4,489 results
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2. 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
3. Addendum to the paper 'A note on weighted Bergman spaces and the Cesàro operator'
- Author
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Stevo Stević and Der-Chen Chang
- Subjects
Pure mathematics ,010308 nuclear & particles physics ,General Mathematics ,Operator (physics) ,010102 general mathematics ,Mathematical analysis ,Weighted Bergman space ,Addendum ,01 natural sciences ,Bergman space ,0103 physical sciences ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,46E15 ,0101 mathematics ,polydisk ,Cesàro operator ,Mathematics ,Bergman kernel ,47B38 - Abstract
Let H(Dn) be the space of holomorphic functions on the unit polydisk Dn, and let , where p, q> 0, α = (α1,…,αn) with αj > -1, j =1,..., n, be the class of all measurable functions f defined on Dn such thatwhere Mp(f,r) denote the p-integral means of the function f. Denote the weighted Bergman space on . We provide a characterization for a function f being in . Using the characterization we prove the following result: Let p> 1, then the Cesàro operator is bounded on the space .
- Published
- 2005
4. 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
5. 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
6. 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
7. 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
8. AN ALMOST SCHUR THEOREM ON 4-DIMENSIONAL MANIFOLDS
- Author
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Guofang Wang, Yuxin Ge, Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM), and Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Pure mathematics ,010308 nuclear & particles physics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Short paper ,01 natural sciences ,Schur's theorem ,Computer Science::Computers and Society ,[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] ,Ricci-flat manifold ,0103 physical sciences ,Sectional curvature ,Mathematics::Differential Geometry ,0101 mathematics ,Mathematics::Symplectic Geometry ,Schur product theorem ,Mathematics ,Scalar curvature - Abstract
International audience; In this short paper we prove that the almost Schur theorem, introduced by De Lellis and Topping, is true on 4-dimensional Riemannian manifolds of nonnegative scalar curvature and discuss some related problems on other dimensional manifolds.
- Published
- 2012
9. 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
10. 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
11. 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
12. 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
13. 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
14. 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
15. 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
16. Graded Bourbaki ideals of graded modules
- Author
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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
17. 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
18. 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.
- Published
- 2020
19. Perturbations of the Continuous Spectrum of a Certain Nonlinear Two-Dimensional Operator Sheaf
- Author
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Denis Borisov
- Subjects
Statistics and Probability ,Pure mathematics ,Dimensional operator ,Applied Mathematics ,General Mathematics ,Operator (physics) ,010102 general mathematics ,Continuous spectrum ,Essential spectrum ,01 natural sciences ,010305 fluids & plasmas ,Bounded function ,0103 physical sciences ,Sheaf ,0101 mathematics ,Laplace operator ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper, we consider the operator sheaf $$ -\Delta +V+\varepsilon {\mathrm{\mathcal{L}}}_{\varepsilon}\left(\lambda \right)+{\lambda}^2 $$ in the space L2(ℝ2), where the real-valued potential V depends only on the first variable x1, e is a small positive parameter, λ is the spectral parameter, $$ {\mathrm{\mathcal{L}}}_{\varepsilon}\left(\lambda \right) $$ is a localized operator bounded with respect to the Laplacian −Δ, and the essential spectrum of this operator is independent of e and contains certain critical points defined as isolated eigenvalues of the operator $$ -\frac{d^2}{dx_1^2}+V\left({x}_1\right) $$ in L2(ℝ). The basic result obtained in this paper states that for small values of e, in neighborhoods of critical points mentioned, isolated eigenvalues of the sheaf considered arise. Sufficient conditions for the existence or absence of such eigenvalues are obtained. The number of arising eigenvalues is determined, and in the case where they exist, the first terms of their asymptotic expansions are found.
