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2. On the paper 'A ‘lost’ notebook of Ramanujan'
- Author
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R.P Agarwal
- Subjects
Algebra ,symbols.namesake ,Mathematics(all) ,General Mathematics ,symbols ,Ramanujan's sum ,Mathematics - Published
- 1984
- Full Text
- View/download PDF
3. Collected papers, vol. I: Combinatorics
- Author
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Paul R Stein
- Subjects
Mathematics(all) ,GEORGE (programming language) ,General Mathematics ,Classics ,Mathematics - Published
- 1979
4. Markov processes and related problems of analysis (selected papers)
- Author
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Gian-Carlo Rota
- Subjects
Discrete mathematics ,symbols.namesake ,Mathematics(all) ,General Mathematics ,symbols ,Markov process ,Mathematics - Published
- 1985
- Full Text
- View/download PDF
5. Key papers in the development of information theory
- Author
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G.-C Rota
- Subjects
Mathematics(all) ,Development (topology) ,General Mathematics ,Information theory ,Mathematical economics ,Mathematics - Published
- 1975
- Full Text
- View/download PDF
6. Selected papers
- Author
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Gian-Carlo Rota
- Subjects
Discrete mathematics ,Mathematics(all) ,General Mathematics ,Humanities ,Mathematics - Published
- 1977
- Full Text
- View/download PDF
7. Correction to my paper on Nakayama R-varieties
- Author
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Idun Reiten
- Subjects
Pure mathematics ,Mathematics(all) ,General Mathematics ,Mathematics - Published
- 1977
- Full Text
- View/download PDF
8. A note on our paper 'theory of decomposition in semigroups'
- Author
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Gábor J. Székely and Imre Z. Ruzsa
- Subjects
Krohn–Rhodes theory ,Algebra ,Mathematics(all) ,General Mathematics ,Decomposition (computer science) ,Mathematics - Published
- 1986
- Full Text
- View/download PDF
9. Zur algebraischen geometrie (selected papers)
- Author
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Gian-Carlo Rota
- Subjects
Discrete mathematics ,Mathematics(all) ,General Mathematics ,Mathematics - Published
- 1985
- Full Text
- View/download PDF
10. The Bellman continuum, a collection of the works of Richard E. Bellman. Selected papers
- Author
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Gian-Carlo Rota
- Subjects
Mathematics(all) ,Continuum (measurement) ,General Mathematics ,Mathematical economics ,Mathematics - Published
- 1989
11. Collected papers
- Author
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Gian-Carlo Rota
- Subjects
Algebra ,Mathematics(all) ,Number theory ,General Mathematics ,Library science ,Humanities ,Classics ,Mathematical physics ,Mathematics - Published
- 1985
12. The Frobenius morphism in invariant theory II
- Author
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Spenko, Spela, Raedschelders, Theo, Van Den Bergh, Michel, RAEDSCHELDERS, Theo, SPENKO, Spela, VAN DEN BERGH, Michel, Algebra and Analysis, Mathematics, and Algebra
- Subjects
Frobenius summand ,General Mathematics ,Invariant theory ,Mathematics - Rings and Algebras ,Frobenius kernel ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,Mathematics - Algebraic Geometry ,Rings and Algebras (math.RA) ,Géométrie algébrique ,FOS: Mathematics ,FFRT ,Representation Theory (math.RT) ,Groupes algébriques ,Algèbre commutative et algèbre homologique ,Algebraic Geometry (math.AG) ,Mathematics - Representation Theory ,Géométrie non commutative - Abstract
Let $R$ be the homogeneous coordinate ring of the Grassmannian $\mathbb{G}=Gr(2,n)$ defined over an algebraically closed field $k$ of characteristic $p \geq \max\{n-2,3\}$. In this paper we give a description of the decomposition of $R$, considered as graded $R^{p^r}$-module, for $r \geq 2$. This is a companion paper to our earlier paper, where the case $r=1$ was treated, and taken together, our results imply that $R$ has finite F-representation type (FFRT). Though it is expected that all rings of invariants for reductive groups have FFRT, ours is the first non-trivial example of such a ring for a group which is not linearly reductive. As a corollary, we show that the ring of differential operators $D_k(R)$ is simple, that $\mathbb{G}$ has global finite F-representation type (GFFRT) and that $R$ provides a noncommutative resolution for $R^{p^r}$., Comment: 52 pages
- Published
- 2022
13. Sharp Hardy-Sobolev-Maz'ya, Adams and Hardy-Adams inequalities on the Siegel domains and complex hyperbolic spaces
- Author
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Lu, Guozhen and Yang, Qiaohua
- Subjects
Mathematics - Differential Geometry ,Mathematics - Analysis of PDEs ,Differential Geometry (math.DG) ,Mathematics - Classical Analysis and ODEs ,Mathematics - Complex Variables ,General Mathematics ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,42B35, 42B15, 42B37, 35J08 ,Complex Variables (math.CV) ,Analysis of PDEs (math.AP) - Abstract
This paper continues the program initiated in the works by the authors [60], [61] and [62] and by the authors with Li [51] and [52] to establish higher order Poincar\'e-Sobolev, Hardy-Sobolev-Maz'ya, Adams and Hardy-Adams inequalities on real hyperbolic spaces using the method of Helgason-Fourier analysis on the hyperbolic spaces. The aim of this paper is to establish such inequalities on the Siegel domains and complex hyperbolic spaces. Firstly, we prove a factorization theorem for the operators on the complex hyperbolic space which is closely related to Geller' operator, as well as the CR invariant differential operators on the Heisenberg group and CR sphere. Secondly, by using, among other things, the Kunze-Stein phenomenon on a closed linear group $SU(1,n)$ and Helgason-Fourier analysis techniques on the complex hyperbolic spaces, we establish the Poincar\'e-Sobolev, Hardy-Sobolev-Maz'ya inequality on the Siegel domain $\mathcal{U}^{n}$ and the unit ball $\mathbb{B}_{\mathbb{C}}^{n}$. Finally, we establish the sharp Hardy-Adams inequalities and sharp Adams type inequalities on Sobolev spaces of any positive fractional order on the complex hyperbolic spaces. The factorization theorem we proved is of its independent interest in the Heisenberg group and CR sphere and CR invariant differential operators therein., Comment: 51 pages
- Published
- 2022
14. On the Goulden–Jackson–Vakil conjecture for double Hurwitz numbers
- Author
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Norman Do and Danilo Lewański
- Subjects
Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,14H10, 14N10, 05A15 ,General Mathematics ,FOS: Mathematics ,FOS: Physical sciences ,Mathematical Physics (math-ph) ,Algebraic Geometry (math.AG) ,Mathematical Physics - Abstract
Goulden, Jackson and Vakil observed a polynomial structure underlying one-part double Hurwitz numbers, which enumerate branched covers of $\mathbb{CP}^1$ with prescribed ramification profile over $\infty$, a unique preimage over 0, and simple branching elsewhere. This led them to conjecture the existence of moduli spaces and tautological classes whose intersection theory produces an analogue of the celebrated ELSV formula for single Hurwitz numbers. In this paper, we present three formulas that express one-part double Hurwitz numbers as intersection numbers on certain moduli spaces. The first involves Hodge classes on moduli spaces of stable maps to classifying spaces; the second involves Chiodo classes on moduli spaces of spin curves; and the third involves tautological classes on moduli spaces of stable curves. We proceed to discuss the merits of these formulas against a list of desired properties enunciated by Goulden, Jackson and Vakil. Our formulas lead to non-trivial relations between tautological intersection numbers on moduli spaces of stable curves and hints at further structure underlying Chiodo classes. The paper concludes with generalisations of our results to the context of spin Hurwitz numbers., Comment: 20 pages. Some corollaries are added in the second version, and software numerical checks are performed
- Published
- 2022
15. Existence and nonexistence of extremal functions for sharp Trudinger-Moser inequalities
- Author
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Lu Zhang, Guozhen Lu, and Nguyen Lam
- Subjects
Pure mathematics ,Inequality ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Mathematics::Analysis of PDEs ,Function (mathematics) ,Type (model theory) ,Space (mathematics) ,01 natural sciences ,Infimum and supremum ,Symmetry (physics) ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics ,media_common - Abstract
Our main purpose in this paper is to establish the existence and nonexistence of extremal functions (also known as maximizers) and symmetry of extremals for several Trudinger-Moser type inequalities on the entire space R n , including both the critical and subcritical Trudinger-Moser inequalities (see Theorems 1.1, 1.2, 1.3, 1.4, 1.5). Most of earlier works on existence of maximizers in the literature rely on the complicated blow-up analysis of PDEs for the associated Euler-Lagrange equations of the corresponding Moser functionals. The new approaches developed in this paper are using the identities and relationship between the supremums of the subcritical Trudinger-Moser inequalities and the critical ones established by the same authors in [25] , combining with the continuity of the supremum function that is observed for the first time in the literature. These allow us to establish the existence and nonexistence of the maximizers for the Trudinger-Moser inequalities in different ranges of the parameters (including those inequalities with the exact growth). This method is considerably simpler and also allows us to study the symmetry problem of the extremal functions and prove that the extremal functions for the subcritical singular Truddinger-Moser inequalities are symmetric. Moreover, we will be able to calculate the exact values of the supremums of the Trudinger-Moser type in certain cases. These appear to be the first results in this direction.
