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Volume 33 Issue 6

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Journal Overview

Publicly Available November 1, 2021

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September 25, 2021

On the topological complexity of manifolds with abelian fundamental group

Daniel C. Cohen, Lucile Vandembroucq

Page range: 1403-1421

Abstract

We find conditions which ensure that the topological complexity of a closed manifold M with abelian fundamental group is nonmaximal, and see through examples that our conditions are sharp. This generalizes results of Costa and Farber on the topological complexity of spaces with small fundamental group. Relaxing the commutativity condition on the fundamental group, we also generalize results of Dranishnikov on the Lusternik–Schnirelmann category of the cofibre of the diagonal map Δ:M→M×M{\Delta:M\to M\times M} for nonorientable surfaces by establishing the nonmaximality of this invariant for a large class of manifolds.

September 25, 2021

Reconstructing étale groupoids from semigroups

Tristan Bice, Lisa Orloff Clark

Page range: 1423-1444

Abstract

We unify various étale groupoid reconstruction theorems such as the following: • Kumjian and Renault’s reconstruction from a groupoid C*-algebra; • Exel’s reconstruction from an ample inverse semigroup; • Steinberg’s reconstruction from a groupoid ring; • Choi, Gardella and Thiel’s reconstruction from a groupoid Lp{L^{p}}-algebra. We do this by working with certain bumpy semigroups S of functions defined on an étale groupoid G . The semigroup structure of S together with the diagonal subsemigroup D then yields a natural domination relation ≺{\prec} on S . The groupoid of ≺{\prec}-ultrafilters is then isomorphic to the original groupoid G .

Open Access September 25, 2021

Fiberwise linear differential operators

Fabrizio Pugliese, Giovanni Sparano, Luca Vitagliano

Page range: 1445-1469

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Abstract

We define a new notion of fiberwise linear differential operator on the total space of a vector bundle E . Our main result is that fiberwise linear differential operators on E are equivalent to (polynomial) derivations of an appropriate line bundle over E∗{E^{\ast}}. We believe this might represent a first step towards a definition of multiplicative (resp. infinitesimally multiplicative) differential operators on a Lie groupoid (resp. a Lie algebroid). We also discuss the linearization of a differential operator around a submanifold.

October 10, 2021

Foliations with isolated singularities on Hirzebruch surfaces

Carlos Galindo, Francisco Monserrat, Jorge Olivares

Page range: 1471-1486

Abstract

We study foliations ℱ{\mathcal{F}} on Hirzebruch surfaces Sδ{S_{\delta}} and prove that, similarly to those on the projective plane, any ℱ{\mathcal{F}} can be represented by a bi-homogeneous polynomial affine 1-form. In case ℱ{\mathcal{F}} has isolated singularities, we show that, for δ=1{\delta=1}, the singular scheme of ℱ{\mathcal{F}} does determine the foliation, with some exceptions that we describe, as is the case of foliations in the projective plane. For δ≠1{\delta\neq 1}, we prove that the singular scheme of ℱ{\mathcal{F}} does not determine the foliation. However, we prove that, in most cases, two foliations ℱ{\mathcal{F}} and ℱ′{\mathcal{F}^{\prime}} given by sections s and s′{s^{\prime}} have the same singular scheme if and only if s′=Φ⁢(s){s^{\prime}=\Phi(s)}, for some global endomorphism Φ of the tangent bundle of Sδ{S_{\delta}}.

September 25, 2021

On a stronger reconstruction notion for monoids and clones

Mike Behrisch, Edith Vargas-García

Page range: 1487-1506

Abstract

Motivated by reconstruction results by Rubin, we introduce a new reconstruction notion for permutation groups, transformation monoids and clones, called automatic action compatibility, which entails automatic homeomorphicity. We further give a characterization of automatic homeomorphicity for transformation monoids on arbitrary carriers with a dense group of invertibles having automatic homeomorphicity. We then show how to lift automatic action compatibility from groups to monoids and from monoids to clones under fairly weak assumptions. We finally employ these theorems to get automatic action compatibility results for monoids and clones over several well-known countable structures, including the strictly ordered rationals, the directed and undirected version of the random graph, the random tournament and bipartite graph, the generic strictly ordered set, and the directed and undirected versions of the universal homogeneous Henson graphs.

