Non-intersecting squared Bessel paths: critical time and double scaling limit

We consider the double scaling limit for a model of $n$ non-intersecting squared Bessel processes in the confluent case: all paths start at time $t=0$ at the same positive value $x=a$, remain positive, and are conditioned to end at time $t=1$ at $x=0$. After appropriate rescaling, the paths fill a region in the $tx$--plane as $n\to \infty$ that intersects the hard edge at $x=0$ at a critical time $t=t^{*}$. In a previous paper (arXiv:0712.1333), the scaling limits for the positions of the paths at time $t\neq t^{*}$ were shown to be the usual scaling limits from random matrix theory. Here, we describe the limit as $n\to \infty$ of the correlation kernel at critical time $t^{*}$ 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 $3\times 3$ 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

Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights

We study a model of $n$ non-intersecting squared Bessel processes in the confluent case: all paths start at time $t = 0$ at the same positive value $x = a$, remain positive, and are conditioned to end at time $t = T$ at $x = 0$. In the limit $n \to \infty$, after appropriate rescaling, the paths fill out a region in the $tx$-plane that we describe explicitly. In particular, the paths initially stay away from the hard edge at $x = 0$, but at a certain critical time $t^*$ the smallest paths hit the hard edge and from then on are stuck to it. For $t \neq t^*$ we obtain the usual scaling limits from random matrix theory, namely the sine, Airy, and Bessel kernels. A key fact is that the positions of the paths at any time $t$ constitute a multiple orthogonal polynomial ensemble, corresponding to a system of two modified Bessel-type weights. As a consequence, there is a $3 \times 3$ matrix valued Riemann-Hilbert problem characterizing this model, that we analyze in the large $n$ limit using the Deift-Zhou steepest descent method. There are some novel ingredients in the Riemann-Hilbert analysis that are of independent interest.Comment: 59 pages, 11 figure

Noncolliding Squared Bessel Processes

We consider a particle system of the squared Bessel processes with index $\nu > -1$ conditioned never to collide with each other, in which if $-1 < \nu < 0$ 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 $J_{\nu}$ 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

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

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

Matrix interpretation of multiple orthogonality

In this work we give an interpretation of a (s(d + 1) + 1)-term recurrence relation in terms of type II multiple orthogonal polynomials.We rewrite this recurrence relation in matrix form and we obtain a three-term recurrence relation for vector polynomials with matrix coefficients. We present a matrix interpretation of the type II multi-orthogonality conditions.We state a Favard type theorem and the expression for the resolvent function associated to the vector of linear functionals. Finally a reinterpretation of the type II Hermite- Padé approximation in matrix form is given

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

Cavity-induced coherence effects in spontaneous emission from pre-Selection of polarization

Spontaneous emission can create coherences in a multilevel atom having close lying levels, subject to the condition that the atomic dipole matrix elements are non-orthogonal. This condition is rarely met in atomic systems. We report the possibility of bypassing this condition and thereby creating coherences by letting the atom with orthogonal dipoles to interact with the vacuum of a pre-selected polarized cavity mode rather than the free space vacuum. We derive a master equation for the reduced density operator of a model four level atomic system, and obtain its analytical solution to describe the interference effects. We report the quantum beat structure in the populations.Comment: 6 pages in REVTEX multicolumn format, 5 figures, new references added, journal reference adde

A framework for increasing the value of predictive data-driven models by enriching problem domain characterization with novel features

The need to leverage knowledge through data mining has driven enterprises in a demand for more data. However, there is a gap between the availability of data and the application of extracted knowledge for improving decision support. In fact, more data do not necessarily imply better predictive data-driven marketing models, since it is often the case that the problem domain requires a deeper characterization. Aiming at such characterization, we propose a framework drawn on three feature selection strategies, where the goal is to unveil novel features that can effectively increase the value of data by providing a richer characterization of the problem domain. Such strategies involve encompassing context (e.g., social and economic variables), evaluating past history, and disaggregate the main problem into smaller but interesting subproblems. The framework is evaluated through an empirical analysis for a real bank telemarketing application, with the results proving the benefits of such approach, as the area under the receiver operating characteristic curve increased with each stage, improving previous model in terms of predictive performance.The work of P. Cortez was supported by FCT within the Project Scope UID/CEC/00319/2013. The authors would like to thank the anonymous reviewers for their helpful comments.info:eu-repo/semantics/publishedVersio