- Published
- 2019
16. Normal crossings singularities for symplectic topology
- Author
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Mark McLean, Aleksey Zinger, and Mohammad Farajzadeh Tehrani
- Subjects
Pure mathematics ,Logarithm ,Divisor ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Mathematics - Algebraic Geometry ,Mathematics - Symplectic Geometry ,0103 physical sciences ,FOS: Mathematics ,Symplectic Geometry (math.SG) ,53D05, 53D45, 14N35 ,Gravitational singularity ,010307 mathematical physics ,0101 mathematics ,Equivalence (formal languages) ,Algebraic Geometry (math.AG) ,Mathematics::Symplectic Geometry ,Symplectic sum ,Symplectic geometry ,Mathematics - Abstract
We introduce topological notions of normal crossings symplectic divisor and variety and establish that they are equivalent, in a suitable sense, to the desired geometric notions. Our proposed concept of equivalence of associated topological and geometric notions fits ideally with important constructions in symplectic topology. This partially answers Gromov's question on the feasibility of defining singular symplectic (sub)varieties and lays foundation for rich developments in the future. In subsequent papers, we establish a smoothability criterion for symplectic normal crossings varieties, in the process providing the multifold symplectic sum envisioned by Gromov, and introduce symplectic analogues of logarithmic structures in the context of normal crossings symplectic divisors., Comment: 65 pages, 4 figures; a number of typos fixed; the exposition has been significantly revised, fixing a technical error in the non-compact case in the process; this paper is now restricted to the simple normal crossings case; the arbitrary normal crossings case will be detailed in a followup paper
- Published
- 2018
17. On the mean field type bubbling solutions for Chern–Simons–Higgs equation
- Author
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Shusen Yan and Chang-Shou Lin
- Subjects
General Mathematics ,010102 general mathematics ,Chern–Simons theory ,Structure (category theory) ,Type (model theory) ,01 natural sciences ,Mean field theory ,0103 physical sciences ,Higgs boson ,010307 mathematical physics ,Uniqueness ,0101 mathematics ,Parallelogram ,Mathematical physics ,Mathematics - Abstract
This paper is the second part of our comprehensive study on the structure of the solutions for the following Chern–Simons–Higgs equation: (0.1) { Δ u + 1 e 2 e u ( 1 − e u ) = 4 π ∑ j = 1 N δ p j , in Ω , u is doubly periodic on ∂ Ω , where Ω is a parallelogram in R 2 and e > 0 is a small parameter. In part 1 [29] , we proved the non-coexistence of different bubbles in the bubbling solutions and obtained an existence result for the Chern–Simons type bubbling solutions under some nearly necessary conditions. Mean field type bubbling solutions for (0.1) have been constructed in [27] . In this paper, we shall study two other important issues for the mean field type bubbling solutions: the necessary conditions for the existence and the local uniqueness. The results in this paper lay the foundation to find the exact number of solutions for (0.1) .
- Published
- 2018
18. Quasi-elliptic cohomology I
- Author
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Zhen Huan
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Elliptic cohomology ,16. Peace & justice ,Space (mathematics) ,Mathematics::Algebraic Topology ,01 natural sciences ,Cohomology ,Mathematics::K-Theory and Homology ,0103 physical sciences ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Equivariant map ,Mathematics - Algebraic Topology ,010307 mathematical physics ,55N34, 55P35 ,0101 mathematics ,Tate curve ,Constant (mathematics) ,Computer Science::Databases ,Quotient ,Orbifold ,Mathematics - Abstract
Quasi-elliptic cohomology is a variant of elliptic cohomology theories. It is the orbifold K-theory of a space of constant loops. For global quotient orbifolds, it can be expressed in terms of equivariant K-theories. Thus, the constructions on it can be made in a neat way. This theory reflects the geometric nature of the Tate curve. In this paper we provide a systematic introduction of its construction and definition., Comment: Final Version. 26 pages. To appear in Advances in Mathematics. In this paper we generalize the construction in arXiv:1612.00930. The subtle point of this generalization is explained in Section 2
- Published
- 2018
19. The restricted content and the d-dimensional Analyst's Travelling Salesman Theorem for general sets
- Author
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Hyde, Matthew
- Subjects
General Mathematics - Abstract
In his 1990 paper, Jones characterized subsets of rectifiable curves in via a multiscale sum of β-numbers, which measure how far a given set deviates from a straight line at each scale and location. This characterization was extended by Okikiolu to subsets of and by Schul to subsets of a Hilbert space.\ud \ud Recently, there has been some interest in characterizing subsets of higher dimensional surfaces in . Using a variant of Jones' β-number introduced by Azzam and Schul, Villa gave a characterization of lower regular subsets of a certain class of topologically stable surfaces – introduced in a 2004 paper of David – via a multiscale sum of these new β-numbers.\ud \ud In this paper we remove the lower regularity condition and prove an analogous result for general d-dimensional subsets of . To do this, we introduce the restricted content, which assigns ‘mass’ to any subset of (even to sets with zero Hausdorff measure), and use it to define new d-dimensional variant of Jones' β-number that is defined for any set in .
- Published
- 2022
20. Balanced derivatives, identities, and bounds for trigonometric and Bessel series
- Author
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Bruce C. Berndt, Sun Kim, Martino Fassina, and Alexandru Zaharescu
- Subjects
symbols.namesake ,Pure mathematics ,Series (mathematics) ,General Mathematics ,symbols ,Trigonometric functions ,Divisor (algebraic geometry) ,Trigonometry ,Upper and lower bounds ,Bessel function ,Gauss circle problem ,Ramanujan's sum ,Mathematics - Abstract
Motivated by two identities published with Ramanujan's lost notebook and connected, respectively, with the Gauss circle problem and the Dirichlet divisor problem, in an earlier paper, three of the present authors derived representations for certain sums of products of trigonometric functions as double series of Bessel functions [8] . These series are generalized in the present paper by introducing the novel notion of balanced derivatives, leading to further theorems. As we will see below, the regions of convergence in the unbalanced case are entirely different than those in the balanced case. From this viewpoint, it is remarkable that Ramanujan had the intuition to formulate entries that are, in our new terminology, “balanced”. If x denotes the number of products of the trigonometric functions appearing in our sums, in addition to proving the identities mentioned above, theorems and conjectures for upper and lower bounds for the sums as x → ∞ are established.
