Arbitrary Analytic Function Research Articles

Derivatives of the Matrix Exponential and Their Computation

Matrix exponentials and their derivatives play an important role in the perturbation analysis, control, and parameter estimation of linear dynamical systems. The well-known integral representation of the derivative of the matrix exponential exp( t A) in the direction V, namely ∫ t 0exp(( t − τ) A) V exp(τ A) dτ, enables us to derive a number of new properties for it, along with spectral, series, and exact representations. Many of these results extend to arbitrary analytic functions of a matrix argument, for which we have also derived a simple relation between the gradients of their entries and the directional derivatives in the elementary directions. Based on these results, we construct and optimize two new algorithms for computing the directional derivative. We have also developed a new algorithm for computing the matrix exponential that is more efficient than direct Padé approximation, which is based on a rational representation of the exponential in terms of the hyperbolic function A coth( A). Finally, these results are illustrated by an application to a biologically important parameter estimation problem which arises in nuclear magnetic resonance spectroscopy.

Invariants at fixed and arbitrary energy. A unified geometric approach

Invariants at arbitrary and fixed energy (strongly and weakly conserved quantities) for two-dimensional Hamiltonian systems are treated in a unified way. This is achieved by utilizing the Jacobi metric geometrization of the dynamics. Using Killing tensors we obtain an integrability condition for quadratic invariants which involves an arbitrary analytic function S(z). For invariants at arbitrary energy the function S(z) is a second-degree polynomial with real second derivative. The integrability condition then reduces to Darboux's condition for quadratic invariants at arbitrary energy. The four types of classical quadratic invariants for positive-definite two-dimensional Hamiltonians are shown to correspond to certain conformal transformations. We derive the explicit relation between invariants in the physical and Jacobi time gauges. In this way knowledge about the invariant in the physical time gauge enables one to write down directly the components of the corresponding Killing tensor for the Jacobi metric. We also discuss the possibility of searching for linear and quadratic invariants at fixed energy and its connection with the problem of the third integral in galactic dynamics. In our approach linear and quadratic invariants at fixed energy can be found by solving a linear ordinary differential equation of the first or second degree, respectively.

Classes and objects of chemical thermodynamics in object-oriented programming. 1. A class of analytical functions of temperature and pressure

The development of principles that might ease writing software for thermodynamic calculations with any participating solutions has been considered as the objective of this paper. To this end, the object-oriented programming is suggested to apply, which, in fact, allows the creation of a new specialised algorithmic language, rather universal and flexible, for programming of sophisticated thermodynamic applications. This also makes it possible to represent any thermodynamic property by arbitrary user-supplied analytical functions and furnishes a new way for the storage of thermodynamic properties in a data bank.

Explicit Solutions of the Steady Two‐Dimensional Navier‐Stokes Equations

A method is described to determine exact general solutions of the steady two‐dimensional Navier‐Stokes equations. The solutions can also be utilized to construct the stream function for a class of unsteady flows, which in turn contain two arbitrary analytic functions of a complex variable defined in the fluid region.

A Note on Interface Cracks With and Without Friction in Contact Zone

A complete set of Comninou’s bimaterial interface crack-tip fields with and without friction in the contact zone (Comninou, 1977a,b) is given in terms of several arbitrary analytic functions. When the bimaterial becomes homogeneous, the complex function representation fully describes the crack-tip field for a cracked body under conditions of crack surface contact and slip, which can occur when the body is subjected to combined compression and shear loadings.

An asymptotic analysis of stationary and moving cracks with frictional contact along bimaterial interfaces and in homogeneous solids

