1,257 results
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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. Corrigendum to "Exact essential norm of generalized Hilbert matrix operators on classical analytic function spaces" [Adv. Math. 408 (2022) 108598].
- Author
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Lindström, M., Miihkinen, S., and Norrbo, D.
- Subjects
- *
FUNCTION spaces , *ANALYTIC functions , *ANALYTIC spaces , *MATHEMATICS , *BERGMAN spaces , *HILBERT transform - Abstract
In this note, we provide a corrected proof of Lemma 3.2 in [4]. We also point out some improvements to Remark 5.5 in the same paper. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
13. Pursuing the double affine Grassmannian II: Convolution
- Author
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Braverman, Alexander and Finkelberg, Michael
- Subjects
- *
MATHEMATICAL series , *GRASSMANN manifolds , *MATHEMATICAL convolutions , *GROUP theory , *NUMERICAL analysis , *MATHEMATICS - Abstract
Abstract: This is the second paper of a series (started by Braverman and Finkelberg, 2010 ) which describes a conjectural analog of the affine Grassmannian for affine Kac–Moody groups (also known as the double affine Grassmannian). The current paper is dedicated to describing a conjectural analog of the convolution diagram for the double affine Grassmannian. In the case when our conjectures can be derived from Nakajima (2009) . [Copyright &y& Elsevier]
- Published
- 2012
- Full Text
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14. Holographic formula for Q-curvature
- Author
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Graham, C. Robin and Juhl, Andreas
- Subjects
- *
GEOMETRY , *DIFFERENTIAL equations , *OPERATOR theory , *MATHEMATICS - Abstract
Abstract: This paper derives an explicit formula for Branson''s Q-curvature in even-dimensional conformal geometry. The ingredients in the formula come from the Poincaré metric in one higher dimension; hence the formula is called holographic. When specialized to the conformally flat case, the holographic formula expresses the Q-curvature as a multiple of the Pfaffian and the divergence of a natural 1-form. The paper also outlines the relation between holographic formulae for Q-curvature and a new theory of conformally covariant families of differential operators due to the second author. [Copyright &y& Elsevier]
- Published
- 2007
- Full Text
- View/download PDF
15. Freiman's inverse problem with small doubling property
- Author
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Jin, Renling
- Subjects
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ARITHMETIC , *ALGEBRAIC number theory , *MATHEMATICS , *NUMBER theory - Abstract
Abstract: Let be the set of all nonnegative integers, be a finite set, and 2A be the set of all numbers of the form for all a and b in A. In [G.A. Freiman, Foundations of a Structural Theory of Set Addition, Transl. Math. Monogr., vol. 37, American Mathematical Society, Providence, RI, 1973 (translated from the Russian)] the arithmetic structure of A was optimally characterized when . 2 In [G.A. Freiman, Foundations of a Structural Theory of Set Addition, Transl. Math. Monogr., vol. 37, American Mathematical Society, Providence, RI, 1973 (translated from the Russian)] the structure of A was also characterized without proof when . Since then the efforts have been made to generalize these results, see [V. Lev, P.Y. Smeliansky, On addition of two distinct sets of integers, Acta Arith. 70 (1) (1995) 85–91; V. Lev, On the structure of sets of integers with small doubling property, unpublished manuscripts, 1995; Y.O. Hamidoune, A. Plagne, A generalization of Freiman''s theorem, Acta Arith. 103 (2) (2002) 147–156] for example. However, no optimal characterization of the structure of A has been obtained without imposing extra conditions, until now, when . In this paper we optimally characterize, with the help of nonstandard analysis, the arithmetic structure of A when , where b is positive but not too large. Precisely, we prove that there is a real number and there is such that if and for , then A is either a subset of an arithmetic progression of length at most or a subset of a bi-arithmetic progression 3 of length at most . An application of this result to the inverse problem for upper asymptotic density is presented near the end of the paper. In the application we improve the most important part of the main theorem in [R. Jin, Solution to the inverse problem for upper asymptotic density, J. Reine Angew. Math. (Crelle''s J.) 595 (2006) 121–166]. [2] One can easily prove that is always true and implies that A is an arithmetic progression. [3] The definition of bi-arithmetic progression can be found in the beginning of Introduction. [Copyright &y& Elsevier]
- Published
- 2007
- Full Text
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16. A -analogue of Kazhdan's property (T)
- Author
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Pavlov, A.A. and Troitsky, E.V.
