Orthogonal polynomials associated with related measures and Sobolev orthogonal polynomials

Connection between two sequences of orthogonal polynomials, where the associated measures are related to each other by a first degree polynomial multiplication (or division), are looked at. The results are applied to obtain information regarding Sobolev orthogonal polynomials associated with certain pairs of measures.

The electrochemical reduction of the purines guanine and adenine at platinum electrodes in several room temperature ionic liquids

The reduction of guanine was studied by microelectrode voltammetry in the room temperature ionic liquids (RTILs) N-hexyltriethylammonium bis (trifluoromethanesulfonyl) imide [N6,2,2,2][N(Tf)2], 1-butyl-3-methylimidazolium hexafluorosphosphate [C4mim][PF6], N-butyl-N-methyl-pyrrolidinium bis(trifluoromethanesulfonyl)imide [C4mpyrr][N(Tf)2], 1-butyl-3-methylimidazolium bis(trifluoromethanesulfonyl)imide [C4mim][N(Tf)2], N-butyl-N-methyl-pyrrolidinium dicyanamide [C4mpyrr][N(NC)2] and tris(P-hexyl)-tetradecylphosphonium trifluorotris(pentafluoroethyl)phosphate [P14,6,6,6][FAP] on a platinum microelectrode. In [N6,2,2,2][NTf2] and [P14,6,6,6][FAP], but not in the other ionic liquids studied, guanine reduction involves a one-electron, diffusion-controlled process at very negative potential to produce an unstable radical anion, which is thought to undergo a dimerization reaction, probably after proton abstraction from the cation of the ionic liquid. The rate of this subsequent reaction depends on the nature of the ionic liquid, and it is faster in the ionic liquid [P14,6,6,6][FAP], in which the formation of the resulting dimer can be voltammetrically monitored at less negative potentials than required for the reduction of the parent molecule. Adenine showed similar behaviour to guanine but the pyrimidines thymine and cytosine did not; thymine was not reduced at potentials less negative than required for solvent (RTIL) decomposition while only a poorly defined wave was seen for cytosine. The possibility for proton abstraction from the cation in [N6,2,2,2][NTf2] and [P14,6,6,6][FAP] is noted and this is thought to aid the electrochemical dimerization process. The resulting rapid reaction is thought to shift the reduction potentials for guanine and adenine to lower values than observed in RTILs where the scope for proton abstraction is not present. Such shifts are characteristic of so-called EC processes where reversible electron transfer is followed by a chemical reaction. © 2009 Elsevier B.V.

Kernel polynomials from L-orthogonal polynomials

A positive measure ψ defined on [a,b] such that its moments μn=∫a btndψ(t) exist for n=0,±1,±2,⋯, is called a strong positive measure on [a,b]. If 0≤apolynomials {Qn}, defined by ∫a bt-n+sQn(t)dψ(t)=0, s=0,1,⋯,n-1, is known to exist. We refer to these polynomials as the L-orthogonal polynomials with respect to the strong positive measure ψ. The purpose of this manuscript is to consider some properties of the kernel polynomials associated with these L-orthogonal polynomials. As applications, we consider the quadrature rules associated with these kernel polynomials. Associated eigenvalue problems and numerical evaluation of the nodes and weights of such quadrature rules are also considered. © 2010 IMACS. Published by Elsevier B.V. All rights reserved.

Interlacing of zeros of orthogonal polynomials under modification of the measure

We investigate the mutual location of the zeros of two families of orthogonal polynomials. One of the families is orthogonal with respect to the measure dμ (x), supported on the interval (a, b) and the other with respect to the measure |x -c|τ|x -d|γdμ (x), where c and d are outside (a, b) We prove that the zeros of these polynomials, if they are of equal or consecutive degrees, interlace when either 0 < τ, γ ≤ 1 or γ = 0 and 0 < τ ≤ 2. This result is inspired by an open question of Richard Askey and it generalizes recent results on some families of orthogonal polynomials. Moreover, we obtain further statements on interlacing of zeros of specific orthogonal polynomials, such as the Askey-Wilson ones. © 2013 Elsevier Inc.

Sieved para-orthogonal polynomials on the unit circle

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Comparative study between powers of sigmoid functions, MLP-backpropagation and polynomials in function approximation problems

Function approximation is a very important task in environments where the computation has to be based on extracting information from data samples in real world processes. So, the development of new mathematical model is a very important activity to guarantee the evolution of the function approximation area. In this sense, we will present the Polynomials Powers of Sigmoid (PPS) as a linear neural network. In this paper, we will introduce one series of practical results for the Polynomials Powers of Sigmoid, where we will show some advantages of the use of the powers of sigmiod functions in relationship the traditional MLP-Backpropagation and Polynomials in functions approximation problems.