The plane strain and plane stress problem of a stationary or steadily moving crack with frictional sliding crack surface contact is investigated, with emphasis on the asymptotic structure of the crack tip fields. The crack is assumed to lie along the interface of an elastic anisotropic bimaterial with an aligned plane of symmetry, which covers special cases where the bimaterial is orthotropic or isotropic, or where the bimaterial becomes homogeneous. A full representation of the asymptotic fields around the interface crack is derived in terms of several arbitrary analytic functions, with explicit expressions for the singular crack tip stress and displacement fields given for a steadily propagating interface crack in an isotropic bimaterial, which are used to predict the direction of possible crack deviation from the interface. For a stationary crack, the singularity of the stresses can be, in general, stronger or weaker than r−½ (where r is the distance to the crack tip) depending on the loading history, while for a steadily growing crack, the singularity must be weaker than r−½, resulting in zero energy release rate at the crack tip. For bimaterials with orthotropic symmetries, the form of the singular stress field is found somewhat similar to that of the classic mode II problem. When these types of materials become homogeneous, and irrespective of the amount of friction between the contacting crack faces, the singular crack tip fields are identical to those of the classic mode II problem. Hence, the solutions are also governed by the conventional stress intensity factor KII, implying a nonzero crack tip energy release rate, which is related to KII in the usual manner. Implications of the above findings will be discussed.

Minimally coupled scalar field wormholes.

We examine conditions for the existence of wormhole solutions in the Einstein equations with a dynamical minimally coupled scalar field matter source with an arbitrary effective potential. We find three wormhole solutions which connect a baby universe with two asymptotically flat spaces through a narrow tube in a constant, cosinelike, and $\ensuremath{\Sigma}{i=0}^{3}{c}_{i}{a}^{\ensuremath{-}i}$ potential, respectively ($a$ being the scale factor). Because of the fact that the scalar field is not constant inside these wormholes, the gravitational constant takes different values in the two asymptotically flat spaces. These wormholes, under a conformal transformation, may be regarded as instanton-type solutions in higher order gravity with a Lagrangian that is an arbitrary analytic function of the scalar curvature.

A Class of Solutions for the Equations Governing the Steady Two‐Dimensional Motion of a Viscous Incompressible Liquid

A general viscosity dependent solution for the stream function is found satisfying the simplest nondegenerate form of the steady flow Navier‐Stokes equations for a viscous incompressible liquid. The solution is two‐dimensional and is expressed in terms of arbitrary analytic functions in the fluid domain. This class of flows is generated by complex stream functions, and the region of definition is restricted by an inequality containing these analytic functions. A general potential flow, and degenerate Stokes or creeping flows are recovered as particular solutions in limiting cases.

Deconvolution of Neutron Scattering Data from a Pulsed Spallation Source

ABSTRACTDiffraction peak shapes generated from high purity, low strain uranium silicide powder have been used to deconvolute diffraction peaks from radiation damaged material. The deconvolution was based on the assumption that defect scattering near a strained Bragg peak could be represented as the summation of a series of discrete peaks with the same convoluted peak shape measured for unirradiated material in the same range of d-spacing. Discrete peaks were taken to have the same density as the measured time of flight data so that a deconvoluted scattering intensity was generated at each experimental point. This deconvolution technique accounts in a natural manner for peak asymmetries arising from the time structure of the neutron pulse shape and instrument broadening without arbitrary analytical functions and fitting parameters.

Explicit Solutions of the Two‐Dimensional Navier Stokes Equations

Explicit solutions are found for the stream function satisfying the Navier Stokes equations representing the steady two‐dimensional motion of a viscous incompressible liquid. The solutions contain two arbitrary analytic functions and in general are confined to certain regions of the x, y plane.

Third-order ODEs for twisting type-N vacuum solutions

Several different `answers' to the question of the best form of a third-order ODE for twisting, type-N, vacuum spacetimes which admit two homothetic motions have already been presented in the literature. The problem of course is quite difficult and these equations have yet to be solved. Therefore, on the basis of our (earlier) understanding of the maximal group of symmetries of the defining equations, we present yet another. We believe that coming to this problem via several different formalisms, arriving at several different forms, will eventually lead to a solution of the problem. In addition, we give a simple presentation of the extension of that maximal group of symmetries to the case where the vacuum has been generalized to allow for a pressureless, null fluid, which introduces an arbitrary analytic function into the parameter space for the group.

Magnetotelluric response of anisotropic 2-D structures

SUMMARY The effect of anisotropy on the distribution of Earth's conductivity is evaluated by calculating the electromagnetic response of multilayered 2-D structures. The electric and magnetic fields are expanded in terms of Fourier series, the coefficients being obtained by applying the corresponding boundary conditions on each interface, given by arbitrary analytical functions. Then the results are used to analyse some particular structures.

