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Two curious q-supercongruences and their extensions 两个奇特的 q 超共轭及其扩展

IF 0.8 3区数学 Q1 MATHEMATICS

Forum Mathematicum

Pub Date : 2024-01-01 DOI: 10.1515/forum-2023-0164

Haihong He, Xiaoxia Wang

We prove two single-parameter q-supercongruences which were recently conjectured by Guo, and establish their further extensions with one more parameter. Crucial ingredients in the proof are the terminating form of the q-binomial theorem, a Karlsson–Minton-type summation formula due to Gasper, and the method of “creative microscoping” developed by Guo and Zudilin. Incidentally, an assertion of Li, Tang and Wang is also confirmed by establishing its q-analogue.

我们证明了郭明錤最近猜想的两个单参数 q-supercongruences ，并在多一个参数的情况下建立了它们的进一步扩展。证明的关键要素是 q 二项式定理的终结形式、Gasper 提出的卡尔松-明顿式求和公式，以及郭和祖迪林提出的 "创造性微观 "方法。顺便提一下，李、唐和王的一个论断也通过建立其 q-analogue 得到了证实。

引用次数: 0

Regularity of fractional heat semigroups associated with Schrödinger operators on Heisenberg groups 与海森堡群上薛定谔算子相关的分数热半群的正则性

IF 0.8 3区数学 Q1 MATHEMATICS

Forum Mathematicum

Pub Date : 2024-01-01 DOI: 10.1515/forum-2023-0285

Chuanhong Sun, Pengtao Li, Zengjian Lou

Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>L</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:mo>-</m:mo> <m:msub> <m:mi mathvariant="normal">Δ</m:mi> <m:msup> <m:mi>ℍ</m:mi> <m:mi>n</m:mi> </m:msup> </m:msub> </m:mrow> <m:mo>+</m:mo> <m:mi>V</m:mi> </m:mrow> </m:mrow> </m:math> {L=-{Delta}_{mathbb{H}^{n}}+V} be a Schrödinger operator on Heisenberg groups <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℍ</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> {mathbb{H}^{n}} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="normal">Δ</m:mi> <m:msup> <m:mi>ℍ</m:mi> <m:mi>n</m:mi> </m:msup> </m:msub> </m:math> {{Delta}_{mathbb{H}^{n}}} is the sub-Laplacian, the nonnegative potential V belongs to the reverse Hölder class <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>B</m:mi> <m:mrow> <m:mi mathvariant="script">𝒬</m:mi> <m:mo>/</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:math> {B_{mathcal{Q}/2}} . Here <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒬</m:mi> </m:math> {mathcal{Q}} is the homogeneous dimension of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℍ</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> {mathbb{H}^{n}} . In this article, we introduce the fractional heat semigroups <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:msup> <m:mi>e</m:mi> <m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mi>t</m:mi> <

设 L = - Δ ℍ n + V {L=-{Delta}_{mathbb{H}^{n}}+V} 是海森堡群 ℍ n {{mathbb{H}^{n}} 上的薛定谔算子，其中 Δ ℍ n {{Delta}_{mathbb{H}^{n}} 是子拉普拉斯。其中 Δ ℍ n {{Delta}_{mathbb{H}^{n}} 是子拉普拉卡，非负势 V 属于反向荷尔德类 B 𝒬 / 2 {B_{mathcal{Q}/2}} 。｝ .这里𝒬 {mathcal{Q}} 是ℍ n {mathbb{H}^{n} 的同次元维度。｝ .在本文中，我们引入分数热半群 { e - t L α } t > 0 {{e^{-tL^{alpha}}}_{t>0}} 。 , α > 0 {alpha>0} , 与 L 相关联。通过热方程的基本解，我们分别估计了分数热核 K α , t L ( ⋅ , ⋅ ) {K_{alpha,t}^{L}(,cdot,,cdot,)} 的梯度和时间分数导数。作为应用，我们通过{ e - t L α } t > 0 {{e^{-tL^{alpha}}}_{t>0}} 来描述空间 BMO L γ ( ℍ n ) {mathrm{BMO}_{L}^{gamma}(mathbb{H}^{n})} 。 .

