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Volume 32 Issue 4

Issue of Forum Mathematicum

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Journal Overview

Publicly Available July 1, 2020

Frontmatter

Page range: i-iv

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March 10, 2020

Decomposition rank of approximately subhomogeneous C^*-algebras

George A. Elliott, Zhuang Niu, Luis Santiago, Aaron Tikuisis

Page range: 827-889

Abstract

It is shown that every Jiang–Su stable approximately subhomogeneous C*{{\mathrm{C}^{*}}}-algebra has finite decomposition rank. This settles a key direction of the Toms–Winter conjecture for simple approximately subhomogeneous C*{{\mathrm{C}^{*}}}-algebras. A key step in the proof is that subhomogeneous C*{{\mathrm{C}^{*}}}-algebras are locally approximated by a certain class of more tractable subhomogeneous algebras, namely a non-commutative generalization of the class of cell complexes. The result is applied, in combination with other recent results, to show classifiability of crossed product C*{{\mathrm{C}^{*}}}-algebras associated to minimal homeomorphisms with mean dimension zero.

February 22, 2020

Classification of indecomposable involutive set-theoretic solutions to the Yang–Baxter equation

Wolfgang Rump

Page range: 891-903

Abstract

Using the theory of cycle sets and braces, non-degenerate indecomposable involutive set-theoretic solutions to the Yang–Baxter equation are classified in terms of their universal coverings and their fundamental group. The universal coverings are characterized as braces with an adjoint orbit generating the additive group. Using this description, all coverings of non-degenerate indecomposable cycle sets are classified. The method is illustrated by examples.

March 20, 2020

Degree bounds for modular covariants

Jonathan Elmer, Müfit Sezer

Page range: 905-910

Abstract

Let V,W{V,W} be representations of a cyclic group G of prime order p over a field 𝕜{\Bbbk} of characteristic p . The module of covariants 𝕜⁢[V,W]G{\Bbbk[V,W]^{G}} is the set of G -equivariant polynomial maps V→W{V\rightarrow W}, and is a module over 𝕜⁢[V]G{\Bbbk[V]^{G}}. We give a formula for the Noether bound β⁢(𝕜⁢[V,W]G,𝕜⁢[V]G){\beta(\Bbbk[V,W]^{G},\Bbbk[V]^{G})}, i.e. the minimal degree d such that 𝕜⁢[V,W]G{\Bbbk[V,W]^{G}} is generated over 𝕜⁢[V]G{\Bbbk[V]^{G}} by elements of degree at most d .

March 20, 2020

The Kobayashi–Royden metric on punctured spheres

Gunhee Cho, Junqing Qian

Page range: 911-918

Abstract

This paper gives an explicit formula of the asymptotic expansion of the Kobayashi–Royden metric on the punctured sphere ℂ⁢ℙ1∖{0,1,∞}{\mathbb{CP}^{1}\setminus\{0,1,\infty\}} in terms of the exponential Bell polynomials. We prove a local quantitative version of the Little Picard’s Theorem as an application of the asymptotic expansion. Furthermore, the approach in the paper leads to the interesting consequence that the coefficients in the asymptotic expansion are rational numbers. Furthermore, the explicit formula of the metric and the conclusion regarding the coefficients apply to more general cases of ℂ⁢ℙ1∖{a1,…,an}{\mathbb{CP}^{1}\setminus\{a_{1},\ldots,a_{n}\}}, n≥3{n\geq 3}, as well, and the metric on ℂ⁢ℙ1∖{0,13,-16±36⁢i}{\mathbb{CP}^{1}\setminus\{0,\frac{1}{3},-\frac{1}{6}\pm\frac{\sqrt{3}}{6}i\}} will be given as a concrete example of our results.

