1,076 results
Search Results
2. Derived Non-archimedean analytic Hilbert space
- Author
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Mauro Porta, Jorge António, Institut de Recherche Mathématique Avancée (IRMA), and Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Pure mathematics ,Fiber (mathematics) ,General Mathematics ,010102 general mathematics ,Short paper ,Formal scheme ,Hilbert space ,Space (mathematics) ,01 natural sciences ,symbols.namesake ,Mathematics - Algebraic Geometry ,Mathematics::Category Theory ,0103 physical sciences ,Localization theorem ,FOS: Mathematics ,symbols ,010307 mathematical physics ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,0101 mathematics ,Algebraic Geometry (math.AG) ,Quotient ,Mathematics - Abstract
In this short paper we combine the representability theorem introduced in [17, 18] with the theory of derived formal models introduced in [2] to prove the existence representability of the derived Hilbert space RHilb(X) for a separated k-analytic space X. Such representability results relies on a localization theorem stating that if X is a quasi-compact and quasi-separated formal scheme, then the \infty-category Coh^+(X^rig) of almost perfect complexes over the generic fiber can be realized as a Verdier quotient of the \infty-category Coh^+(X). Along the way, we prove several results concerning the the \infty-categories of formal models for almost perfect modules on derived k-analytic spaces., 28 pages
- Published
- 2019
3. Nψ,ϕ-type Quotient Modules over the Bidisk
- Author
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Chang Hui Wu and Tao Yu
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Essential spectrum ,Hardy space ,Characterization (mathematics) ,Type (model theory) ,01 natural sciences ,symbols.namesake ,Compact space ,Compression (functional analysis) ,0103 physical sciences ,Quotient module ,symbols ,010307 mathematical physics ,0101 mathematics ,Quotient ,Mathematics - Abstract
Let H2(ⅅ2) be the Hardy space over the bidisk ⅅ2, and let Mψ,ϕ = [(ψ(z) − ϕ(w))2] be the submodule generated by (ψ(z) − ϕ(w))2, where ψ(z) and ϕ(w) are nonconstant inner functions. The related quotient module is denoted by Nψ,ϕ = H2(ⅅ2) ⊖ Mψ,ϕ. In this paper, we give a complete characterization for the essential normality of Nψ,ϕ. In particular, if ψ(z)= z, we simply write Mψ,ϕ and Nψ,ϕ as Mϕ and Nϕ respectively. This paper also studies compactness of evaluation operators L(0)∣nϕ and R(0)ϕnϕ, essential spectrum of compression operator Sz on Nϕ, essential normality of compression operators Sz and Sw on Nϕ.
- Published
- 2020
4. More about singular traces on simply generated operator ideals
- Author
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Albrecht Pietsch
- Subjects
Large class ,Sequence ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Hilbert space ,Extension (predicate logic) ,Space (mathematics) ,01 natural sciences ,symbols.namesake ,Operator (computer programming) ,0103 physical sciences ,symbols ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
During half a century, singular traces on ideals of Hilbert space operators have been constructed by looking for linear forms on associated sequence ideals. Only recently, the author was able to eliminate this auxiliary step by directly applying Banach’s version of the extension theorem; see (Integral Equ. Oper. Theory 91, 21, 2019 and 92, 7, 2020). Of course, the relationship between the new approach and the older ones must be investigated. In the first paper, this was done for $${\mathfrak {L}}_{1,\infty } (H)$$ . To save space, such considerations were postponed in the second paper, which deals with a large class of principal ideals, called simply generated. This omission will now be rectified.
- Published
- 2020
5. Vector-valued q-variational inequalities for averaging operators and the Hilbert transform
- Author
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Tao Ma, Wei Liu, and Guixiang Hong
- Subjects
Mathematics::Functional Analysis ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Banach space ,01 natural sciences ,symbols.namesake ,0103 physical sciences ,Variational inequality ,symbols ,010307 mathematical physics ,Hilbert transform ,0101 mathematics ,Martingale (probability theory) ,Mathematics - Abstract
Recently, the authors have established $$L^p$$ -boundedness of vector-valued q-variational inequalities for averaging operators which take values in the Banach space satisfying the martingale cotype q property in Hong and Ma (Math Z 286(1–2):89–120, 2017). In this paper, we prove that the martingale cotype q property is also necessary for the vector-valued q-variational inequalities, which was a question left open in the previous paper. Moreover, we also prove that the UMD property and the martingale cotype q property can be characterized in terms of vector valued q-variational inequalities for the Hilbert transform.
- Published
- 2020
6. Metaplectic representations of Hecke algebras, Weyl group actions, and associated polynomials
- Author
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Vidya Venkateswaran, Jasper V. Stokman, Siddhartha Sahi, Algebra, Geometry & Mathematical Physics (KDV, FNWI), Quantum Matter and Quantum Information, KdV Other Research (FNWI), Faculty of Science, and KDV (FNWI)
- Subjects
Weyl group ,Polynomial ,Pure mathematics ,Algebraic combinatorics ,Series (mathematics) ,General Mathematics ,010102 general mathematics ,General Physics and Astronomy ,20C08 (Primary), 11F68, 22E50 (Secondary) ,Rational function ,01 natural sciences ,symbols.namesake ,Macdonald polynomials ,Gauss sum ,0103 physical sciences ,symbols ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Representation Theory (math.RT) ,Mathematics::Representation Theory ,Dirichlet series ,Mathematics - Representation Theory ,Mathematics - Abstract
Chinta and Gunnells introduced a rather intricate multi-parameter Weyl group action on rational functions on a torus, which, when the parameters are specialized to certain Gauss sums, describes the functional equations of Weyl group multiple Dirichlet series associated to metaplectic (n-fold) covers of algebraic groups. In subsequent joint work with Puskas, they extended this action to a "metaplectic" representation of the equal parameter affine Hecke algebra, which allowed them to obtain explicit formulas for the p-parts of these Dirichlet series. They have also verified by a computer check the remarkable fact that their formulas continue to define a group action for general (unspecialized) parameters. In the first part of paper we give a conceptual explanation of this fact, by giving a uniform and elementary construction of the "metaplectic" representation for generic Hecke algebras as a suitable quotient of a parabolically induced affine Hecke algebra module, from which the associated Chinta-Gunnells Weyl group action follows through localization. In the second part of the paper we extend the metaplectic representation to the double affine Hecke algebra, which provides a generalization of Cherednik's basic representation. This allows us to introduce a new family of "metaplectic" polynomials, which generalize nonsymmetric Macdonald polynomials. In this paper, we provide the details of the construction of metaplectic polynomials in type A; the general case will be handled in the sequel to this paper., 39 pages. Version 2 is a significant revision. Added second part introducing a new family of "metaplectic" polynomials, which generalize nonsymmetric Macdonald polynomials and metaplectic Iwahori-Whittaker functions. Title has been changed and the introduction has been expanded
- Published
- 2021
7. An effective Chebotarev density theorem for families of number fields, with an application to $$\ell $$-torsion in class groups
- Author
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Lillian B. Pierce, Caroline L. Turnage-Butterbaugh, and Melanie Matchett Wood
- Subjects
Discrete mathematics ,Mathematics - Number Theory ,Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,Algebraic number field ,01 natural sciences ,Riemann hypothesis ,symbols.namesake ,Arbitrarily large ,Number theory ,Discriminant ,Field extension ,0103 physical sciences ,FOS: Mathematics ,symbols ,Torsion (algebra) ,Number Theory (math.NT) ,010307 mathematical physics ,0101 mathematics ,Dedekind zeta function ,Mathematics - Abstract
We prove a new effective Chebotarev density theorem for Galois extensions $L/\mathbb{Q}$ that allows one to count small primes (even as small as an arbitrarily small power of the discriminant of $L$); this theorem holds for the Galois closures of "almost all" number fields that lie in an appropriate family of field extensions. Previously, applying Chebotarev in such small ranges required assuming the Generalized Riemann Hypothesis. The error term in this new Chebotarev density theorem also avoids the effect of an exceptional zero of the Dedekind zeta function of $L$, without assuming GRH. We give many different "appropriate families," including families of arbitrarily large degree. To do this, we first prove a new effective Chebotarev density theorem that requires a zero-free region of the Dedekind zeta function. Then we prove that almost all number fields in our families yield such a zero-free region. The innovation that allows us to achieve this is a delicate new method for controlling zeroes of certain families of non-cuspidal $L$-functions. This builds on, and greatly generalizes the applicability of, work of Kowalski and Michel on the average density of zeroes of a family of cuspidal $L$-functions. A surprising feature of this new method, which we expect will have independent interest, is that we control the number of zeroes in the family of $L$-functions by bounding the number of certain associated fields with fixed discriminant. As an application of the new Chebotarev density theorem, we prove the first nontrivial upper bounds for $\ell$-torsion in class groups, for all integers $\ell \geq 1$, applicable to infinite families of fields of arbitrarily large degree., Comment: 52 pages. This shorter version aligns with the published paper. Note that portions of Section 8 of the longer v1 have been developed as a separate paper with identifier arXiv:1902.02008
- Published
- 2019
8. Type classification of extreme quantized characters
- Author
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Ryosuke Sato
- Subjects
Pure mathematics ,Dynamical systems theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Context (language use) ,01 natural sciences ,Representation theory ,Quantization (physics) ,symbols.namesake ,Character (mathematics) ,Operator algebra ,0103 physical sciences ,symbols ,010307 mathematical physics ,0101 mathematics ,Quantum ,Mathematics ,Von Neumann architecture - Abstract
The notion of quantized characters was introduced in our previous paper as a natural quantization of characters in the context of asymptotic representation theory forquantum groups. As in the case of ordinary groups, the representation associated with any extreme quantized character generates a von Neumann factor. From the viewpoint of operator algebras (and measurable dynamical systems), it is natural to ask what is the Murray–von Neumann–Connes type of the resulting factor. In this paper, we give a complete solution to this question when the inductive system is of quantum unitary groups $U_{q}(N)$.
