The Experts below are selected from a list of 14598 Experts worldwide ranked by ideXlab platform
Victor Kowalenko - One of the best experts on this subject based on the ideXlab platform.
-
Programming the Partition Method for a Power Series Expansion
The Partition Method for a Power Series Expansion, 2017Co-Authors: Victor KowalenkoAbstract:A programming methodology is concerned with: (1) the analysis of a problem by developing algorithms based on modern programming techniques, (2) designing programs in appropriate languages and (3) implementation on a suitable platform. Chapter 5 completes the programming methodology for the partition method for a Power Series Expansion that began with the presentation of the BRCP algorithm in Chapter 3 and continued with the general theory of partition method for a Power Series Expansion in the previous chapter. The results of these chapters are employed to produce two C/C++ codes, which generate the coefficients in a general symbolic form that can be introduced into Mathematica. Both programs are fully listed in Appendix B , where they appear as Programs 1 and 2. The first program calculates the coefficients D k D k and E k E k in Theorem 4.1 for k ranging from unity to a specified number, while the second calculates only one coefficient, which is useful when the number of partitions becomes too large. By assigning values to the coefficients of the inner and outer Power Series, the coefficients of the resulting Power Series Expansion can be calculated using either the integer arithmetic routines in Mathematica, thereby avoiding rounded-off decimal values, or they can be evaluated via the symbolic routines to yield polynomial coefficients.
-
Chapter 4 – General Theory
The Partition Method for a Power Series Expansion, 2017Co-Authors: Victor KowalenkoAbstract:Chapter 4 presents the general theory behind the partition method for a Power Series Expansion, which is required for the development of a programming methodology appearing in the following chapter. The theory begins by introducing the pseudo-composite function g(af(x))g(af(x)), where a is arbitrary and the functions g(x)g(x) and f(x)f(x) are expressed as general Power Series Expansions that are referred to as the outer and inner Series, respectively. Then Theorem 4.1 presents general expressions for the coefficients of the resulting Power Series Expansion for both the quotient of pseudo-composite functions and its inverted form in terms of sums over partitions. Next it is shown that Faa di Bruno's formula or the Bell polynomial approach represents a special case of the partition method for a Power Series Expansion. To demonstrate both the generality and versatility of the partition method for a Power Series Expansion, Theorem 4.1 is used to determine the Power Series Expansion for the function f(z)=exp(azνsinρz)f(z)=exp(azνsinρz). If the quotient of the pseudo-composite functions in Theorem 4.1 is smooth, then the coefficients of its Taylor/Maclaurin Series can also be expressed in terms of a sum over the partitions. In a second corollary the theory is extended by taking an arbitrary Power ρ of the quotient of the pseudo-composite functions of Theorem 4.1. The chapter concludes by deriving the Power Series Expansion for the exponentiated function of the second corollary to Theorem 4.1.
-
Developments from Programming the Partition Method for a Power Series Expansion
arXiv: Combinatorics, 2012Co-Authors: Victor KowalenkoAbstract:Recently, a novel method based on coding partitions [1]-[4] has been used to derive Power Series Expansions to previously intractable problems. In this method the coefficients at $k$ are determined by summing the contributions made by each partition whose elements sum to $k$. These contributions are found by assigning values to each element and multiplying by an appropriate multinomial factor. This work presents a theoretical framework for the partition method for a Power Series Expansion. To overcome the complexity due to the contributions, a programming methodology is created allowing more general problems to be studied than envisaged originally. The methodology uses the bi-variate recursive central partition (BRCP) algorithm, which is based on a tree-diagram approach to scanning partitions. Its main advantage is that partitions are generated in the multiplicity representation. During the development of the theoretical framework, scanning over partitions was seen as a discrete operation with an operator $L_{P,k}[ \cdot]$, whose summand depends on the coefficients of the two Series when the original function is written as a pseudo-composite function. Simple modifications result in programs for other operators of specific types of partitions such as: (1) only odd or even elements, (2) a fixed number of elements, (3) discrete elements, (4) specific elements and (5) those restricted by element size. Another modification generates conjugate partitions by transposing Ferrers diagrams. The operator approach is then applied to the generating functions for both discrete and standard partitions. The main generalisation introduces a parameter $\omega$, whose Powers give the number of elements in the partitions while the coefficients become polynomials in $\omega$. Finally, Power Series Expansions for more advanced infinite products are derived, culminating in Heine's multi-parameter product.
