JournalTOCs

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

[119 journals]

[math]-stability of topological entropy for contactomorphisms
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  Authors: Lucas Dahinden
  Abstract: Communications in Contemporary Mathematics, Volume 23, Issue 06, September 2021.
  Topological entropy is not lower semi-continuous: small perturbation of the dynamical system can lead to a collapse of entropy. In this note we show that for some special classes of dynamical systems (geodesic flows, Reeb flows, positive contactomorphisms) topological entropy at least is stable in the sense that there exists a nontrivial continuous lower bound, given that a certain homological invariant grows exponentially.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2021-04-01T07:00:00Z
  DOI: 10.1142/S0219199721500152
  Issue No: Vol. 23, No. 06 (2021)
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: 2021-10-14T07:00:00Z
  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: 2021-10-14T07:00:00Z
  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: 2021-10-08T07:00:00Z
  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 Núñez-Betancourt, 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: 2021-10-06T07:00:00Z
  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: 2021-10-05T07:00:00Z
  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 “Riemann-like” 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: 2021-09-29T07:00:00Z
  DOI: 10.1142/S0219199721500826
Knudsen type group for time in [math] and related Boltzmann type equations
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  Authors: Jörg-Uwe 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: 2021-09-25T07:00:00Z
  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 Artstein-Avidan 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: 2021-09-25T07:00:00Z
  DOI: 10.1142/S021919972150084X
1-Laplacian 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 non-trivial 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: 2021-09-18T07:00:00Z
  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: 2021-09-17T07:00:00Z
  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 easy-to-read 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: 2021-09-16T07:00:00Z
  DOI: 10.1142/S0219199721500632
On the global existence and time-decay 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 time-decay rates for the constructed global solutions. Our analyses mainly rely on Fourier frequency localization technology and on a refined time-weighted energy inequalities in different frequency regimes.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2021-09-06T07:00:00Z
  DOI: 10.1142/S0219199721500784
Quasi-clean rings and strongly quasi-clean 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 quasi-idempotent 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 quasi-Boolean ring if every element of [math] is quasi-idempotent. A ring [math] is called (strongly) quasi-clean if each of its elements is a sum of a quasi-idempotent 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) quasi-clean rings is conducted. The abundant examples of (strongly) quasi-clean rings state that the class of (strongly) quasi-clean rings is very larger than the class of (strongly) clean rings. We prove that an indecomposable commutative semilocal ring is quasi-clean 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 quasi-clean if and only if [math] is quasi-clean if and only if [math], [math] is a maximal ideal of [math]. For a prime [math] and a positive integer [math], [math] is strongly quasi-clean if and only if [math]. Some open questions are also posed.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2021-09-03T07:00:00Z
  DOI: 10.1142/S0219199721500796
Global Hölder continuity of solutions to quasilinear equations with
Morrey data
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  Authors: Sun-Sig Byun, Dian K. Palagachev, Pilsoo Shin
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  We deal with general quasilinear divergence-form 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 non-regular 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: 2021-08-27T07:00:00Z
  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 functional-valued 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: 2021-08-27T07:00:00Z
  DOI: 10.1142/S0219199721500759
Non-well-ordered 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 second-order differential operators. The assumptions involve lower and upper solutions, which may be either well-ordered, 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: 2021-08-27T07:00:00Z
  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, Trudinger-type 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: 2021-08-24T07:00:00Z
  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 1-string can be virtually homotoped to a minimally filling and crossing-minimal representative by monotonically decreasing both genus and the number of self-intersections. We generalize her result to the case of non-parallel virtual [math]-strings. Cahn also proved that any two crossing-irreducible representatives of a virtual 1-string 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 5-strings. We show that Kadokami’s statement holds for non-parallel [math]-strings and exhibit a counterexample for general virtual 3-strings.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2021-08-24T07:00:00Z
  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: 2021-08-24T07:00:00Z
  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: 2021-08-24T07:00:00Z
  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 semi-regular Sasakian structures. Our method allows us to completely solve the following problem of Boyer and Galicki in the class of semi-regular Sasakian structures: determine which simply connected rational homology 5-spheres admit negative Sasakian structures.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2021-08-24T07:00:00Z