- Published
- 2019
9. Courant-sharp Robin eigenvalues for the square: the case with small Robin parameter
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Katie Gittins, Bernard Helffer, Université de Neuchâtel (Université de Neuchâtel), Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), and Helffer, Bernard
- Subjects
Spectral theory ,General Mathematics ,Courant-sharp ,[MATH] Mathematics [math] ,01 natural sciences ,Domain (mathematical analysis) ,Square (algebra) ,Mathematics - Spectral Theory ,symbols.namesake ,Robin eigenvalues ,0103 physical sciences ,FOS: Mathematics ,[MATH.MATH-SP] Mathematics [math]/Spectral Theory [math.SP] ,Neumann boundary condition ,square ,[MATH]Mathematics [math] ,0101 mathematics ,Spectral Theory (math.SP) ,Eigenvalues and eigenvectors ,Mathematics ,35P99, 58J50, 58J37 ,010102 general mathematics ,Mathematical analysis ,Mathematics::Spectral Theory ,Robin boundary condition ,Number theory ,Dirichlet boundary condition ,symbols ,010307 mathematical physics ,[MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP] - Abstract
International audience; This article is the continuation of our first work on the determination of the cases where there is equality in Courant's Nodal Domain theorem in the case of a Robin boundary condition (with Robin parameter h). For the square, our first paper focused on the case where h is large and extended results that were obtained by Pleijel, Bérard-Helffer, for the problem with a Dirichlet boundary condition. There, we also obtained some general results about the behaviour of the nodal structure (for planar domains) under a small deformation of h, where h is positive and not close to 0. In this second paper, we extend results that were obtained by Helffer-Persson-Sundqvist for the Neumann problem to the case where h > 0 is small. MSC classification (2010): 35P99, 58J50, 58J37.
- Published
- 2019
10. Cubics in 10 variables vs. cubics in 1000 variables: Uniformity phenomena for bounded degree polynomials
- Author
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Daniel Erman, Steven V Sam, and Andrew Snowden
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Pure mathematics ,General Mathematics ,media_common.quotation_subject ,MathematicsofComputing_GENERAL ,Hilbert's basis theorem ,Commutative Algebra (math.AC) ,01 natural sciences ,Mathematics - Algebraic Geometry ,symbols.namesake ,0103 physical sciences ,FOS: Mathematics ,Ideal (ring theory) ,0101 mathematics ,Algebraic number ,Algebraic Geometry (math.AG) ,Mathematics ,media_common ,Conjecture ,Hilbert's syzygy theorem ,Mathematics::Commutative Algebra ,Degree (graph theory) ,Applied Mathematics ,010102 general mathematics ,13A02, 13D02 ,Mathematics - Commutative Algebra ,Infinity ,Bounded function ,symbols ,010307 mathematical physics - Abstract
Hilbert famously showed that polynomials in n variables are not too complicated, in various senses. For example, the Hilbert Syzygy Theorem shows that the process of resolving a module by free modules terminates in finitely many (in fact, at most n) steps, while the Hilbert Basis Theorem shows that the process of finding generators for an ideal also terminates in finitely many steps. These results laid the foundations for the modern algebraic study of polynomials. Hilbert's results are not uniform in n: unsurprisingly, polynomials in n variables will exhibit greater complexity as n increases. However, an array of recent work has shown that in a certain regime---namely, that where the number of polynomials and their degrees are fixed---the complexity of polynomials (in various senses) remains bounded even as the number of variables goes to infinity. We refer to this as Stillman uniformity, since Stillman's Conjecture provided the motivating example. The purpose of this paper is to give an exposition of Stillman uniformity, including: the circle of ideas initiated by Ananyan and Hochster in their proof of Stillman's Conjecture, the followup results that clarified and expanded on those ideas, and the implications for understanding polynomials in many variables., This expository paper was written in conjunction with Craig Huneke's talk on Stillman's Conjecture at the 2018 JMM Current Events Bulletin
- Published
- 2018
11. Eigenfunction Expansions of Ultradifferentiable Functions and Ultradistributions. III. Hilbert Spaces and Universality
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Aparajita Dasgupta and Michael Ruzhansky
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Pure mathematics ,CONVOLUTION ,General Mathematics ,Structure (category theory) ,Boundary (topology) ,Type (model theory) ,Universality ,01 natural sciences ,Mathematics - Spectral Theory ,symbols.namesake ,Mathematics - Analysis of PDEs ,Primary 46F05 ,Tensor (intrinsic definition) ,0103 physical sciences ,FOS: Mathematics ,DISTRIBUTIONS ,Secondary 22E30 ,0101 mathematics ,Spectral Theory (math.SP) ,Mathematics ,Hilbert spaces ,Sequence ,Applied Mathematics ,010102 general mathematics ,Hilbert space ,Universality (philosophy) ,Eigenfunction ,Sequence spaces ,Smooth functions ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Mathematics and Statistics ,Physics and Astronomy ,Komatsu classes ,symbols ,Tensor representations ,010307 mathematical physics ,Primary 46F05, Secondary 22E30 ,Analysis ,Analysis of PDEs (math.AP) - Abstract
In this paper we analyse the structure of the spaces of smooth type functions, generated by elements of arbitrary Hilbert spaces, as a continuation of the research in our previous papers in this series. We prove that these spaces are perfect sequence spaces. As a consequence we describe the tensor structure of sequential mappings on the spaces of smooth type functions and characterise their adjoint mappings. As an application we prove the universality of the spaces of smooth type functions on compact manifolds without boundary., 23 pages
- Published
- 2021
12. Fourier multipliers on graded lie groups
- Author
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Michael Ruzhansky and Veronique Fischer
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Pure mathematics ,Mathematics(all) ,General Mathematics ,Graded nilpotent Lie groups ,Type (model theory) ,01 natural sciences ,symbols.namesake ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Representation Theory (math.RT) ,Analysis on Lie groups ,Mathematics ,Group (mathematics) ,010102 general mathematics ,Lie group ,Dual (category theory) ,Functional Analysis (math.FA) ,Sobolev space ,Mathematics - Functional Analysis ,Nilpotent ,Fourier transform ,symbols ,010307 mathematical physics ,Fourier multipliers ,Mathematics - Representation Theory ,Primary: 43A22, Secondary: 43A15, 22E30 - Abstract
In this paper we study multipliers on graded nilpotent Lie groups defined via group Fourier transform. More precisely, we show that H\"ormander type conditions on the Fourier multipliers imply $L^p$-boundedness. We express these conditions using difference operators and positive Rockland operators. We also obtain a more refined condition using Sobolev spaces on the dual of the group which are defined and studied in this paper., Comment: 23 pages
- Published
- 2020
13. The fully marked surface theorem
- Author
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Mehdi Yazdi and David Gabai
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,General Mathematics ,Taut foliation ,Homology (mathematics) ,01 natural sciences ,symbols.namesake ,Mathematics - Geometric Topology ,Euler characteristic ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics ,010102 general mathematics ,Geometric Topology (math.GT) ,16. Peace & justice ,Surface (topology) ,Mathematics::Geometric Topology ,Cohomology ,Manifold ,57R30, 57K32, 57M50 ,symbols ,Foliation (geology) ,010307 mathematical physics ,Mathematics::Differential Geometry ,Euler class - Abstract
In his seminal 1976 paper Bill Thurston observed that a closed leaf S of a foliation has Euler characteristic equal, up to sign, to the Euler class of the foliation evaluated on [S], the homology class represented by S. The main result of this paper is a converse for taut foliations: if the Euler class of a taut foliation $\mathcal{F}$ evaluated on [S] equals up to sign the Euler characteristic of S and the underlying manifold is hyperbolic, then there exists another taut foliation $\mathcal{F'}$ such that $S$ is homologous to a union of leaves and such that the plane field of $\mathcal{F'}$ is homotopic to that of $\mathcal{F}$. In particular, $\mathcal{F}$ and $\mathcal{F'}$ have the same Euler class. In the same paper Thurston proved that taut foliations on closed hyperbolic 3-manifolds have Euler class of norm at most one, and conjectured that, conversely, any integral cohomology class with norm equal to one is the Euler class of a taut foliation. This is the second of two papers that together give a negative answer to Thurston's conjecture. In the first paper, counterexamples were constructed assuming the main result of this paper., Comment: 36 pages, 16 figures. Portions of this work previously appeared as an appendix to arXiv:1603.03822, but has evolved into its own work and has been accepted for publication separately. Final version to appear in Acta Mathematica
- Published
- 2020
14. On Some Families of Certain Divisible Polynomials and Their Zeta Functions
- Author
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Koji Chinen
- Subjects
12D10 ,Pure mathematics ,Property (philosophy) ,General Mathematics ,010102 general mathematics ,13A50 ,Differential operator ,01 natural sciences ,Riemann hypothesis ,symbols.namesake ,0103 physical sciences ,symbols ,010307 mathematical physics ,0101 mathematics ,Invariant (mathematics) ,11T71 ,Mathematics - Abstract
The formal weight enumerators were first introduced by M. Ozeki, and it was shown in the author's previous paper that there are various families of similar divisible polynomials. Among them, three families are dealt with in this paper and their properties are investigated: they are analogues of the Mallows--Sloane bound, the extremal property, the Riemann hypothesis analogue, etc. In the course of the investigation, some generalizations of the theory of invariant differential operators developed by I. Duursma and T. Okuda are deduced.
