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The advantages and disadvantages of application of Padé approximants to regression analysis with two independent covariates are discussed. The main difficulty of using Padé function is nonlinearity of data fitting. Possible approaches to overcoming the problem are discussed. New formulation of residuals is suggested in the method of least squares. It leads to a system of linear equations in case of rational functions. The possibility of using ridge regularization technique to avoid overfitting is demonstrated in this approach. To illustrate the efficiency of the suggested method, several practical cases from physics and reliability theory are considered. [ABSTRACT FROM AUTHOR]

This article presents a buck converter in which the inductor has been replaced by a transmission line. This approach would be practically conceivable if the power supply and load had a greater spatial distance. Alternatively, the model derived in this way could also be regarded as an intermediate model in order to replace a power coil via discretization with a larger number of smaller coils and capacitors. In the time domain, this new converter can be described by a system of coupled partial and ordinary differential equations. In the frequency domain, a transcendental transfer function is obtained. For comparison with an equivalently parameterized conventional converter, Padé approximants are derived. A linear controller is designed for the converter topology under consideration. [ABSTRACT FROM AUTHOR]

Sea ice regulates heat exchange between the ocean and atmosphere in Earth's polar regions. The thermal conductivity of sea ice governs this exchange, and is a key parameter in climate modelling. However, it is challenging to measure and predict due to its sensitive dependence on temperature, salinity and brine microstructure. Moreover, as temperature increases, sea ice becomes permeable, and fluid can flow through the porous microstructure. While models for thermal diffusion through sea ice have been obtained, advective contributions to transport have not been considered theoretically. Here, we homogenize a multiscale advection–diffusion equation that models thermal transport through porous sea ice when fluid flow is present. We consider two-dimensional models of convective flow and use an integral representation to derive bounds on the thermal conductivity as a function of the Péclet number. These bounds guarantee enhancement in the thermal conductivity due to the added flow. Further, we relate the Péclet number to temperature, making these bounds useful for global climate models. Our analytic approach offers a mathematical theory which can not only improve predictions of atmosphere–ice–ocean heat exchanges in climate models, but can provide a theoretical framework for a range of problems involving advection–diffusion processes in various fields of application. [ABSTRACT FROM AUTHOR]

OBJECTIVE: We developed and evaluated a novel one-shot distributed algorithm for evidence synthesis in distributed research networks with rare outcomes. MATERIALS AND METHODS: Fed-Padé, motivated by a classic mathematical tool, Padé approximants, reconstructs the multi-site data likelihood via Padé approximant whose key parameters can be computed distributively. Thanks to the simplicity of [2,2] Padé approximant, Fed-Padé requests an extremely simple task and low communication cost for data partners. Specifically, each data partner only needs to compute and share the log-likelihood and its first 4 gradients evaluated at an initial estimator. We evaluated the performance of our algorithm with extensive simulation studies and four observational healthcare databases. RESULTS: Our simulation studies revealed that a [2,2]-Padé approximant can well reconstruct the multi-site likelihood so that Fed-Padé produces nearly identical estimates to the pooled analysis. Across all simulation scenarios considered, the median of relative bias and rate of instability of our Fed-Padé are both

This paper is devoted to the continualization of a diatomic lattice, taking into account natural intervals of wavenumber changes. Continualization refers to the replacement of the original pseudo-differential equations by a system of PDEs that provides a good approximation of the dispersion relations. In this regard, the Padé approximants based on the conditions for matching the values of the dispersion relations of the discrete and continuous models at several characteristic points are utilized. As a result, a sixth-order unconditionally stable system with modified inertia is obtained. Appropriate boundary conditions are formulated. The obtained continuous approximation accurately describes the amplitude ratios of neighboring masses. It is also shown that the resulting continuous system provides a good approximation for the natural frequencies. [ABSTRACT FROM AUTHOR]

A generalized Padé-type expression is proposed for Henry's constant of gases in ionic liquids. The constants are determined using an equation of state, and generalized expressions for the Henry's constants of the gases in the ionic liquids are proposed. The solute gases included in the study were oxygen, hydrogen, and carbon monoxide in three solvent ionic liquids ([MDEA][Cl], [Bmim][PF6], and [Hmim][TF2N]). The Valderrama–Patel–Teja equation of state with the mixing rules of Kwak and Mansoori are employed to correlate the solubility data, to examine the thermodynamic consistency of the experimental data, and to determine the fugacity (fi) for each concentration (xi) of the solute gas in the liquid phase. From these data, the fugacity coefficients (fiL/xi) are determined to obtain Henry´s constant as Hi = lim(fiL/xi) when xi→0. The calculated Henry's constants are correlated in terms of the temperature and acentric factor of the gases to finally obtain a generalized expression for Henry´s constant, Hi. [ABSTRACT FROM AUTHOR]