- Subjects
- *
MATHEMATICAL analysis , *MODULES (Algebra) , *MATHEMATICS , *ALGEBRA - Abstract
Abstract: This paper deals with a “naive” way of generalizing Kazhdan''s property (T) to -algebras. Our approach differs from the approach of Connes and Jones, which has already demonstrated its utility. Nevertheless, it turns out that our approach is applicable to a rather subtle question in the theory of -Hilbert modules. Namely, we prove that a separable unital -algebra A has property MI (module infinite—i.e. any countably generated self-dual Hilbert module over A is finitely generated and projective) if and only if A does not satisfy our definition of property (T). The commutative case was studied in an earlier paper. [Copyright &y& Elsevier]
- Published
- 2007
- Full Text
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17. The sub-elliptic obstacle problem: regularity of the free boundary in Carnot groups of step two
- Author
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Danielli, Donatella, Garofalo, Nicola, and Petrosyan, Arshak
- Subjects
- *
MATHEMATICAL analysis , *NUMERICAL analysis , *ASYMPTOTIC expansions , *MATHEMATICS - Abstract
Abstract: The sub-elliptic obstacle problem arises in various branches of the applied sciences, e.g., in mechanical engineering and robotics, mathematical finance, image reconstruction and neurophysiology. In the recent paper [Donatella Danielli, Nicola Garofalo, Sandro Salsa, Variational inequalities with lack of ellipticity. I. Optimal interior regularity and non-degeneracy of the free boundary, Indiana Univ. Math. J. 52 (2) (2003) 361–398; MR1976081 (2004c:35424)] it was proved that weak solutions to the sub-elliptic obstacle problem in a Carnot group belong to the Folland–Stein (optimal) Lipschitz class (the analogue of the well-known interior local regularity for the classical obstacle problem). However, the regularity of the free boundary remained a challenging open problem. In this paper we prove that, in Carnot groups of step , the free boundary is (Euclidean) near points satisfying a certain thickness condition. This constitutes the sub-elliptic counterpart of a celebrated result due to Caffarelli [Luis A. Caffarelli, The regularity of free boundaries in higher dimensions, Acta Math. 139 (3–4) (1977) 155–184; MR0454350 (56 #12601)]. [Copyright &y& Elsevier]
- Published
- 2007
- Full Text
- View/download PDF
18. Chiral equivariant cohomology I
- Author
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Lian, Bong H. and Linshaw, Andrew R.
- Subjects
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MATHEMATICS , *ALGEBRA , *MATHEMATICAL analysis , *HOMOLOGY theory - Abstract
Abstract: We construct a new equivariant cohomology theory for a certain class of differential vertex algebras, which we call the chiral equivariant cohomology. A principal example of a differential vertex algebra in this class is the chiral de Rham complex of Malikov–Schechtman–Vaintrob of a manifold with a group action. The main idea in this paper is to synthesize the algebraic approach to classical equivariant cohomology due to H. Cartan, 2 with [2] Cartan''s theory was further developed by Duflo–Kumar–Vergne [M. Duflo, S. Kumar, M. Vergne, Sur la cohomologie équivariante des variétés différentiables, Astérisque 215 (1993)] and Guillemin–Sternberg [V. Guillemin, S. Sternberg, Supersymmetry and Equivariant de Rham Theory, Springer, 1999]. This paper follows closely the latter approach. the theory of differential vertex algebras, by using an appropriate notion of invariant theory. We also construct the vertex algebra analogues of the Mathai–Quillen isomorphism, the Weil and the Cartan models for equivariant cohomology, and the Chern–Weil map. We give interesting cohomology classes in the new theory that have no classical analogues. [Copyright &y& Elsevier]
- Published
- 2007
- Full Text
- View/download PDF
19. On the existence of isotropic forms of semi-simple algebraic groups over number fields with prescribed local behavior
- Author
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Prasad, Gopal and Rapinchuk, Andrei S.