- Published
- 2020
15. Generic conformally flat hypersurfaces and surfaces in 3-sphere
- Author
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Suyama Yoshihiko
- Subjects
Surface (mathematics) ,Mathematics - Differential Geometry ,Pure mathematics ,Gauss map ,General Mathematics ,010102 general mathematics ,Conformal map ,Space (mathematics) ,01 natural sciences ,3-sphere ,symbols.namesake ,Hypersurface ,Differential Geometry (math.DG) ,Primary 53B25, Secondary 53E40 ,0103 physical sciences ,Euclidean geometry ,Gaussian curvature ,symbols ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
The aim of this paper is to verify that the study of generic conformally flat hypersurfaces in 4-dimensional space forms is reduced to a surface theory in the standard 3-sphere. The conformal structure of generic conformally flat (local-)hypersurfaces is characterized as conformally flat (local-)3-metrics with the Guichard condition. Then, there is a certain class of orthogonal analytic (local-)Riemannian 2-metrics with constant Gauss curvature -1 such that any 2-metric of the class gives rise to a one-parameter family of conformally flat 3-metrics with the Guichard condition. In this paper, we firstly relate 2-metrics of the class to surfaces in the 3-sphere: for a 2-metric of the class, a 5-dimensional set of (non-isometric) analytic surfaces in the 3-sphere is determined such that any surface of the set gives rise to an evolution of surfaces in the 3-sphere issuing from the surface and the evolution is the Gauss map of a generic conformally flat hypersurface in the Euclidean 4-space. Secondly, we characterize analytic surfaces in the 3-sphere which give rise to generic conformally flat hypersurfaces., 39 pages
- Published
- 2020
16. Almost uniform domains and Poincar\'e inequalities
- Author
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Jasun Gong and Sylvester Eriksson-Bique
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,28A80 ,General Mathematics ,31E05 (secondary) ,010102 general mathematics ,28A80, 30L10, 30L99, 31E05, 35A23, 42B25, 46E35 ,30L99 (primary) ,16. Peace & justice ,01 natural sciences ,symbols.namesake ,Mathematics - Metric Geometry ,Mathematics - Classical Analysis and ODEs ,0103 physical sciences ,Poincaré conjecture ,QA1-939 ,symbols ,Mathematics::Metric Geometry ,28A75 ,010307 mathematical physics ,0101 mathematics ,26A45 ,Mathematics - Abstract
Here we show existence of numerous subsets of Euclidean and metric spaces that, despite having empty interior, still support Poincar\'e inequalities. Most importantly, our methods do not depend on any rectilinear or self-similar structure of the underlying space. We instead employ the notion of uniform domain of Martio and Sarvas. Our condition relies on the measure density of such subsets, as well as the regularity and relative separation of their boundary components. In doing so, our results hold true for metric spaces equipped with doubling measures and Poincar\'e inequalities in general, and for the Heisenberg groups in particular. To our knowledge, these are the first examples of such subsets on any step-2 Carnot group. Such subsets also give, in general, new examples of Sobolev extension domains on doubling metric measure spaces. When specialized to the plane, we give general sufficient conditions for planar subsets, possibly with empty interior, to be Ahlfors 2-regular and to satisfy a (1,2)-Poincar\'e inequality. In the Euclidean case, our construction also covers the non-self-similar Sierpi\'nski carpets of Mackay, Tyson, and Wildrick, as well as higher dimensional analogues not treated in the literature. The analysis of the Poincar\'e inequality with exponent p=1, for these carpets and their higher dimensional analogues, includes a new way of proving an isoperimetric inequality on a space without constructing Semmes families of curves., Comment: 67 pages, 3 figures. A number of typos fixed, and Section 5 of the paper was removed. It appears now in a much expanded form in a paper "Isoperimetric and Poincar\'e inequalities on non-self-similar Sierpi\'nski sponges: the borderline case''
- Published
- 2019
17. Statistical Transition of Bose Gas to Fermi Gas
- Author
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Victor Pavlovich Maslov
- Subjects
Condensed Matter::Quantum Gases ,Polylogarithm ,Bose gas ,Distribution (number theory) ,General Mathematics ,010102 general mathematics ,01 natural sciences ,symbols.namesake ,Quantum mechanics ,0103 physical sciences ,symbols ,Jump ,Fermi–Dirac statistics ,010307 mathematical physics ,0101 mathematics ,Fermi gas ,Mathematics ,Sign (mathematics) ,Fermi Gamma-ray Space Telescope - Abstract
It is well known that the formula for the Fermi distribution is obtained from the formula for the Bose distribution if the argument of the polylogarithm, the activity a, the energy, and the number of particles change sign. The paper deals with the behavior of the Bose–Einstein distribution as a → 0; in particular, the neighborhood of the point a = 0 is studied in great detail, and the expansion of both the Bose distribution and the Fermi distribution in powers of the parameter a is used. During the transition from the Bose distribution to the Fermi distribution, the principal term of the distribution for the specific energy undergoes a jump as a → 0. In this paper,we find the value of the parameter a, close to zero, but not equal to zero, for which the Bose distribution (in the statistical sense) becomes zero. This allows us to find the point a, distinct from zero, at which a jump of the specific energy occurs. Using the value of the number of particles on the caustic, we can obtain the jump of the total energy of the Bose system to the Fermi system. Near the value a = 0, the author uses Gentile statistics, whichmakes it possible to study the transition fromthe Bose statistics to the the Fermi statistics in great detail. Here an important role is played by the self-consistent equation obtained by the author earlier.
- Published
- 2018
18. Uniform hyperbolicity revisited: index of periodic points and equidimensional cycles
- Author
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Mário Bessa, Paulo Varandas, Jorge Rocha, and uBibliorum
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,Index (economics) ,General Mathematics ,Finest dominated splitting ,Dynamical Systems (math.DS) ,Lyapunov exponent ,Equidimensional ,01 natural sciences ,symbols.namesake ,Uniform hyperbolicity ,Periodic points ,Simple (abstract algebra) ,0103 physical sciences ,FOS: Mathematics ,Oseledets splitting ,Mathematics - Dynamical Systems ,0101 mathematics ,Mathematics ,Lyapunov spectrum ,010102 general mathematics ,Lyapunov exponents ,Computer Science Applications ,symbols ,010307 mathematical physics - Abstract
In this paper we revisit uniformly hyperbolic basic sets and the domination of Oseledets splittings at periodic points. We prove that periodic points with simple Lyapunov spectrum are dense in non-trivial basic pieces of Cr-residual diffeomorphisms on three-dimensional manifolds (r >= 1). In the case of the C1-topology we can prove that either all periodic points of a hyperbolic basic piece for a diffeomorphism f have simple spectrum C1- robustly (in which case f has a finest dominated splitting into one-dimensional sub-bundles and all Lyapunov exponent functions of f are continuous in the weak*-topology) or it can be C1-approximated by an equidimensional cycle associated to periodic points with robust different signatures. The later can be used as a mechanism to guarantee the coexistence of infinitely many periodic points with different signatures., 17 pages, 3 figures. In this new version, due to a mistake on the proof of Theorem 4 of the first version of the paper, we remove Section 2.4. (Regularity of conjugacy classes). Moreover, we introduce a new result on the Cr-topology (see Theorem 1 on this new version) in the same line of the ones obtained in the C1-topology
- Published
- 2018
19. Kreĭn space representation and Lorentz groups of analytic Hilbert modules
- Author
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Yue Wu, Michio Seto, and Rongwei Yang
- Subjects
Pure mathematics ,General Mathematics ,Lorentz transformation ,010102 general mathematics ,Hilbert space ,Hardy space ,Congruence relation ,01 natural sciences ,Lorentz group ,symbols.namesake ,0103 physical sciences ,symbols ,010307 mathematical physics ,0101 mathematics ,Invariant (mathematics) ,Abelian group ,Analytic function ,Mathematics - Abstract
This paper aims to introduce some new ideas into the study of submodules in Hilbert spaces of analytic functions. The effort is laid out in the Hardy space over the bidisk H2(D2). A closed subspace M in H2(D2) is called a submodule if z i M ⊂ M (i = 1, 2). An associated integral operator (defect operator) C M captures much information about M. Using a Kreĭn space indefinite metric on the range of C M , this paper gives a representation of M. Then it studies the group (called Lorentz group) of isometric self-maps of M with respect to the indefinite metric, and in finite rank case shows that the Lorentz group is a complete invariant for congruence relation. Furthermore, the Lorentz group contains an interesting abelian subgroup (called little Lorentz group) which turns out to be a finer invariant for M.
- Published
- 2018
20. Global solutions of quasilinear wave-Klein–Gordon system in two-space dimension: Technical tools
- Author
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Yue Ma
- Subjects
Successor cardinal ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Space dimension ,01 natural sciences ,Foliation ,symbols.namesake ,Dimension (vector space) ,0103 physical sciences ,symbols ,010307 mathematical physics ,0101 mathematics ,Klein–Gordon equation ,Analysis ,Mathematics - Abstract
In this paper and its successor, we make an application of the hyperboloidal foliation method in [Formula: see text] space-time dimension. After the establishment of some technical tools in this paper, we will prove further the global existence of small regular solution to a class of hyperbolic system composed by a wave equation and a Klein–Gordon equation with null couplings. Our method belongs to vector field method and, more precisely, is a combination of the normal form and the hyperboloidal foliation method.
