1,824 results
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52. Thaine's Method for Circular Units and a Conjecture of Gross
- Author
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Henri Darmon
- Subjects
Stark conjectures ,Conjecture ,Elliott–Halberstam conjecture ,General Mathematics ,010102 general mathematics ,abc conjecture ,Algebraic number field ,01 natural sciences ,Class number formula ,Collatz conjecture ,Combinatorics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Lonely runner conjecture ,Mathematics - Abstract
We formulate a conjecture analogous to Gross' refinement of the Stark conjectures on special values of abelian L-series at s = 0. Some evidence for the conjecture can be obtained, thanks to the fundamental ideas of F. Thaine. 1. Introduction. This paper formulates a refined analogue of the usual class number formula for a real quadratic extension of Q, using circular units. The statement of this conjecture is inspired by an analogous conjecture of Gross (Gr). Strong evidence for this conjecture can be given thanks to F. Thaine's powerful method (Th) for generating relations in ideal class groups using circular units. The first two sections briefly recall Dirichlet's analytic class number formula and Gross's refinement of it; they are there mainly to fix notations and provide motivation. Section 4 states the new conjecture. The remaining sections are devoted to proving various results that support it. ACKNOWLEDGEMENTS. I wish to thank Massimo Bertolini and Benedict Gross for many stimulating conversations on the topics of this paper. NOTATIONS. If K is a number field and w is a place of K lying above a prime v of Q, we denote by Kw the localization of K at w, and let Nw be the order of its residue field. The w-adic norm || || w is normalized so that it is equal to Nw" 1 on uniformizing elements.
- Published
- 1995
53. 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
54. 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
55. Rank conditions for finite group actions on 4-manifolds
- Author
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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
56. 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
57. Strong Boundedness and Strong Convergence in Sequence Spaces
- Author
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Naza Tanović-Miller and Martin Buntinas
- Subjects
Sequence ,Weak convergence ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Convergence (routing) ,Applied mathematics ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Modes of convergence ,Mathematics - Abstract
Strong convergence has been investigated in summability theory and Fourier analysis. This paper extends strong convergence to a topological property of sequence spaces E. The more general property of strong boundedness is also defined and examined. One of the main results shows that for an FK-space E which contains all finite sequences, strong convergence is equivalent to the invariance property E = ℓ ν0. E with respect to coordinatewise multiplication by sequences in the space ℓν0 defined in the paper. Similarly, strong boundedness is equivalent to another invariance E = ℓν.E. The results of the paper are applied to summability fields and spaces of Fourier series.
- Published
- 1991
58. 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
59. Acyclicity of Certain Homeomorphism Groups
- Author
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K. Varadarajan and Parameswaran Sankaran
- Subjects
Combinatorics ,Cantor set ,General Mathematics ,Bounded function ,Simple group ,010102 general mathematics ,0103 physical sciences ,Neighbourhood (graph theory) ,010307 mathematical physics ,0101 mathematics ,Bijection, injection and surjection ,01 natural sciences ,Mathematics - Abstract
Introduction. The concept of a mitotic group was introduced in [3] by Baumslag, Dyer and Heller who showed that mitotic groups were acyclic. In [8] one of the authors introduced the concept of a pseudo-mitotic group, a concept weaker than that of a mitotic group, and showed that pseudo-mitotic groups were acyclic and that the group Gn of homeomorphisms of R n with compact support is pseudo-mitotic. In our present paper we develop techniques to prove pseudomitoticity of certain other homeomorphism groups. In [5] Kan and Thurston observed that the group of set theoretic bijections of Q with bounded support is acyclic. A natural question is to decide whether the group of homeomorphisms of Q (resp. the irrationals / ) with bounded support is acyclic or not. In the present paper we develop techniques to answer this question in the affirmative. Also the techniques developed here enable us to show that the group of homeomorphisms of the Cantor set which are identity in a neighbourhood of 0 and 1 is pseudo-mitotic and hence acyclic. It is worth noticing the contrast between our results and the results of R. D. Anderson in [1] and [2]. Anderson, using his techniques shows that the group of all homeomorphisms of Q, / or the Cantor set is a simple group. He also shows that the group of orientation preserving homeomorphisms of S or S is simple.
