1,747 results
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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. On a Paper of Maurice Sion
- Author
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Mark Mahowald
- Subjects
Combinatorics ,Set (abstract data type) ,Class (set theory) ,Continuous function ,General Mathematics ,Bounded variation ,Open set ,Function (mathematics) ,Real line ,Measure (mathematics) ,Mathematics - Abstract
Let M0 be the set of measures μ on the real line such that open sets are μ*-immeasurable. While attempting to find out whether a set μ*-measurable for all μ in Mo is mapped into a similar set by a continuous function of bounded variation, Maurice Sion develops a theory for what he calls variational measure (4). As an application of the theory, he gets conditions on a function f and a set of measures M in order that f map a set, which is μ*-measurable for all μ ∈ M, into a set of the same kind. In particular he proves for his class M2 (def. 2.5), the following theorem (4, § 8.11).
- Published
- 1959
4. Unitary representations of type B rational Cherednik algebras and crystal combinatorics
- Author
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Emily Norton
- Subjects
Functor ,Unitarity ,General Mathematics ,Type (model theory) ,Unitary state ,Fock space ,Combinatorics ,Irreducible representation ,FOS: Mathematics ,Mathematics - Combinatorics ,Partition (number theory) ,Component (group theory) ,Combinatorics (math.CO) ,Representation Theory (math.RT) ,Mathematics::Representation Theory ,Mathematics - Representation Theory ,Mathematics - Abstract
We compare crystal combinatorics of the level 2 Fock space with the classification of unitary irreducible representations of type B rational Cherednik algebras to study how unitarity behaves under parabolic restriction. First, we show that any finite-dimensional unitary irreducible representation of such an algebra is labeled by a bipartition consisting of a rectangular partition in one component and the empty partition in the other component. This is a new proof of a result that can be deduced from theorems of Montarani and Etingof-Stoica. Second, we show that the crystal operators that remove boxes preserve the combinatorial conditions for unitarity, and that the parabolic restriction functors categorifying the crystals send irreducible unitary representations to unitary representations. Third, we find the supports of the unitary representations., This paper supersedes arXiv:1907.00919 and contains that paper as a subsection. 35 pages, some color figures
- Published
- 2021
5. Note on a Paper by Robinson
- Author
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J. A. Todd
- Subjects
General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematical economics ,Mathematics - Abstract
In a recent paper Robinson has obtained an explicit formula for the expression of an invariant matrix of an invariant matrix as a direct sum of invariant matrices. The object of the present note is to show that this formula may be deduced from known properties of Schur functions, with the aid of a result which the author has proved elsewhere.
- Published
- 1950
6. 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
7. 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
8. Lorentz Estimates for Weak Solutions of Quasi-linear Parabolic Equations with Singular Divergence-free Drifts
- Author
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Tuoc Phan
- Subjects
General Mathematics ,Lorentz transformation ,010102 general mathematics ,Mathematical analysis ,Boundary (topology) ,Space (mathematics) ,01 natural sciences ,Parabolic partial differential equation ,010101 applied mathematics ,symbols.namesake ,Bounded function ,symbols ,Vector field ,Maximal function ,0101 mathematics ,Divergence (statistics) ,Mathematics - Abstract
This paper investigates regularity in Lorentz spaces for weak solutions of a class of divergence form quasi-linear parabolic equations with singular divergence-free drifts. In this class of equations, the principal terms are vector field functions that are measurable in ($x,t$)-variable, and nonlinearly dependent on both unknown solutions and their gradients. Interior, local boundary, and global regularity estimates in Lorentz spaces for gradients of weak solutions are established assuming that the solutions are in BMO space, the John–Nirenberg space. The results are even new when the drifts are identically zero, because they do not require solutions to be bounded as in the available literature. In the linear setting, the results of the paper also improve the standard Calderón–Zygmund regularity theory to the critical borderline case. When the principal term in the equation does not depend on the solution as its variable, our results recover and sharpen known available results. The approach is based on the perturbation technique introduced by Caffarelli and Peral together with a “double-scaling parameter” technique and the maximal function free approach introduced by Acerbi and Mingione.
- Published
- 2019
9. Remark on my Paper 'Generators of Monothetic Groups'
- Author
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D. L. Armacost
- Subjects
Algebra ,General Mathematics ,Mathematics - Published
- 1973
10. 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
11. 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
12. 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
13. 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
14. Isomorphisms of Twisted Hilbert Loop Algebras
- Author
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Timothée Marquis and Karl-Hermann Neeb
- Subjects
17B65, 17B70, 17B22, 17B10 ,General Mathematics ,010102 general mathematics ,Hilbert space ,Mathematics - Rings and Algebras ,01 natural sciences ,Combinatorics ,Loop (topology) ,symbols.namesake ,Isomorphism theorem ,Rings and Algebras (math.RA) ,Affine root system ,Product (mathematics) ,0103 physical sciences ,Lie algebra ,FOS: Mathematics ,symbols ,010307 mathematical physics ,Isomorphism ,Representation Theory (math.RT) ,0101 mathematics ,Invariant (mathematics) ,Mathematics - Representation Theory ,Mathematics - Abstract
The closest infinite dimensional relatives of compact Lie algebras are Hilbert-Lie algebras, i.e. real Hilbert spaces with a Lie algebra structure for which the scalar product is invariant. Locally affine Lie algebras (LALAs) correspond to double extensions of (twisted) loop algebras over simple Hilbert-Lie algebras $\mathfrak{k}$, also called affinisations of $\mathfrak{k}$. They possess a root space decomposition whose corresponding root system is a locally affine root system of one of the $7$ families $A_J^{(1)}$, $B_J^{(1)}$, $C_J^{(1)}$, $D_J^{(1)}$, $B_J^{(2)}$, $C_J^{(2)}$ and $BC_J^{(2)}$ for some infinite set $J$. To each of these types corresponds a "minimal" affinisation of some simple Hilbert-Lie algebra $\mathfrak{k}$, which we call standard. In this paper, we give for each affinisation $\mathfrak{g}$ of a simple Hilbert-Lie algebra $\mathfrak{k}$ an explicit isomorphism from $\mathfrak{g}$ to one of the standard affinisations of $\mathfrak{k}$. The existence of such an isomorphism could also be derived from the classification of locally affine root systems, but for representation theoretic purposes it is crucial to obtain it explicitely as a deformation between two twists which is compatible with the root decompositions. We illustrate this by applying our isomorphism theorem to the study of positive energy highest weight representations of $\mathfrak{g}$. In subsequent work, the present paper will be used to obtain a complete classification of the positive energy highest weight representations of affinisations of $\mathfrak{k}$., Comment: 22 pages; Minor corrections
- Published
- 2017
15. 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
16. 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
17. 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
- View/download PDF
18. 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
19. 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
20. 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
21. 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
22. 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
23. 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
24. 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
25. 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
26. Generalized Beilinson Elements and Generalized Soulé Characters
- Author
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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
27. 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
28. 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
29. 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
30. 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
31. 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
32. A Skolem–Mahler–Lech Theorem for Iterated Automorphisms of K–algebras
- Author
-
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
33. Weighted Carleson Measure Spaces Associated with Different Homogeneities
- Author
-
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
34. Existence of Taut Foliations on Seifert Fibered Homology 3-spheres
- Author
-
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
35. 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
36. 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
37. 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
38. 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
39. 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
40. 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
41. Linear Maps Preserving Matrices of Local Spectral Radius Zero at a Fixed Vector
- Author
-
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
42. 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
43. 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
44. 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.
- Published
- 2019
45. 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
46. 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
47. 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
48. 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
49. 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
50. 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
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