- Published
- 2022
21. Transfer operators and Hankel transforms between relative trace formulas, II: Rankin–Selberg theory
- Author
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Yiannis Sakellaridis
- Subjects
Transfer (group theory) ,Pure mathematics ,Hecke algebra ,symbols.namesake ,Conjecture ,Trace (linear algebra) ,General Mathematics ,Poisson summation formula ,symbols ,Functional equation (L-function) ,Abelian group ,Fundamental lemma ,Mathematics - Abstract
The Langlands functoriality conjecture, as reformulated in the “beyond endoscopy” program, predicts comparisons between the (stable) trace formulas of different groups G 1 , G 2 for every morphism G 1 L → L G 2 between their L-groups. This conjecture can be seen as a special case of a more general conjecture, which replaces reductive groups by spherical varieties and the trace formula by its generalization, the relative trace formula. The goal of this article and its precursor [11] is to demonstrate, by example, the existence of “transfer operators” between relative trace formulas, which generalize the scalar transfer factors of endoscopy. These transfer operators have all properties that one could expect from a trace formula comparison: matching, fundamental lemma for the Hecke algebra, transfer of (relative) characters. Most importantly, and quite surprisingly, they appear to be of abelian nature (at least, in the low-rank examples considered in this paper), even though they encompass functoriality relations of non-abelian harmonic analysis. Thus, they are amenable to application of the Poisson summation formula in order to perform the global comparison. Moreover, we show that these abelian transforms have some structure — which presently escapes our understanding in its entirety — as deformations of well-understood operators when the spaces under consideration are replaced by their “asymptotic cones”. In this second paper we use Rankin–Selberg theory to prove the local transfer behind Rudnick's 1990 thesis (comparing the stable trace formula for SL 2 with the Kuznetsov formula) and Venkatesh's 2002 thesis (providing a “beyond endoscopy” proof of functorial transfer from tori to GL 2 ). As it turns out, the latter is not completely disjoint from endoscopic transfer — in fact, our proof “factors” through endoscopic transfer. We also study the functional equation of the symmetric-square L-function for GL 2 , and show that it is governed by an explicit “Hankel operator” at the level of the Kuznetsov formula, which is also of abelian nature. A similar theory for the standard L-function was previously developed (in a different language) by Jacquet.
- Published
- 2022
22. Decomposition spaces, incidence algebras and Möbius inversion III: The decomposition space of Möbius intervals
- Author
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Joachim Kock, Imma Gálvez-Carrillo, Andrew Tonks, Universitat Politècnica de Catalunya. Departament de Matemàtiques, and Universitat Politècnica de Catalunya. GEOMVAP - Geometria de Varietats i Aplicacions
- Subjects
Pure mathematics ,Mathematics::General Mathematics ,Mathematics::Number Theory ,General Mathematics ,Coalgebra ,18 Category theory [Classificació AMS] ,Structure (category theory) ,18G Homological algebra [homological algebra] ,Combinatorial topology ,55 Algebraic topology::55P Homotopy theory [Classificació AMS] ,Algebraic topology ,Space (mathematics) ,2-Segal space ,01 natural sciences ,Combinatorics ,decomposition space ,18G30, 16T10, 06A11, 18-XX, 55Pxx ,Mathematics::Category Theory ,0103 physical sciences ,Mathematics - Combinatorics ,Mathematics::Metric Geometry ,Matemàtiques i estadística::Topologia::Topologia algebraica [Àrees temàtiques de la UPC] ,Mathematics - Algebraic Topology ,0101 mathematics ,06 Order, lattices, ordered algebraic structures::06A Ordered sets [Classificació AMS] ,Mathematics ,Topologia combinatòria ,CULF functor ,Mathematics::Combinatorics ,Functor ,Mathematics::Complex Variables ,Homotopy ,010102 general mathematics ,Mathematics - Category Theory ,Möbius interval ,Topologia algebraica ,Hopf algebra ,18 Category theory ,homological algebra::18G Homological algebra [Classificació AMS] ,010307 mathematical physics ,Möbius inversion - Abstract
Decomposition spaces are simplicial $\infty$-groupoids subject to a certain exactness condition, needed to induce a coalgebra structure on the space of arrows. Conservative ULF functors (CULF) between decomposition spaces induce coalgebra homomorphisms. Suitable added finiteness conditions define the notion of M\"obius decomposition space, a far-reaching generalisation of the notion of M\"obius category of Leroux. In this paper, we show that the Lawvere-Menni Hopf algebra of M\"obius intervals, which contains the universal M\"obius function (but is not induced by a M\"obius category), can be realised as the homotopy cardinality of a M\"obius decomposition space $U$ of all M\"obius intervals, and that in a certain sense $U$ is universal for M\"obius decomposition spaces and CULF functors., Comment: 35 pages. This paper is one of six papers that formerly constituted the long manuscript arXiv:1404.3202. v3: minor expository improvements. Final version to appear in Adv. Math
- Published
- 2018
23. Nevanlinna theory of the Askey–Wilson divided difference operator
- Author
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Yik-Man Chiang and Shao-Ji Feng
- Subjects
Pure mathematics ,Basic hypergeometric series ,High Energy Physics::Lattice ,General Mathematics ,010102 general mathematics ,Mathematics::Classical Analysis and ODEs ,Zero (complex analysis) ,Infinite product ,01 natural sciences ,Nevanlinna theory ,010101 applied mathematics ,Operator (computer programming) ,0101 mathematics ,Complex plane ,Picard theorem ,Meromorphic function ,Mathematics - Abstract
This paper establishes a version of Nevanlinna theory based on Askey–Wilson divided difference operator for meromorphic functions of finite logarithmic order in the complex plane C . A second main theorem that we have derived allows us to define an Askey–Wilson type Nevanlinna deficiency which gives a new interpretation that one should regard many important infinite products arising from the study of basic hypergeometric series as zero/pole-scarce. That is, their zeros/poles are indeed deficient in the sense of difference Nevanlinna theory. A natural consequence is a version of Askey–Wilson type Picard theorem. We also give an alternative and self-contained characterisation of the kernel functions of the Askey–Wilson operator. In addition we have established a version of unicity theorem in the sense of Askey–Wilson. This paper concludes with an application to difference equations generalising the Askey–Wilson second-order divided difference equation.
- Published
- 2018
24. The Goldman–Turaev Lie bialgebra in genus zero and the Kashiwara–Vergne problem
- Author
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Yusuke Kuno, Anton Alekseev, Florian Naef, and Nariya Kawazumi
- Subjects
Pure mathematics ,Lie bialgebra ,Degree (graph theory) ,General Mathematics ,010102 general mathematics ,Zero (complex analysis) ,Order (ring theory) ,Field (mathematics) ,Mathematics::Geometric Topology ,01 natural sciences ,Bracket (mathematics) ,Mathematics::Quantum Algebra ,Genus (mathematics) ,0103 physical sciences ,010307 mathematical physics ,Lie theory ,0101 mathematics ,Mathematics - Abstract
In this paper, we describe a surprising link between the theory of the Goldman–Turaev Lie bialgebra on surfaces of genus zero and the Kashiwara–Vergne (KV) problem in Lie theory. Let Σ be an oriented 2-dimensional manifold with non-empty boundary and K a field of characteristic zero. The Goldman–Turaev Lie bialgebra is defined by the Goldman bracket { − , − } and Turaev cobracket δ on the K -span of homotopy classes of free loops on Σ. Applying an expansion θ : K π → K 〈 x 1 , … , x n 〉 yields an algebraic description of the operations { − , − } and δ in terms of non-commutative variables x 1 , … , x n . If Σ is a surface of genus g = 0 the lowest degree parts { − , − } − 1 and δ − 1 are canonically defined (and independent of θ). They define a Lie bialgebra structure on the space of cyclic words which was introduced and studied by Schedler [31] . It was conjectured by the second and the third authors that one can define an expansion θ such that { − , − } = { − , − } − 1 and δ = δ − 1 . The main result of this paper states that for surfaces of genus zero constructing such an expansion is essentially equivalent to the KV problem. In [24] , Massuyeau constructed such expansions using the Kontsevich integral. In order to prove this result, we show that the Turaev cobracket δ can be constructed in terms of the double bracket (upgrading the Goldman bracket) and the non-commutative divergence cocycle which plays the central role in the KV theory. Among other things, this observation gives a new topological interpretation of the KV problem and allows to extend it to surfaces with arbitrary number of boundary components (and of arbitrary genus, see [2] ).
