237 results
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2. Phase portraits of separable quadratic systems and a bibliographical survey on quadratic systems
- Author
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Jaume Llibre and Tao Li
- Subjects
Pure mathematics ,Class (set theory) ,Poincaré compactification ,Phase portrait ,General Mathematics ,010102 general mathematics ,Quadratic function ,01 natural sciences ,Separable space ,Quadratic system ,symbols.namesake ,Quadratic equation ,Separable system ,Poincaré conjecture ,symbols ,Compactification (mathematics) ,0101 mathematics ,Quadratic differential ,Mathematics - Abstract
Although planar quadratic differential systems and their applications have been studied in more than one thousand papers, we still have no complete understanding of these systems. In this paper we have two objectives. First we provide a brief bibliographical survey on the main results about quadratic systems. Here we do not consider the applications of these systems to many areas as in Physics, Chemist, Economics, Biology, … Second we characterize the new class of planar separable quadratic polynomial differential systems. For such class of systems we provide the normal forms which contain one parameter, and using the Poincare compactification and the blow up technique, we prove that there exist 10 non-equivalent topological phase portraits in the Poincare disc for the separable quadratic polynomial differential systems.
- Published
- 2021
3. On the singular value decomposition over finite fields and orbits of GU×GU
- Author
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Robert M. Guralnick
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Unitary state ,Nilpotent matrix ,symbols.namesake ,Finite field ,Character (mathematics) ,Kronecker delta ,Singular value decomposition ,Linear algebra ,symbols ,0101 mathematics ,Algebraic number ,Mathematics - Abstract
The singular value decomposition of a complex matrix is a fundamental concept in linear algebra and has proved extremely useful in many subjects. It is less clear what the situation is over a finite field. In this paper, we classify the orbits of GU m ( q ) × GU n ( q ) on M m × n ( q 2 ) (which is the analog of the singular value decomposition). The proof involves Kronecker’s theory of pencils and the Lang–Steinberg theorem for algebraic groups. Besides the motivation mentioned above, this problem came up in a recent paper of Guralnick et al. (2020) where a concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups was studied and bounds on the number of orbits was needed. A consequence of this work determines possible pairs of Jordan forms for nilpotent matrices of the form A A ∗ and A ∗ A over a finite field and A A ⊤ and A ⊤ A over arbitrary fields.
- Published
- 2021
4. Null controllability of semi-linear fourth order parabolic equations
- Author
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K. Kassab, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)
- Subjects
Null controllability ,Observability ,Global Carleman estimate ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Null (mathematics) ,Exact controllability ,01 natural sciences ,Parabolic partial differential equation ,Dirichlet distribution ,Domain (mathematical analysis) ,010101 applied mathematics ,Controllability ,symbols.namesake ,Linear and semi-linear fourth order parabolic equation ,Bounded function ,MSC : 35K35, 93B05, 93B07 ,Neumann boundary condition ,symbols ,[MATH]Mathematics [math] ,0101 mathematics ,Mathematics - Abstract
International audience; In this paper, we consider a semi-linear fourth order parabolic equation in a bounded smooth domain Ω with homogeneous Dirichlet and Neumann boundary conditions. The main result of this paper is the null controllability and the exact controllability to the trajectories at any time T > 0 for the associated control system with a control function acting at the interior.; Dans ce papier, on considère uneéquation parabolique semi-linéaire de quatrième ordre dans un domaine borné régulier Ω avec des conditions aux limites de type Dirichlet et Neumann homogènes. Le résultat principal de ce papier concerne la contrôlabilitéà zéro et la contrôlabilité exacte pour tout T > 0 du système de contrôle associé avec un contrôle agissantà l'interieur.
- Published
- 2020
5. Boundary value problems for the Brinkman system with L∞ coefficients in Lipschitz domains on compact Riemannian manifolds. A variational approach
- Author
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Wolfgang L. Wendland and Mirela Kohr
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Weak solution ,010102 general mathematics ,Mathematics::Analysis of PDEs ,Fixed-point theorem ,Riemannian manifold ,Lipschitz continuity ,01 natural sciences ,Dirichlet distribution ,Physics::Fluid Dynamics ,010101 applied mathematics ,Sobolev space ,Nonlinear system ,symbols.namesake ,symbols ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
The purpose of this paper is to show well-posedness results in L 2 -based Sobolev spaces for transmission, Dirichlet, Neumann, and mixed boundary value problems for the Brinkman system with L ∞ coefficients in Lipschitz domains on a compact Riemannian manifold of dimension m ≥ 2 . The Dirichlet, transmission, and mixed problems for the nonlinear Darcy-Forchheimer-Brinkman system with L ∞ coefficients are also analyzed. First, we focus on the well-posedness of linear transmission, Dirichlet and mixed boundary value problems for the Brinkman system with L ∞ coefficients in Lipschitz domains on compact Riemannian manifolds by using a variational approach that reduces such a boundary value problem to a mixed variational formulation defined in terms of two bilinear continuous forms, one of them satisfying a coercivity condition and another one the inf-sup condition. Further, we show the equivalence between each boundary value problem for the Brinkman system with L ∞ coefficients and its mixed variational counterpart, and then the well posedness in L 2 -based Sobolev spaces by using the Necas-Babuska-Brezzi technique. The second goal of this paper is the construction of the Newtonian and layer potential operators for the Brinkman system with L ∞ coefficients in Lipschitz domains on compact Riemannian manifolds by using the well-posedness results for the analyzed linear transmission problems. Various mapping properties of these operators are also obtained and used to describe the weak solutions of the Poisson problems with Dirichlet and Neumann conditions for the nonsmooth Brinkman system in terms of such potentials. Finally, we combine the well-posedness results of the Poisson problems of Dirichlet, transmission, and mixed type for the nonsmooth Brinkman system with a fixed point theorem in order to show the existence of a weak solution of the Poisson problem of Dirichlet, transmission, or mixed type for the (nonlinear) Darcy-Forchheimer-Brinkman system with L ∞ coefficients in L 2 -based Sobolev spaces in Lipschitz domains on compact Riemannian manifolds of dimension m ∈ { 2 , 3 } .
- Published
- 2019
6. Convergence of boundary layers for the Keller–Segel system with singular sensitivity in the half-plane
- Author
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Qianqian Hou and Zhi-An Wang
- Subjects
Plane (geometry) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Prandtl number ,Boundary (topology) ,Space (mathematics) ,01 natural sciences ,010101 applied mathematics ,Boundary layer ,symbols.namesake ,symbols ,Boundary value problem ,0101 mathematics ,Layer (object-oriented design) ,Degeneracy (mathematics) ,Mathematics - Abstract
Though the boundary layer formation in the chemotactic process has been observed in experiment (cf. [63] ), the mathematical study on the boundary layer solutions of chemotaxis models is just in its infant stage. Apart from the sophisticated theoretical tools involved in the analysis, how to impose/derive physical boundary conditions is a state-of-the-art in studying the boundary layer problem of chemotaxis models. This paper will proceed with a previous work [24] in one dimension to establish the convergence of boundary layer solutions of the Keller–Segel model with singular sensitivity in a two-dimensional space (half-plane) with respect to the chemical diffusion rate denoted by e ≥ 0 . Compared to the one-dimensional boundary layer problem, there are many new issues arising from multi-dimensions such as possible Prandtl type degeneracy, curl-free preservation and well-posedness of large-data solutions. In this paper, we shall derive appropriate physical boundary conditions and gradually overcome these barriers and hence establish the convergence of boundary layer solutions of the singular Keller–Segel system in the half-plane as the chemical diffusion rate vanishes. Specially speaking, we justify that the boundary layer converges to the outer layer (solution with e = 0 ) plus the inner layer as e → 0 , where both outer and inner layer profiles are precisely derived and well understood. By doing this, the structure of boundary layer solutions is clearly characterized. We hope that our results and methods can shed lights on the understanding of underlying mechanisms of the boundary layer patterns observed in the experiment for chemotaxis such as the work by Tuval et al. [63] , and open a new window in the future theoretical study of chemotaxis models.
