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16
Random Mapping Statistics
 IN ADVANCES IN CRYPTOLOGY
, 1990
"... Random mappings from a finite set into itself are either a heuristic or an exact model for a variety of applications in random number generation, computational number theory, cryptography, and the analysis of algorithms at large. This paper introduces a general framework in which the analysis of ..."
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Cited by 77 (6 self)
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Random mappings from a finite set into itself are either a heuristic or an exact model for a variety of applications in random number generation, computational number theory, cryptography, and the analysis of algorithms at large. This paper introduces a general framework in which the analysis of about twenty characteristic parameters of random mappings is carried out: These parameters are studied systematically through the use of generating functions and singularity analysis. In particular, an open problem of Knuth is solved, namely that of finding the expected diameter of a random mapping. The same approach is applicable to a larger class of discrete combinatorial models and possibilities of automated analysis using symbolic manipulation systems ("computer algebra") are also briefly discussed.
Varieties of Increasing Trees
, 1992
"... An increasing tree is a labelled rooted tree in which labels along any branch from the root go in increasing order. Under various guises, such trees have surfaced as tree representations of permutations, as data structures in computer science, and as probabilistic models in diverse applications. We ..."
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Cited by 55 (6 self)
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An increasing tree is a labelled rooted tree in which labels along any branch from the root go in increasing order. Under various guises, such trees have surfaced as tree representations of permutations, as data structures in computer science, and as probabilistic models in diverse applications. We present a unified generating function approach to the enumeration of parameters on such trees. The counting generating functions for several basic parameters are shown to be related to a simple ordinary differential equation which is non linear and autonomous. Singularity analysis applied to the intervening generating functions then permits to analyze asymptotically a number of parameters of the trees, like: root degree, number of leaves, path length, and level of nodes. In this way it is found that various models share common features: path length is O(n log n), the distributions of node levels and number of leaves are asymptotically normal, etc.
Stability of the solution to inverse obstacle scattering problem
 J.Inverse and IllPosed Problems
, 1994
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Modification of AbelPlana formula for functions with nonintegrable branchpoints
, 710
"... Abstract. The AbelPlana formula is a widely used tool for calculations in Casimir type problems. In this note we present a particular explicit modification of the ..."
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Abstract. The AbelPlana formula is a widely used tool for calculations in Casimir type problems. In this note we present a particular explicit modification of the
Stability of the solutions . . . problems with fixedenergy data
, 2002
"... A review of the author’s results is given. Inversion formulas and stability results for the solutions to 3D inverse scattering problems with fixed energy data are obtained. Inversion of exact and noisy data is considered. The inverse potential scattering problem with fixedenergy scattering data is ..."
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A review of the author’s results is given. Inversion formulas and stability results for the solutions to 3D inverse scattering problems with fixed energy data are obtained. Inversion of exact and noisy data is considered. The inverse potential scattering problem with fixedenergy scattering data is discussed in detail, inversion formulas for the exact and for noisy data are derived, error estimates for the inversion formulas are obtained. The inverse obstacle scattering problem is considered for nonsmooth obstacles. Stability estimates are derived for inverse obstacle scattering problem in the class of smooth obstacles. Global estimates for the scttering amplitude are given when the potential grows to infinity in a bounded domain. Inverse geophysical scattering problem is discussed briefly. An algorithm for constructing the DirichlettoNeumann map from the scattering amplitude and vice versa is obtained. An analytical example of nonuniqueness of the solution to a 3D inverse problem of geophysics and a uniqueness theorem for an inverse problem for parabolic equations are given.
325.tex J.Inverse and IllPosed problems 2,N3,(1994),269275. STABILITY OF THE SOLUTION TO INVERSE OBSTACLE SCATTERING PROBLEM
, 2000
"... Abstract. It is proved that if the scattering amplitudes for two obstacles (from a large class of obstacles) differ a little, then the obstacles differ a little, and the rate of convergence is given. An analytical formula for calculating the characteristic function of the obstacle is obtained, given ..."
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Abstract. It is proved that if the scattering amplitudes for two obstacles (from a large class of obstacles) differ a little, then the obstacles differ a little, and the rate of convergence is given. An analytical formula for calculating the characteristic function of the obstacle is obtained, given the scattering amplitude at a fixed frequency. Introduction. Let D ⊂ R 3 be a bounded domain with a smooth boundary Γ, ( ∇ 2 + k 2)u = 0 in D ′: = R 3 \ D, k = const> 0; u = 0 on Γ (1) u = exp(ikα · x) + A(α ′ , α, k)r −1 exp(ikr) + o(r −1), r: = x  → ∞, α ′: = xr −1. (2) Here α ∈ S 2 is a given unit vector, S 2 is the unit sphere in R 3, the function A(α ′ , α, k) is called the
problems with fixedenergy data. ∗†
, 2000
"... Stability of the solutions to 3D inverse scattering ..."
Rotation in classical zeropoint radiation and in quantum vacuum.
, 2006
"... Two reference systems, rotating {µτ} and non rotating {λτ}, are defined and used as the basis for investigating thermal effects of rotation through both random classical zero point radiation and quantum vacuum. Both reference systems consist of an infinite number of inertial reference frames µτ and ..."
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Two reference systems, rotating {µτ} and non rotating {λτ}, are defined and used as the basis for investigating thermal effects of rotation through both random classical zero point radiation and quantum vacuum. Both reference systems consist of an infinite number of inertial reference frames µτ and λτ respectively. The µ and λ reference frames do not accompany the detector and are defined so that at each moment of proper time τ of the detector there are two inertial frames, µτ and λτ, which agree momentarily, are connected by a Lorentz transformation with the detector velocity as a parameter, and with origins at the detector location at the same time τ. The two field correlation functions measured by the observer rotating through a random classical zero point radiation, have been calculated and presented in terms of elementary functions for both electromagnetic and massless scalar fields. If the correlation functions are periodic with a period 2π Ω spectrum which is very similar, but not identical, to Plank spectrum. of rotation the observer finds the If both fields of such a twofield periodic correlation function, for both electromagnetic and massless scalar case, are taken at the same point then its convergent (regularized) part is shown, using AbelPlana summation formula, to have Planck spectrum with the temperature Trot = ¯hΩ