721500255

 Irreducible modules over the mirror Heisenberg–Virasoro algebra

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Authors: Dong Liu, Yufeng Pei, Limeng Xia, Kaiming Zhao
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
In this paper, we study irreducible modules over the mirror Heisenberg–Virasoro algebra [math], which is the semidirect product of the Virasoro algebra and the twisted Heisenberg algebra. We classify all HarishChandra modules over [math], i.e. irreducible modules with finitedimensional weight spaces. Every such module is either an irreducible highest or an irreducible lowest weight module, or an irreducible module of the intermediate series. Furthermore, we use a twisted version of Feigin–Fuchs construction of the Virasoro algebra to establish the simplicity criterion for Verma modules and obtain a classification of unitary irreducible highest weight modules over [math]. Finally, we determine all irreducible restricted [math]modules of level zero.
Citation: Communications in Contemporary Mathematics
PubDate: 20210316T07:00:00Z
DOI: 10.1142/S0219199721500267

 Sobolev–Kantorovich inequalities under [math] condition

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Authors: Vladimir I. Bogachev, Alexander V. Shaposhnikov, FengYu Wang
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We refine and generalize several interpolation inequalities bounding the [math] norm of a probability density with respect to the reference measure [math] by its Sobolev norm and the Kantorovich distance to [math] on a smooth weighted Riemannian manifold satisfying [math] condition.
Citation: Communications in Contemporary Mathematics
PubDate: 20210316T07:00:00Z
DOI: 10.1142/S0219199721500279

 A Pogorelov estimate and a Liouvilletype theorem to parabolic
[math]Hessian equations
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Authors: Yan He, Haoyang Sheng, Ni Xiang, Jiannan Zhang
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We consider Pogorelov estimates and Liouvilletype theorems to parabolic [math]Hessian equations of the form [math] in [math]. We derive that any [math]convexmonotone solution to [math] when [math] satisfies a quadratic growth and [math] must be a linear function of [math] plus a quadratic polynomial of [math].
Citation: Communications in Contemporary Mathematics
PubDate: 20210305T08:00:00Z
DOI: 10.1142/S0219199721500012

 Realvariable characterizations of local Orliczslice Hardy spaces with
application to bilinear decompositions
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Authors: Yangyang Zhang, Dachun Yang, Wen Yuan
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
Recently, both the bilinear decompositions [math] and [math] were established. In this paper, the authors prove in some sense that the former is sharp, while the latter is not. To this end, the authors first introduce the local Orliczslice Hardy space which contains [math], a variant of the local Orlicz Hardy space, introduced by Bonami and Feuto as a special case, and obtain its dual space by establishing its characterizations via atoms, finite atoms, and various maximal functions, which are new even for [math]. The relationship [math] is also clarified.
Citation: Communications in Contemporary Mathematics
PubDate: 20210305T08:00:00Z
DOI: 10.1142/S0219199721500048

 Expansions for distributional solutions of the elliptic equation in two
dimensions
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Authors: Jiayu Li, Fangshu Wan, Yunyan Yang
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
Assume [math] is a planar domain, and [math] is a locally bounded distributional solution to the elliptic equation −Δu = x 2βh(x)f(u)in Ω, where [math] is a constant, [math] and [math] are real analytic functions defined on [math] and the real line [math], respectively. We establish asymptotic expansions of [math] to arbitrary orders near [math], which complements the recent results of Han–Li–Li on the Yamabe equation, Guo–Li–Wanon the weighted Yamabe equation, and partly extends that of Guo–Wan–Yang on the Liouville equation in a punctured disc. Our method is a combination of a priori estimate and mathematical induction.
Citation: Communications in Contemporary Mathematics
PubDate: 20210304T08:00:00Z
DOI: 10.1142/S0219199721500188

 A Berestycki–Lions type result for a class of problems involving the
1Laplacian operator
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Authors: Claudianor O. Alves
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
In this work we prove a Berestycki–Lions type result for the following class of problems: −Δ1u + u u = f(u)inℝN,u ∈ BV (ℝN), where [math] is the [math]Laplacian operator and [math] is a continuous function satisfying some technical conditions. Here we apply variational methods by using [math]Laplacian problems and taking the limit when [math].
Citation: Communications in Contemporary Mathematics
PubDate: 20210304T08:00:00Z
DOI: 10.1142/S021919972150022X

