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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) 0219-1997 - ISSN (Online) 1793-6683
Published by World Scientific

[121 journals]

Author index Volume 24 (2022)
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  Abstract: Communications in Contemporary Mathematics, Volume 24, Issue 10, December 2022.
  
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-11-14T08:00:00Z
  DOI: 10.1142/S0219199722990012
  Issue No: Vol. 24, No. 10 (2022)
The isoperimetric problem on Riemannian manifolds via
Gromov–Hausdorff asymptotic analysis
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  Authors: Gioacchino Antonelli, Mattia Fogagnolo, Marco Pozzetta
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  In this paper, we prove the existence of isoperimetric regions of any volume in Riemannian manifolds with Ricci bounded below assuming Gromov–Hausdorff asymptoticity to the suitable simply connected model of constant sectional curvature. The previous result is a consequence of a general structure theorem for perimeter-minimizing sequences of sets of fixed volume on noncollapsed Riemannian manifolds with a lower bound on the Ricci curvature. We show that, without assuming any further hypotheses on the asymptotic geometry, all the mass and the perimeter lost at infinity, if any, are recovered by at most countably many isoperimetric regions sitting in some (possibly nonsmooth) Gromov–Hausdorff limits at infinity. The Gromov–Hausdorff asymptotic analysis allows us to recover and extend different previous existence theorems. While studying the isoperimetric problem in the smooth setting, the nonsmooth geometry naturally emerges, and thus our treatment combines techniques from both the theories.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-11-19T08:00:00Z
  DOI: 10.1142/S0219199722500687
Deforming vertex algebras by vertex bialgebras
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  Authors: Naihuan Jing, Fei Kong, Haisheng Li, Shaobin Tan
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  This is a continuation of a previous study initiated by the third author on nonlocal vertex bialgebras and smash product nonlocal vertex algebras. In this paper, we study a notion of right [math]-comodule nonlocal vertex algebra for a nonlocal vertex bialgebra [math] and give a construction of deformations of vertex algebras with a right [math]-comodule nonlocal vertex algebra structure and a compatible [math]-module nonlocal vertex algebra structure. We also give a construction of [math]-coordinated quasi modules for smash product nonlocal vertex algebras. As an example, we give a family of quantum vertex algebras by deforming the vertex algebras associated to non-degenerate even lattices.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-11-11T08:00:00Z
  DOI: 10.1142/S0219199722500675
Derivations of Köthe echelon algebras of order zero and infinity
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  Authors: Krzysztof Piszczek
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  We characterize — in terms of the Köthe matrix — amenable Köthe echelon algebras of order zero and infinity.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-11-11T08:00:00Z
  DOI: 10.1142/S0219199722500717
Local Hölder regularity of minimizers for nonlocal variational
problems
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  Authors: Matteo Novaga, Fumihiko Onoue
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  We study the regularity of solutions to a nonlocal variational problem, which is related to the image denoising model, and we show that, in two dimensions, minimizers have the same Hölder regularity as the original image. More precisely, if the datum is (locally) [math]-Hölder continuous for some [math], where [math] is a parameter related to the nonlocal operator, we prove that the solution is also [math]-Hölder continuous.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-11-10T08:00:00Z
  DOI: 10.1142/S0219199722500584
Note on the [math]-logarithmic Sobolev and [math]-Talagrand inequalities
on Carnot groups
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  Authors: Esther Bou Dagher
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  In the setting of Carnot groups, we prove the [math]-logarithmic Sobolev inequality for probability measures as a function of the Carnot–Carathéodory distance. As an application, we use the Hamilton–Jacobi equation in the setting of Carnot groups to prove the [math]-Talagrand inequality and hypercontractivity.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-11-10T08:00:00Z
  DOI: 10.1142/S0219199722500705
The ternary Goldbach problem with two Piatetski–Shapiro primes and a
prime with a missing digit
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  Authors: Helmut Maier, Michael Th. Rassias
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  Let γ∗ = 8 9 + 2 3 log 10 9 log 10 (≈ 0.919…). Let [math], [math] be fixed. Let also [math]. We prove on assumption of the Generalized Riemann Hypothesis that each sufficiently large odd integer [math] can be represented in the form N0 = p1 + p2 + p3, where the [math] are of the form [math], [math], for [math] and the decimal expansion of [math] does not contain the digit [math]. The proof merges methods of Maynard from his paper on the infinitude of primes with restricted digits, results of Balog and Friedlander on Piatetski-Shapiro primes and the Hardy–Littlewood circle method in two variables. This is the first result on the ternary Goldbach problem with primes of mixed type which involves primes with missing digits.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-10-15T07:00:00Z
  DOI: 10.1142/S0219199721501017
Nilpotency of skew braces and multipermutation solutions of the
Yang–Baxter equation
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  Authors: E. Jespers, A. Van Antwerpen, L. Vendramin
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  We study relations between different notions of nilpotency in the context of skew braces and applications to the structure of solutions to the Yang–Baxter equation. In particular, we consider annihilator nilpotent skew braces, an important class that turns out to be a brace-theoretic analog to the class of nilpotent groups. In this vein, several well-known theorems in group theory are proved in the more general setting of skew braces.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-10-15T07:00:00Z
  DOI: 10.1142/S021919972250064X
A Brezis–Oswald approach for mixed local and nonlocal operators
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  Authors: Stefano Biagi, Dimitri Mugnai, Eugenio Vecchi
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  In this paper, we provide necessary and sufficient conditions for the existence of a unique positive weak solution for some sublinear Dirichlet problems driven by the sum of a quasilinear local and a nonlocal operator, i.e. ℒp,s = −Δp + (−Δ)ps. Our main result is resemblant to the celebrated work by Brezis–Oswald [Remarks on sublinear elliptic equations, Nonlinear Anal. 10 (1986) 55–64]. In addition, we prove a regularity result of independent interest.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-10-04T07:00:00Z
  DOI: 10.1142/S0219199722500572
Arbitrarily fast grow-up rates in quasilinear Keller–Segel systems
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  Authors: Michael Winkler
