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2. On Topological Properties of Some Coverings. An Addendum to a Paper of Lanteri and Struppa
- Author
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Jarosław A. Wiśniewski
- Subjects
Surjective function ,Ample line bundle ,Pure mathematics ,Morphism ,Betti number ,General Mathematics ,Embedding ,Projective space ,Projective test ,Space (mathematics) ,Mathematics - Abstract
Let π: X′ → X be a finite surjective morphism of complex projective manifolds which can be factored by an embedding of X′ into the total space of an ample line bundle 𝓛 over X. A theorem of Lazarsfeld asserts that Betti numbers of X and X′ are equal except, possibly, the middle ones. In the present paper it is proved that the middle numbers are actually non-equal if either 𝓛 is spanned and deg π ≥ dim X, or if X is either a hyperquadric or a projective space and π is not a double cover of an odd-dimensional projective space by a hyperquadric.
- Published
- 1992
3. Correction to the Paper*
- Author
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Casper Goffman and G. M. Petersen
- Subjects
Matrix (mathematics) ,Pure mathematics ,General Mathematics ,Arithmetic ,Mathematics - Published
- 1962
4. Non-cocompact Group Actions and -Semistability at Infinity
- Author
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Ross Geoghegan, Michael L. Mihalik, and Craig R. Guilbault
- Subjects
Class (set theory) ,Pure mathematics ,Property (philosophy) ,Group (mathematics) ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Infinity ,01 natural sciences ,Group action ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics ,Counterexample ,media_common - Abstract
A finitely presented 1-ended group $G$ has semistable fundamental group at infinity if $G$ acts geometrically on a simply connected and locally compact ANR $Y$ having the property that any two proper rays in $Y$ are properly homotopic. This property of $Y$ captures a notion of connectivity at infinity stronger than “1-ended”, and is in fact a feature of $G$, being independent of choices. It is a fundamental property in the homotopical study of finitely presented groups. While many important classes of groups have been shown to have semistable fundamental group at infinity, the question of whether every $G$ has this property has been a recognized open question for nearly forty years. In this paper we attack the problem by considering a proper but non-cocompact action of a group $J$ on such an $Y$. This $J$ would typically be a subgroup of infinite index in the geometrically acting over-group $G$; for example $J$ might be infinite cyclic or some other subgroup whose semistability properties are known. We divide the semistability property of $G$ into a $J$-part and a “perpendicular to $J$” part, and we analyze how these two parts fit together. Among other things, this analysis leads to a proof (in a companion paper) that a class of groups previously considered to be likely counter examples do in fact have the semistability property.
- Published
- 2019
5. Corrigendum to: A Galois Correspondence for Reduced Crossed Products of Simple -algebras by Discrete Groups
- Author
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Roger R. Smith and Jan Cameron
- Subjects
Pure mathematics ,Crossed product ,Group (mathematics) ,Simple (abstract algebra) ,General Mathematics ,Unital ,Bimodule ,Mathematics - Abstract
This note corrects an error in our paper “A Galois correspondence for reduced crossed products of unital simple $\text{C}^{\ast }$-algebras by discrete groups”, http://dx.doi.org/10.4153/CJM-2018-014-6. The main results of the original paper are unchanged.
- Published
- 2019
6. Tannakian Categories With Semigroup Actions
- Author
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Michael Wibmer and Alexey Ovchinnikov
- Subjects
Class (set theory) ,Pure mathematics ,Semigroup ,General Mathematics ,010102 general mathematics ,Braid group ,Tannakian category ,Group Theory (math.GR) ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,01 natural sciences ,010101 applied mathematics ,Linear differential equation ,Mathematics::Category Theory ,FOS: Mathematics ,0101 mathematics ,Mathematics - Group Theory ,Finite set ,Differential (mathematics) ,Axiom ,Mathematics - Abstract
Ostrowski's theorem implies that $\log(x),\log(x+1),\ldots$ are algebraically independent over $\mathbb{C}(x)$. More generally, for a linear differential or difference equation, it is an important problem to find all algebraic dependencies among a non-zero solution $y$ and particular transformations of $y$, such as derivatives of $y$ with respect to parameters, shifts of the arguments, rescaling, etc. In the present paper, we develop a theory of Tannakian categories with semigroup actions, which will be used to attack such questions in full generality. Deligne studied actions of braid groups on categories and obtained a finite collection of axioms that characterizes such actions to apply it to various geometric constructions. In this paper, we find a finite set of axioms that characterizes actions of semigroups that are finite free products of semigroups of the form $\mathbb{N}^n\times \mathbb{Z}/n_1\mathbb{Z}\times\ldots\times\mathbb{Z}/n_r\mathbb{Z}$ on Tannakian categories. This is the class of semigroups that appear in many applications., Comment: minor revision
- Published
- 2017
7. Ramification of the Eigencurve at Classical RM Points
- Author
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Adel Betina
- Subjects
Pure mathematics ,Mathematics - Number Theory ,General Mathematics ,Mathematics::Number Theory ,010102 general mathematics ,Modular form ,Local ring ,Weight space ,Subring ,Galois module ,01 natural sciences ,Base change ,Lift (mathematics) ,0103 physical sciences ,FOS: Mathematics ,Quadratic field ,010307 mathematical physics ,Number Theory (math.NT) ,0101 mathematics ,Mathematics - Abstract
J. Bellaïche and M. Dimitrov showed that the $p$-adic eigencurve is smooth but not étale over the weight space at $p$-regular theta series attached to a character of a real quadratic field $F$ in which $p$ splits. In this paper we prove the existence of an isomorphism between the subring fixed by the Atkin–Lehner involution of the completed local ring of the eigencurve at these points and a universal ring representing a pseudo-deformation problem. Additionally, we give a precise criterion for which the ramification index is exactly 2. We finish this paper by proving the smoothness of the nearly ordinary and ordinary Hecke algebras for Hilbert modular forms over $F$ at the overconvergent cuspidal Eisenstein points, being the base change lift for $\text{GL}(2)_{/F}$ of these theta series. Our approach uses deformations and pseudo-deformations of reducible Galois representations.
- Published
- 2019
- Full Text
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8. A Skolem–Mahler–Lech Theorem for Iterated Automorphisms of K–algebras
- Author
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Jeffrey C. Lagarias and Jason P. Bell
- Subjects
Pure mathematics ,Mathematics - Number Theory ,General Mathematics ,Algebraic extension ,Field (mathematics) ,Dynamical Systems (math.DS) ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,16. Peace & justice ,Automorphism ,Mathematics - Algebraic Geometry ,Skolem–Mahler–Lech theorem ,Scheme (mathematics) ,FOS: Mathematics ,Affine space ,Number Theory (math.NT) ,Mathematics - Dynamical Systems ,Primary: 11D45. Secondary: 14R10. 11Y55, 11D88 ,Algebra over a field ,Algebraic Geometry (math.AG) ,Finite set ,Mathematics - Abstract
This paper proves a commutative algebraic extension of a generalized Skolem-Mahler-Lech theorem due to the first author. Let $A$ be a finitely generated commutative $K$-algebra over a field of characteristic $0$, and let $\sigma$ be a $K$-algebra automorphism of $A$. Given ideals $I$ and $J$ of $A$, we show that the set $S$ of integers $m$ such that $\sigma^m(I) \supseteq J$ is a finite union of complete doubly infinite arithmetic progressions in $m$, up to the addition of a finite set. Alternatively, this result states that for an affine scheme $X$ of finite type over $K$, an automorphism $\sigma \in {\rm Aut}_K(X)$, and $Y$ and $Z$ any two closed subschemes of $X$, the set of integers $m$ with $\sigma^m(Z ) \subseteq Y$ is as above. The paper presents examples showing that this result may fail to hold if the affine scheme $X$ is not of finite type, or if $X$ is of finite type but the field $K$ has positive characteristic., Comment: 29 pages; to appear in the Canadian Journal of Mathematics
- Published
- 2015
9. Weighted Carleson Measure Spaces Associated with Different Homogeneities
- Author
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Xinfeng Wu
- Subjects
Carleson measure ,Pure mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, we introduce weighted Carleson measure spaces associated with different homogeneities and prove that these spaces are the dual spaces of weighted Hardy spaces studied in a forthcoming paper. As an application, we establish the boundedness of composition of two Calderón–Zygmund operators with different homogeneities on the weighted Carleson measure spaces; this, in particular, provides the weighted endpoint estimates for the operators studied by Phong–Stein.