-
Applications of the Cosecant and Related Numbers
Acta Applicandae Mathematicae, 2011Co-Authors: Victor KowalenkoAbstract:Power Series Expansions for cosecant and related functions together with a vast number of applications stemming from their coefficients are derived here. The coefficients for the cosecant Expansion can be evaluated by using: (1) numerous recurrence relations, (2) expressions resulting from the application of the partition method for obtaining a Power Series Expansion and (3) the result given in Theorem 3. Unlike the related Bernoulli numbers, these rational coefficients, which are called the cosecant numbers and are denoted by c k , converge rapidly to zero as k??. It is then shown how recent advances in obtaining meaningful values from divergent Series can be modified to determine exact numerical results from the asymptotic Series derived from the Laplace transform of the Power Series Expansion for tcsc?(at). Next the Power Series Expansion for secant is derived in terms of related coefficients known as the secant numbers d k . These numbers are related to the Euler numbers and can also be evaluated by numerous recurrence relations, some of which involve the cosecant numbers. The approaches used to obtain the Power Series Expansions for these fundamental trigonometric functions in addition to the methods used to evaluate their coefficients are employed in the derivation of Power Series Expansions for integer Powers and arbitrary Powers of the trigonometric functions. Recurrence relations are of limited benefit when evaluating the coefficients in the case of arbitrary Powers. Consequently, Power Series Expansions for the Legendre-Jacobi elliptic integrals can only be obtained by the partition method for a Power Series Expansion. Since the Bernoulli and Euler numbers give rise to polynomials from exponential generating functions, it is shown that the cosecant and secant numbers gives rise to their own polynomials from trigonometric generating functions. As expected, the new polynomials are related to the Bernoulli and Euler polynomials, but they are found to possess far more interesting properties, primarily due to the convergence of the coefficients. One interesting application of the new polynomials is the re-interpretation of the Euler-Maclaurin summation formula, which yields a new regularisation formula.
-
Generalizing the Reciprocal Logarithm Numbers by Adapting the Partition Method for a Power Series Expansion
Acta Applicandae Mathematicae, 2008Co-Authors: Victor KowalenkoAbstract:Recently, a novel method based on the coding of partitions was used to determine a Power Series Expansion for the reciprocal of the logarithmic function, viz. z/ln (1+z). Here we explain how this method can be adapted to obtain Power Series Expansions for other intractable functions. First, the method is adapted to evaluate the Bernoulli numbers and polynomials. As a result, new integral representations and properties are determined for the former. Then via another adaptation of the method we derive a Power Series Expansion for the function z s /ln s (1+z), whose polynomial coefficients A k (s) are referred to as the generalized reciprocal logarithm numbers because they reduce to the reciprocal logarithm numbers when s=1. In addition to presenting a general formula for their evaluation, this paper presents various properties of the generalized reciprocal logarithm numbers including general formulas for specific values of s, a recursion relation and a finite sum identity. Other representations in terms of special polynomials are also derived for the A k (s), which yield general formulas for the highest order coefficients. The paper concludes by deriving new results involving infinite Series of the A k (s) for the Riemann zeta and gamma functions and other mathematical quantities.
R.g. Reid - One of the best experts on this subject based on the ideXlab platform.
-
Use of Power Series Expansion for Residual Stress Determination by the Incremental Hole-Drilling Technique
Experimental Mechanics, 2020Co-Authors: T.c. Smit, R.g. ReidAbstract:Background The integral method of incremental hole-drilling is used extensively to determine the residual stress distribution in isotropic materials. When used with Tikhonov regularization, the method is robust and produces accurate results with minimal uncertainty. Alternatively, an optimal hole depth distribution can be found using the method of Zuccarello to improve the conditioning of the calibration matrices. If substantial measurement noise or a steep variation in stress exists, however, considerable uncertainty in, or distortion of, the calculated residual stress distribution can occur. Series Expansion offers an alternative solution, but it has been reported to become unstable before meaningful accuracy can be achieved. Objective Investigate the use of Series Expansion to determine a rapidly changing throughthickness residual stress distribution in an aluminium alloy 7075 plate subjected to laser shock peening treatment. Methods Power Series Expansion of eigenstrains is used in finite element modelling to calculate the calibration coefficients. Monte Carlo simulation is used to determine robust uncertainties in the residual stress distributions. This allows the Series order with the lowest RMS uncertainty in stress to be selected from those Series orders that have converged. The best estimate of the residual stress distribution is thereby obtained. Results Series Expansion is shown to be stable up to 8^th order and convergence to a stress solution can be found before instability dominates. The method is insensitive to measurement errors due to the least-squares approach employed by the inverse solution. Conclusions The use of Series Expansion reduces the RMS uncertainty in stress when compared to the regularized integral and Zuccarello methods.