- Published
- 2017
21. 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
22. Fourier transforms of powers of well-behaved 2D real analytic functions
- Author
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Michael Greenblatt
- Subjects
Class (set theory) ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Newton polygon ,Function (mathematics) ,01 natural sciences ,Subclass ,symbols.namesake ,Fourier transform ,Mathematics - Classical Analysis and ODEs ,0103 physical sciences ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,symbols ,010307 mathematical physics ,42B20 ,0101 mathematics ,Analytic function ,Mathematics - Abstract
This paper is a companion paper to [G4], where sharp estimates are proven for Fourier transforms of compactly supported functions built out of two-dimensional real-analytic functions. The theorems of [G4] are stated in a rather general form. In this paper, we expand on the results of [G4] and show that there is a class of "well-behaved" functions that contains a number of relevant examples for which such estimates can be explicitly described in terms of the Newton polygon of the function. We will further see that for a subclass of these functions, one can prove noticeably more precise estimates, again in an explicitly describable way., 13 pages
- Published
- 2017
23. ANALYSIS OF CONTACT CAUCHY–RIEMANN MAPS II: CANONICAL NEIGHBORHOODS AND EXPONENTIAL CONVERGENCE FOR THE MORSE–BOTT CASE
- Author
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Rui Wang and Yong-Geun Oh
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Cauchy–Riemann equations ,Homology (mathematics) ,01 natural sciences ,Moduli space ,symbols.namesake ,Symplectization ,0103 physical sciences ,symbols ,A priori and a posteriori ,Field theory (psychology) ,010307 mathematical physics ,0101 mathematics ,Exponential decay ,Symplectic geometry ,Mathematics - Abstract
This is a sequel to the papers Oh and Wang (Real and Complex Submanifolds, Springer Proceedings in Mathematics and Statistics 106 (2014), 43–63, eds. by Y.-J. Suh and et al. for ICM-2014 satellite conference, Daejeon, Korea, August 2014; arXiv:1212.4817; Analysis of contact Cauchy–Riemann maps I: a priori$C^{k}$estimates and asymptotic convergence, submitted, preprint, 2012, arXiv:1212.5186v3). In Oh and Wang (Real and Complex Submanifolds, Springer Proceedings in Mathematics and Statistics 106 (2014), 43–63, eds. by Y.-J. Suh and et al. for ICM-2014 satellite conference, Daejeon, Korea, August 2014; arXiv:1212.4817), the authors introduced a canonical affine connection on $M$ associated to the contact triad $(M,\unicode[STIX]{x1D706},J)$. In Oh and Wang (Analysis of contact Cauchy–Riemann maps I: a priori$C^{k}$estimates and asymptotic convergence, submitted, preprint, 2012, arXiv:1212.5186v3), they used the connection to establish a priori$W^{k,p}$-coercive estimates for maps $w:\dot{\unicode[STIX]{x1D6F4}}\rightarrow M$ satisfying $\overline{\unicode[STIX]{x2202}}^{\unicode[STIX]{x1D70B}}w=0$, $d(w^{\ast }\unicode[STIX]{x1D706}\circ j)=0$without involving symplectization. We call such a pair $(w,j)$ a contact instanton. In this paper, we first prove a canonical neighborhood theorem of the locus $Q$ foliated by closed Reeb orbits of a Morse–Bott contact form. Then using a general framework of the three-interval method, we establish exponential decay estimates for contact instantons $(w,j)$ of the triad $(M,\unicode[STIX]{x1D706},J)$, with $\unicode[STIX]{x1D706}$ a Morse–Bott contact form and $J$ a CR-almost complex structure adapted to $Q$, under the condition that the asymptotic charge of $(w,j)$ at the associated puncture vanishes.We also apply the three-interval method to the symplectization case and provide an alternative approach via tensorial calculations to exponential decay estimates in the Morse–Bott case for the pseudoholomorphic curves on the symplectization of contact manifolds. This was previously established by Bourgeois (A Morse–Bott approach to contact homology, Ph.D. dissertation, Stanford University, 2002) (resp. by Bao (On J-holomorphic curves in almost complex manifolds with asymptotically cylindrical ends, Pacific J. Math. 278(2) (2015), 291–324)), by using special coordinates, for the cylindrical (resp. for the asymptotically cylindrical) ends. The exponential decay result for the Morse–Bott case is an essential ingredient in the setup of the moduli space of pseudoholomorphic curves which plays a central role in contact homology and symplectic field theory (SFT).
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- 2017
24. Hamilton–Jacobi theory, symmetries and coisotropic reduction
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Manuel de León, David Martín de Diego, and Miguel Vaquero
- Subjects
Approximations of π ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Hamilton–Jacobi equation ,Hamiltonian system ,Algebra ,symbols.namesake ,Reduction procedure ,0103 physical sciences ,Homogeneous space ,symbols ,010307 mathematical physics ,0101 mathematics ,Hamiltonian (quantum mechanics) ,Symplectic geometry ,Mathematics - Abstract
Reduction theory has played a major role in the study of Hamiltonian systems. Whilst the Hamilton–Jacobi theory is one of the main tools to integrate the dynamics of certain Hamiltonian problems and a topic of research on its own. Moreover, the construction of several symplectic integrators relies on approximations of a complete solution of the Hamilton–Jacobi equation. The natural question that we address in this paper is how these two topics (reduction and Hamilton–Jacobi theory) fit together. We obtain a reduction and reconstruction procedure for the Hamilton–Jacobi equation with symmetries, even in a generalized sense to be clarified below. Several applications and relations to other reduction of the Hamilton–Jacobi theory are shown in the last section of the paper. It is remarkable that as by-product we obtain a generalization of the Ge–Marsden reduction procedure [18] and the results in [17] . Quite surprisingly, the classical ansatze available in the literature to solve the Hamilton–Jacobi equation (see [2] , [19] ) are also particular instances of our framework.
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- 2017
25. Isomorphisms of Twisted Hilbert Loop Algebras
- Author
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Timothée Marquis and Karl-Hermann Neeb
- Subjects
17B65, 17B70, 17B22, 17B10 ,General Mathematics ,010102 general mathematics ,Hilbert space ,Mathematics - Rings and Algebras ,01 natural sciences ,Combinatorics ,Loop (topology) ,symbols.namesake ,Isomorphism theorem ,Rings and Algebras (math.RA) ,Affine root system ,Product (mathematics) ,0103 physical sciences ,Lie algebra ,FOS: Mathematics ,symbols ,010307 mathematical physics ,Isomorphism ,Representation Theory (math.RT) ,0101 mathematics ,Invariant (mathematics) ,Mathematics - Representation Theory ,Mathematics - Abstract
The closest infinite dimensional relatives of compact Lie algebras are Hilbert-Lie algebras, i.e. real Hilbert spaces with a Lie algebra structure for which the scalar product is invariant. Locally affine Lie algebras (LALAs) correspond to double extensions of (twisted) loop algebras over simple Hilbert-Lie algebras $\mathfrak{k}$, also called affinisations of $\mathfrak{k}$. They possess a root space decomposition whose corresponding root system is a locally affine root system of one of the $7$ families $A_J^{(1)}$, $B_J^{(1)}$, $C_J^{(1)}$, $D_J^{(1)}$, $B_J^{(2)}$, $C_J^{(2)}$ and $BC_J^{(2)}$ for some infinite set $J$. To each of these types corresponds a "minimal" affinisation of some simple Hilbert-Lie algebra $\mathfrak{k}$, which we call standard. In this paper, we give for each affinisation $\mathfrak{g}$ of a simple Hilbert-Lie algebra $\mathfrak{k}$ an explicit isomorphism from $\mathfrak{g}$ to one of the standard affinisations of $\mathfrak{k}$. The existence of such an isomorphism could also be derived from the classification of locally affine root systems, but for representation theoretic purposes it is crucial to obtain it explicitely as a deformation between two twists which is compatible with the root decompositions. We illustrate this by applying our isomorphism theorem to the study of positive energy highest weight representations of $\mathfrak{g}$. In subsequent work, the present paper will be used to obtain a complete classification of the positive energy highest weight representations of affinisations of $\mathfrak{k}$., Comment: 22 pages; Minor corrections
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- 2017
26. On the theory of unconditional bases of Hilbert spaces formed by entire vector-functions
- Author
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Gennadiy Gubreev and Anna Tarasenko
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Discrete mathematics ,General Mathematics ,Entire function ,010102 general mathematics ,Hilbert space ,01 natural sciences ,Separable space ,symbols.namesake ,Operator (computer programming) ,0103 physical sciences ,symbols ,010307 mathematical physics ,0101 mathematics ,Vector-valued function ,Mathematics - Abstract
A quite general theorems on unconditional bases in separable Hilbert spaces, given in terms of values of entire operator valued vector-functions, were established in the papers [2, 3, 4]. In the present paper, we give a detailed analysis of the hypothesis of these theorems. We present examples of various classes of vector-functions that satisfy some of the hypothesis of the above theorems. On the other hand, we show that there exist natural classes of vector-functions that do not satisfy some of the hypothesis of Theorem A.
