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The rise of antimicrobial resistance (AMR) in animals is being fuelled by the widespread use of veterinary antimicrobials. China is the largest global consumer of veterinary antimicrobials, and improving AMR surveillance strategies in this region could help prioritize intervention and preserve antimicrobial efficacy. Here we mapped AMR rates in pigs, chickens and cattle in China using 446 surveys of event-based surveillance between 2000 and 2019 for foodborne bacteria, in combination with geospatial models to identify locations where conducting new surveys could have the highest benefits. Using maps of uncertainty, we show that eastern China currently has the highest AMR rates, and southwestern and northeastern China would benefit the most from additional surveillance efforts. Instead of distributing new surveys evenly across administrative divisions, using geographically targeted surveillance could reduce AMR prediction uncertainty by two-fold. In a context of competing disease control priorities, our findings present a feasible option for optimizing surveillance efforts—and slowing the spread of AMR., Nature Food, 2 (8), ISSN:2662-1355

Summary: Background: The inverse care law states that disadvantaged populations need more health care than advantaged populations but receive less. Gaps in COVID-19-related health care and infection control are not well understood. We aimed to examine inequalities in health in the care cascade from testing for SARS-CoV-2 to COVID-19-related hospitalisation, intensive care unit (ICU) admission, and death in Switzerland, a wealthy country strongly affected by the pandemic. Methods: We analysed surveillance data reported to the Swiss Federal Office of Public Health from March 1, 2020, to April 16, 2021, and 2018 population data. We geocoded residential addresses of notifications to identify the Swiss neighbourhood index of socioeconomic position (Swiss-SEP). The index describes 1·27 million small neighbourhoods of approximately 50 households each on the basis of rent per m2, education and occupation of household heads, and crowding. We used negative binomial regression models to calculate incidence rate ratios (IRRs) with 95% credible intervals (CrIs) of the association between ten groups of the Swiss-SEP index defined by deciles (1=lowest, 10=highest) and outcomes. Models were adjusted for sex, age, canton, and wave of the epidemic (before or after June 8, 2020). We used three different denominators: the general population, the number of tests, and the number of positive tests. Findings: Analyses were based on 4 129 636 tests, 609 782 positive tests, 26 143 hospitalisations, 2432 ICU admissions, 9383 deaths, and 8 221 406 residents. Comparing the highest with the lowest Swiss-SEP group and using the general population as the denominator, more tests were done among people living in neighbourhoods of highest SEP compared with lowest SEP (adjusted IRR 1·18 [95% CrI 1·02–1·36]). Among tested people, test positivity was lower (0·75 [0·69–0·81]) in neighbourhoods of highest SEP than of lowest SEP. Among people testing positive, the adjusted IRR was 0·68 (0·62–0·74) for hospitalisation, was 0·54 (0·43–0·70) for ICU admission, and 0·86 (0·76–0·99) for death. The associations between neighbourhood SEP and outcomes were stronger in younger age groups and we found heterogeneity between areas. Interpretation: The inverse care law and socioeconomic inequalities were evident in Switzerland during the COVID-19 epidemic. People living in neighbourhoods of low SEP were less likely to be tested but more likely to test positive, be admitted to hospital, or die, compared with those in areas of high SEP. It is essential to continue to monitor testing for SARS-CoV-2, access and uptake of COVID-19 vaccination and outcomes of COVID-19. Governments and health-care systems should address this pandemic of inequality by taking measures to reduce health inequalities in response to the SARS-CoV-2 pandemic. Funding: Swiss Federal Office of Public Health, Swiss National Science Foundation, EU Horizon 2020, Branco Weiss Foundation.