April 15, 2020

Boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces

Jiao Chen, Wei Ding, Guozhen Lu

Page range: 919-936

Abstract

After the celebrated work of L. Hörmander on the one-parameter pseudo-differential operators, the applications of pseudo-differential operators have played an important role in partial differential equations, geometric analysis, harmonic analysis, theory of several complex variables and other branches of modern analysis. For instance, they are used to construct parametrices and establish the regularity of solutions to PDEs such as the ∂¯{\overline{\partial}} problem. The study of Fourier multipliers, pseudo-differential operators and Fourier integral operators has stimulated further such applications. It is well known that the one-parameter pseudo-differential operators are Lp⁢(ℝn){L^{p}({\mathbb{R}^{n}})} bounded for 1<p<∞{1<p<\infty}, but only bounded on local Hardy spaces hp⁢(ℝn){h^{p}({\mathbb{R}^{n}})} introduced by Goldberg in [D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 1979, 1, 27–42] for 0<p≤1{0<p\leq 1}. Though much work has been done on the Lp⁢(ℝn1×ℝn2){L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for 1<p<∞{1<p<\infty} and Hardy Hp⁢(ℝn1×ℝn2){H^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for 0<p≤1{0<p\leq 1} for multi-parameter Fourier multipliers and singular integral operators, not much has been done yet for the boundedness of multi-parameter pseudo-differential operators in the range of 0<p≤1{0<p\leq 1}. The main purpose of this paper is to establish the boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces hp⁢(ℝn1×ℝn2){h^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} for 0<p≤1{0<p\leq 1} recently introduced by Ding, Lu and Zhu in [W. Ding, G. Lu and Y. Zhu, Multi-parameter local Hardy spaces, Nonlinear Anal. 184 2019, 352–380].

March 10, 2020

Artin L-functions of almost monomial Galois groups

Mircea Cimpoeaş, Florin Nicolae

Page range: 937-940

Abstract

If K/ℚ{K/\mathbb{Q}} is a finite Galois extension with an almost monomial Galois group and if s0∈ℂ∖{1}{s_{0}\in\mathbb{C}\setminus\{1\}} is not a common zero for any two Artin L -functions associated to distinct complex irreducible characters of the Galois group, then all Artin L -functions of K/ℚ{K/\mathbb{Q}} are holomorphic at s0{s_{0}}. We present examples and basic properties of almost monomial groups.

March 19, 2020

Unitary representations with Dirac cohomology: A finiteness result for complex Lie groups

Jian Ding, Chao-Ping Dong

Page range: 941-964

Abstract

Let G be a connected complex simple Lie group, and let G^d{\widehat{G}^{\mathrm{d}}} be the set of all equivalence classes of irreducible unitary representations with non-vanishing Dirac cohomology. We show that G^d{\widehat{G}^{\mathrm{d}}} consists of two parts: finitely many scattered representations, and finitely many strings of representations. Moreover, the strings of G^d{\widehat{G}^{\mathrm{d}}} come from L^d{\widehat{L}^{\mathrm{d}}} via cohomological induction and they are all in the good range. Here L runs over the Levi factors of proper θ-stable parabolic subgroups of G . It follows that figuring out G^d{\widehat{G}^{\mathrm{d}}} requires a finite calculation in total. As an application, we report a complete description of F^4d{\widehat{F}_{4}^{\mathrm{d}}}.

March 19, 2020

Weighted sum formulas of multiple t-values with even arguments

Zhonghua Li, Ce Xu

Page range: 965-976

Abstract

In this paper, we study the weighted sums of multiple t -values and of multiple t -star values at even arguments. Some general weighted sum formulas are given, where the weight coefficients are given by (symmetric) polynomials of the arguments.

April 15, 2020

Long time behavior of solutions to 3D generalized MHD equations

Xiaopeng Zhao

Page range: 977-993

Abstract

In this paper, we consider the long time behavior of solutions for 3D incompressible MHD equations with fractional Laplacian. Firstly, in a periodic bounded domain, we prove the existence of a global attractor. The analysis reveals a relation between the Laplacian exponent and the regularity of the spaces of velocity and magnetic fields. Finally, in the whole space ℝ3{\mathbb{R}^{3}}, we establish the sharp algebraic decay rate of solutions to the generalized MHD system provided that the parameters satisfy α,β∈(0,2]{\alpha,\beta\in(0,2]}.

April 15, 2020

Bilinear forms on non-homogeneous Sobolev spaces

Carme Cascante, Joaquín M. Ortega

Page range: 995-1026

Abstract

In this paper, we show that if b∈L2⁢(ℝn){b\in L^{2}(\mathbb{R}^{n})}, then the bilinear form defined on the product of the non-homogeneous Sobolev spaces Hs2⁢(ℝn)×Hs2⁢(ℝn){H_{s}^{2}(\mathbb{R}^{n})\times H_{s}^{2}(\mathbb{R}^{n})}, 0<s<1{0<s<1}, by (f,g)∈Hs2⁢(ℝn)×Hs2⁢(ℝn)→∫ℝn(Id-Δ)s2⁢(f⁢g)⁢(𝐱)⁢b⁢(𝐱)⁢d⁢𝐱(f,g)\in H_{s}^{2}(\mathbb{R}^{n})\times H_{s}^{2}(\mathbb{R}^{n})\to\int_{% \mathbb{R}^{n}}(\mathrm{Id}-\Delta)^{\frac{s}{2}}(fg)(\mathbf{x})b(\mathbf{x})% \mathop{}\!d\mathbf{x} is continuous if and only if the positive measure |b⁢(𝐱)|2⁢d⁢𝐱{\lvert b(\mathbf{x})\rvert^{2}\mathop{}\!d\mathbf{x}} is a trace measure for Hs2⁢(ℝn){H_{s}^{2}(\mathbb{R}^{n})}.