This paper is devoted to the continualization of a diatomic lattice, taking into account natural intervals of wavenumber changes. Continualization refers to the replacement of the original pseudo-differential equations by a system of PDEs that provides a good approximation of the dispersion relations. In this regard, the Padé approximants based on the conditions for matching the values of the dispersion relations of the discrete and continuous models at several characteristic points are utilized. As a result, a sixth-order unconditionally stable system with modified inertia is obtained. Appropriate boundary conditions are formulated. The obtained continuous approximation accurately describes the amplitude ratios of neighboring masses. It is also shown that the resulting continuous system provides a good approximation for the natural frequencies.

The present paper is strongly motivated by the brilliant work of Wazwaz [9, Sections 4 and 5] computing analytical approximation in the form of a series truncated at t8 and applying [4/4] Pade approximant to the series. In this paper, we make an attempt to workout analytical approximations in the form of the series truncated at t4 and apply suitable Pade approximations as well as asymptotic approximations with the following features: i) The solution contains exact non integral part. ii) The solution exhibits the population rapid rise along logistic curve followed by decay to zero in the long run. iii) The solution is reasonably comparable with that of Wazwaz [9] using the information from the series with terms only upto t4 . [ABSTRACT FROM AUTHOR]

Inverse Langevin function has an extensive use in statistical mechanics, polymer chemistry, and physics. Main struggle is that the inverse Langevin function cannot be expressed in an exact analytical form. To this end, many approaches to estimate the inverse Langevin function have been proposed. A trade-off can be observed between level of accuracy and mathematical complexity in the existing approximants in the literature. In the present contribution, a simple, yet efficient one-pass predictor-corrector algorithm is proposed for the accurate prediction of the inverse Langevin function. The predictor step uses the approximants yp (x) proposed in the literature. The corrector term is based on a single iteration applied to the error function L (yp) – x. The correction term is based on the yp and has the same form irrespective of the specific approximant used in the predictor step. Hence, the additional computational cost is constant irrespective of the approximant used as predictor. In order to demonstrate the accuracy and efficiency of the approach, maximum relative error and the computational cost before and after corrector step are analysed for eight different approximants. The proposed one-pass predictor-corrector approach allows the use of relatively simpler approximants of the inverse Langevin function by improving their accuracy by at least an order of magnitude. [ABSTRACT FROM AUTHOR]

In this work, we have computed and implemented one‐body integrals concerning Gaussian confinement potentials over Gaussian basis functions. Then, we have set an equivalence between Gaussian and Hooke atoms and we have observed that, according to singlet and triplet state energies, both systems are equivalent for large confinement depth for a series of even number of electrons n = 2, 4, 6, 8 and 10. Unlike with harmonic potentials, Gaussian confinement potentials are dissociative for small enough depth parameter; this feature is crucial in order to model phenomena such as ionization. In this case, in addition to corresponding Taylor‐series expansions, the first diagonal and sub‐diagonal Padé approximant were also obtained, useful to compute the upper and lower limits for the dissociation depth. Hence, this method introduces new advantages compared to others. [ABSTRACT FROM AUTHOR]

The transparent boundary condition for the free Schrödinger equation on a rectangular computational domain requires implementation of an operator of the form ∂ t − i △ Γ where △ Γ is the Laplace-Beltrami operator. It is known that this operator is nonlocal in time as well as space which poses a significant challenge in developing an efficient numerical method of solution. The computational complexity of the existing methods scale with the number of time-steps which can be attributed to the nonlocal nature of the boundary operator. In this work, we report an effectively local approximation for the boundary operator such that the resulting complexity remains independent of number of time-steps. At the heart of this algorithm is a Padé approximant based rational approximation of certain fractional operators that handles corners of the domain adequately. For the spatial discretization, we use a Legendre-Galerkin spectral method with a new boundary adapted basis which ensures that the resulting linear system is banded. A compatible boundary-lifting procedure is also presented which accommodates the segments as well as the corners on the boundary. The proposed novel scheme can be implemented within the framework of any one-step time marching schemes. In particular, we demonstrate these ideas for two one-step methods, namely, the backward-differentiation formula of order 1 (BDF1) and the trapezoidal rule (TR). For the sake of comparison, we also present a convolution quadrature based scheme conforming to the one-step methods which is computationally expensive but serves as a golden standard. Finally, several numerical tests are presented to demonstrate the effectiveness of our novel method as well as to verify the order of convergence empirically. • We presented an efficient numerical realization of the transparent boundary operator ∂ t − i △ Γ on a rectangular domain. • We first present a convolution quadrature (CQ) based numerical recipe that turns out to be computationally expensive. • A novel Padé algorithm is then developed which removes all bottlenecks of the CQ approach and also of Menza's approach. • A new boundary adapted basis is presented for the spatial problem that ensures the bandedness of the linear systems. • Note that the computational complexity of our novel Padé algorithm remains independent of the number of time-steps. [ABSTRACT FROM AUTHOR]

