Pub Date : 2024-06-01DOI: 10.4310/arkiv.2024.v62.n1.a4
Ryan Hynd, Simon Larson, Erik Lindgren
We study the decay (at infinity) of extremals of Morrey’s inequality in $mathbb{R}^n$. These are functions satisfying[underset{x neq y}{sup} frac{lvert u(x)-u(y) rvert}{{lvert x-y rvert}^{1-frac{n}{p}}} = C(p,n) {lVert nabla u (mathbb{R}^n rVert}_{L^p (mathbb{R}^n)} ; textrm{,}]where $p gt n$ and $C(p, n)$ is the optimal constant in Morrey’s inequality. We prove that if $n geq 2$ then any extremal has a power decay of order $beta$ for any[beta lt - frac{1}{3} + frac{2}{3(p-1)} + sqrt{left( -frac{1}{3} + frac{2}{3(p-1)} right)^2 + frac{1}{3}} ; textrm{.}]
{"title":"Decay of extremals of Morrey’s inequality","authors":"Ryan Hynd, Simon Larson, Erik Lindgren","doi":"10.4310/arkiv.2024.v62.n1.a4","DOIUrl":"https://doi.org/10.4310/arkiv.2024.v62.n1.a4","url":null,"abstract":"We study the decay (at infinity) of extremals of Morrey’s inequality in $mathbb{R}^n$. These are functions satisfying[underset{x neq y}{sup} frac{lvert u(x)-u(y) rvert}{{lvert x-y rvert}^{1-frac{n}{p}}} = C(p,n) {lVert nabla u (mathbb{R}^n rVert}_{L^p (mathbb{R}^n)} ; textrm{,}]where $p gt n$ and $C(p, n)$ is the optimal constant in Morrey’s inequality. We prove that if $n geq 2$ then any extremal has a power decay of order $beta$ for any[beta lt - frac{1}{3} + frac{2}{3(p-1)} + sqrt{left( -frac{1}{3} + frac{2}{3(p-1)} right)^2 + frac{1}{3}} ; textrm{.}]","PeriodicalId":501438,"journal":{"name":"Arkiv för Matematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141190152","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-01DOI: 10.4310/arkiv.2024.v62.n1.a5
Abel Lacabanne, Daniel Tubbenhauer, Pedro Vaz
We give a closed formula to evaluate exterior webs (also called MOY webs) and the associated Reshetikhin–Turaev link polynomials.
我们给出了一个封闭公式来评估外部网（又称 MOY 网）和相关的雷谢提金-图拉耶夫链路多项式。
{"title":"A formula to evaluate type-A webs and link polynomials","authors":"Abel Lacabanne, Daniel Tubbenhauer, Pedro Vaz","doi":"10.4310/arkiv.2024.v62.n1.a5","DOIUrl":"https://doi.org/10.4310/arkiv.2024.v62.n1.a5","url":null,"abstract":"We give a closed formula to evaluate exterior webs (also called MOY webs) and the associated Reshetikhin–Turaev link polynomials.","PeriodicalId":501438,"journal":{"name":"Arkiv för Matematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141189879","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-01DOI: 10.4310/arkiv.2024.v62.n1.a8
Lukas Nakamura
We show that the Legendrian lift of an exact, displaceable Lagrangian has vanishing Shelukhin–Chekanov–Hofer pseudo-metric by lifting an argument due to Sikorav to the contactization. In particular, this proves the existence of such Legendrians, providing counterexamples to a conjecture of Rosen and Zhang.
{"title":"Legendrians with vanishing Shelukhin–Chekanov–Hofer metric","authors":"Lukas Nakamura","doi":"10.4310/arkiv.2024.v62.n1.a8","DOIUrl":"https://doi.org/10.4310/arkiv.2024.v62.n1.a8","url":null,"abstract":"We show that the Legendrian lift of an exact, displaceable Lagrangian has vanishing Shelukhin–Chekanov–Hofer pseudo-metric by lifting an argument due to Sikorav to the contactization. In particular, this proves the existence of such Legendrians, providing counterexamples to a conjecture of Rosen and Zhang.","PeriodicalId":501438,"journal":{"name":"Arkiv för Matematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141187797","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-01DOI: 10.4310/arkiv.2024.v62.n1.a6
Sara Maad Sasane, Wilhelm Treschow
We investigate the persistance of embedded eigenvalues under perturbations of a certain self-adjoint Schrödinger-type differential operator in $L^2 (mathbb{R} ; mathbb{R}^n)$, with an asymptotically periodic potential. The studied perturbations are small and belong to a certain Banach space with a specified decay rate, in particular, a weighted space of continuous matrix valued functions. Our main result is that the set of perturbations for which the embedded eigenvalue persists forms a smooth manifold with a specified co-dimension. This is done using tools from Floquet theory, basic Banach space calculus, exponential dichotomies and their roughness properties, and Lyapunov- Schmidt reduction. A second result is provided, where under an extra assumption, it can be proved that the first result holds even when the space of perturbations is replaced by a much smaller space, as long as it contains a minimal subspace. In the end, as a way of showing that the investigated setting exists, a concrete example is presented. The example itself relates to a problem from quantum mechanics and represents a system of electrons in an infinite one-dimensional crystal.
