Pub Date : 2022-05-25DOI: 10.1142/S0219199722500754
F. Podestà, Alberto Raffero
. Starting from compact symmetric spaces of inner type, we provide infinite families of compact homogeneous spaces carrying invariant non-flat Bismut connections with vanishing Ricci tensor. These examples turn out to be generalized symmetric spaces of order 4 and (up to coverings) can be realized as minimal submanifolds of the Bismut flat model spaces, namely compact Lie groups. This construction generalizes the standard Cartan embedding of symmetric spaces.
{"title":"Infinite families of homogeneous bismut ricci flat manifolds","authors":"F. Podestà, Alberto Raffero","doi":"10.1142/S0219199722500754","DOIUrl":"https://doi.org/10.1142/S0219199722500754","url":null,"abstract":". Starting from compact symmetric spaces of inner type, we provide infinite families of compact homogeneous spaces carrying invariant non-flat Bismut connections with vanishing Ricci tensor. These examples turn out to be generalized symmetric spaces of order 4 and (up to coverings) can be realized as minimal submanifolds of the Bismut flat model spaces, namely compact Lie groups. This construction generalizes the standard Cartan embedding of symmetric spaces.","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2022-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64399644","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-17DOI: 10.1142/S0219199722500821
Xiaoliang Li, Cong Wang
. We treat the exterior Dirichlet problem for a class of fully nonlinear elliptic equations of the form f ( λ ( D 2 u )) = g ( x ) , with prescribed asymptotic behavior at infinity. The equations of this type had been studied extensively by Caffarelli–Nirenberg–Spruck [8], Trudinger [35] and many others, and there had been significant discussions on the solv- ability of the classical Dirichlet problem via the continuity method, under the assumption that f is a concave function. In this paper, based on the Perron’s method, we establish an exterior existence and uniqueness result for viscosity solutions of the equations, by assuming f to satisfy certain structure condi- tions as in [8, 35] but without requiring the concavity of f . The equations in our setting may embrace the well-known Monge–Amp`ere equations, Hessian equations and Hessian quotient equations as special cases.
。我们处理了一类形式为f (λ (d2 u)) = g (x)的完全非线性椭圆方程的外部Dirichlet问题，该方程在无穷远处具有规定的渐近行为。Caffarelli-Nirenberg-Spruck [8]， Trudinger[8]等人对这类方程进行了广泛的研究，并在假设f为凹函数的情况下，对经典Dirichlet问题用连续性方法的可解性进行了有意义的讨论。本文基于Perron方法，假设f满足[8,35]中的某些结构条件，但不需要f的凹性，建立了方程黏度解的外部存在唯一性结果。我们设置的方程可以包括著名的Monge-Amp 'ere方程，Hessian方程和Hessian商方程作为特殊情况。
{"title":"On the exterior Dirichlet problem for Hessian type fully nonlinear elliptic equations","authors":"Xiaoliang Li, Cong Wang","doi":"10.1142/S0219199722500821","DOIUrl":"https://doi.org/10.1142/S0219199722500821","url":null,"abstract":". We treat the exterior Dirichlet problem for a class of fully nonlinear elliptic equations of the form f ( λ ( D 2 u )) = g ( x ) , with prescribed asymptotic behavior at infinity. The equations of this type had been studied extensively by Caffarelli–Nirenberg–Spruck [8], Trudinger [35] and many others, and there had been significant discussions on the solv- ability of the classical Dirichlet problem via the continuity method, under the assumption that f is a concave function. In this paper, based on the Perron’s method, we establish an exterior existence and uniqueness result for viscosity solutions of the equations, by assuming f to satisfy certain structure condi- tions as in [8, 35] but without requiring the concavity of f . The equations in our setting may embrace the well-known Monge–Amp`ere equations, Hessian equations and Hessian quotient equations as special cases.","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2022-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48193946","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-05DOI: 10.1142/s0219199723500013
Huxiao Luo, B. Ruf, C. Tarsi
: We consider the semilinear elliptic equations where I α is a Riesz potential, p ∈ ( N + αN , N + α N − 2 ), N ≥ 3, and V is continuous periodic. We assume that 0 lies in the spectral gap ( a, b ) of − ∆ + V . We prove the existence of infinitely many geometrically distinct solutions in H 1 ( R N ) for each λ ∈ ( a, b ), which bifurcate from b if N + αN < p < 1 + 2+ αN . Moreover, b is the unique gap-bifurcation point (from zero) in [ a, b ]. When λ = a , we find infinitely many geometrically distinct solutions in H 2 loc ( R N ). Final remarks are given about the eventual occurrence of a bifurcation from infinity in λ = a . 35Q55, 47J35.