- Published
- 2014
10. Existence of Taut Foliations on Seifert Fibered Homology 3-spheres
- Author
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Shanti Caillat-Gibert and Daniel Matignon
- Subjects
Pure mathematics ,General Mathematics ,Taut foliation ,General Topology (math.GN) ,Physics::Physics Education ,Fibered knot ,Geometric Topology (math.GT) ,Homology (mathematics) ,Mathematics::Geometric Topology ,Mathematics - Geometric Topology ,FOS: Mathematics ,Mathematics::Differential Geometry ,Mathematics::Symplectic Geometry ,Mathematics - General Topology ,Mathematics - Abstract
This paper concerns the problem of existence of taut foliations among 3-manifolds. Since the contribution of David Gabai, we know that closed 3-manifolds with non-trivial second homology group admit a taut foliations. The essential part of this paper focuses on Seifert fibered homology 3-spheres. The result is quite different if they are integral or rational but non-integral homology 3-spheres. Concerning integral homology 3-spheres, we prove that all but the 3-sphere and the Poincar\'e 3-sphere admit a taut foliation. Concerning non-integral homology 3-spheres, we prove there are infinitely many which admit a taut foliation, and infinitely many without taut foliation. Moreover, we show that the geometries do not determine the existence of taut foliations on non-integral Seifert fibered homology 3-spheres., Comment: 34 pages, 1 figure
- Published
- 2014
11. The Ample Cone for a K3 Surface
- Author
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Arthur Baragar
- Subjects
Surface (mathematics) ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Lattice (group) ,Divisor (algebraic geometry) ,Algebraic number field ,01 natural sciences ,K3 surface ,Fractal ,Cone (topology) ,Hausdorff dimension ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
In this paper, we give several pictorial fractal representations of the ample or Kahler cone for surfaces in a certain class of K3 surfaces. The class includes surfaces described by smooth (2, 2, 2) forms in P×P×P defined over a sufficiently large number field K, which have a line parallel to one of the axes, and have Picard number four. We relate the Hausdorff dimension of this fractal to the asymptotic growth of orbits of curves under the action of the surface’s group of automorphisms. We experimentally estimate the Hausdorff dimension of the fractal to be 1.296± .010. The ample cone or Kahler cone for a surface is a significant and often complicated geometric object. Though much is known about the ample cone, particularly for K3 surfaces, only a few non-trivial examples have been explicitly described. These include the ample cones with a finite number of sides (see [N1] for n = 3, and [N2, N3] for n ≥ 5; the case n = 4 is attributed to Vinberg in an unpublished work [N1]); the ample cone for a class of K3 surfaces with n = 3 [Ba3]; and the ample cones for several Kummer surfaces, which are K3 surfaces with n = 20 [V, K-K, Kon]. Though the complexity of the problem generically increases with n, the problem for K3 surfaces with maximal Picard number (n = 20) appear to be tractable because of the small size of the transcendental lattice. In this paper, we introduce accurate pictorial representations of the ample cone and the associated fractal for surfaces within a class of K3 surfaces with Picard number n = 4 (see Figures 1, 3, 4, and 5). As far as the author is aware, the associated fractal has not been studied in any great depth for any ample cone for which the fractal has a non-integer dimension, except the one in [Ba3]. The fractal in that case is Cantor-like (it is a subset of S) and rigorous bounds on its Hausdorff dimension are calculated in [Ba1]. The Hausdorff dimension of the fractal of this paper is estimated to be 1.296± .010. Our second main result is to relate the Hausdorff dimension of the fractal to the growth of the height of curves for an orbit of curves on a surface in this class. Precisely, let V be a surface within our class of K3 surfaces and let A = Aut(V/K) be its group of automorphisms over a sufficiently large number field K. Let D be an ample divisor on V and let C be a curve on V . Define NA(C)(t,D) = #{C′ ∈ A(C) : C′ ·D < t}. Here we have abused notation by letting C′ also represent the divisor class that contains C′. The intersection C′ ·D should be thought of as a logarithmic height of 2000 Mathematics Subject Classification. 14J28, 14J50, 11D41, 11D72, 11H56, 11G10, 37F35, 37D05.
- Published
- 2011
12. Locally Indecomposable Galois Representations
- Author
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Eknath Ghate and Vinayak Vatsal
- Subjects
Pure mathematics ,Galois cohomology ,General Mathematics ,Fundamental theorem of Galois theory ,010102 general mathematics ,Galois group ,Galois module ,01 natural sciences ,Normal basis ,Embedding problem ,symbols.namesake ,0103 physical sciences ,symbols ,010307 mathematical physics ,Galois extension ,0101 mathematics ,Indecomposable module ,Mathematics - Abstract
In a previous paper the authors showed that, under some technical conditions, the local Galois representations attached to the members of a non-CM family of ordinary cusp forms are indecomposable for all except possibly finitely many members of the family. In this paper we use deformation theoretic methods to give examples of non-CM families for which every classical member of weight at least two has a locally indecomposable Galois representation. School of Mathematics, Tata Institute of Fundamental Research, Mumbai 400005, India. e-mail: eghate@math.tifr.res.in Department of Mathematics, University of British Columbia, Vancouver, BC e-mail: vatsal@math.ubc.ca Received by the editors August 5, 2008. Published electronically December 29, 2010. AMS subject classification: 11F80. 1
- Published
- 2011
13. Ruled Exceptional Surfaces and the Poles of Motivic Zeta Functions
- Author
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Bart Rodrigues
- Subjects
Surface (mathematics) ,Pure mathematics ,Intersection ,General Mathematics ,Open problem ,Geometry ,A fibers ,Mathematics - Abstract
In this paper we study ruled surfaces which appear as exceptional surface in a succession of blowing-ups. In particular we prove that the e-invariant of such a ruled exceptional surface E is strictly positive whenever its intersection with the other exceptional surfaces does not contain a fiber (of E). This fact immediately enables us to resolve an open problem concerning an intersection configuration on such a ruled exceptional surface consisting of three nonintersecting sections. In the second part of the paper we apply the non-vanishing of e to the study of the poles of the well-known topological, Hodge and motivic zeta functions.
- Published
- 2007
14. On the Curves Associated to Certain Rings of Automorphic Forms
- Author
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Kamal Khuri-Makdisi
- Subjects
Pure mathematics ,Quaternion algebra ,General Mathematics ,010102 general mathematics ,Automorphic form ,Complex multiplication ,Congruence relation ,01 natural sciences ,Algebra ,Elliptic curve ,0103 physical sciences ,Representation ring ,010307 mathematical physics ,Compactification (mathematics) ,0101 mathematics ,Hecke operator ,Mathematics - Abstract
In a 1987 paper, Gross introduced certain curves associated to a definite quaternion algebra B over Q; he then proved an analog of his result with Zagier for these curves. In Gross’ paper, the curves were defined in a somewhat ad hoc manner. In this article, we present an interpretation of these curves as projective varieties arising from graded rings of automorphic forms on B×, analogously to the construction in the Satake compactification. To define such graded rings, one needs to introduce a “multiplication” of automorphic forms that arises from the representation ring of B×. The resulting curves are unions of projective lines equipped with a collection of Hecke correspondences. They parametrize two-dimensional complex tori with quaternionic multiplication. In general, these complex tori are not abelian varieties; they are algebraic precisely when they correspond to CM points on these curves, and are thus isogenous to a product E × E, where E is an elliptic curve with complex multiplication. For these CM points one can make a relation between the action of the p-th Hecke operator and Frobenius at p, similar to the well-known congruence relation of Eichler and Shimura.
