162 research outputs found
Some discrete multiple orthogonal polynomials
27 pages, no figures.-- MSC2000 codes: 33C45, 33C10, 42C05, 41A28.-- Issue title: "Proceedings of the 6th International Symposium on Orthogonal Polynomials, Special Functions and their Applications" (OPSFA-VI, Rome, Italy, 18-22 June 2001).MR#: MR1985676 (2004g:33015)Zbl#: Zbl 1021.33006In this paper, we extend the theory of discrete orthogonal polynomials (on a linear lattice) to polynomials satisfying orthogonality conditions with respect to r positive discrete measures. First we recall the known results of the classical orthogonal polynomials of Charlier, Meixner, Kravchuk and Hahn (T.S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978; R. Koekoek and R.F. Swarttouw, Reports of the Faculty of Technical Mathematics and Informatics No. 98-17, Delft, 1998; A.F. Nikiforov et al., Classical Orthogonal Polynomials of a Discrete Variable, Springer, Berlin, 1991). These polynomials have a lowering and raising operator, which give rise to a Rodrigues formula, a second order difference equation, and an explicit expression from which the coefficients of the three-term recurrence relation can be obtained. Then we consider r positive discrete measures and define two types of multiple orthogonal polynomials. The continuous case (Jacobi, Laguerre, Hermite, etc.) was studied by Van Assche and Coussement (J. Comput. Appl. Math. 127 (2001) 317–347) and Aptekarev et al. (Multiple orthogonal polynomials for classical weights, manuscript). The families of multiple orthogonal polynomials (of type II) that we will study have a raising operator and hence a Rodrigues formula. This will give us an explicit formula for the polynomials. Finally, there also exists a recurrence relation of order r+1 for these multiple orthogonal polynomials of type II. We compute the coefficients of the recurrence relation explicitly when r=2.This research was supported by INTAS project 00-272, Dirección General de Investigación del Ministerio de Ciencia y Tecnología of Spain under grants BFM-2000-0029 and BFM-2000-0206-C04-01, Dirección General de Investigación de la Comunidad Autónoma de Madrid, and by project G.0184.02 of FWO-Vlaanderen.Publicad
Maximize What Matters: Predicting Customer Churn With Decision-Centric Ensemble Selection
Churn modeling is important to sustain profitable customer relationships in saturated consumer markets. A churn model predicts the likelihood of customer defection. This is important to target retention offers to the right customers and to use marketing resources efficiently. The prevailing approach toward churn model development, supervised learning, suffers an important limitation: it does not allow the marketing analyst to account for campaign planning objectives and constraints during model building. Our key proposition is that creating a churn model in awareness of actual business requirements increases the performance of the final model for marketing decision support. To demonstrate this, we propose a decision-centric framework to create churn models. We test our modeling framework on eight real-life churn data sets and find that it performs significantly better than state-of-the-art churn models. Further analysis suggests that this improvement comes directly from incorporating business objectives into model building, which confirms the effectiveness of the proposed framework. In particular, we estimate that our approach increases the per customer profits of retention campaigns by $.47 on average
Non-intersecting squared Bessel paths: critical time and double scaling limit
We consider the double scaling limit for a model of non-intersecting
squared Bessel processes in the confluent case: all paths start at time
at the same positive value , remain positive, and are conditioned to end
at time at . After appropriate rescaling, the paths fill a region in
the --plane as that intersects the hard edge at at a
critical time . In a previous paper (arXiv:0712.1333), the scaling
limits for the positions of the paths at time were shown to be
the usual scaling limits from random matrix theory. Here, we describe the limit
as of the correlation kernel at critical time and in the
double scaling regime. We derive an integral representation for the limit
kernel which bears some connections with the Pearcey kernel. The analysis is
based on the study of a matrix valued Riemann-Hilbert problem by
the Deift-Zhou steepest descent method. The main ingredient is the construction
of a local parametrix at the origin, out of the solutions of a particular
third-order linear differential equation, and its matching with a global
parametrix.Comment: 53 pages, 15 figure
Noncolliding Squared Bessel Processes
We consider a particle system of the squared Bessel processes with index conditioned never to collide with each other, in which if
the origin is assumed to be reflecting. When the number of particles is finite,
we prove for any fixed initial configuration that this noncolliding diffusion
process is determinantal in the sense that any multitime correlation function
is given by a determinant with a continuous kernel called the correlation
kernel. When the number of particles is infinite, we give sufficient conditions
for initial configurations so that the system is well defined. There the
process with an infinite number of particles is determinantal and the
correlation kernel is expressed using an entire function represented by the
Weierstrass canonical product, whose zeros on the positive part of the real
axis are given by the particle-positions in the initial configuration. From the
class of infinite-particle initial configurations satisfying our conditions, we
report one example in detail, which is a fixed configuration such that every
point of the square of positive zero of the Bessel function is
occupied by one particle. The process starting from this initial configuration
shows a relaxation phenomenon converging to the stationary process, which is
determinantal with the extended Bessel kernel, in the long-term limit.Comment: v3: LaTeX2e, 26 pages, no figure, corrections made for publication in
J. Stat. Phy
Direct and inverse spectral transform for the relativistic Toda lattice and the connection with Laurent orthogonal polynomials
We introduce a spectral transform for the finite relativistic Toda lattice
(RTL) in generalized form. In the nonrelativistic case, Moser constructed a
spectral transform from the spectral theory of symmetric Jacobi matrices. Here
we use a non-symmetric generalized eigenvalue problem for a pair of bidiagonal
matrices (L,M) to define the spectral transform for the RTL. The inverse
spectral transform is described in terms of a terminating T-fraction. The
generalized eigenvalues are constants of motion and the auxiliary spectral data
have explicit time evolution. Using the connection with the theory of Laurent
orthogonal polynomials, we study the long-time behaviour of the RTL. As in the
case of the Toda lattice the matrix entries have asymptotic limits. We show
that L tends to an upper Hessenberg matrix with the generalized eigenvalues
sorted on the diagonal, while M tends to the identity matrix.Comment: 24 pages, 9 figure
The Trigonometric Rosen-Morse Potential in the Supersymmetric Quantum Mechanics and its Exact Solutions
The analytic solutions of the one-dimensional Schroedinger equation for the
trigonometric Rosen-Morse potential reported in the literature rely upon the
Jacobi polynomials with complex indices and complex arguments. We first draw
attention to the fact that the complex Jacobi polynomials have non-trivial
orthogonality properties which make them uncomfortable for physics
applications. Instead we here solve above equation in terms of real orthogonal
polynomials. The new solutions are used in the construction of the
quantum-mechanic superpotential.Comment: 16 pages 7 figures 1 tabl
M\"ossbauer Antineutrinos: Recoilless Resonant Emission and Absorption of Electron Antineutrinos
Basic questions concerning phononless resonant capture of monoenergetic
electron antineutrinos (M\"ossbauer antineutrinos) emitted in bound-state
beta-decay in the 3H - 3He system are discussed. It is shown that lattice
expansion and contraction after the transformation of the nucleus will
drastically reduce the probability of phononless transitions and that various
solid-state effects will cause large line broadening. As a possible
alternative, the rare-earth system 163Ho - 163Dy is favoured.
M\"ossbauer-antineutrino experiments could be used to gain new and deep
insights into several basic problems in neutrino physics
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