St Petersburg Mathematical Journal最新文献

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Solvability of a critical semilinear problem with the spectral Neumann fractional Laplacian 一个具有谱Neumann分数拉普拉斯算子的临界双线性问题的可解性

IF 0.8 4区数学 Q2 MATHEMATICS

St Petersburg Mathematical Journal

Pub Date : 2021-12-28 DOI: 10.1090/spmj/1693

N. Ustinov

Sufficient conditions are provided for the existence of a ground state solution for the problem generated by the fractional Sobolev inequality in <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega element-of upper C squared colon"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>:</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Omega in C^2:</mml:annotation> </mml:semantics></mml:math></inline-formula> <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis negative normal upper Delta right-parenthesis Subscript upper S p Superscript s Baseline u left-parenthesis x right-parenthesis plus u left-parenthesis x right-parenthesis equals u Superscript 2 Super Subscript s Super Superscript asterisk Superscript minus 1 Baseline left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mo>−</mml:mo> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:msubsup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>S</mml:mi> <mml:mi>p</mml:mi> </mml:mrow> <mml:mi>s</mml:mi> </mml:msubsup> <mml:mi>u</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>u</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msubsup> <mml:mn>2</mml:mn> <mml:mi>s</mml:mi> <mml:mo>∗</mml:mo> </mml:msubsup> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(-Delta )_{Sp}^s u(x) + u(x) = u^{2^*_s-1}(x)</mml:annotation> </mml:semantics></mml:math></inline-formula>. Here <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis negative normal upper Delta right-parenthesis Subscript upper S p Superscript s"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mo>−</mml:mo> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:msubsup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>S</mml:mi>

给出了Ω∈C2:Omega中的分数阶Sobolev不等式在C^2中产生的问题基态解存在的充分条件：（−Δ）S p S u（x）+u（x=u 2 s*−1（x）（-Δ）_｛Sp｝^s u（x）+u（x）=u ^｛2^*_s-1｝（x）。这里（−Δ）S p S（-Delta）_｛Sp｝^S代表传统Neumann拉普拉斯算子在Ω中的S次幂，s∈（0，1）s在（0，l）中，2s*=2 n/（n−2s）2^*_s=2n/（n-2s）。对于s=1 s=1的局部情况，Neumann-拉普拉斯算子和Neumann-p-拉普拉斯算子的相应结果早些时候得到。

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引用次数: 3

Symmetries of double ratios and an equation for Möbius structures Möbius结构的二重比对称性和一个方程

IF 0.8 4区数学 Q2 MATHEMATICS

St Petersburg Mathematical Journal

Pub Date : 2021-12-28 DOI: 10.1090/spmj/1688

S. Buyalo

Orthogonal representations <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="eta Subscript n Baseline colon upper S Subscript n Baseline clockwise top semicircle arrow double-struck upper R Superscript upper N"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>η</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>↷</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">eta _ncolon S_ncurvearrowright mathbb {R}^N</mml:annotation> </mml:semantics></mml:math></inline-formula> of the symmetric groups <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript n"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">S_n</mml:annotation> </mml:semantics></mml:math></inline-formula>, <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n greater-than-or-equal-to 4"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">nge 4</mml:annotation> </mml:semantics></mml:math></inline-formula>, with <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N equals n factorial slash 8"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mi>n</mml:mi> <mml:mo>!</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>8</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">N=n!/8</mml:annotation> </mml:semantics></mml:math></inline-formula>, emerging from symmetries of double ratios are treated. For <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n equals 5"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n=5</mml:annotation> </mml:semantics></mml:math></inline-formula>, the representation <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="eta 5"> <mml:semantics> <mml:msub> <mml:mi>η</mml:mi> <mml:mn>5</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">eta _5</mml:annotation> </mml:semantics></mml:

正交表示ηn:Sn↷ 对称群S N S_N，N≥4nge4，其中N=N！/8 N=N/8，从双重比率的对称性中出现。对于n=5n=5，表示η5eta_5被分解为不可约分量，并表明某个分量产生了描述亚Möbius结构类中Möbius结构的方程的解。在这个意义上，决定Möbius结构的条件已经隐含在二重比的对称性中。

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引用次数: 0

Schrödinger operator with decreasing potential in a cylinder 圆柱中势递减的Schrödinger算子

