6,631 results
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2. An unpublished paper ‘Über einige durch unendliche Reihen definirte Functionen eines complexen Argumentes’ by Adolf Hurwitz
- Author
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Nicola Oswald
- Subjects
History ,Pure mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Algebra ,symbols.namesake ,Continuation ,0103 physical sciences ,Functional equation ,symbols ,010307 mathematical physics ,0101 mathematics ,Dirichlet series ,Meromorphic function ,Mathematics - Abstract
In 1903, Epstein published his proof of meromorphic continuation and a functional equation for Dirichlet series associated with quadratic forms, now called Epstein zeta-functions. However, already in 1889 (or even earlier) Hurwitz was aware of these results as his mathematical diaries and some unpublished notes (in an almost final form) found in his estate at the ETH Zurich show. In this article we present and analyze Hurwitz's notes and compare his reasoning with Epstein's paper in detail.
- Published
- 2017
3. Notes on the Paper 'On SS-Quasinormal and S-Quasinormally Embedded Subgroups of Finite Groups' of Shen et al
- Author
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Yuemei Mao, Xiaolan Yi, and Changwen Li
- Subjects
Discrete mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Zhàng ,ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,GeneralLiterature_MISCELLANEOUS ,Mathematics - Abstract
We correct an error in the paper of Z. Shen, S. Li, and J. Zhang published in [4]. In addition, we give an answer to a question posed by the authors.
- Published
- 2018
4. On Nash’s unique contribution to analysis in just three of his papers
- Author
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Sergiu Klainerman
- Subjects
Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Published
- 2016
5. Derived length of zero entropy groups acting on projective varieties in arbitrary characteristic — A remark to a paper of Dinh-Oguiso-Zhang
- Author
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Sichen Li
- Subjects
Automorphism group ,Pure mathematics ,Computer Science::Information Retrieval ,General Mathematics ,010102 general mathematics ,Zhàng ,Astrophysics::Instrumentation and Methods for Astrophysics ,Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) ,01 natural sciences ,0103 physical sciences ,Computer Science::General Literature ,Entropy (information theory) ,010307 mathematical physics ,0101 mathematics ,Algebraically closed field ,Projective test ,Projective variety ,Mathematics - Abstract
Let [Formula: see text] be a projective variety of dimension [Formula: see text] over an algebraically closed field of arbitrary characteristic. We prove a Fujiki–Lieberman type theorem on the structure of the automorphism group of [Formula: see text]. Let [Formula: see text] be a group of zero entropy automorphisms of [Formula: see text] and [Formula: see text] the set of elements in [Formula: see text] which are isotopic to the identity. We show that after replacing [Formula: see text] by a suitable finite-index subgroup, [Formula: see text] is a unipotent group of the derived length at most [Formula: see text]. This result was first proved by Dinh et al. for compact Kähler manifolds.
- Published
- 2020
6. 'Graph Paper' Trace Characterizations of Functions of Finite Energy
- Author
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Robert S. Strichartz
- Subjects
Discrete mathematics ,Plane (geometry) ,General Mathematics ,010102 general mathematics ,Voltage graph ,Mathematics::General Topology ,Graph paper ,01 natural sciences ,Sierpinski triangle ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Sobolev space ,Coxeter graph ,Sierpinski carpet ,0103 physical sciences ,String graph ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Analysis ,Mathematics - Abstract
We characterize functions of finite energy in the plane in terms of their traces on the lines that make up "graph paper" with squares of side length $mn$ for all $n$, and certain $\12-$order Sobolev norms on the graph paper lines. We also obtain analogous results for functions of finite energy on two classical fractals: the Sierpinski gasket and the Sierpinski carpet.
- Published
- 2013
- Full Text
- View/download PDF
7. Variations on a Theme in Paper Folding
- Author
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Burkard Polster
- Subjects
General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,Folding (DSP implementation) ,0101 mathematics ,01 natural sciences ,Linguistics ,Theme (narrative) ,Mathematics - Published
- 2004
8. Special Ulrich bundles on regular Weierstrass fibrations
- Author
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Joan Pons-Llopis and Rosa M. Miró-Roig
- Subjects
Pure mathematics ,Class (set theory) ,Mathematics::Commutative Algebra ,General Mathematics ,010102 general mathematics ,Short paper ,Elliptic surfaces ,Ulrich bundles ,01 natural sciences ,Mathematics::Algebraic Geometry ,Simple (abstract algebra) ,0103 physical sciences ,Weierstrass fibrations ,Rank (graph theory) ,010307 mathematical physics ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
The main goal of this short paper is to prove the existence of rank 2 simple and special Ulrich bundles on a wide class of elliptic surfaces: namely, on regular Weierstrass fibrations \(\pi : S\rightarrow \mathbb {P}^1\). Alongside we also show the existence of rank 2 weakly Ulrich sheaves on arbitrary Weierstrass fibrations \(S\rightarrow C_0\) and we deal with the (non-)existence of rank one Ulrich bundles on them.
- Published
- 2019
9. Note on a Paper by Robinson
- Author
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J. A. Todd
- Subjects
General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematical economics ,Mathematics - Abstract
In a recent paper Robinson has obtained an explicit formula for the expression of an invariant matrix of an invariant matrix as a direct sum of invariant matrices. The object of the present note is to show that this formula may be deduced from known properties of Schur functions, with the aid of a result which the author has proved elsewhere.
- Published
- 1950
10. Derived Non-archimedean analytic Hilbert space
- Author
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Mauro Porta, Jorge António, Institut de Recherche Mathématique Avancée (IRMA), and Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Pure mathematics ,Fiber (mathematics) ,General Mathematics ,010102 general mathematics ,Short paper ,Formal scheme ,Hilbert space ,Space (mathematics) ,01 natural sciences ,symbols.namesake ,Mathematics - Algebraic Geometry ,Mathematics::Category Theory ,0103 physical sciences ,Localization theorem ,FOS: Mathematics ,symbols ,010307 mathematical physics ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,0101 mathematics ,Algebraic Geometry (math.AG) ,Quotient ,Mathematics - Abstract
In this short paper we combine the representability theorem introduced in [17, 18] with the theory of derived formal models introduced in [2] to prove the existence representability of the derived Hilbert space RHilb(X) for a separated k-analytic space X. Such representability results relies on a localization theorem stating that if X is a quasi-compact and quasi-separated formal scheme, then the \infty-category Coh^+(X^rig) of almost perfect complexes over the generic fiber can be realized as a Verdier quotient of the \infty-category Coh^+(X). Along the way, we prove several results concerning the the \infty-categories of formal models for almost perfect modules on derived k-analytic spaces., 28 pages
- Published
- 2019
11. A note on gonality of curves on general hypersurfaces
- Author
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Flaminio Flamini, Paola Supino, Ciro Ciliberto, Francesco Bastianelli, Bastianelli, Francesco, Ciliberto, Ciro, Flamini, Flaminio, and Supino, Paola
- Subjects
Series (mathematics) ,Degree (graph theory) ,family of curves ,General Mathematics ,010102 general mathematics ,Short paper ,Birational geometry ,gonality of curves, projective hypersurfaces ,01 natural sciences ,Hypersurfaces ,Combinatorics ,Mathematics::Algebraic Geometry ,Hypersurface ,Product (mathematics) ,0103 physical sciences ,Hypersurfaces, family of curves, gonality ,010307 mathematical physics ,gonality ,Settore MAT/03 - Geometria ,0101 mathematics ,Mathematics - Abstract
This short paper concerns the existence of curves with low gonality on smooth hypersurfaces $$X\subset \mathbb {P}^{n+1}$$ . After reviewing a series of results on this topic, we report on a recent progress we achieved as a product of the Workshop Birational geometry of surfaces, held at University of Rome “Tor Vergata” on January 11th–15th, 2016. In particular, we obtained that if $$X\subset \mathbb {P}^{n+1}$$ is a very general hypersurface of degree $$d\geqslant 2n+2$$ , the least gonality of a curve $$C\subset X$$ passing through a general point of X is $$\mathrm {gon}(C)=d-\left\lfloor \frac{\sqrt{16n+1}-1}{2}\right\rfloor $$ , apart from some exceptions we list.