- Published
- 2001
15. Stable Bi-Period Summation Formula and Transfer Factors
- Author
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Yuval Z. Flicker
- Subjects
Discrete mathematics ,Pure mathematics ,Transfer (group theory) ,Conjugacy class ,Group (mathematics) ,General Mathematics ,Automorphic form ,Fundamental lemma ,Algebraic number field ,Reductive group ,Unit (ring theory) ,Mathematics - Abstract
This paper starts by introducing a bi-periodic summation formula for automorphic forms on a group G(E), with periods by a subgroup G(F), where E/F is a quadratic extension of number fields. The split case, where E = F ! F, is that of the standard trace formula. Then it introduces a notion of stable bi-conjugacy, and stabilizes the geometric side of the bi-period summation formula. Thus weighted sums in the stable bi-conjugacy class are expressed in terms of stable bi-orbital integrals. These stable integrals are on the same endoscopic groups H which occur in the case of standard conjugacy. The spectral side of the bi-period summation formula involves periods, namely integrals overthe group of F-adele points of G ,o f cusp forms on the group ofE-adele points on the group G. Our stabilization suggests that such cusp forms—with non vanishing periods—and the resulting bi-period distributions associated to "periodic" automorphic forms, are related to analogous bi-period distributions associated to "periodic" au- tomorphic forms on the endoscopic symmetric spaces H(E)/H(F). This offers a sharpening of the theory of liftings, where periods play a key role. The stabilization depends on the "fundamental lemma", which conjectures that the unit elements of the Hecke algebras on GandH havematching orbitalintegrals. Evenin stating this conjecture, oneneeds to intro- duce a "transfer factor". A generalization of the standard transfer factor to the bi-periodic case is introduced. The generalization depends on a new definition of the factors even in the standard case. Finally, the fundamental lemma is verified for SL(2). The geometric side of the trace formula for a test function f ! on the group of adele points of a reductive group G over a number field F ,i s as um of orbital integrals off ! parametrized by rational conjugacy classes, in G(F). It is obtained on integrating over the diagonal x = y the kernel Kf ! (x, y )o f ac onvolution operatorr(f ! ). Each such orbital integral can be expressed as an average of weighted sums of such orbital integrals over the stable conjugacy class, which is the set of rational points in the conjugacy class under the points of the group over the algebraic closure. Each such weighted sum is conjecturally related to a stable (a sum where all coefficients are equal to 1) such sum on an endoscopic group H of the group G.T his process of stabilization has been introduced by Langlands to establishliftingofautomorphicandadmissiblerepresentationsfromtheendoscopicgroups H to the original group G. The purpose of this paper is to develop an analogue in the context of the symmetric space G(E)/G(F), where E/F is a quadratic number field extension. Integrating the kernel Kf ! (x, y )o f the convolution operatorr(f ! ) for the test function f ! on the group of E- adele points of the group G over two independent variables x and y in the subgroup of F-adele points of G, we obtain a sum of bi-orbital integrals of f ! over rational bi-conjugacy classes. We introduce a notion of stable bi-conjugacy, and stabilize the geometric side of the bi-period summation formula. Thus we express the weighted sums in the stable bi
- Published
- 1999
16. Ward’s Solitons II: Exact Solutions
- Author
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Christopher Kumar Anand
- Subjects
Surface (mathematics) ,Pure mathematics ,Function field of an algebraic variety ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Holomorphic function ,Dimension of an algebraic variety ,Algebraic geometry ,01 natural sciences ,Algebraic cycle ,0103 physical sciences ,Real algebraic geometry ,010307 mathematical physics ,0101 mathematics ,Differential algebraic geometry ,Mathematics - Abstract
In a previous paper, we gave a correspondence between certain exact solutions to a (2 + 1)-dimensional integrable Chiral Model and holomorphic bundles on a compact surface. In this paper, we use algebraic geometry to derive a closed-form expression for those solutions and show by way of examples how the algebraic data which parametrise the solution space dictates the behaviour of the solutions.
- Published
- 1998
17. The epsilon constant conjecture for higher dimensional unramified twists of (1)
- Author
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Werner Bley and Alessandro Cobbe
- Subjects
Pure mathematics ,Conjecture ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Constant (mathematics) ,01 natural sciences ,Mathematics - Abstract
Let $N/K$ be a finite Galois extension of p-adic number fields, and let $\rho ^{\mathrm {nr}} \colon G_K \longrightarrow \mathrm {Gl}_r({{\mathbb Z}_{p}})$ be an r-dimensional unramified representation of the absolute Galois group $G_K$ , which is the restriction of an unramified representation $\rho ^{\mathrm {nr}}_{{{\mathbb Q}}_{p}} \colon G_{{\mathbb Q}_{p}} \longrightarrow \mathrm {Gl}_r({{\mathbb Z}_{p}})$ . In this paper, we consider the $\mathrm {Gal}(N/K)$ -equivariant local $\varepsilon $ -conjecture for the p-adic representation $T = \mathbb Z_p^r(1)(\rho ^{\mathrm {nr}})$ . For example, if A is an abelian variety of dimension r defined over ${{\mathbb Q}_{p}}$ with good ordinary reduction, then the Tate module $T = T_p\hat A$ associated to the formal group $\hat A$ of A is a p-adic representation of this form. We prove the conjecture for all tame extensions $N/K$ and a certain family of weakly and wildly ramified extensions $N/K$ . This generalizes previous work of Izychev and Venjakob in the tame case and of the authors in the weakly and wildly ramified case.
- Published
- 2021
18. On the triple correlations of fractional parts of
- Author
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Aled Walker and Niclas Technau
- Subjects
Pure mathematics ,010201 computation theory & mathematics ,General Mathematics ,010102 general mathematics ,0102 computer and information sciences ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
For fixed$\alpha \in [0,1]$, consider the set$S_{\alpha ,N}$of dilated squares$\alpha , 4\alpha , 9\alpha , \dots , N^2\alpha \, $modulo$1$. Rudnick and Sarnak conjectured that, for Lebesgue, almost all such$\alpha $the gap-distribution of$S_{\alpha ,N}$is consistent with the Poisson model (in the limit asNtends to infinity). In this paper, we prove a new estimate for the triple correlations associated with this problem, establishing an asymptotic expression for the third moment of the number of elements of$S_{\alpha ,N}$in a random interval of length$L/N$, provided that$L> N^{1/4+\varepsilon }$. The threshold of$\tfrac {1}{4}$is substantially smaller than the threshold of$\tfrac {1}{2}$(which is the threshold that would be given by a naïve discrepancy estimate).Unlike the theory of pair correlations, rather little is known about triple correlations of the dilations$(\alpha a_n \, \text {mod } 1)_{n=1}^{\infty } $for a nonlacunary sequence$(a_n)_{n=1}^{\infty } $of increasing integers. This is partially due to the fact that the second moment of the triple correlation function is difficult to control, and thus standard techniques involving variance bounds are not applicable. We circumvent this impasse by using an argument inspired by works of Rudnick, Sarnak, and Zaharescu, and Heath-Brown, which connects the triple correlation function to some modular counting problems.In Appendix B, we comment on the relationship between discrepancy and correlation functions, answering a question of Steinerberger.