- Subjects
- *
MATHEMATICS , *ARITHMETIC , *ALGEBRAIC fields , *LINEAR algebra - Abstract
Abstract: This note is a follow-up on the paper [A. Borel, G. Harder, Existence of discrete cocompact subgroups of reductive groups over local fields, J. Reine Angew. Math. 298 (1978) 53–64] of A. Borel and G. Harder in which they proved the existence of a cocompact lattice in the group of rational points of a connected semi-simple algebraic group over a local field of characteristic zero by constructing an appropriate form of the semi-simple group over a number field and considering a suitable S-arithmetic subgroup. Some years ago A. Lubotzky initiated a program to study the subgroup growth of arithmetic subgroups, the current stage of which focuses on “counting” (more precisely, determining the asymptotics of) the number of lattices of bounded covolume (the finiteness of this number was established in [A. Borel, G. Prasad, Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups, Publ. Math. Inst. Hautes Études Sci. 69 (1989) 119–171; Addendum: Publ. Math. Inst. Hautes Études Sci. 71 (1990) 173–177] using the formula for the covolume developed in [G. Prasad, Volumes of S-arithmetic quotients of semi-simple groups, Publ. Math. Inst. Hautes Études Sci. 69 (1989) 91–117]). Work on this program led M. Belolipetsky and A. Lubotzky to ask questions about the existence of isotropic forms of semi-simple groups over number fields with prescribed local behavior. In this paper we will answer these questions. A question of similar nature also arose in the work [D. Morris, Real representations of semisimple Lie algebras have -forms, in: Proc. Internat. Conf. on Algebraic Groups and Arithmetic, December 17–22, 2001, TIFR, Mumbai, 2001, pp. 469–490] of D. Morris (Witte) on a completely different topic. We will answer that question too. [Copyright &y& Elsevier]
- Published
- 2006
- Full Text
- View/download PDF
20. On the decomposition of global conformal invariants II
- Author
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Alexakis, Spyros
- Subjects
- *
INVARIANTS (Mathematics) , *COMPLEX variables , *RIEMANNIAN geometry , *MATHEMATICS - Abstract
Abstract: This paper is a continuation of [S. Alexakis, The decomposition of global conformal invariants I, submitted for publication, see also math.DG/0509571], where we complete our partial proof of the Deser–Schwimmer conjecture on the structure of “global conformal invariants.” Our theorem deals with such invariants that locally depend only on the curvature tensor (without covariant derivatives). In [S. Alexakis, The decomposition of global conformal invariants I, Ann. of Math., in press] we developed a powerful tool, the “super divergence formula” which applies to any Riemannian operator that always integrates to zero on compact manifolds. In particular, it applies to the operator that measures the “non-conformally invariant part” of . This paper resolves the problem of using this information we have obtained on the structure of to understand the structure of . [Copyright &y& Elsevier]
- Published
- 2006
- Full Text
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21. Towards the geometry of double Hurwitz numbers
- Author
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Goulden, I.P., Jackson, D.M., and Vakil, R.
