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A note on the 1-D minimization problem related to solenoidal improvement of the uncertainty principle inequality 与不确定性原理不等式的螺线改进有关的一维最小化问题说明

IF 0.6 4区数学 Q3 MATHEMATICS

Archiv der Mathematik

Pub Date : 2024-08-28 DOI: 10.1007/s00013-024-02042-5

Naoki Hamamoto

This paper gives a second way to solve the one-dimensional minimization problem of the form :

$$begin{aligned} min _{fnot equiv 0}frac{displaystyle int limits _0^infty left( f''right) ^2x^{mu +1}dxint limits _0^infty left( {x}^2left( f'right) ^2 -varepsilon f^2right) {{x}}^{mu -1}d{x}}{displaystyle left( int limits _0^infty left( f'right) ^2 {{x}}^{mu }d{x}right) ^2} end{aligned}$$

for scalar-valued functions f on the half line, where (f') and (f'') are its derivatives and (varepsilon ) and (mu ) are positive parameters with (varepsilon < frac{mu ^2}{4}.) This problem plays an essential part of the calculation of the best constant in Heisenberg’s uncertainty principle inequality for solenoidal vector fields. The above problem was originally solved by using (generalized) Laguerre polynomial expansion; however, the calculation was complicated and long. In the present paper, we give a simpler method to obtain the same solution, the essential part of which was communicated in Theorem 5.1 of the preprint (Hamamoto, arXiv:2104.02351v4, 2021).

本文给出了解决形式为：$$begin{aligned}的一维最小化问题的第二种方法。min _{fnot equiv 0}（int limits _0^infty left( f''right) ^2x^{mu +1}dxint limits _0^infty left( {x}^2left( f'right) ^2 -varepsilon f^2right) {{x}}^{mu -1}d{x}}{displaystyle left( （int limits _0infty left( f'right) ^2 {{x}}^{mu }d{x}right) ^2}end{aligned}$$对于半直线上的标量值函数f，其中（f'）和（f''）是它的导数，（varepsilon）和（mu）是正参数，（varepsilon < （frac/mu ^2}{4}。这个问题是计算海森堡不确定性原理不等式中螺线管矢量场最佳常数的重要部分。上述问题最初是通过（广义）拉盖尔多项式展开来解决的，但计算复杂且耗时较长。在本文中，我们给出了一种更简单的方法来获得相同的解，其基本部分已在预印本（Hamamoto, arXiv:2104.02351v4, 2021）的定理 5.1 中给出。

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引用次数: 0

The first two k-invariants of (textrm{Top}/textrm{O}) $$textrm{Top}/textrm{O}$$的前两个k变量

IF 0.5 4区数学 Q3 MATHEMATICS

Archiv der Mathematik

Pub Date : 2024-08-27 DOI: 10.1007/s00013-024-02036-3

Alexander Kupers

We show that the first two k-invariants of (textrm{Top}/textrm{O}) vanish and give some applications.

我们证明了 (textrm{Top}/textrm{O}) 的前两个 k 变量消失，并给出了一些应用。

引用次数: 0

The Chermak–Delgado measure as a map on posets 切尔马克-德尔加多度量作为正集上的映射

IF 0.5 4区数学 Q3 MATHEMATICS

Archiv der Mathematik

Pub Date : 2024-08-22 DOI: 10.1007/s00013-024-02015-8

William Cocke, Ryan McCulloch

The Chermak–Delgado measure of a finite group is a function which assigns to each subgroup a positive integer. In this paper, we give necessary and sufficient conditions for when the Chermak–Delgado measure of a group is actually a map of posets, i.e., a monotone function from the subgroup lattice to the positive integers. We also investigate when the Chermak–Delgado measure, restricted to the centralizers, is increasing.