AimsIn multiflowered species, the architecture of inflorescences is of primary importance in shaping plant attractiveness. The aim of this study was to disentangle the role of inflorescence traits in plant female reproductive success and pollination patterns along the inflorescence in the lax-flowered orchid Anacamptis laxiflora, a terrestrial species exploiting a deceptive pollination strategy. We also evaluated whether the relationship between inflorescence traits and female reproductive success was modified by the height of surrounding vegetation and/or by population density.MethodsWe delimited experimental plots in a natural population of A. laxiflora. We tallied the individuals within each plot and categorized low-density plots and high-density plots; then, in part of the plots we manually removed surrounding grass thus producing an equal number of plots with high grass and low grass. Within these plots, we recorded inflorescence traits and female reproductive success (i.e. the number of fruit and their position along the inflorescence). We analyzed these data using generalized linear mixed-effects models (GLMMs) and calculated selection gradients.Important FindingsWe found that all the investigated inflorescence traits influenced female reproductive success. In particular, our GLMMs showed that ‘average flower distance’ was the best predictor for shaping reproductive success patterns. We detected significant positive selection on the investigated inflorescence traits, but these selective trends were strictly linked to both the height of the surrounding vegetation and the population density, suggesting a significant influence of local environmental context in shaping selective patterns. Female reproductive success was not linked to the position of flowers along the inflorescence, suggesting that pollinators visit flowers randomly along the inflorescence without a detectable preference for a specific part. This study highlights the importance of inflorescence traits in shaping female reproductive success of multiflowered deceptive orchids, and confirms a primary role for the environmental context in modifying pollinator-mediated selection patterns.

Population genetics focuses on the analysis of genetic differences within and between-group of individuals and the inference of the populations’ structure. These analyses are usually carried out using Bayesian clustering or maximum likelihood estimation algorithms that assign individuals to a given population depending on specific genetic patterns. Although several tools were developed to perform population genetics analysis, their standard graphical outputs may not be sufficiently informative for users lacking interactivity and complete information. StructuRly aims to resolve this problem by offering a complete environment for population analysis. In particular, StructuRly combines the statistical power of the R language with the friendly interfaces implemented using the shiny libraries to provide a novel tool for performing population clustering, evaluating several genetic indexes, and comparing results. Moreover, graphical representations are interactive and can be easily personalized. StructuRly is available either as R package on GitHub, with detailed information for its installation and use and as shinyapps.io servers for those users who are not familiar with R and the RStudio IDE. The application has been tested on Linux, macOS and Windows operative systems and can be launched as a shiny app in every web browser., PLoS ONE, 15 (2), ISSN:1932-6203

The global scale-up in demand for animal protein is the most notable dietary trend of our time. Antimicrobial consumption in animals is threefold that of humans and has enabled large-scale animal protein production. The consequences for the development of antimicrobial resistance in animals have received comparatively less attention than in humans. We analyzed 901 point prevalence surveys of pathogens in developing countries to map resistance in animals. China and India represented the largest hotspots of resistance, with new hotspots emerging in Brazil and Kenya. From 2000 to 2018, the proportion of antimicrobials showing resistance above 50% increased from 0.15 to 0.41 in chickens and from 0.13 to 0.34 in pigs. Escalating resistance in animals is anticipated to have important consequences for animal health and, eventually, for human health., SCOPUS: ar.j, info:eu-repo/semantics/published

Morphometric, biochemical and genetic analyses were conducted on Olea europaea L. of Campania, an area of Southern Italy highly suited to the cultivation of olive trees and the production of extra virgin olive oil (EVOO). We aimed to characterize the distribution of morphological, biochemical and genetic diversity in this area and to develop a practical tool to aid traceability of oils. Phenotypes were characterized using morphometric data of drupes and leaves, biochemical and genetic diversity were assessed on the basis of the fatty acid composition of the EVOOs and with microsatellite markers, respectively. We provide an open-source tool as a novel R package titled &lsquo, OliveR&rsquo, useful in performing multivariate data analysis using a point and click interactive approach. These analyses highlight a clear correlation among the morphological, biochemical and genetic profiles of samples with four collection sites, and confirm that Southern Italy represents a wide reservoir of phenotypic and genetic variability.

Double integrals with algebraic and/or logarithmic singularities are of interest in the application of boundary element method, e.g. linear theory of the aerodynamics of slender bodies of revolution and in many other fields, for example computational electromagnetics. Therefore, the numerical evaluation of such type of integrals deserves attention. In this connection we propose here product interpolatory rules based on suitable Jacobi zeros, giving numerical tests to show the goodness of the proposed algorithm as well as from a point of view of convergence also for the simplicity of implementation.

