Lp extremal polynomials. Results and perspectives

2000 Mathematics Subject Classification: 30C40, 30D50, 30E10, 30E15, 42C05.

Smale's Conjecture on Mean Values of Polynomials and Electrostatics

2000 Mathematics Subject Classification: Primary 30C10, 30C15, 31B35.

Even and Old Overdetermined Strata for Degree 6 Hyperbolic Polynomials

2000 Mathematics Subject Classification: 12D10.

Hyperbolic Fourth-R Quadratic Equation and Holomorphic Fourth-R Polynomials

MSC 2010: 30C10, 32A30, 30G35

Density of Polynomials in the L^2 Space on the Real and the Imaginary Axes and in a Sobolev Space

2000 Mathematics Subject Classification: 41A10, 30E10, 41A65.

Special Classes of Orthogonal Polynomials and Corresponding Quadratures of Gaussian Type

MSC 2010: 33C47, 42C05, 41A55, 65D30, 65D32

Convection-diffusion in MHD: Comparison of primitive variable and vector potential solutions of the magnetic induction equation

Abstract not available

The Real-Quaternionic Indicator of Irreducible Self-Conjugate Representations of Real Reductive Algebraic Groups and A Comment on the Local Langlands Correspondence of GL(2, F )

The real-quaternionic indicator, also called the $\delta$ indicator, indicates if a self-conjugate representation is of real or quaternionic type. It is closely related to the Frobenius-Schur indicator, which we call the $\varepsilon$ indicator. The Frobenius-Schur indicator $\varepsilon(\pi)$ is known to be given by a particular value of the central character. We would like a similar result for the $\delta$ indicator. When $G$ is compact, $\delta(\pi)$ and $\varepsilon(\pi)$ coincide. In general, they are not necessarily the same. In this thesis, we will give a relation between the two indicators when $G$ is a real reductive algebraic group. This relation also leads to a formula for $\delta(\pi)$ in terms of the central character. For the second part, we consider the construction of the local Langlands correspondence of $GL(2,F)$ when $F$ is a non-Archimedean local field with odd residual characteristics. By re-examining the construction, we provide new proofs to some important properties of the correspondence. Namely, the construction is independent of the choice of additive character in the theta correspondence.

Fire in relation to primitive agriculture and grazing in the Tropics ; annotated bibliography.

Vol. 2 was presented at the Ninth Pacific Science Congress, Bangkok, Thailand, Nov. 18-30, 1957.

Multivariate orthogonal polynomials and integrable systems

Multivariate orthogonal polynomials in D real dimensions are considered from the perspective of the Cholesky factorization of a moment matrix. The approach allows for the construction of corresponding multivariate orthogonal polynomials, associated second kind functions, Jacobi type matrices and associated three term relations and also Christoffel-Darboux formulae. The multivariate orthogonal polynomials, their second kind functions and the corresponding Christoffel-Darboux kernels are shown to be quasi-determinants as well as Schur complements of bordered truncations of the moment matrix; quasi-tau functions are introduced. It is proven that the second kind functions are multivariate Cauchy transforms of the multivariate orthogonal polynomials. Discrete and continuous deformations of the measure lead to Toda type integrable hierarchy, being the corresponding flows described through Lax and Zakharov-Shabat equations; bilinear equations are found. Varying size matrix nonlinear partial difference and differential equations of the 2D Toda lattice type are shown to be solved by matrix coefficients of the multivariate orthogonal polynomials. The discrete flows, which are shown to be connected with a Gauss-Borel factorization of the Jacobi type matrices and its quasi-determinants, lead to expressions for the multivariate orthogonal polynomials and their second kind functions in terms of shifted quasi-tau matrices, which generalize to the multidimensional realm, those that relate the Baker and adjoint Baker functions to ratios of Miwa shifted tau-functions in the 1D scenario. In this context, the multivariate extension of the elementary Darboux transformation is given in terms of quasi-determinants of matrices built up by the evaluation, at a poised set of nodes lying in an appropriate hyperplane in R^D, of the multivariate orthogonal polynomials. The multivariate Christoffel formula for the iteration of m elementary Darboux transformations is given as a quasi-determinant. It is shown, using congruences in the space of semi-infinite matrices, that the discrete and continuous flows are intimately connected and determine nonlinear partial difference-differential equations that involve only one site in the integrable lattice behaving as a Kadomstev-Petviashvili type system. Finally, a brief discussion of measures with a particular linear isometry invariance and some of its consequences for the corresponding multivariate polynomials is given. In particular, it is shown that the Toda times that preserve the invariance condition lay in a secant variety of the Veronese variety of the fixed point set of the linear isometry.

On polynomials with given Hilbert function and applications

Using Macaulay's correspondence we study the family of Artinian Gorenstein local algebras with fixed symmetric Hilbert function decomposition. As an application we give a new lower bound for the dimension of cactus varieties of the third Veronese embedding. We discuss the case of cubic surfaces, where interesting phenomena occur.

