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2. Corrigendum to the papers on Exceptional orthogonal polynomials: J. Approx. Theory 182 (2014) 29–58, 184 (2014) 176–208 and 214 (2017) 9–48
- Author
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Antonio J. Durán
- Subjects
Numerical Analysis ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Hilbert space ,Approx ,symbols.namesake ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Orthogonal polynomials ,symbols ,Analysis ,Mathematics - Abstract
We complete a gap in the proof that exceptional polynomials are complete orthogonal systems in the associated Hilbert spaces.
- Published
- 2020
3. On Helmholtz' Paper 'Ueber die thatsächlichen Grundlagen der Geometrie'
- Author
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Klaus Volkert
- Subjects
symbols.namesake ,History ,Mathematics(all) ,General Mathematics ,Helmholtz free energy ,Calculus ,symbols ,foundations of geometry ,Foundations of geometry ,Helmholtz ,Computer Science::Digital Libraries ,Mathematics ,Mathematics::Numerical Analysis - Abstract
The date of publication of Helmholtz's first paper on the foundations of geometry is discussed.
- Published
- 1993
- Full Text
- View/download PDF
4. Euler's 1760 paper on divergent series
- Author
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P.J. Leah and Edward J. Barbeau
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History ,Mathematics(all) ,General Mathematics ,Opera ,Divergent series ,Commentarii ,Algebra ,symbols.namesake ,Bibliography ,Euler's formula ,symbols ,Calculus ,Remainder ,Hypergeometric function ,Mathematics ,Exposition (narrative) - Abstract
That Euler was quite aware of the subtleties of assigning a sum to a divergent series is amply demonstrated in his paper De seriebus divergentibus which appeared in Novi commentarii academiae scientiarum Petropolitanae 5 (1754/55), 205–237 (= Opera Omnia (1) 14, 585–617) in the year 1760. The first half of this paper contains a detailed exposition of Euler's views which should be more readily accessible to the mathematical community.The authors present here a translation from Latin of the summary and first twelve sections of Euler's paper with some explanatory comments. The remainder of the paper, treating Wallis' hypergeometric series and other technical matter, is described briefly. Appended is a short bibliography of works concerning Euler which are available to the English-speaking reader.
- Published
- 1976
- Full Text
- View/download PDF
5. On the paper 'A ‘lost’ notebook of Ramanujan'
- Author
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R.P Agarwal
- Subjects
Algebra ,symbols.namesake ,Mathematics(all) ,General Mathematics ,symbols ,Ramanujan's sum ,Mathematics - Published
- 1984
- Full Text
- View/download PDF
6. Markov processes and related problems of analysis (selected papers)
- Author
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Gian-Carlo Rota
- Subjects
Discrete mathematics ,symbols.namesake ,Mathematics(all) ,General Mathematics ,symbols ,Markov process ,Mathematics - Published
- 1985
- Full Text
- View/download PDF
7. 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
8. 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
9. 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)
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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
10. 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
11. 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
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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
12. Reproducing kernel orthogonal polynomials on the multinomial distribution
- Author
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Robert C. Griffiths and Persi Diaconis
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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
13. Balanced derivatives, identities, and bounds for trigonometric and Bessel series
- Author
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Bruce C. Berndt, Sun Kim, Martino Fassina, and Alexandru Zaharescu
- Subjects
symbols.namesake ,Pure mathematics ,Series (mathematics) ,General Mathematics ,symbols ,Trigonometric functions ,Divisor (algebraic geometry) ,Trigonometry ,Upper and lower bounds ,Bessel function ,Gauss circle problem ,Ramanujan's sum ,Mathematics - Abstract
Motivated by two identities published with Ramanujan's lost notebook and connected, respectively, with the Gauss circle problem and the Dirichlet divisor problem, in an earlier paper, three of the present authors derived representations for certain sums of products of trigonometric functions as double series of Bessel functions [8] . These series are generalized in the present paper by introducing the novel notion of balanced derivatives, leading to further theorems. As we will see below, the regions of convergence in the unbalanced case are entirely different than those in the balanced case. From this viewpoint, it is remarkable that Ramanujan had the intuition to formulate entries that are, in our new terminology, “balanced”. If x denotes the number of products of the trigonometric functions appearing in our sums, in addition to proving the identities mentioned above, theorems and conjectures for upper and lower bounds for the sums as x → ∞ are established.