- Published
- 2018
25. Exceptional collections on Dolgachev surfaces associated with degenerations
- Author
-
Yongnam Lee and Yonghwa Cho
- Subjects
Derived category ,Pure mathematics ,Mathematics::Commutative Algebra ,General Mathematics ,010102 general mathematics ,Picard group ,Vector bundle ,Type (model theory) ,01 natural sciences ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,0103 physical sciences ,Simply connected space ,Algebraic surface ,FOS: Mathematics ,Kodaira dimension ,010307 mathematical physics ,0101 mathematics ,Algebraic Geometry (math.AG) ,Quotient ,Mathematics - Abstract
Dolgachev surfaces are simply connected minimal elliptic surfaces with $p_g=q=0$ and of Kodaira dimension 1. These surfaces were constructed by logarithmic transformations of rational elliptic surfaces. In this paper, we explain the construction of Dolgachev surfaces via $\mathbb Q$-Gorenstein smoothing of singular rational surfaces with two cyclic quotient singularities. This construction is based on the paper by Lee-Park. Also, some exceptional bundles on Dolgachev surfaces associated with $\mathbb Q$-Gorenstein smoothing are constructed based on the idea of Hacking. In the case if Dolgachev surfaces were of type $(2,3)$, we describe the Picard group and present an exceptional collection of maximal length. Finally, we prove that the presented exceptional collection is not full, hence there exist a nontrivial phantom category in the derived category., Comment: 35 pages; 3 figures; exposition improved; Adv. Math. (to appear)
- Published
- 2018
26. Irreducible modules over finite simple Lie pseudoalgebras III. Primitive pseudoalgebras of type H
- Author
-
Bakalov, B., D'Andrea, A., and Kac, V. G.
- Subjects
General Mathematics ,Mathematics - Quantum Algebra ,Conformally symplectic geometry ,Hopf algebra ,Lie pseudoalgebra ,Lie–Cartan algebra of vector fields ,FOS: Mathematics ,Quantum Algebra (math.QA) ,Representation Theory (math.RT) ,17B35 (Primary) 16W30 (Secondary) 17B81 ,Mathematics - Representation Theory - Abstract
A Lie conformal algebra is an algebraic structure that encodes the singular part of the operator product expansion of chiral fields in conformal field theory. A Lie pseudoalgebra is a generalization of this structure, for which the algebra of polynomials k[\partial] in the indeterminate is replaced by the universal enveloping algebra U(d) of a finite-dimensional Lie algebra d over the base field k. The finite (i.e., finitely generated over U(d)) simple Lie pseudoalgebras were classified in our 2001 paper [BDK]. The complete list consists of primitive Lie pseudoalgebras of type W, S, H, and K, and of current Lie pseudoalgebras over them or over simple finite-dimensional Lie algebras. The present paper is the third in our series on representation theory of simple Lie pseudoalgebras. In the first paper, we showed that any finite irreducible module over a primitive Lie pseudoalgebra of type W or S is either an irreducible tensor module or the image of the differential in a member of the pseudo de Rham complex. In the second paper, we established a similar result for primitive Lie pseudoalgebras of type K, with the pseudo de Rham complex replaced by a certain reduction, called the contact pseudo de Rham complex. This reduction in the context of contact geometry was discovered by M. Rumin [Rum]. In the present paper, we show that for primitive Lie pseudoalgebras of type H, a similar to type K result holds with the contact pseudo de Rham complex replaced by a suitable complex. However, the type H case in more involved, since the annihilation algebra is not the corresponding Lie-Cartan algebra, as in other cases, but an irreducible central extension. When the action of the center of the annihilation algebra is trivial, this complex is related to work by M. Eastwood [E] on conformally symplectic geometry, and we call it conformally symplectic pseudo de Rham complex., Comment: 62 pages
- Published
- 2021
27. On emergence and complexity of ergodic decompositions
- Author
-
Pierre Berger and Jairo Bochi
- Subjects
Pure mathematics ,Lebesgue measure ,Dynamical systems theory ,General Mathematics ,010102 general mathematics ,Dynamical Systems (math.DS) ,Lebesgue integration ,37A35, 37C05, 37C45, 37C40, 37J40 ,01 natural sciences ,Measure (mathematics) ,010104 statistics & probability ,Metric space ,symbols.namesake ,FOS: Mathematics ,symbols ,Ergodic theory ,Mathematics - Dynamical Systems ,0101 mathematics ,Dynamical system (definition) ,Probability measure ,Mathematics - Abstract
A concept of emergence was recently introduced in the paper [Berger] in order to quantify the richness of possible statistical behaviors of orbits of a given dynamical system. In this paper, we develop this concept and provide several new definitions, results, and examples. We introduce the notion of topological emergence of a dynamical system, which essentially evaluates how big the set of all its ergodic probability measures is. On the other hand, the metric emergence of a particular reference measure (usually Lebesgue) quantifies how non-ergodic this measure is. We prove fundamental properties of these two emergences, relating them with classical concepts such as Kolmogorov's $\epsilon$-entropy of metric spaces and quantization of measures. We also relate the two types of emergences by means of a variational principle. Furthermore, we provide several examples of dynamics with high emergence. First, we show that the topological emergence of some standard classes of hyperbolic dynamical systems is essentially the maximal one allowed by the ambient. Secondly, we construct examples of smooth area-preserving diffeomorphisms that are extremely non-ergodic in the sense that the metric emergence of the Lebesgue measure is essentially maximal. These examples confirm that super-polynomial emergence indeed exists, as conjectured in the paper [Berger]. Finally, we prove that such examples are locally generic among smooth diffeomorphisms., Comment: v3: Final version; to appear in Advances in Mathematics
- Published
- 2021
28. L-improving estimates for Radon-like operators and the Kakeya-Brascamp-Lieb inequality
- Author
-
Philip T. Gressman
- Subjects
Pure mathematics ,Brascamp–Lieb inequality ,Continuum (topology) ,General Mathematics ,010102 general mathematics ,chemistry.chemical_element ,Radon ,Type (model theory) ,01 natural sciences ,Ambient space ,Range (mathematics) ,Quadratic equation ,chemistry ,Dimension (vector space) ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
This paper considers the problem of establishing L p -improving inequalities for Radon-like operators in intermediate dimensions (i.e., for averages overs submanifolds which are neither curves nor hypersurfaces). Due to limitations in existing approaches, previous results in this regime are comparatively sparse and tend to require special numerical relationships between the dimension n of the ambient space and the dimension k of the submanifolds. This paper develops a new approach to this problem based on a continuum version of the Kakeya-Brascamp-Lieb inequality, established by Zhang [28] and extended by Zorin-Kranich [29] , and on recent results for geometric nonconcentration inequalities [11] . As an initial application of this new approach, this paper establishes sharp restricted strong type L p -improving inequalities for certain model quadratic submanifolds in the range k n ≤ 2 k .
- Published
- 2021
29. GIT versus Baily-Borel compactification for K3's which are double covers of P1×P1
- Author
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Radu Laza and Kieran G. O'Grady
- Subjects
Baily–Borel compactification ,Pure mathematics ,Mathematics::Algebraic Geometry ,Simple (abstract algebra) ,General Mathematics ,Quartic function ,Complete intersection ,Birational geometry ,Type (model theory) ,Mathematics ,Moduli ,Moduli space - Abstract
In previous work, we have introduced a program aimed at studying the birational geometry of locally symmetric varieties of Type IV associated to moduli of certain projective varieties of K3 type. In particular, a concrete goal of our program is to understand the relationship between GIT and Baily-Borel compactifications for quartic K3 surfaces, K3's which are double covers of a smooth quadric surface, and double EPW sextics. In our first paper [36] , based on arithmetic considerations, we have given conjectural decompositions into simple birational transformations of the period maps from the GIT moduli spaces mentioned above to the corresponding Baily-Borel compactifications. In our second paper [35] we studied the case of quartic K3's; we have given geometric meaning to this decomposition and we have partially verified our conjectures. Here, we give a full proof of the conjectures in [36] for the moduli space of K3's which are double covers of a smooth quadric surface. The main new tool here is VGIT for ( 2 , 4 ) complete intersection curves.
