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This paper is concerned with a thorough investigation in achieving exact analytical solution for the Van der Pol (VDP) nonlinear oscillators models via Adomian decomposition method (ADM). The models are nonlinear time dependent second order ordinary differential equations. ADM has already been applied, in existing literatures, to obtain approximate results. But, we adapt the method by adjusting the source term; a procedure that is base on the asymptotic Taylor's series expansion on the term that would have resulted to proliferation of terms during the invertible process. Then, the rational Pade Approximant is applied to clarify and get a better understanding of the uniqueness and convergence of our findings. Two models were used as illustrations and their result pictured to indicate their behaviour in the given domains. And, we found that the adaptation on the models yielded exact results which were further displayed in constructed tables.

All-solid-state batteries (ASSBs) have been considered as the next generation of lithium-ion batteries. Physics-based models have the advantage of providing internal electrochemical information. To promote physics-based models in real-time applications, in this study, a series of model reduction methods are applied to obtain a reduced-order model (ROM) for ASSBs. First, analytical solutions of the partial differential equations (PDEs) are derived by the Laplace transform. Then, the Pade approximation method is used to convert the transcendental transfer functions into lower order fractional transfer functions. Next, the concentration distributions in electrodes and electrolytes are approximated by parabolic and cubic functions, respectively. Due to the fast calculation of concentration distributions in real time, the equilibrium potential, overpotentials, and battery voltage can now be directly calculated. Compared with the original PDE-based model, the voltage errors of the proposed ROM are less than 2.6 mV. Compared with the voltage response of experimental data, a good agreement can be observed for the ROM under three large C-rates discharging conditions. The calculation time of ROM per step is within 0.2 ms, which means that it can be integrated into a battery management system. The proposed ROM achieves excellent performance and a better tradeoff between model fidelity and computational complexity.

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.

Resonance via Pade (RVP) is an efficient method for calculating autoionization resonance states. It is based on the stabilization technique in which the basis set is scaled. The scaling can be uniform (i.e., all basis functions are scaled) or partial. Herein, we compare the two RVP scaling schemes for calculating an autoionization eigenvalue; moreover, the effect of freezing the core electrons is intertwined within this comparison. In order to study the different behavior of the RVP schemes, we associate each RVP scaling scheme with a complex contour of integration. Similarities between RVP and other non-Hermitian methods emerge from the generated contours, which suggest that RVP introduces similar outgoing boundary conditions as the complex scaling (CS), complex basis function (CBF), and reflection-free complex absorbing potential (RF-CAP) methods. A uniform-RVP contour, unlike a partial one, immediately penetrates the complex plane and influences the interaction region. Hence, uniform scaling within RVP destroys the description of the core electrons, as well as the description of the reference state, and yields less reliable results than partial scaling. The 1s22p3s1P autoionization state of Be, at the equation-of-motion coupled-cluster level, is used as our case study model.

In this letter, a scalar non-iterative bi-directional finite element based beam propagation method (NI-BiBPM) is proposed for the analysis of light propagation in photonic devices with circular symmetry. We adopt the Blocked Schur algorithm (BSA) for calculating the square root of the characteristic matrix in order to achieve fast and stable results and to avoid the complications and limitations of other computational modeling techniques such as Taylor or Pade approximations. The suggested NI-BiBPM approach does not only offer reduction in execution time and memory requirements but also guarantees the numerical stability.

The topic of this work is on reliable resolving of J-coupled resonances in spectral envelopes from proton nuclear magnetic resonance (NMR) spectroscopy. These resonances appear as multiplets that none of the conventional nonderivative shape estimators can disentangle. However, the recently formulated nonconventional shape estimator, the derivative fast Padé transform (dFPT), has a chance to meet this challenge. In the preceding article with a polyethylene phantom, using the time signals encoded with water suppressed, the nonparametric dFPT was shown to be able to split apart the compound resonances that contain the known J-coupled multiplets. In the present work, we address the same proton NMR theme, but with sharply different initial conditions from encodings. The goal within the nonparametric dFPT is again to accurately resolve the J-coupled resonances with the same polyethylene phantom, but using raw time signals encoded without water suppression. The parallel work on the same problem employing two startlingly unequal time signals, encoded with and without water suppression in the preceding and the current articles, respectively, can offer an answer to a question of utmost practical significance. How much does water suppression during encoding time signals actually perturb the resonances near and farther away from the dominant water peak? This is why it is important to apply the same dFPT estimator to the time signals encoded without water suppression to complement the findings with water suppression. A notable practical side of this inquiry is in challenging the common wisdom, which invariably takes for granted that it is absolutely necessary to subtract water from the encoded time signals in order to extract meaningful information by way of NMR spectroscopy.

Burnup calculations play a very important role in nuclear reactor design and analysis, and solving burnup equations is an essential topic in burnup calculations. In the last decade, several high-ac...

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 $$({\nu}_{a}/{\nu}_{b})^{2}$$ is also determined. One important aspect of this model is that the temperature-dependent characteristic frequencies $$\nu_{a}$$ and $$\nu_{b}$$ 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 $$({\nu}_{a}/{\nu}_{b})^{2}$$ .