- Published
- 2022
14. Transfer operators and Hankel transforms between relative trace formulas, II: Rankin–Selberg theory
- Author
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Yiannis Sakellaridis
- Subjects
Transfer (group theory) ,Pure mathematics ,Hecke algebra ,symbols.namesake ,Conjecture ,Trace (linear algebra) ,General Mathematics ,Poisson summation formula ,symbols ,Functional equation (L-function) ,Abelian group ,Fundamental lemma ,Mathematics - Abstract
The Langlands functoriality conjecture, as reformulated in the “beyond endoscopy” program, predicts comparisons between the (stable) trace formulas of different groups G 1 , G 2 for every morphism G 1 L → L G 2 between their L-groups. This conjecture can be seen as a special case of a more general conjecture, which replaces reductive groups by spherical varieties and the trace formula by its generalization, the relative trace formula. The goal of this article and its precursor [11] is to demonstrate, by example, the existence of “transfer operators” between relative trace formulas, which generalize the scalar transfer factors of endoscopy. These transfer operators have all properties that one could expect from a trace formula comparison: matching, fundamental lemma for the Hecke algebra, transfer of (relative) characters. Most importantly, and quite surprisingly, they appear to be of abelian nature (at least, in the low-rank examples considered in this paper), even though they encompass functoriality relations of non-abelian harmonic analysis. Thus, they are amenable to application of the Poisson summation formula in order to perform the global comparison. Moreover, we show that these abelian transforms have some structure — which presently escapes our understanding in its entirety — as deformations of well-understood operators when the spaces under consideration are replaced by their “asymptotic cones”. In this second paper we use Rankin–Selberg theory to prove the local transfer behind Rudnick's 1990 thesis (comparing the stable trace formula for SL 2 with the Kuznetsov formula) and Venkatesh's 2002 thesis (providing a “beyond endoscopy” proof of functorial transfer from tori to GL 2 ). As it turns out, the latter is not completely disjoint from endoscopic transfer — in fact, our proof “factors” through endoscopic transfer. We also study the functional equation of the symmetric-square L-function for GL 2 , and show that it is governed by an explicit “Hankel operator” at the level of the Kuznetsov formula, which is also of abelian nature. A similar theory for the standard L-function was previously developed (in a different language) by Jacquet.
- Published
- 2022
15. 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
16. 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
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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
17. 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
18. 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
19. 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
20. 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
21. 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
22. 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
23. 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
24. 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
25. 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
26. 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
27. 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
- Full Text
- View/download PDF
28. 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
29. 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
30. On the second inner variations of Allen–Cahn type energies and applications to local minimizers
- Author
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Nam Q. Le
- Subjects
Work (thermodynamics) ,Laplace transform ,Applied Mathematics ,General Mathematics ,Poincaré inequality ,Type (model theory) ,Term (time) ,Constraint (information theory) ,symbols.namesake ,Identity (mathematics) ,symbols ,Limit (mathematics) ,Mathematics ,Mathematical physics - Abstract
In this paper, we obtain an explicit formula for the discrepancy between the limit of the second inner variations of p -Laplace Allen–Cahn energies and the second inner variation of their Γ -limit which is the area functional. Our analysis explains the mysterious discrepancy term found in our previous paper [8] in the case p = 2 . The discrepancy term turns out to be related to the convergence of certain 4-tensors which are absent in the usual Allen–Cahn functional. These (hidden) 4-tensors suggest that, in the complex-valued Ginzburg–Landau setting, we should expect a different discrepancy term which we are able to identify. Along the way, we partially answer a question of Kohn and Sternberg [6] by giving a relation between the limit of second variations of the Allen–Cahn functional and the second inner variation of the area functional at local minimizers. Moreover, our analysis reveals an interesting identity connecting second inner variation and Poincare inequality for area-minimizing surfaces with volume constraint in the work of Sternberg and Zumbrun [16] .