有限群的 Chermak-Delgado 度量是一个函数，它赋予每个子群一个正整数。在本文中，我们给出了当一个群的 Chermak-Delgado 度量实际上是一个 posets 映射（即从子群网格到正整数的单调函数）时的必要条件和充分条件。我们还研究了当 Chermak-Delgado 度量局限于中心子时，它是递增的。

引用次数: 0

Rigidity of solutions to elliptic equations with one uniform limit 具有一个均匀极限的椭圆方程解的刚性

IF 0.5 4区数学 Q3 MATHEMATICS

Archiv der Mathematik

Pub Date : 2024-08-20 DOI: 10.1007/s00013-024-02040-7

Phuong Le

Let (uge -1) be a solution to the semilinear elliptic equation (-Delta u = f(u)) in (mathbb {R}^N) such that (lim _{x_Nrightarrow -infty } u(x',x_N) = -1) uniformly in (x'in mathbb {R}^{N-1}), (lim _{trightarrow +infty } inf _{x_N>t} u(x) > -1), and u is bounded in each half-space ({x_N<lambda }), (lambda in mathbb {R}). Here (f:[-1,+infty )rightarrow mathbb {R}) is a locally Lipschitz continuous function which satisfies some mild assumptions. We show that u is strictly monotonically increasing in the (x_N)-direction. Under some further assumptions on f, we deduce that u depends only on (x_N) and it is unique up to a translation. In particular, such a solution u to the problem (Delta u = u + 1) in (mathbb {R}^N) must have the form (u(x)equiv e^{x_N+alpha }-1) for some (alpha in mathbb {R}).

让 (uge -1) 是半线性椭圆方程 (-Delta u = f(u))在 (mathbb {R}^N) 中的解，使得 (lim _{x_Nrightarrow -infty } u(x'. x_N) = -1) 均匀地在（x'在 mathbb {R}^{N-1}) 中、x_N) = -1) uniformly in （x'in mathbb {R}^{N-1}）, （（lim _{trightarrow +infty }u(x) > -1), and u is bounded in each half-space ({x_N<lambda }), (lambda in mathbb {R}).这里（f:[-1,+infty )（rightarrow mathbb {R}）是一个局部利普希兹连续函数，它满足一些温和的假设。我们证明了 u 在 (x_N) 方向上是严格单调递增的。根据对 f 的一些进一步假设，我们推导出 u 只依赖于 (x_N)，并且它在平移之前是唯一的。特别是，问题 (Delta u = u + 1) in (mathbb {R}^N) 的解 u 对于某个 (alpha in mathbb {R}) 必须具有 (u(x)equiv e^{x_N+alpha }-1) 的形式。

引用次数: 0

A remark on the norm of the parallel sum 关于平行和的规范的评论

IF 0.6 4区数学 Q3 MATHEMATICS

Archiv der Mathematik

Pub Date : 2024-08-16 DOI: 10.1007/s00013-024-02048-z

Ali Zamani

It is shown that if (a!:!b) is the parallel sum of the two positive definite elements a and b of a (C^*)-algebra, then for any (s, tin [0, 1]),

$$begin{aligned} big Vert a!:!bbig Vert le frac{1}{2}left( Vert aVert !:!Vert bVert + frac{Vert aVert :Vert bVert }{Vert aVert +Vert bVert }sqrt{left( Vert aVert -Vert bVert right) ^2 +4left| a^{1-s}b^{t}right| left| a^{s}b^{1-t}right| },right) . end{aligned}$$

This inequality, which is sharper than the inequality (big Vert a!:!bbig Vert le Vert aVert !:!Vert bVert ), generalizes an earlier related inequality.

研究表明，如果 (a!:!b) 是一个 (C^*)- 代数的两个正定元素 a 和 b 的平行和，那么对于任何 (s, tin [0, 1])，$$begin{aligned}（开始{aligned}）。bbig Vert le frac{1}{2}left( Vert aVert !:！bVert + frac{Vert aVert :Vert bVert }{Vert aVert +Vert bVert }sqrt{left( （ Vert aVert -Vert bVert right) ^2 +4left| a^{1-s}b^{t}right| left| a^{s}b^{1-t}right| },right) .end{aligned}$$这个不等式比不等式（(大小ert a!:!bbig Vert le Vert aVert !:!Vert bVert )）更尖锐，它概括了一个早期的相关不等式。

引用次数: 0

One-dimensional quasi-uniform Kronecker sequences 一维准均匀克罗内克序列

IF 0.5 4区数学 Q3 MATHEMATICS

Archiv der Mathematik

Pub Date : 2024-08-14 DOI: 10.1007/s00013-024-02039-0

Takashi Goda

In this short note, we prove that the one-dimensional Kronecker sequence (ialpha bmod 1, i=0,1,2,ldots ,) is quasi-uniform over the unit interval [0, 1] if and only if (alpha ) is a badly approximable number. Our elementary proof relies on a result on the three-gap theorem for Kronecker sequences due to Halton (Proc Camb Philos Soc, 61:665–670, 1965).