Electronic transition dipoles are crucial for investigating light-matter interactions. Transition dipoles between metastable (autoionizing resonance) states become complex within non-Hermitian formalism, analogous to the resonance energies. Herein, we put forward a robust method for evaluating complex transition dipoles based on real ab initio stabilization calculations. The complex transition dipoles are obtained by analytical continuation via the Pade approximant and are identified as stationary solutions in the complex plane. The capability of the new approach is demonstrated for several transition dipoles of the doubly excited helium resonance states, for which exact values are available for comparison. Nevertheless, the method presented here has no inherent limitation and is suitable for polyatomic systems.

Guided wave propagation and its dispersion phenomenon in infinite solid elastic rods are encountered in several applications including, mechanical and civil engineering fields. A robust root findin...

This article presents a two-way parabolic equation (2WPE) method based on the Pade(2,2) approximation form, which can model bidirectional electromagnetic wave (EW) propagation in a sea environment. Compared with other common approximation forms, the Pade(2,2) approximation has better convergence and accuracy. To accurately simulate the reflection and refraction effects of EW propagation near the sea surface, an improved fractal sea surface model is used to construct the geometric rough sea surface. The results of the Pade(2,2) approximation PE model are compared with those of the Miller–Brown model, and good agreement is observed. Numerous experiments prove that the 2WPE model is more accurate than the one-way parabolic equation model for predicting large-scale radio wave propagation in marine environments. Furthermore, we investigate the relationship between the backward-scattered EWs and the wind speed. The polynomial fitting method is adopted to process a large amount of sampled data for backward-scattered EWs with different wind speeds and frequencies. Through accurate judgment of the points of the polynomial fitting curve in a certain frequency range, the wind speed over the sea surface can be precisely determined. Thus, this article provides a novel theoretical method for marine remote sensing.

Cosmography becomes non-predictive when cosmic data span beyond the red shift limit $z\simeq1 $. This leads to a \emph{strong convergence issue} that jeopardizes its viability. In this work, we critically compare the two main solutions of the convergence problem, i.e. the $y$-parametrizations of the redshift and the alternatives to Taylor expansions based on Pad\'e series. In particular, among several possibilities, we consider two widely adopted parametrizations, namely $y_1=1-a$ and $y_2=\arctan(a^{-1}-1)$, being $a$ the scale factor of the Universe. We find that the $y_2$-parametrization performs relatively better than the $y_1$-parametrization over the whole redshift domain. Even though $y_2$ overcomes the issues of $y_1$, we get that the most viable approximations of the luminosity distance $d_L(z)$ are given in terms of Pad\'e approximations. In order to check this result by means of cosmic data, we analyze the Pad\'e approximations up to the fifth order, and compare these series with the corresponding $y$-variables of the same orders. We investigate two distinct domains involving Monte Carlo analysis on the Pantheon Superovae Ia data, $H(z)$ and shift parameter measurements. We conclude that the (2,1) Pad\'e approximation is statistically the optimal approach to explain low and high-redshift data, together with the fifth-order $y_2$-parametrization. At high redshifts, the (3,2) Pad\'e approximation cannot be fully excluded, while the (2,2) Pad\'e one is essentially ruled out., Comment: 17 pages, 3 figures, to appear in MNRAS

В работе приводится уточненная математическая модель вынужденных колебаний разомкнутой оболочки с малой присоединенной массой при шарнирном опирании. Шарнирное опирание рассмотрено как наиболее часто используемое. Уравнения составлены в линейной постановке, решались асимптотическими методами, с помощью аппроксимации Паде. Проведены экспериментальные исследования, для этого изготовлены испытательный стенд, образцы, разработана программа экспериментов. Выполненное сравнение теоретических и экспериментальных данных составляет менее 5%, что говорит о высокой надежности уточненной математической модели. Полученные зависимости частоты колебаний оболочки и параметра волнообразования позволяют производить расчеты разомкнутый тонкостенных цилиндрических оболочек при параметре волнообразования больше 0,3, при проектировании таких конструкций. The paper presents a refined mathematical model of forced oscillations of an open shell with a small attached mass at a hinge support. Hinge support is considered as the most frequently used. The equations are composed in a linear formulation, solved by asymptotic methods, using Pade approximations. Experimental studies were carried out, a test stand and samples were made for this purpose, and a program of experiments was developed. Comparison of theoretical and experimental data it is less than 5%, which indicates a high reliability of the refined mathematical model. The obtained dependences of the shell oscillation frequency and the wave formation parameter make it possible to calculate the open-ended thin-walled cylindrical shells with a wave formation parameter greater than 0.3, when designing such structures.