{"title":"Bifurcation into spectral gaps for strongly indefinite Choquard equations","authors":"Huxiao Luo, B. Ruf, C. Tarsi","doi":"10.1142/s0219199723500013","DOIUrl":"https://doi.org/10.1142/s0219199723500013","url":null,"abstract":": We consider the semilinear elliptic equations where I α is a Riesz potential, p ∈ ( N + αN , N + α N − 2 ), N ≥ 3, and V is continuous periodic. We assume that 0 lies in the spectral gap ( a, b ) of − ∆ + V . We prove the existence of infinitely many geometrically distinct solutions in H 1 ( R N ) for each λ ∈ ( a, b ), which bifurcate from b if N + αN < p < 1 + 2+ αN . Moreover, b is the unique gap-bifurcation point (from zero) in [ a, b ]. When λ = a , we find infinitely many geometrically distinct solutions in H 2 loc ( R N ). Final remarks are given about the eventual occurrence of a bifurcation from infinity in λ = a . 35Q55, 47J35.","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2022-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44658255","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-03DOI: 10.1142/S021919972250064X
E. Jespers, A. V. Antwerpen, L. Vendramin
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.
{"title":"Nilpotency of skew braces and multipermutation solutions of the Yang-Baxter equation","authors":"E. Jespers, A. V. Antwerpen, L. Vendramin","doi":"10.1142/S021919972250064X","DOIUrl":"https://doi.org/10.1142/S021919972250064X","url":null,"abstract":"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.","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2022-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49559366","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-20DOI: 10.1142/s0219199723500281
R. Iagar, Philippe Laurencçot, Ariel G. S'anchez
We study the dynamics of the following porous medium equation with strong absorption $$partial_t u=Delta u^m-|x|^{sigma}u^q,$$ posed for $(t, x) in (0,infty) times mathbb{R}^N$, with $m>1$, $q in (0, 1)$ and $sigma>2(1-q)/(m-1)$. Considering the Cauchy problem with non-negative initial condition $u_0 in L^infty(mathbb{R}^N)$ instantaneous shrinking and localization of supports for the solution $u(t)$ at any $t>0$ are established. With the help of this property, existence and uniqueness of a nonnegative compactly supported and radially symmetric forward self-similar solution with algebraic decay in time are proven. Finally, it is shown that finite time extinction does not occur for a wide class of initial conditions and this unique self-similar solution is the pattern for large time behavior of these general solutions.
我们研究了以下多孔介质强吸收方程$$partial_t u=Delta u^m-|x|^{sigma}u^q,$$对$(t, x) in (0,infty) times mathbb{R}^N$, $m>1$, $q in (0, 1)$和$sigma>2(1-q)/(m-1)$的动力学。考虑具有非负初始条件$u_0 in L^infty(mathbb{R}^N)$的柯西问题，建立了求解$u(t)$在任意$t>0$处的瞬时收缩和支撑局部化。利用这一性质，证明了具有代数时间衰减的非负紧支持径向对称前向自相似解的存在唯一性。最后，证明了有限时间消光不发生在一类广泛的初始条件下，这种独特的自相似解是这些一般解的大时间行为模式。
{"title":"Self-similar shrinking of supports and non-extinction for a nonlinear diffusion equation with spatially inhomogeneous strong absorption","authors":"R. Iagar, Philippe Laurencçot, Ariel G. S'anchez","doi":"10.1142/s0219199723500281","DOIUrl":"https://doi.org/10.1142/s0219199723500281","url":null,"abstract":"We study the dynamics of the following porous medium equation with strong absorption $$partial_t u=Delta u^m-|x|^{sigma}u^q,$$ posed for $(t, x) in (0,infty) times mathbb{R}^N$, with $m>1$, $q in (0, 1)$ and $sigma>2(1-q)/(m-1)$. Considering the Cauchy problem with non-negative initial condition $u_0 in L^infty(mathbb{R}^N)$ instantaneous shrinking and localization of supports for the solution $u(t)$ at any $t>0$ are established. With the help of this property, existence and uniqueness of a nonnegative compactly supported and radially symmetric forward self-similar solution with algebraic decay in time are proven. Finally, it is shown that finite time extinction does not occur for a wide class of initial conditions and this unique self-similar solution is the pattern for large time behavior of these general solutions.","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2022-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45146295","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-18DOI: 10.1142/s0219199722500213
M. Pimenta, Anderson dos Santos Gonzaga
{"title":"Symmetry and symmetry breaking for Henon type problems involving the 1-Laplacian operator","authors":"M. Pimenta, Anderson dos Santos Gonzaga","doi":"10.1142/s0219199722500213","DOIUrl":"https://doi.org/10.1142/s0219199722500213","url":null,"abstract":"","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2022-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43231159","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-18DOI: 10.1142/s0219199722500146
L. Thibault, D. Zagrodny
{"title":"Determining Functions by Slopes","authors":"L. Thibault, D. Zagrodny","doi":"10.1142/s0219199722500146","DOIUrl":"https://doi.org/10.1142/s0219199722500146","url":null,"abstract":"","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2022-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42281945","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-18DOI: 10.1142/s0219199722500110
M. Izydorek, Joanna Janczewska, Pedro Soares