In this paper, a numerical scheme so-called modified differential transfor-mation method (MDTM) based on differential transformation method (DTM), Laplace transform and Pade approximation will be used to obtain accurate approximate solution for a class of boundary value problems (BVP's). The MDTM is employed as an alter-native technique to overcome some difficulties in the behavior of the solution and to be valid for a large region. The numerical results obtained demonstrate the applicability and validity of this technique. Numerical comparison is made with existing exact solution. [ABSTRACT FROM AUTHOR]

This article focuses on analytic solutions for Newtonian fluid flow with slip and mass transpiration on a porous stretching sheet using the differential transform method and Pade approximants of an exceptionally nonlinear differential equation. The impacts of different parameters including mass transpiration (suction/injection), Navier's slip, and Darcy number parameters on the velocity of the liquid and tangential stress are discussed. A comprehensive comparison of our results with the previous one in the literature is made, and the results showed good agreement. An investigation is conducted of a combination of magnetic liquids that are conceivably pertinent for wound medicines, skin repair, and astute coatings for natural gadgets. It is found that there is a decrease in the velocity profiles and the boundary layer thickness for the case of suction. [ABSTRACT FROM AUTHOR]

For the quantum quartic anharmonic oscillator with the Hamiltonian H = 1 2 (p 2 + x 2) + λ x 4 , which is one of the traditional quantum-mechanical and quantum-field-theory models, the summation of its factorially divergent perturbation series is studied on the basis of the proposed method of the averaging of the corresponding Padé approximants. Thus, applying proper averaging weight function, we are able for the first time to construct the Padé-type approximations that possess correct asymptotic behavior at infinity with a rise of the coupling constant λ. The approach gives very essential theoretical and applicatory-computational advantages in applications of the given method. The convergence of the utilized approximations is studied and the values for the ground state energy E 0 (λ) of the anharmonic oscillator are calculated by the proposed method for a wide range of variation of the coupling constant λ. In addition, we perform comparative analysis of the proposed method with the modern Weniger delta-transformation method and show insufficiency of the latter to sum the divergent perturbation series in the region of the superstrong coupling λ ≳ 5. [ABSTRACT FROM AUTHOR]

Many engineered materials display ordered or disordered microstructures. Such materials exhibit transport properties which are unmatched by their single‐phase homogeneous counterparts. These properties are obtained by the mixture of two or more phases typically characterized by a large contrast in their properties. For the development of these materials, it is critical to develop a robust computational framework in order to provide a fundamental understanding of how microstructure affects performance. This hinges on predicting their macroscopic properties, given the constitutive laws and spatial distribution of their constituents. To this end, this work presents a computational framework based on formulating periodic conduction transport problems in terms of boundary integral equations whose kernel is expressed in terms of Weierstrass zeta‐function. The components of the effective conductivity tensor are then sought in the form of power series expansions of a conductivity contrast parameter. To accelerate their convergence, these expansions are transformed into Padé approximants. Presently restricted to the case of two‐dimensional, two‐phase microstructures, this framework is shown to yield accurate results over the entire range of the contrast parameter. Representation of the kernel as a lattice sum allows the use the fast multipole method, thereby making computations significantly more efficient. [ABSTRACT FROM AUTHOR]

Different strategies can be used to acquire dynamic impedance spectra during a cyclic voltammetry experiment. The spectra are then analyzed by fitting them with a model using a weighted non‐linear least‐squares minimization algorithm. The choice of the weighting factors is not trivial and influences the value of the extracted parameters. At variance with the classic electrochemical impedance spectroscopy, dynamic impedance measurements are performed under non‐stationary conditions, making them typically more prone to errors arising from the voltage and current analog‐to‐digital conversion. Under the assumption that the noise in the voltage and current signals have a constant variance along the measurement and that it is uncorrelated between distinct samples, we calculate an expression for the expected variance of the error of the resulting immittances, which considers the specific procedure used to extract the spectra under the time‐varying nature of the measurements. The calculated variance of the error is then used as a rigorous way to evaluate the weighting factors of the least‐squares minimization, assuming that the fitted model is ideally exact and that there are no systematic errors. By exploring two classical electrochemical systems and fitting the measured spectra with a transfer function measurement model, namely with the Padé approximants, we show that the variance evaluated with our method captures the frequency dependence of the resulting residuals and can be used for reliably performing the complex non‐linear least‐squares fitting procedure. [ABSTRACT FROM AUTHOR]