IF 0.8 4区数学 Q2 MATHEMATICS

St Petersburg Mathematical Journal

Pub Date : 2021-12-28 DOI: 10.1090/spmj/1694

N. Filonov

The Schrödinger operator <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="negative normal upper Delta plus upper V left-parenthesis x comma y right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo>−</mml:mo> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mo>+</mml:mo> <mml:mi>V</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">-Delta + V(x,y)</mml:annotation> </mml:semantics></mml:math></inline-formula> is considered in a cylinder <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript m Baseline times upper U"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>m</mml:mi> </mml:msup> <mml:mo>×</mml:mo> <mml:mi>U</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">mathbb {R}^m times U</mml:annotation> </mml:semantics></mml:math></inline-formula>, where <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U"> <mml:semantics> <mml:mi>U</mml:mi> <mml:annotation encoding="application/x-tex">U</mml:annotation> </mml:semantics></mml:math></inline-formula> is a bounded domain in <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript d"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">mathbb {R}^d</mml:annotation> </mml:semantics></mml:math></inline-formula>. The spectrum of such an operator is studied under the assumption that the potential decreases in longitudinal variables, <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper V left-parenthesis x comma y right-parenthesis EndAbsoluteValue less-than-or-equal-to upper C mathematical left-angle x mathematical right-angle Superscript negative rho"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>V</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo>≤</mml:mo> <mm

考虑Schrödinger算子−Δ + V(x,y) - Delta + V(x,y)在圆柱体R m × U mathbb R{^m }times U中，其中U U是R d mathbb R{^d中的有界域。这种算子的谱是在纵向变量的势减小的假设下研究的，|V(x,y)|≤C⟨x⟩- ρ |V(x,y)| }le C langle x rangle ^{-rho。若ρ > 1}rho > 1，则波算符存在且完备;伯曼不变性原理和极限吸收原理成立;绝对连续光谱填充半轴;奇异连续谱是空的;特征值只能累加到阈值。

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引用次数: 2

On the sharpness of assumptions in the Federer theorem 论费德勒定理中假设的尖锐性

IF 0.8 4区数学 Q2 MATHEMATICS

St Petersburg Mathematical Journal

Pub Date : 2021-12-28 DOI: 10.1090/spmj/1691

B. Makarov, A. Podkorytov

The Federer theorem deals with the “massiveness” of the set of critical values for a <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t"> <mml:semantics> <mml:mi>t</mml:mi> <mml:annotation encoding="application/x-tex">t</mml:annotation> </mml:semantics></mml:math></inline-formula>-smooth map acting from <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript m"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>m</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">mathbb R^m</mml:annotation> </mml:semantics></mml:math></inline-formula> to <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript n"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">mathbb R^n</mml:annotation> </mml:semantics></mml:math></inline-formula>: it claims that the Hausdorff <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics></mml:math></inline-formula>-measure of this set is zero for certain <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics></mml:math></inline-formula>. If <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n greater-than-or-equal-to m"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mi>m</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">nge m</mml:annotation> </mml:semantics></mml:math></inline-formula>, it has long been known that the assumption of that theorem relating the parameters <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m comma n comma t comma p"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">m,n,t,p</mml:annotation> </mml:semantics></mml:math></inline-formula> is sharp. Here it is shown by an example that this assumption is also sharp for <inline-formula content-typ

费德勒定理处理的是从R m mathbb R^m到R n mathbb R^n的光滑映射的临界值集合的“海量性”:它声称该集合的Hausdorff p p -测度在特定的p p下为零。如果n≥m nge m，我们早就知道关于参数m,n,t,p m,n,t,p的定理的假设是尖锐的。这里的一个例子表明，这个假设对于n b>00 m和n bb11m也是尖锐的。

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引用次数: 0

Clebsh–Gordan coefficients for the algebra 𝔤𝔩₃ and hypergeometric functions 代数<s:1>𝔩₃和超几何函数的Clebsh-Gordan系数

IF 0.8 4区数学 Q2 MATHEMATICS

St Petersburg Mathematical Journal

Pub Date : 2021-12-28 DOI: 10.1090/spmj/1686

D. Artamonov

The Clebsh–Gordan coefficients for the Lie algebra g l 3 mathfrak {gl}_3 in the Gelfand–Tsetlin base are calculated. In contrast to previous papers, the result is given as an explicit formula. To obtain the result, a realization of a representation in the space of functions on the group G L 3 GL_3 is used. The keystone fact that allows one to carry the calculation of Clebsh–Gordan coefficients is the theorem that says that functions corresponding to the Gelfand–Tsetlin base vectors can be expressed in terms of generalized hypergeometric functions.

