Pub Date : 2022-03-02DOI: 10.1142/s0219199722500018
Weijie Sheng, Mingxin Wang, Zhi-Cheng Wang
This paper is concerned with the propagation phenomena in a diffusion system with the Belousov–Zhabotinskii chemical reaction in [Formula: see text] under the bistability assumption. We prove that there is a new type of entire solution originated from three moving planar traveling fronts, and evolved to a V-shaped traveling front as time changes, which means that the profile of this solution is not invariant at all. Here an entire solution is referred to a solution that is defined for all time [Formula: see text] and in the whole space [Formula: see text]. Furthermore, we show that not only the entire solution but also all transition fronts share the same global mean speed by constructing suitable radially symmetric expanding and retracting super- and subsolutions.
{"title":"Propagation phenomena in a diffusion system with the Belousov–Zhabotinskii chemical reaction","authors":"Weijie Sheng, Mingxin Wang, Zhi-Cheng Wang","doi":"10.1142/s0219199722500018","DOIUrl":"https://doi.org/10.1142/s0219199722500018","url":null,"abstract":"This paper is concerned with the propagation phenomena in a diffusion system with the Belousov–Zhabotinskii chemical reaction in [Formula: see text] under the bistability assumption. We prove that there is a new type of entire solution originated from three moving planar traveling fronts, and evolved to a V-shaped traveling front as time changes, which means that the profile of this solution is not invariant at all. Here an entire solution is referred to a solution that is defined for all time [Formula: see text] and in the whole space [Formula: see text]. Furthermore, we show that not only the entire solution but also all transition fronts share the same global mean speed by constructing suitable radially symmetric expanding and retracting super- and subsolutions.","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2022-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44156635","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 : 2022-02-02DOI: 10.1142/S0219199722500699
Andrew Ballin, Wen-Qi Niu
We study the simplest example of mirror symmetry for 3d N = 4 SUSY gauge theories: the A-twist of a free hypermultiplet and the B-twist of SQED. We particularly focus on the category of line operators in each theory. Using the work of Costello-Gaiotto, we define these categories as appropriate categories of modules for the boundary vertex operator algebras present in each theory. For the A-twist of a free hyper, this will be a certain category of modules for the βγ VOA, properly containing the category previously studied by Allen-Wood. Applying the work of Creutzig-Kanade-McRae and Creutzig-McRae-Yang, we show that the category of line operators on the A side possesses the structure of a braided tensor category, extending the result of Allen-Wood. In addition, we prove that there is a braided tensor equivalence between the categories of line operators on the A side and B side, completing a non-trivial check of the 3d mirror symmetry conjecture. We derive explicit fusion rules as a consequence of this equivalence and obtain interesting relations with associated quantum group representations.
{"title":"3d Mirror Symmetry and the βγ VOA","authors":"Andrew Ballin, Wen-Qi Niu","doi":"10.1142/S0219199722500699","DOIUrl":"https://doi.org/10.1142/S0219199722500699","url":null,"abstract":"We study the simplest example of mirror symmetry for 3d N = 4 SUSY gauge theories: the A-twist of a free hypermultiplet and the B-twist of SQED. We particularly focus on the category of line operators in each theory. Using the work of Costello-Gaiotto, we define these categories as appropriate categories of modules for the boundary vertex operator algebras present in each theory. For the A-twist of a free hyper, this will be a certain category of modules for the βγ VOA, properly containing the category previously studied by Allen-Wood. Applying the work of Creutzig-Kanade-McRae and Creutzig-McRae-Yang, we show that the category of line operators on the A side possesses the structure of a braided tensor category, extending the result of Allen-Wood. In addition, we prove that there is a braided tensor equivalence between the categories of line operators on the A side and B side, completing a non-trivial check of the 3d mirror symmetry conjecture. We derive explicit fusion rules as a consequence of this equivalence and obtain interesting relations with associated quantum group representations.","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2022-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44391592","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 : 2022-01-24DOI: 10.1142/s0219199723500025
Z. Ran
A BSTRACT . On a general hypersurface of degree d ≤ n in P n or P n itself, we prove the existence of curves of any genus and high enough degree depending on the genus passing through the expected number t of general points or incident to a general collection of subvarieties of suitable codimensions. In some cases we also show that the family of curves through t fixed points has general moduli as family of t -pointed curves. These results imply positivity of certain intersection numbers on Kontsevich spaces of stable maps. An arithmetical appendix by M. C. Chang descibes the set of numerical characters ( n , d , curve degree, genus) to which our results apply.