Summary We demonstrate that the asymptotic approximant applied to the Blasius boundary layer flow over a flat plat (Barlow et al., Q. J. Mech. Appl. Math. 70 (2017) 21–48.) yields accurate analytic closed-form solutions to the Falkner–Skan boundary layer equation for flow over a wedge having angle $\beta\pi/2$ to the horizontal. A wide range of wedge angles satisfying $\beta\in[-0.198837735, 1]$ are considered, and the previously established non-unique solutions for $\beta

We report a theoretical–experimental investigation on elastic electron scattering by acetaldehyde (CH $$_3$$ CHO) in the 0.5–800 eV energy range. Theoretically, the Pade approximation and the single-center partial-wave expansion method were applied to solve the Lippmann–Schwinger scattering equation, whereas a molecular complex optical potential was used to describe the electron–molecule interaction. Calculations in the independent-atom framework were also performed. Angular distributions of the elastically scattered electrons were measured in the 10 $$^{\circ }$$ –130 $$^{\circ }$$ and 30–800 eV ranges using a crossed beams geometry. Absolute values of differential cross sections were derived by the relative flow technique. In addition, integral and momentum transfer cross sections were obtained from the experimental and theoretical differential cross sections via a numerical integration procedure. Comparison between our results and previous experimental and theoretical calculations is made.

We calculate the resummed perturbative free energy of$$ \mathcal{N} $$N= 4 supersymmetric Yang-Mills in four spacetime dimensions (SYM4,4) through second order in the ’t Hooft couplingλat finite temperature and zero chemical potential. Our final result is ultraviolet finite and all infrared divergences generated at three-loop level are canceled by summing over SYM4,4ring diagrams. Non-analytic terms at$$ \mathcal{O} $$O(λ3/2) and$$ \mathcal{O} $$O(λ2logλ) are generated by dressing theA0and scalar propagators. The gauge-field Debye massmDand the scalar thermal massMDare determined from their corresponding finite-temperature self-energies. Based on this, we obtain the three-loop thermodynamic functions of SYM4,4to$$ \mathcal{O} $$O(λ2). We compare our final result with prior results obtained in the weak- and strong-coupling limits and construct a generalized Padé approximant that interpolates between the weak-coupling result and the large-Ncstrong-coupling result. Our results suggest that the$$ \mathcal{O} $$O(λ2) weak-coupling result for the scaled entropy density is a quantitatively reliable approximation to the scaled entropy density for 0 ≤λ≲ 2.

A wide-angle Fourier migrator is proposed to more accurately image complex media with strong lateral velocity contrasts. The broadband wave-equation migrator is developed based on the pseudo-Pade approximation, where the Pade coefficients are independent of spatial coordinates, leading to a pure Fourier transform-based matching solution for one-way wavefield extrapolation. We use genetic algorithms to estimate the constant Pade coefficients more accurately than is feasible with conventional least-squares methods. Because of the global features of pure Fourier migrators, we present an angle-partitioning optimization scheme with dip focusing to improve the performance of the Fourier migrator for super-wide-angle waves and strong velocity contrasts. The wavefield gradient is used to calculate propagation angles during dual-domain wavefield extrapolation. Particular attention is paid to the first-order optimized pseudo-Pade Fourier (OPF1) migrator, which significantly improves the split-step Fourier (SSF) method for strong lateral variations at the cost of one additional Fourier transform in each step. Wavefield extrapolation based on the OPF1 method actually constitutes linear interpolation in the wavenumber domain between two split-step terms. We benchmark the OPF1 migrator with other typical migrators based on the exact dispersion equation. Numerical experiments with impulse responses, the SEG/EAGE salt model and 3D field data demonstrate the excellent performance and efficiency of seismic imaging with the OPF1 migrator.

The mathematical method of Pade approximants is put forward to reconstruct the Borel transformed series of the Adler function and extract higher-order corrections in a model-independent way. The use of the D-log Pade approximants provides valuable information to its analytic structure and yields, together with the standard Pade approximants, reliable model-independent predictions of the higher-order coefficients, as for example, c 5 , 1 = 277 ± 51 . Such reconstruction strongly favours the use of fixed-order perturbation theory (FOPT) for the renormalization-scale setting.

Magnetic resonance spectroscopy (MRS), as a powerful and versatile diagnostic modality in physics, chemistry, medicine and other basic and applied sciences, depends critically upon reliable signal processing. It provides time signals by encoding, but cannot quantify on its own. Mathematical methods do so. The signal processor of choice for MRS is the fast Padé transform (FPT). The spectrum in the FPT is the unique polynomial quotient for the given Maclaurin expansion. The parametric FPT (parameter estimator) performs quantification of time signals encoded with MRS by explicitly solving the spectral analysis problem. Thus far, the non-parametric FPT (shape estimator) could not quantify. However, the non-parametric derivative fast Padé transform (dFPT) can quantify despite performing shape estimation alone. The dFPT was successfully benchmarked on synthesized MRS time signals for derivative orders ranging from 1 to 50. It simultaneously improved resolution (by splitting apart tightly overlapped peaks) and enhanced signal-to-noise ratio (by suppressing the background baseline). The same advantageous features of improving both resolution and signal-to-noise ratio are presently found to be upheld with encoded MRS time signals. Moreover, it is demonstrated that the dFPT hugely outperforms the derivative fast Fourier transform even for derivatives of orders as low as four. The clinical implications are discussed.