-
Use of Power Series Expansion for Residual Stress Determination by the Incremental Hole-Drilling Technique
Experimental Mechanics, 2020Co-Authors: T.c. Smit, R.g. ReidAbstract:The integral method of incremental hole-drilling is used extensively to determine the residual stress distribution in isotropic materials. When used with Tikhonov regularization, the method is robust and produces accurate results with minimal uncertainty. Alternatively, an optimal hole depth distribution can be found using the method of Zuccarello to improve the conditioning of the calibration matrices. If substantial measurement noise or a steep variation in stress exists, however, considerable uncertainty in, or distortion of, the calculated residual stress distribution can occur. Series Expansion offers an alternative solution, but it has been reported to become unstable before meaningful accuracy can be achieved. Investigate the use of Series Expansion to determine a rapidly changing throughthickness residual stress distribution in an aluminium alloy 7075 plate subjected to laser shock peening treatment. Power Series Expansion of eigenstrains is used in finite element modelling to calculate the calibration coefficients. Monte Carlo simulation is used to determine robust uncertainties in the residual stress distributions. This allows the Series order with the lowest RMS uncertainty in stress to be selected from those Series orders that have converged. The best estimate of the residual stress distribution is thereby obtained. Series Expansion is shown to be stable up to 8th order and convergence to a stress solution can be found before instability dominates. The method is insensitive to measurement errors due to the least-squares approach employed by the inverse solution. The use of Series Expansion reduces the RMS uncertainty in stress when compared to the regularized integral and Zuccarello methods.
Alexander N. Drozdov - One of the best experts on this subject based on the ideXlab platform.
-
Improved Power Series Expansion for the time evolution operator: Application to two-dimensional systems
Journal of Chemical Physics, 1999Co-Authors: Alexander N. Drozdov, Shigeo HayashiAbstract:The Power Series Expansion formalism is used to construct analytical approximations for the propagator of the partial differential equation of a generic type. The present approach is limited to systems with polynomial coefficients. Three typical two-dimensional examples, a Henon–Heiles anharmonic resonating system, a system–bath Hamiltonian, and a Fokker–Planck chaotic model are considered. All results are in excellent agreement with those of an established numerical scheme in the field. It is found that the Power Series Expansion method accurately describes the dynamics of very anharmonic processes in the whole time domain.
-
exponential Power Series Expansion for the propagator of general diffusion processes
Physica A-statistical Mechanics and Its Applications, 1993Co-Authors: Alexander N. DrozdovAbstract:The propagator of a general diffusion process is determined by expanding its exponent in a Power Series of a time increment t. The Expansion coefficients can be analytically evaluated from recursive relations. We are thus able to construct a covariant short time approximation for the propagator valid to any desired precision in t. Attention is given both to the mathematics and its physical interpretation. This allows us to shed further light on the results already known in the literature on quantum mechanics and theory of continuous Markov processes.
-
an accurate Power Series Expansion for path integrals on curved manifolds
European Physical Journal B, 1993Co-Authors: Alexander N. DrozdovAbstract:A general and simple framework for treating path integrals on curved manifolds is presented. The crucial point is expanding the exponent of the propagator of general diffusion processes in a Power Series in time. The Expansion coefficients are determined by recursive relations and can be analytically evaluated to any desired level of accuracy int. The treatment is both theoretically and numerically advantageous with respect to the other path integral methods known in the literature. Its Power is illustrated on two exactly solvable models. The propagator obtained is shown to be much more accurate over a broad range oft than the standard short time approximation. In view of its numerical application this means significant reducing the number of time steps that are required to evaluate a path integral.
T.c. Smit - One of the best experts on this subject based on the ideXlab platform.