- Published
- 2020
20. Eigenfunctions of the Laplace Operator and Harmonic Functions on Model Riemannian Manifolds
- Author
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E. Mazepa, A. G. Losev, and I. Romanova
- Subjects
Dirichlet problem ,Pure mathematics ,Series (mathematics) ,General Mathematics ,010102 general mathematics ,Separation of variables ,Eigenfunction ,01 natural sciences ,Manifold ,010305 fluids & plasmas ,Harmonic function ,0103 physical sciences ,Mathematics::Differential Geometry ,Boundary value problem ,0101 mathematics ,Laplace operator ,Mathematics - Abstract
This article explores and develops opportunities Fourier method of separation of variables for the study of the asymptotic behavior of harmonic functions on noncompact Riemannian manifolds of a special form. These manifolds generalize spherically symmetric manifold and are called model ones in a series of works. In the first part of the paper, an estimate of the eigenfunctions of the Laplace operator is obtained on compact Riemannian manifolds $$S$$ in the norm $$C^{m}(S)$$ . In the second part of the paper, the conditions for the unique solvability of the Dirichlet problem for harmonic functions on model manifolds with smooth boundary data at ‘‘infinity’’ are found. It was shown that the solution of this boundary value problem converges to the boundary data in the $$C^{1}$$ -norm.
- Published
- 2020
21. On the classification of simple Lie algebras of dimension seven over fields of characteristic 2
- Author
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Wilian Francisco de Araujo, Marinês Guerreiro, and Alexander Grishkov
- Subjects
Pure mathematics ,Rank (linear algebra) ,General Mathematics ,010102 general mathematics ,Dimension (graph theory) ,SUPERÁLGEBRAS DE LIE ,Field (mathematics) ,01 natural sciences ,010305 fluids & plasmas ,Computational Theory and Mathematics ,Simple (abstract algebra) ,0103 physical sciences ,Lie algebra ,0101 mathematics ,Statistics, Probability and Uncertainty ,Algebra over a field ,Algebraically closed field ,Mathematics - Abstract
This paper is the second part of paper (Grishkov and Guerreiro in Sao Paulo J Math Sci v4(1):93–107, 2010) about simple 7-dimensional Lie algebras over an algebraically closed field k of characteristic two. In this paper we prove that all simple 7-dimensional Lie algebras over k of absolute toral rank three are isomorphic to the Cartan algebra $$W_1$$ or the Hamilton algebra $$H_2.$$ We hope to prove that those algebras are the unique simple 7-dimensional Lie algebras over the field k. Observe that in the case of absolute toral rank 2 this fact was proved in [2].
- Published
- 2020
22. Examples of Integrable Systems with Dissipation on the Tangent Bundles of Three-Dimensional Manifolds
- Author
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Maxim V. Shamolin
- Subjects
Statistics and Probability ,Pure mathematics ,Integrable system ,Dynamical systems theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Degrees of freedom ,Tangent ,Dissipation ,01 natural sciences ,Force field (chemistry) ,010305 fluids & plasmas ,0103 physical sciences ,Mathematics::Differential Geometry ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics ,Variable (mathematics) - Abstract
In this paper, we prove the integrability of certain classes of dynamical systems on the tangent bundles of four-dimensional manifolds (systems with four degrees of freedom). The force field considered possessed so-called variable dissipation; they are generalizations of fields studied earlier. This paper continues earlier works of the author devoted to systems on the tangent bundles of two- and three-dimensional manifolds.
- Published
- 2020
23. Comparison of Classifications of Two-Dimensional Local Type II Fields
- Author
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Igor B. Zhukov and O. Yu. Ivanova
- Subjects
Pure mathematics ,Degree (graph theory) ,Mathematics::Number Theory ,General Mathematics ,Ramification (botany) ,010102 general mathematics ,Local parameter ,Field (mathematics) ,Type (model theory) ,01 natural sciences ,010305 fluids & plasmas ,Residue field ,0103 physical sciences ,0101 mathematics ,Invariant (mathematics) ,Constant (mathematics) ,Mathematics - Abstract
The paper contributes to the theory of the elimination of wild ramification for two-dimensional fields and continues the research related to the classification of fields introduced in the work of Masato Kurihara. We consider two-dimensional mixed-characteristic local fields with the characteristic of the finite residue field not equal to 2. The structure of fields that are weakly unramified over their constant subfield, i.e., the so-called standard fields, is well known. It is also known that any field can be extended into the standard one by a finite extension of its constants subfield. In the general case, the question of the minimum degree of this extension remains open. In Kurihara’s paper, two-dimensional fields are subdivided into two types as follows. A linear relation between the differentials of local parameters is considered. If the valuation of the coefficient at the uniformizer is less than that before the second local parameter, the field belongs to type I; otherwise it belongs to type II. This paper is devoted to the fields of type II. For them, we consider an improved Kurihara invariant: for each field, we introduce a quantity Δ equal to the difference between the valuations of the coefficients in the relation for the differentials of the local parameters. The degree of the constant extension that eliminates the ramification is not less for any field than the ramification index over the constant subfield. However, not all the fields have an extension of this degree. It is proved that in order that the extension of the least possible degree may exist, it suffices for the absolute values of Δ to be sufficiently large. The corresponding estimate for Δ depends on the ramification index of the field over its constant subfield.