A unified numerically solvable framework for dispersion relations with an arbitrary number of species drifting at arbitrary directions and with Krook collision is derived in linear uniform/homogenous kinetic plasma model, which greatly extends the standard one given by Stix (1992). The purpose of this work is to provide a kinetic plasma dispersion relation tool not only in the physical model but also the numerical approach as general/powerful as possible. As a general application, we give the dispersion relations which assume the equilibrium distribution function to be bi-Maxwellian including parallel drift, two directions of perpendicular drift (i.e., drift across magnetic field), ring beam and loss-cone. Both electromagnetic and electrostatic versions are provided, with also the Darwin (a.k.a., magnetoinductive or magnetostatic) version. The species can be treated either magnetized or unmagnetized. Later, the equations are transformed to the matrix form that can be solved by using the powerful matrix algorithm (Xie and Xiao, 2016), which is the first approach can give all the important solutions of a linear kinetic plasma system without requiring initial guess for root finding. To the best of our knowledge, the present model is the most comprehensive model in literature for the distribution function constructed from Maxwellian distributions, which thus can be applied widely for studying waves and instabilities in space, astrophysics, fusion and laser plasmas. We limit the present work to non-relativistic case. Program summary Program Title: BO Program Files doi: http://dx.doi.org/10.17632/cvbrftzfy5.1 Licensing provisions: BSD 3-clause Programming language: Matlab Nature of problem: The linear fluid and kinetic waves and instabilities in plasma can be described by dispersion relations. The challenges are to provide a dispersion relation as general as possible and to obtain all the solutions of it, which is the goal of BO tool. The kinetic version of BO tool provides a unified numerically solvable framework for kinetic dispersion relations, which greatly extends the standard one (Stix, 1992), with an arbitrary number of species. It contains many new features, including anisotropic temperature/loss cone/drift in arbitrary direction/ring beam/collision, unmagnetized/magnetized, electrostatic/electromagnetic/Darwin, can support parallel wave vector k z ≤ 0 , etc. Solution method: Approximating the plasma dispersion function in the kinetic dispersion relation with J -pole Pade expansion. Transforming the dispersion relation to an equivalent matrix eigenvalue problem and find all the solutions using standard matrix eigenvalue library function. Additional comments including restrictions and unusual features: Kinetic relativistic effects are not included in the present version yet.

MHD flow of an electrically conducting Newtonian fluid over a super linear stretching sheet in the presence of suction/injection and Navier slip is studied using modified Adomain decomposition method (MADM) and Pade approximants. Governing nonlinear partial differential equations are transformed into nonlinear ordinary differential equations using an appropriate similarity transformation. The transformed equations are solved analytically by the modified ADM and Pade approximation. The modified ADM for solving nonlinear differential equations is purely and solely the traditional Taylor’s series method. Pade approximants are applied to increase the convergence of the given series. The developed analytical technique is verified comprehensively. It is found that Navier’s slip condition can lead to a non-essential growth of the boundary layer thickness and a decrease in the axial and transverse velocities.

Nuclear magnetic resonance spectroscopy originated in physics and quickly found versatile applications of paramount importance in other sciences, including chemistry. Signal processing in this methodology is a key to data analysis and interpretation. Herein, one of the most powerful tools from mathematical theory of approximations, known as rational polynomials, is the prime example of reliable handling of the two stumbling blocks that hamper further progress: noise suppression and resolution improvement. Within this realm resides the fast Pade transform (FPT), which simultaneously solves both these problems. It has a self-correcting procedure, which is automatically built in rational polynomials through noise suppression by pole-zero cancellations in spectra. Moreover, by solving the quantification problem (called spectral analysis in mathematics), the FPT can unequivocally separate overlapped peaks and thereby improve resolution. Further, lineshape estimations are provided by both non-parametric and parametric signal processing in the FPT. Since the FPT includes singularities (poles) of the expanded function, it achieves exponental convergence $$\exp {(-N)}$$ (the so-named spectral resolution) with respect to the size N of the basis set. This is contrasted to merely the inverse-power-law convergence 1 / N in the fast Fourier transform because its basis functions do not describe the singularities of the expanded function. The present investigation reports on practical aspects of all these critical features and gives several representative illustrations for measured time signals heavily contaminated with noise.

The elastic α-12C scattering for l = 0, 1, 2, 3 at low energies is studied by using the scattering amplitudes represented in terms of the effective range expansion where four effective range terms are included for l = 0,1,2, and five for l = 3. The first effective range terms are fixed by using the binding energies of the $$0_{2}^{+}$$ , $$1_{1}^{-}$$ , $$2_{1}^{+}$$ , and $$3_{1}^{-}$$ ( $$l_{i-t h}^{\pi}$$ ) states of 16O, and the other parameters are fitted to the phase shift data. After fitting the parameters to the data, we find that the broad resonant $$1_{2}^{-}$$ and $$3_{2}^{-}$$ states of 16O are well incorporated, without the need to introduce any other parameterization, such as the Breit-Wigner formula or the Pade approximation, in the amplitudes while the narrow resonant $$0_{3}^{+}$$ and $$2_{2}^{+}$$ states are not. We also calculate the asymptotic normalization coefficients for the bound states of 16O and compare them with the previous results in other works.