  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 Margulis-type 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: 2021-08-21T07:00:00Z
  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: 2021-08-21T07:00:00Z
  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: 2021-08-21T07:00:00Z
  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: 2021-08-19T07:00:00Z
  DOI: 10.1142/S0219199721500656
Well-posedness 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 well-posedness in the generalized Hartree equation [math], [math], [math], for low powers of nonlinearity, [math]. We establish the local well-posedness 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 well-posed data with positive energy, which blows up in finite time.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2021-08-19T07:00:00Z
  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 non-zero 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: 2021-08-11T07:00:00Z
  DOI: 10.1142/S0219199721500590
A Liouville-type 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 Liouville-type theorem for cylindrical viscosity solutions of fully nonlinear CR invariant equations on the Heisenberg group. As a by-product, 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: 2021-08-11T07:00:00Z
  DOI: 10.1142/S0219199721500607
Vector-valued 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 Aleksandrov-type inequality for parallelepipeds.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2021-07-26T07:00:00Z
  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: 2021-06-24T07:00:00Z
  DOI: 10.1142/S0219199721500474
Boundary singularities of semilinear elliptic equations with Leray-Hardy
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: 2021-06-21T07:00:00Z
  DOI: 10.1142/S0219199721500516
Relaxed highest-weight 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 highest-weight modules over affine vertex algebras and W-algebras. The first [K. Kawasetsu and D. Ridout, Relaxed highest-weight 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 highest-weight modules for simple affine vertex algebras of arbitrary rank. The key point is that this can be reduced to the classification of highest-weight 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 non-semisimplicity of category [math] in one of these examples.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2021-06-14T07:00:00Z
  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: 2021-06-14T07:00:00Z
  DOI: 10.1142/S0219199721500541
Delta invariant of curves on rational surfaces I. An analytic approach
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  Authors: José Ignacio Cogolludo-Agustín, Tamás László, Jorge Martín-Morales, 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: 2021-06-09T07:00:00Z
  DOI: 10.1142/S0219199721500528
Isospectral finiteness on convex cocompact hyperbolic 3-manifolds
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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 3-manifolds with that specified length spectrum with multiplicity, homotopy equivalent to a given 3-manifold without a handlebody factor, up to orientation preserving isometries.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2021-06-09T07:00:00Z
  DOI: 10.1142/S0219199721500589
On mixed Hodge–Riemann relations for translation-invariant 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 translation-invariant 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 lower-dimensional 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: 2021-06-07T07:00:00Z
  DOI: 10.1142/S0219199721500498
Integrability of close encounters in the spatial restricted three-body
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 Levi-Civita for the planar restricted three-body 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: 2021-06-04T07:00:00Z
  DOI: 10.1142/S0219199721500401
Exotic symplectomorphisms and contact circle actions
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  Authors: Dušan Drobnjak, Igor Uljarević
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  Using Floer-theoretic methods, we prove that the non-existence of an exotic symplectomorphism on the standard symplectic ball, [math] implies a rather strict topological condition on the free contact circle actions on the standard contact sphere, [math] We also prove an analogue for a Liouville domain and contact circle actions on its boundary. Applications include results concerning the symplectic mapping class group and the fundamental group of the group of contactomorphisms.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2021-06-04T07:00:00Z
  DOI: 10.1142/S0219199721500450
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: 2021-06-04T07:00:00Z
  DOI: 10.1142/S0219199721500486
On the geometry of polytopes generated by heavy-tailed random vectors
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  Authors: Olivier Guédon, Felix Krahmer, Christian Kümmerle, Shahar Mendelson, Holger Rauhut
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  We study the geometry of centrally symmetric random polytopes, generated by [math] independent copies of a random vector [math] taking values in [math]. We show that under minimal assumptions on [math], for [math] and with high probability, the polytope contains a deterministic set that is naturally associated with the random vector — namely, the polar of a certain floating body. This solves the long-standing question on whether such a random polytope contains a canonical body. Moreover, by identifying the floating bodies associated with various random vectors, we recover the estimates that were obtained previously, and thanks to the minimal assumptions on [math], we derive estimates in cases that were out of reach, involving random polytopes generated by heavy-tailed random vectors (e.g., when [math] is [math]-stable or when [math] has an unconditional structure). Finally, the structural results are used for the study of a fundamental question in compressive sensing — noise blind sparse recovery.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2021-05-31T07:00:00Z