- Published
- 2020
24. Material Affine Connections for Growing Solids
- Author
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K G Koifman and S. A. Lychev
- Subjects
Pure mathematics ,Parallel transport ,General Mathematics ,010102 general mathematics ,Affine connection ,01 natural sciences ,Manifold ,010305 fluids & plasmas ,Connection (mathematics) ,Volume form ,Teleparallelism ,0103 physical sciences ,Affine transformation ,Tangent vector ,0101 mathematics ,Mathematics - Abstract
The present paper aims to develop geometrical approach for finite incompatible deformations arising in growing solids. The phenomena of incompatibility is modeled by specific affine connection on material manifold, referred to as material connection. It provides complete description of local incompatible deformations for simple materials. Meanwhile, the differential-geometric representation of such connection is not unique. It means that one can choose different ways for analytical definition of connection for single given physical problem. This shows that, in general, affine connection formalism provides greater potential than is required to the theory of simple materials (first gradient theory). For better understanding of this inconsistence it is advisable to study different ways for material connection formalization in details. It is the subject of present paper. Affine connection endows manifolds with geometric properties, in particular, with parallel transport on them. For simple materials the parallel transport is elegant mathematical formalization of the concept of a materially uniform (in particular, a stress-free) non-Euclidean reference shape. In fact, one can obtain a connection of physical space by determining the parallel transport as a transformation of the tangent vector, which corresponds to the structure of the physical space containing shapes of the body. One can alternatively construct affine connection of material manifold by defining parallel transport as the transformation of the tangent vector, in which its inverse image with respect to locally uniform embeddings does not change. Utilizing of the conception of material connections and the corresponding methods of non-Euclidean geometry may significantly simplify formulation of the initial-boundary value problems of the theory of incompatible deformations. Connection on the physical manifold is compatible with metric and Levi-Civita relations holds for it. Connection on the material manifold is considered in three alternative variants. The first leads to Weitzenbock space (the space of absolute parallelism or teleparallelism, i.e., space with zero curvature and nonmetricity, but with non-zero torsion) and gives a clear interpretation of the material connection in terms of the local linear transformations which transform an elementary volume of simple material into uniform state. The second one allows to choose the Riemannian space structure (with zero torsion and nonmetricity, but nonzero curvature) in material manifold and it is the most convenient way for deriving of field equations. The third variant is based on Weyl manifold with specified volume form and non-vanishing nonmetricity.
- Published
- 2020
25. Tailoring a Pair of Pants: The Phase Tropical Version
- Author
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Ilia Zharkov
- Subjects
Statistics and Probability ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Phase (waves) ,01 natural sciences ,010305 fluids & plasmas ,Mathematics - Algebraic Geometry ,0103 physical sciences ,FOS: Mathematics ,Isotopy ,0101 mathematics ,Algebraic Geometry (math.AG) ,Pair of pants ,Mathematics - Abstract
We show that the phase tropical pair-of-pants is (ambient) isotopic to the complex pair-of-pants. This paper can serve as an addendum to the author's joint paper with Ruddat arXiv:2001.08267 where an isotopy between complex and ober-tropical pairs-of-pants was shown. Thus all three versions are isotopic., 10 pages, 8 figures. arXiv admin note: text overlap with arXiv:2001.08267
- Published
- 2020
26. 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.
- Published
- 2020
27. Hyperbolicity and Uniformity of Varieties of Log General type
- Author
-
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
- Published
- 2020
28. Nψ,ϕ-type Quotient Modules over the Bidisk
- Author
-
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.
- Published
- 2020
29. 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 .
- Published
- 2020
30. 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.