In this work, a theoretical investigation on the electron interactions with 1,2-ethylenediamine and dimethyldiazene (also known as azomethane) is presented. Differential cross sections (DCS), integral cross sections (ICS), and momentum transfer cross sections (MTCS) for elastic scattering as well as total cross sections (TCS) and total absorption cross sections were determined in the 1-500 eV range. To describe the electron-target interaction and solve the scattering equations, a molecular complex optical potential (MCOP) approach combined with the Pade approximant technique was employed. Regarding the DCS, 1,2-ethylenediamine presented the f-wave pattern which was not seen in the results of dimethyldiazene. On the other hand, results of dimethyldiazene have shown a sharp feature at 2.0 eV in the ICS, MTCS, and TCS, which was assigned as the π* resonance originating from the nitrogen–nitrogen bond. Interestingly, the feature observed in dimethyldiazene was found to be slightly shifted toward low energy when compared to the resonance reported for the N2 molecule. The description of the polarization effects could be responsible for this discrepancy, and the methylation effect be responsible for the stabilization of the π* resonance of dimethyldiazene with respect to N2. To the best of our knowledge, this work represents the first extensive study regarding electron collisions with these molecules, being presented to motivate experimental investigations to be performed about these important and widely used building blocks.

The enhanced continualization approach proposed in this paper is aimed to overcome some drawbacks observed in the homogenization of beam lattices. To this end an enhanced homogenization technique is proposed and formulated to obtain consistent micropolar continuum models of the beam lattices and able to simulate with good approximation the boundary layer effects and the Floquet–Bloch spectrum of the Lagrangian model. The continualization technique here proposed is based on a transformation of the difference equation of motion of the discrete system via a proper down-scaling law into a pseudo-differential problem; a further McLaurin approximation is applied to obtain a higher order differential problem. The formulation is carried out for simple one-dimensional beam lattices that are, nevertheless, characterized by a rather wide variety of static and dynamic behaviors: the rod lattice, the beam lattice with node rotations and a 1D beam lattice model with generalized displacements. Higher order models may be obtained which are characterized by differential problems involving non-local inertia terms together with spatial high gradient terms. Moreover, the homogenized models obtained by the proposed enhanced continualization technique turn out to be energetically consistent and provide a good simulation of both the static response and of the acoustic spectrum of the original discrete models. The proposed homogenization procedure is first presented for the simple case of monoatomic axial chains. The beam lattice with node rotations and displacement prevented exhibits, in the static regime, decaying oscillations of the nodal rotation in the boundary layer which is well simulated by the homogenized model obtained by the proposed approach. Similar good results are obtained in the simulation of the optical spectrum. It is worth to note that in this case the homogenized model obtained via Pade approximation turns out to be energetically non-consistent. The analysis of the homogenized model derived from the beam lattice with transverse displacement and rotation of the nodes with elastic supports has shown that both the static and the dynamic response are strongly variable on the parameters of the Lagrangian model. Finally, several different cases have been considered and good simulations have been obtained both in describing the static response to prescribed displacements at the end nodes and in representing the Floquet–Bloch spectrum and the polarization vectors.

Detailed transient behaviors of mass and heat transfer processes are required to solve partial differential equations. When those partial differential equations are coupled, they are still difficul...

Stabilisation technique is a traditional approach for studying resonances, such as the metastable autoionisation states. Stabilisation graphs are usually calculated by scaling the employed basis functions. Over the years, many methods for extracting the resonance complex energies from these graphs have been developed, nowadays, they are usually evaluated via analytical continuation schemes. However, many times it is difficult to obtain a well-behaved stabilisation graphs upon basis-set scaling. Herein, we illustrate that it is possible to shape stabilisation graphs to behave physically by adjusting the relevant Gaussian basis function exponents. We illustrate this for the excited Li–He∗ 2Σ and 2Π states [with He∗(1s2p; 1P)], which serve as an excellent example since generating well-behaved graphs in this case is challenging. We show that by controlling and shaping the stabilisation graphs we can obtain converged complex energies, which are calculated by the resonances via Pade approach (a recently...

The predictive power of perturbative QCD (pQCD) depends on two important issues: (1) how to eliminate the renormalization scheme-and-scale ambiguities at fixed order, and (2) how to reliably estimate the contributions of unknown higher-order terms using information from the known pQCD series. The Principle of Maximum Conformality (PMC) satisfies all of the principles of the renormalization group and eliminates the scheme-and-scale ambiguities by the recursive use of the renormalization group equation to determine the scale of the QCD running coupling $\alpha_s$ at each order. Moreover, the resulting PMC predictions are independent of the choice of the renormalization scheme, satisfying the key principle of renormalization group invariance. In this letter, we show that by using the conformal series derived using the PMC single-scale procedure, in combination with the Pad\'e Approximation Approach (PAA), one can achieve quantitatively useful estimates for the unknown higher-order terms from the known perturbative series. We illustrate this procedure for three hadronic observables $R_{e^+e^-}$, $R_{\tau}$, and $\Gamma(H \to b \bar{b})$ which are each known to 4 loops in pQCD. We show that if the PMC prediction for the conformal series for an observable (of leading order $\alpha_s^p$) has been determined at order $\alpha^n_s$, then the $[N/M]=[0/n-p]$ Pad\'e series provides quantitatively useful predictions for the higher-order terms. We also show that the PMC + PAA predictions agree at all orders with the fundamental, scheme-independent Generalized Crewther relations which connect observables, such as deep inelastic neutrino-nucleon scattering, to hadronic $e^+e^-$ annihilation. Thus, by using the combination of the PMC series and the Pad\'e method, the predictive power of pQCD theory can be greatly improved., Comment: 7 pages, 1 figure. Discussions improved and references updated