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  The chemotaxis system ut = ∇⋅ (D(u)∇u) −∇⋅ (L(u)∇v),0 = Δv − μ + u,μ = 1 Ω ∫Ωu, is considered in a ball [math], [math], where the positive function [math] reflects suitably weak diffusion by satisfying [math] for some [math]. It is shown that whenever [math] is positive and satisfies [math] as [math], one can find a suitably regular nonlinearity [math] with the property that at each sufficiently large mass level [math] there exists a globally defined radially symmetric classical solution to a Neumann-type boundary value problem for (⋆) which satisfies ∥u(⋅,t)∥L∞(Ω) ψ(t) → +∞,as t →∞.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-10-04T07:00:00Z
  DOI: 10.1142/S0219199722500626
Sharp Morrey–Sobolev inequalities and eigenvalue problems on
Riemannian–Finsler manifolds with nonnegative Ricci curvature
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  Authors: Alexandru Kristály, Ágnes Mester, Ildikó I. Mezei
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  Combining the sharp isoperimetric inequality established by Z. Balogh and A. Kristály [Math. Ann., in press, doi:10.1007/s00208-022-02380-1] with an anisotropic symmetrization argument, we establish sharp Morrey–Sobolev inequalities on [math]-dimensional Finsler manifolds having nonnegative [math]-Ricci curvature. A byproduct of this method is a Hardy–Sobolev-type inequality in the same geometric setting. As applications, by using variational arguments, we guarantee the existence/multiplicity of solutions for certain eigenvalue problems and elliptic PDEs involving the Finsler–Laplace operator. Our results are also new in the Riemannian setting.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-10-04T07:00:00Z
  DOI: 10.1142/S0219199722500638
Long-time existence for a Whitham–Boussinesq system in two
dimensions
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  Authors: Achenef Tesfahun
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  This paper is concerned with a two-dimensional Whitham–Boussinesq system modeling surface waves of an inviscid incompressible fluid layer. We prove that the associated Cauchy problem is well-posed for initial data of low regularity, with existence time of scale [math], where [math] and [math] are small parameters related to the level of dispersion and nonlinearity, respectively. In particular, in the KdV regime [math]}, the existence time is of order [math]. The main ingredients in the proof are frequency loacalized dispersive estimates and bilinear Strichartz estimates that depend on the parameter [math].
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-10-04T07:00:00Z
  DOI: 10.1142/S0219199722500651
On the equivalence of classical Helmholtz equation and fractional
Helmholtz equation with arbitrary order
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  Authors: Xinyu Cheng, Dong Li, Wen Yang
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  We show the equivalence of the classical Helmholtz equation and the fractional Helmholtz equation with arbitrary order. This improves a recent result of Guan, Murugan and Wei [Helmholtz solutions for the Fractional Laplacian and other related operators, to appear in Comm. Contemp. Math.].
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-09-30T07:00:00Z
  DOI: 10.1142/S0219199722500365
On a Hardy–Sobolev-type inequality and applications
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  Authors: Jonison L. Carvalho, Marcelo F. Furtado, Everaldo S. Medeiros
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  In this paper, we prove a new Friedrich-type inequality. As an application, we derive some existence and non-existence results to the quasilinear elliptic problem with Robin boundary condition −div( ∇u N−2∇u) + h(x) u q−2u = λk(x) u p−2uin Ω, ∇u N−2(∇u ⋅ ν) + u N−2u = 0 on ∂Ω, where [math] is an exterior domain such that [math].
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-09-29T07:00:00Z
  DOI: 10.1142/S0219199722500377
On criticality theory for elliptic mixed boundary value problems in
divergence form
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  Authors: Yehuda Pinchover, Idan Versano
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  The paper is devoted to the study of positive solutions of a second-order linear elliptic equation in divergence form in a domain [math] that satisfy an oblique boundary condition on a portion of [math]. First, we study weak solutions for the degenerate mixed boundary value problem Pu = 0in Ω,Bu = 0 on ∂ΩRob,u = 0 on ∂ΩDir = ∂Ω∖∂ΩRob, (P,B) where [math] is a bounded Lipschitz domain, [math] is a relatively open portion of [math], and [math] is an oblique (Robin) boundary operator defined on [math] in a weak sense. In particular, we discuss the unique solvability of the above problem, the existence of a principal eigenvalue, and the existence of a minimal positive Green function. Then we establish a criticality theory for positive weak solutions of the operator [math] in a general domain [math] with no boundary condition on [math] and no growth condition at infinity. The paper extends results obtained by Pinchover and Saadon for classical solutions of such a problem, where stronger regularity assumptions on the coefficients of [math], and the boundary [math] are assumed.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-09-29T07:00:00Z
  DOI: 10.1142/S0219199722500511
Dimension of tensor network varieties
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  Authors: Alessandra Bernardi, Claudia De Lazzari, Fulvio Gesmundo
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  The tensor network variety is a variety of tensors associated to a graph and a set of positive integer weights on its edges, called bond dimensions. We determine an upper bound on the dimension of the tensor network variety. A refined upper bound is given in cases relevant for applications such as varieties of matrix product states and projected entangled pairs states. We provide a range (the “supercritical range”) of the parameters where the upper bound is sharp.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-09-26T07:00:00Z
  DOI: 10.1142/S0219199722500596
Subregular W-algebras of type [math]
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  Authors: Zachary Fehily
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  Subregular W-algebras are an interesting and increasingly important class of quantum hamiltonian reductions of affine vertex algebras. Here, we show that the [math] subregular W-algebra can be realized in terms of the [math] regular W-algebra and the half lattice vertex algebra [math]. This generalizes the realizations found for [math] and [math] in [D. Adamović, Realizations of simple affine vertex algebras and their modules: The cases [math] and [math], Comm. Math. Phys. 366 (2019) 1025–1067, arXiv:1711.11342 [math.QA]; D. Adamović, K. Kawasetsu and D. Ridout, A realization of the Bershadsky–Polyakov algebras and their relaxed modules, Lett. Math. Phys., 111 (2021) 1–30, arXiv:2007.00396 [math.QA]] and can be interpreted as an inverse quantum hamiltonian reduction in the sense of Adamović. We use this realization to explore the representation theory of [math] subregular W-algebras. Much of the structure encountered for [math] and [math] is also present for [math]. Particularly, the simple [math] subregular W-algebra at nondegenerate admissible levels can be realized purely in terms of the [math] minimal model vertex algebra and [math].