Generators of supersymmetric polynomials in positive characteristic

In Kantor and Trishin (1997) [3], Kantor and Trishin described the algebra of polynomial invariants of the adjoint representation of the Lie superalgebra gl(m vertical bar n) and a related algebra A, of what they called pseudosymmetric polynomials over an algebraically closed field K of characteristic zero. The algebra A(s) was investigated earlier by Stembridge (1985) who in [9] called the elements of A(s) supersymmetric polynomials and determined generators of A(s). The case of positive characteristic p of the ground field K has been recently investigated by La Scala and Zubkov (in press) in [6]. We extend their work and give a complete description of generators of polynomial invariants of the adjoint action of the general linear supergroup GL(m vertical bar n) and generators of A(s).

Comparative analysis of the corps en cerise in several species of Laurencia (Ceramiales, Rhodophyta) from the Atlantic Ocean

Different species of Laurencia have proven to be a rich source of natural products yielding interesting bioactive halogenated secondary metabolites, such as terpenoids and acetogenins. It is shown that such compounds are accumulated in the spherical, reniform to claviform refractive inclusions called corps en cerise (CC), which are intensively osmiophilic and located mainly in the cortical cells of the thalli and also in trichoblast cells. Up to now, it was believed that CC were present only in these two kinds of cells. Recently, however, a species of Laurencia, L. marilzae, with CC in all cells of the thallus, i.e., cortical, medullary, including the pericentral and axial cells, as well as in the trichoblasts, was described from the Canary Islands, and subsequently also reported to Brazil and Mexico. Within the Laurencia complex, only Laurencia species produce CC. Since the species of Laurencia are targets of interest for the prospection of bioactive substances due to their potential antibacterial, antifungal, anticholinesterasic, antileishmanial, cytotoxic, and antioxidant activities, the present paper carries out a comparative analysis of the corps en cerise in several species of Laurencia from the Atlantic Ocean to obtain basic information that can support natural product bioprospection projects. Our results show that the number and size of the CC are constant within a species, independent of the geographical distribution, corroborating their use for taxonomical purposes to differentiate groups of species that present a lower number from those that have a higher number. In this regard, there was a tendency for the number of CC to be higher in some species of Laurencia from the Canary Islands. The presence of CC can also be used to distinguish species in which these organelles are present in all cells of the thallus from those in which CC are restricted to the cortical cells. Among the species analyzed, L. viridis displayed the most varied secondary metabolites composition, such as sesquiterpenes, diterpenes, triterpenes, all of which showed potent antiviral, cytotoxic, and antitumoral activities, including protein phosphatase type 2A (PP2A) inhibitory effects.

Distribution of metals in several vineyards of North Italy: from soil to wine

Nowadays we live in densely populated regions and this leads to many environmental issues. Among all pollutants that human activities originate, metals are relevant because they can be potentially toxic for most of living beings. We studied the fate of Cd, Cr, Cu, Fe, Mn, Ni, Pb and Zn in a vineyard environment analysing samples of plant, wine and soil. Sites were chosen considering the type of wine produced, the type of cultivation (both organic and conventional agriculture) and the geographic location. We took vineyards that cultivate the same grape variety, the Trebbiano). We investigated 5 vineyards located in the Ravenna district (Italy): two on the Lamone Valley slopes, one in the area of river-bank deposits near Ravenna city, then a farm near Lugo and one near Bagnacavallo in interfluve regions. We carried out a very detailed characterization of soils in the sites, including the analysis of: pH, electric conductivity, texture, total carbonate and extimated content of dolomite, active carbonate, iron from ammonium oxalate, Iron Deficiency Chlorosis Index (IDCI), total nitrogen and organic carbon, available phosphorous, available potassium and Cation Exchange Capacity (CEC). Then we made the analysis of the bulk chemical composition and a DTPA extraction to determine the available fraction of elements in soils. All the sites have proper ground to cultivate, with already a good amount of nutrients, such as not needing strong fertilisations, but a vineyard on hills suffers from iron deficiency chlorosis due to the high level of active carbonate. We found some soils with much silica and little calcium oxide that confirm the marly sandstone substratum, while other soils have more calcium oxide and more aluminium oxide that confirm the argillaceous marlstone substratum. We found some critical situations, such as high concentrations of Chromium, especially in the farm near Lugo, and we noticed differences between organic vineyards and conventional ones: the conventional ones have a higher enrichment in soils of some metals (Copper and Zinc). Each metal accumulates differently in every single part of grapevines. We found differences between hill plants and lowland ones: behaviors of plants in metal accumulations seems to have patterns. Metals are more abundant in barks, then in leaves or sometimes in roots. Plants seem trying to remove excesses of metal storing them in bark. Two wines have excess of acetic acid and one conventional farm produces wine with content of Zinc over the Italian law limit. We already found evidence of high values relating them with uncontaminated environments, but more investigations are suggested to link those values to their anthropogenic supplies.