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- 2021
30. Covering with Chang models over derived models
- Author
-
Grigor Sargsyan
- Subjects
Discrete mathematics ,Conjecture ,Current (mathematics) ,General Mathematics ,010102 general mathematics ,Mathematics - Logic ,01 natural sciences ,Mathematics::Logic ,Continuation ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Logic (math.LO) ,Mathematics - Abstract
We present a covering conjecture that we expect to be true below superstrong cardinals. We then show that the conjecture is true in hod mice. This work is a continuation of the work that started in Covering with Universally Baire Functions Advances in Mathematics, and the main conjecture of the current paper is a revision of the UB Covering Conjecture of the aforementioned paper.
- Published
- 2021
31. Partial orders on conjugacy classes in the Weyl group and on unipotent conjugacy classes
- Author
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Jeffrey Adams, Xuhua He, and Sian Nie
- Subjects
Weyl group ,Pure mathematics ,Series (mathematics) ,General Mathematics ,010102 general mathematics ,Unipotent ,Reductive group ,01 natural sciences ,Injective function ,Primary: 20G07, Secondary: 06A07, 20F55, 20E45 ,symbols.namesake ,Conjugacy class ,0103 physical sciences ,FOS: Mathematics ,symbols ,Order (group theory) ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,Algebraically closed field ,Mathematics::Representation Theory ,Mathematics - Representation Theory ,Mathematics - Abstract
Let $G$ be a reductive group over an algebraically closed field and let $W$ be its Weyl group. In a series of papers, Lusztig introduced a map from the set $[W]$ of conjugacy classes of $W$ to the set $[G_u]$ of unipotent classes of $G$. This map, when restricted to the set of elliptic conjugacy classes $[W_e]$ of $W$, is injective. In this paper, we show that Lusztig's map $[W_e] \to [G_u]$ is order-reversing, with respect to the natural partial order on $[W_e]$ arising from combinatorics and the natural partial order on $[G_u]$ arising from geometry., Comment: 25 pages
- Published
- 2021
32. An infinite self-dual Ramsey theorem
- Author
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Dimitris Vlitas
- Subjects
Mathematics::Logic ,Pure mathematics ,Mathematics::Combinatorics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Mathematics::General Topology ,010307 mathematical physics ,Ramsey's theorem ,0101 mathematics ,01 natural sciences ,Dual (category theory) ,Mathematics - Abstract
In a recent paper [5] S. Solecki proved a finite self-dual Ramsey theorem that extends simultaneously the classical finite Ramsey theorem [4] and the Graham–Rothschild theorem [2] . In this paper we prove the corresponding infinite dimensional version of the self-dual theorem. As a consequence, we extend the classical infinite Ramsey theorem [4] and the Carlson–Simpson theorem [1] .
- Published
- 2017
33. Answer to a 1962 question by Zappa on cosets of Sylow subgroups
- Author
-
Marston Conder
- Subjects
0301 basic medicine ,Pure mathematics ,Complement (group theory) ,Finite group ,Janko group ,General Mathematics ,010102 general mathematics ,Sylow theorems ,01 natural sciences ,Combinatorics ,03 medical and health sciences ,Normal p-complement ,030104 developmental biology ,Locally finite group ,Order (group theory) ,0101 mathematics ,Zappa–Szép product ,Mathematics - Abstract
In a paper in 1962, Guido Zappa asked whether a non-trivial coset of a Sylow p-subgroup of a finite group could contain only elements whose orders are powers of p, and also in that case, at least one element of order p. The first question was raised again recently in a 2014 paper by Daniel Goldstein and Robert Guralnick, when generalising an answer by John Thompson in 1967 to a similar question by L.J. Paige. In this paper we give a positive answer to both questions of Zappa, showing somewhat surprisingly that in a number of non-abelian finite simple groups (including PSL ( 3 , 4 ) , PSU ( 5 , 2 ) and the Janko group J 3 ), some non-trivial coset of a Sylow 5-subgroup (of order 5) contains only elements of order 5. Also Zappa's first question is studied in more detail. Various possibilities for the group and its Sylow p-subgroup P are eliminated, and it then follows that | P | ≥ 5 and | P | ≠ 8 . It is an open question as to whether the order of the Sylow p-subgroup can be 7 or 9 or more.
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- 2017
34. Exotic elliptic algebras of dimension 4
- Author
-
Alexandru Chirvasitu and S. Paul Smith
- Subjects
Discrete mathematics ,Ring (mathematics) ,General Mathematics ,010102 general mathematics ,Homogeneous coordinate ring ,Algebraic geometry ,Automorphism ,01 natural sciences ,Combinatorics ,Elliptic curve ,Grassmannian ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Algebraically closed field ,Incidence (geometry) ,Mathematics - Abstract
Let E be an elliptic curve defined over an algebraically closed field k whose characteristic is not 2 or 3. Let τ be a translation automorphism of E that is not of order 2. In a previous paper we studied an algebra A = A ( E , τ ) that depends on this data: A ( E , τ ) = ( S ( E , τ ) ⊗ M 2 ( k ) ) Γ where S ( E , τ ) is the 4-dimensional Sklyanin algebra associated to ( E , τ ) , M 2 ( k ) is the ring of 2 × 2 matrices over k, and Γ is ( Z / 2 ) × ( Z / 2 ) acting in a particular way as automorphisms of S and M 2 ( k ) . The action of Γ on S is compatible with the translation action of the 2-torsion subgroup E [ 2 ] on E. Following the ideas and results in papers of Artin–Tate–Van den Bergh, Smith–Stafford, and Levasseur–Smith, this paper examines the line modules, point modules, and fat point modules, over A, and their incidence relations. The right context for the results is non-commutative algebraic geometry: we view A as a homogeneous coordinate ring of a non-commutative analogue of P 3 that we denote by Proj n c ( A ) . Point modules and fat point modules determine “points” in Proj n c ( A ) . Line modules determine “lines” in Proj n c ( A ) . Line modules for A are in bijection with certain lines in P ( A 1 ⁎ ) ≅ P 3 and therefore correspond to the closed points of a certain subscheme L of the Grassmannian G ( 1 , 3 ) . Shelton–Vancliff call L the line scheme for A. We show that L is the union of 7 reduced and irreducible components, 3 quartic elliptic space curves and 4 plane conics in the ambient Plucker P 5 , and that deg ( L ) = 20 . The union of the lines corresponding to the points on each elliptic curve is an elliptic scroll in P ( A 1 ⁎ ) . Thus, the lines on that elliptic scroll are in natural bijection with a corresponding family of line modules for A.
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- 2017
35. Flattening of CR singular points and analyticity of the local hull of holomorphy II
- Author
-
Wanke Yin and Xiaojun Huang
- Subjects
Pure mathematics ,Mathematics::Complex Variables ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Holomorphic function ,Codimension ,Singular point of a curve ,Submanifold ,01 natural sciences ,Plateau's problem ,Hypersurface ,Complex space ,Bounded function ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
This is the second article of the two papers, in which we investigate the holomorphic and formal flattening problem of a non-degenerate CR singular point of a codimension two real submanifold in C n with n ≥ 3 . The problem is motivated from the study of the complex Plateau problem that looks for the Levi-flat hypersurface bounded by a given real submanifold and by the classical complex analysis problem of finding the local hull of holomorphy of a real submanifold in a complex space. The present article is focused on non-degenerate flat CR singular points with at least one non-parabolic Bishop invariant. We will solve the formal flattening problem in this setting. The results in this paper and those in [23] are taken from our earlier arxiv post [22] . We split [22] into two independent articles to avoid it being too long.
