13,935 results
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2. Chern's contribution to the Hopf problem: An exposition based on Bryant's paper
- Author
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Markus Upmeier and Aleksy Tralle
- Subjects
Mathematics - Differential Geometry ,media_common.quotation_subject ,010102 general mathematics ,01 natural sciences ,Algebra ,Differential Geometry (math.DG) ,Computational Theory and Mathematics ,Originality ,0103 physical sciences ,FOS: Mathematics ,53C ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Analysis ,Differential (mathematics) ,Mathematics ,Exposition (narrative) ,media_common - Abstract
We give a comprehensive account of Chern's Theorem that S^6 admits no omega-compatible almost complex structures. No claim to originality is being made, as the paper is mostly an expanded version of material already in the literature. This article extends the talks that both authors gave in Marburg during the conference "(Non)existence of complex structures on S^6" in April 2017., Misprints corrected
- Published
- 2018
3. An unpublished paper ‘Über einige durch unendliche Reihen definirte Functionen eines complexen Argumentes’ by Adolf Hurwitz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 However, already in 1889 (or even earlier) Hurwitz was aware of these results as his mathematical diaries and some unpublished notes (in an almost final form) found in his estate at the ETH Zurich show.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
4. 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
5. 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
6. A note on the paper 'Norm inequalities in operator ideals' [J. Funct. Anal. 255 (11) (2008), 3208–3228] by G. Larotonda
- Author
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Milan Lazarević, Danko R. Jocić, and Đorđe Krtinić
- Subjects
Algebra ,Inequality ,Norm (mathematics) ,media_common.quotation_subject ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Analysis ,Mathematics ,media_common - Abstract
In this note we show that the presented proof of [1, th. 12] , being based on a false statement appearing in this proof, is not viable for all of the proclaimed values of the involved parameters. We also determine necessary and sufficient conditions for those parameters, which provides that the considered statement, and therefore [1, th. 12] itself, still remains valid.
- Published
- 2019
7. Revisit to Fritz John’s paper on the blow-up of nonlinear wave equations
- Author
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Xin Yang and Zhengfang Zhou
- Subjects
Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Type inequality ,35L70 ,Wave equation ,01 natural sciences ,Physics::History of Physics ,Mathematics - Analysis of PDEs ,Nonlinear wave equation ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
In Fritz John's famous paper (1979), he discovered that for the wave equation $\Box u=|u|^p$, where $1, Comment: 11 pages, 3 figures
- Published
- 2016
8. '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
9. On a paper of Beltrán and Shao about coprime action
- Author
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H. Meng and Adolfo Ballester-Bolinches
- Subjects
Algebra and Number Theory ,Coprime integers ,Mathematics::Number Theory ,010102 general mathematics ,Structure (category theory) ,Automorphism ,01 natural sciences ,Prime (order theory) ,Action (physics) ,Combinatorics ,Mathematics::Group Theory ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
Assume that A and G are finite groups of coprime orders such that A acts on G via automorphisms. Let p be a prime. The following coprime action version of a well-known theorem of Ito about the structure of a minimal non-p-nilpotent groups is proved: if every maximal A-invariant subgroup of G is p-nilpotent, then G is p-soluble. If, moreover, G is not p-nilpotent, then G must be soluble. Some earlier results about coprime action are consequences of this theorem.
- Published
- 2020
10. 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
11. A correction of the decomposability result in a paper by Meyer-Neutsch
- Author
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Vladimir G. Tkachev
- Subjects
Pure mathematics ,Group Theory (math.GR) ,Commutative Algebra (math.AC) ,01 natural sciences ,Griess algebra ,Commutative nonassociative algebras ,Associative bilinear form ,0103 physical sciences ,17A99, 17C27, 20D08 ,FOS: Mathematics ,Metrised algebras ,0101 mathematics ,Commutative property ,Associative property ,Algebra och logik ,Mathematics ,Idempotents ,Algebra and Number Theory ,Mathematics::Rings and Algebras ,010102 general mathematics ,Subalgebra ,Mathematics - Rings and Algebras ,Mathematics - Commutative Algebra ,Algebra and Logic ,Linear map ,Rings and Algebras (math.RA) ,Product (mathematics) ,Idempotence ,010307 mathematical physics ,Indecomposable module ,Mathematics - Group Theory - Abstract
In this short note, it is shown that there is a gap in the proof of Theorem 11 in the paper of Meyer and Neutsch (J. of Algebra, 1993). We prove, nevertheless, that the statement of the theorem is true and fix the proof by using a certain extremal property of idempotents which has an independent interest., 6 pages, submitted 2017
- Published
- 2018
12. A note on O. Frolkina’s paper 'pairwise disjoint moebius bands in space'
- Author
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Sergey A. Melikhov
- Subjects
Algebra and Number Theory ,010102 general mathematics ,Disjoint sets ,Space (mathematics) ,01 natural sciences ,Combinatorics ,symbols.namesake ,Simple (abstract algebra) ,0103 physical sciences ,symbols ,010307 mathematical physics ,Möbius strip ,0101 mathematics ,Algebraic number ,Mathematics - Abstract
We note a simple algebraic proof of Frolkina’s result that [Formula: see text] does not contain uncountably many pairwise disjoint copies of the Möbius band, and of a similar result in higher dimensions.