- Published
- 2018
12. On Beilinson’s equivalence for p-adic cohomology
- Author
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Daniel Caro, Tomoyuki Abe, Institute for the Physics and Mathematics of the Universe (IPMU), The University of Tokyo (UTokyo), Laboratoire de Mathématiques Nicolas Oresme (LMNO), Centre National de la Recherche Scientifique (CNRS)-Université de Caen Normandie (UNICAEN), and Normandie Université (NU)-Normandie Université (NU)
- Subjects
Pure mathematics ,Derived category ,Functor ,Holonomic ,General Mathematics ,010102 general mathematics ,Short paper ,General Physics and Astronomy ,Unipotent ,01 natural sciences ,Cohomology ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,0103 physical sciences ,010307 mathematical physics ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,0101 mathematics ,Equivalence (formal languages) ,Mathematics::Representation Theory ,ComputingMilieux_MISCELLANEOUS ,Mathematics - Abstract
In this short paper, we construct a unipotent nearby cycle functor and show a p-adic analogue of Beilinson’s equivalence comparing two derived categories: the derived category of holonomic arithmetic $${\mathcal {D}}$$ -modules and the derived category of arithmetic $${\mathcal {D}}$$ -modules whose cohomologies are holonomic.
- Published
- 2018
13. Lattice Polygons and the Number 2i + 7
- Author
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Josef Schicho and Christian Haase
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Discrete mathematics ,General Mathematics ,010102 general mathematics ,Integer lattice ,Toric variety ,Graph paper ,Computer Science::Computational Geometry ,01 natural sciences ,Combinatorics ,Lattice (order) ,0103 physical sciences ,Polygon ,010307 mathematical physics ,0101 mathematics ,Algebraic number ,Invariant (mathematics) ,Mathematics - Abstract
0.1. How it all began. When the second author translated a result on algebraic sur faces into the language of lattice polygons using toric geometry, he got a simple inequality for lattice polygons. This inequality had originally been discovered by Scott [12]. The first author then found a third proof. Subsequently, both authors went through a phase of polygon addiction. Once you get started drawing lattice polygons on graph paper and discovering relations between their numerical invariants, it is not so easy to stop! (The gentle reader has been warned.) Thus, it was just unavoidable that the authors came up with new inequalities: Scott's inequality can be sharpened if one takes into account another invariant, which is de fined by peeling off the skins of the polygons like an onion (see Section 3).
- Published
- 2009
14. Iterates of Generic Polynomials and Generic Rational Functions
- Author
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Jamie Juul
- Subjects
Pure mathematics ,Degree (graph theory) ,Mathematics - Number Theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,MathematicsofComputing_GENERAL ,Galois group ,37P05, 11G50, 14G25 ,Rational function ,01 natural sciences ,Unpublished paper ,Generic polynomial ,Number theory ,Symmetric group ,Iterated function ,0103 physical sciences ,FOS: Mathematics ,Number Theory (math.NT) ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
In 1985, Odoni showed that in characteristic 0 0 the Galois group of the n n -th iterate of the generic polynomial with degree d d is as large as possible. That is, he showed that this Galois group is the n n -th wreath power of the symmetric group S d S_d . We generalize this result to positive characteristic, as well as to the generic rational function. These results can be applied to prove certain density results in number theory, two of which are presented here. This work was partially completed by the late R.W.K. Odoni in an unpublished paper.
- Published
- 2014
15. Non-negative Ricci curvature on closed manifolds under Ricci flow
- Author
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Davi Maximo
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Short paper ,Ricci flow ,01 natural sciences ,Mathematics::Geometric Topology ,Mathematics - Analysis of PDEs ,Differential Geometry (math.DG) ,Bounded curvature ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Mathematics::Differential Geometry ,0101 mathematics ,10. No inequality ,Mathematics::Symplectic Geometry ,Ricci curvature ,Mathematics ,Analysis of PDEs (math.AP) - Abstract
In this short note we show that non-negative Ricci curvature is not preserved under Ricci flow for closed manifolds of dimensions four and above, strengthening a previous result of Knopf in \cite{K} for complete non-compact manifolds of bounded curvature. This brings down to four dimensions a similar result B\"ohm and Wilking have for dimensions twelve and above, \cite{BW}. Moreover, the manifolds constructed here are \Kahler manifolds and relate to a question raised by Xiuxiong Chen in \cite{XC}, \cite{XCL}., Comment: New version with added references and corrected typos
- Published
- 2009
- Full Text
- View/download PDF
16. Why a Population Converges to Stability
- Author
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W.B. Arthur
- Subjects
education.field_of_study ,Fundamental theorem ,Age structure ,General Mathematics ,010102 general mathematics ,Short paper ,Population ,Full view ,01 natural sciences ,0103 physical sciences ,Quantitative Biology::Populations and Evolution ,Ergodic theory ,Age distribution ,010307 mathematical physics ,0101 mathematics ,education ,Mathematical economics ,Smoothing ,Mathematics - Abstract
A large part of mathematical demography is built upon one fundamental theorem, the "strong ergodic theorem" of demography. If the fertility and mortality age-schedules of a population remain unchanged over time, its age distribution, no matter what its initial shape, will converge in time to a fixed and stable form. In brief, when demographic behavior remains unchanged, the population, it is said, converges to stability. This short paper presents a new argument for the convergence of the age structure, one that is self-contained, and that brings the mechanism behind convergence into full view. The idea is simple. Looked at directly, the dynamics of the age-distribution say little to our normal intuition. Looked at from a slightly different angle though, population dynamics define a smoothing or averaging process over the generations -- a process comfortable to our intuition. This smoothing and resmoothing turns out to be the mechanism that forces the age structure toward a fixed and final form.
