Integrability and nonintegrability in terms of transcendental functions in dynamics of a rigid body

The results of the presented work are due to the study of the applied problem of the rigid body motion in a resisting medium. More earlier the complete lists of transcendental first integrals expressed through a finite combination of elementary functions were obtained. This circumstance allowed the author to perform a complete analysis of all phase trajectories and highlight those properties of them which exhibit the roughness and preserve for systems of a more general form. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bounds for symmetric elliptic integrals

Lower and upper bounds for the four standard incomplete symmetric elliptic integrals are obtained. The bounding functions are expressed in terms of the elementary transcendental functions. Sharp bounds for the ratio of the complete elliptic integrals of the second kind and the first kind are also derived. These results can be used to obtain bounds for the product of these integrals. It is shown that an iterative numerical algorithm for computing the ratios and products of complete integrals has the second order of convergence.     

Integrals of psi—function

To the author's knowledge, among the so—called special functions, the gamma function is a unique one which is defined by a linear difference equation and is a hyper—transcendental function. There exists an another well—known hyper—transcendental function called the psi function, which is merely the logarithmic derivative of the gamma function. In this paper the author consider an extension of the gamma function and then obtain a series of integrals of the psi function.     

Asymptotic expansions of integrals of two Bessel functions via the generalized hypergeometric and Meijer functions

Asymptotic expansions of certain finite and infinite integrals involving products of two Bessel functions of the first kind are obtained by using the generalized hypergeometric and Meijer functions. The Bessel functions involved are of arbitrary (generally different) orders, but of the same argument containing a parameter which tends to infinity. These types of integrals arise in various contexts, including wave scattering and crystallography, and are of general mathematical interest being related to the Riemann—Liouville and Hankel integrals. The results complete the asymptotic expansions derived previously by two different methods — a straightforward approach and the Mellin-transform technique. These asymptotic expansions supply practical algorithms for computing the integrals. The leading terms explicitly provide valuable analytical insight into the high-frequency behavior of the solutions to the wave-scattering problems.     

Oscillatory integrals for phase functions having certain degenerate critical points

The paper is concerned with oscillatory integrals for phase functions having certain de- generate critical points. Under a finite type condition of phase functions we show the estimate of oscillatory integrals of the first kind. The decay of the oscillatory integral depends on indices of the finite type, the spatial dimension and the symbol.     

Oscillatory integrals and the method of stationary phase in infinitely many dimensions,with applications to the classical limit of quantum mechanics I

We give a theory of oscillatory integrals in infinitely many dimensions which extends, for a class of phase functions, the finite dimensional theory. In particular we extend the method of stationary phase, the theory of Lagrange immersions and the corresponding asymptotic expansions to the infinite dimensional case. A particular application of the theory to the Feyman path integrals defined in previous work by the authors yields asymptotic expansions to all orders of quantum mechanical quantities in powers of Planck's constant.Work supported by the Norwegian Research Council for Science and the Humanities     

Generalised functions and divergent integrals

A method is given for the interpretation of a class of divergent integrals in terms of a sum of function evaluations over an arbitrary partition of the integration interval. The class of integrands considered includes functions continuous on the integration interval, except at a finite number of algebraic or algebraico-logarithmic singularities, and the delta function and related generalised functions, or products of these. The interpretation assigned to such integrals coincides with that of generalised function theory. Possible applications of the method to the computation of functions are discussed.     

On linear extremal problems in the class of schlicht functions

The form of the Schiffer differential equation puts severe restrictions on the class of functions that can occur as extremal functions for arbitrary coefficient-functionals of finite degree. Theorem 1 characterizes the algebraic extremal functions for coefficient-functionals of finite degree. Furthermore it is shown that the extremal function either is an algebraic function or it must possess a non-isolated singularity, or must have a transcendental branch-point. The results are closely related to the Malmquist-Yosida theorems. However Nevanlinna's Theory of Value Distribution is not the mainly used tool but the special form of the Schiffer differential equation and the multiplicity of certain values together with the Great Picard Theorem are exploited.     

A new characterisation of the Eremenko-Lyubich class

The Eremenko-Lyubich class of transcendental entire functions with a bounded set of singular values has been much studied. We give a new characterisation of this class of functions. We also give a new result regarding direct singularities which are not logarithmic.     

使用对称性计算一类有限元基函数的二重积分

本文使用对称性计算一类有限元基函数在非规则区域的二重积分.通过两个算例验证对称性技巧在重积分计算上所带来的极大便利性.本文的内容进一步说明了使用对称性进行重积分计算在其他学科的应用价值.     

On rotationally invariant vector fields in the plane

This work is concerned with global properties of a class of ℂ-valued vector fields in the plane which are rotationally invariant. It is shown that the finite type rotationally invariant vector fields have global first integrals. We also study the global hypoellipticity and global solvability properties of these vector fields.     

Random dynamics of transcendental functions

This work concerns random dynamics of hyperbolic entire and meromorphic functions of finite order whose derivative satisfies some growth condition at ∞. This class contains most of the classical families of transcendental functions and goes much beyond. Based on uniform versions of Nevanlinna’s value distribution theory, we first build a thermodynamical formalism which, in particular, produces unique geometric and fiberwise invariant Gibbs states. Moreover, spectral gap property for the associated transfer operator along with exponential decay of correlations and a central limit theorem are shown. This part relies on our construction of new positive invariant cones that are adapted to the setting of unbounded phase spaces. This setting rules out the use of Hilbert’s metric along with the usual contraction principle. However, these cones allow us to apply a contraction argument stemming from Bowen’s initial approach.     