A fast algorithm related to the generalized minimal residual algorithm (GMRES) is proposed to approximate solution of a nonlinear hypersingular integral equation arising in a crack problem. At first, a collocation method is proposed and developed in weighted Sobolev space. Then, the Newton-Kantorovjch method is used for solving the obtained system of nonlinear equations.

IMPORTANCE Molecular diagnosis is difficult to achieve in disease groups with a highly heterogeneous genetic background, such as cerebellar ataxia (CA). In many patients, candidate gene sequencing or focused resequencing arrays do not allow investigators to reach a genetic conclusion. OBJECTIVES To assess the efficacy of exome-targeted capture sequencing to detect mutations in genes broadly linked to CA in a large cohort of undiagnosed patients and to investigate their prevalence. DESIGN, SETTING, AND PARTICIPANTS Three hundred nineteen index patients with CA and without a history of dominant transmission were included in the this cohort study by the Spastic Paraplegia and Ataxia Network. Centralized storage was in the DNA and cell bank of the Brain and Spine Institute, Salpetriere Hospital, Paris, France. Patients were classified into 6 clinical groups, with the largest being those with spastic ataxia (ie, CA with pyramidal signs [n = 100]). Sequencing was performed from January 1, 2014, through December 31, 2016. Detected variants were classified as very probably or definitely causative, possibly causative, or of unknown significance based on genetic evidence and genotype-phenotype considerations. MAIN OUTCOMES AND MEASURES Identification of variants in genes broadly linked to CA, classified in pathogenicity groups. RESULTS The 319 included patients had equal sex distribution (160 female [50.2%] and 159 male patients [49.8%]; mean [SD] age at onset, 27.9 [18.6] years). The age at onset was younger than 25 years for 131 of 298 patients (44.0%) with complete clinical information. Consanguinity was present in 101 of 298 (33.9%). Very probable or definite diagnoses were achieved for 72 patients (22.6%), with an additional 19 (6.0%) harboring possibly pathogenic variants. The most frequently mutated genes were SPG7 (n = 14), SACS (n = 8), SETX (n = 7), SYNE1 (n = 6), and CACNA1A (n = 6). The highest diagnostic rate was obtained for patients with an autosomal recessive CA with oculomotor apraxia-like phenotype (6 of 17 [35.3%]) or spastic ataxia (35 of 100 [35.0%]) and patients with onset before 25 years of age (41 of 131 [31.3%]). Peculiar phenotypes were reported for patients carrying KCND3 or ERCC5 variants. CONCLUSIONS AND RELEVANCE Exome capture followed by targeted analysis allows the molecular diagnosis in patients with highly heterogeneous mendelian disorders, such as CA, without prior assumption of the inheritance mode or causative gene. Being commonly available without specific design need, this procedure allows testing of a broader range of genes, consequently describing less classic phenotype-genotype correlations, and post hoc reanalysis of data as new genes are implicated in the disease. © 2018 American Medical Association. All rights reserved.

Ten lactic acid bacteria (LAB) strains, previously isolated from human ileal biopsy of healthy volunteers, were screened for production and secretion of molecules having anti-bacterial and anti-biofilm activities. Because many recent reports indicate that LAB secreted molecules may exert immune-modulatory action, we also tested the effect on human intestinal HCT116 cells challenged with bacterial lipopolysaccharides. One of the Lactobacillus gasseri strains, SF1109, strongly inhibited: (1) Pseudomonas aeruginosa growth; (2) Escherichia coli biofilm production; (3) LPS induction of P-ERK1/2 in HCT116 cells, and was selected for further characterisation of the secreted active molecule. Cell-free supernatant of the L. gasseri SF1109 was analysed and one 1.3 kDa peptide has been characterised. Eight out twelve amino acids of this peptide were identified allowing the synthesis of an octa-peptide which still presented the mentioned activities.