October 10, 2021

Cochain level May–Steenrod operations

Ralph M. Kaufmann, Anibal M. Medina-Mardones

Page range: 1507-1526

Abstract

Steenrod defined in 1947 the Steenrod squares on the mod 2 cohomology of spaces using explicit cochain formulae for the cup- i products; a family of coherent homotopies derived from the broken symmetry of Alexander–Whitney’s chain approximation to the diagonal. He later defined his homonymous operations for all primes using the homology of symmetric groups. This approach enhanced the conceptual understanding of the operations and allowed for many advances, but lacked the concreteness of their definition at the even prime. In recent years, thanks to the development of new applications of cohomology, having definitions of Steenrod operations that can be effectively computed in specific examples has become a key issue. Using the operadic viewpoint of May, this article provides such definitions at all primes introducing multioperations that generalize the Steenrod cup- i products on the simplicial and cubical cochains of spaces.

September 25, 2021

Commutative L-algebras and measure theory

Wolfgang Rump

Page range: 1527-1548

Abstract

Measure and integration theory for finitely additive measures, including vector-valued measures, is shown to be essentially covered by a class of commutative L -algebras, called measurable algebras . The domain and range of any measure is a commutative L -algebra. Each measurable algebra embeds into its structure group , an abelian group with a compatible lattice order, and each (general) measure extends uniquely to a monotone group homomorphism between the structure groups. On the other hand, any measurable algebra X is shown to be the range of an essentially unique measure on a measurable space, which plays the role of a universal covering. Accordingly, we exhibit a fundamental group of X , with stably closed subgroups corresponding to a special class of measures with X as target. All structure groups of measurable algebras arising in a classical context are archimedean. Therefore, they admit a natural embedding into a group of extended real-valued continuous functions on an extremally disconnected compact space, the Stone space of the measurable algebra. Extending Loomis’ integration theory for finitely additive measures, it is proved that, modulo null functions, each integrable function can be represented by a unique continuous function on the Stone space.

September 25, 2021

Rankin–Selberg integrals for principal series representations of GL(n)

Dongwen Liu, Feng Su, Binyong Sun

Page range: 1549-1559

Abstract

We prove that the local Rankin–Selberg integrals for principal series representations of the general linear groups agree with certain simple integrals over the Rankin–Selberg subgroups, up to certain constants given by the local gamma factors.

September 25, 2021

A simple proof of the generalized Leibniz rule on bounded Euclidean domains

Quoc-Hung Nguyen, Yannick Sire, Juan-Luis Vázquez

Page range: 1561-1572

Abstract

This paper is devoted to a simple proof of the generalized Leibniz rule in bounded domains. The operators under consideration are the so-called spectral Laplacian and the restricted Laplacian. Equations involving such operators have lately been considered by Constantin and Ignatova in the framework of the SQG equation [P. Constantin and M. Ignatova, Critical SQG in bounded domains, Ann. PDE 2 2016, 2, Article ID 8] in bounded domains, and by two of the authors [Q.-H. Nguyen and J. L. Vázquez, Porous medium equation with nonlocal pressure in a bounded domain, Comm. Partial Differential Equations 43 2018, 10, 1502–1539] in the framework of the porous medium with nonlocal pressure in bounded domains. We will use the estimates in this work in a forthcoming paper on the study of porous medium equations with pressure given by Riesz-type potentials.

October 17, 2021

Construction of a class of maximal commutative subalgebras of prime Leavitt path algebras

Grzegorz Bajor, Leon van Wyk, Michał Ziembowski

Page range: 1573-1590

Abstract

Considering prime Leavitt path algebras LK⁢(E){L_{K}(E)}, with E being an arbitrary graph with at least two vertices, and K being any field, we construct a class of maximal commutative subalgebras of LK⁢(E){L_{K}(E)} such that, for every algebra A from this class, A has zero intersection with the commutative core ℳK⁢(E){\mathcal{M}_{K}(E)} of LK⁢(E){L_{K}(E)} defined and studied in [C. Gil Canto and A. Nasr-Isfahani, The commutative core of a Leavitt path algebra, J. Algebra 511 2018, 227–248]. We also give a new proof of the maximality, as a commutative subalgebra, of the commutative core ℳR⁢(E){\mathcal{M}_{R}(E)} of an arbitrary Leavitt path algebra LR⁢(E){L_{R}(E)}, where E is an arbitrary graph and R is a commutative unital ring.

October 10, 2021

Seshadri constants on some Quot schemes

Chandranandan Gangopadhyay, Krishna Hanumanthu, Ronnie Sebastian

Page range: 1591-1603

Abstract

Let E be a vector bundle of rank n on ℙ1{\mathbb{P}^{1}}. Fix a positive integer d . Let 𝒬⁢(E,d){\mathcal{Q}(E,d)} denote the Quot scheme of torsion quotients of E of degree d and let Gr⁢(E,d){\mathrm{Gr}(E,d)} denote the Grassmann bundle that parametrizes the d -dimensional quotients of the fibers of E . We compute Seshadri constants of ample line bundles on 𝒬⁢(E,d){\mathcal{Q}(E,d)} and Gr⁢(E,d){\mathrm{Gr}(E,d)}.

