On a mathematical model of immune competition

This work deals with the qualitative analysis of a nonlinear integro-differential model of immune competition with special attention to the dynamics of tumor cells contrasted by the immune system. The analysis gives evidence of how initial conditions and parameters influence the asymptotic behavior of the solutions.     

On a mathematical model for laser-induced thermotherapy

We study a mathematical model for laser-induced thermotherapy, a minimally invasive cancer treatment. The model consists of a diffusion approximation of the radiation transport equation coupled to a bio-heat equation and a model to describe the evolution of the coagulated zone. Special emphasis is laid on a refined model of the applicator device, accounting for the effect of coolant flow inside. Comparisons between experiment and simulations show that the model is able to predict the experimentally achieved temperatures reasonably well.     

On a mathematical model of a coupled fluid-structure problem

An existence theorem is proved relative to a mathematical model associated to the fluid circulation in an elastic domain.     

A mathematical model of a mesh system and its implementation

We define and implement a mathematical model for a general 2-d mesh system, which is arrays of processors with a bounded mesh architecture. As one of the simplest distributed architecture with fixed-connection, the 2-d mesh system has found many applications in computer sciences and engineering, particularly in computer communication.

We use mathematical structures to characterize the mesh system and use C to have implemented an executable version of this model. In this paper, we will present the mathematical model itself, discuss some corresponding implementation issues and compare its behaviors with a simulator which we have been using to observe system behaviors.     

On strategies on a mathematical model for leukemia therapy

A mathematical model for leukemia therapy based on the Gompertzian law of cell growth is studied. It is assumed that the chemotherapeutic agents kill leukemic as well as normal cells.Effectiveness of the medicine is described in terms of a therapy function. Two types of therapy functions are considered: monotonic and non-monotonic. In the former case the level of the effect of the chemotherapy directly depends on the quantity of the chemotherapeutic agent. In the latter case the therapy function achieves its peak at a threshold value and then the effect of the therapy decreases. At any given moment the amount of the applied chemotherapeutic is regulated by a control function with a bounded maximum. Additionally, the total quantity of chemotherapeutic agent which can be used during the treatment process is bounded too.The problem is to find an optimal strategy of treatment to minimize the number of leukemic cells while at the same time retaining as many normal cells as possible.With the help of Pontryagin’s Maximum Principle it was proved that the optimal control function has at most one switch point in both monotonic and non-monotonic cases for most relevant parameter values.A control strategy called alternative is suggested. This strategy involves increasing the amount of the chemotherapeutical medicine up to a certain value within the shortest possible period of time, and holding this level until the end of the treatment.The comparison of the results from the numerical calculation using the Pontryagin’s Maximum Principle with the alternative control strategy shows that the difference between the values of cost functions is negligibly small.     

3-D network model and its parameter calibration

A material model, whose framework is parallel spring-bundles oriented in 3-D space, is proposed. Based on a discussion of the discrete schemes and optimum discretization of the solid angles, a 3-D network cell consisted of one-dimensional components is developed with its geometrical and physical parameters calibrated. It is proved that the 3-D network model is able to exactly simulate materials with arbitrary Poisson ratio from 0 to 1/2, breaking through the limit that the previous models in the literature are only suitable for materials with Poisson ratio from 0 to 1/3. A simplified model is also proposed to realize high computation accuracy within low computation cost. Examples demonstrate that the 3-D network model has particular superiority in the simulation of short-fiber reinforced composites.     

On the solvability of a mathematical model for prion proliferation

We show that a model describing the interaction between normal and infectious prion proteins admits global solutions. More precisely, supposing the involved degradation rates to be bounded, we prove global existence and uniqueness of classical solutions. Based on this existence theory, we provide sufficient conditions for the existence of global weak solutions in the case of unbounded splitting rates. Moreover, we prove global stability of the disease-free steady state.     

On a mathematical model for hot carrier injection in semiconductors

We study a collisionless transport model for electrons in a semiconductor, and we perform an asymptotic analysis for low temperatures or large applied biases. We derive analytic relations for the built-in potential and for the current which flows through the structure.     

A mathematical model for pollution in a river and its remediation by aeration

We present a simple mathematical model for river pollution and investigate the effect of aeration on the degradation of pollutant. The model consists of a pair of coupled reaction–diffusion–advection equations for the pollutant and dissolved oxygen concentrations, respectively. The coupling of these equations occurs because of reactions between oxygen and pollutant to produce harmless compounds. Here we consider the steady-state case in one spatial dimension. For simplified cases the model is solved analytically. We also present a numerical approach to the solution in the general case. The extension to the transient spatial model is relatively straightforward. The study is motivated by the crucial problem of water pollution in many countries and specifically within the Tha Chin River in Thailand. For such real situations, simple models can provide decision support for planning restrictions to be imposed on farming and urban practices.     

On the topological structure of a mathematical model of human unconscious

On the basis of two our previous works, in this paper, following Jacques Lacan psychoanalytic theory, we wish to outline some further remarks on the topological structure of a mathematical model of human unconscious.     