李代数gl3mathfrak的Clebsh–Gordan系数{gl}_3计算了Gelfand–Tsetlin基中的。与以前的论文相比，结果是作为一个显式公式给出的。为了得到这一结果，使用了在群GL_3上函数空间中的一个表示的实现。允许计算Clebsh–Gordan系数的关键事实是定理，该定理表明，与Gelfand–Tsetlin基向量对应的函数可以用广义超几何函数表示。

引用次数: 6

A note on the centralizer of a subalgebra of the Steinberg algebra 关于Steinberg代数的一个子代数的扶正器的注记

IF 0.8 4区数学 Q2 MATHEMATICS

St Petersburg Mathematical Journal

Pub Date : 2021-12-28 DOI: 10.1090/spmj/1695

R. Hazrat, Huanhuan Li

For an ample Hausdorff groupoid <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">mathcal {G}</mml:annotation> </mml:semantics></mml:math></inline-formula>, and the Steinberg algebra <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript upper R Baseline left-parenthesis script upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>R</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">G</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">A_R(mathcal {G})</mml:annotation> </mml:semantics></mml:math></inline-formula> with coefficients in the commutative ring <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics></mml:math></inline-formula> with unit, the centralizer is described for the subalgebra <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript upper R Baseline left-parenthesis upper U right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>R</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>U</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">A_R(U)</mml:annotation> </mml:semantics></mml:math></inline-formula> with <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U"> <mml:semantics> <mml:mi>U</mml:mi> <mml:annotation encoding="application/x-tex">U</mml:annotation> </mml:semantics></mml:math></inline-formula> an open closed invariant subset of the unit space of <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">mathcal {G}</mml:annotation> </mml:semantics></mml:math></inline-formula>. In particular, it is shown that the algebra of the interior of the isotropy is indeed the centralizer of the diagonal subalgebra of the Steinberg algebra. This will unify seve

对于一个例子的Hausdorff群G mathcal {G}，以及在交换环R R中具有系数的Steinberg代数A R(G) A_R(mathcal {G})，描述了子代数A R(U) A_R(U)具有U U是G mathcal {G}的单位空间的开闭不变子集的正化器。特别地，证明了各向同性内部的代数确实是Steinberg代数对角线子代数的中心化器。这将统一文献中的几个结果，并给出莱维特路径代数的相应结果。

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引用次数: 2

Summation method in an optimal control problem with delay 一类时滞最优控制问题的求和方法

IF 0.8 4区数学 Q2 MATHEMATICS

St Petersburg Mathematical Journal

Pub Date : 2021-12-26 DOI: 10.1090/spmj/1759

P. Barkhayev, Yu.Lyubarskii

A summation procedure is described for the construction of the optimal solution in the null controllability problem for a differential equation with distributed delay.

描述了一个求和过程，用于构造具有分布时滞的微分方程零可控性问题的最优解。

引用次数: 0

Weighted Littlewood–Paley inequality for arbitrary rectangles in ℝ² 对任意矩形的加权Littlewood-Paley不等式

IF 0.8 4区数学 Q2 MATHEMATICS

St Petersburg Mathematical Journal

Pub Date : 2021-12-01 DOI: 10.1090/SPMJ/1680

Viacheslav Borovitskiy

引用次数: 2

Structure of the maximal ideal space of ^{∞} on the countable disjoint union of open disks 开放盘可数不相交并上^{∞}最大理想空间的结构

IF 0.8 4区数学 Q2 MATHEMATICS

St Petersburg Mathematical Journal

Pub Date : 2021-12-01 DOI: 10.1090/SPMJ/1681

A. Brudnyi

引用次数: 4

Self-similarity and spectral theory: on the spectrum of substitutions 自相似与光谱理论:关于取代的光谱

IF 0.8 4区数学 Q2 MATHEMATICS

St Petersburg Mathematical Journal

Pub Date : 2021-11-01 DOI: 10.1090/spmj/1756

A. Bufetov, B. Solomyak

This survey of the spectral properties of substitution dynamical systems is devoted to primitive aperiodic substitutions and associated dynamical systems: Z mathbb {Z} -actions and R mathbb {R} -actions, the latter viewed as tiling flows. The focus is on the continuous part of the spectrum. For Z mathbb {Z} -actions the maximal spectral type can be represented in terms of matrix Riesz products, whereas for tiling flows, the local dimension of the spectral measure is governed by the spectral cocycle. References are given to complete proofs and emphasize ideas and various links.

本文对置换动力系统的谱性质进行了综述，主要研究了原始非周期置换和相关动力系统：Zmathbb｛Z｝-作用和Rmathbb{R｝-动作，后者被视为平铺流。重点是频谱的连续部分。对于Zmathbb｛Z｝-作用，最大谱类型可以用矩阵Riesz乘积表示，而对于平铺流，谱测度的局部维数由谱共循环控制。参考文献提供了完整的证明，并强调了思想和各个环节。

引用次数: 2

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