- Published
- 2017
36. Bernstein inequality and holonomic modules
- Author
-
Ivan Losev
- Subjects
Pure mathematics ,Holonomic ,General Mathematics ,010102 general mathematics ,Bernstein inequalities ,01 natural sciences ,Representation theory ,0103 physical sciences ,Bimodule ,010307 mathematical physics ,0101 mathematics ,Algebraic number ,Commutative property ,Simple module ,Mathematics ,Symplectic geometry - Abstract
In this paper we study the representation theory of filtered algebras with commutative associated graded whose spectrum has finitely many symplectic leaves. Examples are provided by the algebras of global sections of quantizations of symplectic resolutions, quantum Hamiltonian reductions, and spherical symplectic reflection algebras. We introduce the notion of holonomic modules for such algebras. We show that, provided the algebraic fundamental groups of all leaves are finite, the generalized Bernstein inequality holds for the simple modules and turns into equality for holonomic simples. Under the same finiteness assumption, we prove that the associated variety of a simple holonomic module is equi-dimensional. We also prove that, if the regular bimodule has finite length, then any holonomic module has finite length. This allows one to reduce the Bernstein inequality for arbitrary modules to simple ones. We prove that the regular bimodule has finite length for the global sections of quantizations of symplectic resolutions, for quantum Hamiltonian reductions and for Rational Cherednik algebras. The paper contains a joint appendix by the author and Etingof that motivates the definition of a holonomic module in the case of global sections of a quantization of a symplectic resolution.
- Published
- 2017
37. Extremal function for capacity and estimates of QED constants in Rn
- Author
-
Tao Cheng and Shanshuang Yang
- Subjects
Pure mathematics ,Extremal length ,Geometric function theory ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Boundary (topology) ,Conformal map ,01 natural sciences ,Upper and lower bounds ,Potential theory ,0103 physical sciences ,010307 mathematical physics ,Uniqueness ,0101 mathematics ,Constant (mathematics) ,Mathematics - Abstract
This paper is devoted to the study of some fundamental problems on modulus and extremal length of curve families, capacity, and n-harmonic functions in the Euclidean space R n . One of the main goals is to establish the existence, uniqueness, and boundary behavior of the extremal function for the conformal capacity cap ( A , B ; Ω ) of a capacitor in R n . This generalizes some well known results and has its own interests in geometric function theory and potential theory. It is also used as a major ingredient in this paper to establish a sharp upper bound for the quasiextremal distance (or QED) constant M ( Ω ) of a domain in terms of its local boundary quasiconformal reflection constant H ( Ω ) , a bound conjectured by Shen in the plane. Along the way, several interesting results are established for modulus and extremal length. One of them is a decomposition theorem for the extremal length λ ( A , B ; Ω ) of the curve family joining two disjoint continua A and B in a domain Ω.
- Published
- 2017
38. Generic singularities of nilpotent orbit closures
- Author
-
Fu, Baohua, Juteau, Daniel, Levy, Paul, Sommers, Eric, Academy of Mathematics & Systems Science and Hua Loo-Keng Key Laboratory of Mathe- matics, The Chinese Academy of Sciences, Beijing 100190, CHINA, Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Lancaster University, and Department of Mathematics and Statistics, University of Massachusetts
- Subjects
Mathematics - Algebraic Geometry ,Mathematics(all) ,17B08, 14L30 ,Symplectic singularities ,FOS: Mathematics ,Nilpotent orbits ,Representation Theory (math.RT) ,[MATH]Mathematics [math] ,Mathematics::Representation Theory ,Algebraic Geometry (math.AG) ,Slodowy slice ,Mathematics - Representation Theory - Abstract
According to a well-known theorem of Brieskorn and Slodowy, the intersection of the nilpotent cone of a simple Lie algebra with a transverse slice to the subregular nilpotent orbit is a simple surface singularity. At the opposite extremity of the nilpotent cone, the closure of the minimal nilpotent orbit is also an isolated symplectic singularity, called a minimal singularity. For classical Lie algebras, Kraft and Procesi showed that these two types of singularities suffice to describe all generic singularities of nilpotent orbit closures: specifically, any such singularity is either a simple surface singularity, a minimal singularity, or a union of two simple surface singularities of type $A_{2k-1}$. In the present paper, we complete the picture by determining the generic singularities of all nilpotent orbit closures in exceptional Lie algebras (up to normalization in a few cases). We summarize the results in some graphs at the end of the paper. In most cases, we also obtain simple surface singularities or minimal singularities, though often with more complicated branching than occurs in the classical types. There are, however, six singularities which do not occur in the classical types. Three of these are unibranch non-normal singularities: an $SL_2(\mathbb C)$-variety whose normalization is ${\mathbb A}^2$, an $Sp_4(\mathbb C)$-variety whose normalization is ${\mathbb A}^4$, and a two-dimensional variety whose normalization is the simple surface singularity $A_3$. In addition, there are three 4-dimensional isolated singularities each appearing once. We also study an intrinsic symmetry action on the singularities, in analogy with Slodowy's work for the regular nilpotent orbit., 56 pages (5 figures). Minor corrections. Accepted in Advances in Math
- Published
- 2017
- Full Text
- View/download PDF
39. New pathways and connections in number theory and analysis motivated by two incorrect claims of Ramanujan
- Author
-
Arindam Roy, Atul Dixit, Bruce C. Berndt, and Alexandru Zaharescu
- Subjects
Discrete mathematics ,Series (mathematics) ,General Mathematics ,010102 general mathematics ,Divisor function ,Divisor (algebraic geometry) ,Divergent series ,01 natural sciences ,Ramanujan's sum ,010101 applied mathematics ,symbols.namesake ,Identity (mathematics) ,Number theory ,symbols ,0101 mathematics ,Convergent series ,Mathematics - Abstract
The focus of this paper commences with an examination of three (not obviously related) pages in Ramanujan's lost notebook, pages 336, 335, and 332, in decreasing order of attention. On page 336, Ramanujan proposes two identities, but the formulas are wrong – each is vitiated by divergent series. We concentrate on only one of the two incorrect “identities,” which may have been devised to attack the extended divisor problem. We prove here a corrected version of Ramanujan's claim, which contains the convergent series appearing in it. The convergent series in Ramanujan's faulty claim is similar to one used by G.F. Voronoi, G.H. Hardy, and others in their study of the classical Dirichlet divisor problem. This now brings us to page 335, which comprises two formulas featuring doubly infinite series of Bessel functions, the first being conjoined with the classical circle problem initiated by Gauss, and the second being associated with the Dirichlet divisor problem. The first and fourth authors, along with Sun Kim, have written several papers providing proofs of these two difficult formulas in different interpretations. In this monograph, we return to these two formulas and examine them in more general settings. The aforementioned convergent series in Ramanujan's “identity” is also similar to one that appears in a curious identity found in Chapter 15 in Ramanujan's second notebook, written in a more elegant, equivalent formulation on page 332 in the lost notebook. This formula may be regarded as a formula for ζ ( 1 2 ) , and in 1925, S. Wigert obtained a generalization giving a formula for ζ ( 1 k ) for any even integer k ≥ 2 . We extend the work of Ramanujan and Wigert in this paper. The Voronoi summation formula appears prominently in our study. In particular, we generalize work of J.R. Wilton and derive an analogue involving the sum of divisors function σ s ( n ) . The modified Bessel functions K s ( x ) arise in several contexts, as do Lommel functions. We establish here new series and integral identities involving modified Bessel functions and modified Lommel functions. Among other results, we establish a modular transformation for an infinite series involving σ s ( n ) and modified Lommel functions. We also discuss certain obscure related work of N.S. Koshliakov. We define and discuss two new related classes of integral transforms, which we call Koshliakov transforms, because he first found elegant special cases of each.
