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In this paper, the solution to a bivariate Appell interpolation problem proposed in a previous work is given. Bounds of the truncation error are considered. Ten new interpolants for real, regular, bivariate functions are constructed. Numerical examples and comparisons with bivariate Bernstein polynomials are considered. [ABSTRACT FROM AUTHOR]

In this paper, an approach to orthogonal polynomials based on matrix calculus is proposed. Known and new basic results are given, such as recurrence relations and determinant forms. New algorithms, similar, but not identical, to the Chebyshev one, for practical calculation of the polynomials are presented. The cases of monic and symmetric orthogonal polynomial sequences and the case of orthonormal polynomial sequences have been considered. Some classical and non-classical examples are given. The work is framed in a broader perspective, already started by the authors. It provides for the determination of properties of a general sequence of polynomials and, therefore, their applicability to special classes of the most important polynomials. [ABSTRACT FROM AUTHOR]

Road traffic emissions are often considered the main source of ultrafine particles (UFP, diameter smaller than 100 nm) in urban environments. However, recent studies worldwide have shown that - in high-insolation urban regions at least - new particle formation events can also contribute to UFP. In order to quantify such events we systematically studied three cities located in predominantly sunny environments: Barcelona (Spain), Madrid (Spain) and Brisbane (Australia). Three long-term data sets (1-2 years) of fine and ultrafine particle number size distributions (measured by SMPS, Scanning Mobility Particle Sizer) were analysed. Compared to total particle number concentrations, aerosol size distributions offer far more information on the type, origin and atmospheric evolution of the particles. By applying k-means clustering analysis, we categorized the collected aerosol size distributions into three main categories: "Traffic" (prevailing 44-63% of the time), "Nucleation" (14-19%) and "Background pollution and Specific cases" (7-22%). Measurements from Rome (Italy) and Los Angeles (USA) were also included to complement the study. The daily variation of the average UFP concentrations for a typical nucleation day at each site revealed a similar pattern for all cities, with three distinct particle bursts. A morning and an evening spike reflected traffic rush hours, whereas a third one at midday showed nucleation events. The photochemically nucleated particles' burst lasted 1-4 h, reaching sizes of 30-40 nm. On average, the occurrence of particle size spectra dominated by nucleation events was 16% of the time, showing the importance of this process as a source of UFP in urban environments exposed to high solar radiation. Nucleation events lasting for 2 h or more occurred on 55% of the days, this extending to > 4 h in 28%of the days, demonstrating that atmospheric conditions in urban environments are not favourable to the growth of photochemically nucleated particles. In summary, although traffic remains the main source of UFP in urban areas, in developed countries with high insolation urban nucleation events are also a main source of UFP. If trafficrelated particle concentrations are reduced in the future, nucleation events will likely increase in urban areas, due to the reduced urban condensation sinks. [ABSTRACT FROM AUTHOR]

In this paper an efficient numerical method for solving a class of multipoint boundary value problems with special boundary conditions of Birkhoff-type is presented. After a quick reference to Birkhoff-type interpolation polynomial which satisfies the particular conditions, and a result on the existence and uniqueness of solution of the given problem, an algorithm is introduced to find a polynomial that approximates the solution. It is a general collocation method. Then an a priori estimation of the error of this approximation is given. Finally, to show the efficiency and the applicability of the method, numerical results are presented. These numerical experiments provide favourable comparisons with the NDSolve command of Mathematica. [ABSTRACT FROM AUTHOR]

For the numerical solution of high order boundary value problems with special boundary conditions a general procedure to determine collocation methods is derived and studied. Computation of the integrals which appear in the coefficients is generated by a recurrence formula and no integrals are involved in the calculation. Several numerical examples are presented to demonstrate the practical usefulness of the proposed method. [ABSTRACT FROM AUTHOR]

Recently we have introduced a new technique for combining classical bivariate Shepard operators with three point polynomial interpolation operators (Dell'Accio and Di Tommaso, On the extension of the Shepard-Bernoulli operators to higher dimensions, unpublished). This technique is based on the association, to each sample point, of a triangle with a vertex in it and other ones in its neighborhood to minimize the error of the three point interpolation polynomial. The combination inherits both degree of exactness and interpolation conditions of the interpolation polynomial at each sample point, so that in Caira et al. (J Comput Appl Math 236:1691-1707, ) we generalized the notion of Lidstone Interpolation (LI) to scattered data sets by combining Shepard operators with the three point Lidstone interpolation polynomial (Costabile and Dell'Accio, Appl Numer Math 52:339-361, ). Complementary Lidstone Interpolation (CLI), which naturally complements Lidstone interpolation, was recently introduced by Costabile et al. (J Comput Appl Math 176:77-90, ) and drawn on by Agarwal et al. () and Agarwal and Wong (J Comput Appl Math 234:2543-2561, ). In this paper we generalize the notion of CLI to bivariate scattered data sets. Numerical results are provided. [ABSTRACT FROM AUTHOR]