- Published
- 2019
7. Reproducing kernel orthogonal polynomials on the multinomial distribution
- Author
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Robert C. Griffiths and Persi Diaconis
- Subjects
Numerical Analysis ,Stationary distribution ,Markov chain ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Poisson kernel ,010103 numerical & computational mathematics ,Kravchuk polynomials ,01 natural sciences ,Combinatorics ,symbols.namesake ,Kernel (statistics) ,Orthogonal polynomials ,symbols ,Test statistic ,Multinomial distribution ,0101 mathematics ,Analysis ,Mathematics - Abstract
Diaconis and Griffiths (2014) study the multivariate Krawtchouk polynomials orthogonal on the multinomial distribution. In this paper we derive the reproducing kernel orthogonal polynomials Q n ( x , y ; N , p ) on the multinomial distribution which are sums of products of orthonormal polynomials in x and y of fixed total degree n = 0 , 1 , … , N . The Poisson kernel ∑ n = 0 N ρ n Q n ( x , y ; N , p ) arises naturally from a probabilistic argument. An application to a multinomial goodness of fit test is developed, where the chi-squared test statistic is decomposed into orthogonal components which test the order of fit. A new duplication formula for the reproducing kernel polynomials in terms of the 1-dimensional Krawtchouk polynomials is derived. The duplication formula allows a Lancaster characterization of all reversible Markov chains with a multinomial stationary distribution whose eigenvectors are multivariate Krawtchouk polynomials and where eigenvalues are repeated within the same total degree. The χ 2 cutoff time, and total variation cutoff time is investigated in such chains. Emphasis throughout the paper is on a probabilistic understanding of the polynomials and their applications, particularly to Markov chains.
- Published
- 2019
8. Superconvergence of kernel-based interpolation
- Author
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Robert Schaback
- Subjects
Numerical Analysis ,Applied Mathematics ,General Mathematics ,Open problem ,Hilbert space ,Numerical Analysis (math.NA) ,010103 numerical & computational mathematics ,Positive-definite matrix ,Superconvergence ,Eigenfunction ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Spline (mathematics) ,FOS: Mathematics ,symbols ,Applied mathematics ,Mathematics - Numerical Analysis ,Boundary value problem ,0101 mathematics ,Spline interpolation ,Analysis ,Mathematics - Abstract
From spline theory it is well-known that univariate cubic spline interpolation, if carried out in its natural Hilbert space W 2 2 [ a , b ] and on point sets with fill distance h , converges only like O ( h 2 ) in L 2 [ a , b ] if no additional assumptions are made. But superconvergence up to order h 4 occurs if more smoothness is assumed and if certain additional boundary conditions are satisfied. This phenomenon was generalized in 1999 to multivariate interpolation in Reproducing Kernel Hilbert Spaces on domains Ω ⊂ R d for continuous positive definite Fourier-transformable shift-invariant kernels on R d . But the sufficient condition for superconvergence given in 1999 still needs further analysis, because the interplay between smoothness and boundary conditions is not clear at all. Furthermore, if only additional smoothness is assumed, superconvergence is numerically observed in the interior of the domain, but a theoretical foundation still is a challenging open problem. This paper first generalizes the “improved error bounds” of 1999 by an abstract theory that includes the Aubin–Nitsche trick and the known superconvergence results for univariate polynomial splines. Then the paper analyzes what is behind the sufficient conditions for superconvergence. They split into conditions on smoothness and localization, and these are investigated independently. If sufficient smoothness is present, but no additional localization conditions are assumed, it is numerically observed that superconvergence always occurs in the interior of the domain, and some supporting arguments are provided. If smoothness and localization interact in the kernel-based case on R d , weak and strong boundary conditions in terms of pseudodifferential operators occur. A special section on Mercer expansions is added, because Mercer eigenfunctions always satisfy the sufficient conditions for superconvergence. Numerical examples illustrate the theoretical findings.
- Published
- 2018
9. Computation of the largest positive Lyapunov exponent using rounding mode and recursive least square algorithm
- Author
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Samir A. M. Martins, Márcio J. Lacerda, Márcia L. C. Peixoto, and Erivelton G. Nepomuceno
- Subjects
Logarithm ,Dynamical systems theory ,General Mathematics ,Applied Mathematics ,Computation ,Rounding ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,Lyapunov exponent ,Interval (mathematics) ,01 natural sciences ,Upper and lower bounds ,010305 fluids & plasmas ,symbols.namesake ,0103 physical sciences ,Line (geometry) ,symbols ,Applied mathematics ,010301 acoustics ,Mathematics - Abstract
It has been shown that natural interval extensions (NIE) can be used to calculate the largest positive Lyapunov exponent (LLE). However, the elaboration of NIE are not always possible for some dynamical systems, such as those modelled by simple equations or by Simulink-type blocks. In this paper, we use rounding mode of floating-point numbers to compute the LLE. We have exhibited how to produce two pseudo-orbits by means of different rounding modes; these pseudo-orbits are used to calculate the Lower Bound Error (LBE). The LLE is the slope of the line gotten from the logarithm of the LBE, which is estimated by means of a recursive least square algorithm (RLS). The main contribution of this paper is to develop a procedure to compute the LLE based on the LBE without using the NIE. Additionally, with the aid of RLS the number of required points has been decreased. Eight numerical examples are given to show the effectiveness of the proposed technique.
- Published
- 2018
10. On emergence and complexity of ergodic decompositions
- Author
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Pierre Berger and Jairo Bochi
- Subjects
Pure mathematics ,Lebesgue measure ,Dynamical systems theory ,General Mathematics ,010102 general mathematics ,Dynamical Systems (math.DS) ,Lebesgue integration ,37A35, 37C05, 37C45, 37C40, 37J40 ,01 natural sciences ,Measure (mathematics) ,010104 statistics & probability ,Metric space ,symbols.namesake ,FOS: Mathematics ,symbols ,Ergodic theory ,Mathematics - Dynamical Systems ,0101 mathematics ,Dynamical system (definition) ,Probability measure ,Mathematics - Abstract
A concept of emergence was recently introduced in the paper [Berger] in order to quantify the richness of possible statistical behaviors of orbits of a given dynamical system. In this paper, we develop this concept and provide several new definitions, results, and examples. We introduce the notion of topological emergence of a dynamical system, which essentially evaluates how big the set of all its ergodic probability measures is. On the other hand, the metric emergence of a particular reference measure (usually Lebesgue) quantifies how non-ergodic this measure is. We prove fundamental properties of these two emergences, relating them with classical concepts such as Kolmogorov's $\epsilon$-entropy of metric spaces and quantization of measures. We also relate the two types of emergences by means of a variational principle. Furthermore, we provide several examples of dynamics with high emergence. First, we show that the topological emergence of some standard classes of hyperbolic dynamical systems is essentially the maximal one allowed by the ambient. Secondly, we construct examples of smooth area-preserving diffeomorphisms that are extremely non-ergodic in the sense that the metric emergence of the Lebesgue measure is essentially maximal. These examples confirm that super-polynomial emergence indeed exists, as conjectured in the paper [Berger]. Finally, we prove that such examples are locally generic among smooth diffeomorphisms., Comment: v3: Final version; to appear in Advances in Mathematics
- Published
- 2021
11. Fonctions complètement multiplicatives de somme nulle
- Author
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Eric Saias and Jean-Pierre Kahane
- Subjects
General Mathematics ,010102 general mathematics ,Multiplicative function ,01 natural sciences ,Abelian and tauberian theorems ,010101 applied mathematics ,Combinatorics ,symbols.namesake ,Riemann hypothesis ,Bounded function ,symbols ,Euler's formula ,0101 mathematics ,Invariant (mathematics) ,Well-defined ,Dirichlet series ,Mathematics - Abstract
Completely multiplicative functions whose sum is zero ($CMO$). The paper deals with $CMO$, meaning completely multiplicative ($CM$) functions $f$ such that $f(1)=1$ and $\sum\limits_1^\infty f(n)=0$. $CM$ means $f(ab)=f(a)f(b)$ for all $(a,b)\in \N^{*2}$, therefore $f$ is well defined by the $f(p)$, $p$ prime. Assuming that $f$ is $CM$, give conditions on the $f(p)$, either necessary or sufficient, both is possible, for $f$ being $CMO$ : that is the general purpose of the authors. The $CMO$ character of $f$ is invariant under slight modifications of the sequence $(f(p))$ (theorem~3). The same idea applies also in a more general context (theorem~4). After general statements of that sort, including examples of $CMO$ (theorem~5), the paper is devoted to ``small'' functions, that is, functions of the form $\frac{f(n)}{n}$, where the $f(n)$ are bounded. Here is a typical result : if $|f(p)|\le 1$ and $Re\, f(p)\le0$ for all $p$, a necessary and sufficient condition for $\big(\frac{f(n)}{n}\big)$ to be $CMO$ is $\sum \, Re\, f(p)/p=-\infty$ (theorem~8). Another necessary and sufficient condition is given under the assumption that $|1+f(p)|\le 1$ and $f(2)\not=-2$ (theorem~7). A third result gives only a sufficient condition (theorem~9). The three results apply to the particular case $f(p)=-1$, the historical example of Euler. Theorems 7 and 8 need auxiliary results, coming either from the existing literature (Hal\'asz, Montgomery--Vaughan), or from improved versions of classical results (Ingham, Ska\l ba) about $f(n)$ under assumptions on the $f*1(n)$, * denoting the multiplicative convolution (theorems~10~and~11).