Generalized cosmic no-hair theorems

We show that the quasi-exponential solution exp ( Bt− At 2) of R+ αR 2 gravity is an attractor for all homogeneous and isotropic solutions of higher order gravity theories derived from a lagrangian that is an arbitrary analytic function of the scalar curvature R. This indicates that a generalized form of the cosmic no-hair conjecture is valid in the framework of higher order gravity theories.

AFFINE STRINGS

A new model of bosonic strings is considered. An action of the model is the sum of the standard string action and a term describing an interaction of a metric with a linear (affine) connection. The Lagrangian of this interaction is an arbitrary analytic function f(R) of the scalar curvature. This is a classically integrable model. The space of classical solutions of the theory consists of sectors with constant curvature. In each sector the equations of motion reduce to the standard string equations and to an additional constant curvature equation for the linear connection. A bifurcation in the space of all Lagrangians takes place. Quantization of the model is briefly discussed. In a quasiclassical approximation one gets the standard string model with a fluctuating cosmological constant. The Lagrangian f(R), like Morse function, governs transitions between manifolds with different topologies.

A plane problem of magneto-elasticity for a ferromagnetic medium weakened by cuts

A plane problem of magneto-elasticity is considered for a soft ferromagnetic medium weakened by several cracks, which are treated as mathematical cuts in the initial (non-deformed) state. Since the approach used to analyse the effect of magnetization of the medium on the stress state in the neighbourhood of a rectilinear crack [1–3]is inapplicable in the general case, a new procedure is developed. It is based on representing the mechanical and magnetic quantities in terms of arbitrary analytic functions. The boundary-value problem is reduced to a system of two singular real integral equations of the first kind. Formulae for the stress intensity factors at the crack tip are obtained. An example is presented.

New approximate analytic solutions of the equations of gas dynamics

Two-term approximate solutions of the equations of gas dynamics in variables of the velocity hodograph are constructed both for subsonic flow containing an arbitrary analytic function of the complex variable and a sonic (critical) point and for supersonic motion having two arbitrary functions of the characteristic variables and the sonic point. The solutions are constructed without using the method of approximations. These solutions are matched on the sonic line for transonic motion using an example.

The exact numerical solution of a Schrodinger equation with two-Coulomb centres plus oscillator potential

A numerical technique of solving the Schrodinger equation with two-Coulomb centers plus the oscillator Hamiltonian has been developed. A scheme of evaluating energy levels and wavefunctions of such a Hamiltonian is described in some detail. The proposed algorithm also allows the study of analytical properties of energy levels in a complex plane of the internuclear distance, i.e. to find positions of branch points in the complex plane. The prolate ellipsoidal coordinates enable separation of variables in the equation studied which greatly facilitates the solution. A numerical method for finding branching point positions for an arbitrary analytical function is outlined.

The problem of torsion of a hollow circular cylinder with variable shear moduli

Classes of functions for shear moduli which depend on a cylindrical system of coordinates are singled-out and general solutions for the stress and displacement functions are constructed in the form of a finite integral operator consisting of completely defined variable coefficients and an arbitrary analytic function of a complex argument. An example of the solution of the problem with mixed boundary conditions of the torsion of a hollow cylinder, one end of which is rigidly clamped, is given.

Analysis of components of demographic change.

"There exist a number of approaches to analysis of the components of difference between demographic rates using standardization techniques. A generalization of some of these methods, for an arbitrary number of components and an arbitrary analytical function of them, is shown to be the theoretically best choice when the analytical function is linear in each of the components and when linear change is assumed for each of the components. Also, a very close approximation to an exact decomposition for the most general case can be obtained when dealing with change over time and time series data are available."

ON THE EXPANSION OF ANALYTIC FUNCTIONS IN A SERIES OF ELEMENTARY SOLUTIONS OF A CONVOLUTION EQUATION

Conditions for the representation of arbitrary analytic functions in the form of an exponential series are studied.