Electromagnetic (EM) fields in the extremely low-frequency (ELF) domain that can be excited by the vertical electric dipole (VED) and propagated in the Earth-ionosphere spherical cavity have received tremendous attention owing to their exploitation in many fields such as communication, earthquake prediction, and geophysical investigations. Some researchers use the planar model and curvature correction terms to simulate EM fields in a spherical cavity, while others directly solve the differential functions and estimate the excited EM fields represented by spherical Bessel and associated Legendre functions. Regarding the divergence problem of a series when simulating EM fields, several numerical methods are available such as Watson transformation method, W. K. B. J method, and speeding algorithm. In this study, Pade approximant is utilized to estimate EM fields propagating in the Earth-ionosphere spherical cavity, where new asymptotic expansions of spherical Bessel functions are introduced for estimating their products for high orders. These EM fields are then interpreted as a sum of the dominant incident fields and the increase of multiple scattering fields. The convergence of series expressions of EM fields estimated by Pade approximant is theoretically guaranteed, with its computational complexity lower than that of the speeding algorithm. The EM fields are further verified by numerical tests. The effects of the height and conductivity of the ionosphere, the conductivity of air, the VED height, and the frequencies of the resonance phenomenon are also highlighted. The proposed method is helpful to capture the true nature of the field distribution within a spherical cavity.

We propose a novel Bayesian method to analytically continue observables to real baryochemical potential $\mu_B$ in finite density QCD. Taylor coefficients at $\mu_B=0$ and data at imaginary chemical potential $\mu_B^I$ are treated on equal footing. We consider two different constructions for the Pad\'e approximants, the classical multipoint Pad\'e approximation and a mixed approximation that is a slight generalization of a recent idea in Pad\'e approximation theory. Approximants with spurious poles are excluded from the analysis. As an application, we perform a joint analysis of the available continuum extrapolated lattice data for both pseudocritical temperature $T_c$ at $\mu_B^I$ from the Wuppertal-Budapest Collaboration and Taylor coefficients $\kappa_2$ and $\kappa_4$ from the HotQCD Collaboration. An apparent convergence of $[p/p]$ and $[p/p+1]$ sequences of rational functions is observed with increasing $p.$ We present our extrapolation up to $\mu_B\approx 600$ MeV., Comment: 10 pages, 4 figures, clarifications and references added, published version

We compute the next-to-leading order virtual corrections to the partonic cross-section of the process $gg\to ZH$, in the high-energy and large-$m_t$ limits. We use Pad\'e approximants to increase the radius of convergence of the high-energy expansion in $m_t^2/s$, $m_t^2/t$ and $m_t^2/u$ and show that precise results can be obtained down to energies which are fairly close to the top quark pair threshold. We present results both for the form factors and the next-to-leading order virtual cross-section., Comment: 19 pages

Pad$\acute{e}$ cosmography has been widely used to probe the cosmic evolution and to investigate the nature of dark energy. In this paper, we find that the Pad$\acute{e}$ approximant can describe the cosmic evolution better than the standard cosmography, and if the luminosity distance $d_L(z)$ described by the Pad$\acute{e}$ approximant is used to reconstruct the dark energy equation of state $w(z)$, then the reconstructed $w(z)$ will approach a constant, i.e. $1/3$ or $0$, when the redshift is very high. This result is general since it is independent of the coefficients in the Pad$\acute{e}$ approximant and the value of the present dimensionless matter density parameter. This intrinsic character will bias the $w(z)$ reconstruction and lead to misconception of the property of dark energy. Therefore, one must exercise caution in investigating the property of dark energy from Pad$\acute{e}$ cosmography when the high redshift data, i.e. $z>2$, are included., 13 pages, 5 figures

We derive exact traversable wormhole solutions in the framework of $f(R)$ gravity with no exotic matter and with stable conditions over the geometric fluid entering the throat. For this purpose, we propose power-law $f(R)$ models and two possible approaches for the shape function $b(r)/r$. The first approach makes use of an inverse power law function, namely $b(r)/r\sim r^{-1-\beta}$. The second one adopts Pad\'e approximants, used to characterize the shape function in a model-independent way. We single out the $P(0,1)$ approximant where the fluid perturbations are negligible within the throat, if the sound speed vanishes at $r=r_0$. The former guarantees an overall stability of the geometrical fluid into the wormhole. Finally we get suitable bounds over the parameters of the model for the above discussed cases. In conclusion, we find that small deviations from General Relativity give stable solutions., Comment: 11 pages, 2 figures