- Published
- 2020
31. A Short Proof of a Theorem Due to O. Gabber
- Author
-
Ivan Panin
- Subjects
Statistics and Probability ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Field (mathematics) ,Regular local ring ,Reductive group ,01 natural sciences ,010305 fluids & plasmas ,Finite field ,Scheme (mathematics) ,0103 physical sciences ,Fraction (mathematics) ,0101 mathematics ,Mathematics - Abstract
A very short proof of an unpublished result due to O. Gabber is given. More exactly, let R be a regular local ring containing a finite field k. Let G be a simply-connected reductive group scheme over k. It is proved that a principal G-bundle over R is trivial if it is trivial over the fraction field of R. This is the mentioned unpublished result due to O. Gabber. In this paper, this result is derived from a purely geometric one, proved in another paper of the author and stated in the Introduction.
- Published
- 2020
32. On Some Local Asymptotic Properties of Sequences with a Random Index
- Author
-
Yu. V. Yakubovich, O. V. Rusakov, and B. A. Baev
- Subjects
Rademacher distribution ,Hurst exponent ,Pure mathematics ,Fractional Brownian motion ,Stochastic process ,General Mathematics ,010102 general mathematics ,Poisson distribution ,01 natural sciences ,010305 fluids & plasmas ,Cox process ,symbols.namesake ,0103 physical sciences ,symbols ,0101 mathematics ,Telegraph process ,Random variable ,Mathematics - Abstract
Random sequences with random or stochastic indices controlled by a doubly stochastic Poisson process are considered in this paper. A Poisson stochastic index process (PSI-process) is a random process with the continuous time ψ(t) obtained by subordinating a sequence of random variables (ξj), j = 0, 1, …, by a doubly stochastic Poisson process Π1(tλ) via the substitution ψ(t) = $${{\xi }_{{{{\Pi }_{1}}(t\lambda )}}}$$ , t $$ \geqslant $$ 0, where the random intensity λ is assumed independent of the standard Poisson process Π1. In this paper, we restrict our consideration to the case of independent identically distributed random variables (ξj) with a finite variance. We find a representation of the fractional Ornstein–Uhlenbeck process with the Hurst exponent H ∈ (0, 1/2) introduced and investigated by R. Wolpert and M. Taqqu (2005) in the form of a limit of normalized sums of independent identically distributed PSI-processes with an explicitly given distribution of the random intensity λ. This fractional Ornstein–Uhlenbeck process provides a local, at t = 0, mean-square approximation of the fractional Brownian motion with the same Hurst exponent H ∈ (0, 1/2). We examine in detail two examples of PSI-processes with the random intensity λ generating the fractional Ornstein–Uhlenbeck process in the Wolpert and Taqqu sense. These are a telegraph process arising when ξ0 has a Rademacher distribution ±1 with the probability 1/2 and a PSI-process with the uniform distribution for ξ0. For these two examples, we calculate the exact and the asymptotic values of the local modulus of continuity for a single PSI-process over a small fixed time span.
- Published
- 2020
33. On Finiteness Conditions in Twisted K-Theory
- Author
-
M. A. Gerasimova
- Subjects
Statistics and Probability ,Pure mathematics ,Statement (logic) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Connection (vector bundle) ,Lie group ,Twisted K-theory ,01 natural sciences ,010305 fluids & plasmas ,Elliptic operator ,Mathematics::K-Theory and Homology ,Bundle ,0103 physical sciences ,0101 mathematics ,Special case ,Mathematics - Abstract
The aim of this (mostly expository) article is to show a connection between the finiteness conditions arising in twisted K-theory. There are two different conditions arising naturally in two main approaches to the problem of computing the index of the appropriate family of elliptic operators (the approach of Nistor and Troitsky and the approach of Mathai, Melrose, and Singer). These conditions are formulated absolutely differently, but in some sense they should be close to each other. In this paper, we find this connection and prove the corresponding formal statement. Thereby it is shown that these conditions map to each other. This opens a possibility to synthesize these approaches. It is also shown that the finiteness condition arising in the paper of Nistor and Troitsky is a special case of the finiteness condition that appears in the paper of Emerson and Meyer, where the theorem of Nistor and Troitsky is proved not only for the case of a bundle of Lie groups, but also for the case of a general groupoid.
- Published
- 2020
34. 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.