{"title":"Interpolation of curves on Fano hypersurfaces","authors":"Z. Ran","doi":"10.1142/s0219199723500025","DOIUrl":"https://doi.org/10.1142/s0219199723500025","url":null,"abstract":"A BSTRACT . On a general hypersurface of degree d ≤ n in P n or P n itself, we prove the existence of curves of any genus and high enough degree depending on the genus passing through the expected number t of general points or incident to a general collection of subvarieties of suitable codimensions. In some cases we also show that the family of curves through t fixed points has general moduli as family of t -pointed curves. These results imply positivity of certain intersection numbers on Kontsevich spaces of stable maps. An arithmetical appendix by M. C. Chang descibes the set of numerical characters ( n , d , curve degree, genus) to which our results apply.","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2022-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42539098","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 : 2022-01-23DOI: 10.1142/S0219199722500419
A. Albouy, A. J. Ureña
. Consider the equation of the linear oscillator u ′′ + u = h ( θ ), where the forcing term h : R → R is 2 π -periodic and positive. We show that the existence of a periodic solution implies the existence of a positive solution. To this aim we establish connections between this problem and some separation questions of convex analysis.
． 考虑线性振荡器u ' ' + u = h (θ)的方程，其中强迫项h: R→R是2 π周期且为正的。我们证明了周期解的存在性意味着正解的存在性。为此，我们建立了该问题与凸分析的若干分离问题之间的联系。
{"title":"An antimaximum principle for periodic solutions of a forced oscillator","authors":"A. Albouy, A. J. Ureña","doi":"10.1142/S0219199722500419","DOIUrl":"https://doi.org/10.1142/S0219199722500419","url":null,"abstract":". Consider the equation of the linear oscillator u ′′ + u = h ( θ ), where the forcing term h : R → R is 2 π -periodic and positive. We show that the existence of a periodic solution implies the existence of a positive solution. To this aim we establish connections between this problem and some separation questions of convex analysis.","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2022-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43826889","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 : 2022-01-12DOI: 10.1142/s0219199723500268
Caroline VanBlargan, Yi Wang
In this paper, we establish quantitative Alexandrov-Fenchel inequalities for quermassintegrals on nearly spherical sets. In particular, we bound the $(k,m)$-isoperimetric deficit from below by the Frankael asymmetry. We also find a lower bound on the $(k,m)$-isoperimetric deficit using the spherical deviation.
{"title":"Quantitative Quermassintegral Inequalities for Nearly Spherical Sets","authors":"Caroline VanBlargan, Yi Wang","doi":"10.1142/s0219199723500268","DOIUrl":"https://doi.org/10.1142/s0219199723500268","url":null,"abstract":"In this paper, we establish quantitative Alexandrov-Fenchel inequalities for quermassintegrals on nearly spherical sets. In particular, we bound the $(k,m)$-isoperimetric deficit from below by the Frankael asymmetry. We also find a lower bound on the $(k,m)$-isoperimetric deficit using the spherical deviation.","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2022-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44696291","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 : 2022-01-10DOI: 10.1142/s0219199722500651
Achenef Tesfahun
A bstract . This paper is concerned with a two dimensional Whitham–Boussinesq system modeling surface waves of an inviscid incompressible fluid layer. We prove that the associated Cauchy problem is well-posed for initial data of low regularity, with existence time of scale O (cid:16) µ 3 / 2 − ǫ − 2 + (cid:17) , where µ and ǫ are small parameters related to the level of dispersion and nonlinearity, respectively. In particular, in the KdV regime { µ ∼ ǫ }, the existence time is of order ǫ − 1 / 2 . The main ingredients in the proof are frequency loacalised dispersive estimates and bilinear Strichartz estimates that depend on the parameter µ .