{"title":"A convergence result for mountain pass periodic solutions of perturbed Hamiltonian systems","authors":"M. Izydorek, Joanna Janczewska, Pedro Soares","doi":"10.1142/s0219199722500110","DOIUrl":"https://doi.org/10.1142/s0219199722500110","url":null,"abstract":"","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2022-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49542855","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-01DOI: 10.1142/S0219199722500766
I. Diamantis
. In this paper we develop a braid theoretic approach for computing the Kauffman bracket skein module of the lens spaces L ( p,q ), KBSM( L ( p,q )), for q 6 = 0. For doing this, we introduce a new concept, that of an unoriented braid . Unoriented braids are obtained from standard braids by ignoring the natural top-to-bottom orientation of the strands. We first define the generalized Temperley-Lieb algebra of type B , TL 1 ,n , which is related to the knot theory of the solid torus ST, and we obtain the universal Kauffman bracket type invariant, V , for knots and links in ST, via a unique Markov trace constructed on TL 1 ,n . The universal invariant V is equivalent to the KBSM(ST). For passing now to the KBSM( L ( p, q )), we impose on V relations coming from the band moves (or slide moves), that is, moves that reflect isotopy in L ( p, q ) but not in ST, and which reflect the surgery description of L ( p, q ), obtaining thus, an infinite system of equations. By construction, solving this infinite system of equations is equivalent to computing KBSM( L ( p,q )). We first present the solution for the case q = 1, which corresponds to obtaining a new basis, B p , for KBSM( L ( p, 1)) with ( ⌊ p/ 2 ⌋ +1) elements. We note that the basis B p is different from the one obtained by Hoste & Przytycki. For dealing with the complexity of the infinite system for the case q > 1, we first show how the new basis B p of KBSM( L ( p, 1)) can be obtained using a diagrammatic approach based on unoriented braids, and we finally extend our result to the case q > 1. The advantage of the braid theoretic approach that we propose for computing skein modules of c.c.o. 3-manifolds, is that the use of braids provides more control on the isotopies of knots and links in the manifolds, and much of the diagrammatic complexity is absorbed into the proofs of the algebraic statements.
{"title":"The Kauffman bracket skein module of the lens spaces via unoriented braids","authors":"I. Diamantis","doi":"10.1142/S0219199722500766","DOIUrl":"https://doi.org/10.1142/S0219199722500766","url":null,"abstract":". In this paper we develop a braid theoretic approach for computing the Kauffman bracket skein module of the lens spaces L ( p,q ), KBSM( L ( p,q )), for q 6 = 0. For doing this, we introduce a new concept, that of an unoriented braid . Unoriented braids are obtained from standard braids by ignoring the natural top-to-bottom orientation of the strands. We first define the generalized Temperley-Lieb algebra of type B , TL 1 ,n , which is related to the knot theory of the solid torus ST, and we obtain the universal Kauffman bracket type invariant, V , for knots and links in ST, via a unique Markov trace constructed on TL 1 ,n . The universal invariant V is equivalent to the KBSM(ST). For passing now to the KBSM( L ( p, q )), we impose on V relations coming from the band moves (or slide moves), that is, moves that reflect isotopy in L ( p, q ) but not in ST, and which reflect the surgery description of L ( p, q ), obtaining thus, an infinite system of equations. By construction, solving this infinite system of equations is equivalent to computing KBSM( L ( p,q )). We first present the solution for the case q = 1, which corresponds to obtaining a new basis, B p , for KBSM( L ( p, 1)) with ( ⌊ p/ 2 ⌋ +1) elements. We note that the basis B p is different from the one obtained by Hoste & Przytycki. For dealing with the complexity of the infinite system for the case q > 1, we first show how the new basis B p of KBSM( L ( p, 1)) can be obtained using a diagrammatic approach based on unoriented braids, and we finally extend our result to the case q > 1. The advantage of the braid theoretic approach that we propose for computing skein modules of c.c.o. 3-manifolds, is that the use of braids provides more control on the isotopies of knots and links in the manifolds, and much of the diagrammatic complexity is absorbed into the proofs of the algebraic statements.","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42518298","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-30DOI: 10.1142/s0219199722500109
Grzegorz Graff, R. Ortega, Alfonso Ruiz-Herrera
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.
{"title":"Attractors of dissipative homeomorphisms of the infinite surface homeomorphic to a punctured sphere","authors":"Grzegorz Graff, R. Ortega, Alfonso Ruiz-Herrera","doi":"10.1142/s0219199722500109","DOIUrl":"https://doi.org/10.1142/s0219199722500109","url":null,"abstract":"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.","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2022-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46591310","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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