- Published
- 1981
17. Analyzing the Weyl Construction for Dynamical Cartan Subalgebras
- Author
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Elizabeth Gillaspy, Anna Duwenig, and Rachael Norton
- Subjects
General Mathematics ,01 natural sciences ,Section (fiber bundle) ,Combinatorics ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,Mathematics::Quantum Algebra ,46L05, 22D25, 22A22 ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Twist ,Operator Algebras (math.OA) ,Mathematics::Representation Theory ,Quotient ,Mathematics ,Science & Technology ,Mathematics::Operator Algebras ,010102 general mathematics ,Spectrum (functional analysis) ,Mathematics - Operator Algebras ,Cartan subalgebra ,C-ASTERISK-ALGEBRAS ,Physical Sciences ,010307 mathematical physics ,EQUIVALENCE - Abstract
When the reduced twisted $C^*$-algebra $C^*_r(\mathcal{G}, c)$ of a non-principal groupoid $\mathcal{G}$ admits a Cartan subalgebra, Renault's work on Cartan subalgebras implies the existence of another groupoid description of $C^*_r(\mathcal{G}, c)$. In an earlier paper, joint with Reznikoff and Wright, we identified situations where such a Cartan subalgebra arises from a subgroupoid $\mathcal{S}$ of $\mathcal{G}$. In this paper, we study the relationship between the original groupoids $\mathcal{S}, \mathcal{G}$ and the Weyl groupoid and twist associated to the Cartan pair. We first identify the spectrum $\mathfrak{B}$ of the Cartan subalgebra $C^*_r(\mathcal{S}, c)$. We then show that the quotient groupoid $\mathcal{G}/\mathcal{S}$ acts on $\mathfrak{B}$, and that the corresponding action groupoid is exactly the Weyl groupoid of the Cartan pair. Lastly we show that, if the quotient map $\mathcal{G}\to\mathcal{G}/\mathcal{S}$ admits a continuous section, then the Weyl twist is also given by an explicit continuous $2$-cocycle on $\mathcal{G}/\mathcal{S} \ltimes \mathfrak{B}$., 32 pages
- Published
- 2022
18. Order 3 symplectic automorphisms on K3 surfaces
- Author
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Alice Garbagnati and Yulieth Prieto Montañez
- Subjects
Pure mathematics ,Endomorphism ,General Mathematics ,010102 general mathematics ,Lattice (group) ,Order (ring theory) ,Automorphism ,01 natural sciences ,Cohomology ,14J28, 14J50 ,Mathematics - Algebraic Geometry ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Abelian group ,Algebraic Geometry (math.AG) ,Quotient ,Mathematics ,Symplectic geometry - Abstract
The aim of this paper is to generalize results known for the symplectic involutions on K3 surfaces to the order 3 symplectic automorphisms on K3 surfaces. In particular, we will explicitly describe the action induced on the lattice $\Lambda_{K3}$, isometric to the second cohomology group of a K3 surface, by a symplectic automorphism of order 3; we exhibit the maps $\pi_*$ and $\pi^*$ induced in cohomology by the rational quotient map $\pi:X\dashrightarrow Y$, where $X$ is a K3 surface admitting an order 3 symplectic automorphism $\sigma$ and $Y$ is the minimal resolution of the quotient $X/\sigma$; we deduce the relation between the N\'eron--Severi group of $X$ and the one of $Y$. Applying these results we describe explicit geometric examples and generalize the Shioda--Inose structures, relating Abelian surfaces admitting order 3 endomorphisms with certain specific K3 surfaces admitting particular order 3 symplectic automorphisms., Comment: 28 pages. Version 2: this is the published version of the paper. The last section of the previous version (v1) was erased (the results are only stated) and it is now contained in arXiv:2209.10141
- Published
- 2021
19. Maximal families of nodal varieties with defect
- Author
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REMKE NANNE KLOOSTERMAN
- Subjects
Surface (mathematics) ,Double cover ,Degree (graph theory) ,Plane (geometry) ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Combinatorics ,Mathematics - Algebraic Geometry ,Hypersurface ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,NODAL ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
In this paper we prove that a nodal hypersurface in P^4 with defect has at least (d-1)^2 nodes, and if it has at most 2(d-2)(d-1) nodes and d>6 then it contains either a plane or a quadric surface. Furthermore, we prove that a nodal double cover of P^3 ramified along a surface of degree 2d with defect has at least d(2d-1) nodes. We construct the largest dimensional family of nodal degree d hypersurfaces in P^(2n+2) with defect for d sufficiently large., v2: A proof for the Ciliberto-Di Gennaro conjecture is added (Section 5); Some minor corrections in the other sections. v3: some minor corrections in the abstract v4: The proof for the Ciliberto-Di Gennaro conjecture has been modified; The paper is split into two parts, the complete intersection case will be discussed in a different paper
- Published
- 2021
20. On Nilpotent Extensions of ∞-Categories and the Cyclotomic Trace
- Author
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Elden Elmanto and Vladimir Sosnilo
- Subjects
Trace (semiology) ,Pure mathematics ,Nilpotent ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
We do three things in this paper: (1) study the analog of localization sequences (in the sense of algebraic $K$-theory of stable $\infty $-categories) for additive $\infty $-categories, (2) define the notion of nilpotent extensions for suitable $\infty $-categories and furnish interesting examples such as categorical square-zero extensions, and (3) use (1) and (2) to extend the Dundas–Goodwillie–McCarthy theorem for stable $\infty $-categories that are not monogenically generated (such as the stable $\infty $-category of Voevodsky’s motives or the stable $\infty $-category of perfect complexes on some algebraic stacks). The key input in our paper is Bondarko’s notion of weight structures, which provides a “ring-with-many-objects” analog of a connective $\mathbb{E}_1$-ring spectrum. As applications, we prove cdh descent results for truncating invariants of stacks extending the work by Hoyois–Krishna for homotopy $K$-theory and establish new cases of Blanc’s lattice conjecture.
- Published
- 2021
21. An index theorem for higher orbital integrals
- Author
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Xiang Tang, Peter Hochs, and Yanli Song
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Index (economics) ,General Mathematics ,01 natural sciences ,Mathematics::K-Theory and Homology ,0103 physical sciences ,FOS: Mathematics ,Representation Theory (math.RT) ,0101 mathematics ,Algebra over a field ,Operator Algebras (math.OA) ,Mathematics ,Group (mathematics) ,010102 general mathematics ,Mathematics - Operator Algebras ,Lie group ,K-Theory and Homology (math.KT) ,Elliptic operator ,Differential Geometry (math.DG) ,Mathematics - K-Theory and Homology ,Equivariant map ,010307 mathematical physics ,Atiyah–Singer index theorem ,Mathematics - Representation Theory - Abstract
Recently, two of the authors of this paper constructed cyclic cocycles on Harish-Chandra's Schwartz algebra of linear reductive Lie groups that detect all information in the $K$-theory of the corresponding group $C^*$-algebra. The main result in this paper is an index formula for the pairings of these cocycles with equivariant indices of elliptic operators for proper, cocompact actions. This index formula completely determines such equivariant indices via topological expressions., 40 pages; updates based on referee comments; expanded proof of Proposition 3.3
- Published
- 2021
22. Noncommutative Counting Invariants and Curve Complexes
- Author
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Ludmil Katzarkov and George Dimitrov
- Subjects
Intersection theory ,medicine.medical_specialty ,Functor ,Conjecture ,Group (mathematics) ,General Mathematics ,010102 general mathematics ,Quiver ,Type (model theory) ,01 natural sciences ,Combinatorics ,0103 physical sciences ,medicine ,010307 mathematical physics ,0101 mathematics ,Partially ordered set ,Commutative property ,Mathematics - Abstract
In our previous paper, viewing $D^b(K(l))$ as a noncommutative curve, where $K(l)$ is the Kronecker quiver with $l$-arrows, we introduced categorical invariants via counting of noncommutative curves. Roughly, these invariants are sets of subcategories in a given category and their quotients. The noncommutative curve-counting invariants are obtained by restricting the subcategories to be equivalent to $D^b(K(l))$. The general definition, however, defines a larger class of invariants and many of them behave properly with respect to fully faithful functors. Here, after recalling the definition, we focus on the examples and extend our studies beyond counting. We enrich our invariants with the following structures: the inclusion of subcategories makes them partially ordered sets and considering semi-orthogonal pairs of subcategories as edges amounts to directed graphs. It turns out that the problem for counting $D^b(A_k)$ in $D^b(A_n)$ has a geometric combinatorial parallel - counting of maps between polygons. Estimating the numbers counting noncommutative curves in $D^b({\mathbb P}^2)$ modulo the group of autoequivalences, we prove finiteness and that the exact determining of these numbers leads to a solution of Markov problem. Via homological mirror symmetry, this gives a new approach to this problem. Regarding the structure of a partially ordered set mentioned above, we initiate intersection theory of noncommutative curves focusing on the case of noncommutative genus zero. The above-mentioned structure of a directed graph (and related simplicial complex) is a categorical analogue of the classical curve complex, introduced by Harvey and Harrer. The paper contains pictures of the graphs in many examples and also presents an approach to Markov conjecture via counting of subgraphs in a graph associated with $D^b({{\mathbb{P}}}^2)$. Some of the results proved here were announced in a previous work.