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-09-23T07:00:00Z
  DOI: 10.1142/S0219199722500493
Complete systems of inequalities relating the perimeter, the area and the
Cheeger constant of planar domains
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  Authors: Ilias Ftouhi
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  The object of the paper is to find complete systems of inequalities relating the perimeter [math], the area [math] and the Cheeger constant [math] of planar sets. To do so, we study the so-called Blaschke–Santaló diagram of the triplet [math] for different classes of domains: simply connected sets, convex sets and convex polygons with at most [math] sides. We completely determine the diagram in the latter cases except for the class of convex [math]-gons when [math] is odd: therein, we show that the boundary of the diagram is given by the graphs of two continuous and strictly increasing functions. An explicit formula for the lower one and a numerical method to obtain the upper one is provided. At last, some applications of the results are presented.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-09-21T07:00:00Z
  DOI: 10.1142/S0219199722500547
On profinite groups in which centralizers have bounded rank
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  Authors: Pavel Shumyatsky
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  The paper deals with profinite groups in which centralizers are of finite rank. For a positive integer [math] we prove that if [math] is a profinite group in which the centralizer of every nontrivial element has rank at most [math], then [math] is either a pro-[math] group or a group of finite rank. Further, if [math] is not virtually a pro-[math] group, then [math] is virtually of rank at most [math].
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-09-16T07:00:00Z
  DOI: 10.1142/S0219199722500559
Coupled and uncoupled sign-changing spikes of singularly perturbed
elliptic systems
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  Authors: Mónica Clapp, Mayra Soares
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  We study the existence and asymptotic behavior of solutions having positive and sign-changing components to the singularly perturbed system of elliptic equations − ðœ€2Δu i + ui = μi ui p−2u i +∑j=1j≠iℓλ ijβij uj αij ui βij−2u i,ui ∈ H01(Ω),u i≠0,i = 1,…,ℓ in a bounded domain [math] in [math], with [math], [math], [math], [math], [math], [math], [math] and [math]. If [math] is the unit ball we obtain solutions with a prescribed combination of positive and nonradial sign-changing components exhibiting two different types of asymptotic behavior as [math]: solutions whose limit profile is a rescaling of a solution with positive and nonradial sign-changing components of the limit system − Δui + ui = μi ui p−2u i +∑j=1j≠iℓλ ijβij uj αij ui βij−2u i, ui ∈ H1(ℝN),u i≠0,i = 1,…,ℓ and solutions whose limit profile is a solution of the uncoupled system, i.e. after rescaling and translation, the limit profile of the [math]th component is a positive or a nonradial sign-changing solution to the equation −Δu + u = μi u p−2u,u ∈ H1(ℝN),u≠0.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-09-14T07:00:00Z
  DOI: 10.1142/S0219199722500481
Reaction–diffusion on a time-dependent interval: Refining the notion of
‘critical length’
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  Authors: Jane Allwright
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  A reaction–diffusion equation is studied in a time-dependent interval whose length varies with time. The reaction term is either linear or of KPP type. On a fixed interval, it is well known that if the length is less than a certain critical value then the solution tends to zero. When the domain length may vary with time, we prove conditions under which the solution does and does not converge to zero in long time. We show that, even with the length always strictly less than the ‘critical length’, either outcome may occur. Examples are given. The proof is based on upper and lower estimates for the solution, which are derived in this paper for a general time-dependent interval.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-09-14T07:00:00Z
  DOI: 10.1142/S021919972250050X
Positive solutions for the Schrödinger–Poisson system with
steep potential well
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  Authors: Miao Du
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  In this paper, we consider the following Schrödinger–Poisson system − Δu + λV (x)u + μϕu = u p−2uin ℝ3, − Δϕ = u2 in ℝ3, where [math] are real parameters and [math]. Suppose that [math] represents a potential well with the bottom [math], the system has been widely studied in the case [math]. In contrast, no existence result of solutions is available for the case [math] due to the presence of the nonlocal term [math]. With the aid of the truncation technique and the parameter-dependent compactness lemma, we first prove the existence of positive solutions for [math] large and [math] small in the case [math]. Then we obtain the nonexistence of nontrivial solutions for [math] large and [math] large in the case [math]. Finally, we explore the decay rate of the positive solutions as [math] as well as their asymptotic behavior as [math] and [math].