- Published
- 2017
40. The other dual of MacMahon's theorem on plane partitions
- Author
-
Mihai Ciucu
- Subjects
Combinatorics ,010201 computation theory & mathematics ,General Mathematics ,Lattice (order) ,010102 general mathematics ,Lattice line ,Hexagonal lattice ,0102 computer and information sciences ,0101 mathematics ,Equilateral triangle ,01 natural sciences ,Mathematics - Abstract
In this paper we introduce a counterpart structure to the shamrocks studied in the paper A dual of Macmahon's theorem on plane partitions by M. Ciucu and C. Krattenthaler (2013) [5] , which, just like the latter, can be included at the center of a lattice hexagon on the triangular lattice so that the region obtained from the hexagon by removing it has its number of lozenge tilings given by a simple product formula. The new structure, called a fern, consists of an arbitrary number of equilateral triangles of alternating orientations lined up along a lattice line. The shamrock and the fern seem to be the only such connected structures with this property. It would be interesting to understand the reason for this.
- Published
- 2017
41. Simplicity of inverse semigroup and étale groupoid algebras
- Author
-
Nóra Szakács and Benjamin Steinberg
- Subjects
Pure mathematics ,Mathematics::Operator Algebras ,General Mathematics ,010102 general mathematics ,Mathematics - Operator Algebras ,Field (mathematics) ,Mathematics - Rings and Algebras ,16. Peace & justice ,01 natural sciences ,Inverse semigroup ,Group action ,Mathematics::K-Theory and Homology ,Simple (abstract algebra) ,Mathematics::Category Theory ,Totally disconnected space ,0103 physical sciences ,20M18, 20M25, 16S99, 16S36, 22A22, 18B40 ,Ideal (order theory) ,010307 mathematical physics ,Simple algebra ,0101 mathematics ,Mathematics - Group Theory ,Unit (ring theory) ,Mathematics - Abstract
In this paper, we prove that the algebra of an \'etale groupoid with totally disconnected unit space has a simple algebra over a field if and only if the groupoid is minimal and effective and the only function of the algebra that vanishes on every open subset is the null function. Previous work on the subject required the groupoid to be also topologically principal in the non-Hausdorff case, but we do not. Furthermore, we provide the first examples of minimal and effective but not topologically principal \'etale groupoids with totally disconnected unit spaces. Our examples come from self-similar group actions of uncountable groups. More generally, we show that the essential algebra of an \'etale groupoid (the quotient by the ideal of functions vanishing on every open set), is simple if and only if the groupoid is minimal and topologically free, generalizing to the algebraic setting a recent result for essential $C^*$-algebras. The main application of our work is to provide a description of the simple contracted inverse semigroup algebras, thereby answering a question of Munn from the seventies. Using Galois descent, we show that simplicity of \'etale groupoid and inverse semigroup algebras depends only on the characteristic of the field and can be lifted from positive characteristic to characteristic $0$. We also provide examples of inverse semigroups and \'etale groupoids with simple algebras outside of a prescribed set of prime characteristics., Comment: Revisions after a referee report. We define the essential algebra of an ample groupoid as the quotient of the Steinberg algebra by its ideal of singular functions and we prove that the essential algebra is simple if and only if the groupoid is minimal and topologically free. When the singular ideal vanishes, one recovers the simplicity result of the previous version of the paper
- Published
- 2021
42. On stability of exactness properties under the pro-completion
- Author
-
Pierre-Alain Jacqmin and Zurab Janelidze
- Subjects
General theorem ,General Mathematics ,010102 general mathematics ,Stability (learning theory) ,01 natural sciences ,Algebra ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,0103 physical sciences ,Embedding ,010307 mathematical physics ,0101 mathematics ,Representation (mathematics) ,Stability theorem ,Mathematics - Abstract
In this paper we formulate and prove a general theorem of stability of exactness properties under the pro-completion, which unifies several such theorems in the literature and gives many more. The theorem depends on a formal approach to exactness properties proposed in this paper, which is based on the theory of sketches. Our stability theorem has applications in proving theorems that establish links between exactness properties, as well as in establishing embedding (representation) theorems for classes of categories defined by exactness properties.
- Published
- 2021
43. Global solutions of 3D incompressible MHD system with mixed partial dissipation and magnetic diffusion near an equilibrium
- Author
-
Jiahong Wu and Yi Zhu
- Subjects
Ideal (set theory) ,Small data ,General Mathematics ,010102 general mathematics ,Mechanics ,Dissipation ,01 natural sciences ,Stability (probability) ,Magnetic field ,Mathematics - Analysis of PDEs ,0103 physical sciences ,FOS: Mathematics ,Compressibility ,35A01, 35B35, 35B65, 76D03, 76E25 ,010307 mathematical physics ,Magnetohydrodynamic drive ,0101 mathematics ,Magnetohydrodynamics ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
This paper focuses on the 3D incompressible magnetohydrodynamic (MHD) equations with mixed partial dissipation and magnetic diffusion. Our main result assesses the global stability of perturbations near the steady solution given by a background magnetic field. The stability problem on the MHD equations with partial or no dissipation has attracted considerable interests recently and there are substantial developments. The new stability result presented here is among the very few stability conclusions currently available for ideal or partially dissipated MHD equations. As a special consequence of the techniques introduced in this paper, we obtain the small data global well-posedness for the 3D incompressible Navier-Stokes equations without vertical dissipation., Comment: Submitted to Advances in Mathematics on December 28, 2018
- Published
- 2021
44. Bilinear forms in Weyl sums for modular square roots and applications
- Author
-
Igor E. Shparlinski, Alexander Dunn, Bryce Kerr, and Alexandru Zaharescu
- Subjects
Conjecture ,Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,Bilinear form ,Siegel zero ,01 natural sciences ,Prime (order theory) ,Quadratic residue ,Combinatorics ,Quadratic equation ,Square root ,0103 physical sciences ,Quadratic field ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
Let q be a prime, P ⩾ 1 and let N q ( P ) denote the number of rational primes p ⩽ P that split in the imaginary quadratic field Q ( − q ) . The first part of this paper establishes various unconditional and conditional (under existence of a Siegel zero) lower bounds for N q ( P ) in the range q 1 / 4 + e ⩽ P ⩽ q , for any fixed e > 0 . This improves upon what is implied by work of Pollack and Benli–Pollack. The second part of this paper is dedicated to proving an estimate for a bilinear form involving Weyl sums for modular square roots (equivalently Salie sums). Our estimate has a power saving in the so-called Polya–Vinogradov range, and our methods involve studying an additive energy coming from quadratic residues in F q . This bilinear form is inspired by the recent automorphic motivation: the second moment for twisted L-functions attached to Kohnen newforms has recently been computed by the first and fourth authors. So the third part of this paper links the above two directions together and outlines the arithmetic applications of this bilinear form. These include the equidistribution of quadratic roots of primes, products of primes, and relaxations of a conjecture of Erdős–Odlyzko–Sarkozy.