- Published
- 2021
23. Correction to: Seifert fibrations of lens spaces
- Author
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Christian Lange and Hansjörg Geiges
- Subjects
Lemma (mathematics) ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Lens (geology) ,Astrophysics::Cosmology and Extragalactic Astrophysics ,Base (topology) ,Mathematics::Geometric Topology ,01 natural sciences ,Number theory ,Differential geometry ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Algebra over a field ,Mathematics::Symplectic Geometry ,Orbifold ,Mathematics - Abstract
We classify the Seifert fibrations of lens spaces where the base orbifold is non-orientable. This is an addendum to our earlier paper ‘Seifert fibrations of lens spaces’. We correct Lemma 4.1 of that paper and fill the gap in the classification that resulted from the erroneous lemma.
- Published
- 2021
24. CARLESON INTERPOLATING SEQUENCES FOR BANACH SPACES OF ANALYTIC FUNCTIONS
- Author
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Paweł Mleczko, David Norrbo, Michał Rzeczkowski, Mikael Lindström, and Mieczysław Mastyło
- Subjects
Mathematics::Functional Analysis ,Pure mathematics ,Computer Science::Information Retrieval ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Mathematics::Classical Analysis and ODEs ,Banach space ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics ,Analytic function - Abstract
This paper presents an approach, based on interpolation theory of operators, to the study of interpolating sequences for interpolation Banach spaces between Hardy spaces. It is shown that the famous Carleson result forH∞can be lifted to a large class of abstract Hardy spaces. A description is provided of the range of the Carleson operator defined on interpolation spaces between the classical Hardy spaces in terms of uniformly separated sequences. A key role in this description is played by some general interpolation results proved in the paper. As by-products, novel results are obtained which extend the Shapiro–Shields result on the characterisation of interpolation sequences for the classical Hardy spacesHp. Applications to Hardy–Lorentz, Hardy–Marcinkiewicz and Hardy–Orlicz spaces are presented.
- Published
- 2021
25. Simpson filtration and oper stratum conjecture
- Author
-
Zhi Hu and Pengfei Huang
- Subjects
Mathematics::Dynamical Systems ,Conjecture ,Mathematics::Commutative Algebra ,General Mathematics ,010102 general mathematics ,Dimension (graph theory) ,Vector bundle ,Algebraic geometry ,01 natural sciences ,Moduli space ,Combinatorics ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Number theory ,0103 physical sciences ,FOS: Mathematics ,Filtration (mathematics) ,010307 mathematical physics ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematics::Symplectic Geometry ,Mathematics ,Stratum - Abstract
In this paper, we prove that for the oper stratification of the de Rham moduli space $M_{\mathrm{dR}}(X,r)$, the closed oper stratum is the unique minimal stratum with dimension $r^2(g-1)+g+1$, and the open dense stratum consisting of irreducible flat bundles with stable underlying vector bundles is the unique maximal stratum., Comment: This paper comes from the last section of arXiv:1905.10765v1 as an independent paper. Comments are welcome! To appear in manuscripta mathematica
- Published
- 2021
26. Graded Bourbaki ideals of graded modules
- Author
-
Jürgen Herzog, Dumitru I. Stamate, and Shinya Kumashiro
- Subjects
Noetherian ,Pure mathematics ,Sequence ,Class (set theory) ,Ideal (set theory) ,Mathematics::Commutative Algebra ,Mathematics::General Mathematics ,General Mathematics ,Mathematics::History and Overview ,010102 general mathematics ,Structure (category theory) ,Mathematics::General Topology ,Field (mathematics) ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,01 natural sciences ,Mathematik ,0103 physical sciences ,FOS: Mathematics ,Homomorphism ,13A02, 13A30, 13D02, 13H10 ,010307 mathematical physics ,0101 mathematics ,Rees algebra ,Mathematics - Abstract
In this paper we study graded Bourbaki ideals. It is a well-known fact that for torsionfree modules over Noetherian normal domains, Bourbaki sequences exist. We give criteria in terms of certain attached matrices for a homomorphism of modules to induce a Bourbaki sequence. Special attention is given to graded Bourbaki sequences. In the second part of the paper, we apply these results to the Koszul cycles of the residue class field and determine particular Bourbaki ideals explicitly. We also obtain in a special case the relationship between the structure of the Rees algebra of a Koszul cycle and the Rees algebra of its Bourbaki ideal., Comment: 29 pages
- Published
- 2021
27. Wild Cantor actions
- Author
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Ramón Barral Lijó, Hiraku Nozawa, Jesús A. Álvarez López, and Olga Lukina
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,Group (mathematics) ,General Mathematics ,010102 general mathematics ,Closure (topology) ,Mathematics::General Topology ,Dynamical Systems (math.DS) ,16. Peace & justice ,Equicontinuity ,01 natural sciences ,Centralizer and normalizer ,Cantor set ,Group action ,Wreath product ,0103 physical sciences ,FOS: Mathematics ,Countable set ,2020: 37B05, 37E25, 20E08, 20E15, 20E18, 20E22, 22F05, 22F50 (Primary), 20F22, 57R30, 57R50 (Secondary) ,010307 mathematical physics ,Mathematics - Dynamical Systems ,0101 mathematics ,Mathematics - Abstract
The discriminant group of a minimal equicontinuous action of a group $G$ on a Cantor set $X$ is the subgroup of the closure of the action in the group of homeomorphisms of $X$, consisting of homeomorphisms which fix a given point. The stabilizer and the centralizer groups associated to the action are obtained as direct limits of sequences of subgroups of the discriminant group with certain properties. Minimal equicontinuous group actions on Cantor sets admit a classification by the properties of the stabilizer and centralizer direct limit groups. In this paper, we construct new families of examples of minimal equicontinuous actions on Cantor sets, which illustrate certain aspects of this classification. These examples are constructed as actions on rooted trees. The acting groups are countable subgroups of the product or of the wreath product of groups. We discuss applications of our results to the study of attractors of dynamical systems and of minimal sets of foliations., 20 pages, 1 figure. The condition of finite generation in Thm 1.9 was replaced by countability. The proof of Thm 1.9 has been simplified. The notation used in 5 has been modified. Several minor corrections across the paper
- Published
- 2022
28. A Unified Approach to the Arens Regularity and Related Problems for a Class of Banach Algebras Associated with Locally Compact Groups
- Author
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A. Ülger and Anthony To-Ming Lau
- Subjects
Mathematics::Functional Analysis ,Pure mathematics ,Class (set theory) ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,Locally compact space ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Based on Katznelson–Tzafriri Theorem on power bounded operators, we prove in this paper a theorem, which applies to the most of the classical Banach algebras of harmonic analysis associated with locally compact groups, to deal with the problems when a given Banach algebra A is Arens regular and when A is an ideal in its bidual. In the second part of the paper, we study the topological center of the bidual of a class of Banach algebras with a multiplier bounded approximate identity.