A determinantal form for Δ-Appell sequences is proposed and general properties are obtained by using elementary linear algebra tools. As particular cases of Δ-Appell sequences the sequence of Bernoulli polynomials of second kind and the one of Boole polynomials are considered. A general linear interpolation problem, which generalizes the classical interpolation problem on equidistant points, is proposed. The solution of this problem is expressed by a basis of Δ-Appell polynomials. Numerical examples which justify theoretical results on the interpolation problem are given. [ABSTRACT FROM AUTHOR]

Characterizing chemical and physical aerosol properties is important to understand their sources, effects, and feedback mechanisms in the atmosphere. This study proposes a scheme to classify aerosol populations based on their spectral optical properties (absorption and scattering). The scheme is obtained thanks to the outstanding set of information on particle size and composition these properties contain. The spectral variability of the aerosol single scattering albedo (dSSA), and the extinction, scattering and absorption Angstrom exponents (EAE, SAE and AAE, respectively) were observed on the basis of two-year measurements of aerosol optical properties (scattering and absorption coefficients at blue, green and red wavelengths) performed in the suburbs of Rome (Italy). Optical measurements of various aerosol types were coupled to measurements of particle number size distributions and relevant optical properties simulations (Mie theory). These latter allowed the investigation of the role of the particle size and composition in the bulk aerosol properties observed. The combination of simulations and measurements suggested a general "paradigm" built on dSSA, SAE and AAE to optically classify aerosols. The paradigm proved suitable to identify the presence of key aerosol populations, including soot, biomass burning, organics, dust and marine particles. The work highlights that (i) aerosol populations show distinctive combinations of SAE and dSSA times AAE, these variables being linked by a linear inverse relation varying with varying SSA; (ii) fine particles show EAE>1.5, whilst EAE<2 is found for both coarse particles and ultrafine soot-rich aerosols; (iii) fine and coarse particles both show SSA>0.8, whilst ultrafine urban Aitken mode and soot particles show SSA<0.8. The proposed paradigm agrees with aerosol observations performed during past major field campaigns, this indicating that relations concerning the paradigm have a general validity. [ABSTRACT FROM AUTHOR]

Particulate matter mass concentrations measured in the city of Rome (Italy) in the period 2001-2004 have been cross-analysed with concurrent Saharan dust advection events to infer the impact these natural episodes bear on the standard air quality parameter PM10 observed at two city stations and at one regional background station. Natural events as Saharan dust advections are associated to a definite health risk. At the same time, the Directive 2008/50/EC allows subtraction of PM exceedances caused by natural contributions from statistics used to determine air-quality of EU sites. In this respect, it is important to detect and characterize such advections by means of reliable, operational techniques. To assess the PM10 increase we used both the regional-background method" suggested by EC Guidelines and a "local background" one, demonstrated to be most suited to this central Mediterranean region. The two approaches provided results within 20% from each other. The sequence of Saharan advections over the city has been either detected by Polarization Lidar (laser radar) observations or forecast by the operational numerical regional mineral dust model BSC-DREAM8b of the Barcelona Supercomputing Centre. Lidar observations were also employed to retrieve the average physical properties of the dust clouds as a function of height. Along the four-year period, Lidar measurements 703 evenly distributed days) revealed Saharan plumes transits over Rome on 28.6% of the days, with minimum occurrence in wintertime. Dust was observed to reach the ground on 17.5% of the days totalling 88 episodes. Most (90 %) of these advections lasted up to 5 days, averaging to ~3 days. Median time lag between advections was 7 days. Typical altitude range of the dust plumes was 0-6 km, with centre of mass at ~3 kma.g.l. BSC-DREAM8b model simulations (1461 days) predicted Lidar detectable 532nm extinction coefficient >0.005 km-1) dust advections on 25.9% of the days, with ground contacts on 13% of the days. As in the Lidar case, the average dust centre of mass was forecast at ~3 km. Along the 703-day Lidar dataset, model forecast and Lidar detection of the presence of dust coincided on 80% of the cases, 92% coincidences are found within a ±1-day window. Combination of the BSC-DREAM8b and Lidar records leads to about 21% of the days being affected by presence of Saharan dust at the ground. This combined dataset has been used to compute the increase in PM with respect to dust-unaffected previous days. This analysis has shown Saharan dust events to exert a meaningful impact on the PM10 records, causing average increases of the order of 11.9 µg m-3. Conversely, PM10 increases computed relying only on the Lidar detections (i.e., presence of dust layers actually observed) were of the order of 15.6 µg m-3. Both analyses indicate the annual average contribution of dust advections to the city PM10 mass concentrations to be of the order of 2.35 µg m-3. These results confirm Saharan advections in the central Mediterranean as important modulators of PM10 loads and exceedances. [ABSTRACT FROM AUTHOR]