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-09-14T07:00:00Z
  DOI: 10.1142/S0219199722500560
Existence of heteroclinic and saddle-type solutions for a class of
quasilinear problems in whole [math]
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  Authors: Claudianor O. Alves, Renan J. S. Isneri, Piero Montecchiari
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  In this work, we use variational methods to prove the existence of heteroclinic and saddle-type solutions for a class quasilinear elliptic equations of the form −ΔΦu + A(x,y)V′(u) = 0inℝ2, where [math] is a N-function, [math] is a periodic positive function and [math] is modeled on the Ginzburg–Landau potential. In particular, our main result includes the case of the potential [math], which reduces to the classical double well Ginzburg–Landau potential when [math], that is, when we are working with the Laplacian operator.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-09-14T07:00:00Z
  DOI: 10.1142/S0219199722500614
Existence and asymptotic behavior of non-normal conformal metrics on
[math] with sign-changing [math]-curvature
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  Authors: Chiara Bernardini
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  We consider the following prescribed [math]-curvature problem: Δ2u = (1 − x p)e4u on ℝ4,Λ :=∫ℝ4(1 − x p)e4udx∞.(1) We show that for every polynomial [math] of degree 2 such that [math], and for every [math], there exists at least one solution to problem (1) which assumes the form [math], where [math] behaves logarithmically at infinity. Conversely, we prove that all solutions to (1) have the form [math], where v(x) = 1 8π2∫ℝ4log y x − y (1 − y p)e4udy and [math] is a polynomial of degree at most two bounded from above. Moreover, if [math] is a solution to (1), it has the following asymptotic behavior: u(x) = − Λ 8π2log x + P + o(log x ),as x → +∞. As a consequence, we give a geometric characterization of solutions in terms of the scalar curvature at infinity of the associated conformal metric [math].
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-09-06T07:00:00Z
  DOI: 10.1142/S0219199722500535
Multiple positive and sign-changing solutions for a class of Kirchhoff
equations
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  Authors: Benniao Li, Wei Long, Aliang Xia
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  This paper is concerned with the existence of multiple non-radial positive and sign-changing solutions for the following Kirchhoff equation: −a + b∫ℝ3 ∇u 2 Δu + (1 + λQ(x))u = u p−2u,in ℝ3, (0.1) where [math] are constants, [math], [math] is a parameter, and [math] is a potential function. Under the assumption on [math] with exponential decay at infinity, we construct multi-peak positive and sign-changing solutions for problem (0.1) as [math] (or [math]), where the peaks concentrate at infinity.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-09-06T07:00:00Z
  DOI: 10.1142/S0219199722500602
Twisting functors and Gelfand–Tsetlin modules over semisimple Lie
algebras
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  Authors: Vyacheslav Futorny, Libor Křižka
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  We associate to an arbitrary positive root [math] of a complex semisimple finite-dimensional Lie algebra [math] a twisting endofunctor [math] of the category of [math]-modules. We apply this functor to generalized Verma modules in the category [math] and construct a family of [math]-Gelfand–Tsetlin modules with finite [math]-multiplicities, where [math] is a commutative [math]-subalgebra of the universal enveloping algebra of [math] generated by a Cartan subalgebra of [math] and by the Casimir element of the [math]-subalgebra corresponding to the root [math]. This covers classical results of Andersen and Stroppel when [math] is a simple root and previous results of the authors in the case when [math] is a complex simple Lie algebra and [math] is the maximal root of [math]. The significance of constructed modules is that they are Gelfand–Tsetlin modules with respect to any commutative [math]-subalgebra of the universal enveloping algebra of [math] containing [math]. Using the Beilinson–Bernstein correspondence we give a geometric realization of these modules together with their explicit description. We also identify a tensor subcategory of the category of [math]-Gelfand–Tsetlin modules which contains constructed modules as well as the category [math].
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-08-29T07:00:00Z
  DOI: 10.1142/S0219199722500316
Tightening and reversing the arithmetic-harmonic mean inequality for
symmetrizations of convex sets
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  Authors: René Brandenberg, Katherina von Dichter, Bernardo González Merino
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  This paper deals with four symmetrizations of a convex set [math]: the intersection, the harmonic and the arithmetic mean, and the convex hull of [math] and [math]. A well-known result of Firey shows that those means build up a subset-chain in the given order. On the one hand, we determine the dilatation factors, depending on the asymmetry of [math], to reverse the containments between any of those symmetrizations. On the other hand, we tighten the relations proven by Firey and show a stability result concerning those factors near the simplex.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-08-25T07:00:00Z
  DOI: 10.1142/S0219199722500456
Smoothness of the diffusion coefficients for particle systems in
continuous space
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  Authors: Arianna Giunti, Chenlin Gu, Jean-Christophe Mourrat, Maximilian Nitzschner
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  For a class of particle systems in continuous space with local interactions, we show that the asymptotic diffusion matrix is an infinitely differentiable function of the density of particles. Our method allows us to identify relatively explicit descriptions of the derivatives of the diffusion matrix in terms of correctors.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-08-24T07:00:00Z
  DOI: 10.1142/S0219199722500274
Curvature estimates for the continuity method
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  Authors: Hosea Wondo
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  We obtain curvature estimates for long-time solutions of the continuity method on compact Kähler manifolds with semi-ample canonical line bundles. In this setting, initiated in [G. La Nave and G. Tian, A continuity method to construct canonical metrics, Math. Ann. 365(3) (2016) 911–921; Y. A. Rubinstein, Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kähler metrics, Adv. Math. 218(5) (2008) 1526–1565], we adapt arguments from [F. T.-H. Fong and Y. Zhang, Local curvature estimates of long-time solutions to the Kähler–Ricci flow, Adv. Math. 375 (2020) 107416] for the Kähler–Ricci flow to this setup. As an application, we derive curvature bounds for general metrics on product manifolds.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-08-24T07:00:00Z
  DOI: 10.1142/S0219199722500420
Blowup and ill-posedness for the complex, periodic KdV equation
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  Authors: J. L. Bona, F. B. Weissler
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  This paper is concerned with complex-valued solutions of the Korteweg–de Vries equation. Interest will be focused upon the initial-value problem with initial data that is periodic in space. Derived here are results of local and global well-posedness, singularity formation in finite time and, perhaps surprisingly, results of non-existence. The overall picture is notably different from the situation that obtains for real-valued solutions.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-08-24T07:00:00Z
