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79
The Wilson function transform
 Int. Math. Res. Not. 2003
"... Abstract. Two unitary integral transforms with a verywell poised 7F6function as a kernel are given. For both integral transforms the inverse is the same as the original transform after an involution on the parameters. The 7F6function involved can be considered as a nonpolynomial extension of the ..."
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Cited by 8 (2 self)
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Abstract. Two unitary integral transforms with a verywell poised 7F6function as a kernel are given. For both integral transforms the inverse is the same as the original transform after an involution on the parameters. The 7F6function involved can be considered as a nonpolynomial extension of the Wilson polynomial, and is therefore called a Wilson function. The two integral transforms are called a Wilson function transform of type I and type II. Furthermore, a few explicit transformations of hypergeometric functions are calculated, and it is shown that the Wilson function transform of type I maps a basis of orthogonal polynomials onto a similar basis of polynomials. 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 8 (4 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.
Multiple Gamma Function and Its Application to Computation of Series, preprint
, 2003
"... Abstract. The multiple gamma function Γn, defined by a recurrencefunctional equation as a generalization of the Euler gamma function, was originally introduced by Kinkelin, Glaisher, and Barnes around 1900. Today, due to the pioneer work of Conrey, Katz and Sarnak, interest in the multiple gamma fu ..."
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Cited by 6 (1 self)
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Abstract. The multiple gamma function Γn, defined by a recurrencefunctional equation as a generalization of the Euler gamma function, was originally introduced by Kinkelin, Glaisher, and Barnes around 1900. Today, due to the pioneer work of Conrey, Katz and Sarnak, interest in the multiple gamma function has been revived. This paper discusses some theoretical aspects of the Γn function and their applications to summation of series and infinite products.
Asymptotic variance of random symmetric digital search trees
, 2009
"... Asymptotics of the variances of many cost measures in random digital search trees are often notoriously messy and involved to obtain. A new approach is proposed to facilitate such an analysis for several shape parameters on random symmetric digital search trees. Our approach starts from a more caref ..."
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Cited by 6 (5 self)
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Asymptotics of the variances of many cost measures in random digital search trees are often notoriously messy and involved to obtain. A new approach is proposed to facilitate such an analysis for several shape parameters on random symmetric digital search trees. Our approach starts from a more careful normalization at the level of Poisson generating functions, which then provides an asymptotically equivalent approximation to the variance in question. Several new ingredients are also introduced such as a combined use of the Laplace and Mellin transforms and a simple, mechanical technique for justifying the analytic dePoissonization procedures involved. The methodology we develop can be easily adapted to many other problems with an underlying binomial distribution. In particular, the less expected and somewhat surprising n(log n) 2variance for certain notions of total pathlength is also clarified.
THE SHAPE OF UNLABELED ROOTED RANDOM TREES
"... Abstract. We consider the number of nodes in the levels of unlabeled rooted random trees and show that the stochastic process given by the properly scaled level sizes weakly converges to the local time of a standard Brownian excursion. Furthermore we compute the average and the distribution of the h ..."
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Cited by 6 (3 self)
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Abstract. We consider the number of nodes in the levels of unlabeled rooted random trees and show that the stochastic process given by the properly scaled level sizes weakly converges to the local time of a standard Brownian excursion. Furthermore we compute the average and the distribution of the height of such trees. These results extend existing results for conditioned GaltonWatson trees and forests to the case of unlabeled rooted trees and show that they behave in this respect essentially like a conditioned GaltonWatson process. 1.
Finitegap solutions of the Fuchsian equation
, 2003
"... Abstract. We find a new class of the Fuchsian equations, which have an algebraic geometric solutions with the parameter belonging to a hyperelliptic curve. Methods of calculating the algebraic genus of the curve, and its branching points, are suggested. Numerous examples are given. ..."
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Cited by 5 (0 self)
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Abstract. We find a new class of the Fuchsian equations, which have an algebraic geometric solutions with the parameter belonging to a hyperelliptic curve. Methods of calculating the algebraic genus of the curve, and its branching points, are suggested. Numerous examples are given.
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 5 (4 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.
Selection rules for periodic orbits and scaling laws for a driven damped quartic oscillator, Nonlinear Anal
"... Abstract. In this paper we investigate the conditions under which periodic solutions of the nonlinear oscillator ¨x + x 3 = 0 persist when the differential equation is perturbed so as to become ¨x + x 3 + εx 3 cos t + γ ˙x = 0. For any frequency ω, there exists a threshold for the damping coefficien ..."
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Cited by 5 (5 self)
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Abstract. In this paper we investigate the conditions under which periodic solutions of the nonlinear oscillator ¨x + x 3 = 0 persist when the differential equation is perturbed so as to become ¨x + x 3 + εx 3 cos t + γ ˙x = 0. For any frequency ω, there exists a threshold for the damping coefficient γ, above which there is no periodic orbit with period 2π/ω. We conjecture that this threshold is infinitesimal in the perturbation parameter, with integer order depending on the frequency ω. Some rigorous analytical results toward the proof of this conjecture are given: these results would provide a complete proof if we could rule out the possibility that other periodic solutions arise besides subharmonic solutions. Moreover the relative size and shape of the basins of attraction of the existing stable periodic orbits are investigated numerically, showing that all attractors different from the origin are subharmonic solutions and hence giving further support to the validity of the conjecture. The method we use is different from those usually applied in bifurcation theory, such as Mel ′ nikov’s method or that of Chow and Hale’s, and allows us to investigate situations in which the nondegeneracy assumptions on the perturbation are violated. 1.