- Published
- 2021
19. On Homogeneous Images of Compact Ordered Spaces
- Author
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Jacek Nikiel and E. D. Tymchatyn
- Subjects
Discrete mathematics ,Pure mathematics ,Continuum (topology) ,General Mathematics ,First-countable space ,010102 general mathematics ,Hausdorff space ,Mathematics::General Topology ,Disjoint sets ,01 natural sciences ,Jordan curve theorem ,symbols.namesake ,Metrization theorem ,0103 physical sciences ,Homogeneous space ,symbols ,010307 mathematical physics ,0101 mathematics ,Indecomposable module ,Mathematics - Abstract
We answer a 1975 question of G. R. Gordh by showing that if X is a homogeneous compactum which is the continuous image of a compact ordered space then at least one of the following holds: (i) X is metrizable, (ii) dimX = 0 or (iii) X is a union of finitely many pairwise disjoint generalized simple closed curves. We begin to examine the structure of homogeneous 0-dimensional spaces which are continuous images of ordered compacta. 1. Introduction. The aim of this paper is to investigate homogeneous spaces which are continuous images of ordered compacta. In 1975, G. R. Gordh proved that if a homo geneous and hereditarily unicoherent continuum is the continuous image of an ordered compactum, then it is metrizable, and so indecomposable (7, Theorem 3). Further, he asked if, in general, every homogeneous continuum which is the continuous image of an ordered compactum must be either metrizable or a generalized simple closed curve. Our Theorem 1 provides an affirmative answer to Gordh's question. Moreover, in Theorem 2, we prove that a homogeneous space which is not 0-dimensional and which is the continuous image of an ordered compactum is either metrizable or a union of finitely many pairwise disjoint generalized simple closed curves. Our methods of proof involve characterizations of continuous images of arcs obtained in ( 16) in terms of cyclic elements and T-sets. When dealing with the class A of all homogeneous and 0-dimensional spaces which are the continuous images of ordered compacta, the situation becomes less clear. By a recent theorem of M. Bell, each member of A is first countable. Moreover, by a result of (18), each member of A can be embedded into a dendron. We give a rather simple construction leading to a wide subclass of A. In particular, we show that not all members of A are orderable, and that there exists a strongly homogeneous space X which is the continuous image of an ordered compactum and which is not first countable. It follows that X $ A. Our investigations of the class A led to some natural questions which are stated at the end of the paper. All spaces considered in this paper are Hausdorff.
- Published
- 1993
20. Torsion in thin regions of Khovanov homology
- Author
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Adam M. Lowrance, Alex Chandler, Radmila Sazdanovic, and Victor Summers
- Subjects
Khovanov homology ,Pure mathematics ,Conjecture ,General Mathematics ,010102 general mathematics ,Diagonal ,Geometric Topology (math.GT) ,Torus ,Mathematics::Algebraic Topology ,Mathematics::Geometric Topology ,01 natural sciences ,Mathematics - Geometric Topology ,Mathematics::K-Theory and Homology ,Mathematics::Quantum Algebra ,57M25, 57M27 ,0103 physical sciences ,FOS: Mathematics ,Torsion (algebra) ,010307 mathematical physics ,0101 mathematics ,Link (knot theory) ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
In the integral Khovanov homology of links, the presence of odd torsion is rare. Homologically thin links, that is links whose Khovanov homology is supported on two adjacent diagonals, are known to only contain $\mathbb{Z}_2$ torsion. In this paper, we prove a local version of this result. If the Khovanov homology of a link is supported in two adjacent diagonals over a range of homological gradings and the Khovanov homology satisfies some other mild restrictions, then the Khovanov homology of that link has only $\mathbb{Z}_2$ torsion over that range of homological gradings. These conditions are then shown to be met by an infinite family of 3-braids, strictly containing all 3-strand torus links, thus giving a partial answer to Sazdanovic and Przytycki's conjecture that 3-braids have only $\mathbb{Z}_2$ torsion in Khovanov homology. We also give explicit computations of integral Khovanov homology for all links in this family., Comment: 20 pages, 11 figures. Section 4 has been simplified
- Published
- 2021
21. Rank conditions for finite group actions on 4-manifolds
- Author
-
Semra Pamuk and Ian Hambleton
- Subjects
Pure mathematics ,Finite group ,57M60, 57S17, 20J06 ,General Mathematics ,010102 general mathematics ,Geometric topology ,Geometric Topology (math.GT) ,Algebraic topology ,Rank (differential topology) ,Mathematics::Algebraic Topology ,01 natural sciences ,Cohomology ,Set (abstract data type) ,Mathematics - Geometric Topology ,Mathematics::K-Theory and Homology ,0103 physical sciences ,Spectral sequence ,FOS: Mathematics ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Algebraic Topology (math.AT) ,Mathematics - Algebraic Topology ,010307 mathematical physics ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
Let M be a closed, connected, orientable topological 4-manifold, and G be a finite group acting topologically and locally linearly on M. In this paper we investigate the Borel spectral sequence for the G-equivariant cohomology of M, and establish new bounds on the rank of G for homologically trivial actions with discrete singular set., 22 pages (v2). Accepted for publication in the Canadian Journal of Mathematics
- Published
- 2021
22. Integral Kernels with Reflection Group Invariance
- Author
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Charles F. Dunkl
- Subjects
Pure mathematics ,Group (mathematics) ,General Mathematics ,010102 general mathematics ,Coxeter group ,Mathematical analysis ,Spherical harmonics ,01 natural sciences ,Classical orthogonal polynomials ,Conjugacy class ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Reflection group ,Laplace operator ,Dunkl operator ,Mathematics - Abstract
Root systems and Coxeter groups are important tools in multivariable analysis. This paper is concerned with differential-difference and integral operators, and orthogonality structures for polynomials associated to Coxeter groups. For each such group, the structures allow as many parameters as the number of conjugacy classes of reflections. The classical orthogonal polynomials of Gegenbauer and Jacobi type appear in this theory as two-dimensional cases. For each Coxeter group and admissible choice of parameters there is a structure analogous to spherical harmonics which relies on the connection between a Laplacian operator and orthogonality on the unit sphere with respect to a group-invariant measure. The theory has been developed in several papers of the author [4,5,6,7]. In this paper, the emphasis is on the study of an intertwining operator which allows the transfer of certain results about ordinary harmonic polynomials to those associated to Coxeter groups. In particular, a formula and a bound are obtained for the Poisson kernel.
- Published
- 1991
23. Homotopy Theory of Diagrams and CW-Complexes Over a Category
- Author
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Robert J. Piacenza
- Subjects
Discrete mathematics ,Pure mathematics ,Homotopy category ,Brown's representability theorem ,Model category ,Computer Science::Information Retrieval ,General Mathematics ,Homotopy ,010102 general mathematics ,Whitehead theorem ,Mathematics::Algebraic Topology ,01 natural sciences ,Weak equivalence ,n-connected ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,Homotopy hypothesis ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
The purpose of this paper is to introduce the notion of a CW complex over a topological category. The main theorem of this paper gives an equivalence between the homotopy theory of diagrams of spaces based on a topological category and the homotopy theory of CW complexes over the same base category.A brief description of the paper goes as follows: in Section 1 we introduce the homotopy category of diagrams of spaces based on a fixed topological category. In Section 2 homotopy groups for diagrams are defined. These are used to define the concept of weak equivalence and J-n equivalence that generalize the classical definition. In Section 3 we adapt the classical theory of CW complexes to develop a cellular theory for diagrams. In Section 4 we use sheaf theory to define a reasonable cohomology theory of diagrams and compare it to previously defined theories. In Section 5 we define a closed model category structure for the homotopy theory of diagrams. We show this Quillen type homotopy theory is equivalent to the homotopy theory of J-CW complexes. In Section 6 we apply our constructions and results to prove a useful result in equivariant homotopy theory originally proved by Elmendorf by a different method.