  DOI: 10.1142/S0219199721500565
Direct limit completions of vertex tensor categories
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  Authors: Thomas Creutzig, Robert McRae, Jinwei Yang
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  We show that direct limit completions of vertex tensor categories inherit vertex and braided tensor category structures, under conditions that hold for example for all known Virasoro and affine Lie algebra tensor categories. A consequence is that the theory of vertex operator (super)algebra extensions also applies to infinite-order extensions. As an application, we relate rigid and non-degenerate vertex tensor categories of certain modules for both the affine vertex superalgebra of [math] and the [math] super Virasoro algebra to categories of Virasoro algebra modules via certain cosets.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2021-05-25T07:00:00Z
  DOI: 10.1142/S0219199721500334
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 second-order 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: 2021-05-25T07:00:00Z
  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: 2021-05-21T07:00:00Z
  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 non-invertible 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 self-conformal 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 self-conformal measure, then [math] if and only if the overlap number [math]. Examples are also discussed.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2021-05-21T07:00:00Z
  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: 2021-05-21T07:00:00Z
  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 well-known theorem of Kulikov, Persson and Pinkham states that a degeneration of a family of K3-surfaces 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: 2021-05-21T07:00:00Z
  DOI: 10.1142/S0219199721500553
Optimal non-homogeneous 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: 2021-05-17T07:00:00Z
  DOI: 10.1142/S0219199721500310
Blowing up solutions for supercritical Yamabe problems on manifolds with
non-umbilic boundary
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  Authors: Marco G. Ghimenti, Anna Maria Micheletti
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  We build blowing-up solutions for a supercritical perturbation of the Yamabe problem on manifolds with boundary, provided the dimension of the manifold is [math] and the trace-free part of the second fundamental form is nonzero everywhere on the boundary.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2021-05-17T07:00:00Z
  DOI: 10.1142/S0219199721500358
Compactness of Sobolev-type 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 Sobolev-type 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. Sobolev-type spaces as well as target spaces considered in this paper are built upon general rearrangement-invariant 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 Sobolev-type spaces built upon Lorentz–Zygmund norms are also presented.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2021-05-17T07:00:00Z
  DOI: 10.1142/S021919972150036X
Multiplicity of negative-energy solutions for singular-superlinear
Schrödinger equations with indefinite-sign 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 non-differentiability of the energy functional and the fact that the intersection of the boundaries of the connected components of the Nehari set is non-empty. We overcome these difficulties by exploring topological structures of that boundary to build non-empty sets whose boundaries have empty intersection and minimizing over them by controlling the energy level.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2021-05-17T07:00:00Z
  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: 2021-05-17T07:00:00Z
  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: 2021-05-17T07:00:00Z
  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 Shi-type 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: 2021-05-11T07:00:00Z
  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 Serrin-type 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: 2021-05-10T07:00:00Z
  DOI: 10.1142/S0219199721500322
Schrödinger–Poisson–Proca systems in EMS regime
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  Authors: Emmanuel Hebey
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  We prove existence of solutions to the Schrödinger–Poisson–Proca system, and stability of this system, in the electro-magneto-static regime in closed 3-dimensional Riemannian manifolds.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2021-05-10T07:00:00Z
  DOI: 10.1142/S0219199721500383
Non-existence 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 non-existence of non-negative 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 sign-changing 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 non-negative matrix.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2021-05-10T07:00:00Z
  DOI: 10.1142/S0219199721500395
Exotic periodic points
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  Authors: Duc-Viet Vu
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  We introduce the notion of exotic periodic points of a meromorphic self-map. We then establish the expected asymptotic for the number of isolated or exotic periodic points for holomorphic self-maps with a simple action on the cohomology groups on a compact Kähler manifold.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2021-05-05T07:00:00Z
  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 non-positively curved, we close our paper constructing new examples of rotationally symmetric spacelike translating solitons embedded into [math].
