Monotonicity, interlacing and electrostatic interpretation of zeros of exceptional Jacobi polynomials

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Monotonicity of zeros of Jacobi polynomials

Denote by x(nk)(alpha, beta), k = 1...., n, the zeros of the Jacobi polynornial P-n((alpha,beta)) (x). It is well known that x(nk)(alpha, beta) are increasing functions of beta and decreasing functions of alpha. In this paper we investigate the question of how fast the functions 1 - x(nk)(alpha, beta) decrease as beta increases. We prove that the products t(nk)(alpha, beta) := f(n)(alpha, beta) (1 - x(nk)(alpha, beta), where f(n)(alpha, beta) = 2n(2) + 2n(alpha + beta + 1) + (alpha + 1)(beta + 1) are already increasing functions of beta and that, for any fixed alpha > - 1, f(n)(alpha, beta) is the asymptotically extremal, with respect to n, function of beta that forces the products t(nk)(alpha, beta) to increase. (c) 2007 Elsevier B.V. All rights reserved.

On the behaviour of zeros of Jacobi polynomials

Denote by x(n,k)(alpha, beta) and x(n,k) (lambda) = x(n,k) (lambda - 1/2, lambda - 1/2) the zeros, in decreasing order, of the Jacobi polynomial P-n((alpha, beta))(x) and of the ultraspherical (Gegenbauer) polynomial C-n(lambda)(x), respectively. The monotonicity of x(n,k)(alpha, beta) as functions of a and beta, alpha, beta > - 1, is investigated. Necessary conditions such that the zeros of P-n((a, b)) (x) are smaller (greater) than the zeros of P-n((alpha, beta))(x) are provided. A. Markov proved that x(n,k) (a, b) < x(n,k)(α, β) (x(n,k)(a, b) > x(n,k)(alpha, beta)) for every n is an element of N and each k, 1 less than or equal to k less than or equal to n if a > alpha and b < β (a < alpha and b > beta). We prove the converse statement of Markov's theorem. The question of how large the function could be such that the products f(n)(lambda) x(n,k)(lambda), k = 1,..., [n/2] are increasing functions of lambda, for lambda > - 1/2, is also discussed. Elbert and Siafarikas proved that f(n)(lambda) = (lambda + (2n(2) + 1)/ (4n + 2))(1/2) obeys this property. We establish the sharpness of their result. (C) 2002 Elsevier B.V. (USA).

Inequalities for zeros of Jacobi polynomials via Sturm's theorem: Gautschi's conjectures

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Goppa codes through polynomials of B[X;(1/2^2)Z_0] and its decoding principle

In this paper, we present a decoding principle for Goppa codes constructed by generalized polynomials, which is based on modified Berlekamp-Massey algorithm. This algorithm corrects all errors up to the Hamming weight $t\leq 2r$, i.e., whose minimum Hamming distance is $2^{2}r+1$.

Line and area orthogonality of Jacobi polynomials,

"Supported in part by contract U.S. AEC AT(11-1) 1469 and grant NSF-6J-217".

A Note on a Classical Generating Function for the Jacobi Polynomials

Mathematics Subject Classification: 33C45.

Integral Representations of Functional Series with Members Containing Jacobi Polynomials

MSC 2010: Primary 33C45, 40A30; Secondary 26D07, 40C10

1.° Jacobi Alexandri le Taneur disputatio physico-mathematica de aequabili motus acceleratione, et de motu telluris : in qua Galilaei decreta de motu gravium perpendiculariter cadentium confirmantur ; et ostenditur vulgarem seu Ptolemaïcam de motu solis opinionem non minùs esse contrariam sacris litteris, quam Copernicanam de motu terrae : ac proinde nihil certi ex iis, quoad illam quaestionem, posse deduci : ad Petrum Cazraeum, societatis Jesu. — 2.° Ejusdem epistolae duae ad eundem, de descensu gravium. — 3.° Petri Cazraei, societatis Jesu, ad Alexandrum le Taneur epistola, qua sententiam Galilaei de descensu gravium impugnat. — 4.° Lettre de monsieur le Taneur au Pere Mersenne, Minime, sur la chûte des corps. — 5.° Ejusdem ad eundem epistola contra Fabrianam de incremento motus accelerati opinionem.