Series expansions remain, in many cases, one of the most accurate ways of estimating critical exponents. The high-temperature expansion is an expansion in powers of the inverse temperature. In the low-temperature expansion configurations are counted in order of their importance as the temperature is increased from zero. Starting from the ground state the series is constructed by successively adding terms with and increasing number of flipped spins. Each term in the high- or low-temperature series is represented by a graph on a lattice and constructing the series amounts to counting the graphs belonging to a fixed order in the expansions. With various extrapolations, for example with Pade approximants, we extract the behavior of a lattice system near a critical point and in particular its critical exponents. The expansions are calculated to high orders for the Ising model in two and three dimensions and for the non-linear O(N) lattice model. At the end we consider polymers and self-avoiding walks. In several tables we compare predictions for the critical temperatures and critical exponents of various expansions, renormalization group techniques and lattice simulations.

We will consider the Pade approximation problem associated with the function \(\psi (x)=(1-x)^\alpha (1-\frac{x}{t})^\beta \), and derive a pair of linear differential equations from it. Since the compatibility condition of the pair is the sixth Painleve equation \(P_\mathrm{VI}\), we obtain special solutions of the \(P_\mathrm{VI}\) equation from the Pade problem.

The equations of state (EoS) and other thermodynamic properties of plasmas of the light elements H, He, and Li, are calculated using inverted fugacity expansions. Fugacity expansions are known as an alternative to density expansions but show often an inferior convergence. If, however, the inversion can be solved, the fugacity representations may be very efficient. In particular, the contributions of deeply bound states are included in the fugacity expansion in a very effective way. The mathematical problems on nonlinearity connected with the inversion of fugacities to densities are reduced to solvable algebraic problems. The inversion of fugacities to densities is solved separately for two density regions: (i) In the low density, non-degenerate region we consider ring contributions describing screening effects and ladder contributions describing bound state formation. (ii) In the high density, degenerate region the electrons are described by the Fermi–Dirac distribution. Hartree–Fock contributions and Pauli blocking have to be taken into account. The ions are considered as classical, strongly correlated subsystem eventually forming a Wigner lattice. We solve the inversion problem for each of the regions. Near the crossing point, the separate solutions are connected to each other, either by smooth concatenation at the crossing point or by Padé approximations.

Observations of transient gravitational wave (GW) events with non-negligible orbital eccentricity can be highly rewarding from astrophysical considerations. Ready-to-use fully analytic frequency domain inspiral GW templates are crucial ingredients to construct eccentric inspiral-merger-ringdown waveform families, required for the detection of such GW events. It turns out that a fully analytic, post-Newtonian (PN) accurate frequency domain inspiral template family, which uses certain post-circular approximation, may only be suitable to model events with initial eccentricities $e_0 \leq 0.2$.We here explore the possibility of combining Post-Circular and Pad\'e approximations to obtain fully analytic frequency domain eccentric inspiral templates. The resulting 1PN-accurate approximant is capable of faithfully capturing eccentric inspirals having $e_0 \leq 0.6$ while employing our 1PN extension of a frequency domain template family that does not use post-circular approximation, detailed in Moore, B., et al.\ 2018, Classical and Quantum Gravity, 35, 235006. We also discuss subtleties that arise while combining post-circular and Pad\'e approximations to obtain higher PN order templates for eccentric inspirals., Comment: 25 pages, 6 figures, 1 table; Accepted in Physical Review D

The perturbative QCD static potential and ultrasoft contributions, which together give the static energy, have been calculated to three- and four-loop order respectively, by several authors. Using the renormalization group, and Pad\'e approximants, we estimate the four-loop corrections to the static energy. We also employ the optimal renormalization method and resum the logarithms of the perturbative series in order to reduce sensitivity to the renormalization scale in momentum space. This is the first application of the method to results at these orders. The convergence behaviour of the perturbative series is also improved in position space using the Restricted Fourier Transform scheme. Using optimal renormalization, we have extracted the value of $\Lambda^{\overline{\textrm{MS}}}_{\textrm{QCD}}$ at different scales for two active flavours by matching to the static energy from lattice QCD simulations., Comment: 16 pages, 3 figures, 3 tables

The dynamics of the Universe are revised using high-redshift data from gamma-ray bursts to constrain cosmographic parameters by means of model-independent techniques. Considering samples from four gamma-ray burst correlations and two hierarchies up to $j_0$ and $s_0$, respectively, we derived limits over the expansion history of the Universe. Since cosmic data span outside $z\simeq0$, we investigated additional cosmographic methods such as auxiliary variables and Pad\'e approximations. Bezi\'er polynomials were employed to calibrate our correlations and heal the circularity problem. Several Markov chain Monte Carlo simulations were performed on the model-independently calibrated Amati, Ghirlanda, Yonetoku, and combo correlations to obtain $1$--$\sigma$ and $2$--$\sigma$ confidence levels and to test the standard cosmological model. Reasonable results are found up to $j_0$ and $s_0$ hierarchies, respectively, only partially alleviating the tension on local $H_0$ measurements as $j_0$ hierarchy is considered. Discussions on systematic errors have been extensively reported here. Our findings show that the $\Lambda$CDM model is not fully confirmed using gamma-ray bursts. Indications against a genuine cosmological constant are summarized and commented on in detail., Comment: 15 pages, 7 figures, 7 tables