- Published
- 1990
60. A Schensted Algorithm Which Models Tensor Representations of the Orthogonal Group
- Author
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Robert A. Proctor
- Subjects
Algebra ,General Mathematics ,Tensor (intrinsic definition) ,010102 general mathematics ,0103 physical sciences ,Orthogonal group ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
This paper is concerned with a combinatorial construction which mysteriously “mimics” or “models” the decomposition of certain reducible representations of orthogonal groups. Although no knowledge of representation theory is needed to understand the body of this paper, a little familiarity is necessary to understand the representation theoretic motivation given in the introduction. Details of the proofs will most easily be understood by people who have had some exposure to Schensted's algorithm or jeu de tacquin.
- Published
- 1990
61. Further inequalities and properties of p-inner parallel bodies
- Author
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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
62. On the compositum of orthogonal cyclic fields of the same odd prime degree
- Author
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Radan Kučera and Cornelius Greither
- Subjects
Annihilation ,Generator (category theory) ,Group (mathematics) ,General Mathematics ,010102 general mathematics ,Prime degree ,Ideal class group ,01 natural sciences ,Combinatorics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Class number ,Unit (ring theory) ,Mathematics - Abstract
The aim of this paper is to study circular units in the compositum K of t cyclic extensions of ${\mathbb {Q}}$ ( $t\ge 2$ ) of the same odd prime degree $\ell $ . If these fields are pairwise arithmetically orthogonal and the number s of primes ramifying in $K/{\mathbb {Q}}$ is larger than $t,$ then a nontrivial root $\varepsilon $ of the top generator $\eta $ of the group of circular units of K is constructed. This explicit unit $\varepsilon $ is used to define an enlarged group of circular units of K, to show that $\ell ^{(s-t)\ell ^{t-1}}$ divides the class number of K, and to prove an annihilation statement for the ideal class group of K.
- Published
- 2020
63. Large values of Dirichlet L-functions at zeros of a class of L-functions
- Author
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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
64. Variation of constants formula and exponential dichotomy for nonautonomous non-densely defined Cauchy problems
- Author
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Pierre Magal and Ousmane Seydi
- Subjects
General Mathematics ,Exponential dichotomy ,010102 general mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Cauchy distribution ,Variation of parameters ,01 natural sciences ,010101 applied mathematics ,Homogeneous ,Boundary value problem ,0101 mathematics ,Persistence (discontinuity) ,Hyperbolic partial differential equation ,Mathematics - Abstract
In this paper, we extend to the non-Hille–Yosida case a variation of constants formula for a nonautonomous and nonhomogeneous Cauchy problems first obtained by Gühring and Räbiger. By using this variation of constants formula, we derive a necessary and sufficient condition for the existence of an exponential dichotomy for the evolution family generated by the associated nonautonomous homogeneous problem. We also prove a persistence result of the exponential dichotomy for small perturbations. Finally, we illustrate our results by considering two examples. The first example is a parabolic equation with nonlocal and nonautonomous boundary conditions, and the second example is an age-structured model that is a hyperbolic equation.
- Published
- 2020
65. Khovanov–Rozansky homology for infinite multicolored braids
- Author
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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
66. On the structure of Kac–Moody algebras
- Author
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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
67. Boundedness of Differential Transforms for Heat Semigroups Generated by Schrödinger Operators
- Author
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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
68. Universal Alternating Semiregular Polytopes
- Author
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Barry Monson and Egon Schulte
- Subjects
Automorphism group ,Transitive relation ,General Mathematics ,010102 general mathematics ,Coxeter group ,Semiregular polytope ,Polytope ,02 engineering and technology ,Symmetry group ,01 natural sciences ,Combinatorics ,Convex polytope ,0202 electrical engineering, electronic engineering, information engineering ,Abstract polytope ,020201 artificial intelligence & image processing ,0101 mathematics ,Mathematics - Abstract
In the classical setting, a convex polytope is said to be semiregular if its facets are regular and its symmetry group is transitive on vertices. This paper continues our study of alternating semiregular abstract polytopes, which have abstract regular facets, still with combinatorial automorphism group transitive on vertices and with two kinds of regular facets occurring in an alternating fashion.Our main concern here is the universal polytope ${\mathcal{U}}_{{\mathcal{P}},{\mathcal{Q}}}$, an alternating semiregular $(n+1)$-polytope defined for any pair of regular $n$-polytopes ${\mathcal{P}},{\mathcal{Q}}$ with isomorphic facets. After a careful look at the local structure of these objects, we develop the combinatorial machinery needed to explain how ${\mathcal{U}}_{{\mathcal{P}},{\mathcal{Q}}}$ can be constructed by “freely assembling” unlimited copies of ${\mathcal{P}}$, ${\mathcal{Q}}$ along their facets in alternating fashion. We then examine the connection group of ${\mathcal{U}}_{{\mathcal{P}},{\mathcal{Q}}}$, and from that prove that ${\mathcal{U}}_{{\mathcal{P}},{\mathcal{Q}}}$ covers any $(n+1)$-polytope ${\mathcal{B}}$ whose facets alternate in any way between various quotients of ${\mathcal{P}}$ or ${\mathcal{Q}}$.