The problem with using asymptotic decompositions to solve boundary problems is that the convergence radius is not large enough to cover both boundaries. To solve this problem, it is proposed to use the method of connecting local asymptotics with Pad´e approximants (PAs). Given the connection of asymptotics to Pad´e approximants, power series become an effective tool for ordinary differential equations of boundary problems. The use of PAs allows in many cases to overcome the local nature of the resulting decompositions inherent in the method of matching asymptotics. Our paper discusses the benefits of using PA to solve mixed problems in mathematical physics. [ABSTRACT FROM AUTHOR]

The simulation of the predissociation spectrum of the Na 2 2 3 Π g ∼ 3 3 Π g ← b 3 Π u (v = 14 , J = 14) transition, observed earlier with the Perturbation Facilitated Optical–Optical Double Resonance (PFOODR) experimental method, is done within the Optimizer project with our Split-operator computational package using Padé approximations for all functions involved in the model. This simulation reproduces the experimental spectrum satisfactory well. The parameters of the model functions involved in the simulations are determined and reported. A brief description of our computational methods and programs is presented. [Display omitted] • Simulation of the predissociation spectrum Na 2 2 3 Π g ∼ 3 3 Π g ← b 3 Π u (v = 14 , J = 14) is done. • New version of the multichannel Split-operator computational package is presented. • New version of the Padé-approximant computational package is presented. • All the programs are adapted to the Optimizer project and used for the modeling. • Quantitative parameters of all the functions involved in the model are determined. [ABSTRACT FROM AUTHOR]

Sparse polynomial interpolation, sparse linear system solving or modular rational reconstruction are fundamental problems in Computer Algebra. They come down to computing linear recurrence relations of a sequence with the Berlekamp–Massey algorithm. Likewise, sparse multivariate polynomial interpolation and multidimensional cyclic code decoding require guessing linear recurrence relations of a multivariate sequence. Several algorithms solve this problem. The so-called Berlekamp–Massey–Sakata algorithm (1988) uses polynomial additions and shifts by a monomial. The Scalar-FGLM algorithm (2015) relies on linear algebra operations on a multi-Hankel matrix, a multivariate generalization of a Hankel matrix. The Artinian Gorenstein border basis algorithm (2017) uses a Gram-Schmidt process. We propose a new algorithm for computing the Gröbner basis of the ideal of relations of a sequence based solely on multivariate polynomial arithmetic. This algorithm allows us to both revisit the Berlekamp–Massey–Sakata algorithm through the use of polynomial divisions and to completely revise the Scalar-FGLM algorithm without linear algebra operations. A key observation in the design of this algorithm is to work on the mirror of the truncated generating series allowing us to use polynomial arithmetic modulo a monomial ideal. It appears to have some similarities with Padé approximants of this mirror polynomial. As an addition from the paper published at the ISSAC conference, we give an adaptive variant of this algorithm taking into account the shape of the final Gröbner basis gradually as it is discovered. The main advantage of this algorithm is that its complexity in terms of operations and sequence queries only depends on the output Gröbner basis. All these algorithms have been implemented in Maple and we report on our comparisons. [ABSTRACT FROM AUTHOR]