- Published
- 2020
35. Schrödinger Quantization of Infinite-Dimensional Hamiltonian Systems with a Nonquadratic Hamiltonian Function
- Author
-
N. N. Shamarov and Oleg G. Smolyanov
- Subjects
Hamiltonian mechanics ,Pure mathematics ,Lebesgue measure ,General Mathematics ,010102 general mathematics ,Convex set ,Space (mathematics) ,01 natural sciences ,Measure (mathematics) ,010305 fluids & plasmas ,Hamiltonian system ,symbols.namesake ,Fourier transform ,0103 physical sciences ,symbols ,0101 mathematics ,Hamiltonian (control theory) ,Mathematics - Abstract
According to a theorem of Andre Weil, there does not exist a standard Lebesgue measure on any infinite-dimensional locally convex space. Because of that, Schrodinger quantization of an infinite-dimensional Hamiltonian system is often defined using a sigma-additive measure, which is not translation-invariant. In the present paper, a completely different approach is applied: we use the generalized Lebesgue measure, which is translation-invariant. In implicit form, such a measure was used in the first paper published by Feynman (1948). In this situation, pseudodifferential operators whose symbols are classical Hamiltonian functions are formally defined as in the finite-dimensional case. In particular, they use unitary Fourier transforms which map functions (on a finite-dimensional space) into functions. Such a definition of the infinite-dimensional unitary Fourier transforms has not been used in the literature.
- Published
- 2020
36. 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
- Published
- 2020
37. On Problems of Stability Theory for Weakly Hyperbolic Invariant Sets
- Author
-
Nikita Begun
- Subjects
Pure mathematics ,Conjecture ,Dynamical systems theory ,General Mathematics ,010102 general mathematics ,Invariant (physics) ,Lipschitz continuity ,01 natural sciences ,010305 fluids & plasmas ,Stability theory ,0103 physical sciences ,Uniqueness ,0101 mathematics ,Mathematics - Abstract
This paper presents a brief survey for the theory of stability of weakly hyperbolic invariant sets. It has been proved in several papers that I published along with Pliss and Sell that a weakly hyperbolic invariant set is stable even if the Lipschitz condition fails to hold. However, the uniqueness of leaves of a weakly hyperbolic invariant set of a perturbed system remains an open question. We show that this problem is connected to the so-called plaque expansivity conjecture in the theory of dynamical systems.
- Published
- 2020
38. Mappings with finite length distortion and prime ends on Riemann surfaces
- Author
-
Sergei Volkov and I Vladimir Ryazanov
- Subjects
Statistics and Probability ,Pure mathematics ,Series (mathematics) ,Generalization ,Applied Mathematics ,General Mathematics ,Riemann surface ,010102 general mathematics ,Boundary (topology) ,02 engineering and technology ,01 natural sciences ,Prime (order theory) ,010305 fluids & plasmas ,Sobolev space ,Distortion (mathematics) ,symbols.namesake ,020303 mechanical engineering & transports ,0203 mechanical engineering ,0103 physical sciences ,Euclidean geometry ,symbols ,0101 mathematics ,Mathematics - Abstract
The present paper is a continuation of our research that was devoted to the theory of the boundary behavior of mappings in the Sobolev classes (mappings with generalized derivatives) on Riemann surfaces. Here we develop the theory of the boundary behavior of the mappings in the class of FLD (mappings with finite length distortion) first introduced for the Euclidean spaces in the article of Martio-Ryazanov-Srebro-Yakubov at 2004 and then included in the known book of these authors at 2009 on the modern mapping theory. As was shown in the recent papers of Kovtonyuk-Petkov-Ryazanov at 2017, such mappings, generally speaking, are not mappings in the Sobolev classes, because their first partial derivatives can be not locally integrable. At the same time, this class is a natural generalization of the well-known significant classes of isometries and quasiisometries. We prove here a series of criteria in terms of dilatations for the continuous and homeomorphic extensions to the boundary of the mappings with finite length distortion between domains on Riemann surfaces by Caratheodory prime ends. The criterion for the continuous extension of the inverse mapping to the boundary is turned out to be the very simple condition on the integrability of the dilatations in the first power. The criteria for the continuous extension of the direct mappings to the boundary have a much more refined nature. One of such criteria is the existence of a majorant for the dilatation in the class of functions with finite mean oscillation, i.e., having a finite mean deviation from its mean value over infinitesimal disks centered at boundary points. As consequences, the corresponding criteria for a homeomorphic extension of mappings with finite length distortion to the closures of domains by Caratheodory prime ends are obtained.