  Citation: Communications in Contemporary Mathematics
  PubDate: 2021-05-05T07:00:00Z
  DOI: 10.1142/S0219199721500346
A new cohomology theory for strict Lie [math]-algebras
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  Authors: Camilo Angulo
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  In this paper, we introduce a new cohomology theory associated to a strict Lie 2-algebra. This cohomology theory is shown to also extend the classical cohomology theory of Lie algebras; in particular, we show that the second cohomology group classifies an appropriate type of extension.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2021-03-26T07:00:00Z
  DOI: 10.1142/S0219199721500176
An asymptotic expansion for the fractional [math]-Laplacian and for
gradient-dependent 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 well-known 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 gradient-dependent nonlocal operators.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2021-03-26T07:00:00Z
  DOI: 10.1142/S0219199721500218
Moduli spaces for Lamé functions and Abelian differentials of the
second kind
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  Authors: Alexandre Eremenko, Andrei Gabrielov, Gabriele Mondello, Dmitri Panov
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  The topology of the moduli space for Lamé functions of degree [math] is determined: this is a Riemann surface which consists of two connected components when [math]; we find the Euler characteristics and genera of these components. As a corollary we prove a conjecture of Maier on degrees of Cohn’s polynomials. These results are obtained with the help of a geometric description of these Riemann surfaces, as quotients of the moduli spaces for certain singular flat triangles. An application is given to the study of metrics of constant positive curvature with one conic singularity with the angle [math] on a torus. We show that the degeneration locus of such metrics is a union of smooth analytic curves and we enumerate these curves.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2021-03-26T07:00:00Z
  DOI: 10.1142/S0219199721500280
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: 2021-03-19T07:00:00Z
  DOI: 10.1142/S0219199721500127
The symplectic structure for renormalization of circle diffeomorphisms
with breaks
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  Authors: S. Ghazouani, K. Khanin
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  The main goal of this paper is to reveal the symplectic structure related to renormalization of circle maps with breaks. We first show that iterated renormalizations of [math] circle diffeomorphisms with [math] breaks, [math], with given size of breaks, converge to an invariant family of piecewise Möbius maps, of dimension [math]. We prove that this invariant family identifies with a relative character variety [math] where [math] is a [math]-holed torus, and that the renormalization operator identifies with a sub-action of the mapping class group [math]. This action allows us to introduce the symplectic form which is preserved by renormalization. The invariant symplectic form is related to the symplectic form described by Guruprasad et al. [Group systems, groupoids, and moduli spaces of parabolic bundles, Duke Math. J. 89(2) (1997) 377–412], and goes back to the earlier work by Goldman [The symplectic nature of fundamental groups of surfaces, Adv. Math. 54(2) (1984) 200–225]. To the best of our knowledge the connection between renormalization in the nonlinear setting and symplectic dynamics had not been brought to light yet.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2021-03-19T07:00:00Z
  DOI: 10.1142/S0219199721500164
[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 “semi-direct 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: 2021-03-18T07:00:00Z
  DOI: 10.1142/S0219199721500139
Local null controllability of a model system for strong interaction
between internal solitary waves
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  Authors: Jon Asier Bárcena-Petisco, Sergio Guerrero, Ademir F. Pazoto
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  In this paper, we prove the local null controllability property for a nonlinear coupled system of two Korteweg–de Vries equations posed on a bounded interval and with a source term decaying exponentially on [math]. The system was introduced by Gear and Grimshaw to model the interactions of two-dimensional, long, internal gravity waves propagation in a stratified fluid. We address the controllability problem by means of a control supported on an interior open subset of the domain and acting on one equation only. The proof consists mainly on proving the controllability of the linearized system, which is done by getting a Carleman estimate for the adjoint system. While doing the Carleman, we improve the techniques for dealing with the fact that the solutions of dispersive and parabolic equations with a source term in [math] have a limited regularity. A local inversion theorem is applied to get the result for the nonlinear system.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2021-03-16T07:00:00Z
  DOI: 10.1142/S0219199721500036
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: 2021-03-16T07:00:00Z
  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: 2021-03-16T07:00:00Z
  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: 2021-03-16T07:00:00Z
  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 semi-direct product of the Virasoro algebra and the twisted Heisenberg algebra. We classify all Harish-Chandra modules over [math], i.e. irreducible modules with finite-dimensional 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: 2021-03-16T07:00:00Z
  DOI: 10.1142/S0219199721500267
Sobolev–Kantorovich inequalities under [math] condition
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  Authors: Vladimir I. Bogachev, Alexander V. Shaposhnikov, Feng-Yu 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: 2021-03-16T07:00:00Z
  DOI: 10.1142/S0219199721500279
A Pogorelov estimate and a Liouville-type 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 Liouville-type theorems to parabolic [math]-Hessian equations of the form [math] in [math]. We derive that any [math]-convex-monotone 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: 2021-03-05T08:00:00Z
  DOI: 10.1142/S0219199721500012
Real-variable characterizations of local Orlicz-slice 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 Orlicz-slice 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: 2021-03-05T08:00:00Z
  DOI: 10.1142/S0219199721500048
Multiplicity of positive solutions for [math]-Laplace equations with two
parameters
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  Authors: Vladimir Bobkov, Mieko Tanaka
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  We study the zero Dirichlet problem for the equation [math] in a bounded domain [math], with [math]. We investigate the relation between two critical curves on the [math]-plane corresponding to the threshold of existence of special classes of positive solutions. In particular, in certain neighborhoods of the point [math], where [math] is the first eigenfunction of the [math]-Laplacian, we show the existence of two and, which is rather unexpected, three distinct positive solutions, depending on a relation between the exponents [math] and [math].