- Published
- 1991
24. Further inequalities and properties of p-inner parallel bodies
- Author
-
Dongmeng Xi, Zhenbing Zeng, and Yingying Lou
- Subjects
Pure mathematics ,Inequality ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics ,media_common - Abstract
A. R. Martínez Fernández obtained upper bounds for quermassintegrals of the p-inner parallel bodies: an extension of the classical inner parallel body to the $L_p$ -Brunn-Minkowski theory. In this paper, we establish (sharp) upper and lower bounds for quermassintegrals of p-inner parallel bodies. Moreover, the sufficient and necessary conditions of the equality case for the main inequality are obtained, which characterize the so-called tangential bodies.
- Published
- 2020
25. Large values of Dirichlet L-functions at zeros of a class of L-functions
- Author
-
Junxian Li
- Subjects
Pure mathematics ,symbols.namesake ,Class (set theory) ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,symbols ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Dirichlet distribution ,Mathematics - Abstract
In this paper, we are interested in obtaining large values of Dirichlet L-functions evaluated at zeros of a class of L-functions, that is, $$ \begin{align*}\max_{\substack{F(\rho)=0\\ T\leq \Im \rho \leq 2T}}L(\rho,\chi), \end{align*} $$ where $\chi $ is a primitive Dirichlet character and F belongs to a class of L-functions. The class we consider includes L-functions associated with automorphic representations of $GL(n)$ over ${\mathbb {Q}}$ .
- Published
- 2020
26. Khovanov–Rozansky homology for infinite multicolored braids
- Author
-
Michael Willis
- Subjects
Large class ,Khovanov homology ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Geometric Topology (math.GT) ,Limiting ,Homology (mathematics) ,Mathematics::Geometric Topology ,01 natural sciences ,Mathematics - Geometric Topology ,Tensor product ,Mathematics::Category Theory ,Mathematics::Quantum Algebra ,Mathematics - Quantum Algebra ,57M25, 57M27 ,0103 physical sciences ,FOS: Mathematics ,Braid ,Quantum Algebra (math.QA) ,010307 mathematical physics ,0101 mathematics ,Twist ,Mathematics - Abstract
We define a limiting $\mathfrak{sl}_N$ Khovanov-Rozansky homology for semi-infinite positive multi-colored braids, and we show that this limiting homology categorifies a highest-weight projector for a large class of such braids. This effectively completes the extension of Cautis' similar result for infinite twist braids, begun in our earlier papers with Islambouli and Abel. We also present several similar results for other families of semi-infinite and bi-infinite multi-colored braids., 37 pages, 13 figures
- Published
- 2020
27. On the structure of Kac–Moody algebras
- Author
-
Timothée Marquis and UCL - SST/IRMP - Institut de recherche en mathématique et physique
- Subjects
Nipotent algebras ,Pure mathematics ,010308 nuclear & particles physics ,General Mathematics ,010102 general mathematics ,Structure (category theory) ,Solvable algebras ,Mathematics - Rings and Algebras ,01 natural sciences ,Nilpotent ,Bracket (mathematics) ,Rings and Algebras (math.RA) ,Homogeneous ,Mathematics::Quantum Algebra ,0103 physical sciences ,FOS: Mathematics ,Cartan matrix ,Kac-Moody algebras ,0101 mathematics ,Algebra over a field ,Element (category theory) ,Mathematics::Representation Theory ,17B67, 17B30 ,Mathematics - Abstract
Let $A$ be a symmetrisable generalised Cartan matrix, and let $\mathfrak g(A)$ be the corresponding Kac-Moody algebra. In this paper, we address the following fundamental question on the structure of $\mathfrak g(A)$: given two homogeneous elements $x,y \in \mathfrak g(A)$, when is their bracket $[x,y]$ a nonzero element? As an application of our results, we give a description of the solvable and nilpotent graded subalgebras of $\mathfrak g(A)$., 32 pages. Final version, to appear in Canadian Journal of Mathematics
- Published
- 2020
28. Boundedness of Differential Transforms for Heat Semigroups Generated by Schrödinger Operators
- Author
-
José L. Torrea and Zhang Chao
- Subjects
Pure mathematics ,Sequence ,Series (mathematics) ,Semigroup ,General Mathematics ,010102 general mathematics ,Singular integral ,01 natural sciences ,Operator (computer programming) ,Bounded function ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Differential (infinitesimal) ,Laplace operator ,Mathematics - Abstract
In this paper we analyze the convergence of the following type of series $$\begin{eqnarray}T_{N}^{{\mathcal{L}}}f(x)=\mathop{\sum }_{j=N_{1}}^{N_{2}}v_{j}\big(e^{-a_{j+1}{\mathcal{L}}}f(x)-e^{-a_{j}{\mathcal{L}}}f(x)\big),\quad x\in \mathbb{R}^{n},\end{eqnarray}$$ where ${\{e^{-t{\mathcal{L}}}\}}_{t>0}$ is the heat semigroup of the operator ${\mathcal{L}}=-\unicode[STIX]{x1D6E5}+V$ with $\unicode[STIX]{x1D6E5}$ being the classical laplacian, the nonnegative potential $V$ belonging to the reverse Hölder class $RH_{q}$ with $q>n/2$ and $n\geqslant 3$, $N=(N_{1},N_{2})\in \mathbb{Z}^{2}$ with $N_{1}, ${\{v_{j}\}}_{j\in \mathbb{Z}}$ is a bounded real sequences, and ${\{a_{j}\}}_{j\in \mathbb{Z}}$ is an increasing real sequence.Our analysis will consist in the boundedness, in $L^{p}(\mathbb{R}^{n})$ and in $BMO(\mathbb{R}^{n})$, of the operators $T_{N}^{{\mathcal{L}}}$ and its maximal operator $T^{\ast }f(x)=\sup _{N}T_{N}^{{\mathcal{L}}}f(x)$.It is also shown that the local size of the maximal differential transform operators (with $V=0$) is the same with the order of a singular integral for functions $f$ having local support. Moreover, if ${\{v_{j}\}}_{j\in \mathbb{Z}}\in \ell ^{p}(\mathbb{Z})$, we get an intermediate size between the local size of singular integrals and Hardy–Littlewood maximal operator.
- Published
- 2020
29. Generalized Beilinson Elements and Generalized Soulé Characters
- Author
-
Kenji Sakugawa
- Subjects
Pure mathematics ,Polylogarithm ,Cyclotomic character ,Generalization ,General Mathematics ,010102 general mathematics ,Algebraic number field ,Cyclotomic field ,01 natural sciences ,Image (mathematics) ,Character (mathematics) ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
The generalized Soulé character was introduced by H. Nakamura and Z. Wojtkowiak and is a generalization of Soulé’s cyclotomic character. In this paper, we prove that certain linear sums of generalized Soulé characters essentially coincide with the image of generalized Beilinson elements in K-groups under Soulé’s higher regulator maps. This result generalizes Huber–Wildeshaus’ theorem, which is a cyclotomic field case of our results, to an arbitrary number fields.
- Published
- 2020
30. Maximal Inequalities of Noncommutative Martingale Transforms
- Author
-
Fedor Sukochev, Yong Jiao, and Dejian Zhou
- Subjects
Atomic decomposition ,Pure mathematics ,Mathematics::Operator Algebras ,General Mathematics ,Algebraic number ,Martingale (probability theory) ,Mathematical proof ,Noncommutative geometry ,Mathematics - Abstract
In this paper, we investigate noncommutative symmetric and asymmetric maximal inequalities associated with martingale transforms and fractional integrals. Our proofs depend on some recent advances on algebraic atomic decomposition and the noncommutative Gundy decomposition. We also prove several fractional maximal inequalities.