- Published
- 2020
39. 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
40. 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
41. 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
42. On the local density formula and the Gross–Keating invariant with an Appendix ‘The local density of a binary quadratic form’ by T. Ikeda and H. Katsurada
- Author
-
Cho Sungmun
- Subjects
Pure mathematics ,Mathematics - Number Theory ,Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,Local factor ,01 natural sciences ,Quadratic form ,0103 physical sciences ,FOS: Mathematics ,11E08, 11E95, 14L15, 20G25 ,Binary quadratic form ,Number Theory (math.NT) ,010307 mathematical physics ,0101 mathematics ,Invariant (mathematics) ,Local field ,Fourier series ,Mathematics - Abstract
T. Ikeda and H. Katsurada have developed the theory of the Gross-Keating invariant of a quadratic form in their recent papers [IK1] and [IK2]. In particular, they prove that the local factor of the Fourier coefficients of the Siegel-Eisenstein series is completely determined by the Gross-Keating invariant with extra datum, called the extended GK datum, in [IK2]. On the other hand, such local factor is a special case of the local densities for a pair of two quadratic forms. Thus we propose a general question if the local density can be determined by certain series of the Gross-Keating invariants and the extended GK datums. In this paper, we prove that the answer to this question is affirmative, for the local density of a single quadratic form defined over an unramified finite extension of $\mathbb{Z}_2$. In the appendix, T. Ikeda and H. Katsurada compute the local density formula of a single binary quadratic form defined over any finite extension of $\mathbb{Z}_2$., 32 pages
- Published
- 2020
43. The Wiener Measure on the Heisenberg Group and Parabolic Equations
- Author
-
S. V. Mamon
- Subjects
Statistics and Probability ,Pure mathematics ,Semigroup ,Stochastic process ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Markov process ,01 natural sciences ,Measure (mathematics) ,010305 fluids & plasmas ,Nilpotent ,symbols.namesake ,0103 physical sciences ,Path integral formulation ,Lie algebra ,symbols ,Heisenberg group ,0101 mathematics ,Mathematics - Abstract
In this paper, we study questions related to the theory of stochastic processes on Lie nilpotent groups. In particular, we consider the stochastic process on the Heisenberg group H3(ℝ) whose trajectories satisfy the horizontal conditions in the stochastic sense relative to the standard contact structure on H3 (ℝ). It is shown that this process is a homogeneous Markov process relative to the Heisenberg group operation. There was found a representation in the form of a Wiener integral for a one-parameter linear semigroup of operators for which the Heisenberg sublaplacian generated by basis vector fields of the corresponding Lie algebra L(H3) is producing. The main method of solving the problem in this paper is using the path integrals technique, which indicates the common direction of further development of the results.
- Published
- 2020
44. Zeros of Holomorphic One-Forms and Topology of Kähler Manifolds
- Author
-
Stefan Schreieder
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Holomorphic function ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Topology (chemistry) ,Mathematics - Abstract
A conjecture of Kotschick predicts that a compact Kähler manifold X fibres smoothly over the circle if and only if it admits a holomorphic one-form without zeros. In this paper we develop an approach to this conjecture and verify it in dimension two. In a joint paper with Hao [10], we use our approach to prove Kotschick’s conjecture for smooth projective three-folds.
