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108
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 34 (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
Enhanced Group Analysis and Exact Solutions of Variable Coefficient Semilinear Diffusion Equations with a Power Source
, 708
"... A new approach to group classification problems and more general investigations on transformational properties of classes of differential equations is proposed. It is based on mappings between classes of differential equations, generated by families of point transformations. A class of variable coef ..."
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Cited by 26 (10 self)
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A new approach to group classification problems and more general investigations on transformational properties of classes of differential equations is proposed. It is based on mappings between classes of differential equations, generated by families of point transformations. A class of variable coefficient (1+1)dimensional semilinear reaction–diffusion equations of the general form f(x)ut = (g(x)ux)x + h(x)u m (m ̸ = 0, 1) is studied from the symmetry point of view in the framework of the approach proposed. The singular subclass of the equations with m = 2 is singled out. The group classifications of the entire class, the singular subclass and their images are performed with respect to both the corresponding (generalized extended) equivalence groups and all point transformations. The set of admissible transformations of the imaged class is exhaustively described in the general case m ̸ = 2. The procedure of classification of nonclassical symmetries, which involves mappings between classes of differential equations, is discussed. Wide families of new exact solutions are also constructed for equations from the classes under consideration by the classical method of Lie reductions and by generation of new solutions from known ones for other equations with point transformations of different kinds (such as additional equivalence transformations and mappings between classes of equations). 1
CYCLE INTEGRALS OF THE JFUNCTION AND MOCK MODULAR FORMS
"... In this paper we construct certain mock modular forms of weight 1/2 whose Fourier coefficients are cycle integrals of the modular jfunction and whose shadows are weakly holomorphic forms of weight 3/2. As an application we construct through a Shimuratype lift a holomorphic function that transforms ..."
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Cited by 23 (5 self)
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In this paper we construct certain mock modular forms of weight 1/2 whose Fourier coefficients are cycle integrals of the modular jfunction and whose shadows are weakly holomorphic forms of weight 3/2. As an application we construct through a Shimuratype lift a holomorphic function that transforms with a rational period function having poles at certain real quadratic integers. This function yields a real quadratic analogue of the Borcherds product.
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 22 (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.
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 22 (8 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)...
CONJUGATE AND CUT TIME IN THE SUBRIEMANNIAN PROBLEM ON THE GROUP OF MOTIONS OF A PLANE
, 2009
"... The leftinvariant subRiemannian problem on the group of motions (rototranslations) of a plane SE(2) is studied. Local and global optimality of extremal trajectories is characterized. Lower and upper bounds on the first conjugate time are proved. The cut time is shown to be equal to the first Max ..."
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Cited by 22 (9 self)
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The leftinvariant subRiemannian problem on the group of motions (rototranslations) of a plane SE(2) is studied. Local and global optimality of extremal trajectories is characterized. Lower and upper bounds on the first conjugate time are proved. The cut time is shown to be equal to the first Maxwell time corresponding to the group of discrete symmetries of the exponential mapping. Optimal synthesis on an open dense subset of the state space is described.
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 20 (3 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.
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 16 (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.
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 par ..."
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Cited by 16 (8 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
The extremal solution of a boundary reaction problem
 Commun. Pure Appl. Anal
"... abstract. We consider ∆u = 0 in Ω, ∂u ∂ν = λf(u) on Γ1, u = 0 on Γ2 where λ> 0, f(u) = eu or f(u) = (1 + u)p and Γ1, Γ2 is a partition of ∂Ω and Ω ⊂ RN. We determine sharp conditions on the dimension N and p> 1 such that the extremal solution is bounded, where the extremal solution refers to ..."
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Cited by 15 (4 self)
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abstract. We consider ∆u = 0 in Ω, ∂u ∂ν = λf(u) on Γ1, u = 0 on Γ2 where λ> 0, f(u) = eu or f(u) = (1 + u)p and Γ1, Γ2 is a partition of ∂Ω and Ω ⊂ RN. We determine sharp conditions on the dimension N and p> 1 such that the extremal solution is bounded, where the extremal solution refers to the one associated to the largest λ for which a solution exists.