- Subjects
- *
GEOMETRY , *MATHEMATICS , *ALGEBRA , *SYMMETRIC functions - Abstract
Abstract: Double Hurwitz numbers count branched covers of with fixed branch points, with simple branching required over all but two points 0 and , and the branching over 0 and specified by partitions of the degree (with m and n parts, respectively). Single Hurwitz numbers (or more usually, Hurwitz numbers) have a rich structure, explored by many authors in fields as diverse as algebraic geometry, symplectic geometry, combinatorics, representation theory, and mathematical physics. The remarkable ELSV formula relates single Hurwitz numbers to intersection theory on the moduli space of curves. This connection has led to many consequences, including Okounkov and Pandharipande''s proof of Witten''s conjecture. In this paper, we determine the structure of double Hurwitz numbers using techniques from geometry, algebra, and representation theory. Our motivation is geometric: we give evidence that double Hurwitz numbers are top intersections on a moduli space of curves with a line bundle (a universal Picard variety). In particular, we prove a piecewise-polynomiality result analogous to that implied by the ELSV formula. In the case (complete branching over one point) and n is arbitrary, we conjecture an ELSV-type formula, and show it to be true in genus 0 and 1. The corresponding Witten-type correlation function has a richer structure than that for single Hurwitz numbers, and we show that it satisfies many geometric properties, such as the string and dilaton equations, and an Itzykson–Zuber-style genus expansion ansatz. We give a symmetric function description of the double Hurwitz generating series, which leads to explicit formulae for double Hurwitz numbers with given m and n, as a function of genus. In the case where m is fixed but not necessarily 1, we prove a topological recursion on the corresponding generating series, which leads to closed-form expressions for double Hurwitz numbers and an analogue of the Goulden–Jackson polynomiality conjecture (an early conjectural variant of the ELSV formula). In a later paper (Faber''s intersection number conjecture and genus 0 double Hurwitz numbers, 2005, in preparation), the formulae in genus 0 will be shown to be equivalent to the formulae for “top intersections” on the moduli space of smooth curves . For example, three formulae we give there will imply Faber''s intersection number conjecture (in: Moduli of Curves and Abelian Varieties, Aspects of Mathematics, vol. E33, Vieweg, Braunschweig, 1999, pp. 109–129) in arbitrary genus with up to three points. [Copyright &y& Elsevier]
- Published
- 2005
- Full Text
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22. The Mukai pairing—II: the Hochschild–Kostant–Rosenberg isomorphism
- Author
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Căldăraru, Andrei
- Subjects
- *
SET theory , *MATHEMATICS , *ARITHMETIC , *DIFFERENTIAL geometry - Abstract
Abstract: We continue the study of the Hochschild structure of a smooth space that we began in our previous paper, examining implications of the Hochschild–Kostant–Rosenberg theorem. The main contributions of the present paper are:[] we introduce a generalization of the usual notions of Mukai vector and Mukai pairing on differential forms that applies to arbitrary manifolds; [] we give a proof of the fact that the natural Chern character map becomes, after the HKR isomorphism, the usual one ; and [] we present a conjecture that relates the Hochschild and harmonic structures of a smooth space, similar in spirit to the Tsygan formality conjecture. [Copyright &y& Elsevier]
- Published
- 2005
- Full Text
- View/download PDF
23. On distinct sums and distinct distances.
- Author
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Tardos, Gábor
- Subjects
- *
ARITHMETIC , *SET theory , *MATHEMATICS - Abstract
The paper (Discrete Comput. Geom. 25 (2001) 629) of Solymosi and Tóth implicitly raised the following arithmetic problem. Consider 𝑛 pairwise disjoint s element sets and form all (s2)n sums of pairs of elements of the same set. What is the minimum number of distinct sums one can get this way? This paper proves that the number of distinct sums is at least 𝑛ds, where ds=1/c⌈s/2⌉ is defined in the paper and tends to e−1 as s goes to infinity. Here e is the base of the natural logarithm. As an application we improve the Solymosi–Tóth bound on an old Erdo&x030B;s problem: we prove that n distinct points in the plane determine Ω(n4e5e−1−ϵ) distinct distances, where ϵ>0 is arbitrary. Our bound also finds applications in other related results in discrete geometry. Our bounds are proven through an involved calculation of entropies of several random variables. [Copyright &y& Elsevier]
- Published
- 2003
- Full Text
- View/download PDF
24. Enumerating Representations in Finite Wreath Products II: Explicit Formulae
- Author
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Müller, Thomas W. and Shareshian, John
- Subjects
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MATHEMATICS , *WREATH products (Group theory) - Abstract
We present several contributions to the enumerative theory of wreath product representations developed in a previous paper by the first named author (Adv. in Math.153 (2000), 118–154). Theorem 3.1 of the present paper establishes an explicit formula for one of the key ingredients in the description of the corresponding generating functions given in Mu¨ller (2000) (the exterior function ΦΓ). Building on Theorem 1 in Mu¨ller (2000) and the latter result, we derive explicit formulae for the exponential generating function of the series {∣Hom (Γ,Rn)∣} in the case where Γ is dihedral or a finite abelian group, and the representation sequence {Rn} is any of {H≀Sn} or {H≀An} with a fixed finite group H, or the sequence {∣Hom(G, Wn)}. Moreover, we verify a conjecture concerning the asymptotic behaviour of the sequence \{∣Hom(G,Wn)∣\} for finite groups G made in Mu¨ller (2000) in the case when G is dihedral or abelian. [Copyright &y& Elsevier]
