On Taylor coefficients of Cauchy-type integrals of finite complex measures

Boundary values of Cauchy-type integrals of finite complex measures given on a unit circle, generally speaking, are not Lebesgue integrable, and therefore at expansion of Cauchy-type integrals in Taylor series, the expansion coefficients cannot be expressed by boundary values using the Lebesgue integral. In this paper, using the notion of A-integration and N-integration, we get a formula for calculating the Taylor expansion coefficients of Cauchy-type integrals of finite complex measures.     

Pseudopower expansion of solutions of generalized equations and constrained optimization problems

We show that the solution of a strongly regular generalized equation subject to a scalar perturbation expands in pseudopower series in terms of the perturbation parameter, i.e., the expansion of orderk is the solution of generalized equations expanded to orderk and thus depends itself on the perturbation parameter. In the polyhedral case, this expansion reduces to a usual Taylor expansion. These results are applied to the problem of regular perturbation in constrained optimization. We show that, if the strong regularity condition is satisfied, the property of quadratic growth holds and, at least locally, the solutions of the optimization problem and of the associated optimality system coincide. If, in addition the number of inequality constraints is finite, the solution and the Lagrange multiplier can be expanded in Taylor series. If the data are analytic, the solution and the multiplier are analytic functions of the perturbation parameter.     

D-Optimal Designs for Trigonometric Regression Models on a Partial Circle

In the common trigonometric regression model we investigate the D-optimal design problem, where the design space is a partial circle. It is demonstrated that the structure of the optimal design depends only on the length of the design space and that the support points (and weights) are analytic functions of this parameter. By means of a Taylor expansion we provide a recursive algorithm such that the D-optimal designs for Fourier regression models on a partial circle can be determined in all cases. In the linear and quadratic case the D-optimal design can be determined explicitly.     

The geometric structure of the expected/observed likelihood expansions

Stochastic expansions of likelihood quantities are a basic tool for asymptotic inference. The traditional derivation is through ordinary Taylor expansions, rearranging terms according to their asymptotic order. The resulting expansions are called hereexpected/observed, being expressed in terms of the score vector, the expected information matrix, log likelihood derivatives and their joint moments. Though very convenient for many statistical purposes, expected/observed expansions are not usually written in tensorial form. Recently, within a differential geometric approach to asymptotic statistical calculations, invariant Taylor expansions based on likelihood yokes have been introduced. The resulting formulae are invariant, but the quantities involved are in some respects less convenient for statistical purposes. The aim of this paper is to show that, through an invariant Taylor expansion of the coordinates related to the expected likelihood yoke, expected/observed expansions up to the fourth asymptotic order may be re-obtained from invariant Taylor expansions. This derivation producesinvariant expected/observed expansions.This research was partially supported by the Italian National Research Council grant n.93.00824.CT10.     

Some remarks on holomorphic functions and taylor series in C

Some previous results on convergence of Taylor series in C^n [3] are improved by indicating outside the domain of convergence the points where the series diverges and simplifying some proofs. These results contain the Cauchy-Hadamard theorem in C. Some Cauchy integral formulas of a holomorphic function on a closed ball in C^n are constructed and the Taylor series expansion is deduced.     

Extreme Bloch Functions and Summation Methods for Bloch Functions

In this paper we construct Bloch functions F for which the set {z | e sup _{|ζ| < 1} |F'(ζ)| ( 1 - |ζ| ² ) = |F'(z)| ( 1 - |z| ² )} is an analytic Jordan curve tangential to the unit disk in some points. It is proved that, using such functions, we can derive analogs to the Taylor expansion for Bloch functions in cases where the Taylor expansion does not converge. October 15, 1997. Date revised: March 12, 1998. Date accepted: June 18, 1998.     

Solution of Two-Point Boundary-Value Problems Using the Differential Transformation Method

This paper considers two-point boundary-value problems using the differential transformation method. An iterative procedure is proposed for both the linear and nonlinear cases. Using the proposed approach, an analytic solution of the two-point boundary-value problem, represented by an mth-order Taylor series expansion, can be obtained throughout the prescribed range.     

On the Jacobi series

Summary The Jacobi series of a functionf is an expansion in a series of ascending powers of a prescribed polynomialP of degreen in which the coefficients are polynomials of lesser degree. These coefficients are usually expressed as contour integrals or are determined by their interpolatory properties. We show how they may be expressed as generalized derivatives off with respect toP. In so doing we also show how the Jacobi series may be expressed (in yet another way) as a generalized Taylor series. In addition, we obtain a number of interesting relations among the generalized derivatives.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday.     