- Published
- 2019
31. Maximal Operator for the Higher Order Calderón Commutator
- Author
-
Xudong Lai
- Subjects
42B20, 42B25 ,Mathematics::Functional Analysis ,Pure mathematics ,Multilinear map ,General Mathematics ,010102 general mathematics ,Mathematics::Classical Analysis and ODEs ,Commutator (electric) ,Space (mathematics) ,01 natural sciences ,law.invention ,010101 applied mathematics ,Mathematics - Classical Analysis and ODEs ,law ,Product (mathematics) ,Maximal operator ,Order (group theory) ,0101 mathematics ,Mathematics ,Weighted space - Abstract
In this paper, we investigate the weighted multilinear boundedness properties of the maximal higher order Calder\'on commutator for the dimensions larger than two. We establish all weighted multilinear estimates on the product of the $L^p(\mathbb{R}^d,w)$ space, including some peculiar endpoint estimates of the higher dimensional Calder\'on commutator., Comment: 36 pages, Canadian Journal of Mathematics, to appear. arXiv admin note: text overlap with arXiv:1712.09020
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- 2019
32. Orlicz Addition for Measures and an Optimization Problem for the -divergence
- Author
-
Deping Ye and Shaoxiong Hou
- Subjects
Pure mathematics ,Optimization problem ,General Mathematics ,010102 general mathematics ,f-divergence ,Star (graph theory) ,01 natural sciences ,Dual (category theory) ,Interpretation (model theory) ,010101 applied mathematics ,Affine transformation ,0101 mathematics ,Isoperimetric inequality ,Divergence (statistics) ,Mathematics - Abstract
This paper provides a functional analogue of the recently initiated dual Orlicz–Brunn–Minkowski theory for star bodies. We first propose the Orlicz addition of measures, and establish the dual functional Orlicz–Brunn–Minkowski inequality. Based on a family of linear Orlicz additions of two measures, we provide an interpretation for the famous $f$-divergence. Jensen’s inequality for integrals is also proved to be equivalent to the newly established dual functional Orlicz–Brunn–Minkowski inequality. An optimization problem for the $f$-divergence is proposed, and related functional affine isoperimetric inequalities are established.
- Published
- 2019
33. Slice-torus Concordance Invariants and Whitehead Doubles of Links
- Author
-
Alberto Cavallo and Carlo Collari
- Subjects
Pure mathematics ,General Mathematics ,Concordance ,Computation ,010102 general mathematics ,Geometric Topology (math.GT) ,Torus ,Mathematics::Geometric Topology ,01 natural sciences ,Mathematics - Geometric Topology ,Link concordance ,57M25, 57M27 ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Invariant (mathematics) ,Link (knot theory) ,Mathematics::Symplectic Geometry ,Mathematics ,Slice genus - Abstract
In the present paper we extend the definition of slice-torus invariant to links. We prove a few properties of the newly-defined slice-torus link invariants: the behaviour under crossing change, a slice genus bound, an obstruction to strong sliceness, and a combinatorial bound. Furthermore, we provide an application to the computation of the splitting number. Finally, we use the slice-torus link invariants, and the Whitehead doubling to define new strong concordance invariants for links, which are proven to be independent from the corresponding slice-torus link invariant., 31 pages, 19 figures, 4 tables. Improved exposition, typos fixed, slight improvement of Propositions 2.10 and 3.5, and added a comment on a result of A. Conway related to Theorem 1.4. Comments are welcome!
- Published
- 2019
34. On Annelidan, Distributive, and Bézout Rings
- Author
-
Ryszard Mazurek and Greg Marks
- Subjects
Pure mathematics ,Ring (mathematics) ,Mathematics::Commutative Algebra ,General Mathematics ,010102 general mathematics ,Distributive lattice ,01 natural sciences ,Prime (order theory) ,010101 applied mathematics ,Annihilator ,Chain (algebraic topology) ,Distributive property ,Ideal (ring theory) ,0101 mathematics ,Symmetry (geometry) ,Mathematics - Abstract
A ring is called right annelidan if the right annihilator of any subset of the ring is comparable with every other right ideal. In this paper we develop the connections between this class of rings and the classes of right Bézout rings and rings whose right ideals form a distributive lattice. We obtain results on localization of right annelidan rings at prime ideals, chain conditions that entail left-right symmetry of the annelidan condition, and construction of completely prime ideals.
- Published
- 2019
35. Calabi–Yau Quotients of Hyperkähler Four-folds
- Author
-
Alice Garbagnati, Chiara Camere, Giovanni Mongardi, Camere, Chiara, Garbagnati, Alice, and Mongardi, Giovanni
- Subjects
irreducible holomorphic symplectic manifold, Hyperkähler manifold, Calabi-Yau 4-fold, Borcea-Voisin construction, automorphism, quotient map, non symplectic involution ,automorphism ,Pure mathematics ,quotient map ,General Mathematics ,010102 general mathematics ,Hyperkähler manifold ,irreducible holomorphic symplectic manifold ,Calabi-Yau 4-fold ,Borcea-Voisin construction ,non symplectic involution ,Automorphism ,01 natural sciences ,Mathematics::Algebraic Geometry ,0103 physical sciences ,Calabi–Yau manifold ,Mathematics::Differential Geometry ,010307 mathematical physics ,0101 mathematics ,Mathematics::Symplectic Geometry ,Quotient ,Mathematics - Abstract
The aim of this paper is to construct Calabi–Yau 4-folds as crepant resolutions of the quotients of a hyperkähler 4-fold $X$ by a non-symplectic involution $\unicode[STIX]{x1D6FC}$. We first compute the Hodge numbers of a Calabi–Yau constructed in this way in a general setting, and then we apply the results to several specific examples of non-symplectic involutions, producing Calabi–Yau 4-folds with different Hodge diamonds. Then we restrict ourselves to the case where $X$ is the Hilbert scheme of two points on a K3 surface $S$, and the involution $\unicode[STIX]{x1D6FC}$ is induced by a non-symplectic involution on the K3 surface. In this case we compare the Calabi–Yau 4-fold $Y_{S}$, which is the crepant resolution of $X/\unicode[STIX]{x1D6FC}$, with the Calabi–Yau 4-fold $Z_{S}$, constructed from $S$ through the Borcea–Voisin construction. We give several explicit geometrical examples of both these Calabi–Yau 4-folds, describing maps related to interesting linear systems as well as a rational $2:1$ map from $Z_{S}$ to $Y_{S}$.
- Published
- 2019
36. On the Weak Order of Coxeter Groups
- Author
-
Matthew Dyer
- Subjects
Pure mathematics ,Conjecture ,General Mathematics ,010102 general mathematics ,Coxeter group ,Group Theory (math.GR) ,010103 numerical & computational mathematics ,01 natural sciences ,Power set ,Bruhat order ,Complete lattice ,Lattice (order) ,FOS: Mathematics ,20F55 (Primary) 17B22(Secondary) ,Closure operator ,0101 mathematics ,Mathematics - Group Theory ,Mathematics - Abstract
This paper provides some evidence for conjectural relations between extensions of (right) weak order on Coxeter groups, closure operators on root systems, and Bruhat order. The conjecture focused upon here refines an earlier question as to whether the set of initial sections of reflection orders, ordered by inclusion, forms a complete lattice. Meet and join in weak order are described in terms of a suitable closure operator. Galois connections are defined from the power set of W to itself, under which maximal subgroups of certain groupoids correspond to certain complete meet subsemilattices of weak order. An analogue of weak order for standard parabolic subsets of any rank of the root system is defined, reducing to the usual weak order in rank zero, and having some analogous properties in rank one (and conjecturally in general)., 37 pages, submitted
- Published
- 2019
37. The Steklov Problem on Differential Forms
- Author
-
Mikhail Karpukhin
- Subjects
Pure mathematics ,Differential form ,General Mathematics ,Operator (physics) ,010102 general mathematics ,Spectral properties ,01 natural sciences ,law.invention ,law ,0103 physical sciences ,Shape optimization ,010307 mathematical physics ,0101 mathematics ,Manifold (fluid mechanics) ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper we study spectral properties of the Dirichlet-to-Neumann map on differential forms obtained by a slight modification of the definition due to Belishev and Sharafutdinov. The resulting operator $\unicode[STIX]{x039B}$ is shown to be self-adjoint on the subspace of coclosed forms and to have purely discrete spectrum there. We investigate properties of eigenvalues of $\unicode[STIX]{x039B}$ and prove a Hersch–Payne–Schiffer type inequality relating products of those eigenvalues to eigenvalues of the Hodge Laplacian on the boundary. Moreover, non-trivial eigenvalues of $\unicode[STIX]{x039B}$ are always at least as large as eigenvalues of the Dirichlet-to-Neumann map defined by Raulot and Savo. Finally, we remark that a particular case of $p$-forms on the boundary of a $2p+2$-dimensional manifold shares many important properties with the classical Steklov eigenvalue problem on surfaces.