- Published
- 2021
29. Properties of triangulated and quotient categories arising from n-Calabi–Yau triples
- Author
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Francesca Fedele
- Subjects
Derived category ,Endomorphism ,Triangulated category ,General Mathematics ,010102 general mathematics ,Field (mathematics) ,01 natural sciences ,Cluster algebra ,Combinatorics ,Mathematics::Category Theory ,0103 physical sciences ,FOS: Mathematics ,Homological algebra ,010307 mathematical physics ,Gap theorem ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics - Representation Theory ,Quotient ,Mathematics - Abstract
The original definition of cluster algebras by Fomin and Zelevinsky has been categorified and generalised in several ways over the course of the past 20 years, giving rise to cluster theory. This study lead to Iyama and Yang's generalised cluster categories $\mathcal{T}/\mathcal{T}^{fd}$ coming from $n$-Calabi-Yau triples $(\mathcal{T}, \mathcal{T}^{fd}, \mathcal{M})$. In this paper, we use some classic tools of homological algebra to give a deeper understanding of such categories $\mathcal{T}/\mathcal{T}^{fd}$. Let $k$ be a field, $n\geq 3$ an integer and $\mathcal{T}$ a $k$-linear triangulated category with a triangulated subcategory $\mathcal{T}^{fd}$ and a subcategory $\mathcal{M}=\text{add}(M)$ such that $(\mathcal{T}, \mathcal{T}^{fd}, \mathcal{M})$ is an $n$-Calabi-Yau triple. In this paper, we prove some properties of the triangulated categories $\mathcal{T}$ and $\mathcal{T}/\mathcal{T}^{fd}$. Our first result gives a relation between the Hom-spaces in these categories, using limits and colimits. Our second result is a Gap Theorem in $\mathcal{T}$, showing when the truncation triangles split. Moreover, we apply our two theorems to present an alternative proof to a result by Guo, originally stated in a more specific setup of dg $k$-algebras $A$ and subcategories of the derived category of dg $A$-modules. This proves that $\mathcal{T}/\mathcal{T}^{fd}$ is Hom-finite and $(n-1)$-Calabi-Yau, its object $M$ is $(n-1)$-cluster tilting and the endomorphism algebras of $M$ over $\mathcal{T}$ and over $\mathcal{T}/\mathcal{T}^{fd}$ are isomorphic. Note that these properties make $\mathcal{T}/\mathcal{T}^{fd}$ a generalisation of the cluster category., Comment: 17 pages. Final accepted version to appear in the Pacific Journal of Mathematics
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- 2021
30. The factorisation property ofl∞(Xk)
- Author
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Paul F. X. Müller, Thomas Schlumprecht, Pavlos Motakis, and Richard Lechner
- Subjects
Pure mathematics ,Property (philosophy) ,Basis (linear algebra) ,General Mathematics ,010102 general mathematics ,Diagonal ,Banach space ,01 natural sciences ,Identity (music) ,Bounded operator ,Factorization ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
In this paper we consider the following problem: letXk, be a Banach space with a normalised basis (e(k, j))j, whose biorthogonals are denoted by${(e_{(k,j)}^*)_j}$, for$k\in\N$, let$Z=\ell^\infty(X_k:k\kin\N)$be theirl∞-sum, and let$T:Z\to Z$be a bounded linear operator with a large diagonal,i.e.,$$\begin{align*}\inf_{k,j} \big|e^*_{(k,j)}(T(e_{(k,j)})\big|>0.\end{align*}$$Under which condition does the identity onZfactor throughT? The purpose of this paper is to formulate general conditions for which the answer is positive.
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- 2020
31. ON THE OPTIMAL EXTENSION THEOREM AND A QUESTION OF OHSAWA
- Author
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Xiangyu Zhou, Zhi Li, and Sha Yao
- Subjects
Algebra ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,Extension (predicate logic) ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, we present a version of Guan-Zhou’s optimal $L^{2}$ extension theorem and its application. As a main application, we show that under a natural condition, the question posed by Ohsawa in his series paper VIII on the extension of $L^{2}$ holomorphic functions holds. We also give an explicit counterexample which shows that the question fails in general.
- Published
- 2020
32. On the fill-in of nonnegative scalar curvature metrics
- Author
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Wenlong Wang, Guodong Wei, Jintian Zhu, and Yuguang Shi
- Subjects
Combinatorics ,Conjecture ,Mean curvature ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Invariant (mathematics) ,01 natural sciences ,Mathematics ,Scalar curvature - Abstract
In the first part of this paper, we consider the problem of fill-in of nonnegative scalar curvature (NNSC) metrics for a triple of Bartnik data $$(\varSigma ,\gamma ,H)$$ . We prove that given a metric $$\gamma $$ on $${{\mathbf {S}}}^{n-1}$$ ( $$3\le n\le 7$$ ), $$({{\mathbf {S}}}^{n-1},\gamma ,H)$$ admits no fill-in of NNSC metrics provided the prescribed mean curvature H is large enough (Theorem 4). Moreover, we prove that if $$\gamma $$ is a positive scalar curvature (PSC) metric isotopic to the standard metric on $${{\mathbf {S}}}^{n-1}$$ , then the much weaker condition that the total mean curvature $$\int _{{{\mathbf {S}}}^{n-1}}H\,{{\mathrm {d}}}\mu _\gamma $$ is large enough rules out NNSC fill-ins, giving an partially affirmative answer to a conjecture by Gromov (Four lectures on scalar curvature, 2019, see P. 23). In the second part of this paper, we investigate the $$\theta $$ -invariant of Bartnik data and obtain some sufficient conditions for the existence of PSC fill-ins.
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- 2020
33. Washington units, semispecial units, and annihilation of class groups
- Author
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Radan Kučera and Cornelius Greither
- Subjects
Discrete mathematics ,Class (set theory) ,Group (mathematics) ,Generalization ,Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,Field (mathematics) ,Algebraic geometry ,01 natural sciences ,Number theory ,0103 physical sciences ,Genus field ,010307 mathematical physics ,0101 mathematics ,Abelian group ,Mathematics - Abstract
Special units are a sort of predecessor of Euler systems, and they are mainly used to obtain annihilators for class groups. So one is interested in finding as many special units as possible (actually we use a technical generalization called “semispecial”). In this paper we show that in any abelian field having a real genus field in the narrow sense all Washington units are semispecial, and that a slightly weaker statement holds true for all abelian fields. The group of Washington units is very often larger than Sinnott’s group of cyclotomic units. In a companion paper we will show that in concrete families of abelian fields the group of Washington units is much larger than that of Sinnott units, by giving lower bounds on the index. Combining this with the present paper gives strong annihilation results.
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- 2020
34. Mappings Preserving Relations Definable by Linear Order
- Author
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A. L. Semenov
- Subjects
Pure mathematics ,Current (mathematics) ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Order (group theory) ,010307 mathematical physics ,0101 mathematics ,Relation (history of concept) ,01 natural sciences ,Injective function ,Mathematics - Abstract
The relations ‘‘between’’, ‘‘cycle’’, and ‘‘separation’’ were defined through the relation of linear order in the classical paper of Edward V. Huntington. In the current paper, the criteria for preserving these relations under injective mappings are obtained.
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- 2020
35. THE MINIMAL MODULAR FORM ON QUATERNIONIC
- Author
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Aaron Pollack
- Subjects
Algebra ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Modular form ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Suppose that $G$ is a simple reductive group over $\mathbf{Q}$, with an exceptional Dynkin type and with $G(\mathbf{R})$ quaternionic (in the sense of Gross–Wallach). In a previous paper, we gave an explicit form of the Fourier expansion of modular forms on $G$ along the unipotent radical of the Heisenberg parabolic. In this paper, we give the Fourier expansion of the minimal modular form $\unicode[STIX]{x1D703}_{Gan}$ on quaternionic $E_{8}$ and some applications. The $Sym^{8}(V_{2})$-valued automorphic function $\unicode[STIX]{x1D703}_{Gan}$ is a weight 4, level one modular form on $E_{8}$, which has been studied by Gan. The applications we give are the construction of special modular forms on quaternionic $E_{7},E_{6}$ and $G_{2}$. We also discuss a family of degenerate Heisenberg Eisenstein series on the groups $G$, which may be thought of as an analogue to the quaternionic exceptional groups of the holomorphic Siegel Eisenstein series on the groups $\operatorname{GSp}_{2n}$.
