Numerical analysis of a mathematical model of the myocardial blood-supply system

A hydraulic model of the hemodynamics of the arterial part of the myocardium is considered, and a numerical analysis of the model is conducted. Computer experiments are used to investigate the dependence of blood flows on parametric and structural changes in the system. Translated from Prikladnaya Matematika i Informatika, No. 2, pp. 56–62, 1999.     

Analysis of a mathematical predator-prey model with delay

We study the behavior of dynamic processes in a mathematical predator-prey model and show that the dynamical system may have a periodic solution whose period coincides with the delay. By the bifurcation method for stability analysis of periodic solutions, we establish that this periodic solution is unstable.     

Analysis of a mathematical model for cancer treatment by the nonstandard finite difference methods

We construct two nonstandard finite difference schemes and use them to study a mathematical model of cancer therapy. Several recent studies show various aspects of the immune response against the cancer. Our discrete models emphasize the role of antibodies in any form of therapy by taking into account the development of anticancer therapies (chemotherapy, immunotherapy, radiation therapy). The nonstandard finite difference models are implemented by using Matlab. Numerical simulations show the existence of a separation line between the basins of attraction of cancerous cell-free and the highest equilibrium cancerous cell.     

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.     

Analysis of a delayed mathematical model for tumor growth

In this paper, a delayed mathematical model of a nonlinear reaction–diffusion equations modeling the growth of tumors is studied. The establishment of the model is based on the diffusion of nutrient and mass conservation for the two-process proliferation and apoptosis (cell death due to aging). It is assumed that the process of proliferation is delayed compared to apoptosis.Nonnegativity of the solutions and stability of stationary solutions are studied in the paper. The results show that the dynamical behavior of the solutions of the model is similar to that of the solutions for the corresponding non-retarded problem under some assumptions.     

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.)     

Bistability of rotational modes in a system of coupled pendulums

The main goal of this research is to examine any peculiarities and special modes observed in the dynamics of a system of two nonlinearly coupled pendulums. In addition to steady states, an in-phase rotation limit cycle is proved to exist in the system with both damping and constant external force. This rotation mode is numerically shown to become unstable for certain values of the coupling strength. We also present an asymptotic theory developed for an infinitely small dissipation, which explains why the in-phase rotation limit cycle loses its stability. Boundaries of the instability domain mentioned above are found analytically. As a result of numerical studies, a whole range of the coupling parameter values is found for the case where the system has more than one rotation limit cycle. There exist not only a stable in-phase cycle, but also two out-of phase ones: a stable rotation limit cycle and an unstable one. Bistability of the limit periodic mode is, therefore, established for the system of two nonlinearly coupled pendulums. Bifurcations that lead to the appearance and disappearance of the out-ofphase limit regimes are discussed as well.     

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.     

A mathematical model of pricing in a large system of cash bonds

The purpose of this paper is to give a mathematical model to generalize the classical approach of compound interest and to overcome the time structure problem of the interest rates. We introduce a suitable stochastic process called the ‘gauge’ process such that its product with the value of any security is assumed to be a martingale in an appropriate probability space. The framework of this model gives a stochastic actualization formula for the pricing of general securities with options and includes Black and Schole's formula without using arbitrage arguments. Emphasis has been placed on numerical calculation.     

A mathematical model of a flexible manufacturing system with limited in-process inventory

This paper describes a computationally simple, asymptotic model of a flexible job shop, especially designed for estimating the influence of limited in-process inventory level on the production rate. Its main features make it very similar to the one by Solberg. While Solberg's model consists of a closed queuing network, we propose an open queuing network with a limited amount of inprocess customers; a single customer class is assumed, the various actual processing routes being accounted for by routing probabilities. For such a queuing network, the product form of state probabilities is valid, and the normalization constant can be very simply obtained by a convolution algorithm, close to the one used by Solberg. Various performance indices are calculated, regarding the job shop behaviour over a long period of time. Comparison of analytical results of the model and simulation results are provided in order to estimate the amount of error introduced by assuming exponentially distributed processing times and Poisson inputs in the mathematical representation. Simulations were carried out in FORTRAN-based SLAM language.     

样条曲线光顺的数学模型分析

采用函数三次样条光顺曲线,证明在样条曲线局部转角小,总转角不超过120°情况下,曲线的光顺指示函数y″(1+y′2)3/2可以简化为二阶导数曲线y″(x).由于y″(x)对x是分段折线函数,对y是线性泛函,因而定出不光顺之处及用叠加原理计算调整公式均变得很简单.此样条函数曲线光顺能够采用电脑自动化进行.     

A system model for the utilization of mineral resources

A multiobjective dynamic optimization model is described for the optimal utilization of mineral resources. The special forms of the objective functions, state-transition relations, and additional constraints satisfy the conditions for using a special dynamic multiobjective programming method. A case study is also outlined.     

Analysis of a time-delayed mathematical model for tumour growth with inhibitors

In this article, we study the fully non-stationary version of a mathematical model for tumour growth under indirect effect of inhibitor with time delay in proliferation. The quasi-stationary version has been studied by our previous work [S. Xu and Z. Feng, Analysis of a mathematical model for tumour growth under indirect effect of inhibitors with time delay in proliferation, J. Math. Anal. Appl. 374 (2011), pp. 178–186]. The existence and uniqueness of a global solution are proved and the asymptotic behaviour of the solution is studied. The results show that the dynamical behaviour of solutions of the fully non-stationary and the quasi-stationary version are similar under some conditions.     

A mathematical model of the dynamics of a population affected by harmful substances

A mathematical model is presented of the dynamics of a population with individuals subjected to the effects of pollutants entering with food. We assume that the product of interaction of ingested pollutants is harmful for the individuals and increases the rate of their death. We describe the equations of the model and study the properties of solutions, including the existence and stability of equilibria. The conditions are obtained for the population becoming extinct as well as the conditions which guarantee that the total population is maintained at a nonzero stationary level. Some results of simulation are presented.     

Analysis of numerical schemes of a mathematical model for sickle cell depolymerization

The purpose of this paper is to present and discuss numerical schemes for a mathematical model that describes carbon monoxide mediated sickle cell polymer melting. Two Runge-Kutta methods are analyzed and shown to be unstable by calculating the first failure value of step size and displaying the bifurcation diagram of RK4. Two nonstandard finite difference (NSFD) schemes are proposed and analyzed; one is shown to be stable subject to a predictable bound on step size, while the second one is unconditionally stable.     

Further development of the mathematical model of a snakeboard

This paper gives the further development for the mathematical model of a derivative of a skateboard known as the snakeboard. As against to the model, proposed by Lewis et al. [1] and investigated by various methods in [1–13], our model takes into account an opportunity that platforms of a snakeboard can rotate independently from each other. This assumption has been made earlier only by Golubev [13]. Equations of motion of the model are derived in the Gibbs-Appell form. Analytical and numerical investigations of these equations are fulfilled assuming harmonic excitations of the rotor and platforms angles. The basic snakeboard gaits are analyzed and shown to result from certain resonances in the rotor and platforms angle frequencies. All the obtained theoretical results are confirmed by numerical experiments.      

Studying a mathematical model of parallel computation by algebraic topology methods

Under study is one of the mathematical models of parallel computational processes, i.e., an asynchronous transition system. Some methods are proposed for calculating the homology groups and the Poincaré polynomial of a finite asynchronous transition system. Conditions are obtained for the decomposability of an asynchronous transition system as a parallel product.