- Published
- 2019
38. A CR Analogue of Yau’s Conjecture on Pseudoharmonic Functions of Polynomial Growth
- Author
-
Shu-Cheng Chang, Yingbo Han, Der-Chen Chang, and Jingzhi Tie
- Subjects
Pure mathematics ,Polynomial ,Conjecture ,Degree (graph theory) ,Volume growth ,General Mathematics ,Mean value ,Space (mathematics) ,Heat kernel ,Mathematics ,Sobolev inequality - Abstract
In this paper, we first derive the CR volume doubling property, CR Sobolev inequality, and the mean value inequality. We then apply them to prove the CR analogue of Yau’s conjecture on the space consisting of all pseudoharmonic functions of polynomial growth of degree at most$d$in a complete noncompact pseudohermitian$(2n+1)$-manifold. As a by-product, we obtain the CR analogue of the volume growth estimate and the Gromov precompactness theorem.
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- 2019
39. Boundary Quotient -algebras of Products of Odometers
- Author
-
Dilian Yang and Hui Li
- Subjects
Product system ,Pure mathematics ,Semigroup ,If and only if ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,0101 mathematics ,Zappa–Szép product ,01 natural sciences ,Odometer ,Quotient ,Mathematics - Abstract
In this paper, we study the boundary quotient $\text{C}^{\ast }$-algebras associated with products of odometers. One of our main results shows that the boundary quotient $\text{C}^{\ast }$-algebra of the standard product of $k$ odometers over $n_{i}$-letter alphabets $(1\leqslant i\leqslant k)$ is always nuclear, and that it is a UCT Kirchberg algebra if and only if $\{\ln n_{i}:1\leqslant i\leqslant k\}$ is rationally independent, if and only if the associated single-vertex $k$-graph $\text{C}^{\ast }$-algebra is simple. To achieve this, one of our main steps is to construct a topological $k$-graph such that its associated Cuntz–Pimsner $\text{C}^{\ast }$-algebra is isomorphic to the boundary quotient $\text{C}^{\ast }$-algebra. Some relations between the boundary quotient $\text{C}^{\ast }$-algebra and the $\text{C}^{\ast }$-algebra $\text{Q}_{\mathbb{N}}$ introduced by Cuntz are also investigated.
- Published
- 2019
40. adic -functions for
- Author
-
Daniel Barrera Salazar and Chris Williams
- Subjects
Pure mathematics ,Distribution (number theory) ,General Mathematics ,010102 general mathematics ,Modular form ,Automorphic form ,Function (mathematics) ,Algebraic number field ,01 natural sciences ,0103 physical sciences ,Eigenform ,010307 mathematical physics ,Isomorphism ,Modular symbol ,0101 mathematics ,Mathematics - Abstract
Since Rob Pollack and Glenn Stevens used overconvergent modular symbols to construct$p$-adic$L$-functions for non-critical slope rational modular forms, the theory has been extended to construct$p$-adic$L$-functions for non-critical slope automorphic forms over totally real and imaginary quadratic fields by the first and second authors, respectively. In this paper, we give an analogous construction over a general number field. In particular, we start by proving a control theorem stating that the specialisation map from overconvergent to classical modular symbols is an isomorphism on the small slope subspace. We then show that if one takes the modular symbol attached to a small slope cuspidal eigenform, then one can construct a ray class distribution from the corresponding overconvergent symbol, which moreover interpolates critical values of the$L$-function of the eigenform. We prove that this distribution is independent of the choices made in its construction. We define the$p$-adic$L$-function of the eigenform to be this distribution.
- Published
- 2019
41. On the Pointwise Bishop–Phelps–Bollobás Property for Operators
- Author
-
Sun Kwang Kim, Vladimir Kadets, Miguel Martín, Han Ju Lee, and Sheldon Dantas
- Subjects
Pointwise ,Pure mathematics ,Property (philosophy) ,General Mathematics ,010102 general mathematics ,Banach space ,Regular polygon ,46B04 (Primary), 46B07, 46B20 (Secondary) ,Space (mathematics) ,Compact operator ,01 natural sciences ,Mathematics - Functional Analysis ,Range (mathematics) ,Dimension (vector space) ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
We study approximation of operators between Banach spaces $X$ and $Y$ that nearly attain their norms in a given point by operators that attain their norms at the same point. When such approximations exist, we say that the pair $(X, Y)$ has the pointwise Bishop-Phelps-Bollob\'as property (pointwise BPB property for short). In this paper we mostly concentrate on those $X$, called universal pointwise BPB domain spaces, such that $(X, Y)$ possesses pointwise BPB property for every $Y$, and on those $Y$, called universal pointwise BPB range spaces, such that $(X, Y)$ enjoys pointwise BPB property for every uniformly smooth $X$. We show that every universal pointwise BPB domain space is uniformly convex and that $L_p(\mu)$ spaces fail to have this property when $p>2$. For universal pointwise BPB range space, we show that every simultaneously uniformly convex and uniformly smooth Banach space fails it if its dimension is greater than one. We also discuss a version of the pointwise BPB property for compact operators., Comment: 19 pages, to appear in the Canadian J. Math. In this version, section 6 and the appendix of the previous version have been removed
- Published
- 2018
42. On the First Zassenhaus Conjecture and Direct Products
- Author
-
M.A. Serrano, Andreas Bächle, and Wolfgang Kimmerle
- Subjects
Ring (mathematics) ,Pure mathematics ,16S34, 16U60, 20C05 ,General Mathematics ,010102 general mathematics ,Sylow theorems ,Group Theory (math.GR) ,Mathematics - Rings and Algebras ,01 natural sciences ,Hall subgroup ,Mathematics::Group Theory ,Rings and Algebras (math.RA) ,0103 physical sciences ,FOS: Mathematics ,Order (group theory) ,010307 mathematical physics ,0101 mathematics ,Abelian group ,Frobenius group ,Mathematics - Group Theory ,Direct product ,Group ring ,Mathematics - Abstract
In this paper we study the behavior of the first Zassenhaus conjecture (ZC1) under direct products as well as the General Bovdi Problem (Gen-BP) which turns out to be a slightly weaker variant of (ZC1). Among others we prove that (Gen-BP) holds for Sylow tower groups, so in particular for the class of supersolvable groups. (ZC1) is established for a direct product of Sylow-by-abelian groups provided the normal Sylow subgroups form together a Hall subgroup. We also show (ZC1) for certain direct products with one of the factors a Frobenius group. We extend the classical HeLP method to group rings with coefficients from any ring of algebraic integers. This is used to study (ZC1) for the direct product $G \times A$, where $A$ is a finite abelian group and $G$ has order at most 95. For most of these groups we show that (ZC1) is valid and for all of them that (Gen-BP) holds. Moreover, we also prove that (Gen-BP) holds for the direct product of a Frobenius group with any finite abelian group., 17 pages. Comments welcome!