On the Discretization of Double-Bracket Flows

This paper extends the method of Magnus series to Lie-algebraic equations originating in double-bracket flows. We show that the solution of the isospectral flow Y'=[[Y,N],Y] , Y(0)=Y ₀ ∈\Sym(n) , can be represented in the form Y(t)=e ^Ω(t) Y ₀ e ^-Ω(t) , where the Taylor expansion of Ω can be constructed explicitly, term-by-term, identifying individual expansion terms with certain rooted trees with bicolor leaves. This approach is extended to other Lie-algebraic equations that can be appropriately expressed in terms of a finite ``alphabet.'     

Stability and bifurcation of couette flow in the case of a narrow gap between rotating cylinders : PMM vol. 38, n 6, 1974, pp. 1025–1030

Stability and bifurcation of Couette flow between concentric rotating cylinders are investigated for the case when the ratios of their radii R and angular velocities Ω are nearly equal to unity. The limiting problem in the linear theory when R → 1 and Ω → 1 is the problem of convection stability in the layer [1]. We find that this is also correct in the case of a nonlinear problem. Below we show that solution of the problem of free convection yields the principal term of the expansion of the secondary flow (Taylor vortex) in the powers of a small parameter δ = R − 1. Therefore the results of [2, 3] can be used to provide, in the present case, a strict justification for the use of the Liapunov-Schmidt method to compute the Taylor vortices. The numerical results obtained for the critical Reynolds' number and the amplitude of the secondary flow provide a good illustration of the asymptotic passage as δ → 0.     

Taylor Series Expansion in Discrete Clifford Analysis

Discrete Clifford analysis is a discrete higher-dimensional function theory which corresponds simultaneously to a refinement of discrete harmonic analysis and to a discrete counterpart of Euclidean Clifford analysis. The discrete framework is based on a discrete Dirac operator that combines both forward and backward difference operators and on the splitting of the basis elements $\mathbf{e}_j = \mathbf{e}_j^+ + \mathbf{e}_j^-$ into forward and backward basis elements $\mathbf{e}_j^\pm $ . For a systematic development of this function theory, an indispensable tool is the Taylor series expansion, which decomposes a discrete (monogenic) function in terms of discrete homogeneous (monogenic) building blocks. The latter are the so-called discrete Fueter polynomials. For a discrete function, the authors assumed a series expansion which is formally equivalent to the Taylor series expansion in Euclidean Clifford analysis; however, attention needed to be paid to the geometrical conditions on the domain of the function, the convergence and the equivalence to the given discrete function. We furthermore applied the theory to discrete delta functions and investigated the connection with Shannon sampling theorem (Bell Sys Tech J 27:379–423, 1948). We found that any discrete function admits a series expansion into discrete homogeneous polynomials and any discrete monogenic function admits a Taylor series expansion in terms of the discrete Fueter polynomials, i.e. discrete homogeneous monogenic polynomials. Although formally the discrete Taylor series expansion of a function resembles the continuous Taylor series expansion, the main difference is that there is no restriction on discrete functions to be represented as infinite series of discrete homogeneous polynomials. Finally, since the continuous expansion of the Taylor series expansion of discrete delta functions is a sinc function, the discrete Taylor series expansion lays a link with Shannon sampling.     

非线性发展方程的Taylor展开方法

提出了积分非线性发展方程的新方法,即Taylor展开方法．标准的Galerkin方法可以看作0-阶Taylor展开方法,而非线性Galerkin方法可以看作1-阶修正Taylor展开方法A·D2此外,证明了数值解的存在性及其收敛性．结果表明,在关于严格解的一些正则性假设下,较高阶的Taylor展开方法具有较高阶的收敛速度．最后,给出了用Taylor展开方法求解二维具有非滑移边界条件Navier-Stokes方程的具体例子．     

Asymptotic expansions for a class of zeta-functions

In this paper, we shall reveal the hidden structure in recent results of Katsurada as the Meijer G-function hierarchy. In Sect. 1, we consider the holomorphic Eisenstein series and show that Katsurada’s two new expressions are variants of the classical Chowla–Selberg integral formula (Fourier expansion) with or without the beta-transform of Katsurada being incorporated. In Sect. 2, we treat the Taylor series expansion of the Lipschitz–Lerch transcendent in the perturbation variable. In the proofs, we make an extensive use of the beta-transform (used to be called the Mellin–Barnes formula).     