- Published
- 2020
45. Homological properties of quotient divisible Abelian groups and compact groups dual to them
- Author
-
Nikolay I. Kryuchkov
- Subjects
Pure mathematics ,Fundamental group ,General Mathematics ,010102 general mathematics ,General Physics and Astronomy ,Topological space ,01 natural sciences ,Triviality ,Divisible group ,010305 fluids & plasmas ,Universality (dynamical systems) ,0103 physical sciences ,Homomorphism ,0101 mathematics ,Abelian group ,Quotient ,Mathematics - Abstract
Homological properties of quotient divisible Abelian groups are studied. These groups form an important class of groups, which has been extensively studied in recent years. The first part of the paper is devoted to conditions for the triviality of extension groups in which one of the arguments is a quotient divisible group. Under certain additional assumptions, groups of homomorphisms from quotient divisible groups to reduced Abelian groups are described. Universality properties of quotient divisible Abelian groups are investigated. The second part of the paper considers homological properties of compact Abelian groups dual to quotient divisible groups in the sense of L.S. Pontryagin. Such groups are said to be “quotient toroidal.” Conditions for the triviality of group extensions in which one of the arguments is a quotient toroidal group are studied. Certain groups of continuous homomorphisms in which the second argument is a quotient toroidal group are described. The last part of the paper is devoted to conditions for the triviality of the groups of extensions of quotient divisible groups by compact quotient toroidal ones. The fundamental group of the topological space of a quotient toroidal group is characterized.
- Published
- 2020
46. Ramanujan denesting formulae for cubic radicals
- Author
-
K. I. Pimenov and M. A. Antipov
- Subjects
Pure mathematics ,Polynomial ,Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,General Physics and Astronomy ,Inverse ,Extension (predicate logic) ,Type (model theory) ,01 natural sciences ,010305 fluids & plasmas ,Ramanujan's sum ,symbols.namesake ,0103 physical sciences ,symbols ,0101 mathematics ,Cubic function ,Mathematics - Abstract
This paper contains an explanation of Ramanujan-type formulas with cubic radicals of cubic irrationalities in the situation when these radicals are contained in a pure cubic extension. We give a complete description of formulas of such type, answering the Zippel’s question. It turns out that Ramanujan-type formulas are in some sense unique in this situation. In particular, there must be no more than three summands in the right-hand side and the norm of the irrationality in question must be a cube. In this situation we associate cubic irrationalities with a cyclic cubic polynomial, which is reducible if and only if one can simplify the corresponding cubic radical. This correspondence is inverse to the so-called Ramanujan correspondence defined in the preceding papers, where one associates a pure cubic extension to some cyclic polynomial.
- Published
- 2020
47. On the polar Orlicz-Minkowski problems and the p-capacitary Orlicz-Petty bodies
- Author
-
Xiaokang Luo, Deping Ye, and Baocheng Zhu
- Subjects
Mathematics::Functional Analysis ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Mathematics::Classical Analysis and ODEs ,Metric Geometry (math.MG) ,Type (model theory) ,01 natural sciences ,Measure (mathematics) ,52A20, 52A38, 52A39, 52A40, 53A15 ,Mathematics - Metric Geometry ,0103 physical sciences ,Minkowski space ,FOS: Mathematics ,Mathematics::Metric Geometry ,Polar ,010307 mathematical physics ,Orthogonal matrix ,0101 mathematics ,Isoperimetric inequality ,Mathematics - Abstract
In this paper, we propose and study the polar Orlicz-Minkowski problems: under what conditions on a nonzero finite measure $\mu$ and a continuous function $\varphi:(0,\infty)\rightarrow(0,\infty)$, there exists a convex body $K\in\mathcal{K}_0$ such that $K$ is an optimizer of the following optimization problems: \begin{equation*} \inf/\sup \bigg\{\int_{S^{n-1}}\varphi\big( h_L \big) \,d \mu: L \in \mathcal{K}_{0} \ \text{and}\ |L^\circ|=\omega_{n}\bigg\}. \end{equation*} The solvability of the polar Orlicz-Minkowski problems is discussed under different conditions. In particular, under certain conditions on $\varphi,$ the existence of a solution is proved for a nonzero finite measure $\mu$ on $S^{n-1}$ which is not concentrated on any hemisphere of $S^{n-1}.$ Another part of this paper deals with the $p$-capacitary Orlicz-Petty bodies. In particular, the existence of the $p$-capacitary Orlicz-Petty bodies is established and the continuity of the $p$-capacitary Orlicz-Petty bodies is proved., Comment: This paper has been accepted by Indiana University Mathematics Journal
- Published
- 2020
48. Archimedean non-vanishing, cohomological test vectors, and standard L-functions of $${\mathrm {GL}}_{2n}$$: real case
- Author
-
