Mathematical models and numerical methods for stability analysis of materials subjected to radiation pulses

We consider problems in modeling the interaction between a solid body and radiation pulses having intensity of order 10⁹–10¹⁵ W/m². We construct one-dimensional mathematical models for radiative damage to materials on the basis of the theory of thermal erosion and the kinetic theory of stability.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 70, pp. 67–73, 1990.     

Predictive models of expenditure and over-indebtedness for assessing the affordability of new consumer credit applications

Lenders are under increasing pressure to consider measures of affordability and indebtedness as well as risk, when assessing consumer credit applications. In order to evaluate the affordability of a new credit product, a lender needs information about the applicant's income and outgoings. However, while most lenders obtain information about income and credit commitments many do not have much, if any, pertaining to other expenditure. Therefore, they are not well positioned to determine an individual's ability to fund new borrowing. This paper demonstrates that using only data captured on a typical application form, combined with data from a credit bureau, it is possible to develop good predictive models of expenditure and over-indebtedness that can be used in conjunction with measures of risk to reject applications from individuals who are likely to already be over-indebted, or to restrict the volume of credit advanced to that which the applicant can afford.     

Persistence and extinction in a mathematical model of cell populations affected by radiation

A mathematical model consisting of a system of two ordinary differential equations is formulated to represent the interrelationship between healthy and radiated cells at a given cite. Three different modes of radiation are considered: constant, decaying, and periodic radiation. For the constant case, precise criteria for persistence and extinction are obtained. In the decaying case, it is shown that the radiated cells always become extinct. Finally in the periodic case, criteria are obtained for a perturbed positive periodic solution. Dedicated to the memory of Professor Miklós Farkas     

Mathematical models of thermomechanics of magnetizable and polarizable electrically conducting bodies subject to quasisteady electromagnetic radiation

We give a brief survey of recent research on the problems of mathematical modeling of the thermomechanical behavior of deformable bodies with different electrical conductivity, magnetizability, and polarizability subject to a quasisteady electromagnetic field in the radio and infrared frequency ranges. Translated fromMatematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 1, 1998, pp. 97–107.     

Permanence,extinction and periodic solutions in a mathematical model of cell populations affected by periodic radiation

A periodic mathematical model of cell populations affected by periodic radiation is presented and studied in this paper. We obtain some sufficient conditions on the permanence and extinction of the system. Furthermore, criteria on the existence and global asymptotic stability of unique positive periodic solutions are established. Some numerical examples are shown to verify our results. A discussion is presented for further study.     

Maps serving the combined coupling for use in environmental models and their behaviour in the presence of dynamical noise

Many physical, biological as well as the environmental problems, can be described by the dynamics of driven coupled oscillators. In order to study their behaviour as a function of coupling strength and nonlinearity, we considered dynamics of two maps serving the combined coupling (diffusive and linear) in the above fields. Firstly, we have considered a logistic difference equation on extended domain that is a part of the maps, that is discussed using its bifurcation diagram, Lyapunov exponent, sample as well as the permutation entropy. Secondly we have performed the dynamical analysis of the coupled maps using Lyapunov exponent and cross sample entropy in dependence on two coupling parameters. Further, we investigated how dynamical noise can affects the structure of their bifurcation diagrams. It was done (i) by the noise entering in two specific ways, that disturbs either the logistic parameter on extended domain or (ii) by an additive “shock” to the state variables. Finally, we demonstrated the effect of forcing by parametric noise, introduced in all maps’ parameter, on Lyapunov exponent of coupled maps.     

Damage caused to polymer films by laser radiation and its relation to the nature and structure of the material

The experimental results of an investigation of the damage caused to polymer films by a free-running neodymium laser are described in relation to the nature and structure of the material. It is shown that the damage pattern is determined by the material structure.V. I. Lenin Tadzhik State University, Dushanbe. Translated from Mekhanika Polimerov, No. 6, pp. 1092–1093, November–December, 1971.     

Evaluation of different strategies for building and displaying models to describe bacterial growth in foods

This paper compares the different methods for acquiring datadescribing microorganism growth, the types of model that canbe built from such data, and the different ways in which suchmodels can be presented for use.     

Formulation and application of models for anisotropic hardening in sheet metals subject to complex loading-path changes

In this work material models for the simulation of forming processes involving non-uniform loading path changes are investigated. Materials exhibiting an evolution of the yield surface due to the formation and interaction of dislocation structures on the micro structural level are modeled by employing evolving structural tensors. These account for the evolution of anisotropy and hardening behavior known from technical alloys on a phenomenological basis. In a first step material parameters are identified from experiments for non-uniform loading conditions. In a second step, the evolution of structural tensors for varying loading conditions is investigated and related to the evolution of kinematic quantities. Finally applications relevant to the simulation of sheet metal forming processes are presented. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Three-dimensional models for analyzing viscous flows in tuber by means of generalized prandtl equations

Algorithms are constructed for analyzing stationaryviscous flows in cylindrical tubes. Stable implicit schemes based on generalized Prandtl equations and consisting of unified modules are described. Systems of linear algebraic equations are solved using efficient merthods of pass. Examples of analysis of flows in a tube of square cross-section are given for Reynolds number 20, 50, and 100., Bibliography: 14 titles. Translated fromObchyslywval’na ta Prykladna Matematyka, No. 77, 1993, pp. 120–128.     