- Published
- 2015
31. Theb-adic tent transformation for quasi-Monte Carlo integration using digital nets
- Author
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Takashi Goda, Takehito Yoshiki, and Kosuke Suzuki
- Subjects
Discrete mathematics ,Numerical Analysis ,Polynomial ,Kernel (set theory) ,Applied Mathematics ,General Mathematics ,Lattice (group) ,Hilbert space ,Numerical Analysis (math.NA) ,Prime (order theory) ,Sobolev space ,symbols.namesake ,Rate of convergence ,FOS: Mathematics ,symbols ,Mathematics - Numerical Analysis ,Quasi-Monte Carlo method ,Analysis ,Mathematics - Abstract
In this paper we investigate quasi-Monte Carlo (QMC) integration using digital nets over Z b in reproducing kernel Hilbert spaces. The tent transformation (previously called baker’s transform) was originally used for lattice rules by Hickernell (2002) to achieve higher order convergence of the integration error for smooth non-periodic integrands, and later, has been successfully applied to digital nets over Z 2 by Cristea et al. (2007) and Goda (2015). The aim of this paper is to generalize the latter two results to digital nets over Z b for an arbitrary prime b . For this purpose, we introduce the b -adic tent transformation for an arbitrary positive integer b greater than 1, which is a generalization of the original (dyadic) tent transformation. Further, again for an arbitrary positive integer b greater than 1, we analyze the mean square worst-case error of QMC rules using digital nets over Z b which are randomly digitally shifted and then folded using the b -adic tent transformation in reproducing kernel Hilbert spaces. Using this result, for a prime b , we prove the existence of good higher order polynomial lattice rules over Z b among a smaller number of candidates as compared to the result by Dick and Pillichshammer (2007), which achieve almost the optimal convergence rate of the mean square worst-case error in unanchored Sobolev spaces of smoothness of arbitrary high order.
- Published
- 2015
32. 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
33. 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
34. Noncommutative geometry and conformal geometry. III. Vafa–Witten inequality and Poincaré duality
- Author
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Raphael Ponge and Hang Wang
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,General Mathematics ,Mathematics - Operator Algebras ,Conformal map ,Noncommutative geometry ,Algebra ,symbols.namesake ,Mathematics::K-Theory and Homology ,Conformal symmetry ,symbols ,Noncommutative algebraic geometry ,Noncommutative quantum field theory ,Mathematics::Symplectic Geometry ,Spectral triple ,Conformal geometry ,Poincaré duality ,Mathematics - Abstract
This paper is the the third part of a series of paper whose aim is to use of the framework of \emph{twisted spectral triples} to study conformal geometry from a noncommutive geometric viewpoint. In this paper we reformulate the inequality of Vafa-Witten \cite{VW:CMP84} in the setting of twisted spectral triples. This involves a notion of Poincar\'e duality for twisted spectral triples. Our main results have various consequences. In particular, we obtain a version in conformal geometry of the original inequality of Vafa-Witten, in the sense of an explicit control of the Vafa-Witten bound under conformal changes of metric. This result has several noncommutative manifestations for conformal deformations of ordinary spectral triples, spectral triples associated to conformal weights on noncommutative tori, and spectral triples associated to duals of torsion-free discrete cocompact subgroups satisfying the Baum-Connes conjecture., Comment: Final version. 38 pages
- Published
- 2015
35. Inner product on B∗-algebras of operators on a free Banach space over the Levi-Civita field
- Author
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José Aguayo, Miguel Nova, and Khodr Shamseddine
- Subjects
Discrete mathematics ,Pure mathematics ,Approximation property ,Nuclear operator ,General Mathematics ,Hilbert space ,Spectral theorem ,Operator theory ,Compact operator ,Compact operator on Hilbert space ,symbols.namesake ,symbols ,Operator norm ,Mathematics - Abstract
Let C be the complex Levi-Civita field and let c 0 ( C ) or, simply, c 0 denote the space of all null sequences z = ( z n ) n ∈ N of elements of C . The natural inner product on c 0 induces the sup-norm of c 0 . In a previous paper Aguayo et al. (2013), we presented characterizations of normal projections, adjoint operators and compact operators on c 0 . In this paper, we work on some B ∗ -algebras of operators, including those mentioned above; then we define an inner product on such algebras and prove that this inner product induces the usual norm of operators. We finish the paper with a characterization of closed subspaces of the B ∗ -algebra of all adjoint and compact operators on c 0 which admit normal complements.