Cheng Chen, Fangyang Tian, Dihua Jiang, and Bingchen Lin
- Subjects
Pure mathematics ,Mathematics - Number Theory ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Linear model ,Structure (category theory) ,22E45 (Primary), 11F67 (Secondary) ,Type (model theory) ,Lambda ,Infinity ,01 natural sciences ,Invariant theory ,Linear form ,0103 physical sciences ,FOS: Mathematics ,Number Theory (math.NT) ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics - Representation Theory ,Mathematics ,media_common - Abstract
The standard $L$-functions of $\mathrm{GL}_{2n}$ expressed in terms of the Friedberg-Jacquet global zeta integrals have better structure for arithmetic applications, due to the relation of the linear periods with the modular symbols. The most technical obstacles towards such arithmetic applications are (1) non-vanishing of modular symbols at infinity and (2) the existance or construction of uniform cohomological test vectors. Problem (1) is also called the non-vanishing hypothesis at infinity, which was proved by Binyong Sun, by establishing the existence of certain cohomological test vectors. In this paper, we explicitly construct an archimedean local integral that produces a new type of a twisted linear functional $\Lambda_{s,\chi}$, which, when evaluated with our explicitly constructed cohomological vector, is equal to the local twisted standard $L$-function $L(s,\pi\otimes\chi)$ as a meromorphic function of $s\in \mathbb{C}$. With the relations between linear models and Shalika models, we establish (1) with an explicitly constructed cohomological vector, and hence recovers a non-vanishing result of Binyong Sun via a completely different method. Our main result indicates a complete solution to (2), which will be presented in a paper of Dihua Jiang, Binyong Sun and Fangyang Tian with full details and with applications to the global period relations for the twisted standard $L$-functions at critical places., Comment: 39 pages. The current version of this paper is significantly shorter than the previous one, as the first author pointed out a conceptual intepretation of construction of cohomological test vector in the old version of this paper. Section 4 is completely rewritten. Also fix some inaccuracies
- Published
- 2019
49. A sparse approach to mixed weak type inequalities
- Author
-
Marcela Caldarelli and Israel P. Rivera-Ríos
- Subjects
Pure mathematics ,Inequality ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Novelty ,Singular integral ,Weak type ,01 natural sciences ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,GEOM ,media_common ,Mathematics - Abstract
In this paper we provide some quantitative mixed weak-type estimates assuming conditions that imply that $$uv\in A_{\infty }$$ for Calderon–Zygmund operators, rough singular integrals and commutators. The main novelty of this paper lies in the fact that we rely upon sparse domination results, pushing an approach to endpoint estimates that was introduced in Domingo-Salazar et al. (Bull Lond Math Soc 48(1):63–73, 2016) and extended in Lerner et al. (Adv Math 319:153–181, 2017) and Li et al. (J Geom Anal, 2018).
- Published
- 2019
50. On Extensions of Semigroups and Their Applications to Toeplitz Algebras
- Author
-
Suren A. Grigoryan, R. N. Gumerov, and Ekaterina Lipacheva
- Subjects
Pure mathematics ,Toeplitz algebra ,Mathematics::Operator Algebras ,Semigroup ,General Mathematics ,010102 general mathematics ,Normal extension ,01 natural sciences ,Toeplitz matrix ,010305 fluids & plasmas ,Numerical semigroup ,0103 physical sciences ,Grothendieck group ,0101 mathematics ,Commutative property ,Mathematics ,Additive group - Abstract
The paper deals with the normal extensions of cancellative commutative semigroups and the Toeplitz algebras for those semigroups. By the Toeplitz algebra for a semigroup S one means the reduced semigroup C*-algebra C*(S). We study the normal extensions of cancellative commutative semigroups by the additive group ℤn of integers modulo n. Moreover, we assume that such an extension is generated by one element. We present a general method for constructing normal extensions of semigroups which contain no non-trivial subgroups. The Grothendieck group for a given semigroup and the group of all integers are involved in this construction. Examples of such extensions for the additive semigroup of non-negative integers are given. A criterion for a normal extension generated by an element to be isomorphic to a numerical semigroup is given in number-theoretic terms. The results concerning the Toeplitz algebras are the following. For a cancellative commutative semigroup S and its normal extension L generated by one element, there exists a natural embedding the semigroup C*-algebra C*(S) into C*(L). The semigroup C*-algebra C*(L) is topologically ℤn-graded. The results in the paper are announced without proofs.
- Published
- 2019
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