ResumenObjetivosPresentar nuestra experiencia en el tratamiento mínimamente invasivo de la lumbociatalgia con la inyección de corticoides y anestésicos locales bajo control tomográfico.Materiales y métodosSe realizaron bloqueos selectivos lumbares bajo control tomográfico a 102 pacientes con lumbociatalgia crónica, en un período comprendido entre agosto del 2011 y junio del 2012. Del total de pacientes, se infiltraron 65 a nivel foraminal (64%), 29 a nivel epidural (28%) y 8 a ambos niveles (8%). Los procedimientos se realizaron en forma ambulatoria con anestesia local. Todos los pacientes recibieron tratamiento con antiinflamatorios no esteroides (AINES) vía oral y se utilizó la escala numérica del dolor y el índice de Oswestry (IDO) para medir la discapacidad funcional en cada caso.ResultadosEl 100% de los pacientes mostró disminución significativa de la sintomatología apenas finalizó el procedimiento, sin observarse complicaciones inmediatas durante el mismo. Se hizo un seguimiento clínico posterior con las escalas anteriormente mencionadas a los 7 días, 1, 3 y 6 meses. En 95 pacientes (93%) se observó una mejora significativa de los síntomas y se suspendió o se redujo la medicación oral, mientras que en 6 pacientes existió una mejoría parcial de los síntomas al mes, pero hubo una recaída a los 3 meses. En estos casos se debió reiniciar el tratamiento con AINES, manteniéndose a 4 pacientes dentro de la categoría del IDO anterior (aunque con una disminución de al menos 2 puntos en el score numérico del dolor). Sólo un paciente no presentó mejoría de la sintomatología durante el seguimiento y tuvo reaparición de los síntomas habituales a los 7 días, por lo que se debió reprogramar una segunda infiltración.ConclusiónEn nuestra experiencia el bloqueo nervioso lumbar selectivo bajo control tomográfico, utilizando esteroides y anestésicos locales, resultó un procedimiento efectivo en el control del dolor con un bajo índice de complicaciones.AbstractObjectivesTo present our experience with minimally invasive treatment of low back pain and sciatica with the computed tomography-guided percutaneous injection of steroids and local anaesthetics.Materials and methodsFrom August 2011 to June 2012, 102 patients underwent selective computed tomography-guided foraminal block for low back pain and sciatica treatment. Sixty-five patients received foraminal infiltration (64%), 29 epidural infiltration (28%), and 8 (8%) were subject to combined procedures. All procedures were performed on an outpatient basis with local anaesthetic, with no immediate complications. All patients received oral NSAIDs (non-steroidal anti-inflammatory drugs) prior to the procedure. A numeric scale of pain and the Oswestry index (IDO) was employed to measure local pain and limb disability. All patients showed at least 7 points in the initial evaluation.ResultsAll the patients showed a significant reduction in pain by the end of procedure. A clinical follow-up was made after 7 days, 1, 3, and 6 months after the treatment using the previously mentioned scales. Ninety-five patients (93%) showed a significant improvement in their symptoms, with suspension or decrease in oral medication. Six patients showed only a partial reduction of symptoms during the follow-up after one month, with a recurrence of symptoms after 3 months and restarted oral treatment. Four of these patients remained in the same IDO category with at least a 2 point decrease in the pain scale. Only one patient showed no improvement in symptoms during follow-up with a recurrence of symptoms 7 days after procedure, and for whom a second procedure was reprogrammed.ConclusionIn our experience CT-guided percutaneous lumbar selective nerve block using steroids and local anaesthetics, is an effective method of pain control with a very low incidence of complications.