- Published
- 2019
13. A remark on our paper 'Negative Holomorphic curvature and positive canonical bundle'
- Author
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Shing-Tung Yau and Damin Wu
- Subjects
Statistics and Probability ,Mathematics - Differential Geometry ,Pure mathematics ,Conjecture ,010102 general mathematics ,Holomorphic function ,Curvature ,Mathematical proof ,01 natural sciences ,Canonical bundle ,Continuation ,Differential Geometry (math.DG) ,Argument ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Geometry and Topology ,Sectional curvature ,0101 mathematics ,Statistics, Probability and Uncertainty ,Analysis ,Mathematics - Abstract
This is a continuation of our first paper in [WY16]. There are two purposes of this paper: One is to give a proof of the main result in [WY16] without going through the argument depending on numerical effectiveness. The other one is to provide a proof of our conjecture, mentioned in [TY], where the assumption of negative holomorphic sectional curvature is dropped to quasi-negative. We should note that a solution to our conjecture is also provided by Diverio-Trapani [DT]. Both proofs depend on our argument in [WY16]. But our argument here makes use of the argument given by the second author and Cheng in [CY75].
- Published
- 2016
14. Corrigendum to the paper 'On the K^2 of degenerations of surfaces and the multiple point formula'
- Author
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Alberto Calabri, Ciro Ciliberto, Richard Miranda, and Flaminio Flamini
- Subjects
Pure mathematics ,deformations, degenerations, enumerative problems, singularities ,deformations ,010102 general mathematics ,Minor (linear algebra) ,Socio-culturale ,01 natural sciences ,degenerations ,Combinatorics ,Multiple point ,Section (fiber bundle) ,Mathematics (miscellaneous) ,0103 physical sciences ,enumerative problems ,Gravitational singularity ,010307 mathematical physics ,Settore MAT/03 - Geometria ,0101 mathematics ,Statistics, Probability and Uncertainty ,singularities ,Mathematics - Abstract
We correct an error in the Multiple Point Formula (7.3) in the paper mentioned in the title. This correction propagates to formulas (7.5), (7.6), (7.23) and (8.18), and it affects minor results in Section 8, where few statements require an extra assumption, but it does not affect the main results of Section 8.
- Published
- 2017
15. A commentary on Teichmüller’s paper Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Flächen
- Author
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Athanase Papadopoulos, Annette A'Campo-Neuen, Norbert A'Campo, and Vincent Alberge
- Subjects
Fuchsian group ,Teichmüller space ,Quasiconformal mapping ,Pure mathematics ,Euclidean space ,Riemann surface ,010102 general mathematics ,Surface (topology) ,Mathematics::Geometric Topology ,01 natural sciences ,Algebra ,symbols.namesake ,Invariance of domain ,0103 physical sciences ,Uniformization theorem ,symbols ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
This is a mathematical commentary on Teichmuller's paper ``Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Flachen" (Determination of extremal quasiconformal maps of closed oriented Riemann surfaces). This paper is among the last (and may be the last one) that Teichmuller wrote on the theory of moduli. It contains the proof of the so-called Teichmuller existence theorem for a closed surface of genus at least 2. For this proof, the author defines a mapping between a space of equivalence classes of marked Riemann surfaces (the Teichmuller space) and a space of equivalence classes of certain Fuchsian groups (the so-called Fricke space). After that, he defines a map between the latter and the Euclidean space of dimension 6g-6 Using Brouwer's theorem of invariance of domain, he shows that this map is a homeomorphism. This involves in particular a careful definition of the topologies of Fricke space, the computation of its dimension, and comparison results between hyperbolic distance and quasiconformal dilatation. The use of the invariance of domain theorem is in the spirit of Poincare and Klein's use of the so-called ``continuity principle" in their attempts to prove the uniformization theorem.