April 22, 2020

Weyl and Zariski chambers on projective surfaces

Krishna Hanumanthu, Nabanita Ray

Page range: 1027-1037

Abstract

Let X be a nonsingular complex projective surface. The Weyl and Zariski chambers give two interesting decompositions of the big cone of X . Following the ideas of [T. Bauer and M. Funke, Weyl and Zariski chambers on K3 surfaces, Forum Math. 24 2012, 3, 609–625] and [S. A. Rams and T. Szemberg, When are Zariski chambers numerically determined?, Forum Math. 28 2016, 6, 1159–1166], we study these two decompositions and determine when a Weyl chamber is contained in the interior of a Zariski chamber and vice versa. We also determine when a Weyl chamber can intersect non-trivially with a Zariski chamber.

May 9, 2020

Rankin–Selberg L-functions via good sections

Yeongseong Jo

Page range: 1039-1074

Abstract

In this article, we revisit Rankin–Selberg integrals established by Jacquet, Piatetski-Shapiro and Shalika. We prove the equality of Rankin–Selberg local factors defined with Schwartz–Bruhat functions and the factors attached to good sections, introduced by Piatetski-Shapiro and Rallis. Moreover, we propose a notion of exceptional poles in the framework of good sections. For cases of Rankin–Selberg, Asai and exterior square L -functions, the exceptional poles are consistent with well-known exceptional poles which characterize certain distinguished representations.

April 22, 2020

Archimedean domains of skew generalized power series

Ryszard Mazurek

Page range: 1075-1093

Abstract

A skew generalized power series ring R⁢[[S,ω,≤]]{R[[S,\omega,\leq]]} consists of all functions from a strictly ordered monoid (S,≤){(S,\leq)} to a ring R whose support is artinian and narrow, with pointwise addition, and with multiplication given by convolution twisted by an action ω of the monoid S on the ring R . Special cases of this ring construction are skew polynomial rings, skew Laurent polynomial rings, skew power series rings, skew Laurent series rings, skew monoid rings, skew group rings, skew Mal’cev–Neumann series rings, the “unskewed” versions of all of these, and generalized power series rings. In this paper, we characterize the skew generalized power series rings R⁢[[S,ω,≤]]{R[[S,\omega,\leq]]} that are left (right) Archimedean domains in the case where the order ≤{\leq} is total, or ≤{\leq} is semisubtotal and the monoid S is commutative torsion-free cancellative, or ≤{\leq} is trivial and S is totally orderable. We also answer four open questions posed by Moussavi, Padashnik and Paykan regarding the rings in the title.

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Forum Mathematicum (FORUM) is a general mathematics journal, which is devoted to the publication of research articles in all fields of pure and applied mathematics, including mathematical physics. Forum Mathematicum belongs to the top 50 journals in pure and applied mathematics, as measured by citation impact.

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Volume 32 Issue 4

Issue of Forum Mathematicum

Frontmatter

Decomposition rank of approximately subhomogeneous C*-algebras

Abstract

Classification of indecomposable involutive set-theoretic solutions to the Yang–Baxter equation

Abstract

Degree bounds for modular covariants

Abstract

The Kobayashi–Royden metric on punctured spheres

Abstract

Boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces

Abstract

Artin L-functions of almost monomial Galois groups

Abstract

Unitary representations with Dirac cohomology: A finiteness result for complex Lie groups

Abstract

Weighted sum formulas of multiple t-values with even arguments

Abstract

Long time behavior of solutions to 3D generalized MHD equations

Abstract

Bilinear forms on non-homogeneous Sobolev spaces

Abstract

Weyl and Zariski chambers on projective surfaces

Abstract

Rankin–Selberg L-functions via good sections

Abstract

Archimedean domains of skew generalized power series

Abstract

Decomposition rank of approximately subhomogeneous C^*-algebras