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- 2020
36. Hyperbolicity and Uniformity of Varieties of Log General type
- Author
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Amos Turchet, Kristin DeVleming, Kenneth Ascher, Ascher, Kenneth, Devleming, Kristin, and Turchet, Amos
- Subjects
Pure mathematics ,Conjecture ,Mathematics - Number Theory ,Generalization ,General Mathematics ,010102 general mathematics ,Type (model theory) ,01 natural sciences ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,0103 physical sciences ,FOS: Mathematics ,Sheaf ,Trigonometric functions ,Uniform boundedness ,Cotangent bundle ,Number Theory (math.NT) ,010307 mathematical physics ,0101 mathematics ,Variety (universal algebra) ,Algebraic Geometry (math.AG) ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
Projective varieties with ample cotangent bundle satisfy many notions of hyperbolicity, and one goal of this paper is to discuss generalizations to quasi-projective varieties. A major hurdle is that the naive generalization fails, i.e. the log cotangent bundle is never ample. Instead, we define a notion called almost ample which roughly asks that the log cotangent is as positive as possible. We show that all subvarieties of a quasi-projective variety with almost ample log cotangent bundle are of log general type. In addition, if one assumes globally generated then we obtain that such varieties contain finitely many integral points. In another direction, we show that the Lang-Vojta conjecture implies the number of stably integral points on curves of log general type, and surfaces of log general type with almost ample log cotangent sheaf are uniformly bounded., v5: exposition greatly improved. Previous section on function fields removed, to be expanded upon in a future paper. To appear in IMRN
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- 2020
37. Nψ,ϕ-type Quotient Modules over the Bidisk
- Author
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Chang Hui Wu and Tao Yu
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Essential spectrum ,Hardy space ,Characterization (mathematics) ,Type (model theory) ,01 natural sciences ,symbols.namesake ,Compact space ,Compression (functional analysis) ,0103 physical sciences ,Quotient module ,symbols ,010307 mathematical physics ,0101 mathematics ,Quotient ,Mathematics - Abstract
Let H2(ⅅ2) be the Hardy space over the bidisk ⅅ2, and let Mψ,ϕ = [(ψ(z) − ϕ(w))2] be the submodule generated by (ψ(z) − ϕ(w))2, where ψ(z) and ϕ(w) are nonconstant inner functions. The related quotient module is denoted by Nψ,ϕ = H2(ⅅ2) ⊖ Mψ,ϕ. In this paper, we give a complete characterization for the essential normality of Nψ,ϕ. In particular, if ψ(z)= z, we simply write Mψ,ϕ and Nψ,ϕ as Mϕ and Nϕ respectively. This paper also studies compactness of evaluation operators L(0)∣nϕ and R(0)ϕnϕ, essential spectrum of compression operator Sz on Nϕ, essential normality of compression operators Sz and Sw on Nϕ.
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- 2020
38. Low dimensional orders of finite representation type
- Author
-
Daniel Chan and Colin Ingalls
- Subjects
Ring (mathematics) ,Plane curve ,Root of unity ,General Mathematics ,010102 general mathematics ,14E16 ,Local ring ,Order (ring theory) ,Mathematics - Rings and Algebras ,Type (model theory) ,01 natural sciences ,Noncommutative geometry ,Combinatorics ,Minimal model program ,Mathematics - Algebraic Geometry ,Rings and Algebras (math.RA) ,Mathematics - Quantum Algebra ,0103 physical sciences ,FOS: Mathematics ,Quantum Algebra (math.QA) ,010307 mathematical physics ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
In this paper, we study noncommutative surface singularities arising from orders. The singularities we study are mild in the sense that they have finite representation type or, equivalently, are log terminal in the sense of the Mori minimal model program for orders (Chan and Ingalls in Invent Math 161(2):427–452, 2005). These were classified independently by Artin (in terms of ramification data) and Reiten–Van den Bergh (in terms of their AR-quivers). The first main goal of this paper is to connect these two classifications, by going through the finite subgroups $$G \subset {{{\,\mathrm{GL}\,}}_2}$$ , explicitly computing $$H^2(G,k^*)$$ , and then matching these up with Artin’s list of ramification data and Reiten–Van den Bergh’s AR-quivers. This provides a semi-independent proof of their classifications and extends the study of canonical orders in Chan et al. (Proc Lond Math Soc (3) 98(1):83–115, 2009) to the case of log terminal orders. A secondary goal of this paper is to study noncommutative analogues of plane curves which arise as follows. Let $$B = k_{\zeta } \llbracket x,y \rrbracket $$ be the skew power series ring where $$\zeta $$ is a root of unity, or more generally a terminal order over a complete local ring. We consider rings of the form $$A = B/(f)$$ where $$f \in Z(B)$$ which we interpret to be the ring of functions on a noncommutative plane curve. We classify those noncommutative plane curves which are of finite representation type and compute their AR-quivers.
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- 2020
39. Sobolev regular solutions for the incompressible Navier–Stokes equations in higher dimensions: asymptotics and representation formulae
- Author
-
Weiping Yan and Vicenţiu D. Rădulescu
- Subjects
Pure mathematics ,Regular polyhedron ,General Mathematics ,010102 general mathematics ,Dimension (graph theory) ,01 natural sciences ,Domain (mathematical analysis) ,Sobolev space ,Bounded function ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Asymptotic expansion ,Representation (mathematics) ,Navier–Stokes equations ,Mathematics - Abstract
In this paper, we consider the steady incompressible Navier–Stokes equations in a smooth bounded domain $$\Omega \subset \mathbb R^n$$ Ω ⊂ R n with the dimension $$n\ge 3$$ n ≥ 3 . We first establish asymptotic expansion formulae of Sobolev regular finite energy solutions in $$\Omega$$ Ω . In the second part of this paper, explicit representation formulae of Sobolev regular solutions are showed in the regular polyhedron $$\Omega :=[0,T]^n$$ Ω : = [ 0 , T ] n .
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- 2020
40. More about singular traces on simply generated operator ideals
- Author
-
Albrecht Pietsch
- Subjects
Large class ,Sequence ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Hilbert space ,Extension (predicate logic) ,Space (mathematics) ,01 natural sciences ,symbols.namesake ,Operator (computer programming) ,0103 physical sciences ,symbols ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
During half a century, singular traces on ideals of Hilbert space operators have been constructed by looking for linear forms on associated sequence ideals. Only recently, the author was able to eliminate this auxiliary step by directly applying Banach’s version of the extension theorem; see (Integral Equ. Oper. Theory 91, 21, 2019 and 92, 7, 2020). Of course, the relationship between the new approach and the older ones must be investigated. In the first paper, this was done for $${\mathfrak {L}}_{1,\infty } (H)$$ . To save space, such considerations were postponed in the second paper, which deals with a large class of principal ideals, called simply generated. This omission will now be rectified.
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- 2020
41. On some universal Morse–Sard type theorems
- Author
-
Alba Roviello, Adele Ferone, Mikhail V. Korobkov, Ferone, A., Korobkov, M. V., and Roviello, A.