- Published
- 2016
16. Notes on Perelman’s papers
- Author
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Bruce Kleiner and John Lott
- Subjects
Mathematics - Differential Geometry ,three-manifold ,geometrization theorem ,53C21 ,Mathematical proof ,57M40 ,53C44 ,01 natural sciences ,Physics::Fluid Dynamics ,Perelman ,symbols.namesake ,Ricci flow ,0103 physical sciences ,FOS: Mathematics ,Mathematics::Metric Geometry ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics ,Poincaré Conjecture ,Discrete mathematics ,Ricci flow with surgery ,Conjecture ,010102 general mathematics ,Hyperbolization theorem ,Mathematics::Geometric Topology ,57M50 ,Manifold ,long-term behaviour ,Differential Geometry (math.DG) ,Cover (topology) ,Poincaré conjecture ,symbols ,Mathematics::Differential Geometry ,010307 mathematical physics ,Geometry and Topology ,Geometrization conjecture ,entropy formula - Abstract
These are detailed notes on Perelman's papers "The entropy formula for the Ricci flow and its geometric applications" and "Ricci flow with surgery on three-manifolds"., 216 pages, minor corrections made
- Published
- 2008
17. Cantor sets of arcs in decomposable local Siegel disk boundaries☆☆Portions of this paper were presented by the first author at the SE Sectional Meeting of the AMS (Gainesville, FL, March 1999), and at the Spring Topology and Dynamics Conference (Salt Lake City, UT, March 1999)
- Author
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Andrew O. Maner, Lex Oversteegen, and John C. Mayer
- Subjects
Discrete mathematics ,Pure mathematics ,Mathematics::Dynamical Systems ,Decomposable continuum ,010102 general mathematics ,Mathematics::General Topology ,u-invariant ,Conformal map ,01 natural sciences ,Cantor set ,Siegel disk ,Monotone polygon ,Bounded function ,0103 physical sciences ,Embedding ,Interval (graph theory) ,Point (geometry) ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Tranche ,Mathematics - Abstract
In this paper we construct a family of circle-like continua, each admitting a finest monotone map onto S 1 such that there exists a subset of point inverses which is homeomorphic to the Cantor set cross an interval. We then show how to realize some members of this family as the boundaries ∂U of bounded irreducible local Siegel disks U . These boundaries are geometrically rigid in the following sense: there exist arbitrarily small periodic homeomorphisms of the sphere, conformal on U , which keep U invariant. The embedding portion of this paper follows a flexible construction of Herman. These results provide a partial answer to a question of Rogers and a complete answer to a question of Brechner, Guay, and Mayer.
- Published
- 2001
18. 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
19. Recurrent critical points and typical limit sets for conformal measures☆☆Portions of the paper were presented at the AMS Special Session on Low-Dimensional Dynamics in Milwaukee, Wisconsin, October 1997
- Author
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Lex Oversteegen, Alexander Blokh, and John C. Mayer
- Subjects
Discrete mathematics ,010102 general mathematics ,Julia set ,Conformal map ,Density theorem ,Lebesgue integration ,Postcritical set ,01 natural sciences ,Measure (mathematics) ,symbols.namesake ,0103 physical sciences ,symbols ,Complex dynamics ,Ergodic theory ,ω-limit set ,Point (geometry) ,010307 mathematical physics ,Geometry and Topology ,Limit (mathematics) ,0101 mathematics ,Conformal measure ,Mathematics - Abstract
For a rational f : C → C with a conformal measure μ we show that if there is a subset of the Julia set J(f) of positive μ -measure whose points are not eventual preimages of critical or parabolic points and have limit sets not contained in the union of the limit sets of recurrent critical points, then μ is non-atomic, μ(J(f))=1 , ω(x)=J(f) for μ -a.e. point x∈J(f) and f is conservative, ergodic and exact. The proof uses a version of the Lebesgue Density Theorem valid for Borel measures and conformal balls.
- Published
- 2000
20. Erratum for the paper ‘On the chain-level intersection pairing for PL manifolds’
- Author
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J E McClure
- Subjects
Lemma (mathematics) ,Pure mathematics ,010102 general mathematics ,Partial algebra ,intersection pairing ,01 natural sciences ,Combinatorics ,Chain (algebraic topology) ,Intersection ,18D50 ,Pairing ,0103 physical sciences ,57Q65 ,partial algebra ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,General position ,Mathematics ,Sign (mathematics) ,general position - Abstract
Greg Friedman has pointed out that there are sign errors in our paper ‘On the chain-level intersection pairing for PL manifolds’, linked above, and in particular Lemma 10.5(b) (which is a key step in the proof of the main theorem) is not correct as stated. ¶ The purpose of this note is to provide a correction.
- Published
- 2009
21. Note on a paper by A. Granville and K. Soundararajan
- Author
-
Jie Wu, Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), and Wu, Jie
- Subjects
Discrete mathematics ,Algebra and Number Theory ,Distribution (number theory) ,010102 general mathematics ,Equal probability ,01 natural sciences ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,Combinatorics ,symbols.namesake ,Results on L(1,χ) ,\chi)$ ,0103 physical sciences ,symbols ,Results on $L(1 ,010307 mathematical physics ,11M20 ,0101 mathematics ,Random variable ,Euler product ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] ,Mathematics - Abstract
In this note, we improve some results of Granville and Soundararajan on the distribution of values of the truncated random Euler product L ( 1 , X ; y ) : = ∏ p ⩽ y ( 1 − X ( p ) / p ) −1 , where the X ( p ) are independent random variables, taking the values ±1 with equal probability p / 2 ( p + 1 ) and 0 with probability 1 / ( p + 1 ) .