ResumenObjetivoMostrar nuestra experiencia en ecografía prostática transrectal y biopsia prostática bajo guía ecográfica a lo largo de 10 años, a través de la realización de un análisis retrospectivo, con el fin de evaluar sus beneficios en la detección del adenocarcinoma, identificar la distribución de muestras positivas en la glándula y exponer la utilidad de la punción de imágenes nodulares.Materiales y métodosEntre marzo del 2001 y noviembre del 2011 se realizaron 1.163 ecografías prostáticas transrectales seguidas de biopsias prostáticas bajo control ecográfico. En la muestra se incluyó a pacientes que se realizaron más de una biopsia. Los datos obtenidos de estos procedimientos permitieron identificar la presencia de tres grupos etarios de acuerdo a la incidencia de la patología, conocer la localización más frecuente del adenocarcinoma dentro de la próstata y evaluar la utilidad de la biopsia de los nódulos prostáticos en la detección del adenocarcinoma.ResultadosEl análisis de la muestra arrojó un porcentaje de detección del adenocarcinoma del 16% en los menores de 50 años, del 36% en aquellos entre los 50 y 60 años, y del 48% en los mayores de 65 años.La base izquierda de la próstata fue la localización con el índice más alto de positividad (15%), seguido por la media izquierda (14%), la base derecha (14%), la media derecha (13%), el ápex izquierdo (12%), el medio lateral izquierdo (12%), el ápex derecho (11%) y el medio lateral derecho (10%).Doscientos noventa y nueve pacientes presentaron imágenes nodulares: 118 (Grupo A) tuvieron alguna muestra positiva y 181 presentaron resultados negativos (Grupo B). A su vez, el Grupo A se dividió en 3 subgrupos: el Subgrupo A, cuyo resultado positivo se registró en la muestra del nódulo, pero fue negativo en el resto de la glándula; Subgrupo B, cuyos resultados fueron positivos en el resto de la glándula y negativos en la muestra del nódulo; y Subgrupo C, cuyo resultado fue positivo tanto en las muestras de la glándula como en las del nódulo (combinación más frecuente de los tres subgrupos).ConclusiónLa ecografía y la biopsia prostáticas bajo guía ecográfica juegan un rol importante en la evaluación de la próstata ante la sospecha de un adenocarcinoma. Según nuestra experiencia, la biopsia prostática bajo guía ecográfica es el método de mayor utilidad para la detección, especialmente en pacientes mayores de 65 años, y debería incluirse la muestra del nódulo (si la hubiese) en el esquema por octantes.AbstractPurposeTo present our experience in prostatic transrectal ultrasound and transrectal ultrasound-guided biopsy over a ten-year period by a retrospective analysis in order to determine the detection rate of adenocarcinoma, identify the distribution of positive samples in the gland, and evaluate the usefulness of biopsy of nodules seen on imaging studies.Materials and methodsA total of 1163 ultrasound and ultrasound-guided transrectal prostate biopsies were performed between March 2001 and November 2011. The population sample included patients who had more than one biopsy performed. Data obtained from these procedures enables us to identify three age groups according to the incidence of pathology, to determine the most frequent location of adenocarcinoma within the prostate, and to evaluate the usefulness of prostate nodules biopsy in the detection of adenocarcinoma.ResultsThe data analysis showed a 16% detection rate of adenocarcinoma in men under 50 years of age, 36% in patients between 50 and 65 years, and 48% in patients older than 65 years. The left base of the prostate had the highest detection rate (15%), followed by left medium (14%), right base (14%), right medium (13%), left apex (12%), left lateral medium (12%), right apex (11%) and right lateral medium (10%).Nodular images were found in 299 patients: 118 were positive for adenocarcinoma (Group A) and 181 were negative (Group B). Group A was divided into 3 subgroups: Subgroup A, with a positive result only in the nodule sample, and a negative result in the rest of the gland sample; Subgroup B, with a positive result in the gland samples but negative in the nodule sample; and Subgroup C, with results that were positive both in the gland and nodule samples.ConclusionUltrasound and ultrasound-guided transrectal prostate biopsy play an important role in the evaluation of the prostate when adenocarcinoma is suspected. Based on our experience, ultrasound-guided prostate biopsy is the most useful method for the detection of adenocarcinoma, especially in patients older than 65 years of age, and the nodule sample (if any) should be included in the eight-biopsy scheme.

An account of the error and the convergence theory is given for Gauss-Laguerre and Gauss-Radau-Laguerre quadrature formulae. We develop also truncated models of the original Gauss rules to compute integrals extended over the positive real axis. Numerical examples confirming the theoretical results are given comparing these rules among themselves and with different quadrature formulae proposed by other authors (Evans, Int. J. Comput. Math. 82:721730, 2005; Gautschi, BIT 31:438446, 1991).

