Some unsteady diffusion and absorption problems

Specific problems of the diffusion type, focusing on the absorptionof heat (or matter) at one or more boundary surfaces after itsrelease within the adjacent medium at a localized source, areconsidered, and the measures of absorption or outflow duringarbitrary intervals of time are directly expressed in termsof auxiliary temperature/matter distributions with given sourceson the pertinent surfaces. Detailed applications of these reciprocalrelations are presented, as is a brief analysis that includessome nonlinear or convective aspects in the basic diffusionequations.     

Some methodological problems of polymer science

An attempt was made to discuss some methodological problems of polymer science. An emphasis was made of the necessity to study the methodology problems.Translated from Mekhanika Polimerov, No. 2, pp. 197–202, March–April, 1970.     

Some algebraic structures for diffusion in social networks

Restrictions on influence structures discussed by Buskens and Yamaguchi (1999) and others lead the states of possible distributions of information to form an algebraic structure of a lattice that is closed under union. From an examination of such a lattice of distribution, it is possible to recreate an influence network that is not observed directly. Finally, it is noted that these structures are different from those examined via Galois lattices; therefore, if these diffusion processes occur, Galois lattices do not recreate the actual social structure underlying this diffusion.     

Some fundamental problems of polymer mechanics

A survey of the present state of the science of polymer mechanics is presented with particular reference to the theory of linear and nonlinear viscoelasticity of homogeneous and heterogeneous polymers. Various approaches adopted in studies of this subject are discussed; these include methods based on the theory of creep of metals, on a statistical model composed of viscoelastic elements, and on the representation of functionals which, in a quite general sense, are sufficiently close to being linear.The second of these methods, which is approximate, yields results which always point to the nonlinearity of the properties of solids, even those that consist of linear elements. The third method is a natural generalization of the Boltzmann-Volterra theory.One of the fundamental tasks relating to the linear viscoelasticity of polymers is the application of the temperature-time analogy in studies of nonuniform and nonstationary problems of solids subjected to loading and heating-cooling cycles. The necessity of developing methods of modeling phenomena in solids of complex shape in nonuniform and nonstationary temperature and stress states (particularly in the nonlinear range) is discussed and a new method of solving problems of this type is described.Mekhanika Polimerov, Vol. 1, No. 3, pp 33–42, 1965     

Fractal structures in weakly dissipative kam theory and related problems

This paper presents a number of numerical quantities that admit interpretation in terms of classical and quantum mathematical physics.     

Some problems of simple finite deformation of viscoelastic bodies

Boltzmann's superposition principle is extended to large strains in the case where the principal strain axes do not change with time and are identically directed at all points of the body. The relations obtained are confirmed by experiment. On the basis of these relations the author examines the problem of large strains of a heated viscoelastic cylinder subjected to the action of time-dependent internal and external pressures, and the analogous problem in the presence of a reinforcing cylindrical shell. The solution of the solving nonlinear integral equation of the latter problem is unique and is obtained in the form of a convergent infinite series.Mekhanika Polimerov, Vol. 2, No. 4, pp. 508–518, 1966     

Some first passage problems related to cusum procedures

Let $\text{X}{}_1\text{,}\text{X}{}_2\text{,…}$ be independent random variables, and set W_n = max(0,W_n-1 + X_n), W₀ = 0, n ? 1. The so-called cusum (cumulative sum) procedure uses the first passage time T(h) = inf{n ? 1: W_n?h}for detecting changes in the mean μ of the process. It is shown that lim_h→∞ μET(h)/h = 1 if μ > 0. Also, a cusum procedure for detecting changes in the normal mean is derived when the variance is unknown. An asymptotic approximation to the average run length is given.     

Some problems associated with the control of distributed structures

Control of structures can be carried out conveniently by modal control, whereby the structure is controlled by controlling its modes. Modal control requires the estimation of the modal states for feedback, which can present a problem. One approach that does not require modal state estimation is direct feedback control, which implies collocated sensors and actuators. This paper examines some problems encountered in direct feedback control of distributed structures in conjunction with pole placement. A perturbation technique permits the computation of control gains for multi-input systems. The paper demonstrates that the difficulties experienced in using direct feedback in conjunction with pole placement are endemic to the approach.This research was sponsored by the Air Force Office of Scientific Research Grant No. AFOSR-83-0017, monitored by Dr. A. K. Amos, whose support is fully appreciated. This paper was presented at the Meeting on Optimal Control and Calculus of Variations, Oberwolfach, West Germany, June 15–21, 1986.     

