On the discrete kinetic theory for active particles. Mathematical tools

This paper deals with the development of a mathematical discrete kinetic theory to model the dynamics of large systems of interacting active particles whose microscopic state includes not only geometrical and mechanical variables (typically position and velocity), but also peculiar functions, called activities, which are able to modify laws of classical mechanics. The number of the above particles is sufficiently large to describe the overall state of the system by a suitable probability distribution over the microscopic state, while the microscopic state is discrete. This paper deals with a methodological approach suitable to derive the mathematical tools and structures which can be properly used to model a variety of models in different fields of applied sciences. The last part of the paper outlines some research perspectives towards modelling.     

On the initial value problem of a class of models of the kinetic theory for active particles

This paper deals with the qualitative analysis of the Cauchy problem for a class of systems constituted by a large number of interacting entities called active particles. Their state includes, in addition to geometrical and mechanical variables, also a microscopic state related to their socio-biological behavior, which is called activity. Microscopic interactions are governed by the self-organizing ability, which finalizes the dynamics according to well defined strategies.     

Modelling complex systems in applied sciences; methods and tools of the mathematical kinetic theory for active particles

This paper deals with the modelling of large systems of interacting individuals characterized by a microscopic state which includes both mechanical and socio-biological activities. The first part of the paper is devoted to the derivation and critical analysis of the modelling of microscopic equations and subsequently of the derivation of evolution equations. The second part analyzes how these mathematical structures can be properly used to model a variety of models in different fields of applied sciences: traffic flow, population dynamics, biology. Some research perspectives are analyzed in the last part of the paper.     

On the modeling of nonlinear interactions in large complex systems

This work deals with the modeling of large systems of interacting entities in the framework of the mathematical kinetic theory for active particles. The contents are specifically focused on the modeling of nonlinear interactions which is one of the most important issues in the mathematical approach to modeling and simulating complex systems, and which includes a learning–hiding dynamics. Applications are focused on the modeling of complex biological systems and on immune competition.     

Eigenvalue sensitivity and parametric optimization of the large rotating systems

The contribution presents the modal synthesis method of the mathematical modelling of the large rotating systems. The condensed mathematical model is used for deriving analytical formula for eigenvalue sensitivity calculation. According to the sensitivity analysis suitable design parameters are chosen and the optimization process for the purpose of steady state response minimization in a certain operating speed area is performed. The theory is applied to a simple test‐gearbox. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a class of integro-differential equations modeling complex systems with nonlinear interactions

This work deals with the qualitative analysis of the initial value problem for a class of large systems of interacting entities in the framework of the mathematical kinetic theory for active particles. The contents are specifically focused on the case where the system interacts with the outer environment and the entities are subject to nonlinearly additive interactions.     

Modeling the hiding–learning dynamics in large living systems

This paper deals with the modeling of large systems of interacting entities in the framework of the mathematical kinetic theory for active particles. Various mathematical structures of the hiding–learning dynamics are derived at the mesoscopic scale. Subsequently, these structures are further detailed referring to modeling issues and, in particular, to the learning-hiding competition among tumors and the immune system cells.     

Global solution to the Cauchy problem for discrete velocity models of vehicular traffic

This paper offers the proof of the existence for large times of the Cauchy problem for a class of vehicular traffic models with discrete velocities derived from the mathematical tools of the kinetic theory for active particles. This approach has the ability to capture some basic aspects concerning the complexity of the systems under consideration through an appropriate modeling, via stochastic games, of the interactions at the microscopic scale. The proof is related to specific aspects of the real system and hopefully can contribute towards improving the modeling approach.     

Inexact-Newton methods for semismooth systems of equations with block-angular structure

Systems of equations with block-angular structure have applications in evolution problems coming from physics, engineering and economy. Many times, these systems are time-stage formulations of mathematical models that consist of mathematical programming problems, complementarity, or other equilibrium problems, giving rise to nonlinear and nonsmooth equations. The final versions of these dynamic models are nonsmooth systems with block-angular structure. If the number of state variables and equations is large, it is sensible to adopt an inexact-Newton strategy for solving this type of systems. In this paper we define two inexact-Newton algorithms for semismooth block-angular systems and we prove local and superlinear convergence.     

New methods for proving the existence and stability of periodic solutions in singularly perturbed delay systems

We carry out a detailed analysis of the existence, asymptotics, and stability problems for periodic solutions that bifurcate from the zero equilibrium state in systems with large delay. The account is based on a specific meaningful example given by a certain scalar nonlinear second-order differential-difference equation that is a mathematical model of a single-circuit RCL oscillator with delay in a feedback loop.     