Recently, Mastroianni and Monegato derived error estimates for a numerical approach to evaluate the integral ∫ab ∫-11 f(x, y)/x-y dxdy, where (a,b) ≡ (-1,1) or (a,b) ≡ (a,-1) or (a,b) ≡ (1, b) and f(x,y) is a smooth function (see G. Mastroianni and G. Monegato, Error estimates in the numerical evaluation of some BEM singular integrals, Math. Comp. 70 2001, 251-267). The error bounds for the quadrature rule approximating the inner integral given in Theorems 3, 4 of that paper are not correct according to the proof. However, following a different approach, we are able to improve the pointwise error estimates given in that paper.

To compute integrals on bounded or unbounded intervals we propose a new numerical approach by using weights and nodes of the classical Gauss quadrature rules. An account of the error and the convergence theory is given for the proposed quadrature formulas which have the advantage of reducing the condition number of the linear system arising when applying Nystr ¨ om methods to solve integral equations. Numerical examples confirming the theoretical results are provided to illustrate the accuracy of the introduced method.

Interpolatory quadrature rules have been well studied and widely applied for numerical integration in the presence of simple singularities. However, these rules have been used less for strongly singular integrals due to the fact that divergence and instability problems increase with respect to the order of the singularity. In the present paper we study this aspect proving convergence and stability of suitable interpolatory rules to compute certain singular and hypersingular integrals. The particular case of these integrals for oscillatory functions is also examined. Some numerical tests confirming our theoretical analysis are also presented.

In a recent paper, we investigated the uniform convergence of Lagrange interpolation at the zeros of the orthogonal polynomials with respect to a Freud-type weight in the presence of constraints. We show that by a simple procedure it is always possible to transform the matrices of these zeros into matrices such that the corresponding Lagrange interpolating polynomial with respect to the given constraints well approximates a given function. Here, starting from the interest to construct a suitable interpolation operator with a preassigned node, we introduce an algorithm that allows us to obtain new matrices. For the Lagrange operator related to these new matrices that have the preassigned node among their elements, we prove results about the optimal rate of convergence as well as we apply successfully this method to some applications. © 2014 IMACS.

An algorithm for the approximate evaluation of the Hilbert transform has been proposed. The convergence of the procedure is proved. The stability of the algorithm is considered and some numerical examples are given.

We deal with the numerical evaluation of the Hilbert transform on the real line by a Gauss type quadrature rule. The convergence and the stability of the method are investigated. The goodness of the numerical results for practical applications is examined.

Let W ≔ e − Q , where Q : R → R is even, continuous, and of smooth polynomial growth at infinity. Then we call W 2 = e −2 Q a Freud weight , the most typical examples being W 2 β ( x ) ≔ exp(−| x | β ), β > 1. Corresponding to the weight W 2 , we can form the sequence of orthonormal polynomials { p j ( W 2 , x )} ∞ j =0 . The functions of the second kind are q j ( W 2 , x ) ≔ H [ p j W 2 ]( x ) j ≥ 0, where H denotes the Hilbert transform; that is, for g ∈ L , ( R ), H [ g ]( x ) ≔ P.V. ∫ ∞ −∞ g ( t )/ t − x dt . Here P.V. denotes principal value. For a large class of Freud weights, we obtain bounds on { q j } ∞ j =0 in the L ∞ , and L p norms. We also estimate the generalized function of the second kind q j ( W 2 , v , x ) ≔ H [ p j W v ]( x ), for a fixed function v . We then apply these estimates to investigate the convergence of series of the second kind , which form the basis of Henrici′s method of approximating Cauchy principal value integrals.

We investigate the uniform convergence of Lagrange interpolation at the zeros of Hermite polynomials in the presence of constraints. We show that by a simple procedure it is always possible to transform the matrices of these zeros into matrices such that the corresponding Lagrange interpolating polynomial with respect to the given constraints well approximates a given function. This procedure was, at first, successfully introduced for the polynomial interpolation with constraints on bounded intervals [1]. For this procedure we obtain the same results obtained in [2], where only the zeros of the Hermite polynomial are used.

The authors construct some extended interpolation formulae to approximate the derivatives of a function in uniform norm. They prove theorems on uniform convergence and give estimates of pointwise type and of simultaneous approximation.