- Published
- 2020
69. 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
70. Maximal Inequalities of Noncommutative Martingale Transforms
- Author
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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
71. Addendum to 'Nearly Countable Dense Homogeneous Spaces'
- Author
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Michael Hrušáak and Jan van Mill
- Subjects
Combinatorics ,Normal subgroup ,Product group ,Dense set ,Homogeneous ,Group (mathematics) ,General Mathematics ,Addendum ,Countable set ,Mathematics ,Group operation - Abstract
This paper provides an addendum to M. Hrusak and J. van Mill “Nearly Countable Dense Homogeneous Spaces.” Canad. J. Math., published online 2013-03-08, http://dx.doi.org/10.4153/ CJM-2013-006-8. It was brought to our attention by Su Gao that the proof of Theorem 5.2 in our paper is incomplete. We are indebted to him for this observation. The aim of this note is to correct this. Theorem 5.2 Let G be a closed subgroup of S∞ and let κ be the number of orbits for the canonical action G × 2N → 2N. Then there is an action of a Polish group H on X = N× [0, 1) such that X has κ H-types of countable dense sets. Proof Let G act on X in the following natural way: (g, (n, t)) 7−→ (g(n), t) for g ∈ G, n ∈ N, t ∈ [0, 1). Put F = { f ∈H (X) : (∀ n ∈ N)( f (n, 0) = (n, 0)) } . Then F is a closed normal subgroup of H (X) and hence is Polish. Moreover, for any two countable dense subsets D and E of N × (0, 1) there exists f ∈ F such that f (D) = E. Treat G also as subgroup of H (X). The Polish semi-direct product group H = G o F acts on X as follows: ((g, f ), x) 7→ ( f ◦ g)(x) for f ∈ F, g ∈ G, x ∈ X. Note that topologically, H = G o F is G× F, but its group operation ∗ is given by (g1, f1) ∗ (g2, f2) = (g1g2, f1g1 f2g 1 ). A typical countable dense subset of X has the form D ∪ A, where D is a countable dense subset of N× (0, 1), and A ⊆ N× {0}. By identifying P(N× {0}) and 2N in the standard way, it is clear that we get what we want. Centro de Ciencias Matematicas, UNAM, A.P. 61-3, Xangari, Morelia, Michoacan, 58089, Mexico e-mail: michael@matmor.unam.mx Faculty of Sciences, Department of Mathematics, VU University Amsterdam, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands e-mail: j.van.mill@vu.nl Received by the editors October 28, 2013. Published electronically November 7, 2013. The first author was supported by a PAPIIT grant IN 102311 and CONACyT grant 177758. The second author is pleased to thank the Centro de Ciencias Matematicas at Morelia for generous hospitality and support. AMS subject classification: 54H05, 03E15, 54E50.
- Published
- 2014
72. 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
- Published
- 2019
73. One-Level Density of Low-lying Zeros of Quadratic and Quartic Hecke -functions
- Author
-
Peng Gao and Liangyi Zhao
- Subjects
Field (physics) ,General Mathematics ,Gaussian ,010102 general mathematics ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,Quadratic equation ,Quartic function ,symbols ,Point (geometry) ,0101 mathematics ,Lying ,Mathematical physics ,Mathematics - Abstract
In this paper we prove some one-level density results for the low-lying zeros of families of quadratic and quartic Hecke $L$-functions of the Gaussian field. As corollaries, we deduce that at least 94.27% and 5%, respectively, of the members of the quadratic family and the quartic family do not vanish at the central point.