- Published
- 2015
36. Traces on operator ideals and related linear forms on sequence ideals (part I)
- Author
-
Albrecht Pietsch
- Subjects
Discrete mathematics ,Pure mathematics ,Ideal (set theory) ,Trace (linear algebra) ,Group (mathematics) ,General Mathematics ,Hilbert space ,Separable space ,symbols.namesake ,Fractional ideal ,symbols ,Commutative algebra ,Invariant (mathematics) ,Mathematics - Abstract
The Calkin theorem provides a one-to-one correspondence between all operator ideals A(H) over the separable infinite-dimensional Hilbert space H and all symmetric sequence ideals a(N) over the index set N≔{1,2,…}. The main idea of the present paper is to replace a(N) by the ideal z(N0) that consists of all sequences (αh) indexed by N0≔{0,1,2,…} for which (α0,α1,α1,…,αh,…,αh︷2hterms,…)∈a(N). This new kind of sequence ideals is characterized by two properties: (1) For (αh)∈z(N0) there is a non-increasing (βh)∈z(N0) such that ∣αh∣≤βh. (2) z(N0) is invariant under the operator S+:(α0,α1,α2,…)↦(0,α0,α1,…). Using this modification of the Calkin theorem, we simplify, unify, and complete earlier results of [4,5,7–9,13,14,19–21,25] The central theorem says that there are canonical isomorphisms between the linear spaces of all traces on A(H), all symmetric linear forms on a(N), and all 12S+-invariant linear forms on z(N0). In this way, the theory of linear forms on ideals of a non-commutative algebra that are invariant under the members of a non-commutative group is reduced to the theory of linear forms on ideals of a commutative algebra that are invariant under a single operator. It is hoped that the present approach deserves the rating “streamlined”. Our main objects are linear forms in the purely algebraic sense. Only at the end of this paper continuity comes into play, when the case of quasi-normed ideals is considered. We also sketch a classification of operator ideals according to the existence of various kinds of traces. Details will be discussed in a subsequent publication.
- Published
- 2014
37. Implicit iterative method for approximating a common solution of split equilibrium problem and fixed point problem for a nonexpansive semigroup
- Author
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S. H. Rizvi and K.R. Kazmi
- Subjects
Discrete mathematics ,Nonexpansive semigroup ,Semigroup ,Iterative method ,General Mathematics ,Minimization problem ,Secondary 47J25 65J15 90C33 ,Hilbert space ,Averaged mapping ,Fixed point ,symbols.namesake ,Fixed point problem ,Implicit iterative method ,Variational inequality ,QA1-939 ,symbols ,Primary 65K15 ,Fixed-point problem ,Applied mathematics ,Equilibrium problem ,Split equilibrium problem ,Mathematics - Abstract
In this paper, we introduce and study an implicit iterative method to approximate a common solution of split equilibrium problem and fixed point problem for a nonexpansive semigroup in real Hilbert spaces. Further, we prove that the nets generated by the implicit iterative method converge strongly to the common solution of split equilibrium problem and fixed point problem for a nonexpansive semigroup. This common solution is the unique solution of a variational inequality problem and is the optimality condition for a minimization problem. Furthermore, we justify our main result through a numerical example. The results presented in this paper extend and generalize the corresponding results given by Plubtieng and Punpaeng [S. Plubtieng, R. Punpaeng, Fixed point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces, Math. Comput. Model. 48 (2008) 279–286] and Cianciaruso et al. [F. Cianciaruso, G. Marino, L. Muglia, Iterative methods for equilibrium and fixed point problems for nonexpansive semigroups in Hilbert space, J. Optim. Theory Appl. 146 (2010) 491–509].
