Narrow Results

The problem of extrapolating the series in powers of small variables to the region of large variables is addressed. Such a problem is typical of quantum theory and statistical physics. A method of extrapolation is developed based on self-similar factor and root approximants, suggested earlier by the authors. It is shown that these approximants and their combinations can effectively extrapolate power series to the region of large variables, even up to infinity. Several examples from quantum and statistical mechanics are analyzed, illustrating the approach.

Power series method is used to analyze the distribution of stresses of the transversely isotropic cylinder under arbitrary axisymmetrical normal load. The stress function and normal load are expanded as the form of Fourier series. Unknown parameters in equations are determined using boundary conditions and final solutions of stress and displacement are obtained accordingly. The analysis of stresses obtained from different material parameters shows that the axial stress is significantly affected by Young's modulus and the circumference stress is sensitive to the variation of Poisson's ratio.

This paper describes a new reliable numerical method for computing chaotic solutions of dynamical systems and, in special cases, is applied to Chen strange attractor. The numerical precision of the computation is finely mastered. We introduce a modification of the method of power series for the construction of approximate solutions of the Chen system together with forward/backward control of the precision. As a test for the method, we obtained the region of convergence of series and researched the behavior of the trajectories on this attractor. The results of a numerical experiment are presented.

This research offers a precise analytical solution for initial value problems of both linear and nonlinear fractional order partial differential equations. A new approach called the limit residual function method is developed and utilized to generate a series solution for the equations. The key tools of this method are the concepts of power series, residual function, and the limit at zero. The new method is characterized by the speed of determining the series solution coefficients and the limited mathematical operations used. The study examines five significant applications of fractional partial differential equations using this new method. To validate the accuracy of the results, they are compared with exact solutions in the classical case for the discussed applications. We conclude that the proposed approach is straightforward, effective, and successful in solving linear and nonlinear fractional differential equations.

In our previous work, we proposed a general fourth order partial differential equation (PDE) for the generation of blending surfaces together with an accurate closed form solution for special boundary conditions where closed form solutions exist. In general, however, a closed form solution of a PDE is often either not existent or not obtainable depending on the boundary conditions and the coefficients of the PDE. In order to solve a wider range of surface blending problems using the new form of PDE, we propose in this paper a much more general resolution method using power series expansion. The main idea is to decompose the boundary conditions into a number of basic functions and separate them into two groups, one with and the other without closed form solutions. For the first group, the closed form solutions are produced with our previously proposed method. For the second group, a composite function is constructed which combines these basic functions with the power series of one parametric variable. The unknown constants of the composite function are determined by making use of both the boundary conditions and the weighted residual method, which minimizes the surface errors. Compared with existing resolution methods, the proposed method exhibits a number of advantages: 1) it is a general resolution method and hence able to solve a much wider class of surface blending problems; 2) the boundary conditions are always satisfied exactly that is crucial to the quality of blending surfaces; and 3) only a small number of collocation points and unknown constants are needed, making our method computationally efficient. Several examples given in the paper have demonstrated that this method is not only accurate, but fast enough for interactive applications.

Let A be an integral domain, X an analytic indeterminate over A and I a proper ideal (not necessarily prime) of A. In this paper, we study the ring

Next, we study the Krull dimension of R. We give a necessary condition for R to be of finite Krull dimension. In particular, if R is of finite dimension then I must be an SFT ideal of A. Then we determine bounds for dim(R). Examples are given to indicate the sharpness of the results. In case I is a maximal ideal of A and A is either a Noetherian ring, SFT Prüfer domain or A[[X]] is catenarian and I SFT, we establish that dim(R) = dim(A[[X]]) = dim(A) + 1.

Finally, we examine the possible transfer of the LFD property and the catenarity between the rings A, A[[X]] and R in case I is a maximal ideal of A.

The work of Helmer [Divisibility properties of integral functions, Duke Math. J. 6(2) (1940) 345–356] applied algebraic methods to the field of complex analysis when he proved the ring of entire functions on the complex plane is a Bezout domain (i.e. all finitely generated ideals are principal). This inspired the work of Henriksen [On the ideal structure of the ring of entire functions, Pacific J. Math. 2(2) (1952) 179–184. On the prime ideals of the ring of entire functions, Pacific J. Math. 3(4) (1953) 711–720] who proved a correspondence between the maximal ideals within the ring of entire functions and ultrafilters on sets of zeroes as well as a correspondence between the prime ideals and growth rates on the multiplicities of zeroes. We prove analogous results on rings of analytic functions in the non-Archimedean context: all finitely generated ideals in the ring of analytic functions on an annulus of a characteristic zero non-Archimedean field are two-generated but not guaranteed to be principal. We also prove the maximal and prime ideal structure in the non-Archimedean context is similar to that of the ordinary complex numbers; however, the methodology has to be significantly altered to account for the failure of Weierstrass factorization on balls of finite radius in fields which are not spherically complete, which was proven by Lazard [Les zeros d’une function analytique d’une variable sur un corps value complet, Publ. Math. l’IHES 14(1) (1942) 47–75].