- Subjects
Uncertainty principle ,Dubovitskii-Federer theorems ,Near critical ,Morse-Sard theorem ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Algebraic geometry ,Morse code ,Sobolev-Lorentz mapping ,Holder mapping ,01 natural sciences ,law.invention ,Sobolev space ,Combinatorics ,law ,0103 physical sciences ,010307 mathematical physics ,Differentiable function ,Bessel potential space ,0101 mathematics ,Critical set ,Mathematics - Abstract
The classical Morse–Sard theorem claims that for a mapping v : R n → R m + 1 of class C k the measure of critical values v ( Z v , m ) is zero under condition k ≥ n − m . Here the critical set, or m-critical set is defined as Z v , m = { x ∈ R n : rank ∇ v ( x ) ≤ m } . Further Dubovitskiĭ in 1957 and independently Federer and Dubovitskiĭ in 1967 found some elegant extensions of this theorem to the case of other (e.g., lower) smoothness assumptions. They also established the sharpness of their results within the C k category. Here we formulate and prove a bridge theorem that includes all the above results as particular cases: namely, if a function v : R n → R d belongs to the Holder class C k , α , 0 ≤ α ≤ 1 , then for every q > m the identity H μ ( Z v , m ∩ v − 1 ( y ) ) = 0 holds for H q -almost all y ∈ R d , where μ = n − m − ( k + α ) ( q − m ) . Intuitively, the sense of this bridge theorem is very close to Heisenberg's uncertainty principle in theoretical physics: the more precise is the information we receive on measure of the image of the critical set, the less precisely the preimages are described, and vice versa. The result is new even for the classical C k -case (when α = 0 ); similar result is established for the Sobolev classes of mappings W p k ( R n , R d ) with minimal integrability assumptions p = max ( 1 , n / k ) , i.e., it guarantees in general only the continuity (not everywhere differentiability) of a mapping. However, using some N-properties for Sobolev mappings, established in our previous paper, we obtained that the sets of nondifferentiability points of Sobolev mappings are fortunately negligible in the above bridge theorem. We cover also the case of fractional Sobolev spaces. The proofs of the most results are based on our previous joint papers with J. Bourgain and J. Kristensen (2013, 2015). We also crucially use very deep Y. Yomdin's entropy estimates of near critical values for polynomials (based on algebraic geometry tools).
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- 2020
42. The regularity theory for the parabolic double obstacle problem
- Author
-
Jinwan Park and Ki-Ahm Lee
- Subjects
Operator (computer programming) ,General Mathematics ,Obstacle ,010102 general mathematics ,0103 physical sciences ,Obstacle problem ,Mathematical analysis ,Boundary (topology) ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Nonlinear operators ,Mathematics - Abstract
In this paper, we study the regularity of the free boundaries of the parabolic double obstacle problem for the heat operator and fully nonlinear operator. The result in this paper are generalizations of the theory for the elliptic problem in Lee et al. (Calc Var Partial Differ Equ 58(3):104, 2019) and Lee and Park (The regularity theory for the double obstacle problem for fully nonlinear operator, , 2018) to parabolic case and also the theory for the parabolic single obstacle problem in Caffarelli et al. (J Am Math Soc 17(4):827–869, 2004) to double obstacle case. New difficulties in the theory which are generated by the characteristic of parabolic PDEs and the existence of the upper obstacle are discussed in detail. Furthermore, the thickness assumptions to have the regularity of the free boundary are carefully considered.
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- 2020
43. Masur–Veech volumes, frequencies of simple closed geodesics, and intersection numbers of moduli spaces of curves
- Author
-
Vincent Delecroix, Elise Goujard, Peter Zograf, Anton Zorich, Groupe Sociétés, Religions, Laïcités (GSRL), Centre National de la Recherche Scientifique (CNRS)-École pratique des hautes études (EPHE), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), École pratique des hautes études (EPHE), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), École Pratique des Hautes Études (EPHE), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), and ANR-19-CE40-0021,Phymath,physique mathématique(2019)
- Subjects
Teichmüller space ,Surface (mathematics) ,Pure mathematics ,Geodesic ,General Mathematics ,[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] ,Dynamical Systems (math.DS) ,Algebraic geometry ,01 natural sciences ,Mathematics - Geometric Topology ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] ,[MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT] ,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] ,0103 physical sciences ,FOS: Mathematics ,Mathematics - Combinatorics ,Mathematics - Dynamical Systems ,0101 mathematics ,Algebraic Geometry (math.AG) ,Quadratic differential ,Mathematics ,Meromorphic function ,010102 general mathematics ,Geometric Topology (math.GT) ,Mathematics::Geometric Topology ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,Mapping class group ,Moduli space ,[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] ,Combinatorics (math.CO) ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,010307 mathematical physics - Abstract
We express the Masur-Veech volume and the area Siegel-Veech constant of the moduli space $\mathcal{Q}_{g,n}$ of genus $g$ meromorphic quadratic differentials with $n$ simple poles as polynomials in the intersection numbers of $\psi$-classes with explicit rational coefficients. The formulae obtained in this article result from lattice point counts involving the Kontsevich volume polynomials that also appear in Mirzakhani's recursion for the Weil-Petersson volumes of the moduli spaces of bordered hyperbolic surfaces with geodesic boundaries. A similar formula for the Masur-Veech volume (though without explicit evaluation) was obtained earlier by Mirzakhani via completely different approach. Furthermore, we prove that the density of the mapping class group orbit of any simple closed multicurve $\gamma$ inside the ambient set of integral measured laminations computed by Mirzakhani coincides with the density of square-tiled surfaces having horizontal cylinder decomposition associated to $\gamma$ among all square-tiled surfaces in $\mathcal{Q}_{g,n}$. We study the resulting densities (or, equivalently, volume contributions) in more detail in the special case $n=0$. In particular, we compute the asymptotic frequencies of separating and non-separating simple closed geodesics on a closed hyperbolic surface of genus $g$ for small $g$ and we show that for large genera the separating closed geodesics are $\sqrt{\frac{2}{3\pi g}}\cdot\frac{1}{4^g}$ times less frequent., Comment: The current paper (as well as the companion paper arXiv:2007.04740) has grown from arxiv:1908.08611. The conjectures stated in arXiv:1908.08611 are proved by A. Aggarwal in arXiv:2004.05042
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- 2021
44. Vector-valued q-variational inequalities for averaging operators and the Hilbert transform
- Author
-
Tao Ma, Wei Liu, and Guixiang Hong
- Subjects
Mathematics::Functional Analysis ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Banach space ,01 natural sciences ,symbols.namesake ,0103 physical sciences ,Variational inequality ,symbols ,010307 mathematical physics ,Hilbert transform ,0101 mathematics ,Martingale (probability theory) ,Mathematics - Abstract
Recently, the authors have established $$L^p$$ -boundedness of vector-valued q-variational inequalities for averaging operators which take values in the Banach space satisfying the martingale cotype q property in Hong and Ma (Math Z 286(1–2):89–120, 2017). In this paper, we prove that the martingale cotype q property is also necessary for the vector-valued q-variational inequalities, which was a question left open in the previous paper. Moreover, we also prove that the UMD property and the martingale cotype q property can be characterized in terms of vector valued q-variational inequalities for the Hilbert transform.