- Published
- 2018
43. Mixed Perverse Sheaves on Flag Varieties for Coxeter Groups
- Author
-
Cristian Vay, Simon Riche, and Pramod N. Achar
- Subjects
Hecke algebra ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Coxeter group ,16. Peace & justice ,01 natural sciences ,Diagrammatic reasoning ,Perverse sheaf ,Mathematics::Category Theory ,0103 physical sciences ,Grothendieck group ,010307 mathematical physics ,Abelian category ,0101 mathematics ,Mathematics - Abstract
In this paper we construct an abelian category of mixed perverse sheaves attached to any realization of a Coxeter group, in terms of the associated Elias–Williamson diagrammatic category. This construction extends previous work of the first two authors, where we worked with parity complexes instead of diagrams, and we extend most of the properties known in this case to the general setting. As an application we prove that the split Grothendieck group of the Elias–Williamson diagrammatic category is isomorphic to the corresponding Hecke algebra, for any choice of realization.
- Published
- 2020
- Full Text
- View/download PDF
44. Spherical Fundamental Lemma for Metaplectic Groups
- Author
-
Caihua Luo
- Subjects
Pure mathematics ,Metaplectic group ,Formalism (philosophy) ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Fundamental lemma ,Topology ,01 natural sciences ,Mathematics - Abstract
In this paper, we prove the spherical fundamental lemma for metaplectic group Mp2n based on the formalism of endoscopy theory by J. Adams, D. Renard, and W.-W. Li.
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- 2018
45. Local Dimensions of Measures of Finite Type II: Measures Without Full Support and With Non-regular Probabilities
- Author
-
Kevin G. Hare, Michael Ka Shing Ng, and Kathryn E. Hare
- Subjects
Pure mathematics ,Class (set theory) ,General Mathematics ,010102 general mathematics ,Interval (mathematics) ,Absolute continuity ,01 natural sciences ,Measure (mathematics) ,010305 fluids & plasmas ,Cantor set ,Isolated point ,Dimension (vector space) ,0103 physical sciences ,Hausdorff measure ,0101 mathematics ,Mathematics - Abstract
Consider a finite sequence of linear contractions Sj(x) = px + dj and probabilities pj > 0 with ∑Pj = 1. We are interested in the self-similar measure , of finite type. In this paper we study the multi-fractal analysis of such measures, extending the theory to measures arising from non-regular probabilities and whose support is not necessarily an interval.Under some mild technical assumptions, we prove that there exists a subset of supp μ of full μ and Hausdorff measure, called the truly essential class, for which the set of (upper or lower) local dimensions is a closed interval. Within the truly essential class we show that there exists a point with local dimension exactly equal to the dimension of the support. We give an example where the set of local dimensions is a two element set, with all the elements of the truly essential class giving the same local dimension. We give general criteria for these measures to be absolutely continuous with respect to the associated Hausdorff measure of their support, and we show that the dimension of the support can be computed using only information about the essential class.To conclude, we present a detailed study of three examples. First, we show that the set of local dimensions of the biased Bernoulli convolution with contraction ratio the inverse of a simple Pisot number always admits an isolated point. We give a precise description of the essential class of a generalized Cantor set of finite type, and show that the k-th convolution of the associated Cantor measure has local dimension at x ∊ (0,1) tending to 1 as ft: tends to infinity. Lastly, we show that within a maximal loop class that is not truly essential, the set of upper local dimensions need not be an interval. This is in contrast to the case for finite type measures with regular probabilities and full interval support.
- Published
- 2018
46. The Algebraic de Rham Cohomology of Representation Varieties
- Author
-
Eugene Z. Xia
- Subjects
Pure mathematics ,Astrophysics::High Energy Astrophysical Phenomena ,General Mathematics ,Parameterized complexity ,Torus ,Mathematics - Algebraic Geometry ,General Relativity and Quantum Cosmology ,13D03, 14F40, 14L24, 14Q10, 14R20 ,Natural family ,FOS: Mathematics ,De Rham cohomology ,Variety (universal algebra) ,Algebraic number ,Connection (algebraic framework) ,Representation (mathematics) ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
The SL(2,C)-representation varieties of punctured surfaces form natural families parameterized by holonomies at the punctures. In this paper, we first compute the loci where these varieties are singular for the cases of one-holed and two-holed tori and the four-holed sphere. We then compute the de Rham cohomologies of these varieties of the one-holed torus and the four-holed sphere when the varieties are smooth via the Grothendieck theorem. Furthermore, we produce the explicit Gauss-Manin connection on the natural family of the smooth SL(2,C)-representation variety of the one-holed torus., Comment: Minor stylistic revision from version 1, 21 pages
- Published
- 2018
47. The ER(z)-cohomology of Bℤ/(2q) and ℂℙn
- Author
-
Vitaly Lorman, Nitu Kitchloo, and W. Stephen Wilson
- Subjects
Pure mathematics ,Series (mathematics) ,General Mathematics ,Complex projective space ,010102 general mathematics ,Eilenberg–MacLane space ,01 natural sciences ,Cohomology ,Atiyah–Hirzebruch spectral sequence ,0103 physical sciences ,Spectral sequence ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
The ER(2)-cohomology of Bℤ/(2q) and ℂℙn are computed along with the Atiyah–Hirzebruch spectral sequence for ER(2)*(ℂℙ∞). This, along with other papers in this series, gives us the ER(2)-cohomology of all Eilenberg–MacLane spaces.
- Published
- 2018
48. A Class of Abstract Linear Representations for Convolution Function Algebras over Homogeneous Spaces of Compact Groups
- Author
-
Arash Ghaani Farashahi
- Subjects
Pure mathematics ,Linear representation ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,Convolution power ,01 natural sciences ,Algebra ,Compact group ,Homogeneous ,Homogeneous space ,Invariant measure ,0101 mathematics ,Mathematics - Abstract
This paper introduces a class of abstract linear representations on Banach convolution function algebras over homogeneous spaces of compact groups. LetGbe a compact group andHa closed subgroup ofG. Letμbe the normalizedG-invariant measure over the compact homogeneous spaceG/Hassociated with Weil's formula and. We then present a structured class of abstract linear representations of the Banach convolution function algebrasLp(G/H,μ).
- Published
- 2018
49. Closed Convex Hulls of Unitary Orbits in Certain Simple Real Rank Zero C* -algebras
- Author
-
Ping Wong Ng and Paul Skoufranis
- Subjects
Pure mathematics ,46L05 ,Simplex ,Rank (linear algebra) ,Mathematics::Operator Algebras ,General Mathematics ,010102 general mathematics ,Mathematics - Operator Algebras ,Zero (complex analysis) ,010103 numerical & computational mathematics ,01 natural sciences ,Upper and lower bounds ,Unitary state ,Functional Analysis (math.FA) ,Separable space ,Mathematics - Functional Analysis ,FOS: Mathematics ,Convex combination ,0101 mathematics ,Operator Algebras (math.OA) ,Majorization ,Mathematics - Abstract
In this paper, we characterize the closures of convex hulls of unitary orbits of self-adjoint operators in unital, separable, simple C* -algebras with non-trivial tracial simplex, real rank zero, stable rank one, and strict comparison of projections with respect to tracial states. In addition, an upper bound for the number of unitary conjugates in a convex combination needed to approximate a self-adjoint are obtained.
- Published
- 2017
50. Absolute Continuity of Wasserstein Barycenters Over Alexandrov Spaces
- Author
-
Yin Jiang
- Subjects
010101 applied mathematics ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Hausdorff measure ,0101 mathematics ,Absolute continuity ,Space (mathematics) ,Curvature ,01 natural sciences ,Probability measure ,Mathematics - Abstract
In this paper, we prove that on a compact, n-dimensional Alexandrov space with curvature at least −1, the Wasserstein barycenter of Borel probability measures μ1 ,… , μm is absolutely continuous with respect to the n-dimensional Hausdorff measure if one of them is.
- Published
- 2017
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