Rational Approximation Research Articles

On the symmetric 2-adic complexity of periodic binary sequences

Symmetric 2-adic complexity is a better measure for assessing the strength of a periodic binary sequence against the rational approximation attack. We study the symmetric 2-adic complexity of the sequences with known 2-adic complexity. To this end, we propose an equivalent definition of the symmetric 2-adic complexity and two sufficient conditions for a periodic sequence to achieve the maximum symmetric 2-adic complexity. As a first application, several types of sequences with maximum 2-adic complexity are easily shown to have the maximum symmetric 2-adic complexity. In addition, with the new definition, by analysing the algebraic structure of the sequences and applying the method for the calculation of their 2-adic complexity, we determine the symmetric 2-adic complexity of the Ding-Helleseth-Martinsen sequence and the sequence constructed by interleaving a pair of Legendre sequences.

Uniform Rational Approximation of Even and Odd Continuations of Functions

Исследуется поведение наилучших рациональных приближений нечетного продолжения функции. Показано, что без дополнительных условий на гладкость функции оценить наилучшее рациональное приближение нечетного продолжения функции на $[-1,1]$ через наилучшее рациональное приближение исходной функции на $[0,1]$ невозможно. Найдена точная верхняя оценка для наилучших рациональных приближений четного (нечетного) продолжения функции через нечетное (четное) продолжение и экстремальное произведение Бляшке. Библиография: 12 названий.

Runge–Kutta type Time Stepping Methods for Space Fractional Reaction Diffusion Model with Restricted Padé Approximation

A new numerical method that solves a time-space-fractional reaction-diffusion equation effectively is presented. The space-fractional Riesz operator is first discretized using fourth order compact finite differences, resulting in a system with a linear stiff term. The system of differential equations is then solved through time integration using an exponential time difference method, which explicitly handles the nonlinear non-stiff term. Restricted Padé rational approximation of a single real pole is used to approximate the matrix exponential e-A. The problems concerning the stability and computational efficiency of this new approach are tackled by means of a splitting technique. Based on compact finite differences with third-order accuracy in time and restricted Padé approximation, this novel efficient technique reduces computing cost and effectively tackles the problems with non-smooth initial data. The superiority of new method in terms of accuracy, reliability, and computational efficiency is finally shown by numerical examples.

Rational approximations of the Arrhenius and general temperature integrals, expansion of the incomplete gamma function

When analyzing materials non-isothermally using the Arrhenius equation under linear heating, a temperature integration is necessary. While the frequency factor in this equation is typically assumed to be constant, it can actually vary with temperature for certain solid-state reactions. The resulting temperature integral, known as the Arrhenius or general temperature integrals, usually have no analytical solutions. Therefore, special functions and approximation functions are often used to estimate them. In this particular study, new rational approximations for the Arrhenius and general temperature integrals were derived through the expansion of the incomplete gamma function. Two sets of these rational approximations, which exhibit excellent accuracy, are presented. One set of approximations for the Arrhenius integral matches the widely used Senum and Yang’s approximations, while the other set, which offers even greater accuracy, has not been previously reported. An obtained rational approximation has been utilized to simulate the thermal degradation of a commercially available PMMA, illustrating a practical application example.

Regularity and numerical approximation of fractional elliptic differential equations on compact metric graphs

The fractional differential equation L β u = f L^\beta u = f posed on a compact metric graph is considered, where β > 0 \beta >0 and L = κ 2 − ∇ ( a ∇ ) L = \kappa ^2 - \nabla (a\nabla ) is a second-order elliptic operator equipped with certain vertex conditions and sufficiently smooth and positive coefficients κ , a \kappa ,a . We demonstrate the existence of a unique solution for a general class of vertex conditions and derive the regularity of the solution in the specific case of Kirchhoff vertex conditions. These results are extended to the stochastic setting when f f is replaced by Gaussian white noise. For the deterministic and stochastic settings under generalized Kirchhoff vertex conditions, we propose a numerical solution based on a finite element approximation combined with a rational approximation of the fractional power L − β L^{-\beta } . For the resulting approximation, the strong error is analyzed in the deterministic case, and the strong mean squared error as well as the L 2 ( Γ × Γ ) L_2(\Gamma \times \Gamma ) -error of the covariance function of the solution are analyzed in the stochastic setting. Explicit rates of convergences are derived for all cases. Numerical experiments for L = κ 2 − Δ , κ > 0 {L = \kappa ^2 - \Delta , \kappa >0} are performed to illustrate the results.