Dissection of a mathematical model

Interdisciplinary models of broad systems with environmental, economic and sociopolitical interacting components are progressively developed to accept the challenge of dramatic global changes affecting the planet Earth and its ever increasing population. This paper, based on the author's experience in the interdisciplinary—physical, chemical, biological, economic and sociopolitical—modelling of the marine system, and using specific references to such models to provide concrete illustrations of the basic concepts, is an attempt to bring out the general substructural features of mathematical models.     

On one mathematical model of creep in superalloys

In [Sv1] a new micromechanical approach to the prediction of creep flow in composites with perfect matrix/particle interfaces, based on the nonlinear Maxwell viscoelastic model, taking into account a finite number of discrete slip systems in the matrix, has been suggested; high-temperature creep in such composites is conditioned by the dynamic recovery of the dislocation structure due to slip/climb motion of dislocations along the matrix/particle interfaces. In this article the proper formulation of the system of PDE's generated by this model is presented, some existence results are obtained and the convergence of Rothe sequences, applied in the specialized software CDS, is studied.     

Trends in researching the socioeconomic influences on mathematical achievement

We introduce the topic of socioeconomic influences on mathematical achievement through an overview of existing research reports and articles. International trends in the way the topic has emerged and become increasingly important in the international field of mathematics education research are outlined. From this review, there is a discussion about what appears to be neglected in previous work in this area and how the papers in this issue of ZDM provide information about some of these neglected areas. The main argument in this article is that socioeconomic influences on mathematical achievement should not be considered as a taken-for-granted fact that is accepted uncritically. Instead, it is suggested that the relationship between multiple socioeconomic influences and various understandings of mathematical achievement are historically contingent ways of understanding exclusions and inclusions in mathematics education practices. Research is not simply “evidencing” the facts of these relationships; research is also implicated in constructing the ways in which we think about these. Thus, mathematics education researchers could devise more nuanced approaches for understanding the social, political and historical constitution of these relationships.     

A mathematical programming approach to clusterwise regression model and its extensions

The clusterwise regression model is used to perform cluster analysis within a regression framework. While the traditional regression model assumes the regression coefficient (β) to be identical for all subjects in the sample, the clusterwise regression model allows β to vary with subjects of different clusters. Since the cluster membership is unknown, the estimation of the clusterwise regression is a tough combinatorial optimization problem. In this research, we propose a “Generalized Clusterwise Regression Model” which is formulated as a mathematical programming (MP) problem. A nonlinear programming procedure (with linear constraints) is proposed to solve the combinatorial problem and to estimate the cluster membership and β simultaneously. Moreover, by integrating the cluster analysis with the discriminant analysis, a clusterwise discriminant model is developed to incorporate parameter heterogeneity into the traditional discriminant analysis. The cluster membership and discriminant parameters are estimated simultaneously by another nonlinear programming model.     

Salt pile stability: a mathematical model

The maximum height for the salt pile in a circular dome with a 4ft retaining wall was determined by two methods. The first method used rigid-body physics; in this model, the critical angel, the maximum angle of inclination allowed while maintaining static equilibrium, was determined using only the external coefficient of friction for salt. Because the static equilibrium also depended upon internal friction, a second model was developed. Development of the second model utilized particle physics, fluid mechanics and soil stress analysis. Mohr's circle, the internal coefficient of friction for salt and its angle of repose were used to determine the critical angle. These results were combined to form our solution model, Model II, which consisted of two submodels:Model II(a) provides a general solution where the front-end loader is allowed to freely travel to any location on the salt pile. This model yields a maximum height of 17.4ft for a symmetric cone with a critical angle of 14.6°.Model II(b) provides a volume-maximizing solution if the loader's travel is restricted. This model yields a maximum height of 23.7ft for a wedge shape with a ramp slope of 14.6° and a back edge slope of 35.9°, where the loader must not cross the peak.Therefore, the authors recommend that Model II(a) be used in the general situation, since the loader is allowed to drive anywhere on the salt pile in this case. When the maximum volume provided is insufficient, Model II(b) can be utilized to increase the capacity of the dome. (Note: The loader must not cross the peak in this model.)     

On the qualitative analysis of the solutions of a mathematical model of social dynamics

This work deals with a family of dynamical systems which were introduced in [M.L. Bertotti, M. Delitala, From discrete kinetic and stochastic game theory to modelling complex systems in applied sciences, Math. Models Methods Appl. Sci. 7 (2004) 1061–1084], modelling the evolution of a population of interacting individuals, distinguished by their social state. The existence of certain uniform distribution equilibria is proved and the asymptotic trend is investigated.     

Existence and uniqueness for a mathematical model in superfluidity

In this paper we propose a model to study superfluidity by considering as state variables the order parameter, describing the concentration of the superfluid phase, the velocity of the superfluid and the absolute temperature. We assume that the order parameter satisfies a Ginzburg–Landau equation and that the velocity is decomposed as the sum of a normal and a superfluid component. The heat equation provides the evolution equation for the temperature. We prove that this model is consistent with the principles of thermodynamics. Well‐posedness of the resulting initial and boundary value problem is shown. Copyright © 2008 John Wiley & Sons, Ltd.