引用次数: 0

Multiplicity of solutions for a singular system with sign-changing potential 具有符号变化势能的奇异系统解的多重性

IF 0.8 3区数学 Q1 MATHEMATICS

Forum Mathematicum

Pub Date : 2024-01-01 DOI: 10.1515/forum-2023-0345

Wentao Lin, Yilan Wei

This paper focuses on a singular system with a sign-changing potential in Γ, a bounded domain with a Lipschitz boundary in ℝ d

{mathbb{R}^{d}}

. By imposing appropriate conditions on the weight potential, which is allowed to change sign, we establish the existence of multiple solutions using the shape optimization approach. This study represents one of the earliest endeavors to explore and analyze the occurrence of multiple solutions in fractional singular systems involving sign-changing potentials. By explicitly addressing this particular aspect, our paper contributes significantly to the limited body of literature that exists in this specific field.

本文的研究重点是一个在 Γ 中具有符号变化势能的奇异系统，Γ 是一个在 ℝ d {mathbb{R}^{d} 中具有 Lipschitz 边界的有界域。｝ .通过对允许改变符号的权势施加适当的条件，我们利用形状优化方法确定了多解的存在性。这项研究是探索和分析涉及符号变化势的分数奇异系统中多解现象的最早尝试之一。通过明确探讨这一特定方面，我们的论文为这一特定领域现有的有限文献做出了重大贡献。

引用次数: 0

On the modular isomorphism problem for groups of nilpotency class 2 with cyclic center 关于具有循环中心的无钾类 2 群的模态同构问题

IF 0.8 3区数学 Q1 MATHEMATICS

Forum Mathematicum

Pub Date : 2024-01-01 DOI: 10.1515/forum-2023-0237

Diego García-Lucas, Leo Margolis

We show that the modular isomorphism problem has a positive answer for groups of nilpotency class 2 with cyclic center, i.e., that for such p-groups G and H an isomorphism between the group algebras FG and FH implies an isomorphism of the groups G and H for F the field of p elements. For groups of odd order this implication is also proven for F being any field of characteristic p. For groups of even order we need either to make an additional assumption on the groups or on the field.

我们证明了模同构问题对于具有循环中心的无幂级数 2 的群有一个肯定的答案，即对于这样的 p 群 G 和 H，群代数 FG 和 FH 之间的同构意味着群 G 和 H 对于 p 元素域 F 的同构。对于奇数阶群，F 是任何特征 p 的域时，这一蕴涵也可得到证明。对于偶数阶群，我们需要对群或域做一个额外的假设。

引用次数: 0

Donoho–Stark and Price uncertainty principles for a class of q-integral transforms with bounded kernels 一类有界核 q 积分变换的 Donoho-Stark 和 Price 不确定性原理

IF 0.8 3区数学 Q1 MATHEMATICS

Forum Mathematicum

Pub Date : 2024-01-01 DOI: 10.1515/forum-2023-0244

Luis P. Castro, Rita C. Guerra

We consider a very global q-integral transform, essentially characterized by having a bounded kernel and satisfying a set of natural and useful properties for the realization of applications. The main ambition of this work is to seek conditions that guarantee uncertainty principles of the Donoho–Stark type for that class of q-integral transforms. It should be noted that the global character of the q-integral transform in question allows one to immediately deduce corresponding Donoho–Stark uncertainty principles for q-integral operators that are its particular cases. These particular cases are very well-known operators, namely: a q-cosine-Fourier transform, a q-sine-Fourier transform, a q-Fourier transform, a q-Bessel–Fourier transform and a q-Dunkl transform. Moreover, generalizations of the local uncertainty principle of Price for the q-cosine-Fourier transform, q-sine-Fourier transform, q-Fourier transform, q-Bessel–Fourier transform and q-Dunkl transform are also obtained.