- Published
- 2016
27. Bergman kernels on punctured Riemann surfaces
- Author
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Xiaonan Ma, George Marinescu, and Hugues Auvray
- Subjects
Mathematics - Differential Geometry ,Mathematics(all) ,Pure mathematics ,General Mathematics ,Poincaré metric ,Holomorphic function ,01 natural sciences ,symbols.namesake ,Uniform norm ,Line bundle ,0103 physical sciences ,FOS: Mathematics ,Hermitian manifold ,Number Theory (math.NT) ,Tensor ,Complex Variables (math.CV) ,0101 mathematics ,Mathematics ,Bergman kernel ,Mathematics - Number Theory ,Mathematics - Complex Variables ,Mathematics::Complex Variables ,Riemann surface ,010102 general mathematics ,General Medicine ,16. Peace & justice ,Differential Geometry (math.DG) ,Metric (mathematics) ,symbols ,010307 mathematical physics - Abstract
In this paper we consider a punctured Riemann surface endowed with a Hermitian metric which equals the Poincar\'e metric near the punctures and a holomorphic line bundle which polarizes the metric. We show that the Bergman kernel can be localized around the singularities and its local model is the Bergman kernel of the punctured unit disc endowed with the standard Poincar\'e metric. As a consequence, we obtain an optimal uniform estimate of the supremum norm of the Bergman kernel, involving a fractional growth order of the tensor power., Comment: 42 pages, 2 figures; v.2 is a final update to agree with the published paper
- Published
- 2016
28. On Invariant Subspaces for the Shift Operator
- Author
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Junfeng Liu
- Subjects
Physics and Astronomy (miscellaneous) ,Function space ,General Mathematics ,reducing subspace ,Shift operator ,01 natural sciences ,hyperinvariant subspace ,Combinatorics ,symbols.namesake ,invariant subspace ,0103 physical sciences ,Computer Science (miscellaneous) ,0101 mathematics ,Invariant (mathematics) ,Mathematics ,Mathematics::Functional Analysis ,shift operator ,lcsh:Mathematics ,010102 general mathematics ,Invariant subspace ,Hardy space ,lcsh:QA1-939 ,Linear subspace ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Chemistry (miscellaneous) ,Bergman space ,lebesgue space ,symbols ,Standard probability space ,010307 mathematical physics ,hardy space - Abstract
In this paper, we improve two known invariant subspace theorems. More specifically, we show that a closed linear subspace M in the Hardy space H p ( D ) ( 1 &le, p <, &infin, ) is invariant under the shift operator M z on H p ( D ) if and only if it is hyperinvariant under M z , and that a closed linear subspace M in the Lebesgue space L 2 ( &part, D ) is reducing under the shift operator M e i &theta, on L 2 ( &part, D ) if and only if it is hyperinvariant under M e i &theta, At the same time, we show that there are two large classes of invariant subspaces for M e i &theta, that are not hyperinvariant subspaces for M e i &theta, and are also not reducing subspaces for M e i &theta, Moreover,we still show that there is a large class of hyperinvariant subspaces for M z that are not reducing subspaces for M z . Furthermore, we gave two new versions of the formula of the reproducing function in the Hardy space H 2 ( D ) , which are the analogue of the formula of the reproducing function in the Bergman space A 2 ( D ) . In addition, the conclusions in this paper are interesting now, or later if they are written into the literature of invariant subspaces and function spaces.
- Published
- 2019
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- View/download PDF
29. Motivic euler characteristics and witt-valued characteristic classes
- Author
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Marc Levine
- Subjects
Pure mathematics ,General Mathematics ,01 natural sciences ,Mathematics - Algebraic Geometry ,symbols.namesake ,Mathematics::K-Theory and Homology ,Euler characteristic ,0103 physical sciences ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Mathematics - Algebraic Topology ,0101 mathematics ,Algebraic Geometry (math.AG) ,Splitting principle ,Mathematics ,010102 general mathematics ,14F42, 55N20, 55N35 ,K-Theory and Homology (math.KT) ,Reductive group ,Cohomology ,Characteristic class ,Transfer (group theory) ,Scheme (mathematics) ,Mathematics - K-Theory and Homology ,Mathematik ,Euler's formula ,symbols ,010307 mathematical physics - Abstract
This paper examines a number of related questions about Euler characteristics and characteristic classes with values in Witt cohomology. We establish a motivic version of the Becker-Gottllieb transfer, generalizing a construction of Hoyois. Ananyevskiy's splitting principle reduces questions about characteristic classes of vector bundles in $\text{SL}$-oriented, $\eta$-invertible theories to the case of rank two bundles. We refine the torus-normalizer splitting principle for $\text{SL}_2$ to help compute the characteristic classes in Witt cohomology of symmetric powers of a rank two bundle, and then generalize this to develop a general calculus of characteristic classes with values in Witt cohomology., Comment: This is a final version, the paper is published online by the Nagoya Math. Journal
- Published
- 2019
30. Conical metrics on Riemann surfaces, II: spherical metrics
- Author
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Rafe Mazzeo and Xuwen Zhu
- Subjects
Mathematics - Differential Geometry ,General Mathematics ,Riemann surface ,010102 general mathematics ,Mathematical analysis ,Friedrichs extension ,Deformation theory ,01 natural sciences ,Moduli space ,Constant curvature ,symbols.namesake ,Mathematics - Analysis of PDEs ,Differential Geometry (math.DG) ,Conic section ,0103 physical sciences ,symbols ,FOS: Mathematics ,Gravitational singularity ,010307 mathematical physics ,0101 mathematics ,Laplace operator ,Mathematics ,Analysis of PDEs (math.AP) - Abstract
We continue our study, initiated in our earlier paper, of Riemann surfaces with constant curvature and isolated conic singularities. Using the machinery developed in that earlier paper of extended configuration families of simple divisors, we study the existence and deformation theory for spherical conic metrics with some or all of the cone angles greater than $2\pi$. Deformations are obstructed precisely when the number $2$ lies in the spectrum of the Friedrichs extension of the Laplacian. Our main result is that, in this case, it is possible to find a smooth local moduli space of solutions by allowing the cone points to split. This analytic fact reflects geometric constructions in papers by Mondello and Panov., Comment: Final version accepted by Int. Math. Res. Not
- Published
- 2019
- Full Text
- View/download PDF
31. Compactness of conformally compact Einstein 4-manifolds II
- Author
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Yuxin Ge, Sun-Yung Alice Chang, Jie Qing, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,53C80 ,General Mathematics ,[PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph] ,53C21 ,Curvature ,01 natural sciences ,53C25 ,58J05 ,symbols.namesake ,0103 physical sciences ,53C25, 53A30, 53C21, 58J05, 53C80 ,FOS: Mathematics ,Uniqueness ,0101 mathematics ,Einstein ,Mathematics ,Compactness ,53A30 ,010102 general mathematics ,Conformally compact Einstein manifold ,Compact space ,math.DG ,Differential Geometry (math.DG) ,symbols ,010307 mathematical physics - Abstract
In this paper, we establish compactness results of some class of conformally compact Einstein 4-manifolds. In the first part of the paper, we improve the earlier results obtained by Chang-Ge. In the second part of the paper, as applications, we derive some compactness results under perturbation conditions when the L^2-norm of the Weyl curvature is small. We also derive the global uniqueness of conformally compact Einstein metrics on the 4-Ball constructed in the earlier work of Graham-Lee., Comment: 28 pages
- Published
- 2018
32. Poincaré duality for spaces with isolated singularities
- Author
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Mathieu Klimczak
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Algebraic geometry ,01 natural sciences ,57P10, 55N33, 55P62 ,symbols.namesake ,Formalism (philosophy of mathematics) ,Number theory ,0103 physical sciences ,symbols ,Gravitational singularity ,Mathematics - Algebraic Topology ,010307 mathematical physics ,0101 mathematics ,Poincaré duality ,Mathematics - Abstract
In this paper we assign, under reasonable hypothesis, to each pseudomanifold with isolated singularities a rational Poincar\'e duality space. These spaces are constructed with the formalism of intersections spaces defined by Markus Banagl and are indeed related to them in the even dimensional case., Comment: New version of the paper formerly known as "Spatialization of self dual complexes for spaces with isolated singularities". Final version to appear in manuscripta mathematica
- Published
- 2016
33. Corrigendum: On generalizing Lutz twists
- Author
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Dishant M. Pancholi and John B. Etnyre
- Subjects
Lemma (mathematics) ,General Mathematics ,010102 general mathematics ,Torus ,Disjoint sets ,01 natural sciences ,Combinatorics ,symbols.namesake ,Mathematics Subject Classification ,0103 physical sciences ,symbols ,Isotopy ,Vector field ,010307 mathematical physics ,0101 mathematics ,Lagrangian ,Mathematics - Abstract
In this note we point out an error in [2]. We show how to repair the proof in dimension 5. The results are true in general as can easily be seen from recent work of Borman, Eliashberg and Murphy [1]. The proof of Lemma 3.4 in [2] is incorrect. Below we will describe the problem with the proof and then show how it can easily be repaired in dimension 5. We then observe that Lemma 3.4, and thus the main results of the paper, is true in all dimensions based on recent work of Borman, Eliashberg and Murphy [1]. However this approach does not give an explicit construction and hence goes against the sprit of the original paper and in addition all the results of [2] follow directly from [1]. Acknowledgement: We thank Yasha Eliashberg for pointing out the error in the proof of Lemma 3.4 in [2]. The first author was partially supported by a grant from the Simons Foundation (#342144) and NSF grant DMS-1309073. 1. Exact Lagrangians, Liouville flows, and the error in the proof of Lemma 3.4 We begin by recalling the statement of Lemma 3.4 from [2]. To state the lemma we first establish some notation (that is slightly different that what was used in [2]). Consider T 2 × [0, 1] with coordinates (θ, φ, r) and the contact structure ξi = kerαi, i = 1, 2, where αi = ki(r) dθ + li(r) dφ. Here we have k1(r) = cos π 2 r and l(r) = sin π 2 r, and for i = 2 we have k2 and l2 agreeing with k1 and l1 near r = 0 and 1, and the curve (k2(r), l2(r)) in R has 5π/2 winding about the origin. In particular notice that ξ2 is obtained from ξ1 by adding Giroux torsion. Lemma 3.4 from [2] now reads as follows. Lemma 1. Let W be a manifold with contact form λ, there is a contact structure on W × [0, 1] × ([0, 1]× T ) such that the following properties are satisfied: (1) near W×{0}× [0, 1]×T 2 and W× [0, 1]×{0, 1}×T 2 the contact structure is contactomorphic to λ+ eα1, and (2) near W × {1} × [0, 1]× T 2 the contact structure is contactomorphic to λ+ eα2. Here t is the coordinate on the first [0, 1] factor. See [2] for details on how the main constructions and theorems of the paper follow from this lemma. The strategy of the proof in [2] was: (1) To construct a contact structure on W×[0, 1]×T 3 that near W×{0}×T 3 is given by λ+eβ0 and near W ×{1}× T 3 is given by λ+ e × β1, where βi is the contact structure on T 3 with Giroux torsion i and we are thinking of T 3 as S×T 2 with the S-factors Legendrian curves. (2) Then cut W × [0, 1]×T 3 along W × [0, 1]× ({θ0, θ1}×T ) so that one of the resulting pieces is as described in the lemma. To try to arrange this let β = p1 dθ1+p1 dθ2 be the Liouville form on T ∗T 2 = R×T 2 with coordinates (p1, p2, θ1, θ2). Notice that α = λ + β is a contact form on W × T ∗T . We will see below that we can arrange the two items above that are needed for our proof if there is a radial vector field v in R centered at a point p whose flow expands dβ (that is, Lvdβ = dβ) and a Lagrangian torus T 2 in a small neighborhood of {q}×T 2 ⊂ T ∗T 2 that is exact with respect to ιvdβ that is isotopic to {q}×T 2 by an isotopy disjoint from {p} × T . One may easily arrange all of this except for either the last 2010 Mathematics Subject Classification. 57R17 (primary), and 53D35(secondary).