- Published
- 2002
- Full Text
- View/download PDF
25. Rigidity of inversive distance circle packings revisited.
- Author
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Xu, Xu
- Subjects
- *
COMBINATORICS , *MATHEMATICAL analysis , *INFINITESIMAL geometry , *NONNEGATIVE matrices , *MATHEMATICS - Abstract
Inversive distance circle packing metric was introduced by P Bowers and K Stephenson [7] as a generalization of Thurston's circle packing metric [34] . They conjectured that the inversive distance circle packings are rigid. For nonnegative inversive distance, Guo [22] proved the infinitesimal rigidity and then Luo [27] proved the global rigidity. In this paper, based on an observation of Zhou [37] , we prove this conjecture for inversive distance in ( − 1 , + ∞ ) by variational principles. We also study the global rigidity of a combinatorial curvature introduced in [14,16,19] with respect to the inversive distance circle packing metrics where the inversive distance is in ( − 1 , + ∞ ) . [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
26. Enriched Stone-type dualities.
- Author
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Hofmann, Dirk and Nora, Pedro
- Subjects
- *
MATHEMATICAL equivalence , *MATHEMATICS , *EQUIVALENCE relations (Set theory) , *BOOLEAN algebra , *ALGEBRAIC logic - Abstract
A common feature of many duality results is that the involved equivalence functors are liftings of hom-functors into the two-element space resp. lattice. Due to this fact, we can only expect dualities for categories cogenerated by the two-element set with an appropriate structure. A prime example of such a situation is Stone's duality theorem for Boolean algebras and Boolean spaces, the latter being precisely those compact Hausdorff spaces which are cogenerated by the two-element discrete space. In this paper we aim for a systematic way of extending this duality theorem to categories including all compact Hausdorff spaces. To achieve this goal, we combine duality theory and quantale-enriched category theory. Our main idea is that, when passing from the two-element discrete space to a cogenerator of the category of compact Hausdorff spaces, all other involved structures should be substituted by corresponding enriched versions. Accordingly, we work with the unit interval [ 0 , 1 ] and present duality theory for ordered and metric compact Hausdorff spaces and (suitably defined) finitely cocomplete categories enriched in [ 0 , 1 ] . [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
27. Ranks of Maharam algebras.
- Author
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Perović, Žikica and Veličković, Boban
- Subjects
- *
SEMIGROUP algebras , *ALGEBRA , *MATHEMATICS , *ABSTRACT algebra , *MATHEMATICAL analysis - Abstract
Solving a well-known problem of Maharam, Talagrand [18] constructed an exhaustive non uniformly exhaustive submeasure, thus also providing the first example of a Maharam algebra that is not a measure algebra. To each exhaustive submeasure one can canonically assign a certain countable ordinal, its exhaustivity rank. In this paper, we use carefully constructed Schreier families and norms derived from them to provide examples of exhaustive submeasures of arbitrary high exhaustivity rank. This gives rise to uncountably many non isomorphic separable atomless Maharam algebras. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
