Nonsymmetric Askey-Wilson polynomials as vector-valued polynomials

Nonsymmetric Askey-Wilson polynomials are usually written as Laurent polynomials. We write them equivalently as 2-vector-valued symmetric Laurent polynomials. Then the Dunkl-Cherednik operator of which they are eigenfunctions, is represented as a 2x2 matrix-valued operator. As a new result made possible by this approach we obtain positive definiteness of the inner product in the orthogonality relations, under certain constraints on the parameters. A limit transition to nonsymmetric little q-Jacobi polynomials also becomes possible in this way. Nonsymmetric Jacobi polynomials are considered as limits both of the Askey-Wilson and of the little q-Jacobi case.Comment: 16 pages. Dedicated to Paul Butzer on the occasion of his 80th birthday. v4: minor correction in (4.14

Transformation design and nonlinear Hamiltonians

We study a class of nonlinear Hamiltonians, with applications in quantum optics. The interaction terms of these Hamiltonians are generated by taking a linear combination of powers of a simple `beam splitter' Hamiltonian. The entanglement properties of the eigenstates are studied. Finally, we show how to use this class of Hamiltonians to perform special tasks such as conditional state swapping, which can be used to generate optical cat states and to sort photons.Comment: Accepted for publication in Journal of Modern Optic

Laws relating runs, long runs, and steps in gambler's ruin, with persistence in two strata

Define a certain gambler's ruin process \mathbf{X}_{j}, \mbox{ \ }j\ge 0, such that the increments $\varepsilon_{j}:=\mathbf{X}_{j}-\mathbf{X}_{j-1}$ take values $\pm1$ and satisfy $P(\varepsilon_{j+1}=1|\varepsilon_{j}=1, |\mathbf{X}_{j}|=k)=P(\varepsilon_{j+1}=-1|\varepsilon_{j}=-1,|\mathbf{X}_{j}|=k)=a_k$, all $j\ge 1$, where $a_k=a$ if $0\le k\le f-1$, and $a_k=b$ if $f\le k<N$. Here $0<a, b <1$ denote persistence parameters and $f ,N\in \mathbb{N}$ with $f<N$. The process starts at $\mathbf{X}_0=m\in (-N,N)$ and terminates when $|\mathbf{X}_j|=N$. Denote by ${\cal R}'_N$, ${\cal U}'_N$, and ${\cal L}'_N$, respectively, the numbers of runs, long runs, and steps in the meander portion of the gambler's ruin process. Define $X_N:=\left ({\cal L}'_N-\frac{1-a-b}{(1-a)(1-b)}{\cal R}'_N-\frac{1}{(1-a)(1-b)}{\cal U}'_N\right )/N$ and let $f\sim\eta N$ for some $0<\eta <1$. We show $\lim_{N\to\infty} E\{e^{itX_N}\}=\hat{\varphi}(t)$ exists in an explicit form. We obtain a companion theorem for the last visit portion of the gambler's ruin.Comment: Presented at 8th International Conference on Lattice Path Combinatorics, Cal Poly Pomona, Aug., 2015. The 2nd version has been streamlined, with references added, including reference to a companion document with details of calculations via Mathematica. The 3rd version has 2 new figures and improved presentatio

A solution to the Al-Salam--Chihara moment problem

We study the $q$-hypergeometric difference operator $L$ on a particular Hilbert space. In this setting $L$ can be considered as an extension of the Jacobi operator for $q^{-1}$-Al-Salam--Chihara polynomials. Spectral analysis leads to unitarity and an explicit inverse of a $q$-analog of the Jacobi function transform. As a consequence a solution of the Al-Salam--Chihara indeterminate moment problem is obtained.Comment: 22 page

The impact of Stieltjes' work on continued fractions and orthogonal polynomials

Stieltjes' work on continued fractions and the orthogonal polynomials related to continued fraction expansions is summarized and an attempt is made to describe the influence of Stieltjes' ideas and work in research done after his death, with an emphasis on the theory of orthogonal polynomials

Symmetric diffusions with polynomial eigenvectors

25 pagesInternational audienceWe describe symmetric diffusion operators where the spectral decomposition is given through a family of orthogonal polynomials. In dimension one, this reduces to the case of Hermite, Laguerre and Jacobi polynomials. In higher dimension, some basic examples arise from compact Lie groups. We give a complete description of the bounded sets on which such operators may live. We then provide a classification of those sets when the polynomials are ordered according to their usual degree

Staphylococcus aureus bacteriuria as a prognosticator for outcome of Staphylococcus aureus bacteremia: a case-control study