- Published
- 2016
34. Euler, the clothoid and
- Author
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Nick Lord
- Subjects
Laplace transform ,General Mathematics ,010102 general mathematics ,Substitution (logic) ,Function (mathematics) ,Expression (computer science) ,Notation ,01 natural sciences ,symbols.namesake ,0103 physical sciences ,History of mathematics ,Euler's formula ,symbols ,Calculus ,010307 mathematical physics ,0101 mathematics ,Complex number ,Mathematics - Abstract
One of the many definite integrals that Euler was the first to evaluate was(1)He did this, almost as an afterthought, at the end of his short, seven-page paper catalogued as E675 in [1] and with the matter-of-fact title, On the values of integrals from x = 0 to x = ∞. It is a beautiful Euler miniature which neatly illustrates the unexpected twists and turns in the history of mathematics. For Euler's derivation of (1) emerges as the by-product of a solution to a problem in differential geometry concerning the clothoid curve which he had first encountered nearly forty years earlier in his paper E65, [1]. As highlighted in the recent Gazette article [2], E675 is notable for Euler's use of a complex number substitution to evaluate a real-variable integral. He used this technique in about a dozen of the papers written in the last decade of his life. The rationale for this manoeuvre caused much debate among later mathematicians such as Laplace and Poisson and the technique was only put on a secure footing by the work of Cauchy from 1814 onwards on the foundations of complex function theory, [3, Chapter 1]. Euler's justification was essentially pragmatic (in agreement with numerical evidence) and by what Dunham in [4, p. 68] characterises as his informal credo, ‘Follow the formulas, and they will lead to the truth.’ Smithies, [3, p. 187], contextualises Euler's approach by noting that, at that time, ‘a function was usually thought of as being defined by an analytic expression; by the principle of the generality of analysis, which was widely and often tacitly accepted, such an expression was expected to be valid for all values, real or complex, of the independent variable’. In this article, we examine E675 closely. We have tweaked notation and condensed the working in places to reflect modern usage. At the end, we outline what is, with hindsight, needed to make Euler's arguments watertight: it is worth noting that all of his conclusions survive intact and that the intermediate functions of one and two variables that he introduces in E675 remain the key ingredients for much subsequent work on these integrals.
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- 2016
35. $K$-homology and Fredholm operators II: elliptic operators
- Author
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Erik van Erp and Paul Baum
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Index (economics) ,General Mathematics ,010102 general mathematics ,Dirac (software) ,K-homology ,Dirac operator ,01 natural sciences ,Elliptic operator ,symbols.namesake ,Operator (computer programming) ,Differential Geometry (math.DG) ,19K56, 58J20 ,0103 physical sciences ,FOS: Mathematics ,symbols ,010307 mathematical physics ,0101 mathematics ,Atiyah–Singer index theorem ,Mathematics - Abstract
This is an expository paper which gives a proof of the Atiyah-Singer index theorem for elliptic operators. Specifcally, we compute the geometric K-cycle that corresponds to the analytic K-cycle determined by the operator. This paper and its companion ("K-homology and index theory II: Dirac Operators") was written to clear up basic points about index theory that are generally accepted as valid, but for which no proof has been published. Some of these points are needed for the solution of the Heisenberg-elliptic index problem in our paper "K-homology and index theory on contact manifolds" Acta. Math. 2014., Comment: 13 pages
- Published
- 2016
36. Generalized Fourier Transforms Arising from the Enveloping Algebras of 𝔰𝔩(2) and 𝔬𝔰𝔭(1∣2)
- Author
-
Joris Van der Jeugt, Roy Oste, and Hendrik De Bie
- Subjects
Pure mathematics ,Uncertainty principle ,General Mathematics ,Operator (physics) ,010102 general mathematics ,42B10, 13F20, 17B60 ,Universal enveloping algebra ,Dirac operator ,01 natural sciences ,symbols.namesake ,Kernel (algebra) ,Fourier transform ,Mathematics - Classical Analysis and ODEs ,Helmholtz free energy ,0103 physical sciences ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,symbols ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics - Representation Theory ,Mathematics ,Dual pair - Abstract
The Howe dual pair (sl(2),O(m)) allows the characterization of the classical Fourier transform (FT) on the space of rapidly decreasing functions as the exponential of a well-chosen element of sl(2) such that the Helmholtz relations are satisfied. In this paper we first investigate what happens when instead we consider exponentials of elements of the universal enveloping algebra of sl(2). This leads to a complete class of generalized Fourier transforms, that all satisfy properties similar to the classical FT. There is moreover a finite subset of transforms which very closely resemble the FT. We obtain operator exponential expressions for all these transforms by making extensive use of the theory of integer-valued polynomials. We also find a plane wave decomposition of their integral kernel and establish uncertainty principles. In important special cases we even obtain closed formulas for the integral kernels. In the second part of the paper, the same problem is considered for the dual pair (osp(1|2),Spin(m)), in the context of the Dirac operator. This connects our results with the Clifford-Fourier transform studied in previous work., Comment: Second version, changes in title, introduction and section 2
- Published
- 2015
37. Existence and nonexistence of extremals for critical Adams inequalities in R4 and Trudinger-Moser inequalities in R2
- Author
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Guozhen Lu, Maochun Zhu, and Lu Chen
- Subjects
Pure mathematics ,Current (mathematics) ,Inequality ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Function (mathematics) ,Space (mathematics) ,01 natural sciences ,symbols.namesake ,Fourier transform ,0103 physical sciences ,Domain (ring theory) ,symbols ,Order (group theory) ,010307 mathematical physics ,0101 mathematics ,Symmetry (geometry) ,Mathematics ,media_common - Abstract
Though much progress has been made with respect to the existence of extremals of the critical first order Trudinger-Moser inequalities in W 1 , n ( R n ) and higher order Adams inequalities on finite domain Ω ⊂ R n , whether there exists an extremal function for the critical higher order Adams inequalities on the entire space R n still remains open. The current paper represents the first attempt in this direction by considering the critical second order Adams inequality in the entire space R 4 . The classical blow-up procedure cannot apply to solving the existence of critical Adams type inequality because of the absence of the Polya-Szego type inequality. In this paper, we develop some new ideas and approaches based on a sharp Fourier rearrangement principle (see [31] ), sharp constants of the higher-order Gagliardo-Nirenberg inequalities and optimal poly-harmonic truncations to study the existence and nonexistence of the maximizers for the Adams inequalities in R 4 of the form S ( α ) = sup ‖ u ‖ H 2 = 1 ∫ R 4 ( exp ( 32 π 2 | u | 2 ) − 1 − α | u | 2 ) d x , where α ∈ ( − ∞ , 32 π 2 ) . We establish the existence of the threshold α ⁎ , where α ⁎ ≥ ( 32 π 2 ) 2 B 2 2 and B 2 ≥ 1 24 π 2 , such that S ( α ) is attained if 32 π 2 − α α ⁎ , and is not attained if 32 π 2 − α > α ⁎ . This phenomenon has not been observed before even in the case of first order Trudinger-Moser inequality. Therefore, we also establish the existence and non-existence of an extremal function for the Trudinger-Moser inequality on R 2 . Furthermore, the symmetry of the extremal functions can also be deduced through the Fourier rearrangement principle.
- Published
- 2020
38. Truncated Hecke-Rogers type series
- Author
-
Ae Ja Yee and Chun Wang
- Subjects
Pure mathematics ,Series (mathematics) ,Differential equation ,General Mathematics ,010102 general mathematics ,Type (model theory) ,Mathematical proof ,01 natural sciences ,symbols.namesake ,GEORGE (programming language) ,Pentagonal number theorem ,0103 physical sciences ,Euler's formula ,symbols ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
The recent work of George Andrews and Mircea Merca on the truncated version of Euler's pentagonal number theorem has opened up a new study on truncated theta series. Since then several papers on the topic have followed. The main purpose of this paper is to generalize the study to Hecke-Rogers type double series, which are associated with some interesting partition functions. Our proofs heavily rely on a formula from the work of Zhi-Guo Liu on the q-partial differential equations and q-series.