28. Corrigendum to "Successive minima of line bundles" [Adv. Math. 365 (2020) 107045].
- Author
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Ambro, Florin and Ito, Atsushi
- Subjects
- *
MATHEMATICS , *LINEAR systems - Abstract
The proof of Theorem 0.4 in "Successive minima of line bundles" contains an error. We give a modified statement of Theorem 0.4, which is weaker than the original one. Since Theorem 0.1 in the paper is a special case of Theorem 0.4, we give a new proof of Theorem 0.1. This proof improves a lower bound in Theorem 0.1. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
29. Multifractal analysis of some multiple ergodic averages.
- Author
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Fan, Ai-Hua, Schmeling, Jörg, and Wu, Meng
- Subjects
- *
ERGODIC theory , *MULTIFRACTALS , *INTEGERS , *MAXIMAL functions , *MATHEMATICS - Abstract
In this paper we study the multiple ergodic averages 1 n ∑ k = 1 n φ ( x k , x k q , ⋯ , x k q ℓ − 1 ) , ( x n ) ∈ Σ m on the symbolic space Σ m = { 0 , 1 , ⋯ , m − 1 } N ⁎ where m ≥ 2 , ℓ ≥ 2 , q ≥ 2 are integers. We give a complete solution to the problem of multifractal analysis of the limit of the above multiple ergodic averages. Actually we develop a non-invariant and non-linear version of thermodynamic formalism that is of its own interest. We study a large class of measures (called telescopic product measures). The special case of telescopic product measures defined by the fixed points of some non-linear transfer operators plays a crucial role in studying the level sets of the limit, which are not shift-invariant. These measures share many properties with Gibbs measures in the classical thermodynamic formalism. Our work also concerns variational principle, pressure function and Legendre transform in this new setting. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
30. 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
31. 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
32. 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
33. Quasi-elliptic cohomology I
- Author
-
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
34. The Webster scalar curvature flow on CR sphere. Part II.
- Author
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Ho, Pak Tung
- Subjects
- *
CURVATURE , *EXISTENCE theorems , *MATHEMATICAL functions , *MATHEMATICAL analysis , *MATHEMATICS - Abstract
This is the second of two papers, in which we study the problem of prescribing Webster scalar curvature on the CR sphere as a given function f . Using the Webster scalar curvature flow, we prove an existence result under suitable assumptions on the Morse indices of f . [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
35. Balanced derivatives, identities, and bounds for trigonometric and Bessel series
- Author
-
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
36. Transfer operators and Hankel transforms between relative trace formulas, II: Rankin–Selberg theory
- Author
-
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
37. Decomposition spaces, incidence algebras and Möbius inversion III: The decomposition space of Möbius intervals
- Author
-
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
38. Nevanlinna theory of the Askey–Wilson divided difference operator
- Author
-
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
39. The Goldman–Turaev Lie bialgebra in genus zero and the Kashiwara–Vergne problem
- Author
-
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
40. 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
41. 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
42. 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
43. A Hardy–Moser–Trudinger inequality
- Author
-
Wang, Guofang and Ye, Dong
- Subjects
- *
MATHEMATICAL inequalities , *DIMENSIONAL analysis , *PROOF theory , *FUNCTION spaces , *MATHEMATICAL analysis , *MATHEMATICS - Abstract
Abstract: In this paper we obtain an inequality on the unit disk B in , which improves the classical Moser–Trudinger inequality and the classical Hardy inequality at the same time. Namely, there exists a constant such that where This inequality is a two-dimensional analog of the Hardy–Sobolev–Mazʼya inequality in higher dimensions, which has been intensively studied recently. We also prove that the supremum is achieved in a suitable function space, which is an analog of the celebrated result of Carleson–Chang for the Moser–Trudinger inequality. [Copyright &y& Elsevier]
- Published
- 2012
- Full Text
- View/download PDF
44. GIT versus Baily-Borel compactification for K3's which are double covers of P1×P1
- Author
-
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.