- Published
- 2017
12. 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
13. Ill-posedness of the Prandtl equations in Sobolev spaces around a shear flow with general decay
- Author
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Cheng-Jie Liu and Tong Yang
- Subjects
Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Prandtl number ,Mathematics::Analysis of PDEs ,01 natural sciences ,Physics::Fluid Dynamics ,010101 applied mathematics ,Sobolev space ,symbols.namesake ,Inviscid flow ,symbols ,0101 mathematics ,Exponential decay ,Shear flow ,Approximate solution ,Ill posedness ,Mathematics ,Variable (mathematics) - Abstract
Motivated by the paper Gerard-Varet and Dormy (2010) [6] [JAMS, 2010] about the linear ill-posedness for the Prandtl equations around a shear flow with exponential decay in normal variable, and the recent study of well-posedness on the Prandtl equations in Sobolev spaces, this paper aims to extend the result in [6] to the case when the shear flow has general decay. The key observation is to construct an approximate solution that captures the initial layer to the linearized problem motivated by the precise formulation of solutions to the inviscid Prandtl equations.
- Published
- 2017
14. Hamilton–Jacobi theory, symmetries and coisotropic reduction
- Author
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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.
- Published
- 2017
15. New pathways and connections in number theory and analysis motivated by two incorrect claims of Ramanujan
- Author
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Arindam Roy, Atul Dixit, Bruce C. Berndt, and Alexandru Zaharescu
- Subjects
Discrete mathematics ,Series (mathematics) ,General Mathematics ,010102 general mathematics ,Divisor function ,Divisor (algebraic geometry) ,Divergent series ,01 natural sciences ,Ramanujan's sum ,010101 applied mathematics ,symbols.namesake ,Identity (mathematics) ,Number theory ,symbols ,0101 mathematics ,Convergent series ,Mathematics - Abstract
The focus of this paper commences with an examination of three (not obviously related) pages in Ramanujan's lost notebook, pages 336, 335, and 332, in decreasing order of attention. On page 336, Ramanujan proposes two identities, but the formulas are wrong – each is vitiated by divergent series. We concentrate on only one of the two incorrect “identities,” which may have been devised to attack the extended divisor problem. We prove here a corrected version of Ramanujan's claim, which contains the convergent series appearing in it. The convergent series in Ramanujan's faulty claim is similar to one used by G.F. Voronoi, G.H. Hardy, and others in their study of the classical Dirichlet divisor problem. This now brings us to page 335, which comprises two formulas featuring doubly infinite series of Bessel functions, the first being conjoined with the classical circle problem initiated by Gauss, and the second being associated with the Dirichlet divisor problem. The first and fourth authors, along with Sun Kim, have written several papers providing proofs of these two difficult formulas in different interpretations. In this monograph, we return to these two formulas and examine them in more general settings. The aforementioned convergent series in Ramanujan's “identity” is also similar to one that appears in a curious identity found in Chapter 15 in Ramanujan's second notebook, written in a more elegant, equivalent formulation on page 332 in the lost notebook. This formula may be regarded as a formula for ζ ( 1 2 ) , and in 1925, S. Wigert obtained a generalization giving a formula for ζ ( 1 k ) for any even integer k ≥ 2 . We extend the work of Ramanujan and Wigert in this paper. The Voronoi summation formula appears prominently in our study. In particular, we generalize work of J.R. Wilton and derive an analogue involving the sum of divisors function σ s ( n ) . The modified Bessel functions K s ( x ) arise in several contexts, as do Lommel functions. We establish here new series and integral identities involving modified Bessel functions and modified Lommel functions. Among other results, we establish a modular transformation for an infinite series involving σ s ( n ) and modified Lommel functions. We also discuss certain obscure related work of N.S. Koshliakov. We define and discuss two new related classes of integral transforms, which we call Koshliakov transforms, because he first found elegant special cases of each.
- Published
- 2017
16. Geometry of slow–fast Hamiltonian systems and Painlevé equations
- Author
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E. I. Yakovlev and L. M. Lerman
- Subjects
General Mathematics ,010102 general mathematics ,Submanifold ,01 natural sciences ,Manifold ,Hamiltonian system ,010101 applied mathematics ,symbols.namesake ,Slow manifold ,Tangent space ,symbols ,0101 mathematics ,Hamiltonian (quantum mechanics) ,Mathematics::Symplectic Geometry ,Symplectic manifold ,Mathematical physics ,Mathematics ,Symplectic geometry - Abstract
In the first part of the paper we introduce some geometric tools needed to describe slow–fast Hamiltonian systems on smooth manifolds. We start with a smooth bundle p : M → B where ( M , ω ) is a C ∞ -smooth presymplectic manifold with a closed constant rank 2-form ω and ( B , λ ) is a smooth symplectic manifold. The 2-form ω is supposed to be compatible with the structure of the bundle, that is the bundle fibers are symplectic manifolds with respect to the 2-form ω and the distribution on M generated by kernels of ω is transverse to the tangent spaces of the leaves and the dimensions of the kernels and of the leaves are supplementary. This allows one to define a symplectic structure Ω e = ω + e − 1 p ∗ λ on M for any positive small e , where p ∗ λ is the lift of the 2-form λ to M . Given a smooth Hamiltonian H on M one gets a slow–fast Hamiltonian system with respect to Ω e . We define a slow manifold S M for this system. Assuming S M is a smooth submanifold, we define a slow Hamiltonian flow on S M . The second part of the paper deals with singularities of the restriction of p to S M . We show that if dim M = 4 , dim B = 2 and Hamilton function H is generic, then the behavior of the system near a singularity of fold type is described, to the main order, by the equation Painleve-I, and if this singularity is a cusp, then the related equation is Painleve-II.
- Published
- 2016
17. 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
18. Exponential tractability of linear weighted tensor product problems in the worst-case setting for arbitrary linear functionals
- Author
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Peter Kritzer, Henryk Woźniakowski, and Friedrich Pillichshammer
- Subjects
Statistics and Probability ,Discrete mathematics ,Numerical Analysis ,Polynomial ,Control and Optimization ,Algebra and Number Theory ,Logarithm ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Hilbert space ,010103 numerical & computational mathematics ,01 natural sciences ,Exponential polynomial ,Exponential function ,Singular value ,symbols.namesake ,Tensor product ,Bounded function ,symbols ,0101 mathematics ,Mathematics - Abstract
We study the approximation of compact linear operators defined over certain weighted tensor product Hilbert spaces. The information complexity is defined as the minimal number of arbitrary linear functionals needed to obtain an e -approximation for the d -variate problem which is fully determined in terms of the weights and univariate singular values. Exponential tractability means that the information complexity is bounded by a certain function that depends polynomially on d and logarithmically on e − 1 . The corresponding unweighted problem was studied in Hickernell et al. (2020) with many negative results for exponential tractability. The product weights studied in the present paper change the situation. Depending on the form of polynomial dependence on d and logarithmic dependence on e − 1 , we study exponential strong polynomial, exponential polynomial, exponential quasi-polynomial, and exponential ( s , t ) -weak tractability with max ( s , t ) ≥ 1 . For all these notions of exponential tractability, we establish necessary and sufficient conditions on weights and univariate singular values for which it is indeed possible to achieve the corresponding notion of exponential tractability. The case of exponential ( s , t ) -weak tractability with max ( s , t ) 1 is left for future study. The paper uses some general results obtained in Hickernell et al. (2020) and Kritzer and Woźniakowski (2019).