我们考虑的是一种非常全局的 q 积分变换，其基本特征是具有有界内核，并满足实现应用的一系列自然而有用的特性。这项工作的主要目标是为该类 q 积分变换寻求保证多诺霍-斯塔克类型不确定性原理的条件。应该指出的是，相关 q 积分变换的全局特性允许我们立即为作为其特殊情况的 q 积分算子推导出相应的 Donoho-Stark 不确定性原理。这些特例是非常著名的算子，即：q-余弦-傅里叶变换、q-正弦-傅里叶变换、q-傅里叶变换、q-贝塞尔-傅里叶变换和 q-敦克尔变换。此外，还获得了普赖斯局部不确定性原理对 q-余弦-傅里叶变换、q-正弦-傅里叶变换、q-傅里叶变换、q-贝塞尔-傅里叶变换和 q-Dunkl 变换的概括。

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引用次数: 0

A note on the post quantum-Sheffer polynomial sequences 关于后量子谢弗多项式序列的说明

IF 0.8 3区数学 Q1 MATHEMATICS

Forum Mathematicum

Pub Date : 2024-01-01 DOI: 10.1515/forum-2023-0004

Subuhi Khan, Mehnaz Haneef

In this article, the post quantum analogue of Sheffer polynomial sequences is introduced using concepts of post quantum calculus. The series representation, recurrence relations, determinant expression and certain other properties of this class are established. Further, the 2D-post quantum-Sheffer polynomials are introduced via generating function and their properties are established. Certain identities and integral representations for the 2D-post quantum-Hermite polynomials, 2D-post quantum-Laguerre polynomials, and 2D-post quantum-Bessel polynomials are also considered.

本文利用后量子微积分的概念，介绍了谢弗多项式序列的后量子类似物。本文建立了该类多项式的序列表示、递推关系、行列式表达和某些其他性质。此外，还通过生成函数引入了二维后量子谢弗多项式，并建立了它们的性质。此外，还考虑了二维后量子-赫米特多项式、二维后量子-拉盖尔多项式和二维后量子-贝塞尔多项式的某些等式和积分表示。

引用次数: 0

The globally smooth solutions and asymptotic behavior of the nonlinear wave equations in dimension one with multiple speeds 一维多速度非线性波方程的全局平稳解和渐近行为

IF 0.8 3区数学 Q1 MATHEMATICS

Forum Mathematicum

Pub Date : 2024-01-01 DOI: 10.1515/forum-2023-0139

Changhua Wei

We are interested in the one-dimensional nonlinear wave equations with multiple wave speeds by the energy method. By choosing different multipliers corresponding to the different wave speeds, we show that the one-dimensional nonlinear wave equations also have globally smooth solutions provided that the nonlinearities satisfy certain structural conditions when the initial data are small. Furthermore, we can prove that the global solutions will converge to the solutions of the linearized system based on the decay properties of the nonlinearities.

我们感兴趣的是用能量法研究具有多种波速的一维非线性波方程。通过选择与不同波速相对应的不同乘数，我们证明了在初始数据较小时，只要非线性满足一定的结构条件，一维非线性波方程也有全局平稳解。此外，根据非线性的衰减特性，我们可以证明全局解将收敛于线性化系统的解。

引用次数: 0

Tilings of the sphere by congruent quadrilaterals III: Edge combination a 3 b with general angles 全等四边形的球面平分线 III：带一般角的边组合 a 3 b

IF 0.8 3区数学 Q1 MATHEMATICS

Forum Mathematicum

Pub Date : 2024-01-01 DOI: 10.1515/forum-2023-0209

Yixi Liao, Pinren Qian, Erxiao Wang, Yingyun Xu

Edge-to-edge tilings of the sphere by congruent quadrilaterals are completely classified in a series of three papers. This last one classifies the case of a 3 ⁢ b

{a^{3}b}

-quadrilaterals with some irrational angle: there are a sequence of 1-parameter families of quadrilaterals admitting 2-layer earth map tilings together with their basic flip modifications under extra condition, and 5 sporadic quadrilaterals each admitting a special tiling. A summary of the full classification is presented in the end.