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- 2020
45. Bernoulliness of when is an irrational rotation: towards an explicit isomorphism
- Author
-
Christophe Leuridan
- Subjects
Rational number ,Lebesgue measure ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Diophantine approximation ,01 natural sciences ,Irrational rotation ,Combinatorics ,0103 physical sciences ,010307 mathematical physics ,Bernoulli scheme ,Isomorphism ,0101 mathematics ,Real number ,Unit interval ,Mathematics - Abstract
Let $\unicode[STIX]{x1D703}$ be an irrational real number. The map $T_{\unicode[STIX]{x1D703}}:y\mapsto (y+\unicode[STIX]{x1D703})\!\hspace{0.6em}{\rm mod}\hspace{0.2em}1$ from the unit interval $\mathbf{I}= [\!0,1\![$ (endowed with the Lebesgue measure) to itself is ergodic. In a short paper [Parry, Automorphisms of the Bernoulli endomorphism and a class of skew-products. Ergod. Th. & Dynam. Sys.16 (1996), 519–529] published in 1996, Parry provided an explicit isomorphism between the measure-preserving map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ and the unilateral dyadic Bernoulli shift when $\unicode[STIX]{x1D703}$ is extremely well approximated by the rational numbers, namely, if $$\begin{eqnarray}\inf _{q\geq 1}q^{4}4^{q^{2}}~\text{dist}(\unicode[STIX]{x1D703},q^{-1}\mathbb{Z})=0.\end{eqnarray}$$ A few years later, Hoffman and Rudolph [Uniform endomorphisms which are isomorphic to a Bernoulli shift. Ann. of Math. (2)156 (2002), 79–101] showed that for every irrational number, the measure-preserving map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ is isomorphic to the unilateral dyadic Bernoulli shift. Their proof is not constructive. In the present paper, we relax notably Parry’s condition on $\unicode[STIX]{x1D703}$: the explicit map provided by Parry’s method is an isomorphism between the map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ and the unilateral dyadic Bernoulli shift whenever $$\begin{eqnarray}\inf _{q\geq 1}q^{4}~\text{dist}(\unicode[STIX]{x1D703},q^{-1}\mathbb{Z})=0.\end{eqnarray}$$ This condition can be relaxed again into $$\begin{eqnarray}\inf _{n\geq 1}q_{n}^{3}~(a_{1}+\cdots +a_{n})~|q_{n}\unicode[STIX]{x1D703}-p_{n}| where $[0;a_{1},a_{2},\ldots ]$ is the continued fraction expansion and $(p_{n}/q_{n})_{n\geq 0}$ the sequence of convergents of $\Vert \unicode[STIX]{x1D703}\Vert :=\text{dist}(\unicode[STIX]{x1D703},\mathbb{Z})$. Whether Parry’s map is an isomorphism for every $\unicode[STIX]{x1D703}$ or not is still an open question, although we expect a positive answer.
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- 2020
46. On Counting Certain Abelian Varieties Over Finite Fields
- Author
-
Chia-Fu Yu and Jiangwei Xue
- Subjects
Isogeny ,Pure mathematics ,Class (set theory) ,Current (mathematics) ,Mathematics - Number Theory ,Series (mathematics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Connection (mathematics) ,Finite field ,Simple (abstract algebra) ,0103 physical sciences ,FOS: Mathematics ,Number Theory (math.NT) ,010307 mathematical physics ,0101 mathematics ,Abelian group ,Mathematics - Abstract
This paper contains two parts toward studying abelian varieties from the classification point of view. In a series of papers, the current authors and T.-C. Yang obtain explicit formulas for the numbers of superspecial abelian surfaces over finite fields. In this paper, we give an explicit formula for the size of the isogeny class of simple abelian surfaces with real Weil number $\sqrt{q}$. This establishes a key step that one may extend our previous explicit calculations of superspecial abelian surfaces to those of supersingular abelian surfaces.The second part is to introduce the notion of genera and ideal complexes of abelian varieties with additional structures in a general setting. The purpose is to generalize the results of Yu on abelian varieties with additional structures to similitude classes, which establishes more results on the connection between geometrically defined and arithmetically defined masses for further investigation., Comment: 23 pages. Section 5.4 corrected
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- 2020
47. Discrete series multiplicities for classical groups over $\mathbf {Z}$ and level 1 algebraic cusp forms
- Author
-
Olivier Taïbi and Gaëtan Chenevier
- Subjects
Classical group ,Pure mathematics ,Discrete series representation ,General Mathematics ,Computation ,010102 general mathematics ,Automorphic form ,Multiplicity (mathematics) ,01 natural sciences ,Number theory ,0103 physical sciences ,Test functions for optimization ,010307 mathematical physics ,0101 mathematics ,Algebraic number ,Mathematics - Abstract
The aim of this paper is twofold. First, we introduce a new method for evaluating the multiplicity of a given discrete series representation in the space of level 1 automorphic forms of a split classical group $G$ over $\mathbf {Z}$ , and provide numerical applications in absolute rank $\leq 8$ . Second, we prove a classification result for the level one cuspidal algebraic automorphic representations of $\mathrm{GL}_{n}$ over $\mathbf {Q}$ ( $n$ arbitrary) whose motivic weight is $\leq 24$ . In both cases, a key ingredient is a classical method based on the Weil explicit formula, which allows to disprove the existence of certain level one algebraic cusp forms on $\mathrm{GL}_{n}$ , and that we push further on in this paper. We use these vanishing results to obtain an arguably “effortless” computation of the elliptic part of the geometric side of the trace formula of $G$ , for an appropriate test function. Thoses results have consequences for the computation of the dimension of the spaces of (possibly vector-valued) Siegel modular cuspforms for $\mathrm{Sp}_{2g}(\mathbf {Z})$ : we recover all the previously known cases without relying on any, and go further, by a unified and “effortless” method.
- Published
- 2020
48. Remarks on the geodesic-Einstein metrics of a relative ample line bundle
- Author
-
Xueyuan Wan and Xu Wang
- Subjects
Ample line bundle ,Pure mathematics ,Geodesic ,General Mathematics ,010102 general mathematics ,Holomorphic function ,Fibration ,Type (model theory) ,01 natural sciences ,Mathematics::Algebraic Geometry ,Flow (mathematics) ,Bounded function ,Bundle ,0103 physical sciences ,Mathematics::Differential Geometry ,010307 mathematical physics ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
In this paper, we introduce the associated geodesic-Einstein flow for a relative ample line bundle L over the total space $$\mathcal {X}$$ of a holomorphic fibration and obtain a few properties of that flow. In particular, we prove that the pair $$(\mathcal {X}, L)$$ is nonlinear semistable if the associated Donaldson type functional is bounded from below and the geodesic-Einstein flow has long-time existence property. We also define the associated S-classes and C-classes for $$(\mathcal {X}, L)$$ and obtain two inequalities between them when L admits a geodesic-Einstein metric. Finally, in the appendix of this paper, we prove that a relative ample line bundle is geodesic-Einstein if and only if an associated infinite rank bundle is Hermitian–Einstein.
- Published
- 2020
49. Rectifying and Osculating Curves on a Smooth Surface
- Author
-
Absos Ali Shaikh and Pinaki Ranjan Ghosh
- Subjects
Applied Mathematics ,General Mathematics ,Numerical analysis ,010102 general mathematics ,Mathematical analysis ,Osculating curve ,01 natural sciences ,Smooth surface ,0103 physical sciences ,Mathematics::Metric Geometry ,Mathematics::Differential Geometry ,010307 mathematical physics ,Tangent vector ,0101 mathematics ,Invariant (mathematics) ,Mathematics ,Geodesic curvature ,Osculating circle - Abstract
The main motive of the paper is to look on rectifying and osculating curves on a smooth surface. In this paper we find the normal and geodesic curvature for a rectifying curve on a smooth surface and we also prove that geodesic curvature is invariant under the isometry of surfaces such that rectifying curves remain. We find a sufficient condition for which an osculating curve on a smooth surface remains invariant under isometry of surfaces and also we prove that the component of the position vector of an osculating curve α(s) on a smooth surface along any tangent vector to the surface at α(s) is invariant under such isometry.
- Published
- 2020
50. SNC Log Symplectic Structures on Fano Products
- Author
-
Katsuhiko Okumura
- Subjects
Pure mathematics ,Mathematics::Algebraic Geometry ,General Mathematics ,Poisson manifold ,010102 general mathematics ,0103 physical sciences ,Projective space ,010307 mathematical physics ,Fano plane ,0101 mathematics ,01 natural sciences ,Symplectic geometry ,Mathematics - Abstract
This paper classifies Poisson structures with the reduced simple normal crossing divisor on a product of Fano varieties of Picard number 1. The characterization of even-dimensional projective spaces from the viewpoint of Poisson structures is given by Lima and Pereira. In this paper, we generalize the characterization of projective spaces to any dimension.
- Published
- 2020
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