{"title":"Embedded eigenvalues for asymptotically periodic ODE systems","authors":"Sara Maad Sasane, Wilhelm Treschow","doi":"10.4310/arkiv.2024.v62.n1.a6","DOIUrl":"https://doi.org/10.4310/arkiv.2024.v62.n1.a6","url":null,"abstract":"We investigate the persistance of embedded eigenvalues under perturbations of a certain self-adjoint Schrödinger-type differential operator in $L^2 (mathbb{R} ; mathbb{R}^n)$, with an asymptotically periodic potential. The studied perturbations are small and belong to a certain Banach space with a specified decay rate, in particular, a weighted space of continuous matrix valued functions. Our main result is that the set of perturbations for which the embedded eigenvalue persists forms a smooth manifold with a specified co-dimension. This is done using tools from Floquet theory, basic Banach space calculus, exponential dichotomies and their roughness properties, and Lyapunov- Schmidt reduction. A second result is provided, where under an extra assumption, it can be proved that the first result holds even when the space of perturbations is replaced by a much smaller space, as long as it contains a minimal subspace. In the end, as a way of showing that the investigated setting exists, a concrete example is presented. The example itself relates to a problem from quantum mechanics and represents a system of electrons in an infinite one-dimensional crystal.","PeriodicalId":501438,"journal":{"name":"Arkiv för Matematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141189988","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-01DOI: 10.4310/arkiv.2024.v62.n1.a10
Roswitha Rissner, Irena Swanson
We count the numbers of associated primes of powers of ideals as defined in $href{https://dx.doi.org/10.1007/s11512-013-0184-1}{[2]}$. We generalize those ideals to monomial ideals $operatorname{BHH}(m, r, s)$ for $r geq 2, m, s geq1$; we establish partially the associated primes of powers of these ideals, and we establish completely the depth function of quotients by powers of these ideals: the depth function is periodic of period $r$ repeated $m$ times on the initial interval before settling to a constant value. The number of needed variables for these depth functions are lower than those from general constructions in $href{https://doi.org/10.1090/proc/15083}{[6]}$.
我们计算了 $href{https://dx.doi.org/10.1007/s11512-013-0184-1}{[2]}$ 中定义的幂的相关素数。我们把这些理想归纳为 $r geq 2, m, s geq1$ 的单项式理想 $operatorname{BHH}(m,r,s)$；我们部分地建立了这些理想的幂的相关素数，并完全建立了这些理想的幂的商的深度函数：深度函数是周期为 $r$ 的周期性函数，在初始区间上重复 $m$ 次，然后稳定为一个常值。这些深度函数所需的变量数目低于 $href{https://doi.org/10.1090/proc/15083}{[6]}$ 中一般构造的变量数目。
{"title":"Fluctuations in depth and associated primes of powers of ideals","authors":"Roswitha Rissner, Irena Swanson","doi":"10.4310/arkiv.2024.v62.n1.a10","DOIUrl":"https://doi.org/10.4310/arkiv.2024.v62.n1.a10","url":null,"abstract":"We count the numbers of associated primes of powers of ideals as defined in $href{https://dx.doi.org/10.1007/s11512-013-0184-1}{[2]}$. We generalize those ideals to monomial ideals $operatorname{BHH}(m, r, s)$ for $r geq 2, m, s geq1$; we establish partially the associated primes of powers of these ideals, and we establish completely the depth function of quotients by powers of these ideals: the depth function is periodic of period $r$ repeated $m$ times on the initial interval before settling to a constant value. The number of needed variables for these depth functions are lower than those from general constructions in $href{https://doi.org/10.1090/proc/15083}{[6]}$.","PeriodicalId":501438,"journal":{"name":"Arkiv för Matematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141187798","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-01DOI: 10.4310/arkiv.2024.v62.n1.a9
Magnus C. Ørke
We study traveling waves for a class of fractional Korteweg–De Vries and fractional Degasperis–Procesi equations with a parametrized Fourier multiplier operator of order $s in (-1, 0)$. For both equations there exist local analytic bifurcation branches emanating from a curve of constant solutions, consisting of smooth, even and periodic traveling waves. The local branches extend to global solution curves. In the limit we find a highest, cusped traveling-wave solution and prove its optimal $s$-Hölder regularity, attained in the cusp.