Association between trends in clinical variables and outcome in intensive care patients with faecal peritonitis: analysis of the GenOSept cohort.

INTRODUCTION Patients admitted to intensive care following surgery for faecal peritonitis present particular challenges in terms of clinical management and risk assessment. Collaborating surgical and intensive care teams need shared perspectives on prognosis. We aimed to determine the relationship between dynamic assessment of trends in selected variables and outcomes. METHODS We analysed trends in physiological and laboratory variables during the first week of intensive care unit (ICU) stay in 977 patients at 102 centres across 16 European countries. The primary outcome was 6-month mortality. Secondary endpoints were ICU, hospital and 28-day mortality. For each trend, Cox proportional hazards (PH) regression analyses, adjusted for age and sex, were performed for each endpoint. RESULTS Trends over the first 7 days of the ICU stay independently associated with 6-month mortality were worsening thrombocytopaenia (mortality: hazard ratio (HR) = 1.02; 95% confidence interval (CI), 1.01 to 1.03; P <0.001) and renal function (total daily urine output: HR =1.02; 95% CI, 1.01 to 1.03; P <0.001; Sequential Organ Failure Assessment (SOFA) renal subscore: HR = 0.87; 95% CI, 0.75 to 0.99; P = 0.047), maximum bilirubin level (HR = 0.99; 95% CI, 0.99 to 0.99; P = 0.02) and Glasgow Coma Scale (GCS) SOFA subscore (HR = 0.81; 95% CI, 0.68 to 0.98; P = 0.028). Changes in renal function (total daily urine output and renal component of the SOFA score), GCS component of the SOFA score, total SOFA score and worsening thrombocytopaenia were also independently associated with secondary outcomes (ICU, hospital and 28-day mortality). We detected the same pattern when we analysed trends on days 2, 3 and 5. Dynamic trends in all other measured laboratory and physiological variables, and in radiological findings, changes inrespiratory support, renal replacement therapy and inotrope and/or vasopressor requirements failed to be retained as independently associated with outcome in multivariate analysis. CONCLUSIONS Only deterioration in renal function, thrombocytopaenia and SOFA score over the first 2, 3, 5 and 7 days of the ICU stay were consistently associated with mortality at all endpoints. These findings may help to inform clinical decision making in patients with this common cause of critical illness.

Trends in climatic variables and future reference evapotranspiration in Duero Valley (Spain)

The impact of climate change and its relation with evapotranspiration was evaluated in the Duero River Basin (Spain). The study shows possible future situations 50 yr from now from the reference evapotranspiration (ETo). The maximum temperature (Tmax), minimum temperature (Tmin), dew point (Td), wind speed (U) and net radiation (Rn) trends during the 1980–2009 period were obtained and extrapolated with the FAO-56 Penman-Montheith equation to estimate ETo. Changes in stomatal resistance in response to increases in CO2 were also considered. Four scenarios were done, taking the concentration of CO2 and the period analyzed (annual or monthly) into consideration. The scenarios studied showed the changes in ETo as a consequence of the annual and monthly trends in the variables Tmax, Tmin, Td, U and Rn with current and future CO2 concentrations (372 ppm and 550 ppm). The future ETo showed increases between 118 mm (11 %) and 55 mm (5 %) with respect to the current situation of the river basin at 1042 mm. The months most affected by climate change are May, June, July, August and September, which also coincide with the maximum water needs of the basin’s crops

Bernstein polynomials in element-free Galerkin method

In the recent decades, meshless methods (MMs), like the element-free Galerkin method (EFGM), have been widely studied and interesting results have been reached when solving partial differential equations. However, such solutions show a problem around boundary conditions, where the accuracy is not adequately achieved. This is caused by the use of moving least squares or residual kernel particle method methods to obtain the shape functions needed in MM, since such methods are good enough in the inner of the integration domains, but not so accurate in boundaries. This way, Bernstein curves, which are a partition of unity themselves,can solve this problem with the same accuracy in the inner area of the domain and at their boundaries.