Modern biomedical and tribological systems are increasingly deploying combinations of nanofluids and bioconvecting microorganisms which enable improved control of thermal management. Motivated by these developments, in this study, a new mathematical model is developed for the combined nanofluid bioconvection axisymmetric squeezing flow between rotating circular plates (an important configuration in, for example, rotating bioreactors and lubrication systems). The Buongiorno two-component nanoscale model is deployed, and swimming gyrotactic microorganisms are considered which do not interact with the nanoparticles. Thermal radiation is also included, and a Rosseland diffusion flux approximation is utilized. Appropriate similarity transformations are implemented to transform the nonlinear, coupled partial differential conservation equations for mass, momentum, energy, nanoparticle species and motile microorganism species under suitable boundary conditions from a cylindrical coordinate system into a dimensionless nonlinear ordinary differential boundary value problem. An efficient scheme known as differential transform method (DTM) combined with Padé-approximations is then applied to solve the emerging nonlinear similarity equations. The impact of different non-dimensional parameters i.e. squeezing Reynolds number, rotational Reynolds number, Prandtl number, thermophoresis parameter, Brownian dynamics parameter, thermal radiation parameter, Schmidt number, bioconvection number and Péclet number on velocity, temperature, nanoparticle concentration and motile gyrotactic microorganism density number distributions is computed and visualized graphically. The torque effects on both plates, i.e. the lower and the upper plate, are also determined. From the graphical results, it is seen that momentum in the squeezing regime is suppressed clearly as the upper disk approaches the lower disk. This inhibits the axial flow and produces axial flow retardation. Similarly, by enhancing the value of squeezing Reynolds number, the tangential velocity distribution also decreases. More rigorous squeezing clearly therefore also inhibits tangential momentum development in the regime and leads to tangential flow deceleration. Tables are also provided for multiple values of flow parameters. The numerical values obtained by DTM-Padé computation show very good agreement with shooting quadrature. DTM-Padé is shown to be a precise and stable semi-numerical methodology for studying rotating multi-physical flow problems. Radiative heat transfer has an important influence on the transport characteristics. When radiation is neglected, different results are obtained. It is important therefore to include radiative flux in models of rotating bioreactors and squeezing lubrication dual disk damper technologies since high temperatures associated with radiative flux can impact significantly on combined nanofluid bioconvection which enables more accurate prediction of actual thermofluidic characteristics. Corrosion and surface degradation effects may therefore be mitigated in designs. [ABSTRACT FROM AUTHOR]

In this research article, a numerical scheme namely modified differential transformation method (MDTM) is employed successfully to obtain accurate approximate solutions for classes of nonlinear differential equations. This scheme based on differential transform method (DTM), Laplace transform and Padé approximants. Validity and efficiency of MDTM are tested upon several examples and comparisons are made to demonstrate that. The results lead to conclude that the MDTM is effective, explicit and easy to use. [ABSTRACT FROM AUTHOR]

Inequalities play a fundamental role in both theoretical and applied mathematics and contain many patterns of symmetries. In many studies, inequalities have been used to provide estimates of some functions based on the properties of their symmetry. In this paper, we present the following new asymptotic expansion related to the ordinary gamma function Γ(1+w)∼2πw(w/e)ww2+760w2−120w/2exp∑r=1∞μrwr,w→∞, with the recurrence relation of coefficients μr. Furthermore, we use Padé approximants and our new asymptotic expansion to deduce the new bounds of Γ(w) better than some of its recent ones.

The Hurwitz zeta function ζ (s , a) admits a well-known (divergent) asymptotic expansion in powers of 1 / a involving the Bernoulli numbers. Using Wilson orthogonal polynomials, we determine an effective bound for the error made when this asymptotic series is replaced by nearly diagonal Padé approximants. By specialization, we obtain new fast converging sequences of rational approximations to the values of the Riemann zeta function at every integers ≥2. The latter can be viewed, in a certain sense, as analogues of Apéry's celebrated sequences of rational approximations to ζ (2) and ζ (3). [ABSTRACT FROM AUTHOR]

For the quantum quartic anharmonic oscillator with the Hamiltonian H = (p²+x²)/2+λx4, which is one of the traditional quantum-mechanical and quantum-field-theory models, we study summation of its factorially divergent perturbation series by the proposed method of averaging of the corresponding Padé approximants. Thus, for the first time, we are able to construct the Padé-type approximations that possess correct asymptotic behaviour at infinity with a rise of the coupling constant λ. The approach gives very essential theoretical and applicatory-computational advantages in applications of the given method. We also study convergence of the applied approximations and calculate by the proposed method the ground state energy E0(λ) of the anharmonic oscillator for a wide range of variation of the coupling constant λ. [ABSTRACT FROM AUTHOR]

The Digamma function Γ ′ / Γ admits a well-known (divergent) asymptotic expansion involving the Bernoulli numbers. Using Touchard-type orthogonal polynomials, we determine an effective bound for the error made when this asymptotic expansion is replaced by its nearly diagonal Padé approximant. By specialization, we obtain new fast converging sequences of approximations to Euler's constant γ . Even though these approximations are not strong enough to prove the putative irrationality of γ , we explain why they can be viewed, in some sense, as analogs of Apéry's celebrated sequences of approximations to ζ (2) and ζ (3) . Similar ideas applied to the asymptotic expansion log Γ enable us to obtain a refined version of Stirling's formula. [ABSTRACT FROM AUTHOR]

This paper presents an iterative technique based on homotopy analysis method for solving system of Volterra integro-differential equations. The technique provides us series solutions to the problems which are combined with the diagonal Padé approximants and Laplace transform to obtain closed-form solutions. The technique is effectively applied on system of linear and nonlinear Volterra integro-differential equations which eventually yield closed-form solutions of the problems and this technique is also extended to boundary value problem for the integro-differential equation related to Blasius problem. An interesting comparison of the present solution is made with solutions of other methods and it is observed that the results are in excellent agreement with other methods in the literature. [ABSTRACT FROM AUTHOR]

In this paper, by making use of the generating function methods and Padé approximation techniques, we establish some new recurrence formulas for the Hermite polynomials and their squares. These results presented here are the corresponding extensions of some known formulas.