- Published
- 2007
22. ADDITIONS AND CORRECTIONS TO THE PAPER 'ISOTRIVIAL FAMILIES OF CURVES ON AFFINE SURFACES AND CHARACTERIZATION OF THE AFFINE PLANE'
- Author
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M G Zaĭdenberg, Institut Fourier (IF), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), Institut Fourier (IF ), and Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])
- Subjects
Pure mathematics ,Plane curve ,010102 general mathematics ,Geometry ,General Medicine ,Characterization (mathematics) ,16. Peace & justice ,01 natural sciences ,Affine plane ,Affine involution ,Affine geometry of curves ,0103 physical sciences ,010307 mathematical physics ,Affine transformation ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,0101 mathematics ,ComputingMilieux_MISCELLANEOUS ,Mathematics - Abstract
International audience
- Published
- 1992
23. Quantitative Diophantine approximation with congruence conditions
- Author
-
Anish Ghosh, Shucheng Yu, and Mahbub Alam
- Subjects
Pure mathematics ,Algebra and Number Theory ,Mathematics - Number Theory ,Mathematics::Number Theory ,010102 general mathematics ,Short paper ,Variance (accounting) ,Diophantine approximation ,Space (mathematics) ,01 natural sciences ,Argument ,0103 physical sciences ,FOS: Mathematics ,Congruence (manifolds) ,Number Theory (math.NT) ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
In this short paper we prove a quantitative version of the Khintchine-Groshev Theorem with congruence conditions. Our argument relies on a classical argument of Schmidt on counting generic lattice points, which in turn relies on a certain variance bound on the space of lattices.
- Published
- 2021
24. A Conditional Proof of the Leopoldt Conjecture for CM Fields
- Author
-
Preda Mihailescu
- Subjects
Combinatorics ,Conjecture ,Mathematics::Number Theory ,010102 general mathematics ,0103 physical sciences ,Short paper ,010307 mathematical physics ,0101 mathematics ,Conditional proof ,01 natural sciences ,Mathematics - Abstract
In this short paper, we reduce the proof of the Leopoldt Conjecture to the proof of the fact that Iwasawa’s \(\mu \)-constant vanishes for all CM \(\mathbb {Z}_p\)-extensions, an assumption that will be proved in a separate paper.
- Published
- 2021
25. On a paper by A. I. Lee and J. M. Hill
- Author
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Piero De Mottoni and Roberta Dal Passo
- Subjects
Applied Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Humanities ,Analysis ,Mathematics - Published
- 1985
- Full Text
- View/download PDF
26. Short-Time Heat Content Asymptotics via the Wave and Eikonal Equations
- Author
-
Nathanael Schilling
- Subjects
Eikonal equation ,010102 general mathematics ,Short paper ,Boundary (topology) ,Function (mathematics) ,01 natural sciences ,ddc ,Combinatorics ,Mathematics - Analysis of PDEs ,Differential geometry ,0103 physical sciences ,Content (measure theory) ,FOS: Mathematics ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
In this short paper, we derive an alternative proof for some known (van den Berg & Gilkey 2015) short-time asymptotics of the heat content in a compact full-dimensional submanifolds S with smooth boundary. This includes formulae like $$\begin{aligned} \int _{S} \exp (t\Delta ) (f \mathbb {1}_{S}) \,\mathrm {d}V= \int _S f \,\mathrm {d}V- \sqrt{\frac{t}{\pi }} \int _{\partial S} f \,\mathrm {d}A+ o(\sqrt{t}),\quad t \rightarrow 0^+, \end{aligned}$$ ∫ S exp ( t Δ ) ( f 1 S ) d V = ∫ S f d V - t π ∫ ∂ S f d A + o ( t ) , t → 0 + , and explicit expressions for similar expansions involving other powers of $$\sqrt{t}$$ t . By the same method, we also obtain short-time asymptotics of $$\int _S \exp (t^m\Delta ^m)(f \mathbb {1}_S)\,\mathrm {d}V$$ ∫ S exp ( t m Δ m ) ( f 1 S ) d V , $$m \in \mathbb N$$ m ∈ N , and more generally for one-parameter families of operators $$t \mapsto k(\sqrt{-t\Delta })$$ t ↦ k ( - t Δ ) defined by an even Schwartz function k.