  Citation: Communications in Contemporary Mathematics
  PubDate: 2021-03-05T08:00:00Z
  DOI: 10.1142/S0219199721500085
An ambient approach to conformal geodesics
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  Authors: Joel Fine, Yannick Herfray
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  Conformal geodesics are distinguished curves on a conformal manifold, loosely analogous to geodesics of Riemannian geometry. One definition of them is as solutions to a third-order differential equation determined by the conformal structure. There is an alternative description via the tractor calculus. In this article, we give a third description using ideas from holography. A conformal [math]-manifold [math] can be seen (formally at least) as the asymptotic boundary of a Poincaré–Einstein [math]-manifold [math]. We show that any curve [math] in [math] has a uniquely determined extension to a surface [math] in [math], which we call the ambient surface of [math]. This surface meets the boundary [math] in right angles along [math] and is singled out by the requirement that it be a critical point of renormalized area. The conformal geometry of [math] is encoded in the Riemannian geometry of [math]. In particular, [math] is a conformal geodesic precisely when [math] is asymptotically totally geodesic, i.e. its second fundamental form vanishes to one order higher than expected. We also relate this construction to tractors and the ambient metric construction of Fefferman and Graham. In the [math]-dimensional ambient manifold, the ambient surface is a graph over the bundle of scales. The tractor calculus then identifies with the usual tensor calculus along this surface. This gives an alternative compact proof of our holographic characterization of conformal geodesics.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2021-03-05T08:00:00Z
  DOI: 10.1142/S0219199721500097
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: 2021-03-04T08:00:00Z
  DOI: 10.1142/S0219199721500188
A Berestycki–Lions type result for a class of problems involving the
1-Laplacian 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: 2021-03-04T08:00:00Z
  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, non-degenerate, 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: 2021-03-04T08:00:00Z
  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: 2021-03-04T08:00:00Z
  DOI: 10.1142/S0219199721500243
The Neumann problem for degenerate Hessian quotient equations
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  Authors: Xinqun Mei
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  In this paper, we prove the existence of the [math]-solution to the classical Neumann problem for the degenerate elliptic Hessian quotient equation [math] under the condition that [math].
  Citation: Communications in Contemporary Mathematics
  PubDate: 2021-02-26T08:00:00Z
  DOI: 10.1142/S0219199721500061
Note on linear relations in Galois cohomology and étale [math]-theory
of curves
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  Authors: Piotr Krasoń
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  In this paper, we investigate a local to global principle for Galois cohomology of number fields with coefficients in the Tate module of an abelian variety. In [G. Banaszak and P. Krasoń, On a local to global principle in étale K-groups of curves, J. K-Theory Appl. Algebra Geom. Topol. 12 (2013) 183–201], G. Banaszak and the author obtained the sufficient condition for the validity of the local to global principle for étale [math]-theory of a curve. This condition in fact has been established by means of an analysis of the corresponding problem in the Galois cohomology. We show that in some cases, this result is the best possible i.e. if this condition does not hold we obtain counterexamples. We also give some examples of curves and their Jacobians. Finally, we prove the dynamical version of the local to global principle for étale [math]-theory of a curve. The dynamical local to global principle for the groups of Mordell–Weil type has recently been considered by S. Barańczuk in [S. Barańczuk, On a dynamical local-global principle in Mordell-Weil type groups, Expo. Math. 35(2) (2017) 206–211]. We show that all our results remain valid for Quillen [math]-theory of [math] if the Bass and Quillen–Lichtenbaum conjectures hold true for [math]
  Citation: Communications in Contemporary Mathematics
  PubDate: 2021-02-26T08:00:00Z
  DOI: 10.1142/S0219199721500103
Elliptic classes on Langlands dual flag varieties
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  Authors: Richárd Rimányi, Andrzej Weber
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  Characteristic classes of Schubert varieties can be used to study the geometry and the combinatorics of homogeneous spaces. We prove a relation between elliptic classes of Schubert varieties on a generalized full flag variety and those on its Langlands dual. This new symmetry is motivated by 3D mirror symmetry, and it is only revealed if Schubert calculus is elevated from cohomology or K theory to the elliptic level.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2021-02-26T08:00:00Z