- Published
- 2020
19. Instability of high dimensional Hamiltonian systems: Multiple resonances do not impede diffusion
- Author
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Amadeu Delshams, Rafael de la Llave, Tere M. Seara, Universitat Politècnica de Catalunya. Departament de Matemàtiques, and Universitat Politècnica de Catalunya. SD - Sistemes Dinàmics de la UPC
- Subjects
Pure mathematics ,Mathematics(all) ,General Mathematics ,Dynamical Systems (math.DS) ,Scattering map ,01 natural sciences ,010305 fluids & plasmas ,Hamiltonian system ,symbols.namesake ,Arnold diffusion ,0103 physical sciences ,FOS: Mathematics ,Sistemes hamiltonians ,Mathematics - Dynamical Systems ,Hamiltonian systems ,0101 mathematics ,Mathematics ,Scattering ,010102 general mathematics ,Mathematical analysis ,Instability ,Matemàtiques i estadística [Àrees temàtiques de la UPC] ,Resonance ,Torus ,Codimension ,37J40 ,Hamiltonian ,Resonances ,symbols ,Hamiltonian (quantum mechanics) ,Symplectic geometry - Abstract
We consider models given by Hamiltonians of the form H ( I , φ , p , q , t ; e ) = h ( I ) + ∑ j = 1 n ± ( 1 2 p j 2 + V j ( q j ) ) + e Q ( I , φ , p , q , t ; e ) where I ∈ I ⊂ R d , φ ∈ T d , p , q ∈ R n , t ∈ T 1 . These are higher dimensional analogues, both in the center and hyperbolic directions, of the models studied in [28] , [29] , [43] and are usually called “a-priori unstable Hamiltonian systems”. All these models present the large gap problem. We show that, for 0 e ≪ 1 , under regularity and explicit non-degeneracy conditions on the model, there are orbits whose action variables I perform rather arbitrary excursions in a domain of size O ( 1 ) . This domain includes resonance lines and, hence, large gaps among d-dimensional KAM tori. This phenomenon is known as Arnold diffusion. The method of proof follows closely the strategy of [28] , [29] . The main new phenomenon that appears when the dimension d of the center directions is larger than one is the existence of multiple resonances in the space of actions I ∈ I ⊂ R d . We show that, since these multiple resonances happen in sets of codimension greater than one in the space of actions I, they can be contoured. This corresponds to the mechanism called diffusion across resonances in the Physics literature. The present paper, however, differs substantially from [28] , [29] . On the technical details of the proofs, we have taken advantage of the theory of the scattering map developed in [31] —notably the symplectic properties—which were not available when the above papers were written. We have analyzed the conditions imposed on the resonances in more detail. More precisely, we have found that there is a simple condition on the Melnikov potential which allows us to conclude that the resonances are crossed. In particular, this condition does not depend on the resonances. So that the results are new even when applied to the models in [28] , [29] .
- Published
- 2016
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20. On the strong divergence of Hilbert transform approximations and a problem of Ul’yanov
- Author
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Holger Boche and Volker Pohl
- Subjects
Numerical Analysis ,Sequence ,Conjecture ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,020206 networking & telecommunications ,02 engineering and technology ,01 natural sciences ,Combinatorics ,symbols.namesake ,Uniform norm ,Subsequence ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,Hilbert transform ,0101 mathematics ,Divergence (statistics) ,Finite set ,Fourier series ,Analysis ,Mathematics - Abstract
This paper studies the approximation of the Hilbert transform f ? = H f of continuous functions f with continuous conjugate f ? based on a finite number of samples. It is known that every sequence { H N f } N ? N which approximates f ? from samples of f diverges (weakly) with respect to the uniform norm. This paper conjectures that all of these approximation sequences even contain no convergent subsequence. A property which is termed strong divergence.The conjecture is supported by two results. First it is proven that the sequence of the sampled conjugate Fejer means diverges strongly. Second, it is shown that for every sample based approximation method { H N } N ? N there are functions f such that ? H N f ? ∞ exceeds any given bound for any given number of consecutive indices N .As an application, the later result is used to investigate a problem associated with a question of Ul'yanov on Fourier series which is related to the possibility to construct adaptive approximation methods to determine the Hilbert transform from sampled data. This paper shows that no such approximation method with a finite search horizon exists.
- Published
- 2016
21. Bifurcation and multiplicity results for critical nonlocal fractional Laplacian problems
- Author
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Raffaella Servadei, Giovanni Molica Bisci, Alessio Fiscella, Fiscella, A, Molica Bisci, G, and Servadei, R
- Subjects
Discrete mathematics ,Pure mathematics ,General Mathematics ,variational techniques ,010102 general mathematics ,Multiplicity (mathematics) ,integrodifferential operators ,01 natural sciences ,Dirichlet distribution ,Fractional Laplacian ,010101 applied mathematics ,Sobolev space ,symbols.namesake ,critical nonlinearities ,Operator (computer programming) ,Fractional Laplacian, critical nonlinearities, best fractional critical Sobolev constant, variational techniques, integrodifferential operators ,Bounded function ,best fractional critical Sobolev constant ,fractional Laplacian, critical nonlinearities, best fractional critical Sobolev constant, variational techniques, integrodifferential operators ,symbols ,Exponent ,0101 mathematics ,Bifurcation ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper we consider the following critical nonlocal problem { − L K u = λ u + | u | 2 ⁎ − 2 u in Ω u = 0 in R n ∖ Ω , where s ∈ ( 0 , 1 ) , Ω is an open bounded subset of R n , n > 2 s , with continuous boundary, λ is a positive real parameter, 2 ⁎ : = 2 n / ( n − 2 s ) is the fractional critical Sobolev exponent, while L K is the nonlocal integrodifferential operator L K u ( x ) : = ∫ R n ( u ( x + y ) + u ( x − y ) − 2 u ( x ) ) K ( y ) d y , x ∈ R n , whose model is given by the fractional Laplacian − ( − Δ ) s . Along the paper, we prove a multiplicity and bifurcation result for this problem, using a classical theorem in critical points theory. Precisely, we show that in a suitable left neighborhood of any eigenvalue of − L K (with Dirichlet boundary data) the number of nontrivial solutions for the problem under consideration is at least twice the multiplicity of the eigenvalue. Hence, we extend the result got by Cerami, Fortunato and Struwe in [14] for classical elliptic equations, to the case of nonlocal fractional operators.
- Published
- 2016
22. Calculating the spectral factorization and outer functions by sampling-based approximations—Fundamental limitations
- Author
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Volker Pohl and Holger Boche
- Subjects
Numerical Analysis ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Sampling (statistics) ,Spectral density ,010103 numerical & computational mathematics ,Function (mathematics) ,Dirichlet's energy ,Spectral theorem ,Hardy space ,Singular integral ,01 natural sciences ,symbols.namesake ,symbols ,0101 mathematics ,Closed-form expression ,Analysis ,Mathematics - Abstract
This paper considers the problem of approximating the spectral factor of continuous spectral densities with finite Dirichlet energy based on finitely many samples of these spectral densities. Although there exists a closed form expression for the spectral factor, this formula shows a very complicated behavior because of the non-linear dependency of the spectral factor from spectral density and because of a singular integral in this expression. Therefore approximation methods are usually applied to calculate the spectral factor. It is shown that there exists no sampling-based method which depends continuously on the samples and which is able to approximate the spectral factor for all densities in this set. Instead, to any sampling-based approximation method there exists a large set of spectral densities so that the approximation method does not converge to the spectral factor for every spectral density in this set as the number of available sampling points is increased. The paper will also show that the same results hold for sampling-based algorithms for the calculation of the outer function in the theory of Hardy spaces.
