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Large deviations of combinatorial distributions II: Local limit theorems
, 1997
"... This paper is a sequel to our paper [17] where we derived a general central limit theorem for probabilities of large deviations applicable to many classes of combinatorial structures and arithmetic functions; we consider corresponding local limit theorems in this paper. More precisely, given a seq ..."
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Cited by 32 (5 self)
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This paper is a sequel to our paper [17] where we derived a general central limit theorem for probabilities of large deviations applicable to many classes of combinatorial structures and arithmetic functions; we consider corresponding local limit theorems in this paper. More precisely, given a sequence of integral random variables n#1 each of maximal span 1 (see below for definition), we are interested in the asymptotic behavior of the probabilities n = m} (m N, m = n x n # n , n := n , # n := n ), ##, where x n can tend to with n at a rate that is restricted to O(# n ). Our interest here is not to derive asymptotic expression for n = m} valid for the widest possible range of m, but to show that for m lying in the interval n O(# n ), very precise asymptotic formulae can be obtained. These formulae are in close connection with our results in [17]. Although local limit theorems receive a constant research interest [2, 3, 7, 14, 13, 24], our approach and results, especially Theorem 1, seem rarely discussed in a systematic manner. Recall that a lattice random variable X is said to be of maximal span h if X takes only values of the form b + hk, k Z, for some constants b and h > 0; and there does not exist b # and h # > h such that X takes only values of the form b # + h # k
The Incomplete Gamma Functions Since Tricomi
 In Tricomi's Ideas and Contemporary Applied Mathematics, Atti dei Convegni Lincei, n. 147, Accademia Nazionale dei Lincei
, 1998
"... The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asy ..."
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Cited by 15 (1 self)
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The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asymptotic expansions, zeros, inequalities, computational methods, and applications.
Asymptotic Estimates of Elementary Probability Distributions
 Studies in Applied Mathematics
, 1996
"... Several new asymptotic estimates (with precise error bounds) are derived for Poisson and binomial distributions as the parameters tend to infinity. The analytic methods used are also applicable to other discrete distribution functions. ..."
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Cited by 12 (6 self)
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Several new asymptotic estimates (with precise error bounds) are derived for Poisson and binomial distributions as the parameters tend to infinity. The analytic methods used are also applicable to other discrete distribution functions.
Measures of Distinctness for Random Partitions and Compositions of an Integer
, 1997
"... This paper is concerned with problems of the following type: # Accepted for publication in Advances in Applied Mathematics. Given a random (under a suitable probability model) partition or composition, study quantitatively the measures of the degree of distinctness of its parts ..."
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Cited by 12 (3 self)
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This paper is concerned with problems of the following type: # Accepted for publication in Advances in Applied Mathematics. Given a random (under a suitable probability model) partition or composition, study quantitatively the measures of the degree of distinctness of its parts
strata in Euler’s elastic problem
 Journal of Dynamical and Control Systems
, 2007
"... Abstract. The classical Euler problem on stationary configurations of elastic rod in the plane is studied in detail by geometric control techniques as a leftinvariant optimal control problem on the group of motions of a twodimensional plane E(2). The attainable set is described, the existence and ..."
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Cited by 11 (2 self)
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Abstract. The classical Euler problem on stationary configurations of elastic rod in the plane is studied in detail by geometric control techniques as a leftinvariant optimal control problem on the group of motions of a twodimensional plane E(2). The attainable set is described, the existence and boundedness of optimal controls are proved. Extremals are parametrized by the Jacobi elliptic functions of natural coordinates induced by the flow of the mathematical pendulum on fibers of the cotangent bundle of E(2). The group of discrete symmetries of the Euler problem generated by reflections in the phase space of the pendulum is studied. The corresponding Maxwell points are completely described via the study of fixed points of this group. As a consequence, an upper bound on cut points in the Euler problem is obtained. 1.
Asymptotics Of Jack Polynomials As The Number Of Variables Goes To Infinity
 Math. Res. Notices
, 1998
"... In this paper we study the asymptotic behavior of the Jack rational functions P (z 1 ; : : : ; zn ; `) as the number of variables n and the signature grow to infinity. Our results generalize the results of A. Vershik and S. Kerov [VK2] obtained in the Schur function case (` = 1). For ` = 1=2; 2 ou ..."