- Published
- 2019
74. The Genus of a Random Bipartite Graph
- Author
-
Yifan Jing and Bojan Mohar
- Subjects
Random graph ,General Mathematics ,010102 general mathematics ,0102 computer and information sciences ,01 natural sciences ,05C10, 57M15 ,Combinatorics ,Integer ,010201 computation theory & mathematics ,Genus (mathematics) ,FOS: Mathematics ,Bipartite graph ,Mathematics - Combinatorics ,Almost surely ,Combinatorics (math.CO) ,0101 mathematics ,Constant (mathematics) ,Mathematics - Abstract
Archdeacon and Grable (1995) proved that the genus of the random graph $G\in\mathcal{G}_{n,p}$ is almost surely close to $pn^2/12$ if $p=p(n)\geq3(\ln n)^2n^{-1/2}$. In this paper we prove an analogous result for random bipartite graphs in $\mathcal{G}_{n_1,n_2,p}$. If $n_1\ge n_2 \gg 1$, phase transitions occur for every positive integer $i$ when $p=\Theta((n_1n_2)^{-\frac{i}{2i+1}})$. A different behaviour is exhibited when one of the bipartite parts has constant size, $n_1\gg1$ and $n_2$ is a constant. In that case, phase transitions occur when $p=\Theta(n_1^{-1/2})$ and when $p=\Theta(n_1^{-1/3})$., Comment: 19 pages
- Published
- 2019
75. 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
76. Slice-torus Concordance Invariants and Whitehead Doubles of Links
- Author
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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
77. 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
78. Eigenvalue Optimisation on Flat Tori and Lattice Points in Anisotropically Expanding Domains
- Author
-
Jean Lagacé
- Subjects
General Mathematics ,Dimension (graph theory) ,0211 other engineering and technologies ,02 engineering and technology ,35P20, 11H06, 52C07 ,01 natural sciences ,Dirichlet distribution ,Mathematics - Spectral Theory ,symbols.namesake ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,Neumann boundary condition ,Number Theory (math.NT) ,0101 mathematics ,Remainder ,Spectral Theory (math.SP) ,Eigenvalues and eigenvectors ,Mathematics ,021103 operations research ,Mathematics - Number Theory ,010102 general mathematics ,Mathematical analysis ,Torus ,Mathematics::Spectral Theory ,symbols ,Cube ,Laplace operator ,Analysis of PDEs (math.AP) - Abstract
This paper is concerned with the maximisation of the k'th eigenvalue of the Laplacian amongst flat tori of unit volume in dimension d as k goes to infinity. We show that in any dimension maximisers exist for any given k, but that any sequence of maximisers degenerates as k goes to infinity when the dimension is at most 10. Furthermore, we obtain specific upper and lower bounds for the injectivity radius of any sequence of maximisers. We also prove that flat Klein bottles maximising the k'th eigenvalue of the Laplacian exhibit the same behaviour. These results contrast with those obtained recently by Gittins and Larson, stating that sequences of optimal cuboids for either Dirichlet or Neumann boundary conditions converge to the cube no matter the dimension. We obtain these results via Weyl asymptotics with explicit control of the remainder in terms of the injectivity radius. We reduce the problem at hand to counting lattice points inside anisotropically expanding domains, where we generalise methods of Yu. Kordyukov and A. Yakovlev by considering domains that expand at different rates in various directions., Comment: 20 pages
- Published
- 2019
79. 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
80. 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
81. Titchmarsh’s Method for the Approximate Functional Equations for , , and
- Author
-
Yoshio Tanigawa, T. Makoto Minamide, and Jun Furuya
- Subjects
010101 applied mathematics ,Exponential sum ,General Mathematics ,010102 general mathematics ,Applied mathematics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Let $\unicode[STIX]{x1D701}(s)$ be the Riemann zeta function. In 1929, Hardy and Littlewood proved the approximate functional equation for $\unicode[STIX]{x1D701}^{2}(s)$ with error term $O(x^{1/2-\unicode[STIX]{x1D70E}}((x+y)/|t|)^{1/4}\log |t|)$, where $-1/2. Later, in 1938, Titchmarsh improved the error term by removing the factor $((x+y)/|t|)^{1/4}$. In 1999, Hall showed the approximate functional equations for $\unicode[STIX]{x1D701}^{\prime }(s)^{2},\unicode[STIX]{x1D701}(s)\unicode[STIX]{x1D701}^{\prime \prime }(s)$, and $\unicode[STIX]{x1D701}^{\prime }(s)\unicode[STIX]{x1D701}^{\prime \prime }(s)$ (in the range $0) whose error terms contain the factor $((x+y)/|t|)^{1/4}$. In this paper we remove this factor from these three error terms by using the method of Titchmarsh.