Some topological structures of extensions of abstract reachability problems

We consider the problem of the reachability of states that are elements of a topological space under constraints of asymptotic nature on the choice of an argument of a given objective mapping. We study constructions that have the sense of extensions of the original space and are implemented with the use of methods that are natural for applied mathematics but employ elements of extensions used in general topology. The study is oriented towards the application in the problem on the construction and investigation of properties of reachability sets for control systems.Constructions involving an approximate observation of constraints in control problems, as well as various generalized regimes, were widely used by N.N. Krasovskii and his students. In particular, this approach was applied in the proof of N.N. Krasovskii and A.I. Subbotin’s fundamental theorem of the alternative, which made it possible to establish the existence of a saddle point in a nonlinear differential game. In the investigation of impulse control problems, Krasovskii used techniques from the theory of generalized functions, which formed the basis for many studies in this direction. A number of A.B. Kurzhanski’s papers are devoted to the solution of control problems related in one way or another to the construction of reachability sets. Control problems with incomplete information, duality issues for control and observation problems, and team control problems constitute a far from exhaustive list of research areas where Kurzhanskii obtained profound results. These studies are characterized by the use of a wide range of tools and methods from applied mathematics and various constructions as well as by the combination of theoretical investigations and procedures related to the possibility of computer modeling.The research direction developed in the present paper mainly concerns the problem of constraint observation (including “asymptotic” constraints) and involves other issues. Nevertheless, the idea of constructing generalized elements of various nature (in particular, generalized controls) seems to be useful for the purpose of asymptotic analysis of control problems that do not possess stability as well as problems on the comparison of different tendencies in the choice of control in the form of dependences on a complex of factors inherent in the original real-life problem. The use of such tools as the Stone–?ech compactification and Wallman’s extension is, of course, oriented toward the study of qualitative issues. In the authors’ opinion, the combined application of the approaches to the construction of extensions used in control theory and in general topology holds promise from the point of view of both pure and applied mathematics. Apparently, the present paper can be considered as a certain step in this direction.     

Inverse problems of anomalous diffusion theory: An artificial neural network approach

The results are presented of computer simulation of the operation of a three-layer perceptron trained for solving inverse problems of anomalous diffusion theory. Several types of inverse problems are considered, including the problem of determining the Hurst exponent of a selfsimilar medium.     

Some results on divisor problems related to cusp forms

The Ramanujan Journal - Let $$\lambda _{f}(n)$$ be the normalized Fourier coefficients of a holomorphic Hecke cusp form of full level. We study a generalized divisor problem with $$\lambda...     

Some radii problems related to certain p-valent meromorphic functions

The purpose of the present paper is to introduce several new classes of p-valent meromorphic functions defined by Ruscheweyh derivative operator for meromorphic multivalent functions. Further, we investigate the radii problems of these analytic classes in the punctured unit disk.     

Some computationally relevant group theoretic structures of fixed charge problems

Among the most commonly occurring mixed-integer problems in operations research are linear programs with fixed charge objective functions. In this paper special structures of the equivalent form of such problems obtained from optimal solutions to their continuous relaxations are characterized and exploited in a series of penalty procedures for branch-and-bound type algorithms. The selection, construction, and solution of such penalty problems are discussed, and computational experience with the procedures is presented.     

Some applications of the Cauchy integral in problems of the deformation of thin plates

Three complex potentials are used to obtain exact solutions of problems of the deformation of infinite thin plates of constant thickness with a circular or elliptical hole under uniaxial tension or under the action of internal pressure in three-dimensional setting. Solutions are sought using the Cauchy integral, which enables the solutions to be presented in finite form. The solutions obtained are compared with known solutions.     

Some radius problems related to a certain subclass of analytic functions

For real parameters α and β such that 0≤α1β,we denote by S(α,β) the class of normalized analytic functions which satisfy the following two-sided inequality:αR(zf′(z)/f(z))β,z∈U,where U denotes the open unit disk.We find a sufficient condition for functions to be in the class S(α,β) and solve several radius problems related to other well-known function classes.     

Excitation of spatiotemporal structures in elastic electroactive contractile fibers

The spatiotemporal structures induced by a weak localized stimulus in excitable contractile fibers are studied by using a two-component reaction–diffusion model with global coupling. The character of the induced structures is analyzed, and the regimes of excitation spreading over the fiber are determined depending on the global coupling strength and the contraction type (associated with the mechanical fiber fixation conditions). It is shown that the global coupling can lead to long-lasting transient dynamics and new oscillatory attractor modes.