- Published
- 2020
27. 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
28. Ruelle Zeta Function from Field Theory
- Author
-
Charles Hadfield, Santosh Kandel, and Michele Schiavina
- Subjects
Nuclear and High Energy Physics ,FOS: Physical sciences ,Dynamical Systems (math.DS) ,01 natural sciences ,Interpretation (model theory) ,Ruelle zeta function ,symbols.namesake ,0103 physical sciences ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Analytic torsion ,Field theory (psychology) ,Mathematics - Algebraic Topology ,Mathematics - Dynamical Systems ,0101 mathematics ,Mathematics::Symplectic Geometry ,Equivalence (measure theory) ,Mathematical Physics ,Mathematical physics ,Mathematics ,Original Paper ,Partition function (quantum field theory) ,Conjecture ,010102 general mathematics ,37C30, 81T70, 81T45 ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,Mathematics - Symplectic Geometry ,symbols ,Symplectic Geometry (math.SG) ,010307 mathematical physics ,Lagrangian - Abstract
We propose a field-theoretic interpretation of Ruelle zeta function and show how it can be seen as the partition function forBFtheory when an unusual gauge-fixing condition on contact manifolds is imposed. This suggests an alternative rephrasing of a conjecture due to Fried on the equivalence between Ruelle zeta function and analytic torsion, in terms of homotopies of Lagrangian submanifolds., Annales Henri Poincaré, 21 (12), ISSN:1424-0661, ISSN:1424-0637
- Published
- 2020
29. Special Ulrich bundles on regular Weierstrass fibrations
- Author
-
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
30. Derived Non-archimedean analytic Hilbert space
- Author
-
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
31. Convex plumbings in closed hyperbolic 4-manifolds
- Author
-
Leone Slavich, Bruno Martelli, Stefano Riolo, Martelli B., Riolo S., and Slavich L.
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,Hyperbolic geometry ,Hyperbolic 4-manifold ,Intersection form ,Plumbing ,Algebraic geometry ,01 natural sciences ,Mathematics - Geometric Topology ,Simple (abstract algebra) ,Hyperbolic set ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics ,Original Paper ,010102 general mathematics ,Regular polygon ,Geometric Topology (math.GT) ,Submanifold ,Mathematics::Geometric Topology ,57M50 ,Differential geometry ,010307 mathematical physics ,Geometry and Topology ,Mathematics::Differential Geometry - Abstract
We show that every plumbing of disc bundles over surfaces whose genera satisfy a simple inequality may be embedded as a convex submanifold in some closed hyperbolic four-manifold. In particular its interior has a geometrically finite hyperbolic structure that covers a closed hyperbolic four-manifold., 18 pages, 11 figures
- Published
- 2021
32. Spectral Properties of Schrödinger Operators Associated with Almost Minimal Substitution Systems
- Author
-
Benjamin Eichinger and Philipp Gohlke
- Subjects
Nuclear and High Energy Physics ,Pure mathematics ,Mathematics::Dynamical Systems ,37B10 ,Dynamical systems theory ,Context (language use) ,01 natural sciences ,Measure (mathematics) ,81Q10, 37B10, 52C23 ,0103 physical sciences ,Ergodic theory ,Mathematics - Dynamical Systems ,0101 mathematics ,Schrödinger operators ,Mathematical Physics ,Eigenvalues and eigenvectors ,Mathematics ,Original Paper ,010102 general mathematics ,Spectrum (functional analysis) ,Ergodicity ,Substitution (algebra) ,dinger operators ,Statistical and Nonlinear Physics ,52C23 ,81Q10 ,Non-primitive substitutions ,010307 mathematical physics ,Schrö - Abstract
We study the spectral properties of ergodic Schr\"{o}dinger operators that are associated to a certain family of non-primitive substitutions on a binary alphabet. The corresponding subshifts provide examples of dynamical systems that go beyond minimality, unique ergodicity and linear complexity. In some parameter region, we are naturally in the setting of an infinite ergodic measure. The almost sure spectrum is singular and contains an interval. Some criteria for the exclusion of eigenvalues are fully characterized, including the existence of strongly palindromic sequences. Many of our structural insights rely on return word decompositions in the context of non-uniformly recurrent sequences. We introduce an associated induced system that is conjugate to an odometer., Comment: Included a result on eigenvalues, added another case distinction (Erratum)
- Published
- 2020
33. On the quasi-isometric and bi-Lipschitz classification of 3D Riemannian Lie groups
- Author
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Katrin Fässler and Enrico Le Donne
- Subjects
Pure mathematics ,Dimension (graph theory) ,Quasi-isometric ,isometric ,53C23 ,01 natural sciences ,differentiaaligeometria ,0103 physical sciences ,Simply connected space ,Mathematics::Metric Geometry ,0101 mathematics ,Isometric ,20F65 ,bi-Lipschitz ,Mathematics ,Transitive relation ,Original Paper ,Lie groups ,Riemannian manifold ,010102 general mathematics ,22D05 ,ryhmäteoria ,Lie group ,Bi-Lipschitz ,Classification ,Lipschitz continuity ,metriset avaruudet ,quasi-isometric ,classification ,Differential geometry ,geometria ,010307 mathematical physics ,Geometry and Topology ,Mathematics::Differential Geometry ,Counterexample - Abstract
This note is concerned with the geometric classification of connected Lie groups of dimension three or less, endowed with left-invariant Riemannian metrics. On the one hand, assembling results from the literature, we give a review of the complete classification of such groups up to quasi-isometries and we compare the quasi-isometric classification with the bi-Lipschitz classification. On the other hand, we study the problem whether two quasi-isometrically equivalent Lie groups may be made isometric if equipped with suitable left-invariant Riemannian metrics. We show that this is the case for three-dimensional simply connected groups, but it is not true in general for multiply connected groups. The counterexample also demonstrates that ‘may be made isometric’ is not a transitive relation.