{"title":"Highest waves for fractional Korteweg–De Vries and Degasperis–Procesi equations","authors":"Magnus C. Ørke","doi":"10.4310/arkiv.2024.v62.n1.a9","DOIUrl":"https://doi.org/10.4310/arkiv.2024.v62.n1.a9","url":null,"abstract":"We study traveling waves for a class of fractional Korteweg–De Vries and fractional Degasperis–Procesi equations with a parametrized Fourier multiplier operator of order $s in (-1, 0)$. For both equations there exist local analytic bifurcation branches emanating from a curve of constant solutions, consisting of smooth, even and periodic traveling waves. The local branches extend to global solution curves. In the limit we find a highest, cusped traveling-wave solution and prove its optimal $s$-Hölder regularity, attained in the cusp.","PeriodicalId":501438,"journal":{"name":"Arkiv för Matematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141188165","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-01DOI: 10.4310/arkiv.2024.v62.n1.a1
Shahroud Azami
In this paper, we investigate continuity, differentiability and monotonicity for the first nonzero eigenvalue of the Wentzell–Laplace operator along the conformal mean curvature flow on $n$-dimensional compact manifolds with boundary for $n geq 3$ under a boundary condition. In especial, we show that the first nonzero eigenvalue of the Wentzell–Laplace operator is monotonic under the conformal mean curvature flow and we find some monotonic quantities dependent to the first nonzero eigenvalue along the conformal mean curvature flow.
{"title":"Evolution of eigenvalue of the Wentzell–Laplace operator along the conformal mean curvature flow","authors":"Shahroud Azami","doi":"10.4310/arkiv.2024.v62.n1.a1","DOIUrl":"https://doi.org/10.4310/arkiv.2024.v62.n1.a1","url":null,"abstract":"In this paper, we investigate continuity, differentiability and monotonicity for the first nonzero eigenvalue of the Wentzell–Laplace operator along the conformal mean curvature flow on $n$-dimensional compact manifolds with boundary for $n geq 3$ under a boundary condition. In especial, we show that the first nonzero eigenvalue of the Wentzell–Laplace operator is monotonic under the conformal mean curvature flow and we find some monotonic quantities dependent to the first nonzero eigenvalue along the conformal mean curvature flow.","PeriodicalId":501438,"journal":{"name":"Arkiv för Matematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141190021","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-01DOI: 10.4310/arkiv.2024.v62.n1.a7
Eoghan McDowell
This paper identifies all pairs of ordinary irreducible characters of the alternating group which agree on conjugacy classes of elements of order not divisible by a fixed integer $l$, for $l^prime = 3$. We do likewise for spin characters of the symmetric and alternating groups. We find that the only such characters are the conjugate or associate pairs labelled by partitions with a certain parameter divisible by $l$. When $l$ is prime, this implies that the rows of the $l$-modular decomposition matrix are distinct except for the rows labelled by these pairs. When $l=3$ we exhibit many additional examples of such pairs of characters.
{"title":"Characters and spin characters of alternating and symmetric groups determined by values on $l^prime$-classes","authors":"Eoghan McDowell","doi":"10.4310/arkiv.2024.v62.n1.a7","DOIUrl":"https://doi.org/10.4310/arkiv.2024.v62.n1.a7","url":null,"abstract":"This paper identifies all pairs of ordinary irreducible characters of the alternating group which agree on conjugacy classes of elements of order not divisible by a fixed integer $l$, for $l^prime = 3$. We do likewise for spin characters of the symmetric and alternating groups. We find that the only such characters are the conjugate or associate pairs labelled by partitions with a certain parameter divisible by $l$. When $l$ is prime, this implies that the rows of the $l$-modular decomposition matrix are distinct except for the rows labelled by these pairs. When $l=3$ we exhibit many additional examples of such pairs of characters.","PeriodicalId":501438,"journal":{"name":"Arkiv för Matematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141187792","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-01DOI: 10.4310/arkiv.2024.v62.n1.a3
Shoutao Guo, Li Liang
In this paper, we first introduce stable functors with respect to a pre-enveloping / pre-covering subcategory and investigate some of their properties. Using that we then introduce and study a relative complete cohomology theory in abelian categories. Some properties of the cohomology including vanishing are given. As applications, we give some characterizations of objects of finite homological dimensions including the flat dimension, cotorsion dimension, Gorenstein injective/flat dimension and projectively coresolved Gorenstein flat dimension.
