Mathematical models for organizing the tax service

Game-theoretical models are proposed for describing the interaction of tax inspectors and taxpayers. A return-auditing probability that maximizes net tax revenue is calculated for a model without corruption. A dynamic audit strategy is described, and it is shown to converge to the optimal model under complete information. For a model with corruption, we also calculate the optimal probability of secondary audit and carry out a comparative analysis of net tax revenue assuming various tax rates and various penalties for evasion and negligent audit. The study was partially supported by the Russian Program for Economic Studies (grant 97-3011) and Russian Foundation of Basic Research (grant 96-01-0999). Useful comments by Mikhail Alekseev, Jim Leitzel, and Francisco Marhuenda are acknowledged. Translated from Chislennye Metody i Vychislitel'nyi Eksperiment, Moscow State University, pp. 64–81, 1998.     

Heuristic optimization methods for motion planning of autonomous agricultural vehicles

In this paper, two heuristic optimization techniques are tested and compared in the application of motion planning for autonomous agricultural vehicles: Simulated Annealing and Genetic Algorithms. Several preliminary experimentations are performed for both algorithms, so that the best neighborhood definitions and algorithm parameters are found. Then, the two tuned algorithms are run extensively, but for no more than 2000 cost function evaluations, as run-time is the critical factor for this application. The comparison of the two algorithms showed that the Simulated Annealing algorithm achieves the better performance and outperforms the Genetic Algorithm. The final optimum found by the Simulated Annealing algorithm is considered to be satisfactory for the specific motion planning application.     

Mathematical models for polygamous mating systems

In this paper, two-sex models for populations with polygamous mating systems are developed. These models depend on the mating ratio and are applicable to both polygynous and polyandrous populations. Models for the population dynamics of populations with and without competition are constructed. Also, marriage models are discussed. The models developed describe situations common to mathematical ecology and mathematical demography. All of the models developed here are based on systems of linear and nonlinear autonomous differential equations.     

Mathematical modeling for colloidal dispersion undergoing Brownian motion

An averaged motion approach for modeling Brownian dynamics for suspension systems of electrically charged particles in liquid is developed. The continuum model for the motion of particles consists of a system of integral equations coupled with a degenerate parabolic equation. Existence and uniqueness of global solution for the coupled system are established, and numerical results for the non-Newtonian viscosity of the mixture in terms of shear rate or Pechlet number are obtained. The model reveals some non-Newtonian properties such as the well-known shear thinning phenomenon for the viscosity of colloidal dispersions.     

Mathematical models of service

As service has become the predominant part of all advanced economies, increasing attention has been paid to conceptualizing and formulating its mathematical structure. Models that arise from service and the management of service may be broadly group d into three main categories: (1) customer behavior models, that explain how customers react to service, (2) service quality impact models, that address the business consequences of service quality, and (3) normative service models, that prescribe how organizations should organize and manage their service. We briefly discuss the most important models in each category, which arise primarily from the Marketing, Operations, and Operations Research literatures.     

Mathematical models for the elastic beam with structural damping

In this paper the existence, regularity and sharp estimates for the solutions of an abstract second order evolution equation are proved and applications to models of a (possibly non homogeneous) elastic beam with a frequency-proportional damping are given.     

Mathematical programming models for piecewise-linear discriminant analysis

Mathematical programming (MP) discriminant analysis models are widely used to generate linear discriminant functions that can be adopted as classification models. Nonlinear classification models may have better classification performance than linear classifiers, but although MP methods can be used to generate nonlinear discriminant functions, functions of specified form must be evaluated separately. Piecewise-linear functions can approximate nonlinear functions, and two new MP methods for generating piecewise-linear discriminant functions are developed in this paper. The first method uses maximization of classification accuracy (MCA) as the objective, while the second uses an approach based on minimization of the sum of deviations (MSD). The use of these new MP models is illustrated in an application to a test problem and the results are compared with those from standard MCA and MSD models.     

Mathematical programming models for some smallest-world problems

Given a weighted graph G, in the minimum-cost-edge-selection problem (MCES), a minimum weighted set of edges is chosen subject to an upper bound on the diameter of graph G. Similarly, in the minimum-diameter-edge-selection problem (MDES), a set of edges is chosen to minimize the diameter subject to an upper bound on their total weight. These problems are shown to be equivalent and proven to be NP-complete. MCES is then formulated as a 0–1 integer programming problem. The problems MCES and MDES provide models for determining smallest-world networks and for measuring the “small-worldness” of graphs.     