The rational GARCH (RGARCH) model has been proposed as an alternative GARCHmodel that captures the asymmetric property of volatility. In addition to the previously proposedRGARCH model, we propose an alternative RGARCH model called the RGARCH-Exp model thatis more stable when dealing with outliers. We measure the performance of the volatility estimationby a loss function calculated using realized volatility as a proxy for true volatility and compare theRGARCH-type models with other asymmetric type models such as the EGARCH and GJR models.We conduct empirical studies of six stocks on the Tokyo Stock Exchange and find that a volatilityestimation using the RGARCH-type models outperforms the GARCH model and is comparable toother asymmetric GARCH models.

The performed numerical analysis reveals that Wynn's identity for the compass 1 / (N − C) + 1 / (S − C) = 1 / (W − C) + 1 / (E − C) = 1 / η (here C stands for centre, the other letters correspond to the four directions of the compass) gives the long sought criterion, the minimal | η | or Image 1 -criterion, for the choice of the optimal Padé approximant. The work of this method is illustrated by calculation of multipoint Padé approximation by a new formula for calculation of this best rational approximation. The calculation of the optimal Padé approximant by this criterion is demonstrated in calculation of series summation – frequently encountered problem in theoretical physics. This study originates from a magneto-hydrodynamic problem of heating of solar corona by Alfvén waves, where the present method is used for a predictor in solution of differential equations. In such a way, an efficient and valuable control mechanism for N -point Padé approximant calculation is proposed. We believe that the suggested method and criterion can be useful for many applied problems in numerous areas not only in physics but in any scientific application where differential equations are solved. The obtained new solution of the Cauchy-Jacobi problem is illustrated by a Fortran program. The algorithm is generalized for the case of the first K -derivatives at N -nodal points. The numerical analysis of probability distribution function of the errors will review, which formula for extrapolation of functions deserves to be implemented in the applied mathematics software like Mathematica and Maple. [ABSTRACT FROM AUTHOR]

First-principles calculations of density functional theory (DFT) using the generalized gradient approximation (GGA) and generalized gradient approximation with the Hubbard U (GGA + U) for the exchange-correlation energy functional are used to investigate the structural, electronic, and magnetic properties of the spinel system C d x N i 1 − x F e 2 O 4 in the range 0 ≤ x ≤ 1. High-temperature series expansions (HTSE) combined with the Padé approximants (PA) method and mean field theory (MFT) are applied to the C d x N i 1 − x F e 2 O 4 systems. The lattice parameter and magnetic moment of Fe and Ni ions increase when Cd content x increases. Density of States (DOS) calculations show that the C d x N i 1 − x F e 2 O 4 has a semiconductor behaviour. The obtained theoretical results show that the Curie temperature decreases with x. These obtained results are in agreement with experimental ones. [ABSTRACT FROM AUTHOR]

The properties of a model based on the analogy between binary stratifying liquid mixtures and coupled self-oscillating systems is studied. In this paper, an expression representing a curve mapping in the phase diagram of a binary system is derived. The mapping of a boundary curve is presented in the form of the dependence between the ratio of characteristic frequencies assigned to pure components and the mixture concentration. In the framework of this model, the temperature dependence of the squared ratio of frequencies is also determined. One important aspect of this model is that the temperature-dependent characteristic frequencies and reflect the properties of pure components (instead of solutions). The problem is posed of investigating the physical aspects of the model proposed in this paper in more detail. In the present study, it has been shown that the use of Pade approximants makes it possible to determine the character of the temperature dependence of each characteristic frequency separately on the basis of the temperature dependence derived by linking to experimental data for the ratio . [ABSTRACT FROM AUTHOR]

First-principles calculations of density functional theory (DFT) are used to investigate the structural, electronic, and magnetic properties of the spinel system CdxCo1−xFe2O4 (x = 0, 0.5, and 1). High-temperature series expansions (HTSE) combined with the Padé approximants (PA) method and mean field theory (MFT) have been used to calculate the critical temperature (TC). The lattice parameter and the local magnetic moments of Co and Fe increased when Cd content x increases. The obtained theoretical results show that the Curie temperature decrease with x. Density of states (DOS) calculations show a half-metallic behavior for x = 0.5 and semiconducting behavior for x = 0 and 1. This work, provide useful information for the substitution effect of a nonmagnetic ion on the properties of spinel ferrites, based on first-principles calculations and HTSE. [ABSTRACT FROM AUTHOR]