Chemo-physical and biological mechanisms behind the anticancer activity of plasma activated Ringer's Lactate solution for the treatment of peritoneal carcinosis from primitive epithelial ovarian/tubular tumor

Cancer research and development of targeting agents in this field is based on robust studies using preclinical models. The failure rate of standardized treatment approaches for several solid tumors has led to the urgent need to fine-tune more sophisticated and faithful preclinical models able to recapitulate the features of in vivo human tumors, with the final aim to shed light on new potential therapeutic targets. Epithelial Ovarian Cancer (EOC) serous histotype (HGSOC) is one of the most lethal diseases in women due to its high aggressiveness (75% of patients diagnosed at FIGO III-IV state) and poor prognosis (less of 50% in 5 years), whose therapy often fails as chemoresistance sets in. This thesis aimed at using the novel perfusion-based bioreactor U-CUP that provides direct perfusion throughout the tumor tissue seeking to obtain an EOC 3D ex vivo model able to recapitulate the features of the original tumor including the tumor microenvironment and maintaining its cellular heterogeneity. Moreover, we optimized this approach so that it can be successfully applied to slow-frozen tumoral tissues, further extending the usefulness of this tool. We also investigated the effectiveness of Plasma Activated Ringer’s Lactate solution (PA-RL) against Epithelial Ovarian Cancer (EOC) serous histotype in both 2D and 3D cultures using ex-vivo specimens from HGSOC patients. We propose PA-RL as a novel therapy with local intraperitoneal administration, which could act on primary or metastatic ovarian tumors inducing a specific cancer cell death with reduced damage on the surrounding healthy tissues.

Characterization of an international tetraploid wheat germplasm including landraces and primitive wheat towards improved resilience to abiotic and biotic stresses and quality

This thesis aimed to characterise two large tetraploid germplasm collections. The Global Durum Panel, involving modern cultivars and landrances and the Tetraploid Global Collection which comprises all the tetraploid wheat subgroups. Two distinct parallel studies were carried out. The first is focused on the characterisation of both collection for yield and quality related traits. The panel were phenotyped for two consecutive years each. In this phase the following traits were collected: the number of fertile spikelets per spike, the number of fertile florets of central spikelet for the spike-related traits. The following grain related traits were also phenotyped: the thousand kernel weight, the average grain area, average grain length, average grain width, grain brightness, grain redness, grain yellowness. GWAS analysis were performed for each collected trait and major QTLs were subjected to candidate gene analysis. Major QTLs emerging from GWA study were located on chromosome 2A with a strong bibliographic evidence for grain number-related traits such as the fertile spikelet number, the number of fertile florets per central spikelet. On the other hand two evident peaks were detected on chromosomes 6A and 7B for grain size and weight related traits. The second work was focused on the characterisation of the Global Durum Panel for root system architecture components, namely the root growth angle. GWAS analysis was perfomed and three major QTLs were detected on chromosome 2A, 6A and 7A. These three QTLs all have a bibliographic evidence.

Automorphisms of irreducible holomorphic symplectic manifolds and related problems

We study automorphisms and the mapping class group of irreducible holomorphic symplectic (IHS) manifolds. We produce two examples of manifolds of K3[2] type with a symplectic action of the alternating group A7. Our examples are realized as double EPW-sextics, the large cardinality of the group allows us to prove the irrationality of the associated families of Gushel-Mukai threefolds. We describe the group of automorphisms of double EPW-cubes. We give an answer to the Nielsen realization problem for IHS manifolds in analogy to the case of K3 surfaces, determining when a finite group of mapping classes fixes an Einstein (or Kähler-Einstein) metric. We describe, for some deformation classes, the mapping class group and its representation in second cohomology. We classify non-symplectic involutions of manifolds of OG10 type determining the possible invariant and coinvariant lattices. We study non-symplectic involutions on LSV manifolds that are geometrically induced from non-symplectic involutions on cubic fourfolds.

Transmission Of Intra-cellular Genetic Information: A System Proposal.

One of the great challenges of the scientific community on theories of genetic information, genetic communication and genetic coding is to determine a mathematical structure related to DNA sequences. In this paper we propose a model of an intra-cellular transmission system of genetic information similar to a model of a power and bandwidth efficient digital communication system in order to identify a mathematical structure in DNA sequences where such sequences are biologically relevant. The model of a transmission system of genetic information is concerned with the identification, reproduction and mathematical classification of the nucleotide sequence of single stranded DNA by the genetic encoder. Hence, a genetic encoder is devised where labelings and cyclic codes are established. The establishment of the algebraic structure of the corresponding codes alphabets, mappings, labelings, primitive polynomials (p(x)) and code generator polynomials (g(x)) are quite important in characterizing error-correcting codes subclasses of G-linear codes. These latter codes are useful for the identification, reproduction and mathematical classification of DNA sequences. The characterization of this model may contribute to the development of a methodology that can be applied in mutational analysis and polymorphisms, production of new drugs and genetic improvement, among other things, resulting in the reduction of time and laboratory costs.