在这篇短文中，我们证明一维克朗内克序列 (ialpha bmod 1, i=0,1,2,ldots ,) 在单位区间 [0, 1] 上是准均匀的，当且仅当(alpha )是一个坏的可近似数。我们的基本证明依赖于哈尔顿（Halton）关于克朗内克序列三缺口定理的结果（Proc Camb Philos Soc, 61:665-670, 1965）。

引用次数: 0

Common substring with shifts in b-ary expansions b-ary 扩展中带有移位的共同子串

IF 0.5 4区数学 Q3 MATHEMATICS

Archiv der Mathematik

Pub Date : 2024-08-08 DOI: 10.1007/s00013-024-02038-1

Xin Liao, Dingding Yu

Denote by (S_n(x,y)) the length of the longest common substring of x and y with shifts in their first n digits of the b-ary expansions. We show that the sets of pairs (x, y), for which the growth rate of (S_n(x,y)) is (alpha log n) with (0le alpha le infty ), have full Hausdorff dimension. Our method relies upon some estimation of the spectral radius of matrices.

用 (S_n(x,y))表示 x 和 y 的最长公共子串的长度，它们的 b-ary 展开的前 n 位有移位。我们证明，对于(S_n(x,y))的增长率为(alpha log n) with (0 le alpha le infty )的成对集合（x, y），具有全豪斯多夫维。我们的方法依赖于对矩阵谱半径的一些估计。

引用次数: 0

A note on parallel mean curvature surfaces and Codazzi operators 关于平行平均曲率曲面和科达齐算子的说明

IF 0.5 4区数学 Q3 MATHEMATICS

Archiv der Mathematik

Pub Date : 2024-08-08 DOI: 10.1007/s00013-024-02043-4

Felippe Guimarães

We use an intrinsic Klotz–Osserman type result for surfaces in terms of Codazzi operators to study surfaces with parallel mean curvature and non-positive Gaussian curvature in product spaces.

我们利用科达齐算子对曲面的内在克洛茨-奥斯曼类型结果，来研究乘积空间中具有平行平均曲率和非正高斯曲率的曲面。

引用次数: 0

Rotationally symmetric gradient Yamabe solitons 旋转对称梯度山叶孤子

IF 0.5 4区数学 Q3 MATHEMATICS

Archiv der Mathematik

Pub Date : 2024-08-06 DOI: 10.1007/s00013-024-02032-7

Antonio W. Cunha, Rong Mi

This short note deals with compact and complete and non-compact gradient Yamabe solitons (M, g, f) such that it has metric of constant scalar curvature. Firstly, we give a new proof of triviality for gradient compact Yamabe solitons. Also, under some integral conditions, we are able to improve a result due to Ma and Miquel (Ann Global Anal Geom 42:195–205, 2012). Finally, we obtain that the Yamabe metric becomes rotationally symmetric. Results for k-Yamabe solitons are also obtained here.

这篇短文论述了具有恒定标量曲率度量的紧凑、完整和非紧凑梯度山边孤子（M, g, f）。首先，我们给出了梯度紧凑山边孤子的新的三性证明。此外，在一些积分条件下，我们还能改进 Ma 和 Miquel 的一个结果（Ann Global Anal Geom 42:195-205, 2012）。最后，我们得到山边公设变得旋转对称了。这里还得到了 k-Yamabe 孤子的结果。

引用次数: 0

An improvement of the sharp Li–Yau bound on closed manifolds 封闭流形上尖锐李-尤约束的改进

IF 0.5 4区数学 Q3 MATHEMATICS

Archiv der Mathematik

Pub Date : 2024-08-06 DOI: 10.1007/s00013-024-02027-4

Jia-Yong Wu

In this paper, we give a generalization of Zhang’s recent work about a sharp Li–Yau gradient bound on compact manifolds by extending Hamilton’s gradient estimates. In particular, we take a special auxiliary function to indicate that our estimate is a slight improvement of Zhang’s result.

在本文中，我们通过扩展汉密尔顿的梯度估计，给出了张最近关于紧凑流形上尖锐李-尤梯度约束的工作的概括。特别是，我们用一个特殊的辅助函数来表示我们的估计是对张的结果的轻微改进。

引用次数: 0

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