- Published
- 2014
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38. On some geometric properties of generalized Orlicz–Lorentz sequence spaces
- Author
-
Paweł Foralewski
- Subjects
Sequence ,Pure mathematics ,Mathematics(all) ,General Mathematics ,Lorentz transformation ,Mathematical analysis ,Monotonic function ,Function (mathematics) ,Type (model theory) ,Space (mathematics) ,Linear subspace ,symbols.namesake ,symbols ,Order (group theory) ,Mathematics - Abstract
In this paper, we continue investigations concerning generalized Orlicz–Lorentz sequence spaces λ φ initiated in the papers of Foralewski et al. (2008) [16] , [17] (cf. also Foralewski (2011) [11] , [12] ). As we will show in Example 1.1 , Example 1.2 , Example 1.3 the class of generalized Orlicz–Lorentz sequence spaces is much more wider than the class of classical Orlicz–Lorentz sequence spaces. Moreover, it is shown that if a Musielak–Orlicz function φ satisfies condition δ 2 λ , then λ φ has the coordinatewise Kadec–Klee property. Next, monotonicity properties are considered. In order to get sufficient conditions for uniform monotonicity of the space λ φ , a strong condition of δ 2 type and the notion of regularity of function φ are introduced. Finally, criteria for non-squareness of λ φ , of their subspaces of order continuous elements ( λ φ ) a as well as of finite dimensional subspaces λ φ n of λ φ are presented.
- Published
- 2013
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- View/download PDF
39. The structure of AS-Gorenstein algebras
- Author
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Hiroyuki Minamoto and Izuru Mori
- Subjects
Mathematics(all) ,Pure mathematics ,Mathematics::Commutative Algebra ,Quantum group ,Graded Frobenius algebras ,General Mathematics ,Fano algebras ,Mathematics::Rings and Algebras ,Non-associative algebra ,Preprojective algebras ,Algebra ,Quadratic algebra ,Cayley–Dickson construction ,symbols.namesake ,Interior algebra ,Trivial extensions ,Frobenius algebra ,symbols ,Nest algebra ,CCR and CAR algebras ,AS-regular algebras ,Mathematics - Abstract
In this paper, we define a notion of AS-Gorenstein algebra for N -graded algebras, and show that symmetric AS-regular algebras of Gorenstein parameter 1 are exactly preprojective algebras of quasi-Fano algebras. This result can be compared with the fact that symmetric graded Frobenius algebras of Gorenstein parameter −1 are exactly trivial extensions of finite-dimensional algebras. The results of this paper suggest that there is a strong interaction between classification problems in noncommutative algebraic geometry and those in representation theory of finite-dimensional algebras.
- Published
- 2011
40. The semiclassical Sobolev orthogonal polynomials: A general approach
- Author
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Roberto S. Costas-Santos and Juan J. Moreno-Balcázar
- Subjects
33C45, 33D45, 42C05 ,Mathematics(all) ,nonstandard inner product ,Orthogonal polynomials ,General Mathematics ,Semiclassical orthogonal polynomials ,Classical orthogonal polynomials ,symbols.namesake ,operator theory ,Wilson polynomials ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Nonstandard inner product ,Mathematics ,Discrete mathematics ,Numerical Analysis ,Discrete orthogonal polynomials ,Applied Mathematics ,Biorthogonal polynomial ,Operator theory ,Sobolev orthogonal polynomials ,Difference polynomials ,Mathematics - Classical Analysis and ODEs ,Hahn polynomials ,semiclassical orthogonal polynomials ,symbols ,Jacobi polynomials ,Analysis - Abstract
We say that the polynomial sequence $(Q^{(\lambda)}_n)$ is a semiclassical Sobolev polynomial sequence when it is orthogonal with respect to the inner product $$ _S= +\lambda , $$ where ${\bf u}$ is a semiclassical linear functional, ${\mathscr D}$ is the differential, the difference or the $q$--difference operator, and $\lambda$ is a positive constant. In this paper we get algebraic and differential/difference properties for such polynomials as well as algebraic relations between them and the polynomial sequence orthogonal with respect to the semiclassical functional $\bf u$. The main goal of this article is to give a general approach to the study of the polynomials orthogonal with respect to the above nonstandard inner product regardless of the type of operator ${\mathscr D}$ considered. Finally, we illustrate our results by applying them to some known families of Sobolev orthogonal polynomials as well as to some new ones introduced in this paper for the first time., Comment: 23 pages, special issue dedicated to Professor Guillermo Lopez lagomasino on the occasion of his 60th birthday, accepted in Journal of Approximation Theory