October 21, 2021

Hörmander Fourier multiplier theorems with optimal Besov regularity on multi-parameter Hardy spaces

Jiao Chen, Liang Huang, Guozhen Lu

Page range: 1605-1627

Abstract

We will establish the boundedness of the Fourier multiplier operator Tm⁢fT_{m}f on multi-parameter Hardy spaces Hp⁢(Rn1×⋯×Rnr)H^{p}(\mathbb{R}^{n_{1}}\times\cdots\times\mathbb{R}^{n_{r}}) (0<p≤10<p\leq 1) when the multiplier 𝑚 is of optimal smoothness in multi-parameter Besov spaces B2,q(s1,…,sr)⁢(Rn1×⋯×Rnr)B^{{(s_{1},\ldots,s_{r})}}_{2,q}(\mathbb{R}^{n_{1}}\times\cdots\times\mathbb{R}^{n_{r}}), where Tm⁢f⁢(x)=∫Rn1×⋯×Rnrm⁢(ξ)⁢f^⁢(ξ)⁢e2⁢π⁢i⁢x⋅ξ⁢dξT_{m}f(x)=\int_{\mathbb{R}^{n_{1}}\times\cdots\times\mathbb{R}^{n_{r}}}m(\xi)\hat{f}(\xi)e^{2\pi ix\cdot\xi}\,d\xi for x∈Rn1×⋯×Rnrx\in{\mathbb{R}^{n_{1}}\times\cdots\times\mathbb{R}^{n_{r}}}. We will show ∥Tm∥Hp→Hp≲supj1,…,jr∈Z⁡∥mj1,…,jr∥B2,q(s1,…,sr),\lVert T_{m}\rVert_{H^{p}\to H^{p}}\lesssim\sup_{j_{1},\ldots,j_{r}\in\mathbb{Z}}\lVert m_{j_{1},\ldots,j_{r}}\rVert_{B^{{(s_{1},\ldots,s_{r})}}_{2,q}}, where 0<q<∞0<q<\infty and si>ni⁢(1p-12)s_{i}>n_{i}\bigl{(}\frac{1}{p}-\frac{1}{2}\bigr{)}. Here we have used the notation mj1,…,jr⁢(ξ)=m⁢(2j1⁢ξ1,…,2jr⁢ξr)⁢ψ(1)⁢(ξ1)⁢⋯⁢ψ(r)⁢(ξr),m_{j_{1},\ldots,j_{r}}(\xi)=m(2^{j_{1}}\xi_{1},\ldots,2^{j_{r}}\xi_{r})\psi^{(1)}(\xi_{1})\cdots\psi^{(r)}(\xi_{r}), and ψ(i)⁢(ξi)\psi^{(i)}(\xi_{i}) is a suitable cut-off function on Rni\mathbb{R}^{n_{i}} for 1≤i≤r1\leq i\leq r. This multi-parameter Hörmander multiplier theorem is in the spirit of the earlier work of Baernstein and Sawyer in the one-parameter setting and sharpens our recent result of Hörmander multiplier theorem in the bi-parameter case which was established using R. Fefferman’s boundedness criterion. Because R. Fefferman’s boundedness criterion fails in the cases of three or more parameters, it is substantially more difficult to establish such Hörmander multiplier theorems in three or more parameters than in the bi-parameter case. To assume only the optimal smoothness on the multipliers, delicate and hard analysis on the sharp estimates of the square functions on arbitrary atoms are required. Our main theorems give the boundedness on the multi-parameter Hardy spaces under the smoothness assumption of the multipliers in multi-parameter Besov spaces and show the regularity conditions to be sharp.

October 10, 2021

On spectral and non-spectral problem for the planar self-similar measures with four element digit sets