- Published
- 2020
23. 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
24. Truncated Hecke-Rogers type series
- Author
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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
25. 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
26. The Lebesgue measure of the algebraic difference of two random Cantor sets
- Author
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Boris Solomyak, Péter Móra, and Károly Simon
- Subjects
Discrete mathematics ,Mathematics(all) ,Lebesgue measure ,General Mathematics ,010102 general mathematics ,Cantor function ,Random fractals ,01 natural sciences ,Point process ,Cantor set ,Combinatorics ,Null set ,010104 statistics & probability ,symbols.namesake ,Difference of Cantor sets ,Palis conjecture ,Branching processes with random environment ,symbols ,Random compact set ,Almost surely ,0101 mathematics ,Cantor's diagonal argument ,Mathematics - Abstract
In this paper we consider a family of random Cantor sets on the line. We give some sufficient conditions when the Lebesgue measure of the arithmetic difference is positive. Combining this with the main result of a recent joint paper of the second author with M. Dekking we construct random Cantor sets F1, F2 such that the arithmetic difference set F2 − F1 does not contain any intervals but ℒeb(F2 − F1)> 0 almost surely, conditioned on non-extinction.
- Published
- 2009
- Full Text
- View/download PDF
27. The rate of convergence for the cyclic projections algorithm III: Regularity of convex sets
- Author
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Hein Hundal and Frank Deutsch
- Subjects
Mathematics(all) ,Alternating projections ,General Mathematics ,Convex feasibility problem ,0211 other engineering and technologies ,02 engineering and technology ,01 natural sciences ,Cyclic projections ,POCS ,Combinatorics ,symbols.namesake ,Intersection ,Angle between subspaces ,Projections onto convex sets ,0101 mathematics ,Mathematics ,Numerical Analysis ,021103 operations research ,Series (mathematics) ,Applied Mathematics ,010102 general mathematics ,Hilbert space ,Regular polygon ,Orthogonal projections ,Angle between convex sets ,Rate of convergence ,The strong conical hull intersection property (strong CHIP) ,Norm of nonlinear operators ,Iterated function ,Norm (mathematics) ,symbols ,Algorithm ,Analysis ,Regularity properties of convex sets: regular, linearly regular, boundedly regular, boundedly linearly regular, normal, weakly normal, uniformly normal - Abstract
The cyclic projections algorithm is an important method for determining a point in the intersection of a finite number of closed convex sets in a Hilbert space. That is, for determining a solution to the ''convex feasibility'' problem. This is the third paper in a series on a study of the rate of convergence for the cyclic projections algorithm. In the first of these papers, we showed that the rate could be described in terms of the ''angles'' between the convex sets involved. In the second, we showed that these angles often had a more tractable formulation in terms of the ''norm'' of the product of the (nonlinear) metric projections onto related convex sets. In this paper, we show that the rate of convergence of the cyclic projections algorithm is also intimately related to the ''linear regularity property'' of Bauschke and Borwein, the ''normal property'' of Jameson (as well as Bakan, Deutsch, and Li's generalization of Jameson's normal property), the ''strong conical hull intersection property'' of Deutsch, Li, and Ward, and the rate of convergence of iterated parallel projections. Such properties have already been shown to be important in various other contexts as well.
- Published
- 2008
28. The sub-elliptic obstacle problem: C1,α regularity of the free boundary in Carnot groups of step two
- Author
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Arshak Petrosyan, Donatella Danielli, and Nicola Garofalo
- Subjects
Pure mathematics ,General Mathematics ,Open problem ,010102 general mathematics ,Mathematical analysis ,Boundary (topology) ,Carnot group ,01 natural sciences ,010305 fluids & plasmas ,symbols.namesake ,0103 physical sciences ,Variational inequality ,Euclidean geometry ,Obstacle problem ,Free boundary problem ,symbols ,0101 mathematics ,Carnot cycle ,Mathematics - Abstract
The sub-elliptic obstacle problem arises in various branches of the applied sciences, e.g., in mechanical engineering and robotics, mathematical finance, image reconstruction and neurophysiology. In the recent paper [Donatella Danielli, Nicola Garofalo, Sandro Salsa, Variational inequalities with lack of ellipticity. I. Optimal interior regularity and non-degeneracy of the free boundary, Indiana Univ. Math. J. 52 (2) (2003) 361?398; MR1976081 (2004c:35424)] it was proved that weak solutions to the sub-elliptic obstacle problem in a Carnot group belong to the Folland?Stein (optimal) Lipschitz class (the analogue of the well-known interior local regularity for the classical obstacle problem). However, the regularity of the free boundary remained a challenging open problem. In this paper we prove that, in Carnot groups of step r=2, the free boundary is (Euclidean) C1,a near points satisfying a certain thickness condition. This constitutes the sub-elliptic counterpart of a celebrated result due to Caffarelli [Luis A. Caffarelli, The regularity of free boundaries in higher dimensions, Acta Math. 139 (3?4) (1977) 155?184; MR0454350 (56 #12601)].
- Published
- 2007
29. Boundary stabilization of a 3-dimensional structural acoustic model
- Author
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Irena Lasiecka
- Subjects
Mathematics(all) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Boundary (topology) ,Wave equation ,01 natural sciences ,uniform decay rates ,Euler equations ,nonlinear dissipation ,010101 applied mathematics ,Nonlinear system ,Bernoulli's principle ,symbols.namesake ,trace estimates ,Free boundary problem ,symbols ,Acoustic wave equation ,wave equation ,plate equation ,0101 mathematics ,Structural acoustics ,structural acoustic model ,Mathematics - Abstract
The main result of this paper provides uniform decay rates obtained for the energy function associated with a three-dimensional structural acoustic model described by coupled system consisting of the wave equation and plate equation with the coupling on the interface between the acoustic chamber and the wall. The uniform stabilization is achieved by introducing a nonlinear dissipation acting via boundary forces applied at the edge of the plate and viscous or boundary damping applied to the wave equation. The results obtained in this paper extend, to the non-analytic, hyperbolic-like setting, the results obtained previously in the literature for acoustic problems modeled by structurally damped plates (governed by analytic semigroups). As a bypass product, we also obtain optimal uniform decay rates for the Euler Bernoulli plate equations with nonlinear boundary dissipation acting via shear forces only and without (i) any geometric conditions imposed on the domain ,(ii) any growth conditions at the origin imposed on the nonlinear function. This is in contrast with the results obtained previously in the literature ([22] and references therein).
- Published
- 1999
30. Ground states of nonlinear Schrödinger systems with mixed couplings
- Author
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Yuanze Wu and Juncheng Wei
- Subjects
Interaction forces ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Block (permutation group theory) ,01 natural sciences ,Measure (mathematics) ,010101 applied mathematics ,Nonlinear system ,symbols.namesake ,symbols ,0101 mathematics ,Schrödinger's cat ,Mathematics ,Mathematical physics - Abstract
We consider the following k-coupled nonlinear Schrodinger systems: { − Δ u j + λ j u j = μ j u j 3 + ∑ i = 1 , i ≠ j k β i , j u i 2 u j in R N , u j > 0 in R N , u j ( x ) → 0 as | x | → + ∞ , j = 1 , 2 , ⋯ , k , where N ≤ 3 , k ≥ 3 , λ j , μ j > 0 are constants and β i , j = β j , i ≠ 0 are parameters. There have been intensive studies for the above systems when k = 2 or the systems are purely attractive ( β i , j > 0 , ∀ i ≠ j ) or purely repulsive ( β i , j 0 , ∀ i ≠ j ); however very few results are available for k ≥ 3 when the systems admit mixed couplings and the components are organized into groups, i.e., there exist ( i 1 , j 1 ) and ( i 2 , j 2 ) such that β i 1 , j 1 > 0 and β i 2 , j 2 0 . In this paper we give the first systematic and an (almost) complete study on the existence of ground states when the systems admit mixed couplings and the components are organized into groups. We first divide these systems into repulsive-mixed and total-mixed cases. In the first case we prove nonexistence of ground states. In the second case we give a necessary condition for the existence of ground states and also provide estimates for Morse index. The key idea is the block decomposition of the systems (optimal block decompositions, eventual block decompositions), and the measure of total interaction forces between different blocks. Finally the assumptions on the existence of ground states are shown to be optimal in some special cases.