Stages and main tasks of the century-long control theory and system identification development. Part III. The problem of complex systems identifying by inaccurate data

The article presents the main most noteworthy research results on the identification problem and methods that are widely used in solving identification problems. The most significant here is the problem of modeling especially complex systems, for which it is impossible, even in general, but in constructive form for identification, to write down a class of models in which the system under study has the most accurate description in which there is one of them adequate to the available data obtained from experiments. If the available knowledge on the object under study does not allow us to write down a suitable class of models in mathematical form, then asymptotic modeling should be used, as is done in computational mathematics. Instead of differential, integral and other equations, for example, algebraic equations approximating them are solved, the solutions of which approach the exact ones with increasing dimension and discretization step. Various forms of asymptotic expansions are used in identification, including rational approximation as applied to irrational and infinite-dimensional systems. However, in the presence of errors in the data, the use of asymptotic modeling leads to the fact that, with intentions to increase the accuracy of the resulting model due to its large dimension, in many cases the identification problems becomes incorrectly posed. Therefore, we have to look for a compromise or trade-off between bias and variance, which should determine the best-quality model. In addition to the modeling problem, the article discusses two concepts on the basis of which most existing methods for solving identification problems are based. Both of them give their own interpretation of the errors present in the data, namely, stochastic and nonstochastic. With the stochastic concept, errors are treated as a random process or sequence. The concept of estimation consistency is introduced, which makes it possible to use widely the theory of statistics in justifying methods for solving identification problems. The nonstochastic identification paradigm allows for arbitrary but bounded uncertainties in the data, i.e. belonging to some bounded sets. The largest number of different methods, as well as algorithms and software implementations, are made within the framework of the stochastic concept. Within the framework of nonstochastic identification, we mainly consider various modifications of the so-called subspace methods. In relation to complex systems, problems were noted that did not allow the development of sufficiently universal and effective methods for solving identification problems. A number of research areas have been pointed out that need further development in order to obtain more accurate models of complex systems.

Approximation of the Lebesgue constant of the Fourier operator by a logarithmic-fractional-rational function

The Lebesgue constant of the classical Fourier operator is uniformly approximated by a logarithmic-fractional-rational function depending on three parameters; they are defined using the specific properties of logarithmic and rational approximations. A rigorous study of the corresponding residual term having an indefinite (non-monotonic) behavior has been carried out. The obtained approximation results strengthen the known results by more than two orders of magnitude.

Infinite Dimensional Design Approach of Robust Disturbance Observer for a Piezo-Actuated Nano-Positioner With Measurement Delays

In this article, we consider the disturbance observer (DOB) design supporting high performance nano-positioning with sensor-induced time delays, where the standard robust disturbance observer design methodology is extended to time-delay systems by employing an infinite dimensional approach. In particular, the dynamic model including the piezo-induced hysteresis and the measurement delays are investigated and the infinite dimensional <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}^\infty$</tex-math></inline-formula> optimization problem is formulated to achieve an optimal Q-filter design in the DOB control framework. The fragility resulting from unstable pole-zero cancellations in the infinite dimensional Q-filter is further discussed and the direct-moment-matching-based rational approximation is developed for stable digital implementations of the proposed method. Comprehensive experimental studies are conducted on a piezo-actuated stage, where the proposed method successfully demonstrates excellent performances on hysteresis compensation, disturbance suppression, and trajectory tracking, significantly outperforming existing approaches.

Realization of fractional order lowpass filter using different approximation techniques

Many research groups are starting to pay serious attention to the problem of fractional-order circuits. In this paper, a new approach to designing fractional order low-pass filter (FOLPF) is presented. Finding a rational approximation of the fractional Laplace operator sα is a crucial step in the design of fractional order filters. A comparative study of the most widely used approximation techniques named continued fraction expansion (CFE) method and Biquadratic Approximation (RE) method is performed. Then the transfer function of the proposed FOLPF is calculated. Using operational amplifier, the proposed filter is synthesized. The proposed circuit is simulated using Texas instruments TINA software. The results obtained outperform the existing methods.

A Rigorous Explicit Expression for the Mutual Inductance of Two Co-Axial Thin-Wire Coil Antennas Placed above a Layered Ground

This paper presents a quasi-analytical method that allows the derivation of a rigorous series-form representation for the mutual inductance of two co-axial coil antennas located above an arbitrarily layered earth structure. Starting from Biot–Savart law, which gives the integral representation for the primary vector potential generated by the source coil, the potential reflected by the layered ground is derived, and the resulting total vector potential is then integrated along the external circumference of the receiving coil to give the mutual inductance of the two antennas. The obtained representation for the flux is then evaluated analytically through the usage of the Gegenbauer addition theorem once an accurate, rational approximation is used in place of the factor of the integrand that exhibits branch cuts. It is shown how the resulting explicit solution exhibits the same degree of accuracy as purely numerical approaches like the finite-difference time-domain (FDTD) method and conventional numerical quadrature schemes, while it is less time-demanding than the latter methods.