- Published
- 2020
45. Quantum toroidal and shuffle algebras
- Author
-
Andrei Neguţ
- Subjects
Pure mathematics ,General Mathematics ,01 natural sciences ,Shuffle algebra ,Mathematics - Algebraic Geometry ,Factorization ,Physics::Plasma Physics ,Mathematics::Quantum Algebra ,Mathematics - Quantum Algebra ,0103 physical sciences ,FOS: Mathematics ,Quantum Algebra (math.QA) ,Representation Theory (math.RT) ,0101 mathematics ,Algebra over a field ,Mathematics::Representation Theory ,Algebraic Geometry (math.AG) ,Quantum ,Mathematics ,Toroid ,010102 general mathematics ,Quiver ,Torus ,K-theory ,010307 mathematical physics ,Mathematics - Representation Theory - Abstract
In this paper, we prove that the quantum toroidal algebra of gl_n is isomorphic to the double shuffle algebra of Feigin and Odesskii for the cyclic quiver. The shuffle algebra viewpoint will allow us to prove a factorization formula for the universal R-matrix of the quantum toroidal algebra., The previous version of this paper was broken into two parts: the present version contains the representation-theoretic half (to which we added a number of additional results) and the geometric half has been moved to arXiv:1811.01011
- Published
- 2020
46. Minimal surfaces near short geodesics in hyperbolic 3-manifolds
- Author
-
Laurent Mazet, Harold Rosenberg, Instituto Nacional de Matemática Pura e Aplicada (IMPA), Instituto Nacional de matematica pura e aplicada, Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA), Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS), and Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM)
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Minimal surface ,Finite volume method ,Geodesic ,General Mathematics ,010102 general mathematics ,Hyperbolic manifold ,01 natural sciences ,Infimum and supremum ,Continuity property ,Differential Geometry (math.DG) ,[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] ,0103 physical sciences ,Convergence (routing) ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
If $M$ is a finite volume complete hyperbolic $3$-manifold, the quantity $\mathcal A_1(M)$ is defined as the infimum of the areas of closed minimal surfaces in $M$. In this paper we study the continuity property of the functional $\mathcal A_1$ with respect to the geometric convergence of hyperbolic manifolds. We prove that it is lower semi-continuous and even continuous if $\mathcal A_1(M)$ is realized by a minimal surface satisfying some hypotheses. Understanding the interaction between minimal surfaces and short geodesics in $M$ is the main theme of this paper, Comment: 35 pages, 4 figures
- Published
- 2020
47. Instability of high dimensional Hamiltonian systems: Multiple resonances do not impede diffusion
- Author
-
Amadeu Delshams, Rafael de la Llave, Tere M. Seara, Universitat Politècnica de Catalunya. Departament de Matemàtiques, and Universitat Politècnica de Catalunya. SD - Sistemes Dinàmics de la UPC
- Subjects
Pure mathematics ,Mathematics(all) ,General Mathematics ,Dynamical Systems (math.DS) ,Scattering map ,01 natural sciences ,010305 fluids & plasmas ,Hamiltonian system ,symbols.namesake ,Arnold diffusion ,0103 physical sciences ,FOS: Mathematics ,Sistemes hamiltonians ,Mathematics - Dynamical Systems ,Hamiltonian systems ,0101 mathematics ,Mathematics ,Scattering ,010102 general mathematics ,Mathematical analysis ,Instability ,Matemàtiques i estadística [Àrees temàtiques de la UPC] ,Resonance ,Torus ,Codimension ,37J40 ,Hamiltonian ,Resonances ,symbols ,Hamiltonian (quantum mechanics) ,Symplectic geometry - Abstract
We consider models given by Hamiltonians of the form H ( I , φ , p , q , t ; e ) = h ( I ) + ∑ j = 1 n ± ( 1 2 p j 2 + V j ( q j ) ) + e Q ( I , φ , p , q , t ; e ) where I ∈ I ⊂ R d , φ ∈ T d , p , q ∈ R n , t ∈ T 1 . These are higher dimensional analogues, both in the center and hyperbolic directions, of the models studied in [28] , [29] , [43] and are usually called “a-priori unstable Hamiltonian systems”. All these models present the large gap problem. We show that, for 0 e ≪ 1 , under regularity and explicit non-degeneracy conditions on the model, there are orbits whose action variables I perform rather arbitrary excursions in a domain of size O ( 1 ) . This domain includes resonance lines and, hence, large gaps among d-dimensional KAM tori. This phenomenon is known as Arnold diffusion. The method of proof follows closely the strategy of [28] , [29] . The main new phenomenon that appears when the dimension d of the center directions is larger than one is the existence of multiple resonances in the space of actions I ∈ I ⊂ R d . We show that, since these multiple resonances happen in sets of codimension greater than one in the space of actions I, they can be contoured. This corresponds to the mechanism called diffusion across resonances in the Physics literature. The present paper, however, differs substantially from [28] , [29] . On the technical details of the proofs, we have taken advantage of the theory of the scattering map developed in [31] —notably the symplectic properties—which were not available when the above papers were written. We have analyzed the conditions imposed on the resonances in more detail. More precisely, we have found that there is a simple condition on the Melnikov potential which allows us to conclude that the resonances are crossed. In particular, this condition does not depend on the resonances. So that the results are new even when applied to the models in [28] , [29] .
- Published
- 2016
- Full Text
- View/download PDF
48. Existence of optimal ultrafilters and the fundamental complexity of simple theories
- Author
-
Saharon Shelah and Maryanthe Malliaris
- Subjects
0301 basic medicine ,Model theory ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Supercompact cardinal ,Ultraproduct ,16. Peace & justice ,01 natural sciences ,Mathematics::Logic ,03 medical and health sciences ,030104 developmental biology ,Stability theory ,Global theory ,0101 mathematics ,Global structure ,Mathematics - Abstract
In the first edition of Classification Theory, the second author characterized the stable theories in terms of saturation of ultrapowers. Prior to this theorem, stability had already been defined in terms of counting types, and the unstable formula theorem was known. A contribution of the ultrapower characterization was that it involved sorting out the global theory, and introducing nonforking, seminal for the development of stability theory. Prior to the present paper, there had been no such ultrapower characterization of an unstable class. In the present paper, we first establish the existence of so-called optimal ultrafilters on (suitable) Boolean algebras, which are to simple theories as Keisler's good ultrafilters [15] are to all (first-order) theories. Then, assuming a supercompact cardinal, we characterize the simple theories in terms of saturation of ultrapowers. To do so, we lay the groundwork for analyzing the global structure of simple theories, in ZFC, via complexity of certain amalgamation patterns. This brings into focus a fundamental complexity in simple unstable theories having no real analogue in stability.
- Published
- 2016
49. Bridgeland stability conditions on the acyclic triangular quiver
- Author
-
Ludmil Katzarkov and George Dimitrov
- Subjects
Pure mathematics ,010308 nuclear & particles physics ,General Mathematics ,010102 general mathematics ,Quiver ,Mathematics - Category Theory ,Space (mathematics) ,01 natural sciences ,Contractible space ,Mathematics - Algebraic Geometry ,Stability conditions ,Mathematics::Category Theory ,0103 physical sciences ,FOS: Mathematics ,Category Theory (math.CT) ,Representation Theory (math.RT) ,0101 mathematics ,Algebraically closed field ,Mathematics::Representation Theory ,Focus (optics) ,Algebraic Geometry (math.AG) ,Mathematics - Representation Theory ,Topology (chemistry) ,Mathematics - Abstract
Using results in a previous paper "Non-semistable exceptional objects in hereditary categories", we focus here on studying the topology of the space of Bridgeland stability conditions on $D^b(Rep_k(Q ))$, where $Q$ is the acyclic triangular quiver (the underlying graph is the extended Dynkin diagram $\widetilde{\mathbb A}_2$). In particular, we prove that this space is contractible (in the previous paper it was shown that it is connected)., Comment: 51 pages
- Published
- 2016
50. Archimedean non-vanishing, cohomological test vectors, and standard L-functions of GL2: Complex case
- Author
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Bingchen Lin and Fangyang Tian
- Subjects
Pure mathematics ,General Mathematics ,Existential quantification ,010102 general mathematics ,Linear model ,Expression (computer science) ,01 natural sciences ,Period relation ,Character (mathematics) ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Representation (mathematics) ,Mathematics - Abstract
The purpose of this paper is to study the local zeta integrals of Friedberg-Jacquet at complex place and to establish similar results to the recent work [4] joint with C. Chen and D. Jiang. In this paper, we will (1) give a necessary and sufficient condition on an irreducible essentially tempered cohomological representation π of GL 2 n ( C ) with a non-zero Shalika model; (2) construct a new twisted linear period Λ s , χ and give a different expression of the linear model for π; (3) give a necessary and sufficient condition on the character χ such that there exists a uniform cohomological test vector v ∈ V π (which we construct explicitly) for Λ s , χ . As a consequence, we obtain the non-vanishing of local Friedberg-Jacquet integral at complex place. All of the above are essential preparations for attacking a global period relation problem in the forthcoming paper ( [11] ).
- Published
- 2020
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