- Published
- 2019
34. A Few Comments on Ado’s Theorem and Non-Linear Lie Groups
- Author
-
Pierre Anglès
- Subjects
Spin group ,Statement (logic) ,Applied Mathematics ,010102 general mathematics ,Clifford algebra ,Short paper ,Lie group ,Ado's theorem ,01 natural sciences ,Algebra ,Nonlinear system ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
There are claims in the literature on Clifford algebras that every Lie group can be represented as a spin group. In a letter, Pertti Lounesto emphasized, by explicit counter-examples, that this statement is false. This self-contained short paper intends to present a survey of publications of well-known scientists where explicitly developed counter-examples prove the importance of Lounesto’s letter.
- Published
- 2018
35. 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
36. 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
37. Relating relative entropy, optimal transport and Fisher information: a quantum HWI inequality
- Author
-
Nilanjana Datta, Cambyse Rouzé, and Apollo - University of Cambridge Repository
- Subjects
Nuclear and High Energy Physics ,Pure mathematics ,Kullback–Leibler divergence ,math-ph ,FOS: Physical sciences ,math.FA ,Riemannian geometry ,Information theory ,01 natural sciences ,Convexity ,symbols.namesake ,math.MP ,quant-ph ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Fisher information ,Mathematical Physics ,Mathematics ,Original Paper ,Quantum Physics ,Concentration of measure ,Semigroup ,010102 general mathematics ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,Functional Analysis (math.FA) ,ddc ,Sobolev space ,Mathematics - Functional Analysis ,symbols ,010307 mathematical physics ,Quantum Physics (quant-ph) - Abstract
Quantum Markov semigroups characterize the time evolution of an important class of open quantum systems. Studying convergence properties of such a semigroup, and determining concentration properties of its invariant state, have been the focus of much research. Quantum versions of functional inequalities (like the modified logarithmic Sobolev and Poincar\'{e} inequalities) and the so-called transportation cost inequalities, have proved to be essential for this purpose. Classical functional and transportation cost inequalities are seen to arise from a single geometric inequality, called the Ricci lower bound, via an inequality which interpolates between them. The latter is called the HWI-inequality, where the letters I, W and H are, respectively, acronyms for the Fisher information (arising in the modified logarithmic Sobolev inequality), the so-called Wasserstein distance (arising in the transportation cost inequality) and the relative entropy (or Boltzmann H function) arising in both. Hence, classically, all the above inequalities and the implications between them form a remarkable picture which relates elements from diverse mathematical fields, such as Riemannian geometry, information theory, optimal transport theory, Markov processes, concentration of measure, and convexity theory. Here we consider a quantum version of the Ricci lower bound introduced by Carlen and Maas, and prove that it implies a quantum HWI inequality from which the quantum functional and transportation cost inequalities follow. Our results hence establish that the unifying picture of the classical setting carries over to the quantum one., Comment: 29 pages, 2 figures
- Published
- 2017
38. Lattice Polygons and the Number 2i + 7
- Author
-
Josef Schicho and Christian Haase
- Subjects
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
39. Iterates of Generic Polynomials and Generic Rational Functions
- Author
-
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
40. Stillman's Question for Exterior Algebras and Herzog's Conjecture on Betti Numbers of Syzygy Modules
- Author
-
Jason McCullough
- Subjects
Pure mathematics ,13D02, 15A75 ,Algebra and Number Theory ,Conjecture ,Hilbert's syzygy theorem ,Mathematics::Commutative Algebra ,Betti number ,Polynomial ring ,010102 general mathematics ,Short paper ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,01 natural sciences ,Mathematics::Algebraic Geometry ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Counterexample ,Mathematics - Abstract
Let K be a field of characteristic 0 and consider exterior algebras of finite dimensional K-vector spaces. In this short paper we exhibit principal quadric ideals in a family whose Castelnuovo-Mumford regularity is unbounded. This negatively answers the analogue of Stillman's Question for exterior algebras posed by I. Peeva. We show that these examples are dual to modules over polynomial rings that yield counterexamples to a conjecture of J. Herzog on the Betti numbers in the linear strand of syzygy modules., 7 pages - Complete rewrite of previous draft with new section and shorter proofs
- Published
- 2013
41. 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