- Published
- 2019
82. Linear Maps Preserving Matrices of Local Spectral Radius Zero at a Fixed Vector
- Author
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Abdellatif Bourhim and Constantin Costara
- Subjects
Matrix (mathematics) ,Local spectrum ,Spectral radius ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Zero (complex analysis) ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, we characterize linear maps on matrix spaces that preserve matrices of local spectral radius zero at some fixed nonzero vector.
- Published
- 2019
83. 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
84. Integral Formula for Spectral Flow for -Summable Operators
- Author
-
Magdalena Cecilia Georgescu
- Subjects
General Mathematics ,010102 general mathematics ,Mathematical analysis ,Microlocal analysis ,Spectral flow ,Spectral theorem ,Operator theory ,01 natural sciences ,Fourier integral operator ,0103 physical sciences ,010307 mathematical physics ,Integral formula ,0101 mathematics ,Mathematics - Abstract
Fix a von Neumann algebra ${\mathcal{N}}$ equipped with a suitable trace $\unicode[STIX]{x1D70F}$. For a path of self-adjoint Breuer–Fredholm operators, the spectral flow measures the net amount of spectrum that moves from negative to non-negative. We consider specifically the case of paths of bounded perturbations of a fixed unbounded self-adjoint Breuer–Fredholm operator affiliated with ${\mathcal{N}}$. If the unbounded operator is $p$-summable (that is, its resolvents are contained in the ideal $L^{p}$), then it is possible to obtain an integral formula that calculates spectral flow. This integral formula was first proved by Carey and Phillips, building on earlier approaches of Phillips. Their proof was based on first obtaining a formula for the larger class of $\unicode[STIX]{x1D703}$-summable operators, and then using Laplace transforms to obtain a $p$-summable formula. In this paper, we present a direct proof of the $p$-summable formula that is both shorter and simpler than theirs.
- Published
- 2019
85. A CR Analogue of Yau’s Conjecture on Pseudoharmonic Functions of Polynomial Growth
- Author
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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.
- Published
- 2019
86. 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
87. 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
88. A Special Case of Completion Invariance for thec2Invariant of a Graph
- Author
-
Karen Yeats
- Subjects
Combinatorics ,symbols.namesake ,Computer Science::Information Retrieval ,General Mathematics ,symbols ,Feynman diagram ,Invariant (physics) ,Special case ,Graph property ,Graph ,Mathematics - Abstract
Thec2invariant is an arithmetic graph invariant defined by Schnetz. It is useful for understanding Feynman periods. Brown and Schnetz conjectured that thec2invariant has a particular symmetry known as completion invariance. This paper will prove completion invariance of thec2invariant in the case where we are over the field with 2 elements and the completed graph has an odd number of vertices. The methods involve enumerating certain edge bipartitions of graphs; two different constructions are needed.
- Published
- 2018
89. 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
90. 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
91. 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
92. 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.
- Published
- 2018
93. 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
94. On a Class of Fully Nonlinear Elliptic Equations Containing Gradient Terms on Compact Hermitian Manifolds
- Author
-
Rirong Yuan
- Subjects
Hermitian symmetric space ,Hessian equation ,Class (set theory) ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,01 natural sciences ,Hermitian matrix ,Sasakian manifold ,Nonlinear system ,0103 physical sciences ,Hermitian manifold ,Applied mathematics ,A priori and a posteriori ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
In this paper we study a class of second order fully nonlinear elliptic equations containing gradient terms on compact Hermitian manifolds and obtain a priori estimates under proper assumptions close to optimal. The analysis developed here should be useful to deal with other Hessian equations containing gradient terms in other contexts.