- Published
- 2020
31. Lelong–Poincaré formula in symplectic and almost complex geometry
- Author
-
Alexandre Sukhov and Emmanuel Mazzilli
- Subjects
Pure mathematics ,Almost complex manifold ,Mathematics::Complex Variables ,General Mathematics ,010102 general mathematics ,Vector bundle ,01 natural sciences ,General family ,symbols.namesake ,Mathematics::Algebraic Geometry ,Complex geometry ,Convergence (routing) ,Poincaré conjecture ,symbols ,Mathematics::Differential Geometry ,0101 mathematics ,Mathematics::Symplectic Geometry ,Symplectic geometry ,Mathematics - Abstract
In this paper, we present two applications of the theory of singular connections developed by Harvey and Lawson (1993). The first one is a version of the Lelong–Poincare formula with estimates for sections of vector bundles over an almost complex manifold. The second one is a convergence theorem for divisors associated to a general family of symplectic submanifolds constructed by Donaldson (1996) (the case of hypersurfaces) and by Auroux in (1997) (for arbitrary dimensional submanifolds).
- Published
- 2020
32. On the structure of variable exponent spaces
- Author
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Francisco L. Hernández, Mauro Sanchiz, César Ruiz, and Julio Flores
- Subjects
46E30, 47B60 ,Pure mathematics ,Variable exponent ,General Mathematics ,010102 general mathematics ,Structure (category theory) ,010103 numerical & computational mathematics ,Disjoint sets ,Cantor function ,01 natural sciences ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,symbols.namesake ,Singularity ,Computer Science::Systems and Control ,FOS: Mathematics ,symbols ,0101 mathematics ,Mathematics - Abstract
The first part of this paper surveys several results on the lattice structure of variable exponent Lebesgue function spaces (or Nakano spaces) L p ( ⋅ ) ( Ω ) . In the second part strictly singular and disjointly strictly singular operators between spaces L p ( ⋅ ) ( Ω ) are studied. New results on the disjoint strict singularity of the inclusions L p ( ⋅ ) ( Ω ) ↪ L q ( ⋅ ) ( Ω ) are given.
- Published
- 2020
33. Long range scattering for the complex-valued Klein-Gordon equation with quadratic nonlinearity in two dimensions
- Author
-
Kota Uriya, Jun Ichi Segata, and Satoshi Masaki
- Subjects
Logarithm ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Gauge (firearms) ,35L71 ,01 natural sciences ,Term (time) ,010101 applied mathematics ,symbols.namesake ,Nonlinear system ,Range (mathematics) ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,symbols ,0101 mathematics ,Invariant (mathematics) ,Constant (mathematics) ,Klein–Gordon equation ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
In this paper, we study large time behavior of complex-valued solutions to nonlinear Klein-Gordon equation with a gauge invariant quadratic nonlinearity in two spatial dimensions. To find a possible asymptotic behavior, we consider the final value problem. It turns out that one possible behavior is a linear solution with a logarithmic phase correction as in the real-valued case. However, the shape of the logarithmic correction term has one more parameter which is also given by the final data. In the real case the parameter is constant so one cannot see its effect. However, in the complex case it varies in general. The one dimensional case is also discussed., Comment: 25 papges, 2 figures
- Published
- 2020
34. Hardy type inequalities and parametric Lamb equation
- Author
-
R. G. Nasibullin and R. V. Makarov
- Subjects
Pure mathematics ,Euclidean space ,General Mathematics ,010102 general mathematics ,Open set ,Regular polygon ,Boundary (topology) ,010103 numerical & computational mathematics ,Type (model theory) ,01 natural sciences ,Domain (mathematical analysis) ,symbols.namesake ,symbols ,0101 mathematics ,Bessel function ,Parametric statistics ,Mathematics - Abstract
This paper is devoted to Hardy type inequalities with remainders for compactly supported smooth functions on open sets in the Euclidean space. We establish new inequalities with weight functions depending on the distance function to the boundary of the domain. One-dimensional L 1 and L p inequalities and their multidimensional analogues are proved. We consider spatial inequalities in open convex domains with the finite inner radius. Constants in these inequalities depend on the roots of parametric Lamb equation for the Bessel function and turn out to be sharp in some particular cases.
- Published
- 2020
35. Norm Hilbert spaces over G-modules with a convex base
- Author
-
Herminia Ochsenius and Elena Olivos
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Regular polygon ,Hilbert space ,Banach space ,Analogy ,010103 numerical & computational mathematics ,01 natural sciences ,symbols.namesake ,Norm (mathematics) ,symbols ,0101 mathematics ,Ordered subsets ,Subspace topology ,Mathematics - Abstract
By analogy with the classical definition, a Norm Hilbert space E is defined as a Banach space over a valued field K in which each closed subspace has an orthocomplement. In the rank one case (that is, the value group as well as the set of norms of the space are contained in [ 0 , ∞ ) ), they were described by van Rooij in his classical book of 1978, but the name itself was introduced in 1999 by Ochsenius and Schikhof for the case of spaces with an infinite rank valuation. Here we shall only consider spaces over fields with value groups contained in ( R + , ⋅ ) . Yet for the set of their norms we borrow, from the infinite rank case, the notion of a G -module. That structure allows for a greater complexity than what is offered by ordered subsets of R . In this paper we describe a new class of Norm Hilbert spaces, those in which the G -module has a convex base. Their characteristics will be the focus of our study.
- Published
- 2020
36. Central limit theorems for multivariate Bessel processes in the freezing regime II: The covariance matrices
- Author
-
Michael Voit and Sergio Andraus
- Subjects
Pure mathematics ,General Mathematics ,Gaussian ,Mathematics::Classical Analysis and ODEs ,FOS: Physical sciences ,010103 numerical & computational mathematics ,01 natural sciences ,60F05, 60J60, 60B20, 70F10, 82C22, 33C67 ,symbols.namesake ,FOS: Mathematics ,Limit (mathematics) ,Representation Theory (math.RT) ,0101 mathematics ,Mathematical Physics ,Eigenvalues and eigenvectors ,Central limit theorem ,Mathematics ,Numerical Analysis ,Hermite polynomials ,Applied Mathematics ,Probability (math.PR) ,010102 general mathematics ,Mathematical Physics (math-ph) ,Covariance ,symbols ,Laguerre polynomials ,Mathematics - Probability ,Mathematics - Representation Theory ,Analysis ,Bessel function - Abstract
Bessel processes $(X_{t,k})_{t\ge0}$ in $N$ dimensions are classified via associated root systems and multiplicity constants $k\ge0$. They describe interacting Calogero-Moser-Suther\-land particle systems with $N$ particles and are related to $\beta$-Hermite and $\beta$-Laguerre ensembles. Recently, several central limit theorems were derived for fixed $t>0$, fixed starting points, and $k\to\infty$. In this paper we extend the CLT in the A-case from start in 0 to arbitrary starting distributions by using a limit result for the corresponding Bessel functions. We also determine the eigenvalues and eigenvectors of the covariance matrices of the Gaussian limits and study applications to CLTs for the intermediate particles for $k\to\infty$ and then $N\to\infty$., Comment: 20 pages
- Published
- 2019
37. Some generalizations of Ahlfors Lemma
- Author
-
Manabu Ito
- Subjects
Comparison theorem ,Pure mathematics ,Lemma (mathematics) ,Mathematics::Complex Variables ,General Mathematics ,Riemann surface ,010102 general mathematics ,Poincaré metric ,Holomorphic function ,Conformal map ,010103 numerical & computational mathematics ,Curvature ,01 natural sciences ,Unit disk ,symbols.namesake ,symbols ,Mathematics::Metric Geometry ,0101 mathematics ,Mathematics - Abstract
One of the most influential versions of the classical Schwarz–Pick Lemma is probably that of Ahlfors. Pulling back a conformal semimetric on a Riemann surface under any holomorphic map from the open unit disk equipped with a Poincare metric, the curvature of which is assumed to bound from above the curvature of the Riemann surface, he successfully showed that a conformal semimetric to be compared with the Poincare metric is obtained. In the present paper, we give a comparison theorem between two conformal semimetrics of variable curvature in the same spirit. Our main theorem is a local one by its nature, but global results can be derived therefrom.
