Orthogonal polynomials on the unit circle via a polynomial mapping on the real line

Let {[Phi]n}n[greater-or-equal, slanted]0 be a sequence of monic orthogonal polynomials on the unit circle (OPUC) with respect to a symmetric and finite positive Borel measure d[mu] on [0,2[pi]] and let -1,[alpha]0,[alpha]1,[alpha]2,... be the associated sequence of Verblunsky coefficients. In this paper we study the sequence of monic OPUC whose sequence of Verblunsky coefficients iswhere b1,b2,...,bN-1 are N-1 fixed real numbers such that bj[set membership, variant](-1,1) for all j=1,2,...,N-1, so that is also orthogonal with respect to a symmetric and finite positive Borel measure on the unit circle. We show that the sequences of monic orthogonal polynomials on the real line (OPRL) corresponding to {[Phi]n}n[greater-or-equal, slanted]0 and (by Szegö's transformation) are related by some polynomial mapping, giving rise to a one-to-one correspondence between the monic OPUC on the unit circle and a pair of monic OPRL on (a subset of) the interval [-1,1]. In particular we prove thatd[mu][small tilde]([theta])=[zeta]N-1([theta])sin[theta]sin[theta]N([theta])d[mu]([theta]N([theta]))[theta]N'([theta]),supported on (a subset of) the union of 2N intervals contained in [0,2[pi]] such that any two of these intervals have at most one common point, and where, up to an affine change in the variable, [zeta]N-1 and cos[theta]N are algebraic polynomials in cos[theta] of degrees N-1 and N (respectively) defined only in terms of [alpha]0,b1,...,bN-1. This measure induces a measure on the unit circle supported on the union of 2N arcs, pairwise symmetric with respect to the real axis. The restriction to symmetric measures (or real Verblunsky coefficients) is needed in order that Szegö's transformation may be applicable.http://www.sciencedirect.com/science/article/B6TYH-4NNWCG5-1/1/5cc167c4a58817de62d99d3dd5c88e3

Antitumor activity of splicing inhibitor pladienolide B in erythroleukemia: a study in cell lines

Trabalho final de mestrado integrado em Medicina (Hematologia/Biologia Molecular), apresentado à Faculdade de Medicina da Universidade de Coimbra.The splicing of pre-mRNA into functional mRNA, carried out by the spliceosome, represents a crucial step for the cell's genetic expression. Mutations in some of the spliceosome’s components have been identified in several hematological malignancies, including myelodysplastic syndromes and acute myeloid leukemia (AML), which could constitute a potential therapeutic target to be explored. In this context, we evaluated the therapeutic potential of a splicing inhibitor, Pladienolide B (Pla-B), in two erythroleukemia cell-lines. K562 and HEL cells were incubated in the absence or presence of increasing concentrations of Pla-B in single dose (from 0.25 to 100 nM) and in daily administration (for 0.5 nM). Cell viability and density were evaluated using the trypan blue method. Cell death was determined by optical microscopy (May-Grunwald Giemsa staining) and flow cytometry (FC). Cell cycle analysis was evaluated by FC, using a PI/RNAse solution. DNA sequencing was performed to assess the presence of SF3B1 mutations in exons 14 and 15. Treatment with Pla-B significantly decreased the viability and proliferation of the K562 and HEL cells in a time, concentration and administration schedule dependent manner. HEL cells were more sensible to Pla-B than K562 cells (after 72 hours of incubation the IC50 was 1.5 nM and 25 nM, respectively), which may be due to different cell genetic backgrounds. In fact, K562 cells present the BCR-ABL fusion gene and HEL cells the JAK2 V617F mutation. However, SF3B1 mutations in exons 14 or 15 were not detected in any cell model used, suggesting that the observed cytotoxic effect is not dependent on this spliceosome mutation. Pla-B induced cell death preferentially by apoptosis and induced also an accumulation of cells in the G0/G1 phase of the cell cycle. Our results show that Pla-B induces a cytostatic and cytotoxic effect in K562 and HEL cells, suggesting that Pla-B could represent a new therapeutic approach in the treatment of erythroleukemia.O splicing de pre-mRNA em mRNA funcional, mediado pelo spliceossoma, representa uma etapa fundamental na expressão genética da célula. Nos últimos anos, mutações em alguns dos componentes do spliceossoma foram identificadas em várias neoplasias hematológicas, incluindo síndromes mielodisplásicos e leucemia mielóide aguda, representando alvos terapêuticos por explorar. Neste contexto, avaliámos o potencial terapêutico de um inibidor do splicing, Pladienolide B (Pla-B), em duas linhas celulares de eritroleucemia. As células K562 e HEL foram incubadas na presença ou ausência de concentrações crescentes de Pla-B, em dose única (0.25 nM a 100 nM) e dose diária (0.5 nM). A viabilidade e a densidade celulares foram avaliadas pelo método de azul tripano. A morte celular foi determinada por microscopia óptica (coloração de May-Grunwald Giemsa) e citometria de fluxo (FC). A análise do ciclo celular foi realizada por FC, usando uma solução de PI/RNAse. Mutações nos exões 14 ou 15 do gene SF3B1 foram pesquisadas através de sequenciação do DNA. O tratamento das células K562 e HEL com Pla-B reduziu significativamente a viabilidade e proliferação celulares, de um modo dependente de tempo, concentração e modo de administração do fármaco. As células HEL mostraram-se mais sensíveis ao fármaco do que as células K562 (após 72 horas de incubação, o IC50 foi de 1.5 nM e 25 nM, respectivamente), o que pode ser devido a diferenças genéticas. De facto, as células K562 apresentam o gene de fusão BCR-ABL, enquanto as HEL apresentam a mutação do JAK2 V617F. No entanto, não foram encontradas mutações, em nenhum dos modelos, nos exões 14 ou 15 do gene SF3B1, o que sugere que o efeito citotóxico observado não é dependente desta mutação. Verificámos que o Pla-B induz morte celular preferencialmente por apoptose, bem como induz uma acumulação das células na fase G0/G1 do ciclo celular. Mutações nos exões 14 e 15 do gene SF3B1 foram excluídas. Os nossos resultados sugerem que o Pla-B apresenta um efeito anti-proliferativo e citotóxico em ambas as linhas celulares, e que poderá representar uma nova abordagem terapêutica no tratamento da eritroleucemia