全等四边形对球面的边到边倾斜在一系列三篇论文中得到了完整的分类。最后一篇论文对具有某种无理角的 a 3 b {a^{3}b} - 四边形进行了分类：有一系列可进行 2 层地球映射平铺的 1 参数四边形族，以及它们在额外条件下的基本翻转修正，还有 5 个零星四边形，每个都可进行特殊平铺。最后是完整分类的摘要。

引用次数: 0

Octonionic monogenic and slice monogenic Hardy and Bergman spaces 八元单源和切片单源哈代和伯格曼空间

IF 0.8 3区数学 Q1 MATHEMATICS

Forum Mathematicum

Pub Date : 2024-01-01 DOI: 10.1515/forum-2023-0039

Fabrizio Colombo, Rolf Sören Kraußhar, Irene Sabadini

In this paper we discuss some basic properties of octonionic Bergman and Hardy spaces. In the first part we review some fundamental concepts of the general theory of octonionic Hardy and Bergman spaces together with related reproducing kernel functions in the monogenic setting. We explain how some of the fundamental problems in well-defining a reproducing kernel can be overcome in the non-associative setting by looking at the real part of an appropriately defined para-linear octonion-valued inner product. The presence of a weight factor of norm 1 in the definition of the inner product is an intrinsic new ingredient in the octonionic setting. Then we look at the slice monogenic octonionic setting using the classical complex book structure. We present explicit formulas for the slice monogenic reproducing kernels for the unit ball, the right octonionic half-space and strip domains bounded in the real direction. In the setting of the unit ball we present an explicit sequential characterization which can be obtained by applying the special Taylor series representation of the slice monogenic setting together with particular octonionic calculation rules that reflect the property of octonionic para-linearity.

在本文中，我们将讨论八离子伯格曼和哈代空间的一些基本性质。在第一部分中，我们回顾了八离子哈代和伯格曼空间一般理论的一些基本概念，以及单元环境中的相关重现核函数。我们解释了如何通过研究适当定义的准线性八离子值内积的实部，来克服在非联立环境中妥善定义重现核的一些基本问题。内积定义中存在规范为 1 的权重因子是八离子环境中一个固有的新要素。然后，我们使用经典的复书结构来研究切片单原八离子环境。我们给出了单位球、右八离子半空间和实方向有界条域的切片单原重现核的明确公式。在单位球设置中，我们提出了一个明确的序列特征，通过应用切片单原设置的特殊泰勒级数表示，以及反映八离子副线性性质的特殊八离子计算规则，可以获得该特征。

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引用次数: 0

Generalized orthogonal measures on the space of unital completely positive maps 单位完全正映射空间上的广义正交测量

IF 0.8 3区数学 Q1 MATHEMATICS

Forum Mathematicum

Pub Date : 2024-01-01 DOI: 10.1515/forum-2023-0330

Angshuman Bhattacharya, Chaitanya J. Kulkarni

A classical result by Effros connects the barycentric decomposition of a state on a C*-algebra to the disintegration theory of the GNS representation of the state with respect to an orthogonal measure on the state space of the C*-algebra. In this note, we take this approach to the space of unital completely positive maps on a C*-algebra with values in B ⁢ ( H )

{B(H)}

, connecting the barycentric decomposition of the unital completely positive map and the disintegration theory of the minimal Stinespring dilation of the same. This generalizes Effros’ work in the non-commutative setting. We do this by introducing a special class of barycentric measures which we call generalized orthogonal measures. We end this note by mentioning some examples of generalized orthogonal measures.

埃夫罗斯（Effros）的一个经典结果将 C* 代数上的状态的重心分解与状态的 GNS 表示的解体理论联系起来，而 GNS 表示是关于 C* 代数的状态空间上的正交度量的。在本注释中，我们将这一方法应用于 C* 代数上在 B ( H ) {B(H)}中取值的单元全正映射空间，将单元全正映射的重心分解与同一映射的最小施蒂尼斯普林扩张的解体理论联系起来。这概括了埃弗罗斯在非交换背景下的工作。为此，我们引入了一类特殊的重心度量，我们称之为广义正交度量。最后，我们举几个广义正交度量的例子来结束本说明。

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