  DOI: 10.1142/S0219199721500140
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 p-Laplacian 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: 2021-02-19T08:00:00Z
  DOI: 10.1142/S021919972150005X
The method of Puiseux series and invariant algebraic curves
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  Authors: Maria V. Demina
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  An explicit expression for the cofactor related to an irreducible invariant algebraic curve of a polynomial dynamical system in the plane is derived. A sufficient condition for a polynomial dynamical system in the plane to have a finite number of irreducible invariant algebraic curves is obtained. All these results are applied to Liénard dynamical systems [math], [math] with [math]. The general structure of their irreducible invariant algebraic curves and cofactors is found. It is shown that Liénard dynamical systems with [math] can have at most two distinct irreducible invariant algebraic curves simultaneously and, consequently, are not integrable with a rational first integral.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2021-02-18T08:00:00Z
  DOI: 10.1142/S0219199721500073
On solutions for a class of fractional Kirchhoff-type 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 Kirchhoff-type 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 sign-changing) 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: 2021-02-09T08:00:00Z
  DOI: 10.1142/S0219199721500024
Regularity for the fully nonlinear parabolic thin obstacle problem
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  Authors: Georgiana Chatzigeorgiou
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  We prove [math] regularity (in the parabolic sense) for the viscosity solution of a boundary obstacle problem with a fully nonlinear parabolic equation in the interior. Following the method which was first introduced for the harmonic case by L. Caffarelli in 1979, we extend the results of I. Athanasopoulos (1982) who studied the linear parabolic case and the results of E. Milakis and L. Silvestre (2008) who treated the fully nonlinear elliptic case.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2021-02-09T08:00:00Z
  DOI: 10.1142/S0219199721500115
A general method to construct invariant PDEs on homogeneous manifolds
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  Authors: Dmitri V. Alekseevsky, Jan Gutt, Gianni Manno, Giovanni Moreno
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  Let [math] be an [math]-dimensional homogeneous manifold and [math] be the manifold of [math]-jets of hypersurfaces of [math]. The Lie group [math] acts naturally on each [math]. A [math]-invariant partial differential equation of order [math] for hypersurfaces of [math] (i.e., with [math] independent variables and [math] dependent one) is defined as a [math]-invariant hypersurface [math]. We describe a general method for constructing such invariant partial differential equations for [math]. The problem reduces to the description of hypersurfaces, in a certain vector space, which are invariant with respect to the linear action of the stability subgroup [math] of the [math]-prolonged action of [math]. We apply this approach to describe invariant partial differential equations for hypersurfaces in the Euclidean space [math] and in the conformal space [math]. Our method works under some mild assumptions on the action of [math], namely: A1) the group [math] must have an open orbit in [math], and A2) the stabilizer [math] of the fiber [math] must factorize via the group of translations of the fiber itself.
  Citation: Communications in Contemporary Mathematics
  PubDate: 2021-01-07T08:00:00Z
  DOI: 10.1142/S0219199720500893
Codimension bounds and rigidity of ancient mean curvature flows by the
tangent flow at [math]
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  Authors: Douglas Stryker, Ao Sun
  Abstract: Communications in Contemporary Mathematics, Ahead of Print.
  Motivated by the limiting behavior of an explicit class of compact ancient curve shortening flows, by adapting the work of Colding–Minicozzi [11], we prove codimension bounds for ancient mean curvature flows by their tangent flow at [math]. In the case of the [math]-covered circle, we apply this bound to prove a strong rigidity theorem. Furthermore, we extend this paradigm by showing that under the assumption of sufficiently rapid convergence, a compact ancient mean curvature flow is identical to its tangent flow at [math].
  Citation: Communications in Contemporary Mathematics
  PubDate: 2021-01-06T08:00:00Z
  DOI: 10.1142/S0219199720500881