We analyze summertime photochemistry near the surface in Beijing, China, using a 1-D photochemical model (Regional chEmical and trAnsport Model, REAM-1D) constrained by in situ observations, focusing on the budgets of ROx (OH + HO2 + RO2) radicals and O3 formation. While the modeling analysis focuses on near-surface photochemical budgets, the implications for the budget of O3 in the planetary boundary layer are also discussed. In terms of daytime average, the total ROx primary production rate near the surface in Beijing is 6.6ppbv per hour (ppbv h-1, among the highest found in urban atmospheres. The largest primary ROx source in Beijing is photolysis of oxygenated volatile organic compounds (OVOCs), which produces HO2 and RO2 at 2.5ppbv h-11 and 1.7ppbv h-1, respectively. Photolysis of excess HONO from an unknown heterogeneous source is the predominant primary OH source at 2.2ppbv h-1, much larger than that of O¹D+H2O (0.4 ppbv h-11). The largest ROx sink is via OH + NO2 reaction (1.6 ppbv h-11), followed by formation of RO2NO2 (1.0 ppbv h-1) and RONO2 (0.7 ppbv h-11). Due to the large aerosol surface area, aerosol uptake of HO2 appears to be another important radical sink, although the estimate of its magnitude is highly variable depending on the uptake coefficient value used. The daytime average O3 production and loss rates near the surface are 32ppbv h-1 and 6.2ppbv h-1, respectively. Assuming NO2 to be the source of excess HONO, the NO2 to HONO trans formation leads to considerable O3 loss and reduction of its lifetime. Our observation-constrained modeling analysis suggests that oxidation of VOCs (especially aromatics) and heterogeneous reactions (e.g. HONO formation and aerosol uptake HO2) play potentially critical roles in the primary radical budget and O3 formation in Beijing. One important ramification is that O3 production is neither NOx nor VOC limited, but in a transition regime where reduction of either NOx or VOCs could result in reduction of O3 production. The transition regime implies more flexibility in the O3 control strategies than a binary system of either NOx or VOC limited regime. The co-benefit of concurrent reduction of both NOx and VOCs in reducing column O3 production integrated in the planetary boundary layer is significant. Further research on the spatial extent of the transition regime over the polluted eastern China is critically important for controlling regional O3 pollution. [ABSTRACT FROM AUTHOR]

Characterizing chemical and physical aerosol properties is important to understand their sources, effects, and feedback mechanisms in the atmosphere. This study proposes a scheme to classify aerosol populations based on their spectral optical properties (absorption and scattering). The scheme is obtained thanks to the outstanding set of information on particle size and composition these properties contain. The spectral variability of the aerosol Single Scattering Albedo (dSSA), and the Scattering and Absorption Angstrom Exponents (SAE and AAE, respectively) were observed on the basis of two-year measurements of aerosol optical properties (scattering and absorption coefficients at blue, green and red wavelengths) performed in the suburbs of Rome (Italy). Optical measurements of various aerosol types were coupled to measurements of particle number size distributions and relevant optical properties simulations (Mie theory). These latter allowed to investigate the role of the particle size and composition in the bulk aerosol properties observed. The combination of simulations and measurements suggested a general "paradigm" built on dSSA, SAE and AAE to optically classify aerosols. The paradigm proved suitable to identify the presence of key aerosol populations, including soot, biomass burning, organics, dust and marine particles. The work highlights that: (i) aerosol populations show distinctive combinations of SAE and dSSA times AAE, these variables being linked by a linear inverse relation varying with varying SSA; (ii) fine particles show SAE > 1.5, whilst SAE < 1 is found for both coarse particles and ultrafine soot-rich aerosols; (iii) fine and coarse particles both show SSA > 0.8, whilst ultrafine urban Aitken mode and soot particles show SSA < 0.8. A strict agreement was found when comparing the proposed paradigm to aerosol observations performed during past major field campaigns. [ABSTRACT FROM AUTHOR]