  DOI: 10.1142/S0219199722500444
An antimaximum principle for periodic solutions of a forced oscillator
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  Authors: Alain Albouy, Antonio J. Ureña
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  Consider the equation of the linear oscillator [math], where the forcing term [math] is [math]-periodic and positive. We show that the existence of a periodic solution implies the existence of a positive solution. To this aim we establish connections between this problem and some separation questions of convex analysis.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-08-17T07:00:00Z
  DOI: 10.1142/S0219199722500419
Higher depth false modular forms
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  Authors: Kathrin Bringmann, Jonas Kaszian, Antun Milas, Caner Nazaroglu
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  False theta functions are functions that are closely related to classical theta functions and mock theta functions. In this paper, we study their modular properties at all ranks by forming modular completions analogous to modular completions of indefinite theta functions of any signature and thereby develop a structure parallel to the recently developed theory of higher depth mock modular forms. We then demonstrate this theoretical base on a number of examples up to depth three coming from characters of modules for the vertex algebra [math], [math], and from [math]-invariants of three-manifolds associated with gauge group [math].
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-08-17T07:00:00Z
  DOI: 10.1142/S0219199722500432
Complex-analytic intermediate hyperbolicity, and finiteness properties
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  Authors: Antoine Etesse
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  Motivated by the finiteness of the set of automorphisms [math] of a projective manifold of general type [math], and by Kobayashi–Ochiai’s conjecture that a projective manifold [math]-analytically hyperbolic (also known as strongly measure hyperbolic) should be of general type, we investigate the finiteness properties of [math] for a complex manifold satisfying a (pseudo-) intermediate hyperbolicity property. We first show that a complex manifold [math] which is [math]-analytically hyperbolic has indeed finite automorphisms group. We then obtain a similar statement for a pseudo-[math]-analytically hyperbolic, strongly measure hyperbolic projective manifold [math], under an additional hypothesis on the size of the degeneracy set. Some of the properties used during the proofs lead us to introduce a notion of intermediate Picard hyperbolicity, which we last discuss.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-08-11T07:00:00Z
  DOI: 10.1142/S0219199722500468
Four-dimensional closed manifolds admit a weak harmonic Weyl metric
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  Authors: Giovanni Catino, Paolo Mastrolia, Dario D. Monticelli, Fabio Punzo
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  On four-dimensional closed manifolds we introduce a class of canonical Riemannian metrics, that we call weak harmonic Weyl metrics, defined as critical points in the conformal class of a quadratic functional involving the norm of the divergence of the Weyl tensor. This class includes Einstein and, more in general, harmonic Weyl manifolds. We prove that every closed four-manifold admits a weak harmonic Weyl metric, which is the unique (up to dilations) minimizer of the corresponding functional in a suitable conformal class. In general the problem is degenerate elliptic due to possible vanishing of the Weyl tensor. In order to overcome this issue, we minimize the functional in the conformal class determined by a reference metric, constructed by Aubin, with nowhere vanishing Weyl tensor. Moreover, we show that anti-self-dual metrics with positive Yamabe invariant can be characterized by pinching conditions involving suitable quadratic Riemannian functionals.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-08-10T07:00:00Z
  DOI: 10.1142/S021919972250047X
On the Steiner property for planar minimizing clusters. The isotropic case
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  Authors: Valentina Franceschi, Aldo Pratelli, Giorgio Stefani
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  We consider the isoperimetric problem for clusters in the plane with a double density, that is, perimeter and volume depend on two weights. In this paper, we consider the isotropic case, in the parallel paper [V. Franceschi, A. Pratelli and G. Stefani, On the Steiner property for planar minimizing clusters. The anisotropic case, preprint (2020)] the anisotropic case is studied. Here we prove that, in a wide generality, minimal clusters enjoy the “Steiner property”, which means that the boundaries are made by [math] regular arcs, meeting in finitely many triple points with the [math] property.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-07-29T07:00:00Z
  DOI: 10.1142/S0219199722500407
Parameter-dependent multiplicity results of sign-changing solutions for
quasilinear elliptic equations
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  Authors: Yongtao Jing, Zhaoli Liu, Zhi-Qiang Wang
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  Existence of sign-changing solutions to quasilinear elliptic equations of the form −∑i,j=1ND j(aij(x,u)Diu) + 1 2∑i,j=1ND saij(x,u)DiuDju = λf(x,u)in Ω under the Dirichlet boundary condition, where [math] ([math]) is a bounded domain with smooth boundary and [math] is a parameter, is studied. In particular, we examine how the number of sign-changing solutions depends on the parameter [math]. In the case considered here, there exists no nontrivial solution for [math] sufficiently small. We prove that, as [math] becomes large, there exist both arbitrarily many sign-changing solutions with negative energy and arbitrarily many sign-changing solutions with positive energy. The results are proved via a variational perturbation method. We construct new invariant sets of descending flow so that sign-changing solutions to the perturbed equations outside of these sets are obtained, and then we take limits to obtain sign-changing solutions to the original equation.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-07-28T07:00:00Z
  DOI: 10.1142/S0219199722500390
Weak dual pairs in Dirac–Jacobi geometry
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  Authors: Jonas Schnitzer, Alfonso Giuseppe Tortorella
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  Adopting the omni-Lie algebroid approach to Dirac–Jacobi structures, we propose and investigate a notion of weak dual pairs in Dirac–Jacobi geometry. Their main motivating examples arise from the theory of multiplicative precontact structures on Lie groupoids. Among other properties of weak dual pairs, we prove two main results. (1) We show that the property of fitting in a weak dual pair defines an equivalence relation for Dirac–Jacobi manifolds. So, in particular, we get the existence of self-dual pairs and this immediately leads to an alternative proof of the normal form theorem around Dirac–Jacobi transversals. (2) We prove the characteristic leaf correspondence theorem for weak dual pairs paralleling and extending analogous results for symplectic and contact dual pairs. Moreover, the same ideas of this proof apply to get a presymplectic leaf correspondence for weak dual pairs in Dirac geometry (not yet present in literature).