- Published
- 2021
45. 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
46. Partial orders on conjugacy classes in the Weyl group and on unipotent conjugacy classes
- Author
-
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
47. Multiple closed geodesics on 3-spheres
- Author
-
Long, Yiming and Duan, Huagui
- Subjects
- *
GEODESICS , *DIFFERENTIAL geometry , *GLOBAL analysis (Mathematics) , *MATHEMATICS - Abstract
Abstract: This paper is devoted to a study on closed geodesics on Finsler and Riemannian spheres. We call a prime closed geodesic on a Finsler manifold rational, if the basic normal form decomposition (cf. [Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math. 187 (1999) 113–149]) of its linearized Poincaré map contains no rotation matrix with rotation angle which is an irrational multiple of π, or irrational otherwise. We prove that if there exists only one prime closed geodesic on a d-dimensional irreversible Finsler sphere with , it cannot be rational. Then we further prove that there exist always at least two distinct prime closed geodesics on every irreversible Finsler 3-dimensional sphere. Our method yields also at least two geometrically distinct closed geodesics on every reversible Finsler as well as Riemannian 3-dimensional sphere. We prove also such results hold for all compact simply connected 3-dimensional manifolds with irreversible or reversible Finsler as well as Riemannian metrics. [Copyright &y& Elsevier]
- Published
- 2009
- Full Text
- View/download PDF
48. Combinatorial lemmas and applications to dynamics
- Author
-
Huang, Wen and Ye, Xiangdong
- Subjects
- *
DYNAMICS , *ENTROPY , *THERMODYNAMICS , *MATHEMATICS - Abstract
Abstract: The well-known combinatorial lemma of Karpovsky, Milman and Alon and a very recent one of Kerr and Li are extended. The obtained lemmas are applied to study the maximal pattern entropy introduced in the paper. It turns out that the maximal pattern entropy is equal to the supremum of sequence entropies over all sequences both in topological and measure-theoretical settings. Moreover, it is shown the maximal pattern entropy of any topological system is for some with k the maximal length of intrinsic sequence entropy tuples; and a zero-dimensional system has zero sequence entropy for any sequence if and only if the maximal pattern with respect to any open cover is of polynomial order. [Copyright &y& Elsevier]
- Published
- 2009
- Full Text
- View/download PDF
49. G-functions and multisum versus holonomic sequences
- Author
-
Garoufalidis, Stavros
- Subjects
- *
MATHEMATICS , *H-functions , *HYPERGEOMETRIC functions , *NUMERICAL analysis - Abstract
Abstract: The purpose of the paper is three-fold: (a) we prove that every sequence which is a multidimensional sum of a balanced hypergeometric term has an asymptotic expansion of Gevrey type-1 with rational exponents, (b) we construct a class of G-functions that come from enumerative combinatorics, and (c) we give a counterexample to a question of Zeilberger that asks whether holonomic sequences can be written as multisums of balanced hypergeometric terms. The proofs utilize the notion of a G-function, introduced by Siegel, and its analytic/arithmetic properties shown recently by André. [Copyright &y& Elsevier]
- Published
- 2009
- Full Text
- View/download PDF
50. Toric degeneration of branching algebras
- Author
-
Howe, Roger, Jackson, Steven, Lee, Soo Teck, Tan, Eng-Chye, and Willenbring, Jeb
- Subjects
- *
MATHEMATICS , *ALGEBRA , *LATTICE theory , *MATHEMATICAL analysis - Abstract
Abstract: For each classical symmetric pair , there is a naturally defined multi-graded algebra , called the branching algebra for , which encodes the branching rule from G to H. This algebra has a natural family of subalgebras, depending on integer parameters. For a certain range of the parameters, the subalgebras have a particularly simple structure and are called stable branching algebras. In this paper, we show that the stable branching algebras for eight out of the ten families of classical symmetric pairs are flat deformations of the semigroup algebras of explicitly described lattice cones. [Copyright &y& Elsevier]
- Published
- 2009
- Full Text
- View/download PDF
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