- Published
- 2011
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41. Functional inequalities for modified Bessel functions
- Author
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Matti Vuorinen, Saminathan Ponnusamy, and Árpád Baricz
- Subjects
Mathematics(all) ,Distribution (number theory) ,General Mathematics ,Cumulative distribution function ,ta111 ,Mean value ,Functional inequalities ,Modified Bessel functions ,Geometrical convexity ,39B62, 33C10, 62H10 ,Mathematical proof ,symbols.namesake ,Mathematics - Classical Analysis and ODEs ,Gamma–gamma distribution ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Log-convexity ,symbols ,Applied mathematics ,Turán-type inequality ,Logarithmic derivative ,Convexity with respect to Hölder means ,Bessel function ,Mathematics - Abstract
In this paper our aim is to show some mean value inequalities for the modified Bessel functions of the first and second kinds. Our proofs are based on some bounds for the logarithmic derivatives of these functions, which are in fact equivalent to the corresponding Tur\'an type inequalities for these functions. As an application of the results concerning the modified Bessel function of the second kind we prove that the cumulative distribution function of the gamma-gamma distribution is log-concave. At the end of this paper several open problems are posed, which may be of interest for further research., Comment: 14 pages
- Published
- 2011
42. The Gierer–Meinhardt system on a compact two-dimensional Riemannian manifold: Interaction of Gaussian curvature and Green's function
- Author
-
Juncheng Wei, Wang Hung Tse, and Matthias Winter
- Subjects
Mathematics(all) ,Singular perturbation ,Riemannian manifold ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Thermal diffusivity ,symbols.namesake ,Mathematical biology ,Green's function ,Pattern formation ,symbols ,Gaussian curvature ,Convex combination ,Eigenvalues and eigenvectors ,Scalar curvature ,Mathematics - Abstract
In this paper, we rigorously prove the existence and stability of single-peaked patterns for the singularly perturbed Gierer–Meinhardt system on a compact two-dimensional Riemannian manifold without boundary which are far from spatial homogeneity. Throughout the paper we assume that the activator diffusivity ϵ 2 is small enough. We show that for the threshold ratio D ∼ 1 ϵ 2 of the activator diffusivity ϵ 2 and the inhibitor diffusivity D, the Gaussian curvature and the Green's function interact. A convex combination of the Gaussian curvature and the Green's function together with their derivatives are linked to the peak locations and the o ( 1 ) eigenvalues. A nonlocal eigenvalue problem (NLEP) determines the O ( 1 ) eigenvalues which all have negative part in this case.
- Published
- 2010
43. Conformal deformations of integral pinched 3-manifolds
- Author
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Zindine Djadli and Giovanni Catino
- Subjects
Conformal geometry ,Riemann curvature tensor ,Pure mathematics ,Mathematics(all) ,Curvature of Riemannian manifolds ,Fully nonlinear equation ,General Mathematics ,Prescribed scalar curvature problem ,Yamabe flow ,Mathematical analysis ,Curvature ,Geometry of 3-manifolds ,symbols.namesake ,Rigidity ,symbols ,Sectional curvature ,Mathematics::Differential Geometry ,Ricci curvature ,Scalar curvature ,Mathematics - Abstract
In this paper we prove that, under an explicit integral pinching assumption between the L 2 -norm of the Ricci curvature and the L 2 -norm of the scalar curvature, a closed 3-manifold with positive scalar curvature admits a conformal metric of positive Ricci curvature. In particular, using a result of Hamilton, this implies that the manifold is diffeomorphic to a quotient of S 3 . The proof of the main result of the paper is based on ideas developed in an article by M. Gursky and J. Viaclovsky.