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-07-16T07:00:00Z
  DOI: 10.1142/S0219199722500353
On the characterization of constant functions through nonlocal functionals
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  Authors: Massimo Gobbino, Nicola Picenni
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  We address a classical open question by H. Brezis and R. Ignat concerning the characterization of constant functions through double integrals that involve difference quotients. Our first result is a counterexample to the question in its full generality. This counterexample requires the construction of a function whose difference quotients avoid a sequence of intervals with endpoints that diverge to infinity. Our second result is a positive answer to the question when restricted either to functions that are bounded and approximately differentiable almost everywhere, or to functions with bounded variation. We also present some related open problems that are motivated by our positive and negative results.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-07-16T07:00:00Z
  DOI: 10.1142/S0219199722500389
Reduced arc schemes for Veronese embeddings and global Demazure modules
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  Authors: Ilya Dumanski, Evgeny Feigin
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  We consider arc spaces for the compositions of Plücker and Veronese embeddings of the flag varieties for simple Lie groups of types ADE. The arc spaces are not reduced and we consider the homogeneous coordinate rings of the corresponding reduced schemes. We show that each graded component of a homogeneous coordinate ring is a cocyclic module over the current algebra and is acted upon by the algebra of symmetric polynomials. We show that the action of the polynomial algebra is free and that the fiber at the special point of a graded component is isomorphic to an affine Demazure module whose level is the degree of the Veronese embedding. In type A1 (which corresponds to the Veronese curve), we give the precise list of generators of the reduced arc space. In general type, we introduce the notion of global higher level Demazure modules, which generalizes the standard notion of the global Weyl modules, and identify the graded components of the homogeneous coordinate rings with these modules.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-07-14T07:00:00Z
  DOI: 10.1142/S0219199722500341
Symmetric locally free resolutions and rationality problems
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  Authors: Gilberto Bini, Grzegorz Kapustka, Michał Kapustka
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  We show that the birationality class of a quadric surface bundle over [math] is determined by its associated cokernel sheaves. As an application, we discuss stable-rationality of very general quadric bundles over [math] with discriminant curves of fixed degree. In particular, we construct explicit models of these bundles for some discriminant data. Among others, we obtain various birational models of a nodal Gushel–Mukai fourfold, as well as of a cubic fourfold containing a plane. Finally, we prove stable irrationality of several types of quadric surface bundles.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-07-11T07:00:00Z
  DOI: 10.1142/S021919972250033X
Determining functions by slopes
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  Authors: Lionel Thibault, Dariusz Zagrodny
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  It is shown that by knowing the slope of a bounded from below lower semicontinuous convex function at each point of a Banach space, we know the function up to an additive constant.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-06-25T07:00:00Z
  DOI: 10.1142/S0219199722500146
Lower semicontinuity and pointwise behavior of supersolutions for some
doubly nonlinear nonlocal parabolic [math]-Laplace equations
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  Authors: Agnid Banerjee, Prashanta Garain, Juha Kinnunen
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  We discuss pointwise behavior of weak supersolutions for a class of doubly nonlinear parabolic fractional [math]-Laplace equations which includes the fractional parabolic [math]-Laplace equation and the fractional porous medium equation. More precisely, we show that weak supersolutions have lower semicontinuous representative. We also prove that the semicontinuous representative at an instant of time is determined by the values at previous times. This gives a pointwise interpretation for a weak supersolution at every point. The corresponding results hold true also for weak subsolutions. Our results extend some recent results in the local parabolic case, and in the nonlocal elliptic case, to the nonlocal parabolic case. We prove the required energy estimates and measure theoretic De Giorgi type lemmas in the fractional setting.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-06-20T07:00:00Z
  DOI: 10.1142/S0219199722500328
Effective surface energies in nematic liquid crystals as homogenized
rugosity effects
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  Authors: Razvan-Dumitru Ceuca, Jamie M. Taylor, Arghir Zarnescu
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  We study the effect of boundary rugosity in nematic liquid crystalline systems. We consider a highly general formulation of the problem, able to simultaneously deal with several liquid crystal theories. We use techniques of Gamma convergence and demonstrate that the effect of fine-scale surface oscillations may be replaced by an effective homogenized surface energy on a simpler domain. The homogenization limit is then quantitatively studied in a simplified setting, obtaining convergence rates.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-06-10T07:00:00Z
  DOI: 10.1142/S0219199722500201
Kähler tori with almost non-negative scalar curvature
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  Authors: Jianchun Chu, Man-Chun Lee
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  Motivated by the torus stability problem, in this work, we study Kähler metrics with almost non-negative scalar curvature on complex torus. We prove that after passing to a subsequence, non-collapsing sequence of Kähler metrics with almost non-negative scalar curvature will converge to flat torus weakly.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-06-10T07:00:00Z
  DOI: 10.1142/S0219199722500304
A convergence result for mountain pass periodic solutions of perturbed
Hamiltonian systems
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  Authors: Marek Izydorek, Joanna Janczewska, Pedro Soares
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  In this work, we study second-order Hamiltonian systems under small perturbations. We assume that the main term of the system has a mountain pass structure, but do not suppose any condition on the perturbation. We prove the existence of a periodic solution. Moreover, we show that periodic solutions of perturbed systems converge to periodic solutions of the unperturbed systems if the perturbation tends to zero. The assumption on the potential that guarantees the mountain pass geometry of the corresponding action functional is of independent interest as it is more general than those by Rabinowitz [Homoclinic orbits for a class of Hamiltonian systems, Proc. R. Soc. Edinburgh A 114 (1990) 33–38] and the authors [M. Izydorek and J. Janczewska, Homoclinic solutions for a class of the second-order Hamiltonian systems, J. Differ. Equ. 219 (2005) 375–389].