- Published
- 2018
95. 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
96. Fixed Point Theorems for Maps With Local and Pointwise Contraction Properties
- Author
-
Jakub Jasinski and Krzysztof Ciesielski
- Subjects
Pointwise ,Discrete mathematics ,Social connectedness ,General Mathematics ,010102 general mathematics ,Periodic point ,Fixed-point theorem ,Fixed point ,01 natural sciences ,Metric space ,Compact space ,0103 physical sciences ,010307 mathematical physics ,Differentiable function ,0101 mathematics ,Mathematics - Abstract
This paper constitutes a comprehensive study of ten classes of self-maps on metric spaces ⟨X, d⟩ with the pointwise (i.e., local radial) and local contraction properties. Each of these classes appeared previously in the literature in the context of fixed point theorems.We begin with an overview of these fixed point results, including concise self contained sketches of their proofs. Then we proceed with a discussion of the relations among the ten classes of self-maps with domains ⟨X, d⟩ having various topological properties that often appear in the theory of fixed point theorems: completeness, compactness, (path) connectedness, rectifiable-path connectedness, and d-convexity. The bulk of the results presented in this part consists of examples of maps that show non-reversibility of the previously established inclusions between these classes. Among these examples, the most striking is a differentiable auto-homeomorphism f of a compact perfect subset X of ℝ with f′ ≡ 0, which constitutes also a minimal dynamical system. We finish by discussing a few remaining open problems on whether the maps with specific pointwise contraction properties must have the fixed points.
- Published
- 2018
97. Euler-type Relative Equilibria and their Stability in Spaces of Constant Curvature
- Author
-
Juan Manuel Sánchez-Cerritos and Ernesto Pérez-Chavela
- Subjects
Geodesic ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Space (mathematics) ,01 natural sciences ,Stability (probability) ,Measure (mathematics) ,010305 fluids & plasmas ,Constant curvature ,symbols.namesake ,0103 physical sciences ,Euler's formula ,symbols ,Algebraic curve ,0101 mathematics ,Curved space ,Mathematics - Abstract
We consider three point positivemasses moving onS2andH2. An Eulerian-relative equilibrium is a relative equilibrium where the three masses are on the same geodesic. In this paper we analyze the spectral stability of these kind of orbits where the mass at the middle is arbitrary and the masses at the ends are equal and located at the same distance from the central mass. For the case of S2, we found a positive measure set in the set of parameters where the relative equilibria are spectrally stable, and we give a complete classiûcation of the spectral stability of these solutions, in the sense that, except on an algebraic curve in the space of parameters, we can determine if the corresponding relative equilibriumis spectrally stable or unstable. OnH2, in the elliptic case, we prove that generically all Eulerian-relative equilibria are unstable; in the particular degenerate case when the two equal masses are negligible, we get that the corresponding solutions are spectrally stable. For the hyperbolic case we consider the system where the mass in the middle is negligible; in this case the Eulerian-relative equilibria are unstable.
- Published
- 2018
98. 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
99. 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
100. Anisotropic Hardy-Lorentz Spaces with Variable Exponents
- Author
-
Jorge J. Betancor, Lourdes Rodríguez-Mesa, and Víctor Almeida
- Subjects
Mathematics::Functional Analysis ,Physics::General Physics ,Variable exponent ,Mathematics::Complex Variables ,General Mathematics ,Lorentz transformation ,010102 general mathematics ,Mathematics::Classical Analysis and ODEs ,010103 numerical & computational mathematics ,Physics::Classical Physics ,01 natural sciences ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,symbols.namesake ,Atomic decomposition ,Mathematics - Classical Analysis and ODEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,symbols ,Maximal function ,0101 mathematics ,Anisotropy ,Mathematical physics ,Mathematics ,Variable (mathematics) - Abstract
In this paper we introduceHardy-Lorentz spaces with variable exponents associated with dilations in ℝn. We establishmaximal characterizations and atomic decompositions for our variable exponent anisotropic Hardy-Lorentz spaces.
- Published
- 2017
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