René Schott
Centre national de la recherche scientifique
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SIAM Journal on Computing | 1997
Guy Louchard; Claire Kenyon; René Schott
The purpose of this paper is to analyze the maxima properties (value and position) of some data structures. Our theorems concern the distribution of these random variables. Previously known results usually dealt with the mean and sometimes the variance of the random variables. Many of our results rely on diffusion techniques. This is a very powerful tool that has already been used with some success in algorithm complexity analysis.
Archive | 1991
Philip Feinsilver; René Schott
Some general remarks on random walks and martingales for finite probability distributions are presented. Orthogonal systems for the multinomial distribution arise. In particular, a class of generalized Krawtchouk polynomials is determined by a random walk generated by roots of unity. Relations with hypergeometric functions and some limit theorems are discussed.
Information Processing Letters | 1997
Laurent Alonso; Jean-Luc Rémy; René Schott
We present a simple O(n) algorithm that generates uniformly a Schroder tree of size n. The basic idea is to choose a slightly enlarged probability space where uniformity can be achieved.
Archive | 1996
Philip Feinsilver; René Schott
In this chapter, the theory is illustrated by looking at examples from the Hermitian symmetric spaces of types I-III. They are considered in their matrix realizations as domains in complex matrix spaces via the Harish-Chandra embedding as in Chapter 4. The fermion algebra is discussed in the context of the spaces of type II, so(2p). Examples of MAPLE output for su(2,2), sp(4), and so(6) have been included in the MAPLE section at the back of the book.
Random Structures and Algorithms | 1996
Laurent Alonso; Philippe Chassaing; René Schott
Given a set of n coins, some of them weighing H, the others weighing h, h < H, we prove that to determine the set of heavy coins, an optimal algorithm requires an average of 1+ρ 1+ρ+ρ n + O(1) comparisons, using a beam balance, in which ρ denotes the ratio of the probabilities of being light and heavy. A simple quasi-optimal algorithm is described. Similar results are derived for the majority problem.
fundamentals of computation theory | 1991
Guy Louchard; Claire Kenyon; René Schott
The purpose of this paper is to analyse the maxima properties (value and position) of some data structures. Our theorems concern the distribution of the random variables. Previously known results usually dealt with the mean and sometimes the variance of these random variables. Many of our results rely on diffusion techniques. That is a very powerful tool, which has already been used with some success in the analysis of algorithms.
Discrete Mathematics | 1998
Philip Feinsilver; René Schott
This paper presents an operator calculus approach to computing with non-commutative variables. First, we recall the product formulation of formal exponential series. Then we show how to formulate canonical boson calculus on formal series. This calculus is used to represent the action of a Lie algebra on its universal enveloping algebra. As applications, Hamiltons equations for a general Hamiltonian, given as a formal series, are found using a double-dual representation, and a formulation of the exponential of the adjoint representation is given. With these techniques one can represent the Volterra product acting on the enveloping algebra. We illustrate with a three-step nilpotent Lie algebra.
Archive | 1996
Philip Feinsilver; René Schott
In this chapter we focus on Lie algebras of symmetric type, described in the first section. From these algebras, the quotient representations yield homogeneous spaces, in particular symmetric spaces. The generating functions for the basis of the associated representations are called coherent states. There are connections with the multinomial distribution in probability theory and associated orthogonal polynomials generalizing the Meixner classes to several variables. We conclude with a Fourier technique for constructing new classes of orthogonal systems from certain given systems.
Archive | 1996
Philip Feinsilver; René Schott
In this brief chapter, we put together the basic special function properties of the theory. Starting from the right dual, the principal formula produces the matrix elements. They satisfy an addition formula according to the group law. They satisfy recurrence relations as well, which we will find in section two of this chapter. For quotient representations, we have previously seen a generating function for the matrix elements that can be found via the group law, and in section three we give a summation formula expressing the matrix elements for the quotient representation in terms of the general matrix elements
Archive | 1996
Philip Feinsilver; René Schott
In this and the next three chapters, we will look at a variety of examples. Basic algebras include affl, Heisenberg algebra (3-dimensional), finite-difference algebra, sl(2),e2, and a particular two-step nilpotent algebra that we call the ‘ABCD’ algebra (why — to be explained). And then some examples of higher-dimensional algebras.