{"title":"Long-time existence for a whitham boussinesq system in two dimensions","authors":"Achenef Tesfahun","doi":"10.1142/s0219199722500651","DOIUrl":"https://doi.org/10.1142/s0219199722500651","url":null,"abstract":"A bstract . This paper is concerned with a two dimensional Whitham–Boussinesq system modeling surface waves of an inviscid incompressible fluid layer. We prove that the associated Cauchy problem is well-posed for initial data of low regularity, with existence time of scale O (cid:16) µ 3 / 2 − ǫ − 2 + (cid:17) , where µ and ǫ are small parameters related to the level of dispersion and nonlinearity, respectively. In particular, in the KdV regime { µ ∼ ǫ }, the existence time is of order ǫ − 1 / 2 . The main ingredients in the proof are frequency loacalised dispersive estimates and bilinear Strichartz estimates that depend on the parameter µ .","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2022-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46832753","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 : 2022-01-01DOI: 10.1142/S021919972250016X
Vincent Guan, M. Murugan, Juncheng Wei
We show that the bounded solutions to the fractional Helmholtz equation, $(-Delta)^s u= u$ for $0
我们证明了分数阶亥姆霍兹方程$(-Delta)^s u= u$ ($mathbb{R}^n$中$0
{"title":"Helmholtz Solutions for the Fractional Laplacian and Other Related Operators","authors":"Vincent Guan, M. Murugan, Juncheng Wei","doi":"10.1142/S021919972250016X","DOIUrl":"https://doi.org/10.1142/S021919972250016X","url":null,"abstract":"We show that the bounded solutions to the fractional Helmholtz equation, $(-Delta)^s u= u$ for $0<s<1$ in $mathbb{R}^n$, are given by the bounded solutions to the classical Helmholtz equation $(-Delta)u= u$ in $mathbb{R}^n$ for $n ge 2$ when $u$ is additionally assumed to be vanishing at $infty$. When $n=1$, we show that the bounded fractional Helmholtz solutions are again given by the classical solutions $Acos{x} + Bsin{x}$. We show that this classification of fractional Helmholtz solutions extends for $1<s le 2$ and $sin mathbb{N}$ when $u in C^infty(mathbb{R}^n)$. Finally, we prove that the classical solutions are the unique bounded solutions to the more general equation $psi(-Delta) u= psi(1)u$ in $mathbb{R}^n$, when $psi$ is complete Bernstein and certain regularity conditions are imposed on the associated weight $a(t)$.","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44497873","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 : 2021-12-27DOI: 10.1142/s0219199722500456
René Brandenberg, Katherina von Dichter, Bernardo González Merino
This paper deals with four symmetrizations of a convex set C: the intersection, the harmonic and the arithmetic mean, and the convex hull of C and −C. A well-known result of Firey shows that those means build up a subset-chain in the given order. On the one hand, we determine the dilatation factors, depending on the asymmetry of C, to reverse the containments between any of those symmetrizations. On the other hand, we tighten the relations proven by Firey and show a stability result concerning those factors near the simplex.
