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":" ","pages":""},"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":" ","pages":""},"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":" ","pages":""},"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":" ","pages":""},"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":" ","pages":""},"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":" ","pages":""},"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":" ","pages":""},"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}
Pub Date : 2022-03-23DOI: 10.1142/s0219199722500365
Xinyu Cheng, Dong Li, Wen Yang
: We show the equivalence of the classical Helmholtz equation and the fractional Helmholtz equation with arbitrary order. This improves a recent result of Guan, Murugan and Wei [5].
{"title":"On the equivalence of classical helmholtz equation and fractional helmholtz equation with arbitrary order","authors":"Xinyu Cheng, Dong Li, Wen Yang","doi":"10.1142/s0219199722500365","DOIUrl":"https://doi.org/10.1142/s0219199722500365","url":null,"abstract":": We show the equivalence of the classical Helmholtz equation and the fractional Helmholtz equation with arbitrary order. This improves a recent result of Guan, Murugan and Wei [5].","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2022-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43771219","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 : 2022-03-16DOI: 10.1142/s0219199722500067
Yuxia Guo, Shaolong Peng
In this paper, we are mainly concerned with the following system in an exterior domains: [Formula: see text] where [Formula: see text], [Formula: see text] is an integer, [Formula: see text], and [Formula: see text] is the polyharmonic operator. We prove the nonexistence of positive solutions to the above system for [Formula: see text] if [Formula: see text], and [Formula: see text] if [Formula: see text]. The novelty of the paper is that we do not ask [Formula: see text] satisfy any symmetry and asymptotic conditions at infinity. By proving the superharmonic properties of the solutions, we establish the equivalence between systems of partial differential equations (PDEs) and integral equations (IEs), then the method of scaling sphere in integral form can be applied to prove the nonexistence of the solutions.
{"title":"Liouville-type theorems for higher-order Lane–Emden system in exterior domains","authors":"Yuxia Guo, Shaolong Peng","doi":"10.1142/s0219199722500067","DOIUrl":"https://doi.org/10.1142/s0219199722500067","url":null,"abstract":"In this paper, we are mainly concerned with the following system in an exterior domains: [Formula: see text] where [Formula: see text], [Formula: see text] is an integer, [Formula: see text], and [Formula: see text] is the polyharmonic operator. We prove the nonexistence of positive solutions to the above system for [Formula: see text] if [Formula: see text], and [Formula: see text] if [Formula: see text]. The novelty of the paper is that we do not ask [Formula: see text] satisfy any symmetry and asymptotic conditions at infinity. By proving the superharmonic properties of the solutions, we establish the equivalence between systems of partial differential equations (PDEs) and integral equations (IEs), then the method of scaling sphere in integral form can be applied to prove the nonexistence of the solutions.","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2022-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49430432","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 : 2022-03-10DOI: 10.1142/s0219199723500104
Julien Dardennes, J. Gutt, Jun Zhang
We investigate the convexity up to symplectomorphism (called symplectic convexity) of star-shaped toric domains in $mathbb R^4$. In particular, based on the criterion from Chaidez-Edtmair via Ruelle invariant and systolic ratio of the boundary of star-shaped toric domains, we provide elementary operations on domains that can kill the symplectic convexity. These operations only result in $C^0$-small perturbations in terms of domains' volume. Moreover, one of the operations is a systematic way to produce examples of dynamically convex but not symplectically convex toric domains. Finally, we are able to provide concrete bounds for the constants that appear in Chaidez-Edtmair's criterion.
{"title":"Symplectic non-convexity of toric domains","authors":"Julien Dardennes, J. Gutt, Jun Zhang","doi":"10.1142/s0219199723500104","DOIUrl":"https://doi.org/10.1142/s0219199723500104","url":null,"abstract":"We investigate the convexity up to symplectomorphism (called symplectic convexity) of star-shaped toric domains in $mathbb R^4$. In particular, based on the criterion from Chaidez-Edtmair via Ruelle invariant and systolic ratio of the boundary of star-shaped toric domains, we provide elementary operations on domains that can kill the symplectic convexity. These operations only result in $C^0$-small perturbations in terms of domains' volume. Moreover, one of the operations is a systematic way to produce examples of dynamically convex but not symplectically convex toric domains. Finally, we are able to provide concrete bounds for the constants that appear in Chaidez-Edtmair's criterion.","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2022-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45909570","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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