- Published
- 2020
39. Representations of mock theta functions
- Author
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Dandan Chen and Liuquan Wang
- Subjects
Pure mathematics ,Mathematics - Number Theory ,Series (mathematics) ,General Mathematics ,010102 general mathematics ,Parameterized complexity ,01 natural sciences ,Ramanujan theta function ,symbols.namesake ,Identity (mathematics) ,Mathematics - Classical Analysis and ODEs ,0103 physical sciences ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,symbols ,Mathematics - Combinatorics ,05A30, 11B65, 33D15, 11E25, 11F11, 11F27, 11P84 ,Number Theory (math.NT) ,Combinatorics (math.CO) ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
Motivated by the works of Liu, we provide a unified approach to find Appell-Lerch series and Hecke-type series representations for mock theta functions. We establish a number of parameterized identities with two parameters $a$ and $b$. Specializing the choices of $(a,b)$, we not only give various known and new representations for the mock theta functions of orders 2, 3, 5, 6 and 8, but also present many other interesting identities. We find that some mock theta functions of different orders are related to each other, in the sense that their representations can be deduced from the same $(a,b)$-parameterized identity. Furthermore, we introduce the concept of false Appell-Lerch series. We then express the Appell-Lerch series, false Appell-Lerch series and Hecke-type series in this paper using the building blocks $m(x,q,z)$ and $f_{a,b,c}(x,y,q)$ introduced by Hickerson and Mortenson, as well as $\overline{m}(x,q,z)$ and $\overline{f}_{a,b,c}(x,y,q)$ introduced in this paper. We also show the equivalences of our new representations for several mock theta functions and the known representations., Comment: 87 pages, comments are welcome. We have extended the previous version
- Published
- 2020
40. The Two Hyperplane Conjecture
- Author
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David Jerison
- Subjects
Pure mathematics ,Conjecture ,35B35, 35A15 ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Scalar (mathematics) ,Eigenfunction ,01 natural sciences ,symbols.namesake ,Mathematics - Analysis of PDEs ,Hyperplane ,0103 physical sciences ,Poincaré conjecture ,FOS: Mathematics ,symbols ,Convex body ,010307 mathematical physics ,0101 mathematics ,Isoperimetric inequality ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
We introduce a conjecture that we call the {\it Two Hyperplane Conjecture}, saying that an isoperimetric surface that divides a convex body in half by volume is trapped between parallel hyperplanes. The conjecture is motivated by an approach we propose to the {\it Hots Spots Conjecture} of J. Rauch using deformation and Lipschitz bounds for level sets of eigenfunctions. We will relate this approach to quantitative connectivity properties of level sets of solutions to elliptic variational problems, including isoperimetric inequalities, Poincar\'e inequalities, Harnack inequalities, and NTA (non-tangentially accessibility). This paper mostly asks questions rather than answering them, while recasting known results in a new light. Its main theme is that the level sets of least energy solutions to scalar variational problems should be as simple as possible., Comment: 22 pages, this new version corrects one word in the introduction, Aguilera is the name of the first author of a paper cited (not Athanosopoulos). Thanks to Joel Spruck for pointing out this error
- Published
- 2018
41. A comparison principle for convolution measures with applications
- Author
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René Quilodrán and Diogo Oliveira e Silva
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Pointwise ,Paraboloid ,Computer Science::Information Retrieval ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Parabola ,Regular polygon ,01 natural sciences ,Measure (mathematics) ,Projection (linear algebra) ,Convolution ,symbols.namesake ,Fourier transform ,Mathematics - Classical Analysis and ODEs ,0103 physical sciences ,symbols ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
We establish the general form of a geometric comparison principle for $n$-fold convolutions of certain singular measures in $\mathbb{R}^d$ which holds for arbitrary $n$ and $d$. This translates into a pointwise inequality between the convolutions of projection measure on the paraboloid and a perturbation thereof, and we use it to establish a new sharp Fourier extension inequality on a general convex perturbation of a parabola. Further applications of the comparison principle to sharp Fourier restriction theory are discussed in a companion paper., Comment: 17 pages, v2: updated reference to companion paper
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- 2018
42. Transfer operators and Hankel transforms between relative trace formulas, I: character theory
- Author
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Yiannis Sakellaridis
- Subjects
Hecke algebra ,Pure mathematics ,11F70 ,Trace (linear algebra) ,Mathematics - Number Theory ,General Mathematics ,010102 general mathematics ,Poisson summation formula ,Character theory ,Fundamental lemma ,16. Peace & justice ,01 natural sciences ,Transfer (group theory) ,symbols.namesake ,Langlands program ,0103 physical sciences ,symbols ,FOS: Mathematics ,010307 mathematical physics ,Number Theory (math.NT) ,0101 mathematics ,Abelian group ,Representation Theory (math.RT) ,Mathematics - Representation Theory ,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 ${^LG}_1\to {^LG}_2$ 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 continuation is to demonstrate, by example, the existence of "transfer operators" betweeen relative trace formulas, that 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 first paper we study (relative) characters for the Kunzetsov formula and the stable trace formula for $\operatorname{SL}_2$ and their degenerations (as well as for the relative trace formula for torus periods in $\operatorname{PGL}_2$), and we show how they correspond to each other under explicit transfer operators., 68pp, submitted. This is a slightly updated part 1 (of 2) of what was previously posted as a single paper
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- 2018
43. Intuitionism and effective descriptive set theory
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Joan Rand Moschovakis and Yiannis N. Moschovakis
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medicine.medical_specialty ,General Mathematics ,010102 general mathematics ,Lebesgue integration ,Mathematical proof ,01 natural sciences ,Pure Mathematics ,Algebra ,symbols.namesake ,Effective descriptive set theory ,Intuitionism ,Simple (abstract algebra) ,0103 physical sciences ,Calculus ,symbols ,medicine ,010307 mathematical physics ,Set theory ,0101 mathematics ,Borel set ,Descriptive set theory ,Mathematics - Abstract
Our very eloquent charge from Jan van Mill was to “draw a line to Brouwer from descriptive set theory, but this proved elusive: in fact there are few references to Brouwer, in Lusin (1928) [36] and Lusin (1930), none of them substantial; and even though Brouwer refers to Borel, Lebesgue and Hadamard in his early papers, it does not appear that he was influenced by their work in any substantive way. 1 We have not found any references by him to more developed work in descriptive set theory, after the critical Lebesgue (1905). So instead of looking for historical connections or direct influences (in either direction), we decided to identify and analyze some basic themes, problems and results which are common to these two fields; and, as it turns out, the most significant connections are between intuitionistic analysis and effective descriptive set theory, hence the title. ⇒ We will outline our approach and (limited) aims in Section 1 , marking with an arrow (like this one) those paragraphs which point to specific parts of the article. Suffice it to say here that our main aim is to identify a few, basic results of descriptive set theory which can be formulated and justified using principles that are both intuitionistically and classically acceptable; that we will concentrate on the mathematics of the matter rather than on history or philosophy; and that we will use standard, classical terminology and notation. This is an elementary, mostly expository paper, broadly aimed at students of logic and set theory who also know the basic facts about recursive functions but need not know a lot about either intuitionism or descriptive set theory. The only (possibly) new result is Theorem 6.1 , which justifies simple definitions and proofs by induction in Kleene’s Basic System of intuitionistic analysis, and is then used in Theorem 7.1 , Theorem 7.2 to give in the same system a rigorous definition of the Borel sets and prove that they are analytic; the formulation and proof of this last result is one example where methods from effective descriptive set theory are used in an essential way.
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- 2018
44. Spaces with polynomial hulls that contain no analytic discs
- Author
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Alexander J. Izzo
- Subjects
Polynomial (hyperelastic model) ,Dense set ,Euclidean space ,Mathematics - Complex Variables ,General Mathematics ,010102 general mathematics ,Space (mathematics) ,01 natural sciences ,law.invention ,Cantor set ,Combinatorics ,symbols.namesake ,Invertible matrix ,law ,0103 physical sciences ,symbols ,FOS: Mathematics ,Uncountable set ,010307 mathematical physics ,0101 mathematics ,Complex Variables (math.CV) ,Lebesgue covering dimension ,32E20, 46J10, 46J15 ,Mathematics - Abstract
Extensions of the notions of polynomially and rationally convex hulls are introduced. Using these notions, a generalization of a result of Duval and Levenberg on polynomially convex hulls containing no analytic discs is presented. As a consequence it is shown that there exists a Cantor set $X$ in ${\mathbb C}^3$ with a nontrivial polynomially convex hull that contains no analytic discs. Using this Cantor set, it is shown that there exist arcs and curves in ${\mathbb C}^4$ with nontrivial polynomially convex hulls that contain no analytic discs. This answers a question raised a few years ago by Bercovici and can be regarded as a partial answer to a question raised by Wermer over 60 years ago. More generally, it is shown that every uncountable, compact subspace of a Euclidean space can be embedded as a subspace $X$ of ${\mathbb C}^N$, for some N, in such a way as to have a nontrivial polynomially convex hull that contains no analytic discs. In the case when the topological dimension of the space is at most one, $X$ can be chosen so as to have the stronger property that $P(X)$ has a dense set of invertible elements., Several revisions have been made to make the paper more succinct and focused. In particular, an entire section has been removed and made into a separate paper with the title "A doubly generated uniform algebra with a one-point Gleason part off its Shilov boundary"
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- 2018
45. The structure of doubly non-commuting isometries
- Author
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Marcel de Jeu and Paulo R. Pinto
- Subjects