- Published
- 2019
38. Dirichlet boundary values on Euclidean balls with infinitely many solutions for the minimal surface system
- Author
-
Xiaowei Xu, Ling Yang, and Yongsheng Zhang
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,General Mathematics ,Mathematics::Analysis of PDEs ,Boundary (topology) ,01 natural sciences ,Dirichlet distribution ,symbols.namesake ,Mathematics - Analysis of PDEs ,0103 physical sciences ,Euclidean geometry ,FOS: Mathematics ,0101 mathematics ,Mathematics ,Dirichlet problem ,Minimal surface ,Applied Mathematics ,010102 general mathematics ,Codimension ,Lipschitz continuity ,Differential Geometry (math.DG) ,symbols ,Mathematics::Differential Geometry ,010307 mathematical physics ,Unit (ring theory) ,Analysis of PDEs (math.AP) - Abstract
We make systematic developments on Lawson-Osserman constructions relating to the Dirichlet problem (over unit disks) for minimal surfaces of high codimension in their 1977 Acta paper. In particular, we show the existence of boundary functions for which infinitely many analytic solutions and at least one nonsmooth Lipschitz solution exist simultaneously. This newly-discovered amusing phenomenon enriches the understanding on the Lawson-Osserman philosophy., Supercedes arXiv:1610.08162. To appear in the Journal de Math\'ematiques Pures et Appliqu\'ees
- Published
- 2019
39. Fourier–Dunkl system of the second kind and Euler–Dunkl polynomials
- Author
-
Antonio J. Durán, Mario Pérez, and Juan L. Varona
- Subjects
Numerical Analysis ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Generating function ,010103 numerical & computational mathematics ,Function (mathematics) ,Partial fraction decomposition ,01 natural sciences ,Exponential function ,symbols.namesake ,Fourier transform ,symbols ,Euler's formula ,0101 mathematics ,Analysis ,Quotient ,Bessel function ,Mathematics - Abstract
We prove a partial fraction decomposition of a quotient of two functions E α ( i t x ) and I α ( i t ) which are defined in terms of the Bessel functions J α and J α + 1 of the first kind. This expansion leads naturally to the introduction of an orthonormal system with respect to the measure | x | 2 α + 1 d x 2 α + 1 Γ ( α + 1 ) in [ − 1 , 1 ] , which we call the Fourier–Dunkl system of the second kind. Euler–Dunkl polynomials E n , α ( x ) of degree n are defined by considering E α ( t x ) ∕ I α ( t ) as a generating function. It is shown that the sum ∑ m = 1 ∞ 1 ∕ j m , α 2 k , where j m , α are the positive zeros of J α , is equal (up to an explicit factor) to E 2 k − 1 , α ( 1 ) . For α = 1 ∕ 2 this leads to classical results of Euler since the function E 1 ∕ 2 ( x ) is the exponential function and E n , 1 ∕ 2 ( x ) are (essentially) the Euler polynomials. In the second part of the paper a sampling theorem of Whittaker–Shannon–Kotel’nikov type is established which is strongly related to the above-mentioned partial decomposition and which holds for all functions in the Payley–Wiener space defined by the Dunkl transform in [ − 1 , 1 ] .
- Published
- 2019
40. On the strong restricted isometry property of Bernoulli random matrices
- Author
-
Ran Lu
- Subjects
Numerical Analysis ,Applied Mathematics ,General Mathematics ,Gaussian ,Probability (math.PR) ,010102 general mathematics ,62G35, 42C15 ,010103 numerical & computational mathematics ,16. Peace & justice ,01 natural sciences ,Restricted isometry property ,Combinatorics ,Matrix (mathematics) ,Bernoulli's principle ,symbols.namesake ,Robustness (computer science) ,FOS: Mathematics ,symbols ,Erasure ,0101 mathematics ,Random matrix ,Random variable ,Mathematics - Probability ,Analysis ,Mathematics - Abstract
The study of the restricted isometry property (RIP) of corrupted random matrices is particularly important in the field of compressed sensing (CS) with corruptions. If a matrix still satisfies the RIP after that a certain portion of rows are erased, then we say that this matrix has the strong restricted isometry property (SRIP). In the field of compressed sensing, random matrices which satisfy certain moment conditions are of particular interest. Among these matrices, those with entries generated from i.i.d. Gaussian or symmetric Bernoulli random variables are often typically considered. Recent studies have shown that matrices with entries generated from i.i.d. Gaussian random variables satisfy the SRIP under arbitrary erasure of rows with high probability. In this paper, we study the erasure robustness property of Bernoulli random matrices. Our main result shows that with overwhelming probability, the SRIP holds for Bernoulli random matrices. Moreover, our analysis leads to a robust version of the famous Johnson–Lindenstrauss lemma for Bernoulli random matrices.
- Published
- 2019
41. The scattering problem for Hamiltonian ABCD Boussinesq systems in the energy space
- Author
-
Chulkwang Kwak, Felipe Poblete, Claudio Muñoz, and Juan C. Pozo
- Subjects
Scattering ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Scalar (mathematics) ,Conservative vector field ,01 natural sciences ,Virial theorem ,010101 applied mathematics ,symbols.namesake ,Quadratic equation ,Light cone ,symbols ,Compressibility ,0101 mathematics ,Hamiltonian (quantum mechanics) ,Mathematical physics ,Mathematics - Abstract
The Boussinesq a b c d system is a 4-parameter set of equations posed in R t × R x , originally derived by Bona, Chen and Saut [11] , [12] as first order 2-wave approximations of the incompressible and irrotational, two dimensional water wave equations in the shallow water wave regime, in the spirit of the original Boussinesq derivation [17] . Among many particular regimes, depending each of them in terms of the value of the parameters ( a , b , c , d ) present in the equations, the generic regime is characterized by the setting b , d > 0 and a , c 0 . If additionally b = d , the a b c d system is Hamiltonian. The equations in this regime are globally well-posed in the energy space H 1 × H 1 , provided one works with small solutions [12] . In this paper, we investigate decay and the scattering problem in this regime, which is characterized as having (quadratic) long-range nonlinearities, very weak linear decay O ( t − 1 / 3 ) because of the one dimensional setting, and existence of non scattering solutions (solitary waves). We prove, among other results, that for a sufficiently dispersive a b c d systems (characterized only in terms of parameters a , b and c), all small solutions must decay to zero, locally strongly in the energy space, in proper subset of the light cone | x | ≤ | t | . We prove this result by constructing three suitable virial functionals in the spirit of works [27] , [28] , and more precisely [42] (valid for the simpler scalar “good Boussinesq” model), leading to global in time decay and control of all local H 1 × H 1 terms. No parity nor extra decay assumptions are needed to prove decay, only small solutions in the energy space.
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- 2019
42. Frame decomposition and radial maximal semigroup characterization of Hardy spaces associated to operators
- Author
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Lixin Yan, Liang Song, Xuan Thinh Duong, and Ji Li
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Analytic semigroup ,Numerical Analysis ,Semigroup ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Holomorphic functional calculus ,010103 numerical & computational mathematics ,Hardy space ,01 natural sciences ,Functional calculus ,Combinatorics ,symbols.namesake ,Mathematics - Analysis of PDEs ,Bounded function ,Norm (mathematics) ,FOS: Mathematics ,symbols ,0101 mathematics ,Lp space ,Analysis ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
Let $L$ be the generator of an analytic semigroup whose kernels satisfy Gaussian upper bounds and H\"older's continuity. Also assume that $L$ has a bounded holomorphic functional calculus on $L^2(\mathbb{R}^n)$. In this paper, we construct a frame decomposition for the functions belonging to the Hardy space $H_{L}^{1}(\mathbb{R}^n)$ associated to $L$, and for functions in the Lebesgue spaces $L^p$, $1, Comment: 37 pages, to appear in Journal of Approximation Theory
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- 2019
43. Local polar invariants for plane singular foliations
- Author
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Rogério Mol, Felipe Cano, and Nuria Corral
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Pure mathematics ,General Mathematics ,010102 general mathematics ,Holomorphic function ,Dynamical Systems (math.DS) ,Mathematical proof ,01 natural sciences ,symbols.namesake ,32S65, 14C21 ,0103 physical sciences ,Poincaré conjecture ,FOS: Mathematics ,symbols ,Foliation (geology) ,Polar ,010307 mathematical physics ,Mathematics - Dynamical Systems ,0101 mathematics ,Invariant (mathematics) ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
In this survey paper, we take the viewpoint of polar invariants to the local and global study of non-dicritical holomorphic foliation in dimension two and their invariant curves. It appears a characterization of second type foliations and generalized curve foliations as well as a description of the G S V -index in terms of polar curves. We also interpret the proofs concerning the Poincare problem with polar invariants.