Large deviation approach to non equilibrium processes in stochastic lattice gases

We present a review of recent work on the statistical mechanics of non equilibrium processes based on the analysis of large deviations properties of microscopic systems. Stochastic lattice gases are non trivial models of such phenomena and can be studied rigorously providing a source of challenging mathematical problems. In this way, some principles of wide validity have been obtained leading to interesting physical consequences.     

Minimum Cost Allocation of Tenders

In a big organization like the Australian Post Office, where large sums of money are spent to purchase a very large number of items every financial year, the problem is that of allocating the orders of the different items to the competing tenderers at a minimum cost. We have dealt with this in two sections; one is devoted to quantity discounts, the other to order value discounts. We have shown the mathematical formulation using integer and mixed integer linear programming, the mathematical programming systems used, the benefits gained and costs incurred, and the effects of the implementation of minimum cost allocation techniques on the organizational structure of some sections of the A.P.O. We have also shown that these techniques could be used to analyse the relative economic efficiencies of the competing tenderers with the objective of formulating a suitable economic purchasing policy for the organization concerned.     

On the stabilizability problem in Banach space

The model is a linear system defined on Banach (state and control) spaces, with the operator acting on the state only the infinitesimal generator of a strongly continuous semigroup. The stabilizability problem of expressing the control through a bounded operator acting on the state as to make the resulting feedback system globally asymptotically stable is considered. On the negative side, and in contrast with the finite dimensional theory, a few counter examples are given of systems which are densely controllable in the space and yet are not stabilizable, even if some further “nice properties” hold. Use is made of the notion of essential spectrum and its stability under relatively compact perturbations. On the positive side, it is shown, however, that for large classes of systems of physical interest (classical selfadjoint boundary value problems, delay equations, etc.) controllability on a suitable finite dimensional subspace still yields stabilizability on the whole space.     

考虑潜伏期的关联信用风险传染机理研究

关联信用风险及其传染是现代信用风险管理的热点和难点问题。本文基于复杂网络的平均场理论,运用传染病模型,来刻画关联主体网络中关联信用风险的传染机理;并引入潜伏主体及其潜伏期,分析了网络中关联信用风险的传染过程;进一步,在关联主体BA无标度网络中,探讨了关联信用风险特征对传染过程中稳定状态的影响。研究结果表明:潜伏主体的平均潜伏期、感染主体的传染率和恢复率及传染力、信用主体的新生率和死亡率,均会影响关联信用风险传染阈值和稳定状态时感染主体密度。     

Non-gaussian Test Models for Prediction and State Estimation with Model Errors

Turbulent dynamical systems involve dynamics with both a large dimensional phase space and a large number of positive Lyapunov exponents. Such systems are ubiquitous in applications in contemporary science and engineering where the statistical ensemble prediction and the real time filtering/state estimation are needed despite the underlying complexity of the system. Statistically exactly solvable test models have a crucial role to provide firm mathematical underpinning or new algorithms for vastly more complex scientific phenomena. Here, a class of statistically exactly solvable non-Gaussian test models is introduced, where a generalized Feynman-Kac formulation reduces the exact behavior of conditional statistical moments to the solution to inhomogeneous Fokker-Planck equations modified by linear lower order coupling and source terms. This procedure is applied to a test model with hidden instabilities and is combined with information theory to address two important issues in the contemporary statistical prediction of turbulent dynamical systems: the coarse-grained ensemble prediction in a perfect model and the improving long range forecasting in imperfect models. The models discussed here should be useful for many other applications and algorithms for the real time prediction and the state estimation.     

A two-scale model for the periodic homogenization of the wave equation

We present a new mathematical object designed to analyze the oscillations occurring on both microscopic and macroscopic scales in a wave equation with oscillating coefficients and data. Through a Bloch wave homogenization method, our study addresses typical problems of two-scale convergence in the interior of the domain, and sheds some light on the behavior near the boundary. A decoupled system of (systems of) transport equations is derived in each energy band, and the total energy field is approximated. We also recover previously known results in homogenization as a restricted part of our model.     

Scheduling of production and logistics operations of steam production systems in food industries: a case study of the tomato processing industry

This study presents an optimization approach by mathematical modelling to support some of the main operational decisions in steam production systems with multiple industrial boilers. Decisions are related to boiler operations scheduling (start-up, warm-up and shutdown time), fuel replenishment (transportation and inventory management) and fuel composition consumed by each piece of equipment. These decisions are often taken based on practical experience of people involved, instead of any decision support tool using optimization techniques; as a consequence, unnecessary costs are likely to be incurred. The optimization approach is based on mixed integer programming and parameters experimental adjustment procedures. A case study of a large tomato processing plant in Brazil was carried out along 1 year using a 3-year database. Owing to the reasonably good outcomes achieved (annually potential savings around 10%), we consider the proposed approach as a suitable tool to support some of the key decisions in boiler scheduling and fuel logistics in steam production systems for tomato processing and other similar industries.