Derived Operations Satisfy Standard IdentitiesDotsenko, Vladimir
doi: 10.1007/s00574-025-00494-zpmid: N/A
A derived operation is a bilinear operation on a commutative associative algebra A defined intrinsically out of its product and several derivations of the product. We show that operators of left (or right) multiplications of a derived operation always satisfy a “standard identity” of certain order. In particular, it implies that each Rankin–Cohen bracket of modular forms, as well as each higher bracket of Kontsevich’s universal deformation quantization formula for Poisson structures on Rn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {R}^n$$\end{document}, satisfies standard identities.
On Generating Mapping Class Groups by Pseudo-Anosov ElementsHirose, Susumu; Monden, Naoyuki
doi: 10.1007/s00574-026-00501-xpmid: N/A
Wajnryb proved that the mapping class group of a closed oriented surface can be generated by two elements. In this paper we show that it can be generated by two pseudo-Anosov elements. In particular, if the genus is at least nine, the generators may be chosen to be two conjugate pseudo-Anosov elements with arbitrarily large dilatations. We also prove that, for genus at least eight, the mapping class group is generated by two conjugate reducible elements of infinite order. We also obtain analogous generation results by two pseudo-Anosov elements and by two conjugate reducible elements of infinite order for the hyperelliptic mapping class group.
On Immanants of the Cayley Table of Finite Abelian GroupsWang, Xuan; Zhang, Hanbin; Zhang, Shiwen
doi: 10.1007/s00574-026-00495-6pmid: N/A
Let G be a finite abelian group of order n. Let MG\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {M}}_G$$\end{document} be the Cayley table of G and per(MG)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textsf{per}({\mathcal {M}}_G)$$\end{document} the permanent of MG\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {M}}_G$$\end{document}. An interesting result of Hall provided a one to one correspondence between the monomials in per(MG)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textsf{per}({\mathcal {M}}_G)$$\end{document} and zero-sum sequences over G of length n. Generalizing Hall’s result, Panyushev conjectured an analogous correspondence concerning the generalized Cayley table of G. In this paper, we disprove Panyushev’s conjecture and provide a general characterization of the aforementioned correspondence. As the permanent and determinant matrix functions are special cases of immanants (which are very important objects in algebraic combinatorics), we also provide some discussions on the immanants of MG\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {M}}_G$$\end{document} and propose some interesting conjectures.
Averaging Principle for a General Class of Periodic Functions in Discrete SpacesBohner, Martin; Mesquita, Jaqueline G.; Streipert, Sabrina
doi: 10.1007/s00574-025-00485-0pmid: N/A
In this work, we develop a periodic averaging principle for arbitrary discrete time domains, leveraging a novel definition of periodicity. This definition does not rely on the classical requirement for the time domain itself to be periodic. We implement this averaging principle across diverse discrete time domains and explore a range of periodic functions within this extended context. The paper contains several examples with numerical simulations, providing visual demonstrations of our results. This highlights the versatility of our averaging principle and its potential to understand dynamics of nonautonomous recurrences with complex temporal patterns.
A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}$$\end{document}-Compact Holomorphic Lipschitz Mappings on the Unit Ball of a Banach SpaceJiménez-Vargas, A.; Ruiz-Casternado, D.
doi: 10.1007/s00574-026-00500-ypmid: N/A
Let X and Y be complex Banach spaces, BX\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$B_X$$\end{document} be the open unit ball of X and HL0(BX,Y)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {H}L_0(B_X,Y)$$\end{document} be the Banach space of all holomorphic Lipschitz maps f:BX→Y\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f:B_X\rightarrow Y$$\end{document} such that f(0)=0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f(0)=0$$\end{document}, endowed with the Lipschitz norm. Given a Banach operator ideal A\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {A}$$\end{document}, we use the property of A\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {A}$$\end{document}-compactness by Carl and Stephani to introduce and study the subclass of those functions in HL0(BX,Y)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {H}L_0(B_X,Y)$$\end{document} for which its Lipschitz image is a relatively A\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {A}$$\end{document}-compact subset of Y. We focus our attention on its structure as a composition Banach holomorphic Lipschitz ideal by using its connection with A\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {A}$$\end{document}-compact linear operators through linearization/transposition techniques.