{"title":"Tightening and reversing the arithmetic-harmonic mean inequality for symmetrizations of convex sets","authors":"René Brandenberg, Katherina von Dichter, Bernardo González Merino","doi":"10.1142/s0219199722500456","DOIUrl":"https://doi.org/10.1142/s0219199722500456","url":null,"abstract":"This paper deals with four symmetrizations of a convex set C: the intersection, the harmonic and the arithmetic mean, and the convex hull of C and −C. A well-known result of Firey shows that those means build up a subset-chain in the given order. On the one hand, we determine the dilatation factors, depending on the asymmetry of C, to reverse the containments between any of those symmetrizations. On the other hand, we tighten the relations proven by Firey and show a stability result concerning those factors near the simplex.","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2021-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46109344","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 : 2021-12-20DOI: 10.1142/S0219199722500535
C. Bernardini
We consider the following prescribed $Q$-curvature problem begin{equation}label{uno} begin{cases} Delta^2 u=(1-|x|^p)e^{4u}, quadtext{on},,mathbb{R}^4 Lambda:=int_{mathbb{R}^4}(1-|x|^p)e^{4u}dx
{"title":"Existence and asymptotic behavior of non-normal conformal metrics on ℝ4 with sign-changing Q-curvature","authors":"C. Bernardini","doi":"10.1142/S0219199722500535","DOIUrl":"https://doi.org/10.1142/S0219199722500535","url":null,"abstract":"We consider the following prescribed $Q$-curvature problem begin{equation}label{uno} begin{cases} Delta^2 u=(1-|x|^p)e^{4u}, quadtext{on},,mathbb{R}^4 Lambda:=int_{mathbb{R}^4}(1-|x|^p)e^{4u}dx<infty. end{cases} end{equation} We show that for every polynomial $P$ of degree 2 such that $limlimits_{|x|to+infty}P=-infty$, and for every $Lambdain(0,Lambda_mathrm{sph})$, there exists at least one solution which assume the form $u=w+P$, where $w$ behaves logarithmically at infinity. Conversely, we prove that all solutions have the form $v+P$, where $$v(x)=frac{1}{8pi^2}intlimits_{mathbb{R}^4}logleft(frac{|y|}{|x-y|}right)(1-|y|^p)e^{4u}dy$$ and $P$ is a polynomial of degree at most 2 bounded from above. Moreover, if $u$ is a solution to the previous problem, it has the following asymptotic behavior $$u(x)=-frac{Lambda}{8pi^2}log|x|+P+o(log|x|),quadtext{as},,|x|to+infty.$$ As a consequence, we give a geometric characterization of solutions in terms of the scalar curvature at infinity of the associated conformal metric $e^{2u}|dx|^2$.","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2021-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47667094","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 : 2021-12-15DOI: 10.1142/S0219199723500037
Xiaolong Li
. We investigate the curvature operator of the second kind on Riemannian manifolds and prove several classification results. The first one asserts that a closed Riemannian manifold with three-positive curvature operator of the second kind is diffeomorphic to a spherical space form, improving a recent result of Cao-Gursky-Tran assuming two-positivity. The second one states that a closed Riemannian manifold with three-nonnegative curvature operator of the second kind is either diffeomorphic to a spherical space form, or flat, or isometric to a quotient of a compact irreducible symmetric space. This settles the nonnegativity part of Nishikawa’s conjecture under a weaker assumption.
{"title":"Manifolds with nonnegative curvature operator of the second kind","authors":"Xiaolong Li","doi":"10.1142/S0219199723500037","DOIUrl":"https://doi.org/10.1142/S0219199723500037","url":null,"abstract":". We investigate the curvature operator of the second kind on Riemannian manifolds and prove several classification results. The first one asserts that a closed Riemannian manifold with three-positive curvature operator of the second kind is diffeomorphic to a spherical space form, improving a recent result of Cao-Gursky-Tran assuming two-positivity. The second one states that a closed Riemannian manifold with three-nonnegative curvature operator of the second kind is either diffeomorphic to a spherical space form, or flat, or isometric to a quotient of a compact irreducible symmetric space. This settles the nonnegativity part of Nishikawa’s conjecture under a weaker assumption.","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2021-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48017934","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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