42. 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
43. K3 surfaces with maximal finite automorphism groups containing M 20
- Author
-
Alessandra Sarti, Cédric Bonnafé, Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques et Applications (LMA-Poitiers), Université de Poitiers-Centre National de la Recherche Scientifique (CNRS), ANR-16-CE40-0010,GeRepMod,Méthodes géométriques en théorie des représentations modulaires des groupes réductifs finis(2016), and ANR-18-CE40-0024,CATORE,CATEGORIFICATIONS EN TOPOLOGIE ET EN THEORIE DES REPRESENTATIONS(2018)
- Subjects
Finite group ,Algebra and Number Theory ,Group (mathematics) ,010102 general mathematics ,Group Theory (math.GR) ,Kummer surface ,Automorphism ,01 natural sciences ,[MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR] ,K3 surface ,Combinatorics ,Mathematics - Algebraic Geometry ,0103 physical sciences ,FOS: Mathematics ,Order (group theory) ,Mathieu group ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematics - Group Theory ,Symplectic geometry ,Mathematics - Abstract
It was shown by Mukai that the maximum order of a finite group acting faithfully and symplectically on a K3 surface is $960$ and that the group is isomorphic to the group $M\_{20}$. Then Kondo showed that the maximum order of a finite group acting faithfully on a K3 surface is $3\,840$ and this group contains the Mathieu group $M\_{20}$ with index four. Kondo also showed that there is a unique K3 surface on which this group acts faithfully, which is the Kummer surface $\Km(E\_i\times E\_i)$. In this paper we describe two more K3 surfaces admitting a big finite automorphism group of order $1\,920$, both groups contains $M\_{20}$ as a subgroup of index 2. We show moreover that these two groups and the two K3 surfaces are unique. This result was shown independently by S. Brandhorst and K. Hashimoto in a forthcoming paper, with the aim of classifying all the finite groups acting faithfully on K3 surfaces with maximal symplectic part., 15 pages
- Published
- 2021
44. Analyzing the Weyl Construction for Dynamical Cartan Subalgebras
- Author
-
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
45. The transitive groups of degree 48 and some applications
- Author
-
Derek F. Holt, Gareth Tracey, and Gordon F. Royle
- Subjects
Transitive relation ,Algebra and Number Theory ,Degree (graph theory) ,Cayley graph ,010102 general mathematics ,Magma (algebra) ,Permutation group ,01 natural sciences ,Combinatorics ,Conjugacy class ,Symmetric group ,0103 physical sciences ,Enumeration ,010307 mathematical physics ,0101 mathematics ,QA ,Mathematics - Abstract
The primary purpose of this paper is to report on the successful enumeration in Magma of representatives of the 195 826 352 conjugacy classes of transitive subgroups of the symmetric group S 48 of degree 48. In addition, we have determined that 25707 of these groups are minimal transitive and that 713 of them are elusive. The minimal transitive examples have been used to enumerate the vertex-transitive groups of degree 48, of which there are 1 538 868 366 , all but 0.1625% of which arise as Cayley graphs. We have also found that the largest number of elements required to generate any of these groups is 10, and we have used this fact to improve previous general bounds of the third author on the number of elements required to generate an arbitrary transitive permutation group of a given degree. The details of the proof of this improved bound will be published as a separate paper.
- Published
- 2022
46. Order 3 symplectic automorphisms on K3 surfaces
- Author
-
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
47. The Benson - Symonds invariant for ordinary and signed permutation modules
- Author
-
Aparna Upadhyay
- Subjects
Finite group ,Mathematics::Combinatorics ,Algebra and Number Theory ,Generalization ,010102 general mathematics ,Primary 20C30, 20C20, Secondary 05E10 ,Group Theory (math.GR) ,01 natural sciences ,Representation theory ,Combinatorics ,Permutation ,Symmetric group ,0103 physical sciences ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,Invariant (mathematics) ,Mathematics - Group Theory ,Mathematics - Representation Theory ,Mathematics - Abstract
The signed permutation modules are a simultaneous generalization of the ordinary permutation modules and the twisted permutation modules of the symmetric group. In a recent paper Dave Benson and Peter Symonds defined a new invariant $\gamma_G(M)$ for a finite dimensional module $M$ of a finite group $G$ which attempts to quantify how close a module is to being projective. In this paper, we determine this invariant for all the signed permutation modules of the symmetric group using tools from representation theory and combinatorics., Comment: 14 pages. arXiv admin note: substantial text overlap with arXiv:2012.00341
- Published
- 2021