Spatio-temporal averaging for a class of hybrid systems and application to conductance-based neuron models

We obtain a limit theorem endowed with quantitative estimates for a general class of infinite dimensional hybrid processes with intrinsically two different time scales and including a population. As an application, we consider a large class of conductance-based neuron models describing the nerve impulse propagation along a neural cell at the scales of ion channels.     

The integrality of the lot size in the basic EOQ and EPQ models: Applications to other production-inventory models

In this paper we present a method to obtain the solution of the classic economic order quantity (EOQ) and economic production quantity (EPQ) models when the lot size must be an integer quantity. This approach is operatively very simple and allows obtaining a rule to discriminate between the situation in which the optimal solution is unique and when there are two optimal solutions. Also, this method is applicable to the resolution of other production-inventory models. We expose some of them and illustrate the use of the method with numerical examples.     

Spectral theory of one-channel operators and application to absolutely continuous spectrum for Anderson type models

A one-channel operator is a self-adjoint operator on ${?}^2(\mathbb{G})$ for some countable set $\mathbb{G}$ with a rank 1 transition structure along the sets of a quasi-spherical partition of $\mathbb{G}$. Jacobi operators are a very special case. In essence, there is only one channel through which waves can travel across the shells to infinity. This channel can be described with transfer matrices which include scattering terms within the shells and connections to neighboring shells. Not all of the transfer matrices are defined for some countable set of energies. Still, many theorems from the world of Jacobi operators are translated to this setup. The results are then used to show absolutely continuous spectrum for the Anderson model on certain finite dimensional graphs with a one-channel structure. This result generalizes some previously obtained results on antitrees.     

Coupling strategies for the numerical simulation of blood flow in deformable arteries by 3D and 1D models

The fluid structure interaction mechanism in vascular dynamics can be described by either 3D or 1D models, depending on the level of detail of the flow and pressure patterns needed for analysis. A successful strategy that has been proposed in the past years is the so-called geometrical multiscale approach, which consists of coupling both 3D and 1D models so as to use the former only in those regions where details of the fluid flow are needed and describe the remaining part of the vascular network by the simplified 1D model.In this paper we review recently proposed strategies to couple the 3D and 1D models, and within the 3D model, to couple the fluid and structure sub-problems. The 3D/1D coupling strategy relies on the imposition of the continuity of flow rate and total normal stress at the interface. On the other hand, the fluid–structure coupling strategy employs Robin transmission conditions. We present some numerical results and show the effectiveness of the new approaches.     

Gradient bounds for minimizers of free discontinuity problems related to cohesive zone models in fracture mechanics

In this note we consider a free discontinuity problem for a scalar function, whose energy depends also on the size of the jump. We prove that the gradient of every smooth local minimizer never exceeds a constant, determined only by the data of the problem.     

Chaos and fractal properties induced by noninvertibility of models in the form of maps

Many classes of discrete dynamical systems give rise to models in the form of noninvertible maps. With respect to invertible maps, noninvertible maps introduce a singularity of a different nature: the critical set of rank-one, as the geometrical locus of points having at least two coincident preimages. Such new singularities play a fundamental role in the definition of attractors, basins and their bifurcations. The purpose of this paper is a survey of some fundamental results related to two-dimensional noninvertible maps leading to specific chaotic behaviors, as fractal sets, characterizing irreversibility properties of a class of discrete systems.     

Solution of some simple problems in the conduction and radiation of heat by invariant imbedding

In recent years the use of invariant imbedding in the solution of a variety of problems has been increasing. In this paper, application of this method to problems in heat conduction and radiation is demonstrated. No previous knowledge of invariant imbedding is assumed. The method is applied to several relatively simple problems. The initial value problem obtained by the method is numerically stable. Sample calculations are presented which demonstrate the accuracy of the algorithm.     

Approximation to the distribution of the least squares estimators in two dimensional cosine models by randomly weighted bootstrap

Recently, Kundu and Gupta （Metrika, 48：83 C 97, 1998） established the asymptotic normality of the least squares estimators in the two dimensional cosine model. In this paper, we give the approximation to the general least squares estimators by using random weights which is called the Bayesian bootstrap or the random weighting method by Rubin （Annals of Statistics, 9：130 C 134, 1981） and Zheng （Acta Math. Appl. Sinica （in Chinese）, 10（2）： 247 C 253, 1987）. A simulation study shows that this approximation works very well.