<p>Abstract</p> <p>Background</p> <p>When <it>Staphylococcus aureus </it>is isolated in urine, it is thought to usually represent hematogenous spread. Because such spread might have special clinical significance, we evaluated predictors and outcomes of <it>S. aureus </it>bacteriuria among patients with <it>S. aureus </it>bacteremia.</p> <p>Methods</p> <p>A case-control study was performed at John H. Stroger Jr. Hospital of Cook County among adult inpatients during January 2002-December 2006. Cases and controls had positive and negative urine cultures, respectively, for <it>S. aureus</it>, within 72 hours of positive blood culture for <it>S. aureus</it>. Controls were sampled randomly in a 1:4 ratio. Univariate and multivariable logistic regression analyses were done.</p> <p>Results</p> <p>Overall, 59% of patients were African-American, 12% died, 56% of infections had community-onset infections, and 58% were infected with methicillin-susceptible <it>S. aureus </it>(MSSA). Among 61 cases and 247 controls, predictors of <it>S. aureus </it>bacteriuria on multivariate analysis were urological surgery (OR = 3.4, p = 0.06) and genitourinary infection (OR = 9.2, p = 0.002). Among patients who died, there were significantly more patients with bacteriuria than among patients who survived (39% vs. 17%; p = 0.002). In multiple Cox regression analysis, death risks in bacteremic patients were bacteriuria (hazard ratio 2.9, CI 1.4-5.9, p = 0.004), bladder catheter use (2.0, 1.0-4.0, p = 0.06), and Charlson score (1.1, 1.1-1.3, p = 0.02). Neither length of stay nor methicillin-resistant <it>Staphylococcus aureus </it>(MRSA) infection was a predictor of <it>S. aureus </it>bacteriuria or death.</p> <p>Conclusions</p> <p>Among patients with <it>S. aureus </it>bacteremia, those with <it>S. aureus </it>bacteriuria had 3-fold higher mortality than those without bacteriuria, even after adjustment for comorbidities. Bacteriuria may identify patients with more severe bacteremia, who are at risk of worse outcomes.</p

Primary gastric non-Hodgkin's lymphoma in Chinese patients: clinical characteristics and prognostic factors

<p>Abstract</p> <p>Background</p> <p>Optimal management and outcome of primary gastric lymphoma (PGL) have not been well defined in the rituximab era. This study aimed to analyze the clinical characteristics, prognostic factors, and roles of different treatment modalities in Chinese patients with PGL.</p> <p>Methods</p> <p>The clinicopathological features of 83 Chinese patients with PGL were retrospectively reviewed. Staging was performed according to the Lugano staging system for gastrointestinal non-Hodgkin's lymphoma.</p> <p>Results</p> <p>The predominant pathologic subtype among Chinese patients with PGL in our study was diffuse large B cell lymphoma (DLBCL), followed by mucosa-associated lymphoid tissue (MALT) lymphoma. Among the 57 patients with gastric DLBCL, 20 patients (35.1%) were classified as the germinal center B cell-like (GCB) subtype and 37 patients (64.9%) as the non-GCB subtype. The 83 patients had a five-year overall survival (OS) and event-free survival (EFS) of 52% and 59%, respectively. Cox regression analysis showed that stage-modified international prognostic index (IPI) and performance status (PS) were independent predictors of survival. In the 67 B-cell lymphoma patients who received chemotherapy, 36 patients treated with rituximab (at least 3 cycles) had a mean OS of 72 months (95% CI 62-81) versus 62 months (95% CI 47-76) for patients without rituximab treatment (P = 0.021).</p> <p>Conclusion</p> <p>The proportion of Chinese gastric DLBCL cases with non-GCB subtype was higher than the GCB subtype. Stage-modified IPI and PS were effective prognostic factors in Chinese patients with PGL. Our data suggested that primary gastric B-cell lymphoma might have an improved outcome with rituximab in addition to chemotherapy. More studies are necessary, preferentially large prospective randomized clinical trials to obtain more information on the impact of the rituximab in the primary gastric B-cell lymphoma.</p

Orthogonal polynomial interpretation of q-Toda and q-Volterra equations

The correspondences between dynamics of q-Toda and q-Volterra equations for the coefficients of the Jacobi operator and its resolvent function are established. The orthogonal polynomials associated with these Jacobi operators satisfy an Appell condition, with respect to the q-difference operator Dq . Lax type theorems for the point spectrum of the Jacobi operators associated with these equations are obtained. Examples related with the big q-Legendre, discrete q-Hermite I, and little q-Laguerre orthogonal polynomials and q-Toda and q-Volterra equations are given