{"title":"Stable functors and cohomology theory in abelian categories","authors":"Shoutao Guo, Li Liang","doi":"10.4310/arkiv.2024.v62.n1.a3","DOIUrl":"https://doi.org/10.4310/arkiv.2024.v62.n1.a3","url":null,"abstract":"In this paper, we first introduce stable functors with respect to a pre-enveloping / pre-covering subcategory and investigate some of their properties. Using that we then introduce and study a relative complete cohomology theory in abelian categories. Some properties of the cohomology including vanishing are given. As applications, we give some characterizations of objects of finite homological dimensions including the flat dimension, cotorsion dimension, Gorenstein injective/flat dimension and projectively coresolved Gorenstein flat dimension.","PeriodicalId":501438,"journal":{"name":"Arkiv för Matematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141190019","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-01DOI: 10.4310/arkiv.2024.v62.n1.a2
Laura Cossu, Salvatore Tringali
$defF{mathscr{F}}defz{mathfrak{z}}defzprime{mathfrak{z}^prime}$ Let $H$ be a monoid and $pi_H$ be the unique extension of the identity map on $H $ to a monoid homomorphism $F(H) to H$, where we denote by $F(X)$ the free monoid on a set $X$. Given $A subseteq H$, an $A$-word $z$ (i.e., an element of F(A)) is minimal if $pi_H (z) neq pi_H (zprime)$ for every permutation $zprime$ of a proper subword of $z$. The minimal $A$-elasticity of $H$ is then the supremum of all rational numbers $m/n$ with $m, n in mathbb{N}^+$ such that there exist minimal $A$-words $mathfrak{a}$ and $mathfrak{b}$ of length $m$ and $n$, resp., with $pi_H (mathfrak{a}) = pi_H (mathfrak{b})$. Among other things, we show that if $H$ is commutative and $A$ is finite, then the minimal $A$-elasticity of $H$ is finite. This provides a non-trivial generalization of the finiteness part of a classical theorem of Anderson et al. from the case where $H$ is cancellative, commutative, and finitely generated modulo units, and $A$ is the set $mathscr{A} (H)$ of atoms of $H$. We also demonstrate that commutativity is somewhat essential here, by proving the existence of an atomic, cancellative, finitely generated monoid with trivial group of units whose minimal $mathscr{A} (H)$-elasticity is infinite.
{"title":"On the finiteness of certain factorization invariants","authors":"Laura Cossu, Salvatore Tringali","doi":"10.4310/arkiv.2024.v62.n1.a2","DOIUrl":"https://doi.org/10.4310/arkiv.2024.v62.n1.a2","url":null,"abstract":"$defF{mathscr{F}}defz{mathfrak{z}}defzprime{mathfrak{z}^prime}$ Let $H$ be a monoid and $pi_H$ be the unique extension of the identity map on $H $ to a monoid homomorphism $F(H) to H$, where we denote by $F(X)$ the free monoid on a set $X$. Given $A subseteq H$, an $A$-word $z$ (i.e., an element of F(A)) is minimal if $pi_H (z) neq pi_H (zprime)$ for every permutation $zprime$ of a proper subword of $z$. The minimal $A$-elasticity of $H$ is then the supremum of all rational numbers $m/n$ with $m, n in mathbb{N}^+$ such that there exist minimal $A$-words $mathfrak{a}$ and $mathfrak{b}$ of length $m$ and $n$, resp., with $pi_H (mathfrak{a}) = pi_H (mathfrak{b})$. Among other things, we show that if $H$ is commutative and $A$ is finite, then the minimal $A$-elasticity of $H$ is finite. This provides a non-trivial generalization of the finiteness part of a classical theorem of Anderson <i>et al</i>. from the case where $H$ is cancellative, commutative, and finitely generated modulo units, and $A$ is the set $mathscr{A} (H)$ of atoms of $H$. We also demonstrate that commutativity is somewhat essential here, by proving the existence of an atomic, cancellative, finitely generated monoid with trivial group of units whose minimal $mathscr{A} (H)$-elasticity is infinite.","PeriodicalId":501438,"journal":{"name":"Arkiv för Matematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141190023","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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