Extended balance of linear momentum from higher gradient stress field theory

We present the equation of linear momentum considering higher gradients for stress and body force. Both are approximated via Taylor series expansion within a finite Cauchy cube of dimensions L_c. Whereas linear terms of the series expansion result to the classical balance of linear momentum, terms up to third order yield an extended balance equation. The extension includes an internal length scale L²_c caused by surface integrals on the cube. The approach makes use of Cauchy's theorem and standard Newtonian mechanics but constitutive assumptions are not applied. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

非线性算子方程的泰勒展式算法

本文的目的是给出一种解Ｈｉｌｂｅｒｔ空间中非线性方程的ｋ阶泰勒展式算法（ｋ１）．标准Ｇａｌｅｒｋｉｎ方法可以看作１阶泰勒展式算法，而最优非线性Ｇａｌｅｒｋｉｎ方法可视为２阶泰勒展式算法．我们应用这种算法于定常的Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程的数值逼近．在一定情景下，最优非线性Ｇａｌｅｒｋｉｎ方法提供比标准Ｇａｌｅｒｋｉｎ方法和非线性Ｇａｌｅｒｋｉｎ方法更高阶的收敛速度．     

The essence of the generalized Taylor theorem as the foundation of the homotopy analysis method

The generalized Taylor theorem is the foundation of the homotopy analysis method proposed by Liao. This theorem is interesting but hard to understand from the mathematical point of view. Especially, there is a key parameter h whose meaning is still unknown. In the paper, we derive the generalized Taylor theorem from a usual way, that is, we prove that the generalized Taylor expansion is equivalent to a different representation of the usual Taylor expansion at different points. Therefore the meaning of the auxiliary parameter h is clarified. These results give a reasonable explanation of the parameter h and uncover the essence of the generalized Taylor theorem from which we can deeply understand the homotopy analysis method. Through the detailed analysis of some examples, we compare the series solution at the different points with the generalized Taylor series solution obtained by the homotopy analysis method.     

Equiconvergence of some complex interpolatory polynomials

Summary Walsh showed the close relation between the Lagrange interpolant in then ^th roots of unity and the corresponding Taylor expansion for functions belonging to a certain class of analytic functions. Recent extensions of this phenomena to Hermite interpolation and other linear processes of interpolation have been surveyed in [3, 5]. Following a recent idea of L. Yuanren [7], we show how new relations between other linear operators can be derived which exhibit Walsh equiconvergence.Dedicated to R. S. Varga on the occasion of his sixtieth birthdayThese authors were supported by NSERC A3094     

A representation theorem for smooth Brownian martingales

We show that, under certain smoothness conditions, a Brownian martingale, when evaluated at a fixed time, can be represented via an exponential formula at a later time. The time-dependent generator of this exponential operator only depends on the second order Malliavin derivative operator evaluated along a ‘frozen path’. The exponential operator can be expanded explicitly to a series representation, which resembles the Dyson series of quantum mechanics. Our continuous-time martingale representation result can be proven independently by two different methods. In the first method, one constructs a time-evolution equation, by passage to the limit of a special case of a backward Taylor expansion of an approximating discrete-time martingale. The exponential formula is a solution of the time-evolution equation, but we emphasize in our article that the time-evolution equation is a separate result of independent interest. In the second method, we use the property of denseness of exponential functions. We provide several applications of the exponential formula, and briefly highlight numerical applications of the backward Taylor expansion.     

A Degenerate Fourth-Order Parabolic Equation Modeling Bose-Einstein Condensation Part II: Finite-Time Blow-Up

A degenerate fourth-order parabolic equation modeling condensation phenomena related to Bose-Einstein particles is analyzed. The model can be motivated from the spatially homogeneous and isotropic Boltzmann-Nordheim equation by a formal Taylor expansion of the collision integral. It maintains some of the main features of the kinetic model, namely mass and energy conservation and condensation at zero energy. The existence of local-in-time weak solutions satisfying a certain entropy inequality is proven. The main result asserts that if a weighted L ¹ norm of the initial data is sufficiently large and the initial data satisfies some integrability conditions, the solution blows up with respect to the L ^∞ norm in finite time. Furthermore, the set of all such blow-up enforcing initial functions is shown to be dense in the set of all admissible initial data. The proofs are based on approximation arguments and interpolation inequalities in weighted Sobolev spaces. By exploiting the entropy inequality, a nonlinear integral inequality is proved which implies the finite-time blow-up property.