Semi-analytical method for solving a model of the evolution of smoking habit using global rational approximants

Semi-analytical method for solving a model of the evolution of smoking habit using global rational approximants

A fast third order algorithm for two dimensional inhomogeneous fractional parabolic partial differential equations

A computationally fast third order numerical algorithm is developed for inhomogeneous parabolic partial differential equations. The algorithm is based on a third order method developed by using a rational approximation with single Gaussian quadrature pole to avoid complex arithmetic and to achieve high efficiency and accuracy. Difficulties with computational efficiency and accuracy are addressed using partial fraction decomposition technique. Third order accuracy and convergence of the method is proved analytically and verified numerically. Several classical as well as more challenging fractional and distributed order inhomogeneous problems are considered to perform numerical experiments. Computational efficiency of the method is demonstrated through central processing unit (CPU) time and is given in the convergence tables.

Numerical simulation of the time-fractional Fokker–Planck equation and applications to polymeric fluids

We introduce a new approach to the numerical approximation of the time-fractional Navier–Stokes–Fokker–Planck (TFNSFP) system, which involves the coupling of the incompressible Navier–Stokes equations to a time-fractional Fokker–Planck equation. The model arises in the context of dilute polymeric fluids, and it enhances the standard integer-derivative version of the model by including memory effects. The key challenge associated with the numerical solution of the TFNSFP system is that, in addition to it being nonlocal in time, it is, even in its simplest form, posed on a spatial domain that is the Cartesian product of two d-dimensional domains, for d∈{2,3}, the d-dimensional flow domain and the d-dimensional configuration space domain. We combine a kernel compression technique based on rational approximation of the integral kernel of the time-fractional derivative with a space-splitting method. By doing so, we transform the time-fractional partial differential equation into a fixed number of integer-order in time partial differential equations. The Fokker–Planck equation posed on the 2d-dimensional domain is decoupled into two d-dimensional problems, and a standard combination of the Hermite spectral method on the configuration space domain and a finite element method on the flow domain is then applied. Finally, we combine our numerical scheme for the time-fractional Fokker–Planck equation (TFFPE) with a standard Navier–Stokes solver. We propose an efficient implementation strategy based on the algebraic structure of the discretized time-fractional Fokker–Planck equation, which significantly reduces the computational cost.

A generalisation of the Pucci–Saccomandi model of rubber elasticity

The Pucci–Saccomandi model of rubber elasticity, also known as the Gent–Gent model, is generalised in this manuscript for a more enhanced applicability to a wider range of finite deformation behaviours, rubber-like material types, and an increased modelling accuracy. The generalising of interest is developed by devising a higher order rational approximant, i.e., of order [1/1], of the response functions. The proposed model is shown to be a parent to some of the celebrated limiting chain extensibility, classical WI1,I2 and generalised neo-Hookean models. The application of the proposed generalised model to the finite deformation of various isotropic incompressible rubber-like materials is demonstrated, including natural unfilled and filled rubbers, commercial polymers and biomaterials. The considered datasets exhibit various deformation behaviours from the classical large deformation of natural rubbers to the downward concavity of filled rubbers, strain/shear softening behaviours of polymers and the softening behaviour of biomaterials up to the onset of failure. The modelling results are directly compared with those of the Gent–Gent model, where the higher accuracy of the proposed model and its capability to favourably capture the aforementioned challenging and varied behaviours is established. In particular, the Gent–Gent model is shown unsuitable for capturing such downward concavity, shear/strain softening and continuous softening to failure behaviours. Given the improved modelling predictions, simplicity of the functional form of the devised model, and its relatively low number of model parameters, the proposed model is presented as a more general, comprehensive and versatile modelling tool for application to the finite deformation of various isotropic incompressible soft materials.

Finite A2-Continued Fractions in the Problems of Rational Approximations of Real Numbers

Finite A2-Continued Fractions in the Problems of Rational Approximations of Real Numbers

N-Best kernel approximation in reproducing kernel Hilbert spaces

By making a seminal use of the maximum modulus principle of holomorphic functions we prove existence of n-best kernel approximation for a wide class of reproducing kernel Hilbert spaces of holomorphic functions in the unit disc, and for the corresponding class of Bochner type spaces of stochastic processes. This study thus generalizes the classical result of n-best rational approximation for the Hardy space and a recent result of n-best kernel approximation for the weighted Bergman spaces of the unit disc. The type of approximations have significant applications to signal and image processing and system identification, as well as to numerical solutions of the classical and the stochastic type integral and differential equations.