Structure constants ,Non-commutative torus ,General Mathematics ,01 natural sciences ,Unitary state ,Universal $C^\ast$-algebra ,Wold decomposition ,Combinatorics ,symbols.namesake ,0103 physical sciences ,FOS: Mathematics ,Dilation theorem ,0101 mathematics ,Operator Algebras (math.OA) ,Mathematics ,Mathematics::Combinatorics ,Mathematics::Operator Algebras ,Doubly non-commuting isometries ,010102 general mathematics ,Mathematics - Operator Algebras ,Hilbert space ,Torus ,47A45, 47A20 ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,symbols ,010307 mathematical physics - Abstract
Suppose that $n\geq 1$ and that, for all $i$ and $j$ with $1\leq i,j\leq n$ and $i\neq j$, $z_{ij}\in{\mathbb T}$ are given such that $z_{ji}=\overline{z}_{ij}$ for all $i\neq j$. If $V_1,\dotsc, V_n$ are isometries on a Hilbert space such that $V_i^\ast V_j^{\phantom{\ast}}\!=\overline{z}_{ij} V_j^{\phantom{\ast}}\!V_i^\ast$ for all $i\neq j$, then $(V_1,\dotsc,V_n)$ is called an $n$-tuple of doubly non-commuting isometries. The generators of non-commutative tori are well-known examples. In this paper, we establish a simultaneous Wold decomposition for $(V_1,\dotsc,V_n)$. This decomposition enables us to classify such $n$-tuples up to unitary equivalence. We show that the joint listing of a unitary equivalence class of a representation of each of the $2^n$ non-commutative tori that are naturally associated with the structure constants is a classifying invariant. A dilation theorem is also established, showing that an $n$-tuple of doubly non-commuting isometries can be extended to an $n$-tuple of doubly non-commuting unitary operators on an enveloping Hilbert space., Comment: A remark on the relation between the dilation theorem in this paper and other dilation theorems for multiple operators in the literature has been added. Otherwise, there are only a few minor editorial changes compared to the first version; some typos have also been corrected. Final version, to appear in Advances in Mathematics
- Published
- 2018
- Full Text
- View/download PDF
46. Low regularity blowup solutions for the mass-critical NLS in higher dimensions
- Author
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Chenmin Sun and Jiqiang Zheng
- Subjects
Series (mathematics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Dimension (graph theory) ,Mathematics::Analysis of PDEs ,Commutator (electric) ,01 natural sciences ,law.invention ,Schrödinger equation ,symbols.namesake ,Nonlinear system ,Mathematics - Analysis of PDEs ,law ,0103 physical sciences ,symbols ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Nonlinear Schrödinger equation ,Mathematics ,Mathematical physics ,Analysis of PDEs (math.AP) - Abstract
In this paper, we study the $H^s$-stability of the log-log blowup regime (which has been completely described in a series of recent works by Merle and Raphael) for the focusing mass-critical nonlinear Schr\"odinger equations $i\partial_tu+\Delta u+|u|^\frac4du=0$ in $\mathbb{R}^d$ with $d\geq3$. We aim to extend the result in [Colliander and Raphael, Rough blowup solutions to the $L^2$ critical NLS, Math. Anna., 345(2009), 307-366.] for dimension two to the higher dimensions cases $d\geq3$, where we use the bootstrap argument in the above paper and the commutator estimates in [M. Visan and X. Zhang, On the blowup for the $L^2$-critical focusing nonlinear Schr\"odinger equation in higher dimensions below the energy class. SIAM J. Math. Anal., 39(2007), 34-56.]., Comment: Comments are welcome. arXiv admin note: text overlap with arXiv:0907.5047 by other authors
- Published
- 2018
- Full Text
- View/download PDF
47. A refined Poisson summation formula for certain Braverman-Kazhdan spaces
- Author
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Jayce R. Getz and Baiying Liu
- Subjects
Pure mathematics ,Conjecture ,Mathematics - Number Theory ,General Mathematics ,010102 general mathematics ,Poisson summation formula ,01 natural sciences ,Connection (mathematics) ,symbols.namesake ,Fourier transform ,Schwartz space ,11F70 (Primary), 11F66 (Secondary) ,0103 physical sciences ,FOS: Mathematics ,symbols ,Number Theory (math.NT) ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,Special case ,Mathematics::Representation Theory ,Mathematics - Representation Theory ,Meromorphic function ,Mathematics - Abstract
Braverman and Kazhdan (2000) introduced influential conjectures aimed at generalizing the Fourier transform and the Poisson summation formula. Their conjectures should imply that quite general Langlands L-functions have meromorphic continuations and functional equations as predicted by Langlands’ functoriality conjecture. As an evidence for their conjectures, Braverman and Kazhdan (2002) considered a setting related to the so-called doubling method in a later paper and proved the corresponding Poisson summation formula under restrictive assumptions on the functions involved. The connection between the two papers is made explicit in the work of Li (2018). In this paper, we consider a special case of the setting in Braverman and Kazhdan’s later paper and prove a refined Poisson summation formula that eliminates the restrictive assumptions of that paper. Along the way we provide analytic control on the Schwartz space we construct; this analytic control was conjectured to hold (in a slightly different setting) in the work of Braverman and Kazhdan (2002).
- Published
- 2017
48. GAP PHENOMENA AND CURVATURE ESTIMATES FOR CONFORMALLY COMPACT EINSTEIN MANIFOLDS
- Author
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Yuguang Shi, Gang Li, and Jie Qing
- Subjects
Mathematics - Differential Geometry ,gap phenomena ,Primary 53C25 ,General Mathematics ,Conformal map ,Einstein manifold ,Curvature ,01 natural sciences ,curvature estimates ,symbols.namesake ,Relative Volume ,Ricci-flat manifold ,0103 physical sciences ,FOS: Mathematics ,Gap theorem ,0101 mathematics ,Einstein ,Mathematical physics ,Mathematics ,Quantitative Biology::Biomolecules ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Conformally compact Einstein manifolds ,16. Peace & justice ,Pure Mathematics ,math.DG ,Differential Geometry (math.DG) ,rigidity ,Yamabe constants ,Secondary 58J05 ,symbols ,renormalized volumes ,010307 mathematical physics ,Mathematics::Differential Geometry ,Yamabe invariant ,Primary 53C25, Secondary 58J05 - Abstract
In this paper we first use the result in $[12]$ to remove the assumption of the $L^2$ boundedness of Weyl curvature in the gap theorem in $[9]$ and then obtain a gap theorem for a class of conformally compact Einstein manifolds with very large renormalized volume. We also uses the blow-up method to derive curvature estimates for conformally compact Einstein manifolds with large renormalized volume. The second part of this paper is on conformally compact Einstein manifolds with conformal infinities of large Yamabe constants. Based on the idea in $[15]$ we manage to give the complete proof of the relative volume inequality $(1.9)$ on conformally compact Einstein manifolds. Therefore we obtain the complete proof of the rigidity theorem for conformally compact Einstein manifolds in general dimensions with no spin structure assumption (cf. $[29, 15]$) as well as the new curvature pinch estimates for conformally compact Einstein manifolds with conformal infinities of very large Yamabe constant. We also derive the curvature estimates for conformally compact Einstein manifolds with conformal infinities of large Yamabe constant., Comment: 28 pages, 1 figure(with one sentence added)
- Published
- 2017
49. Differential geometry of immersed surfaces in three-dimensional normed spaces
- Author
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Horst Martini, Vitor Balestro, and Ralph Teixeira
- Subjects
Mathematics - Differential Geometry ,Surface (mathematics) ,Gauss map ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Curvature ,01 natural sciences ,symbols.namesake ,Differential geometry ,Differential Geometry (math.DG) ,0103 physical sciences ,Minkowski space ,Gaussian curvature ,symbols ,FOS: Mathematics ,Vector field ,010307 mathematical physics ,Mathematics::Differential Geometry ,0101 mathematics ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper we study curvature types of immersed surfaces in three-dimensional (normed or) Minkowski spaces. By endowing the surface with a normal vector field, which is a transversal vector field given by the ambient Birkhoff orthogonality, we get an analogue of the Gauss map. Then we can define concepts of principal, Gaussian, and mean curvatures in terms of the eigenvalues of the differential of this map. Considering planar sections containing the normal field, we also define normal curvatures at each point of the surface, and with respect to each tangent direction. We investigate the relations between these curvature types. Further on we prove that, under an additional hypothesis, a compact, connected surface without boundary whose Minkowski Gaussian curvature is constant must be a Minkowski sphere. Since existing literature on the subject of our paper is widely scattered, in the introductory part also a survey of related results is given.
- Published
- 2017
- Full Text
- View/download PDF
50. Equivariant Dirac operators and differentiable geometric invariant theory
- Author
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Michèle Vergne, Paul-Emile Paradan, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Jussieu (IMJ), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques et de Modélisation de Montpellier ( I3M ), Université Montpellier 2 - Sciences et Techniques ( UM2 ) -Université de Montpellier ( UM ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Mathématiques de Jussieu ( IMJ ), Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Diderot - Paris 7 ( UPD7 ) -Centre National de la Recherche Scientifique ( CNRS ), and Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,General Mathematics ,[ MATH.MATH-SG ] Mathematics [math]/Symplectic Geometry [math.SG] ,Dirac operator ,01 natural sciences ,Mathematics::Algebraic Topology ,symbols.namesake ,Mathematics::K-Theory and Homology ,0103 physical sciences ,[ MATH.MATH-KT ] Mathematics [math]/K-Theory and Homology [math.KT] ,FOS: Mathematics ,Differentiable function ,0101 mathematics ,Spin^c structures ,Mathematics::Symplectic Geometry ,Mathematics ,010102 general mathematics ,Multiplicity (mathematics) ,K-Theory and Homology (math.KT) ,[MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG] ,[ MATH.MATH-DG ] Mathematics [math]/Differential Geometry [math.DG] ,Equivariant index ,Transversally elliptic symbol ,Differential Geometry (math.DG) ,Multiplicity ,Mathematics - Symplectic Geometry ,[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] ,Mathematics - K-Theory and Homology ,symbols ,[MATH.MATH-KT]Mathematics [math]/K-Theory and Homology [math.KT] ,Equivariant map ,Symplectic Geometry (math.SG) ,Condensed Matter::Strongly Correlated Electrons ,010307 mathematical physics ,Geometric invariant theory - Abstract
In this paper, we give a geometric expression for the multiplicities of the equivariant index of a spin-c Dirac operator., Comment: The authors have decided to divide the preprint "Multiplicities of equivariant Spin-c Dirac operators" into two separate publications. The present paper is the first of them. The other part is the preprint arXiv:1512.02367 entitled "Admissible orbits for compact Lie group"
- Published
- 2017
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