We analyze summertime photochemistry near the surface over Beijing, China, using a 1-D photochemical model (Regional chEmical and trAnsport Model, REAM-1D) constrained by in situ observations, focusing on the budgets of ROx (OH + HO2 + RO2) radicals and O3 formation. The daytime average of total ROx primary production rate in Beijing is ~6.6 ppbv h-1 , among the highest found in urban atmospheres. The largest primary ROx source in Beijing is photolysis of oxygenated volatile organic compounds (OVOCs), which produces HO2 and RO2 at average daytime rates of 2.5 ppbv h-1 and 1.7 ppbv h-1 , respectively. Photolysis of excess HONO from the unknown heterogeneous source is a predominant primary OH source at 2.2 ppbv h-1 , much larger than that of O¹D+H2O (0.4 ppbv h-1). The largest ROx sink is via OH+NO2 reaction (1.6 ppbv h-1), followed by formation of RO2NO2 (1.0 ppbv h-1) and RONO2 (0.7 ppbv h-1). Due to the large aerosol surface area, aerosol uptake of HO2 appears to be another important radical sink, although the estimate of its magnitude is highly variable depending on the reactive uptake coeficient value used. The daytime average O3 production and loss rates are 32 ppbv h-1 and 6.2 ppbv h-1 , respectively. Assuming NO2 to be the source of excess HONO, the NO2 to HONO transformation leads to significant O3 loss and reduction of its lifetime. Our observation-based modeling analyses suggest that VOCs and heterogeneous reactions (e.g. HONO formation and aerosol uptake HO2) play major roles in the primary radical budget and O3 formation in Beijing. Among the VOC precursors for OVOCs, which strongly affect ROx budgets and O3 formation, aromatics are the largest contributor. One important ramification is that O3 production is neither NOx nor VOC limited, but in a transition regime, where reduction of either NOx or VOCs could result in reduction of O3 production. The transition regime implies more flexibility in the O3 control strategies than a binary system of either NOx or VOC limited regime. Further research on the spatial extent of the transition regime over the polluted eastern China is critically important for controlling regional O3 pollution. [ABSTRACT FROM AUTHOR]

A class of polynomials Lk(x), k=1, 2, ... of degree 2k satisfying the differential equation L''k(x)=Lk-1(x) and the initial condition L'k(0)=0 is introduced. A relationship with a particular sequence of Appell polynomials of even degree is given. For any choice of one more boundary condition, a special sequence of polynomials is obtained. Special sequences of these polynomials are the basis of an interpolatory problem related to a fixed linear functional. Some of these polynomials are well known in literature. Finally, the problem of the polynomial expansion in these basis of real C∞ functions is posed. Some examples are given. [ABSTRACT FROM AUTHOR]

The class of polynomials Lk(x), k=1, 2, ..., of degree 2k+1 satisfying the differential equation L''k(x)=Lk-1(x) and the initial condition Lk(0)=0 is introduced. For any choice of one more condition, we have a definite and special sequence of polynomials. The sequences of polynomials related to a linear functional F on C2k+1 are interesting, since they are the basis of a particular non-classic interpolatory problem. [ABSTRACT FROM AUTHOR]

A correct description of fine (diameter <1μm) and ultrafine (<0.1μm) aerosol particles in urban areas is of interest for particle exposure assessment but also basic atmospheric research. We examined the spatio-temporal variability of atmospheric aerosol particles (size range 3-800 nm) using concurrent number size distribution measurements at a maximum of eight observation sites in and around Leipzig, a city in Central Europe. Two main experiments were conducted with different time span and number of observation sites (2 years at 3 sites; 1 month at 8 sites). A general observation was that the particle number size distribution varied in time and space in a complex fashion as a result of interaction between local and far-range sources, and the meteorological conditions. To identify statistically independent factors in the urban aerosol, different runs of principal component (PC) analysis were conducted encompassing aerosol, gas phase, and meteorological parameters from the multiple sites. Several of the resulting PCs, outstanding with respect to their temporal persistence and spatial coverage, could be associated with aerosol particle modes: a first accumulation mode ("droplet mode", 300-800 nm), considered to be the result of liquid phase processes and far-range transport; a second accumulation mode (centered around diameters 90-250 nm), considered to result from primary emissions as well as aging through condensation and coagulation; an Aitken mode (30-200 nm) linked to urban traffic emissions in addition to an urban and a rural Aitken mode; a nucleation mode (5-20 nm) linked to urban traffic emissions; nucleation modes (3-20 nm) linked to photochemically induced particle formation; an aged nucleation mode (10-50 nm). Additional PCs represented only local sources at a single site, or infrequent phenomena. In summary, the analysis of size distributions of high time and size resolution yielded a surprising wealth of statistical aerosol components occurring in the urban atmosphere over one single city. A paradigm on the behaviour of sub-μm urban aerosol particles is proposed, with recommendations how to efficiently monitor individual sub-fractions across an entire city. [ABSTRACT FROM AUTHOR]