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-05-28T07:00:00Z
  DOI: 10.1142/S0219199722500110
Asymptotic properties of generalized D-solutions to the stationary axially
symmetric Navier–Stokes equations
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  Authors: Zijin Li, Xinghong Pan
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  In this paper, we derive asymptotic properties of both the velocity and the vorticity fields to the 3-dimensional axially symmetric Navier–Stokes equations at infinity under the generalized D-solution assumption [math] for [math]. We do not impose any zero or non-zero constant vector asymptotic assumption to the solution at infinity. Our results generalize those in [H. Choe and B. Jin, Asymptotic properties of axis-symmetric D-solutions of the Navier–Stokes equations, J. Math. Fluid Mech. 11(2) (2009) 208–232; S. Weng, Decay properties of axially symmetric D-solutions to the steady Navier–Stokes equations, J. Math. Fluid Mech. 20(1) (2018) 7–25; B. Carrillo, X. Pan and Q. S. Zhang, Decay and vanishing of some axially symmetric D-solutions of the Navier–Stokes equations, J. Funct. Anal. 279(1) (2020) 108504], where the authors focused on the case [math] and the velocity field approaches zero at infinity. Meanwhile, when [math] and the velocity field approaches zero at infinity, our results coincide with the results in [H. Choe and B. Jin, Asymptotic properties of axis-symmetric D-solutions of the Navier–Stokes equations, J. Math. Fluid Mech. 11(2) (2009) 208–232; S. Weng, Decay properties of axially symmetric D-solutions to the steady Navier–Stokes equations, J. Math. Fluid Mech. 20(1) (2018) 7–25; B. Carrillo, X. Pan and Q. S. Zhang, Decay and vanishing of some axially symmetric D-solutions of the Navier–Stokes equations, J. Funct. Anal. 279(1) (2020) 108504].
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-05-28T07:00:00Z
  DOI: 10.1142/S0219199722500134
Helmholtz solutions for the fractional Laplacian and other related
operators
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  Authors: Vincent Guan, Mathav Murugan, Juncheng Wei
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  We show that the bounded solutions to the fractional Helmholtz equation, [math] for [math] in [math], are given by the bounded solutions to the classical Helmholtz equation [math] in [math] for [math] when [math] is additionally assumed to be vanishing at [math]. When [math], we show that the bounded fractional Helmholtz solutions are again given by the classical solutions [math]. We show that this classification of fractional Helmholtz solutions extends for [math] and [math] when [math]. Finally, we prove that the classical solutions are the unique bounded solutions to the more general equation [math] in [math], when [math] is complete Bernstein and certain regularity conditions are imposed on the associated weight [math].
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-05-28T07:00:00Z
  DOI: 10.1142/S021919972250016X
On Rogers–Shephard-type inequalities for the lattice point
enumerator
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  Authors: David Alonso-Gutiérrez, Eduardo Lucas, Jesús Yepes Nicolás
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  In this paper, we study various Rogers–Shephard-type inequalities for the lattice point enumerator [math] on [math]. In particular, for any non-empty convex bounded sets [math], we show that Gn(K + L)Gn(K ∩ (−L)) ≤ 2n n Gn(K + (−1, 1)n)G n(L + (−1, 1)n) and Gn−k(PH⊥K)Gk(K ∩ H) ≤ n kGn(K + (−1, 1)n), for [math], [math]. Additionally, a discrete counterpart to a classical result by Berwald for concave functions, from which other discrete Rogers–Shephard-type inequalities may be derived, is shown. Furthermore, we prove that these new discrete analogues for [math] imply the corresponding results involving the Lebesgue measure.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-05-28T07:00:00Z
  DOI: 10.1142/S0219199722500225
Hyperbolic Mass via Horospheres
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  Authors: Hyun Chul Jang, Pengzi Miao
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  We derive geometric formulas for the mass of asymptotically hyperbolic manifolds using coordinate horospheres. As an application, we obtain a new rigidity result of hyperbolic space: if a complete asymptotically hyperbolic manifold has scalar curvature lower bound [math] and is isometric to hyperbolic space outside a coordinate horosphere, then the manifold is isometric to hyperbolic space. In addition, we apply our formula to investigate regions near infinity that do not contribute to the mass quantity, which leads to improved rigidity results of hyperbolic space.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-05-28T07:00:00Z
  DOI: 10.1142/S0219199722500237
Exact and optimal controllability for scalar conservation laws with
discontinuous flux
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  Authors: Adimurthi, Shyam Sundar Ghoshal
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  This paper describes the reachable set and resolves an optimal control problem for the scalar conservation laws with discontinuous flux. We give a necessary and sufficient criteria for the reachable set. A new backward resolution has been described to obtain the reachable set. Regarding the optimal control problem, we first prove the existence of a minimizer and then the backward algorithm allows us to compute it. The same method also applies to compute the initial data control for an exact control problem. Our methodology for the proof relies on the explicit formula for the conservation laws with the discontinuous flux and finer properties of the characteristics curves.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-05-28T07:00:00Z
  DOI: 10.1142/S0219199722500249
H-space 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: 2022-05-20T07:00:00Z
  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: 2022-05-20T07:00:00Z
  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: 2022-05-20T07:00:00Z
  DOI: 10.1142/S0219199722500195
Symmetry and symmetry breaking for Hénon-type problems involving the
1-Laplacian 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 Hénon-type 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 non-radial 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: 2022-05-20T07:00:00Z
  DOI: 10.1142/S0219199722500213
Four-dimensional 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 four-manifolds 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 finite-dimensional family of solutions. The results are new for Ricci flow.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-05-20T07:00:00Z