Given the scalar sequence { f m } m = 0 ∞ that satisfies f m = ∑ i = 1 k a i ζ i m , m = 0 , 1 , … , where a i , ζ i ∈ ℂ and ζi are distinct, the algorithm of Prony concerns the determination of the ai and the ζi from a finite number of the fm. This algorithm is also related to Padé approximants from the infinite power series ∑ j = 0 ∞ f j z j . In this work, we discuss ways of extending Prony’s algorithm to sequences of vectors { f m } m = 0 ∞ in ℂ N that satisfy f m = ∑ i = 1 k a i ζ i m , m = 0 , 1 , … , where a i ∈ ℂ N and ζ i ∈ ℂ . Two distinct problems arise depending on whether the vectors ai are linearly independent or not. We consider different approaches that enable us to determine the ai and ζi for these two problems, and develop suitable methods. We concentrate especially on extensions that take into account the possibility of the components of the ai being coupled. One of the applications we consider concerns the case in which f m = ∑ i = 1 r a i ζ i m , m = 0 , 1 , … , r large , and we would like to approximate/determine of a number of the pairs (ζi, ai) for which |ζi| are largest. We present the related theory and provide numerical examples that confirm this theory. This application can be extended to the more general case in which f m = ∑ i = 1 r p i (m) ζ i m , m = 0 , 1 , … , where p i (m) ∈ ℂ N are some (vector-valued) polynomials in m, and ζ i ∈ ℂ are distinct. Finally, the methods suggested here can be extended to vector sequences in infinite dimensional spaces in a straightforward manner. [ABSTRACT FROM AUTHOR]

We study a reliable pole selection for the rational approximation of the resolvent of fractional powers of operators in both the finite and infinite dimensional setting. The analysis exploits the representation in terms of hypergeometric functions of the error of the Padé approximation of the fractional power. We provide quantitatively accurate error estimates that can be used fruitfully for practical computations. We present some numerical examples to corroborate the theoretical results. The behavior of rational Krylov methods based on this theory is also presented. [ABSTRACT FROM AUTHOR]

The present work deals with rational model order reduction methods based on the single-point Least-Square (LS) Padé approximation techniques introduced in Bonizzoni et al. (ESAIM Math. Model. Numer. Anal., 52(4), 1261–1284 2018, Math. Comput. 89, 1229–1257 2020). Algorithmical aspects concerning the construction of rational LS-Padé approximants are described. In particular, we show that the computation of the Padé denominator can be carried out efficiently by solving an eigenvalue-eigenvector problem involving a Gramian matrix. The LS-Padé techniques are employed to approximate the frequency response map associated with two parametric time-harmonic acoustic wave problems, namely a transmission-reflection problem and a scattering problem. In both cases, we establish the meromorphy of the frequency response map. The Helmholtz equation with stochastic wavenumber is also considered. In particular, for Lipschitz functionals of the solution and their corresponding probability measures, we establish weak convergence of the measure derived from the LS-Padé approximant to the true one. 2D numerical tests are performed, which confirm the effectiveness of the approximation methods. [ABSTRACT FROM AUTHOR]

This paper presents an interval approach for the Holomorphic Embedding Load Flow Method (HELM) considering load and generation uncertainties. In the proposed method, power flow equations are considered as holomorphic complex functions, being their solutions determined recursively based on the expansion in Maclaurin series and analytical continuation using Padé approximants. The concepts of the interval arithmetic are applied into the HELM equations assuming active and reactive powers as uncertain data to obtain interval values associated with the voltage phasors for all the system buses. The procedures to calculate interval results are described along the paper including the use of interval arithmetic concepts applied into the HELM equations. IEEE test systems such as 6, 14, 30 and 57-bus are used for computational simulations, being the results compared with Monte Carlo (MC) approach, Krawczyk method, Taylor series, affine arithmetic and the traditional interval power flow. As important contribution, a novel interval power flow technique is presented allowing to compute interval values closer to the ones determined by MC simulations than other alternative methods. Results are obtained in a computational time approximately 500 times faster than the MC approach considering 20,000 simulations and the interval sensitivities are, at least, twice lower than other interval methods. • The Interval Holomorphic Embedding Load Flow Method (IHELM) is presented. • Load and generation uncertainties are considered. • Results are obtained in an advantageous computational time. • Voltage intervals are determined by a direct method. [ABSTRACT FROM AUTHOR]