- Published
- 2010
- Full Text
- View/download PDF
44. Multi-wing hyperchaotic attractors from coupled Lorenz systems
- Author
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Giuseppe Grassi, D.A. Miller, Frank L. Severance, Grassi, Giuseppe, Frank L., Severance, and Damon A., Miller
- Subjects
Mathematics::Dynamical Systems ,Wing ,Similarity (geometry) ,General Mathematics ,Applied Mathematics ,Mathematical analysis ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,Lorenz system ,Nonlinear Sciences::Chaotic Dynamics ,symbols.namesake ,Coupling (physics) ,Jacobian matrix and determinant ,Homogeneous space ,Attractor ,symbols ,Chaos ,Astrophysics::Solar and Stellar Astrophysics ,Eigenvalues and eigenvectors ,Mathematics - Abstract
This paper illustrates an approach to generate multi-wing attractors in coupled Lorenz systems. In particular, novel four-wing (eight-wing) hyperchaotic attractors are generated by coupling two (three) identical Lorenz systems. The paper shows that the equilibria of the proposed systems have certain symmetries with respect to specific coordinate planes and the eigenvalues of the associated Jacobian matrices exhibit the property of similarity. In analogy with the original Lorenz system, where the two-wings of the butterfly attractor are located around the two equilibria with the unstable pair of complex-conjugate eigenvalues, this paper shows that the four-wings (eight-wings) of these attractors are located around the four (eight) equilibria with two (three) pairs of unstable complex-conjugate eigenvalues.
- Published
- 2009
45. Spatiotemporal structure of pulsating solitons in the cubic–quintic Ginzburg–Landau equation: A novel variational formulation
- Author
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Stefan C. Mancas and S. Roy Choudhury
- Subjects
Hopf bifurcation ,Integrable system ,General Mathematics ,Applied Mathematics ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,symbols.namesake ,Nonlinear system ,Dissipative soliton ,Classical mechanics ,Quasiperiodic function ,Attractor ,symbols ,Dissipative system ,Soliton ,Nonlinear Sciences::Pattern Formation and Solitons ,Mathematics - Abstract
Comprehensive numerical simulations (reviewed in Dissipative Solitons, Akhmediev and Ankiewicz (Eds.), Springer, Berlin, 2005) of pulse solutions of the cubic–quintic Ginzburg–Landau Equation (CGLE), a canonical equation governing the weakly nonlinear behavior of dissipative systems in a wide variety of disciplines, reveal various intriguing and entirely novel classes of solutions. In particular, there are five new classes of pulse or solitary waves solutions, viz. pulsating, creeping, snake, erupting, and chaotic solitons. In contrast to the regular solitary waves investigated in numerous integrable and non-integrable systems over the last three decades, these dissipative solitons are not stationary in time. Rather, they are spatially confined pulse-type structures whose envelopes exhibit complicated temporal dynamics. The numerical simulations also reveal very interesting bifurcations sequences of these pulses as the parameters of the CGLE are varied. In this paper, we address the issues of central interest in the area, i.e., the conditions for the occurrence of the five categories of dissipative solitons, as well the dependence of both their shape and their stability on the various parameters of the CGLE, viz. the nonlinearity, dispersion, linear and nonlinear gain, loss and spectral filtering parameters. Our predictions on the variation of the soliton amplitudes, widths and periods with the CGLE parameters agree with simulation results. First, we elucidate the Hopf bifurcation mechanism responsible for the various pulsating solitary waves, as well as its absence in Hamiltonian and integrable systems where such structures are absent. Next, we develop and discuss a variational formalism within which to explore the various classes of dissipative solitons. Given the complex dynamics of the various dissipative solutions, this formulation is, of necessity, significantly generalized over all earlier approaches in several crucial ways. Firstly, the starting formulation for the Lagrangian is recent and not well explored. Also, the trial functions have been generalized considerably over conventional ones to keep the shape relatively simple (and the trial function integrable!) while allowing arbitrary temporal variation of the amplitude, width, position, speed and phase of the pulses. In addition, the resulting Euler–Lagrange equations are treated in a completely novel way. Rather than consider the stable fixed points which correspond to the well-known stationary solitons or plain pulses, we use dynamical systems theory to focus on more complex attractors viz. periodic, quasiperiodic, and chaotic ones. Periodic evolution of the trial function parameters on stable periodic attractors yield solitons whose amplitudes and widths are non-stationary or time dependent. In particular, pulsating and snake dissipative solitons may be treated in this manner. Detailed results are presented here for the pulsating solitary waves – their regimes of occurrence, bifurcations, and the parameter dependences of the amplitudes, widths, and periods agree with simulation results. Snakes and chaotic solitons will be addressed in subsequent papers. This overall approach fails only to address the fifth class of dissipative solitons, viz. the exploding or erupting solitons.
- Published
- 2009
46. 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.