-
Use of Power Series Expansion for Residual Stress Determination by the Incremental Hole-Drilling Technique
Experimental Mechanics, 2020Co-Authors: T.c. Smit, R.g. ReidAbstract:Background The integral method of incremental hole-drilling is used extensively to determine the residual stress distribution in isotropic materials. When used with Tikhonov regularization, the method is robust and produces accurate results with minimal uncertainty. Alternatively, an optimal hole depth distribution can be found using the method of Zuccarello to improve the conditioning of the calibration matrices. If substantial measurement noise or a steep variation in stress exists, however, considerable uncertainty in, or distortion of, the calculated residual stress distribution can occur. Series Expansion offers an alternative solution, but it has been reported to become unstable before meaningful accuracy can be achieved. Objective Investigate the use of Series Expansion to determine a rapidly changing throughthickness residual stress distribution in an aluminium alloy 7075 plate subjected to laser shock peening treatment. Methods Power Series Expansion of eigenstrains is used in finite element modelling to calculate the calibration coefficients. Monte Carlo simulation is used to determine robust uncertainties in the residual stress distributions. This allows the Series order with the lowest RMS uncertainty in stress to be selected from those Series orders that have converged. The best estimate of the residual stress distribution is thereby obtained. Results Series Expansion is shown to be stable up to 8^th order and convergence to a stress solution can be found before instability dominates. The method is insensitive to measurement errors due to the least-squares approach employed by the inverse solution. Conclusions The use of Series Expansion reduces the RMS uncertainty in stress when compared to the regularized integral and Zuccarello methods.
-
Use of Power Series Expansion for Residual Stress Determination by the Incremental Hole-Drilling Technique
Experimental Mechanics, 2020Co-Authors: T.c. Smit, R.g. ReidAbstract:The integral method of incremental hole-drilling is used extensively to determine the residual stress distribution in isotropic materials. When used with Tikhonov regularization, the method is robust and produces accurate results with minimal uncertainty. Alternatively, an optimal hole depth distribution can be found using the method of Zuccarello to improve the conditioning of the calibration matrices. If substantial measurement noise or a steep variation in stress exists, however, considerable uncertainty in, or distortion of, the calculated residual stress distribution can occur. Series Expansion offers an alternative solution, but it has been reported to become unstable before meaningful accuracy can be achieved. Investigate the use of Series Expansion to determine a rapidly changing throughthickness residual stress distribution in an aluminium alloy 7075 plate subjected to laser shock peening treatment. Power Series Expansion of eigenstrains is used in finite element modelling to calculate the calibration coefficients. Monte Carlo simulation is used to determine robust uncertainties in the residual stress distributions. This allows the Series order with the lowest RMS uncertainty in stress to be selected from those Series orders that have converged. The best estimate of the residual stress distribution is thereby obtained. Series Expansion is shown to be stable up to 8th order and convergence to a stress solution can be found before instability dominates. The method is insensitive to measurement errors due to the least-squares approach employed by the inverse solution. The use of Series Expansion reduces the RMS uncertainty in stress when compared to the regularized integral and Zuccarello methods.
Youcef Ferdi - One of the best experts on this subject based on the ideXlab platform.
-
Computation of Fractional Order Derivative and Integral via Power Series Expansion and Signal Modelling
Nonlinear Dynamics, 2006Co-Authors: Youcef FerdiAbstract:The three techniques of s -to- z transform, Power Series Expansion (PSE) and signal modelling are combined to develop a new procedure for efficiently computing the fractional order derivatives and integrals of discrete-time signals. A mapping function between the s -plane and the z -plane is first chosen, and then a PSE of this mapping function raised to fractional order is performed to get the desired infinite impulse response of the ideal digital fractional operator. Finally, the desired impulse response is modelled as the impulse response of a linear invariant system whose rational transfer function is determined using deterministic signal modelling techniques. Three non-iterative techniques, namely Padé, Prony and Shanks’ methods have been considered in this paper. Using Al-Alaoui’s rule as s -to- z transform, computation examples show that both Prony and Shanks’ method can achieve more accurate fractional differentiation and integration than Padé method which is equivalent to continued fraction Expansion technique.
-
computation of fractional order derivative and integral via Power Series Expansion and signal modelling
Nonlinear Dynamics, 2006Co-Authors: Youcef FerdiAbstract:The three techniques of s-to-z transform, Power Series Expansion (PSE) and signal modelling are combined to develop a new procedure for efficiently computing the fractional order derivatives and integrals of discrete-time signals. A mapping function between the s-plane and the z-plane is first chosen, and then a PSE of this mapping function raised to fractional order is performed to get the desired infinite impulse response of the ideal digital fractional operator. Finally, the desired impulse response is modelled as the impulse response of a linear invariant system whose rational transfer function is determined using deterministic signal modelling techniques. Three non-iterative techniques, namely Pade, Prony and Shanks’ methods have been considered in this paper. Using Al-Alaoui’s rule as s-to-z transform, computation examples show that both Prony and Shanks’ method can achieve more accurate fractional differentiation and integration than Pade method which is equivalent to continued fraction Expansion technique.