Dynamics of coupled systems

A survey in the form of a review of the author’s research results in the area of dynamics of coupled rotations and coupled systems is presented. The theorem on the existence of a trigger of the coupled singularities and the separatrix in the form of the number eight is presented with a few examples of coupled rotations of the simple systems with debalances. Also, a survey of models and dynamics of coupled systems composed of a number of deformable bodies (plates, beams or belts) with different properties of materials and discrete layer properties is presented and mathematically described. The constitutive stress–strain relations for materials of the coupled sandwich structure elements are presented for different properties: elastic, viscoelastic and creeping. The characteristic modes of the coupled system vibrations are obtained and analyzed for different kinds of materials and structure composition. The visualization of the characteristic numbers and modes and eigenamplitude forms are presented. Structural analysis of sandwich structure vibrations is done.     

Initial-value problems and singularities in general relativity

Linearized solution of Datta in a non-symmetric and isentropic motion of a perfect fluid is studied by dealing with a Cauchy problem in co-moving coordinates in the framework of general relativity. The problem of singularities is discussed from the standpoint of a local observer both for rotating and non-rotating fluids. It is shown that, whatever the distribution of matter, a singularity which occurred in the past in both the rotating and non-rotating parts of the universe must occur again later after some finite proper time, if the universe is closed. A modification is incorporated in Penrose’s theorem by explicitly exhibiting that the universe defined by Penrose can possess a Cauchy hypersurface.     

Non-Autonomous Evolution Equations on Nonsmooth Domair

The solution of a stationary boundary value problem on a domain with conical points has singularities near these points. Here we first consider existence results in appropriate weighted Sobolev spaces in order to incorporate the singularities. We secondly use these results to prove existence, uniqueness and regularity of solutions of non-autonomous second order evolution equations on such domains.     

Griffiths' singularities in diluted ising models on the Cayley tree

The Griffiths singularities are fully exhibited for a class of diluted ferromagnetic Ising models defined on the Cayley tree (Bethe lattice). For the deterministic model the Lee-Yang circle theorem is explicitly proven for the magnetization at the origin and it is shown that, in the thermodynamic limit, the Lee-Yang singularities become dense in the entire unit circle for the whole ferromagnetic phase. Smoothness (infinite differentiability) of the quenched magnetizationm at the origin with respect to the external magnetic field is also proven for convenient choices of temperature and disorder. From our analysis we also conclude that the existence of metastable states is impossible for the random models under consideration.     

On the resolution of singularities of ordinary differential systems

We show how some differential geometric ideas help to resolve some singularities of ordinary differential systems. Hence a singular problem is replaced by a regular one, which facilitates further analysis of the system. The methods employed are constructive and the regularized systems can also be used for numerical computations. This revised version was published online in June 2006 with corrections to the Cover Date.     

Laurent–Padé Approximants to Four Kinds of Chebyshev Polynomial Expansions. Part I. Maehly Type Approximants

Laurent Padé–Chebyshev rational approximants, A _m (z,z ^–1)/B _n (z,z ^–1), whose Laurent series expansions match that of a given function f(z,z ^–1) up to as high a degree in z,z ^–1 as possible, were introduced for first kind Chebyshev polynomials by Clenshaw and Lord [2] and, using Laurent series, by Gragg and Johnson [4]. Further real and complex extensions, based mainly on trigonometric expansions, were discussed by Chisholm and Common [1]. All of these methods require knowledge of Chebyshev coefficients of f up to degree m+n. Earlier, Maehly [5] introduced Padé approximants of the same form, which matched expansions between f(z,z ^–1)B _n (z,z ^–1) and A _m (z,z ^–1). The derivation was relatively simple but required knowledge of Chebyshev coefficients of f up to degree m+2n. In the present paper, Padé–Chebyshev approximants are developed not only to first, but also to second, third and fourth kind Chebyshev polynomial series, based throughout on Laurent series representations of the Maehly type. The procedures for developing the Padé–Chebyshev coefficients are similar to that for a traditional Padé approximant based on power series [8] but with essential modifications. By equating series coefficients and combining equations appropriately, a linear system of equations is successfully developed into two sub-systems, one for determining the denominator coefficients only and one for explicitly defining the numerator coefficients in terms of the denominator coefficients. In all cases, a type (m,n) Padé–Chebyshev approximant, of degree m in the numerator and n in the denominator, is matched to the Chebyshev series up to terms of degree m+n, based on knowledge of the Chebyshev coefficients up to degree m+2n. Numerical tests are carried out on all four Padé–Chebyshev approximants, and results are outstanding, with some formidable improvements being achieved over partial sums of Laurent–Chebyshev series on a variety of functions. In part II of this paper [7] Padé–Chebyshev approximants of Clenshaw–Lord type will be developed for the four kinds of Chebyshev series and compared with those of the Maehly type.     

Domain sensitivity for singular elliptic problems: Harmonic transformation approach

A method for computing the sensitivities of functionals depending on the solutions of elliptic equations defined over variable domains is presented. It is based on the material derivative approach and allows the uniform treatment of both singular and nonsingular cases. The novelty consists in defining the vector field connected with the domain transformation as the solution of an auxiliary elliptic equation. Such a choice does not restrict the range of admissible goal functionals and has many advantages from the numerical point of view. It allows one also to consider singular domain variations.     

Pole-type singularities and the numerical conformal mapping of doubly-connected domains

Let f be the function which maps conformally a given doubly-connected domain Ω onto a circular annulus, and let $\text{H(z)=}\text{f′(z)}\text{f(z)}\text{?}\text{1}\text{z}$. In this paper we consider the problem of determining the main singularities of the function H in compl($\text{Ω∪?Ω}$). Our purpose is to provide information regarding the location and nature of such singularities, and to explain how this information can be used to improve the efficiency of certain expansion methods for numerical conformal mapping.