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- 2019
44. Some Hecke-Rogers type identities
- Author
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Liuquan Wang and Ae Ja Yee
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Partition function (quantum field theory) ,Pure mathematics ,Series (mathematics) ,General Mathematics ,010102 general mathematics ,Theta function ,Type (model theory) ,Mathematical proof ,01 natural sciences ,Ramanujan's sum ,Identity (mathematics) ,symbols.namesake ,0103 physical sciences ,symbols ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
Recently, a new partition function associated with Ramanujan's third order mock theta function ω ( q ) was discovered, and subsequently its overpartition analogue was introduced. In this paper, we prove an intriguing identity arising from the study of that analogue, which involves a double series of Hecke-Rogers type. In addition, two further identities will be given. Our proofs heavily rely on formulas from the work of Liu [12] .
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- 2019
45. On the Bures–Wasserstein distance between positive definite matrices
- Author
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Rajendra Bhatia, Tanvi Jain, and Yongdo Lim
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,Positive-definite matrix ,Riemannian geometry ,01 natural sciences ,Manifold ,symbols.namesake ,Perspective (geometry) ,Fixed-point iteration ,Metric (mathematics) ,symbols ,Matrix analysis ,0101 mathematics ,Quantum information ,Mathematics - Abstract
The metric d ( A , B ) = tr A + tr B − 2 tr ( A 1 ∕ 2 B A 1 ∕ 2 ) 1 ∕ 2 1 ∕ 2 on the manifold of n × n positive definite matrices arises in various optimisation problems, in quantum information and in the theory of optimal transport. It is also related to Riemannian geometry. In the first part of this paper we study this metric from the perspective of matrix analysis, simplifying and unifying various proofs. Then we develop a theory of a mean of two, and a barycentre of several, positive definite matrices with respect to this metric. We explain some recent work on a fixed point iteration for computing this Wasserstein barycentre. Our emphasis is on ideas natural to matrix analysis.
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- 2019
46. On differentiability in the Wasserstein space and well-posedness for Hamilton–Jacobi equations
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Adrian Tudorascu and Wilfrid Gangbo
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Classical theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Hilbert space ,01 natural sciences ,Hamilton–Jacobi equation ,010101 applied mathematics ,symbols.namesake ,Probability space ,symbols ,Differentiable function ,0101 mathematics ,Hamiltonian (quantum mechanics) ,Random variable ,Well posedness ,Mathematics - Abstract
In this paper we elucidate the connection between various notions of differentiability in the Wasserstein space: some have been introduced intrinsically (in the Wasserstein space, by using typical objects from the theory of Optimal Transport) and used by various authors to study gradient flows, Hamiltonian flows, and Hamilton–Jacobi equations in this context. Another notion is extrinsic and arises from the identification of the Wasserstein space with the Hilbert space of square-integrable random variables on a non-atomic probability space. As a consequence, the classical theory of well-posedness for viscosity solutions for Hamilton–Jacobi equations in infinite-dimensional Hilbert spaces is brought to bear on well-posedness for Hamilton–Jacobi equations in the Wasserstein space.
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- 2019
47. Non-universality of the Riemann zeta function and its derivatives when σ≥1
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Takashi Nakamura and Hirofumi Nagoshi
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Numerical Analysis ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Universality theorem ,010103 numerical & computational mathematics ,01 natural sciences ,Riemann zeta function ,Bohr model ,Universality (dynamical systems) ,symbols.namesake ,symbols ,0101 mathematics ,Analysis ,Mathematics ,Mathematical physics - Abstract
Let ζ ( s ) be the Riemann zeta function. In 1911, Bohr showed that the set { ζ ( σ + i τ ) : σ > 1 , τ ∈ R } is dense in ℂ . By Voronin’s denseness theorems in 1972, the sets { ( ζ ( σ + i λ 1 + i τ ) , … , ζ ( σ + i λ n + i τ ) ) : σ ≥ 1 , τ ∈ R } with distinct λ 1 , … , λ n ∈ R and { ( ζ ( σ + i τ ) , ζ ′ ( σ + i τ ) , … , ζ ( n − 1 ) ( σ + i τ ) ) : σ ≥ 1 , τ ∈ R } are dense in ℂ n . By Voronin’s universality theorem, for any fixed 1 ∕ 2 σ 1 and any non-negative integer k , the set { ζ σ , τ ( k ) : τ ∈ R } is dense in C [ a , b ] , where ζ σ , τ ( k ) ( t ) ≔ ζ ( k ) ( σ + i t + i τ ) , t ∈ [ a , b ] . In the present paper, we prove that the set { ζ σ , τ ( k ) : σ ≥ 1 , τ ∈ R } ∩ C [ a , b ] is not dense in C [ a , b ] .
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- 2019
48. Uniqueness of Coxeter structures on Kac–Moody algebras
- Author
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Valerio Toledano Laredo and Andrea Appel
- Subjects
Pure mathematics ,Lie bialgebra ,General Mathematics ,Braid group ,Category O ,01 natural sciences ,symbols.namesake ,Mathematics::Category Theory ,Mathematics::Quantum Algebra ,Mathematics - Quantum Algebra ,0103 physical sciences ,FOS: Mathematics ,Quantum Algebra (math.QA) ,Category Theory (math.CT) ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics::Representation Theory ,Mathematics ,Discrete mathematics ,Weyl group ,Functor ,Quantum group ,010102 general mathematics ,Coxeter group ,Mathematics - Category Theory ,Monodromy ,symbols ,010307 mathematical physics ,Mathematics - Representation Theory - Abstract
Let g be a symmetrisable Kac-Moody algebra, and U_h(g) the corresponding quantum group. We showed in arXiv:1610.09744 and arXiv:1610.09741 that the braided quasi-Coxeter structure on integrable, category O representations of U_h(g) which underlies the R-matrix actions arising from the Levi subalgebras of U_h(g) and the quantum Weyl group action of the generalised braid group B_g can be transferred to integrable, category O representations of g. We prove in this paper that, up to unique equivalence, there is a unique such structure on the latter category with prescribed restriction functors, R--matrices, and local monodromies. This extends, simplifies and strengthens a similar result of the second author valid when g is semisimple, and is used in arXiv:1512.03041 to describe the monodromy of the rational Casimir connection of g in terms of the quantum Weyl group operators of U_h(g). Our main tool is a refinement of Enriquez's universal algebras, which is adapted to the PROP describing a Lie bialgebra graded by the non-negative roots of g., Expanded Introduction and Sec. 5 to discuss convolution product (5.11), cosimplicial structure on basis elements (5.13) and module structure on coinvariants (5.15). Minor revisions in Sec. 7.1 (gradings), 7.4 (deformation DY modules), 9.7 (exposition), 15.7 (Drinfeld double) and 15.15 (rigidity for diagrammatic KM algebras). Final version, to appear in Adv. Math. 81 pages
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- 2019
49. Arithmetic Siegel–Weil formula on X0(N)
- Author
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Tonghai Yang and Tuoping Du
- Subjects
Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,Dirac delta function ,Theta function ,Square-free integer ,Kronecker limit formula ,01 natural sciences ,Modular curve ,symbols.namesake ,0103 physical sciences ,Eisenstein series ,symbols ,010307 mathematical physics ,0101 mathematics ,Arithmetic ,Mathematics - Abstract
In this paper, we proved an arithmetic Siegel–Weil formula and the modularity of some arithmetic theta function on the modular curve X 0 ( N ) when N is square free. In the process, we also constructed some generalized Delta function for Γ 0 ( N ) and proved some explicit Kronecker limit formula for Eisenstein series on X 0 ( N ) .
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- 2019
50. Exceptional Jacobi polynomials
- Author
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Niels Bonneux
- Subjects
Numerical Analysis ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,symbols.namesake ,Mathematics - Classical Analysis and ODEs ,TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,symbols ,Jacobi polynomials ,0101 mathematics ,Analysis ,Mathematics - Abstract
In this paper we present a systematic way to describe exceptional Jacobi polynomials via two partitions. We give the construction of these polynomials and restate the known aspects of these polynomials in terms of their partitions. The aim is to show that the use of partitions is an elegant way to label these polynomials. Moreover, we prove asymptotic results according to the regular and exceptional zeros of these polynomials., 40 pages, 1 figure
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- 2019
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