Simulation of random medium of fuel pebble with liquid salt coolant using PebMC code

Pebble-bed High Temperature reactors (HTR) are proposed to generate heat for power production and process heat application such as Hydrogen (H2) production. A HTR fuel pebble consists of random distribution of several thousands of fuel particles, called Tri-Structural Isotropic (TRISO), in fuel zone followed by a graphite layer. This configuration along with coolant layer results in double heterogeneity effect. A multi-group Monte Carlo code PebMC has been developed to accurately analyse molten salt cooled HTR lattice cell. Neutron transport in such a complex fuel distribution requires resonance self-shielded cross-section which is generated using rational approximation and Dancoff correction. Dancoff factor accounts for those neutrons leaving a fuel particle and directly colliding with another fuel particle, without any collision with moderator nuclei.In this study, an analytical expression based on a two-region model developed for gas cooled HTRs is adopted to calculate Dancoff factor for pebble with salt coolant. The implementation of these analytical expressions with 172groups WIMS library in PebMC code have been discussed. Subsequently, paper describes neutron tracking in random medium of HTR lattice using Monte Carlo method. The influence of fuel salt on Dancoff factor for various packing of TRISO particles and subsequent effect on infinite multiplication factor (K-inf) calculation have been presented. In addition, effect of temperature of molten salt on Dancoff factor have been studied.

Rational approximation for solving Fredholm integro-differential equations by new algorithm

Abstract In this article, we used a novel semi-analytical approach, named the optimal auxiliary function method (OAFM), to solve integro-differential equations (IDEs). The OAFM includes an auxiliary function and convergence control parameters, which expedite the convergence of the method. To apply the proposed method, some assumptions regarding small or large parameters in the problem are necessary. We present numerical outcomes acquired via the OAFM alongside those obtained from other numerical techniques in tables. Furthermore, we demonstrate the efficacy and ease of implementing the proposed method for various IDEs using 2D graphs.

Second virial coefficient, Boyle temperature and equation of state of van Hove fluids with a downward concavity attractive parabolic-well

The second virial coefficient, the Boyle temperature and the equation of state of van Hove fluids with a relatively short ranged attractive parabolic-well of downward concavity are considered. The analytic second virial coefficient for this fluid is obtained explicitly and it is used to compute the Boyle temperature of the fluid as a function of the range of the potential. Further, an equation of state is derived using the second-order thermodynamic perturbation theory of Barker and Henderson in the macroscopic compressibility approximation, with the hard-sphere fluid being the reference fluid. For this latter we profit from the fully analytical expression of the radial distribution function, consistent with the Carnahan-Starling equation state, derived within the so-called rational function approximation method up to a range twice the size of the hard-core diameter. The results for the reduced pressure of the fluid as a function of the packing fraction and two values of the range of the potential well at different temperatures are compared with Monte Carlo simulation data. Estimates of the values of the critical temperature are also provided.

Hand activity classification based on perturbed nearfield radiation and augmented impedance of a wearable textile antenna

Diagnosis of mobility disorder has become increasingly important in recent years. This paper investigates the feasibility of classifying six hand activities including two crucially Tremored and Non-Tremored motions, based on perturbed near-field radiation of the single body worn textile (BWT) antenna. The deployed antenna shows excellent biocompatibility and less vulnerability to electromagnetic absorption. Time-domain conversion of activities set using rational function approximation is investigated. Window cropping technique is exploited for data augmentation. Seven young and one elderly volunteer were used to assess the system’s performance. The measurements were performed in a chamber with an anechoic ceiling and negligible electromagnetic emanation. The Discrete Wavelet Transform (DWT) is adopted to extract the time domain features based on the fluctuation of impedance along with sliding window. Support Vector Machine (SVM) is utilized for classification. Two more algorithms namely k-Nearest Neighbor (k-NN) and Naive Bayes (NB) also been applied to test the accuracy of the classification and it is found that SVM produces best outcome in every cases under investigation. The performance of the different participants for the wrist mount and shoulder mount are tested individually. The wrist mount performs quite well, except for the elderly subject, which obtains low accuracy in both scenarios but the shoulder wins by a margin of 0.3%. Clear distinction between the results of tremored and non-tremored activities is seen. The combined augmented data for wrist and shoulder mount is observed to be 2.1% and 1.2% higher than the non-augmented data set, respectively.