48. Polyvector fields and polydifferential operators associated with Lie pairs
- Author
-
Mathieu Stiénon, Ruggero Bandiera, and Ping Xu
- Subjects
lie algebroids ,Lie algebroid ,Statistics::Theory ,Gerstenhaber algebra ,01 natural sciences ,Combinatorics ,Mathematics::Probability ,Mathematics::K-Theory and Homology ,Mathematics::Quantum Algebra ,Mathematics - Quantum Algebra ,0103 physical sciences ,FOS: Mathematics ,Quantum Algebra (math.QA) ,Gerstenhaber algebras ,homotopy lie algebras ,0101 mathematics ,Mathematics::Representation Theory ,Mathematical Physics ,Mathematics ,Algebra and Number Theory ,Homotopy ,010102 general mathematics ,Manifold ,Foliation ,Transfer (group theory) ,010307 mathematical physics ,Geometry and Topology ,Isomorphism - Abstract
We prove that the spaces $\operatorname{tot}\big(\Gamma(\Lambda^\bullet A^\vee \otimes_R\mathcal{T}_{\operatorname{poly}}^{\bullet}\big)$ and $\operatorname{tot}\big(\Gamma(\Lambda^\bullet A^\vee)\otimes_R\mathcal{D}_{\operatorname{poly}}^{\bullet}\big)$ associated with a Lie pair $(L,A)$ each carry an $L_\infty$ algebra structure canonical up to an $L_\infty$ isomorphism with the identity map as linear part. These two spaces serve, respectively, as replacements for the spaces of formal polyvector fields and formal polydifferential operators on the Lie pair $(L,A)$. Consequently, both $\mathbb{H}^\bullet_{\operatorname{CE}}(A,\mathcal{T}_{\operatorname{poly}}^{\bullet})$ and $\mathbb{H}^\bullet_{\operatorname{CE}}(A,\mathcal{D}_{\operatorname{poly}}^{\bullet})$ admit unique Gerstenhaber algebra structures. Our approach is based on homotopy transfer and the construction of a Fedosov dg Lie algebroid (i.e. a dg foliation on a Fedosov dg manifold)., Comment: [v2] 50 pages, paper was expanded; [v1] Paper arXiv:1605.09656v1 was expended and split into two papers. The first part is arXiv:1605.09656v2. The second part is the present paper. A new result addressing uniqueness of the constructed structures has been added
- Published
- 2021
49. Free objects and Gröbner-Shirshov bases in operated contexts
- Author
-
Zihao Qi, Guodong Zhou, Kai Wang, and Yufei Qin
- Subjects
Pure mathematics ,Polynomial ,Algebra and Number Theory ,Functor ,13P10(Primary), 03C05, 08B20, 12H05, 16S10 ,010102 general mathematics ,Mathematics - Rings and Algebras ,Basis (universal algebra) ,Type (model theory) ,01 natural sciences ,Operator (computer programming) ,0103 physical sciences ,Physics::Accelerator Physics ,Universal algebra ,010307 mathematical physics ,Free object ,0101 mathematics ,Algebraic number ,Mathematics - Abstract
This paper investigates algebraic objects equipped with an operator, such as operated monoids, operated algebras etc. Various free object functors in these operated contexts are explicitly constructed. For operated algebras whose operator satisfies a set $\Phi$ of relations (usually called operated polynomial identities (aka. OPIs)), Guo defined free objects, called free $\Phi$-algebras, via universal algebra. Free $\Phi$-algebras over algebras are studied in details. A mild sufficient condition is found such that $\Phi$ together with a Gr\"obner-Shirshov basis of an algebra $A$ form a Gr\"obner-Shirshov basis of the free $\Phi$-algebra over algebra $A$ in the sense of Guo et al.. Ample examples for which this condition holds are provided, such as all Rota-Baxter type OPIs, a class of differential type OPIs, averaging OPIs and Reynolds OPI., Comment: Slightly revised version of the published paper in Journal of Algebra
- Published
- 2021
50. Generalised Igusa-Todorov functions and Lat-Igusa-Todorov algebras
- Author
-
José Vivero, Diego Bravo, Marcelo Lanzilotta, and Octavio Mendoza
- Subjects
Class (set theory) ,Pure mathematics ,Algebra and Number Theory ,Conjecture ,Mathematics::Number Theory ,010102 general mathematics ,Dimension (graph theory) ,01 natural sciences ,Mathematics::Algebraic Geometry ,Mathematics::K-Theory and Homology ,0103 physical sciences ,FOS: Mathematics ,Order (group theory) ,Computer Science::Symbolic Computation ,010307 mathematical physics ,Representation Theory (math.RT) ,16E05, 16E10, 16G10 (Primary) ,0101 mathematics ,Mathematics::Representation Theory ,Mathematics - Representation Theory ,Mathematics - Abstract
In this paper we study a generalisation of the Igusa-Todorov functions which gives rise to a vast class of algebras satisfying the finitistic dimension conjecture. This class of algebras is called Lat-Igusa-Todorov and includes, among others, the Igusa-Todorov algebras (defined by J. Wei) and the self-injective algebras which in general are not Igusa-Todorov algebras. Finally, some applications of the developed theory are given in order to relate the different homological dimensions which have been discussed through the paper., 18 pages, submitted to a peer-reviewed journal
- Published
- 2021
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