Melchisedechi Thevenotii

Short stature and primary ovarian insufficiency possibly due to chromosomal position effect in a balanced X;1 translocation.

BACKGROUND: Primary ovarian insufficiency (POI) is defined as a primary ovarian defect characterized by absent menarche (primary amenorrhea), a decrease in the initial primordial follicle number, high follicle-stimulating hormone (FSH) levels and hypoestrogenism. Although the etiology of a majority of POI cases is not yet identified, several data suggest that POI has a strong genetic component. Conventional cytogenetic and molecular analyses have identified regions of the X chromosome that are associated with ovarian function, as well as POI candidate genes, such as FMR1 and DIAPH2. Here we describe a 10.5-year-old girl presenting with high FSH and luteinizing hormone (LH) levels, pathologic GH stimulation arginine and clonidine tests, short stature, pterygium, ovarian dysgenesis, hirsutism and POI. RESULTS: Cytogenetic analysis demonstrated a balanced reciprocal translocation between the q arms of chromosomes X and 1, with breakpoints falling in Xq21 and 1q41 bands. Molecular studies did not unravel any chromosome microdeletion/microduplication, and no XIST-mediated inactivation was found on the derivative chromosome 1. Interestingly, through immunofluorescence assays, we found that part of the Xq21q22 trait, translocated to chromosome 1q41, was late replicating and therefore possibly inactivated in 30 % metaphases both in lymphocytes and skin fibroblasts, in addition to a skewed 100 % inactivation of the normal X chromosome. These findings suggest that a dysregulation of gene expression might occur in this region. Two genes mapping to the Xq translocated region, namely DIAPH2 and FMR1, were found overexpressed if compared with controls. CONCLUSIONS: We report a case in which gonadal dysgenesis and POI are associated with over-expression of DIAPH2 gene and of FMR1 gene in wild type form. We hypothesize that this over-expression is possibly due to a phenomenon known as "chromosomal position effect", which accounts for gene expression variations depending on their localization within the nucleus. For the same effect a double mosaic inactivation of genes mapping to the Xq21-q22 region, demonstrated by immunofluorescence assays, may be the cause of a functional Xq partial monosomy leading to most Turner traits of the proband's phenotype.

Analysis Of The A11a22 X 11a11 And A 33a22 X 1 electronic transitions in selenocarbonyl difluoride

The absorption spectrum of F2CSe in the 18800-21900 cm-1 region has been recorded at -770 C and 220 C under the conditions of medium resolution. The responsible electronic promotion is TI* + n excitation which leads to 3A2 and lA2 excited states. Progressions in vI', v2', v3" v4' and v4" have been identified in the spectrum and have been analyzed in terms of vibronic transitions between a planar ground state and a nQnplanar excited state. The - 3 - 1 - 1 - 1 origins of the a A2 + X Al and A A2 + X Al systems were assigned to the bands at 19018 cm-l and 19689 cm-l . This has given a singlet-triplet splittl. n g lA2 - 3A2 P f 671 cm -1 The out-of-plane wagging levels were found to be anharmonic. 1 -1 Barrier heights of 2483 cm- and 2923 cm were obtained for the lA2 and 3A2 upper states from a fitting of the energy levels of a Lorentzian-quadratic function to the observed levels in the out-of-plane wagging modes. 1 3 For the A2 and A2 states nonplanar equilibrium angles of 30.10 and 31.40 have been evaluated respectively. i

Estudio electroquímico de la inserción de litio en los vanadatos MxV2O5+x[M=Ni,Zn; x=1,2,3:M=Cd;x=1,2]

Tesis (Maestría en Ciencias con Especialidad en Ingeniería Cerámica) UANL