Orthogonal polynomials and quadratic transformations

33 pages, no figures.-- MSC1991 code: Primary 42C05.MR#: MR1680116 (2000b:42021)Zbl#: Zbl 0936.42012Starting from a sequence $\{P_n\}_{n\geq 0}$ of monic polynomials orthogonal with respect to a linear functional ${\bf u}$, we find a linear functional ${\bf v}$ such that $\{Q_n\}_{\geq 0}$, with either $Q_{2n}(x)=P_n(T(x))$ or $Q_{2n+1}(x)=(x-a)\,P_n(T(x))$ where $T$ is a monic quadratic polynomial and a\in\C, is a sequence of monic orthogonal polynomials with respect to ${\bf v}$. In particular, we discuss the case when ${\bf u}$ and ${\bf v}$ are both positive definite linear functionals. Thus, we obtain a solution for an inverse problem which is a converse, for quadratic mappings, of one analyzed in [11].This paper was finished with financial support of a Grant from Junta Nacional de Investigação Científica e Tecnológica (JNITC) - BD976 - and Centro de Matemática da Universidade de Coimbra (CMUC) of Portugal. The work of the first author was supported by the Dirección General de Enseñanza Superior (DGES) of Spain - PB96-0120-C03-01.Publicad

On linearly related sequences of derivatives of orthogonal polynomials

We discuss an inverse problem in the theory of (standard) orthogonal polynomials involving two orthogonal polynomial families (Pn)n and (Qn)n whose derivatives of higher orders m and k (resp.) are connected by a linear algebraic structure relation such as for all n=0,1,2,..., where M and N are fixed nonnegative integer numbers, and ri,n and si,n are given complex parameters satisfying some natural conditions. Let u and v be the moment regular functionals associated with (Pn)n and (Qn)n (resp.). Assuming 0[less-than-or-equals, slant]m[less-than-or-equals, slant]k, we prove the existence of four polynomials [Phi]M+m+i and [Psi]N+k+i, of degrees M+m+i and N+k+i (resp.), such that the (k-m)th-derivative, as well as the left-product of a functional by a polynomial, being defined in the usual sense of the theory of distributions. If k=m, then u and v are connected by a rational modification. If k=m+1, then both u and v are semiclassical linear functionals, which are also connected by a rational modification. When k>m, the Stieltjes transform associated with u satisfies a non-homogeneous linear ordinary differential equation of order k-m with polynomial coefficients.http://www.sciencedirect.com/science/article/B6WK2-4SSNDCC-2/1/2844d686d4f273e5ddf7b4d7146c9ee

On the Krall-type discrete polynomials

In this paper we present a unified theory for studying the so-called Krall-type discrete orthogonal polynomials. In particular, the three-term recurrence relation, lowering and raising operators as well as the second order linear difference equation that the sequences of monic orthogonal polynomials satisfy are established. Some relevant examples of q-Krall polynomials are considered in detail.http://www.sciencedirect.com/science/article/B6WK2-4CC2YF9-1/1/1bbcf94cc1184e679b497c3b8e754b2

Explicit inverses of some tridiagonal matrices

We give explicit inverses of tridiagonal 2-Toeplitz and 3-Toeplitz matrices which generalize some well-known results concerning the inverse of a tridiagonal Toeplitz matrix.http://www.sciencedirect.com/science/article/B6V0R-42KDHCJ-2/1/d6207c3bf9c66184210a9296f72fe1e