Qian Li, Zhi-Yi Wu

Page range: 1629-1639

Abstract

We consider the self-similar measure μM,𝒟{\mu_{M,{\mathcal{D}}}} generated by an expanding real matrix M=(ρ-100ρ-1)∈M2⁢(ℝ){M=\begin{pmatrix}\rho^{-1}&0\\ 0&\rho^{-1}\end{pmatrix}\in M_{2}({\mathbb{R}})} and a digit set 𝒟={(00),(ab),(cd),(a+cb+d)}⊆ℤ2.{{\mathcal{D}}=\Biggl{\{}\begin{pmatrix}0\\ 0\end{pmatrix},\begin{pmatrix}a\\ b\end{pmatrix},\begin{pmatrix}c\\ d\end{pmatrix},\begin{pmatrix}a+c\\ b+d\end{pmatrix}\Biggr{\}}\subseteq{\mathbb{Z}}^{2}}. In this paper, we study the spectral and non-spectral problems of μM,𝒟{\mu_{M,{\mathcal{D}}}}. In this case that (ab){(\begin{smallmatrix}a\\ b\end{smallmatrix})} and (cd){(\begin{smallmatrix}c\\ d\end{smallmatrix})} are two independent vectors, we prove that if ρ-1∈ℤ{\rho^{-1}\in{\mathbb{Z}}}, then μM,𝒟{\mu_{M,{\mathcal{D}}}} is a spectral measure if and only if ρ-1∈2⁢ℤ{\rho^{-1}\in 2{\mathbb{Z}}}. For the case that (ab){(\begin{smallmatrix}a\\ b\end{smallmatrix})} and (cd){(\begin{smallmatrix}c\\ d\end{smallmatrix})} are two dependent vectors, we first give the sufficient and necessary condition for L2⁢(μM,𝒟){L^{2}(\mu_{M,{\mathcal{D}}})} to contain an infinite orthogonal set of exponential functions. Based on this result, we can give the exact cardinality of orthogonal exponential functions in L2⁢(μM,𝒟){L^{2}(\mu_{M,{\mathcal{D}}})} when L2⁢(μM,𝒟){L^{2}(\mu_{M,{\mathcal{D}}})} does not admit any infinite orthogonal set of exponential functions by classifying the values of ρ.

October 26, 2021

C-minimal topological groups

Wenfei Xi, Menachem Shlossberg

Page range: 1641-1650

Abstract

In this paper, we study topological groups having all closed subgroups (totally) minimal and we call such groups c-(totally) minimal . We show that a locally compact c-minimal connected group is compact. Using a well-known theorem of [P. Hall and C. R. Kulatilaka, A property of locally finite groups, J. Lond. Math. Soc. 39 1964, 235–239] and a characterization of a certain class of Lie groups, due to [S. K. Grosser and W. N. Herfort, Abelian subgroups of topological groups, Trans. Amer. Math. Soc. 283 1984, 1, 211–223], we prove that a c-minimal locally solvable Lie group is compact. It is shown that a topological group G is c-(totally) minimal if and only if G has a compact normal subgroup N such that G/NG/N is c-(totally) minimal. Applying this result, we prove that a locally compact group G is c-totally minimal if and only if its connected component c⁢(G)c(G) is compact and G/c⁢(G)G/c(G) is c-totally minimal. Moreover, a c-totally minimal group that is either complete solvable or strongly compactly covered must be compact. Negatively answering [D. Dikranjan and M. Megrelishvili, Minimality conditions in topological groups, Recent Progress in General Topology. III, Atlantis Press, Paris 2014, 229–327, Question 3.10 (b)], we find, in contrast, a totally minimal solvable (even metabelian) Lie group that is not compact.

October 26, 2021

Cevian properties in ideal lattices of Abelian ℓ-groups

Miroslav Ploščica

Page range: 1651-1658

Abstract

We consider the problem of describing the lattices of compact ℓ{\ell}-ideals of Abelian lattice-ordered groups. (Equivalently, describing the spectral spaces of Abelian lattice-ordered groups.) It is known that these lattices have countably based differences and admit a Cevian operation. Our first result says that these two properties are not sufficient: there are lattices having both countably based differences and Cevian operations, which are not representable by compact ℓ{\ell}-ideals of Abelian lattice-ordered groups. As our second result, we prove that every completely normal distributive lattice of cardinality at most ℵ1{\aleph_{1}} admits a Cevian operation. This complements the recent result of F. Wehrung, who constructed a completely normal distributive lattice having countably based differences, of cardinality ℵ2{\aleph_{2}}, without a Cevian operation.

October 23, 2021

Study of twisted Bargmann transform via Bargmann transform

Shubham R. Bais, Venku Naidu D

Page range: 1659-1670

Abstract

In the present article, we give an alternate and easier proof for the image characterization of L2⁢(ℝ2⁢n){L^{2}(\mathbb{R}^{2n})} under the twisted Bargmann transform which was earlier studied by Krontz, Thangavelu and Xu. As a consequence, we study some properties of the twisted Bergman spaces for 0<p≤∞{0<p\leq\infty} and the Lp{L^{p}}-boundedness of the twisted Bargmann transform, 1≤p≤∞{1\leq p\leq\infty}. We also study Lp{L^{p}}-boundedness of the twisted Bargmann projection Pt{P_{t}} and the duality relations between the spaces Btp⁢(ℂ2⁢n){B_{t}^{p}(\mathbb{C}^{2n})}, 1<p<∞{1<p<\infty}.

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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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