Recently, a new set of positivity bounds with $t$ derivatives have been discovered. We explore the generic features of these generalized positivity bounds with loop amplitudes and apply these bounds to constrain the parameters in chiral perturbation theory up to the next-to-next-to-leading order. We show that the generalized positivity bounds give rise to stronger constraints on the $\bar l_i$ constants, compared to the existing axiomatic bounds. The parameter space of the $b_i$ constants is constrained by the generalized positivity bounds to be a convex region that is enclosed for many sections of the total space. We also show that the improved version of these positivity bounds can further enhance the constraints on the parameters. The often used Pad\'e unitarization method however does not improve the analyticity of the amplitudes in the chiral perturbation theory at low energies., Comment: 31 pages, 15 figures; JHEP version

The narrow-angle parabolic equation (NAPE) with the effective sound speed approximation (ESSA) is widely used for sound and infrasound propagation in a moving medium such as the atmosphere. However, it is valid only for angles less than 20° with respect to the nominal propagation direction. In this paper, the wave equation and extra-wide-angle parabolic equation (EWAPE) for high-frequency (short-wavelength) sound waves in a moving medium with arbitrary Mach numbers are derived without the ESSA. For relatively smooth variations in the medium velocity, the EWAPE is valid for propagation angles up to 90°. Using the Pade (n,n) series expansion and narrow-angle approximation, the EWAPE is reduced to the wide-angle parabolic equation (WAPE) and NAPE. Versions of these equations are then formulated for low Mach numbers, which is the case that is usually considered in the literature. The phase errors pertinent to the equations considered are studied. It is shown that the equations for low Mach numbers and the WAPE with the ESSA are applicable only under rather restrictive conditions on the medium velocity. An effective numerical implementation of the WAPE for arbitrary Mach numbers in the Pade (1,1) approximation is developed and applied to sound propagation in the atmosphere.

It is shown that the Prony and matrix beam methods of exponential approximation, which recover the function image basing on a time-solution (an original), and the numerical method of the Laplace-Pade inverse transformation can be considered as a pair of direct and inverse Laplace numerical transformation. The simulation examples of dynamic systems by the combined Prony-Laplace-Pade and matrix beam methods in the time and frequency domains are presented, which are typical for tasks of the circuit theory and radio-electronics. In particular, radio pulse passage with the intra-pulse angular modulation is investigated; the systems with phase and frequency shift keying are studied. The matrix beam method is used to lower the approximation order of circuit functions with a given error value.

Negative-ion resonances are important metastable states that result from the collision between an electron and a neutral target. The course of many chemical processes in nature is often dictated by how an intermediate resonance state falls apart. This article reports on the development of an electron propagator (EP) based on a Hamiltonian Ĥ perturbed by a complex absorbing potential (CAP) and a multiconfigurational self-consistent field (MCSCF) initial state to study these resonances. Perturbation of Ĥ by a CAP makes the resonances amenable to a bound-state method like MCSCF. Resonances stand out among the non-resonant states as persistent complex eigenvalues of the perturbed Ĥ when the strength (η) of the CAP is varied. The MCSCF method gives a reliable and accurate description of the target states, especially when the non-dynamical correlations are dominant. The resonance energies are obtained from the poles of the EP. We propose three variants of our EP depending on how the effect of the CAP is introduced. We find that the computationally most efficient variant is the one in which the reference state of the EP is an unperturbed MCSCF wavefunction and a non-zero CAP is defined only on the virtual orbital subspace of the reference state. The onset of the CAP is carefully optimized in order to minimize the artifacts due to reflections from the CAP. An extrapolation method (based on a Pade approximant) and a de-perturbation method are adopted in order to account for the limitations of finite basis sets used and determine the resonance energy in the limit of η → 0. 2P Be-, 2Πg N2-, and 2Π CO- shape resonances are investigated. The position and width of these resonances computed in this study agree well with those reported earlier in the literature.

The symmetry energy is an invaluable tool for studying dense nuclear matter. Unfortunately, its definition is somewhat implicit, and therefore, phenomenological methods are necessary to describe experimental facts. This paper discusses the differences arising from the use of Taylor series expansion and Pad&eacute, approximation to determine theoretically the symmetry energy and the possible consequences for neutron stars. For this purpose, a form of the nuclear matter equation of state that explicitly depends on the symmetry energy is used. The obtained results point out that the applied approximations lead to modifications of the equilibrium proton fractions and equation of state, especially in their high-density limit. However, this effect is small near the saturation density n 0 .

International audience; Pseudodifferential mode parabolic equations in curvilinear coordinates are derived from the horizontal refraction equations. An energy-conserving split-step Padé method for their numerical solution is developed. Several examples are considered including the propagation of sound in a perfect wedge, a whispering gallery formation near circular isobath, and the breaking of the whispering gallery waveguide in the vicinity of an inflection point of the isobath.