Using complete orthogonal ${{ℒ}}^{(p_l^{*})}$-Self-Friction Polynomials (${{ℒ}}^{(p_l^{*})}$-SFPs) introduced by one of the authors, the analytical and power series formulas for SF atomic nuclear attraction integrals over $\chi$-noninteger Slater type orbitals ($\chi$-NISTOs) and $V$-noninteger Coulomb–Yukawa-like potentials ($V$-NICYPs) are presented, where ${\alpha }^{*}$ are the integer (${\alpha }^{*}=\alpha ,-\infty <\alpha \le 2)$ or noninteger (${\alpha }^{*}\ne \alpha ,-\infty <{\alpha }^{*}<3)$ SF quantum numbers and $p_l^{*}=2l+2-{\alpha }^{*}$. As an application, the computer calculations for dependence of the atomic nuclear attraction integrals over $\chi$-NISTOs and $V$-NICYPs functions are presented.

Inverse-kinematics is an emphasis and difficulty in the design and application of the humanoid robot hand with coupled joints because of nonlinearity induced by trigonometric transcendental function. In this paper, a power series based inverse-kinematics algorithm is presented, by which the transcendental equation including trigonometric function can be converted into an algebraic equation. An approximate solution is derived first by means of power series expansions; with 1D linear interpolation for errors compensating, the final solution with small error can then be achieved. For robot with linearly coupled joints, the algorithms based on power series expanded to quadratic and quartic terms are used to calculate the accurate joint angles. For robot with nonlinearly coupled joints, the specific procedures are proposed to select appropriate transmission ratio. Simulation and experimental results demonstrate effectiveness of the proposed inverse kinematics method.

In this paper, we use Padé approximation, to solve a boundary-value problem for an isothermal gas sphere, which is a special form of Lane–Emden equation. However, in general, the solution of Lane–Emden equation cannot be given analytically. This type of non-linear differential equations can be solved only numerically. An example is presented to show the ability of the method for Lane–Emden differential equation. Numerical solutions to the isothermal gas sphere have been discussed. We use MAPLE computer algebra systems for numerical calculations [Frank (1996)].

This paper systematically investigates the Lie symmetry analysis of the time-fractional Buckmaster equation in the sense of Riemann–Liouville fractional derivative. With the aid of infinitesimal symmetries, this equation is transformed into a nonlinear ordinary differential equation of fractional order (FODE), where the fractional derivatives are in Erdelyi–Kober sense. The reduced FODE is solved with the explicit power series method and some figures for the obtained power series solutions are also depicted. Finally, Ibragimov’s method and Noether’s theorem have been employed to conclude the conservation laws of this equation.

We define and investigate (structural) row-finite matrices, lower triangular matrices, and power series in near-rings by using inverse limits of some special classes of matrices. Our approach is different from that described in [8]. We show that polynomials can be embedded in power series and power series can be embedded in lower triangular matrices as in rings. We extend the concept of order of a polynomial (or a power series) from rings to near-rings. A natural topology is defined on lower triangular matrices by generalizing the concept of order of power series. We also obtain some results concerning prime and completely prime ideals in lower triangular matrix near-rings.

We consider the two Diophantine equations y^m = F(x) and G(y) = F(x) under the assumption that gcd(m, deg F) > 1 and gcd (deg G, deg F) > 1, respectively. We prove that the bounds for the denominator of the coefficients of the power series arising from the above two situations can be improved considerably and thus we establish improved upper bounds for the size of the solutions (namely for |x| and |y|). We also give explicit upper bounds for the integer solutions of equations of the form

We consider polynomials with integer coefficients and discuss their factorization properties in ℤ[[x]], the ring of formal power series over ℤ. We treat polynomials of arbitrary degree and give sufficient conditions for their reducibility as power series. Moreover, if a polynomial is reducible over ℤ[[x]], we provide an explicit factorization algorithm. For polynomials whose constant term is a prime power, our study leads to the discussion of p-adic integers.

We express products of Lambert series as power series. We use these Lambert series-to-power series identities to obtain new Liouville identities with two functions. Many of the known Liouville identities follow from our new identities. We explore the relationships between Lambert series, new Liouville identities, and convolution sums. We then obtain recursive formulae for various convolution sums.