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- 2009
- Full Text
- View/download PDF
47. 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
48. Long time approximations for solutions of wave equations via standing waves from quasimodes
- Author
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Eugenia Pérez
- Subjects
Mathematics(all) ,General Mathematics ,Asymptotic analysis ,Applied Mathematics ,Mathematical analysis ,Hilbert space ,Standing waves ,Function (mathematics) ,Spectral analysis ,Eigenfunction ,Wave equation ,Compact operator ,symbols.namesake ,Operator (computer programming) ,symbols ,Quasimodes ,Linear combination ,Eigenvalues and eigenvectors ,Mathematics - Abstract
A quasimode for a positive, symmetric and compact operator on a Hilbert space could be defined as a pair ( u , λ ), where u is a function approaching a certain linear combination of eigenfunctions associated with the eigenvalues of the operator in a “small interval” [ λ − r , λ + r ] . Its value in describing asymptotics for low and high frequency vibrations in certain singularly perturbed spectral problems, which depend on a small parameter e, has been made clear recently in many papers. In this paper, considering second order evolution problems, we provide estimates for the time t in which standing waves of the type e i λ t u approach their solutions u ( t ) when the initial data deal with quasimodes of the associated operators. We establish a general abstract framework and we extended it to the case where operators and spaces depend on the small parameter e: now λ and u can depend on e and also perform the estimates for t. We apply the results to vibrating systems with concentrated masses.
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- 2008
- Full Text
- View/download PDF
49. Hadamard products for generalized Rogers–Ramanujan series
- Author
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Tim Huber
- Subjects
Pure mathematics ,Mathematics(all) ,Generalized Stieltjes–Wigert polynomials ,Mathematics::General Mathematics ,General Mathematics ,Mathematics::Number Theory ,Ramanujan's Eisenstein series ,Ramanujan's sum ,symbols.namesake ,Hadamard transform ,q-Bessel function ,Eisenstein series ,q-Airy function ,Mathematics ,Sequence ,Rogers–Ramanujan series ,Numerical Analysis ,Series (mathematics) ,Applied Mathematics ,Mathematics::History and Overview ,Zero (complex analysis) ,Hadamard products ,Algebra ,Product (mathematics) ,Orthogonal polynomials ,symbols ,Analysis - Abstract
The purpose of this paper is to derive product representations for generalizations of the Rogers–Ramanujan series. Special cases of the results presented here were first stated by Ramanujan in the “Lost Notebook” and proved by George Andrews. The analysis used in this paper is based upon the work of Andrews and the broad contributions made by Mourad Ismail and Walter Hayman. Each series considered is related to an extension of the Rogers–Ramanujan continued fraction and corresponds to an orthogonal polynomial sequence generalizing classical orthogonal sequences. Using Ramanujan's differential equations for Eisenstein series and corresponding analogues derived by V. Ramamani, the coefficients in the series representations of each zero are expressed in terms of certain Eisenstein series.
- Published
- 2008
- Full Text
- View/download PDF
50. Embedding Sn into Rn+1 with given integral Gauss curvature and optimal mass transport on Sn
- Author
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Vladimir Oliker
- Subjects
Pure mathematics ,Mathematics::Complex Variables ,Euclidean space ,General Mathematics ,Mathematical analysis ,Regular polygon ,Convex set ,Polytope ,symbols.namesake ,Polyhedron ,Hypersurface ,Convex polytope ,Gaussian curvature ,symbols ,Mathematics::Metric Geometry ,Mathematics - Abstract
In [A.D. Aleksandrov, Convex Polyhedra, GITTL, Moscow, USSR, 1950 (in Russian); English translation: A.D. Aleksandrov, Convex Polyhedra, Springer, Berlin-New York, 2005] A.D. Aleksandrov raised a general question of finding variational statements and proofs of existence of polytopes with given geometric data. The first goal of this paper is to give a variational solution to the problem of existence and uniqueness of a closed convex hypersurface in Euclidean space with prescribed integral Gauss curvature. Our solution includes the case of a convex polytope. This problem was also first considered by Aleksandrov and below it is referred to as Aleksandrov's problem. The second goal of this paper is to show that in variational form the Aleksandrov problem is closely connected with the theory of optimal mass transport on a sphere with cost function and constraints arising naturally from geometric considerations.
- Published
- 2007
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