Mathematical models of bipolar disorder

We use limit cycle oscillators to model bipolar II disorder, which is characterized by alternating hypomanic and depressive episodes and afflicts about 1% of the United States adult population. We consider two non-linear oscillator models of a single bipolar patient. In both frameworks, we begin with an untreated individual and examine the mathematical effects and resulting biological consequences of treatment. We also briefly consider the dynamics of interacting bipolar II individuals using weakly-coupled, weakly-damped harmonic oscillators. We discuss how the proposed models can be used as a framework for refined models that incorporate additional biological data. We conclude with a discussion of possible generalizations of our work, as there are several biologically-motivated extensions that can be readily incorporated into the series of models presented here.     

Mathematical models of suspension bridges

In this work we try to explain various mathematical models describing the dynamical behaviour of suspension bridges such as the Tacoma Narrows bridge. Our attention is concentrated on the derivation of these models, an interpretation of particular parameters and on a discussion of their advantages and disadvantages. Our work should be a starting point for a qualitative study of dynamical structures of this type and that is why we have a closer look at the models, which have not been studied in literature yet. We are also trying to find particular conditions for unique solutions of some models.     

Mathematical models of tumour angiogenesis

We first study a parabolic–ODE system modelling tumour growth proposed by Othmer and Stevens [Aggregation, blowup, and collapse: the ABC's of taxis in reinforced random walks, SIAM J. Appl. Math. 57 (4) (1997) 1044–1081]. According to Levine and Sleeman [A system of reaction and diffusion equations arising in the theory of reinforced random walks, SIAM J. Appl. Math. 57 (3) (1997) 683–730], we reduced it to a hyperbolic equation and showed the existence of collapse in [A. Kubo, T. Suzuki, Asymptotic behavior of the solution to a parabolic ODE system modeling tumour growth, Differential Integral Equations 17 (2004) 721–736]. We also deal with the system in case the reduced equation is elliptic and show the existence of collapse analogously. Next we apply the above result to another model proposed by Anderson and Chaplain arising from tumour angiogenesis and show the existence of collapse. Further we investigate a contact point between these two models and a common property to them.     

Mathematical models of urban growth

The article models the distribution of cities by population. Two approaches are considered to mathematical modeling of urban growth: a probability model in which the number of cities depends on the population and the rank model of distribution of cities by their population. Five population censuses are analyzed for Russia’s cities. The probability density function n(x, α) for the number of cities as a function of their population x is fitted to all the available censuses with a time-dependent coefficient α . The function α(t ) is approximated and a prediction for the nearest future is computed. In particular, it is shown that in 2010 compared with 2002 the number of large cities should increase, while the number of small town should decrease. A model is also proposed for the interaction of urban areas linked into a single hierarchical system. The model is based on a system of ordinary differential equations describing the change in urban population. Independently of the initial distribution, all the cities and town line up by the rank–size law and deviations from this law, as in real life, are observed only for some large and very small cities. Model parameters are fitted for Russia’s cities.     

Mathematical models for drug delivery to chronic patients

Mathematically, analysis of drug delivery kinetics involves two moving boundary problems: diffusion front and eroding front. In this paper, we have models for drug delivery for the sites which can be enclosed by spherical shaped matrices covered by membranes and these problems are helpful for designing the drug delivery devices to deliver the drug inside from outside and a corresponding device supplying drug from inside. Once the time required for treatment and rate of drug delivery is known from medical diagnosis, this analysis can design a device releasing the drug/active agent over a long period of time. The purpose of such drug delivery is to achieve more effective therapies while eliminating the effect of over dosing and maintaining drug levels within the desired levels. The device may work on optimal use of drug and increase the patient’s convenience. The proposed models provide design for eroding tumor or chemotherapy to cancerous regions. The results have been obtained for steady state release rate, zero order release time and life time of the device and discussed. It has been observed that zero order time and life time increase by introducing a membrane of uniform thickness.     

Mathematical models for explaining the emergence of specialization in performing tasks

In an evolving community consisting of many individuals, it is often the case that the individuals tend, over time, to become more specialized in performing the tasks necessary for survival and growth of the community as a whole. The contribution in this work is a collection of linear and nonlinear mathematical models that provide insights as to when and why functional specialization emerges in general, rather than specific, settings. The results from these models, which are based on an evolutionary approach, apply to communities in which individuals allocate their time in the best interest of the community as a whole. © 2004 Wiley Periodicals, Inc. Complexity 10:37–48, 2004     

Mathematical models and control for systems with segregation

We consider mathematical models and optimal control problems for systems that consist of a large number of uncontrolled aggregates and an environment interacting with them. The evolution of each aggregate is described by an ordinary differential equation with a random parameter; the evolution of the environment is defined by the averaged interaction with aggregates and the values of control parameters. We obtain optimality conditions and develop an approach to the calculation of the distribution densities of the residence time of aggregates in the system.