 Uprolling unrolled quantum groups

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Authors: Thomas Creutzig, Matthew Rupert
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We construct families of commutative (super) algebra objects in the category of weight modules for the unrolled restricted quantum group [math] of a simple Lie algebra [math] at roots of unity, and study their categories of local modules. We determine their simple modules and derive conditions for these categories being finite, nondegenerate, and ribbon. Motivated by numerous examples in the [math] case, we expect some of these categories to compare nicely to categories of modules for vertex operator algebras. We focus in particular on examples expected to correspond to the higher rank triplet vertex algebra [math] of Feigin and Tipunin and the [math] algebras.
Citation: Communications in Contemporary Mathematics
PubDate: 20210304T08:00:00Z
DOI: 10.1142/S0219199721500231

 Higher dimensional elliptic fibrations and Zariski decompositions

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Authors: Antonella Grassi, David Wen
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
We study the existence and properties of birationally equivalent models for elliptically fibered varieties. In particular these have either the structure of Mori fiber spaces or, assuming some standard conjectures, minimal models with a Zariski decomposition compatible with the elliptic fibration. We prove relations between the birational invariants of the elliptically fibered variety, the base of the fibration and of its Jacobian.
Citation: Communications in Contemporary Mathematics
PubDate: 20210304T08:00:00Z
DOI: 10.1142/S0219199721500243

 Monotonicity and symmetry of positive solutions to fractional
[math]Laplacian equation
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Authors: Wei Dai, Zhao Liu, Pengyan Wang
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
In this paper, we are concerned with the following Dirichlet problem for nonlinear equations involving the fractional [math]Laplacian: (−Δ)pαu = f(x,u,∇u),u> 0in Ω,u ≡ 0 in ℝn∖Ω, where [math] is a bounded or an unbounded domain which is convex in [math]direction, and [math] is the fractional [math]Laplacian operator defined by (−Δ)pαu(x) = C n,α,pP.V.∫ℝn u(x) − u(y) p−2[u(x) − u(y)] x − y n+αp dy. Under some mild assumptions on the nonlinearity [math], we establish the monotonicity and symmetry of positive solutions to the nonlinear equations involving the fractional [math]Laplacian in both bounded and unbounded domains. Our results are extensions of Chen and Li [Maximum principles for the fractional pLaplacian and symmetry of solutions, Adv. Math. 335 (2018) 735–758] and Cheng et al. [The maximum principles for fractional Laplacian equations and their applications, Commun. Contemp. Math. 19(6) (2017) 1750018].
Citation: Communications in Contemporary Mathematics
PubDate: 20210219T08:00:00Z
DOI: 10.1142/S021919972150005X

 On solutions for a class of fractional Kirchhofftype problems with
Trudinger–Moser nonlinearity
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Authors: Manassés de Souza, Uberlandio B. Severo, Thiago Luiz do Rêgo
Abstract: Communications in Contemporary Mathematics, Ahead of Print.
In this paper, we prove the existence of at least three nontrivial solutions for the following class of fractional Kirchhofftype problems: (1 + b∥u∥2)[(−Δ)1/2u + V (x)u] = f(u)in Ω,u = 0 in ℝ∖Ω, where [math] is a constant, [math] is a bounded open interval, [math] is a continuous potential, the nonlinear term [math] has exponential growth of Trudinger–Moser type, [math] and [math] denotes the standard Gagliardo seminorm of the fractional Sobolev space [math]. More precisely, by exploring a minimization argument and the quantitative deformation lemma, we establish the existence of a nodal (or signchanging) solution and by means of the Mountain Pass Theorem, we get one nonpositive and one nonnegative ground state solution. Moreover, we show that the energy of the nodal solution is strictly larger than twice the ground state level. When we regard [math] as a positive parameter, we study the behavior of the nodal solutions as [math].
Citation: Communications in Contemporary Mathematics
PubDate: 20210209T08:00:00Z
DOI: 10.1142/S0219199721500024