一类Hamilton系统的Abel积分比值单调性的判别条件

用直接计算的方法对一类Hamilton系统的两个Abel积分比值的单调性进行讨论,指出该单词性条件可由两个判定函数直接确定．     

Residue integrals and Waring’s formulas for algebraic or even transcendental systems

The present article is focused on the study of a special class of systems of non-linear transcendental equations for which classical algebraic and symbolic methods are inapplicable. For the purpose of study of such systems we develop a method for computing residue integrals with integration over certain cycles. We describe conditions under which the mentioned residue integrals coincide with power sums of the inverses to the roots of a system of equations (i.e. multidimensional Waring’s formulas). Based on this, we develop an algorithm that computes such power sums without computing the roots. As an application of the suggested method, we consider a problem of finding sums of multi-variable number series.     

The Incomplete Gamma Function Expressed as a Sum of Macdonald Functions

The incomplete gamma function is rewritten as a finite sum of Macdonald functions (modified Bessel functions of the second kind), K_n+1/2(z), allowing a class of integrals appearing in atomic physics to be easily evaluated.     

Solutions Almost Periodic at Infinity to Differential Equations With Unbounded Operator Coefficients

The new class of functions almost periodic at infinity is defined using the subspace of functions with integrals decreasing at infinity. We obtain spectral criteria for almost periodicity at infinity of bounded solutions to differential equations with unbounded operator coefficients. For the new class of asymptotically finite operator semigroups we prove the almost periodicity at infinity of their orbits.     

On computer implementation of the Hertz elastic contact model and its simplifications

The paper is concerned with an elastic contact model of rigid bodies which is developed in the framework of the Hertz model. For this new model, we suggest more effective algorithms with reduced computational time. We also present an algorithm for representation of the geometry of the contacting surfaces in the local contact coordinate system. This coordinate system tracks permanently the surfaces of the bodies, which are able to contact. An approach to computation of the normal elastic force is presented. It is based on the reduction to a single transcendental scalar equation that includes the complete elliptic integrals of the first and second kinds. The computational time in the Hertz-model simulation was considerably reduced due to the use of the differential technique for computation of the complete elliptic integrals and due to the replacement of the implicit transcendental equation by a differential one. Using the classic solution of the Hertz contact problem, we then present an invariant form for the force function, which depends on the geometric properties of the intersection of the undeformed volumes occupied with the rigid bodies (so-called volumetric model). The reduced expression for the force function obtained is shown to be different from that accepted in the classic contact theory hypotheses. Our expression has been tested in several examples dealing with bodies that contact elastically including Hertz’s classical model. In the context of the Hertz contact problem, an approximate model for computation the resulting wrench of the dry friction tangent forces is set up. The wrench consists of the total friction force and the drilling friction torque. The approach under consideration naturally extends the contact model constructed earlier. The dry friction forces and torque are integrated over the contact elliptic spot. Generally an analytic computation of the integrals mentioned is bulk and cumbersome leading to decades of terms that include rational functions depending on complete elliptic integrals. To be able to implement a fast computer model of elastically contacting bodies, one should first set up an approximate model in the way initially proposed by Contensou. To verify the model developed, we have used results obtained by several authors. First we test our method on the Tippe-Top dynamic model. Simulations show that the top’s evolution can be verified with a high quality compared with the use of the theory of set-valued functions. In addition, the ball bearing dynamic model has been also used for a detailed verification of different approaches to the computation of tangent forces. Then the friction model of the regularized Coulomb type and the approximate Contensou one, each embedded into the whole bearing dynamic model, were thoroughly tested and compared. It turned out that the simplified Contensou approach provides a computer model that runs even faster compared with the case of the point contact. In addition, the volumetric model demonstrated a reliable behavior and an acceptable accuracy.     

Generalized Fresnel integrals

A general class of (finite dimensional) oscillatory integrals with polynomially growing phase functions is studied. A representation formula of the Parseval type is proven as well as a formula giving the integrals in terms of analytically continued absolutely convergent integrals. Their asymptotic expansion for “strong oscillations” is given. The expansion is in powers of ?1/2M, where ? is a small parameters and 2M is the order of growth of the phase function. Additional assumptions on the integrands are found which are sufficient to yield convergent, resp. Borel summable, expansions.     

Indefinite integrals of products of special functions

A method is given for deriving indefinite integrals involving squares and other products of functions which are solutions of second-order linear differential equations. Several variations of the method are presented, which applies directly to functions which obey homogeneous differential equations. However, functions which obey inhomogeneous equations can be incorporated into the products and examples are given of integrals involving products of Bessel functions combined with Lommel, Anger and Weber functions. Many new integrals are derived for a selection of special functions, including Bessel functions, associated Legendre functions, and elliptic integrals. A number of integrals of products of Gauss hypergeometric functions are also presented, which seem to be the first integrals of this type. All results presented have been numerically checked with Mathematica.