For the numerical solution of the hypersingular integral equation of a notched half-plane problem we propose collocation methods which look for an approximation of the derivative of the solution of the original equation. This derivative is the solution of a Cauchy singular integral equation with additional fixed singularities. We also give a solvability analysis of the original equation which motivates the suggested numerical methods.

We examine the convergence of product quadrature formulas of interpolatory type, based on the zeros of certain generalized Jacobi polynomials, for the discretization of integrals of the type \[ ∫ − 1 1 K ( x , y ) f ( x ) d x , − 1 ≤ y ≤ 1 , \int _{ - 1}^1 {K(x,y)f(x)\,dx,} \quad - 1 \leq y \leq 1, \] where the kernel K ( x , y ) K(x,y) is weakly singular and the function f ( x ) f(x) has singularities only at the endpoints ± 1 \pm 1 . In particular, when K ( x , y ) = log ⁡ | x − y | K(x,y) = \log |x - y| , K ( x , y ) = | x − y | v K(x,y) = |x - y{|^v} , v > − 1 v > - 1 , and f ( x ) f(x) has algebraic singularities of the form ( 1 ± x ) σ {(1 \pm x)^\sigma } , σ > − 1 \sigma > - 1 , we prove that the uniform rate of convergence of the rules is O ( m − 2 − 2 σ log 2 m ) O({m^{ - 2 - 2\sigma }}{\log ^2}m) in the case of the first kernel, and O ( m − 2 − 2 σ − 2 v log ⁡ m ) O({m^{ - 2 - 2\sigma - 2v}}\log m) if v ≤ 0 v \leq 0 , or O ( m − 2 − 2 σ log ⁡ m ) O({m^{ - 2 - 2\sigma }}\log m) if v > 0 v > 0 , for the second, where m is the number of points in the quadrature rule.

We discuss solvability properties of a nonlinear hypersingular integral equation of Prandtl’s type using monotonicity arguments together with different collocation iteration schemes for the numerical solution of such equations.

After some remarks on the convergence order of the classical gaussian formula for the numerical evaluation of integrals on unbounded interval, the authors develop a new quadrature rule for the approximation of such integrals of interest in the practical applications. The convergence of the proposed algorithm is considered and some numerical examples are given.

The authors develop an algorithm for the numerical evaluation of Cauchy principal value integrals of oscillatory functions. The method is based on an interpolatory procedure at the zeros of the orthogonal polynomials with respect to a Jacobi weight. A numerically stable procedure is obtained and the corresponding algorithm can be implemented in a fast way yielding satisfactory numerical results. Bounds of the error and of the amplification factor are also provided.

The mathematical model of some environmental physics problems is represented by a singular integral equation with an oscillatory kernel. We investigate a method for the numerical evaluation of Cauchy principal value integrals of oscillatory functions. The method is based on an interpolatory procedure at the zeros of the orthogonal polynomials with respect to a Jacobi weight. In this way, we obtain a procedure that is numerically stable and the algorithm can be implemented in a fast way yielding satisfactory numerical results. Bounds of the error and of the amplification factor are also provided.

An algorithm for the approximate evaluation of integrals defined by Cauchy principal value or by Hadamard finite part has been proposed. The convergence of the procedure is proved. The stability of the algorithm is considered and some numerical examples are given.

The authors study the error of the product quadrature rules to compute the integral\(\int_{ - 1}^1 {f\left( x \right) u\left( x \right)dx} \). Estimates inL 1-weighted norm are established whenu is a weight with algebraic and/or logarithmic singularities and the quadrature points are classical Jacobi zeros. Upper bounds for the generalized functions of second kind are also given.

Uniform convergence of Lagrange interpolation at the zeros of Jacobi polynomials in the presence of constraints is investigated. We show that by a simple procedure it is always possible to transform the matrices of these zeros into matrices such that the corresponding Lagrange interpolating polynomial respecting the given constraints well approximates a given function and its derivatives.

Weighted L(p) convergence of derivatives of extended Lagrange interpolation at the union of zeros of generalized Jacobi polynomials and some additional points is investigated.