The volatility of stock return does not follow the classical Brownian motion, but instead it follows a form that is closely related to fractional Brownian motion. Taking advantage of this information, the rough version of classical Heston model also known as rough Heston model has been derived as the macroscopic level of microscopic Hawkes process where it acts as a high-frequency price process. Unlike the pricing of options under the classical Heston model, it is significantly harder to price options under rough Heston model due to the large computational cost needed. Previously, some studies have proposed a few approximation methods to speed up the option computation. In this study, we calibrate five different approximation methods for pricing options under rough Heston model to SPX options, namely a third-order Padé approximant, three variants of fourth-order Padé approximant, and an approximation formula made from decomposing the option price. The main purpose of this study is to fill in the gap on lack of numerical study on real market options. The numerical experiment includes calibration of the mentioned methods to SPX options before and after the Lehman Brothers collapse.

This paper is devoted to an overview of the basic properties of the Padé transformation and its generalizations. The merits and limitations of the described approaches are discussed. Particular attention is paid to the application of Padé approximants in the mechanics of liquids and gases. One of the disadvantages of asymptotic methods is that the standard ansatz in the form of a power series in some parameter usually does not reflect the symmetry of the original problem. The search for asymptotic ansatzes that adequately take into account this symmetry has become one of the most important problems of asymptotic analysis. The most developed technique from this point of view is the Padé approximation.

We consider the problem of finding approximate analytical solutions for nonlinear equations typical of physics applications. The emphasis is on the modification of the method of Padé approximants that are known to provide the best approximation for the class of rational functions, but do not provide sufficient accuracy or cannot be applied at all for those nonlinear problems, whose solutions exhibit behavior characterized by irrational functions. In order to improve the accuracy, we suggest a method of self-similarly corrected Padé approximants taking into account the irrational functional behavior. The idea of the method is in representing the sought solution as a product of two factors, one of which is given by a self-similar root approximant, responsible for irrational functional behavior, and the other being a Padé approximant corresponding to a rational function. The efficiency of the method is illustrated by constructing very accurate solutions for nonlinear differential equations. A thorough investigation is given proving that the suggested method is more accurate than the method of standard Padé approximants. [ABSTRACT FROM AUTHOR]

Propagation of elastic waves through discrete and continuous periodically heterogeneous media is studied. A two-scale asymptotic procedure allows us to derive macroscopic dynamic equations applicable at frequencies close to the resonant frequencies of the unit cells. Matching the asymptotic solutions by two-point Padé approximants, we obtain new higher-order equations that describe the dynamic behaviour of the medium both in the low and in the high frequency limits. An advantage of the proposed approach is that all the macroscopic parameters can be determined explicitly in terms of the microscopic properties of the medium. Dispersion diagrams are evaluated and the propagation of transient waves induced by pulse and harmonic loads is considered. The developed analytical models are verified by comparison with data of numerical simulations. For high-contrast media, we can observe an analogy between the propagation of waves in heterogeneous solids and in thin-walled waveguides. It is also shown that different combinations of cell resonances may result in some additional types of waves that do not appear in the classical continuous theory. [ABSTRACT FROM AUTHOR]

Detailed The analytical solution of the unsteady boundary layer flow due to impulsive porous stretching sheet is solved by means of an Adomian Decomposition Method (ADM). The ADM alongside Pade approximants are connected to tackle the nonlinear partial differential equation with different boundary conditions got in demonstrating the unsteady boundary layer flow. Apparently, this yields better accuracy when compared to results from other prime methods. The present results are in good match with the pioneering work of other authors. The impact of parameters like mass suction/injection parameter and Darcy number on the velocity profile were examined. The present results can be found very useful in a wide range of applications in extrusion process and related process in fluid dynamics and heat transfer problems. [ABSTRACT FROM AUTHOR]

An analytical algorithm is proposed for studying the stability of a cylindrical shell subjected to nonuniform external pressure. The algorithm is based on a perturbation procedure and Padé approximants. Unlike the commonly used perturbation method, this approach is not limited by the smallness of the perturbation parameter. The approximate formulas obtained are sufficiently accurate and can be used in engineering practice. [ABSTRACT FROM AUTHOR]

We perform a further investigation for the Eulerian polynomials of type B and type D. By making use of the generating function methods and Pade approximation techniques, we establish some new recurrence formulas for the Eulerian polynomials of type B and type D. Some of these results presented here are the corresponding extensions of some known formulas. [ABSTRACT FROM AUTHOR]