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Cited by 11 (5 self)
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In this paper we study the asymptotic behavior of the Jack rational functions P (z 1 ; : : : ; zn ; `) as the number of variables n and the signature grow to infinity. Our results generalize the results of A. Vershik and S. Kerov [VK2] obtained in the Schur function case (` = 1). For ` = 1=2; 2 our results describe approximation of the spherical functions of the infinitedimensional symmetric spaces U(1)=O(1) and U(21)=Sp(1) by the spherical functions of the corresponding finitedimensional symmetric spaces. Contents 1.1. Statement of the main result 1.2. Regular and infinitesimally regular sequences 1.3. Extremality of the limit functions 1.4. Related results 1.5. Acknowledgments 2. Jack polynomials and shifted Jack polynomials 2.1. Orthogonality 2.2. Interpolation 2.3. Branching rules 2.4. Binomial formula 2.5. Generating functions 2.6. Partitions and signatures 2.7. Extended symmetric functions 3. Asymptotic properties of VershikKerov sequences of signatures 4. Sufficient conditions of regularity 5. Necessary conditions of regularity 5.1. The "only if " part of Theorem 1.1 5.2. A growth estimate for jf()j, f 2 7. Appendix. A direct proof of the formula (2.10) for generating functions The authors were supported by the Russian Basic Research Foundation grant 950100814. The first author's stay at IAS in Princeton and MSRI in Berkeley was supported by NSF grants DMS9304580 and DMS9022140 respectively. Typeset by A M ST E X 1 A. OKOUNKOV AND G. OLSHANSKI 1.1 Statement of the main result. Jack symmetric functions P (z 1 ; : : : ; z n ; `) 2 Q(`)[z \Sigma1 S(n) which are indexed by decreasing sequences of integers (called signatures) = ( 1 \Delta \Delta \Delta n ) 2 Z are eigenfunctions of the quantum CalogeroSutherland Hamiltonian [C,Su] (1.1)...
Resonance tongues and instability pockets in the quasiperiodic HillSchrödinger equation
"... This paper concerns Hill's equation with a (parametric) forcing that is real analytic and quasiperiodic with frequency vector! 2 Rd and a `frequency' (or `energy') parameter a and a small parameter b: The 1dimensional Schrödinger equation with quasiperiodic potential occurs as a particular case. I ..."
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Cited by 11 (5 self)
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This paper concerns Hill's equation with a (parametric) forcing that is real analytic and quasiperiodic with frequency vector! 2 Rd and a `frequency' (or `energy') parameter a and a small parameter b: The 1dimensional Schrödinger equation with quasiperiodic potential occurs as a particular case. In the parameter plane R² = {a; b}; for small values of b we show the following.The resonance `tongues' with rotation number 1 2 hk;!i; k 2 Zd have C1boundary curves. Our arguments are based on reducibility and certain properties of the Schrödinger operator with quasiperiodic potential. Analogous to the case of Hill's equation with periodic forcing (i.e., d = 1),several further results are obtained with respect to the geometry of the tongues. One result regards transversality of the boundaries at b = 0: Another result concerns the generic occurrence of instability pockets in the tongues in a reversible nearMathieu case, that may depend on several deformation parameters. These pockets describe the generic opening and closing behaviour of spectral gaps of the Schrödinger operator in dependence of the parameter b: This result uses a refined averaging technique. Also consequences are given for the behaviour of Lyapunov exponent
APPLICATIONS OF DIFFERENTIAL SUBORDINATION TO CERTAIN SUBCLASSES OF MEROMORPHICALLY MULTIVALENT FUNCTIONS
"... ABSTRACT. By making use of the principle of differential subordination, the authors investigate several inclusion relationships and other interesting properties of certain subclasses of meromorphically multivalent functions which are defined here by means of a linear operator. They also indicate rel ..."
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Cited by 10 (1 self)
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ABSTRACT. By making use of the principle of differential subordination, the authors investigate several inclusion relationships and other interesting properties of certain subclasses of meromorphically multivalent functions which are defined here by means of a linear operator. They also indicate relevant connections of the various results presented in this paper with those obtained in earlier works.
Root asymptotics of spectral polynomials for the Lamé operator
 Commun. Math. Phys
"... Abstract. The study of polynomial solutions to the classical Lamé equation in its algebraic form, or equivalently, of doubleperiodic solutions of its Weierstrass form has a long history. Such solutions appear at integer values of the spectral parameter and their respective eigenvalues serve as the ..."
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Cited by 9 (4 self)
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Abstract. The study of polynomial solutions to the classical Lamé equation in its algebraic form, or equivalently, of doubleperiodic solutions of its Weierstrass form has a long history. Such solutions appear at integer values of the spectral parameter and their respective eigenvalues serve as the ends of bands in the boundary value problem for the corresponding Schrödinger equation with finite gap potential given by the Weierstrass ℘function on the real line. In this paper we establish several natural (and equivalent) formulas in terms of hypergeometric and elliptic type integrals for the density of the appropriately scaled asymptotic distribution of these eigenvalues when the integervalued spectral parameter tends to infinity. We also show that this density satisfies a Heun differential equation with four singularities.
Conjugate points in Euler’s elastic problem
 Journal of Dynamical and Control Systems (accepted), available at: arXiv:0705.1003
"... Abstract. For the classical Euler elastic problem, conjugate points are described. Inflexional elasticas admit the first conjugate point between the first and third inflexion points. All other elasticas do not have conjugate points. As a result, the problem of stability of Euler elasticas is solved. ..."
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Cited by 8 (4 self)
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Abstract. For the classical Euler elastic problem, conjugate points are described. Inflexional elasticas admit the first conjugate point between the first and third inflexion points. All other elasticas do not have conjugate points. As a result, the problem of stability of Euler elasticas is solved. 1.