  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 bi-Lagrangian structure on [math] and a compatible linear connection on [math]. We prove that the split-signature metric given by the almost bi-Lagrangian structure is Einstein with nonzero scalar curvature, provided that the parabolic geometry is torsion-free 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 non-degeneracy 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: 2022-05-20T07:00:00Z
  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 non-expansive (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: 2022-05-20T07:00:00Z
  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: 2022-05-18T07:00:00Z
  DOI: 10.1142/S0219199722500158
On the dynamics of charged particles in an incompressible flow: From
kinetic-fluid to fluid–fluid models
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  Authors: Young-Pil 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 kinetic-fluid model, we study the asymptotic regime corresponding to strong local alignment and diffusion forces. Under suitable assumptions on well-prepared 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 global-in-time strong solvability for the isothermal/pressureless EPNS system. In particular, this global-in-time solvability gives the estimates of hydrodynamic limit hold for all times.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-05-10T07:00:00Z
  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: 2022-05-10T07:00:00Z
  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: 2022-04-18T07:00:00Z
  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 time-fractional 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 non-vanishing 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 time-analyticity of the PDEs under consideration.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-04-02T07:00:00Z
  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: 2022-03-30T07:00:00Z
  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 Ruiz-Herrera
  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 so-called inverse saddle, impacts the topology of the attractor — it cannot be arcwise connected.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-03-30T07:00:00Z
  DOI: 10.1142/S0219199722500109
Positive solutions for a Minkowski-curvature equation with indefinite
weight and super-exponential 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 Minkowski-curvature equations with indefinite weight and nonlinear term having superlinear growth at zero and super-exponential growth at infinity. As an example, for the equation ( u′ 1 − (u′ )2)′ + a(t)(eup − 1) = 0, where [math] and [math] is a sign-changing function satisfying the mean-value 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: 2022-03-28T07:00:00Z
  DOI: 10.1142/S0219199722500055
Liouville-type theorems for higher-order 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: 2022-03-16T07:00:00Z
  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 non-unirational 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: 2022-03-10T08:00:00Z
  DOI: 10.1142/S0219199722500080
Propagation phenomena in a diffusion system with the
Belousov–Zhabotinskii chemical reaction
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  Authors: Wei-Jie Sheng, Mingxin Wang, Zhi-Cheng 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 V-shaped 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: 2022-03-02T08:00:00Z
  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 Borel-measurable 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: 2022-02-25T08:00:00Z
  DOI: 10.1142/S0219199722500031
Geometric inequalities for anti-blocking bodies
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  Authors: Shiri Artstein-Avidan, Shay Sadovsky, Raman Sanyal
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  We study the class of (locally) anti-blocking 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, near-optimal bounds on Mahler volumes, Saint-Raymond-type inequalities on mixed volumes, and reverse Kleitman inequalities for mixed volumes. We apply our results to the combinatorics of posets and prove Sidorenko-type inequalities for linear extensions of pairs of [math]-dimensional posets. The results rely on some elegant decompositions of differences of anti-blocking bodies, which turn out to hold for anti-blocking bodies with respect to general polyhedral cones.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-02-21T08:00:00Z
  DOI: 10.1142/S0219199721501133
Global well-posedness 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 quasi-uniform 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 time-space cylinder of unit size, and showing that the solution is sup-norm 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: 2022-02-21T08:00:00Z
  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, Kyeong-Dong 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: 2022-02-17T08:00:00Z
  DOI: 10.1142/S0219199721501121
Determinant of Friedrichs Dirichlet Laplacians on 2-dimensional 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 2-dimensional hyperbolic (Gaussian curvature [math]) cones in terms of the cone angle and the geodesic radius of the boundary.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-01-24T08:00:00Z
  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 non-uniform 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 non-Gaussian trial state for the Dirichlet magnetic Laplacian.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2022-01-21T08:00:00Z
  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 two-way propagation of small amplitude gravity waves on the surface of an ideal fluid, the so-called Boussinesq system of the Korteweg–de Vries type. We use a Gramian-based 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: 2022-01-20T08:00:00Z
  DOI: 10.1142/S021919972150111X
Functional convergence of continuous-time random walks with continuous
paths
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  Authors: Marcin Magdziarz, Piotr Zebrowski
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  Continuous-time 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: 2022-01-19T08:00:00Z
  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 power-type 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: 2022-01-19T08:00:00Z
  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: 2022-01-19T08:00:00Z
  DOI: 10.1142/S0219199721501108