For the numerical solution of the second order nonlinear two-point boundary value problems a family of polynomial global methods is derived. Numerical examples provide favorable comparisons with other existing methods. [ABSTRACT FROM AUTHOR]

It is well-known that Muller’s method for the computation of the zeros of continuous functions has order ≈ 1.84 [10], and does not have the character of global convergence. Muller’s method is based on the interpolating polynomial built on the last three points of the iterative sequence. In this paper the authors take as nodes of the interpolating polynomial the last two points of the sequence and the middle point between them. The resulting method has order p=2 for regular functions. This method leads to a globally convergent algorithm because it uses dichotomic techniques. Many numerical examples are given to show how the proposed code improves on Muller’s method. [ABSTRACT FROM AUTHOR]

In [1] there is an expansion in Bernoulli polynomials for sufficiently smooth real functions in an interval [ a, b]⊂ R that has useful applications to numerical analysis. An analogous result in a 2-dimensional context is derived in [2] in the case of rectangle. In this note we generalize the above-mentioned one-dimensional expansion to the case of C m-functions on a 2-dimensional simplex; a method to generalize the expansion on an N-dimensional simplex is also discussed. This new expansion is applied to find new cubature formulas for 2-dimensional simplex. [ABSTRACT FROM AUTHOR]

Recently, by Costabile, Gualtieri and Serra (1999), an iterative method was presented for the computation of zeros of C1 functions. This method combines the assured convergence of the bisection-like algorithms with a superlinear convergence speed which characterizes Newton-like methods. The order of the method and the cost per iteration is exactly equivalent to the Newton method. In this paper we present a new iterative method for the computation of the zeros of C1 functions with the same properties of convergence as the method proposed by Costabile, Gualtieri and Serra (1999) but with order 1+ $$\sqrt 2 $$ ≐2.41 for C3 functions. Compared with the methods of order 1+ $$\sqrt 2 $$ presented by Traub (1964), our methods ensure global convergence. Then we consider a generalization of this procedure which gives a class of methods of order ( n+ $$\sqrt {n^2 + 4} $$ )/2, where n is the degree of the approximating polynomial, with one-point iteration functions with memory. Finally a number of numerical tests are performed. The numerical results seem to show that, at least on a set of problems, the new methods work better than the methods proposed, and, therefore, than both the Newton and Alefeld and Potra (1992) methods. [ABSTRACT FROM AUTHOR]

For the numerical solution of the initial value problem a parallel, global integration method is derived and studied. It is a collocation method. If f( x, y)≡ f( x) the method coincides with the Filippi's modified Clenshaw–Curtis quadrature [11]. Two numerical algorithms are considered and implemented, one of which is the application of the new method to Picard iterations, so it is a waveform relaxation technique [3]. Numerical experiments are favourably compared with the ones given by the known GAM [2], GBS [14] and Sarafyan [18] methods. [ABSTRACT FROM AUTHOR]

A six order formula of an explicit method for the numerical initial value problem, proposed in [1], [3], is determined. This formula which for f(x,y)≡f(x) is the Gauss quadrature formula with three points, requires only three stages by comparison with seven stages required by an explicit six order Runge-Kutta method. Convergence and stability are considered and numerical example are also given. [ABSTRACT FROM AUTHOR]

In this paper a detailed study of the convergence and stability of a numerical method for the differential problem has carried out and its truncation error estimated. Some numerical experiments are described. [ABSTRACT FROM AUTHOR]

Copyright of Calcolo is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Calcolo is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Calcolo is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Calcolo is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Calcolo is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Calcolo is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Two new general classes of polynomial sequences called respectively odd and even Lidstone-type polynomials are considered. These classes include classic Lidstone polynomials of first and second kind. Some characterizations of the two classes are given, including matrix form, conjugate sequences, generating function, recurrence relations, and determinant forms. Some examples are presented and some applications are sketched. [ABSTRACT FROM AUTHOR]