Infinite Quasimonotone Systems of Ordinary Differential Equations with Applications to Stochastic Processes

We deal with the initial value problem for countably infinite linear systems of ordinary differential equations of the form y '( t ) = A ( t ) y ( t ) where A ( t ) = ( a ij ( t ): i , j S 1) is a measurable, infinite and essentially positive matrix, i.e., a ij ( t ) S 0 for i p j . The main novelty of our approach is the systematic use of a classical comparison theorem for finite linear systems which leads easily to the existence of a nonnegative minimal solution and its properties. Application to generalized stochastic birth and death processes produces criteria for honest and dishonest probability distributions. A short proof of the Kolmogorov and Chapman-Kolmogorov equations for stochastic processes follows. The results hold for L 1 -coefficients. Our method extends to nonlinear infinite systems of quasimonotone type and can be used for numerical procedures that yield exact results; cf. the Addendum.     

Differential Inequalities Applied to The Stochastic Birth and Death Process

We describe an approach to a countably infinite system of ordinary differential equations belonging to the theory of the stochastic birth and death process. The main novelty in our method is the systematic use of a classical theorem on sub- and supersolutions for finite linear systems of the form y '( t ) = Ay ( t ). It leads in a simple way to the minimal solution and some of its properties. For convenience a proof of the theorem is given at the end.     

Frequency and Magnitude Response Design Approximation Problems for Nonlinear-phase Nonrecursive Digital Filters

In this article special (possibly constrained) problems of linear and nonlinear complex approximation are studied with respect to the existence and uniqueness of solutions and the convergence of the approximation errors, where the errors are measured by an arbitrary L p and l p norm respectively. The problems arise in connection with the frequency and magnitude response approximation at the design of nonrecursive digital filters in the frequency domain. Two main results of the article concern the completeness of the functions exp ( m ik y ), k = 0,1,2,… , with respect to a certain space of continuous functions. These results imply that, under usual assumptions and with increasing number of approximating functions exp ( m ik y ), the errors in the frequency and magnitude response approximation problems converge to zero when the design region is not the total interval [0, ~ ] (in case of real coefficients) and not [ m ~ , ~ ] (in case of complex coefficients) which is given for the majority of filter design problems, but that the frequency response errors may not converge to zero when the design region equals [0, ~ ] or [ m ~ , ~ ] respectively.     

On Detecting Solutions of Polynomial Nonlinear Difference Equations

In this paper a method for discovering solutions of nonlinear polynomial difference equations is presented. It is based on the concepts of i -operator and star-product. These notions create a proper algebraic background by means of which we can find linear equations "included" into the original nonlinear one and to seek for solutions among them. A corresponding algorithm and some examples are also provided.     

Necessary and sufficient conditions for robust global asymptotic stabilization of discrete-time systems

We give Lyapunov-like conditions for non-uniform in time output stabilization of discrete-time systems. Particularly, it is proved that for a discrete-time control system there exists a (continuous) output stabilizing feedback if and only if there exists a (strong) output control Lyapunov function (OCLF). Moreover, strategies for the construction of continuous robust feedback stabilizers are presented.     

Interface Motion in Disordered Ferromagnets

We consider numerically the depinning transition in the random-field Ising model. Our analysis reveals that the three and four dimensional model displays a simple scaling behavior whereas the five dimensional scaling behavior is affected by logarithmic corrections. This suggests that d = 5 is the upper critical dimension of the depinning transition in the random-field Ising model. Furthermore, we investigate the so-called creep regime (small driving fields and temperatures) where the interface velocity is given by an Arrhenius law.     

Sign-Solvable Cone-Systems

Qualitative properties shared by the solutions to the family of linear equations of the form $$Ax=b, \quad (A\in C, b \in B),\eqno(1)$$ where C is a cone in ${\shadR}^{m\times n}$ and B is a cone in ${\shadR}^m$ are studied. In particular, the cones C and B for which the sign patterns of the solutions to (1) are independent of the choice of A ε C and b ε B are characterized.     

Towards a new framework for evaluating systemic problem structuring methods

Operational researchers and social scientists often make significant claims for the value of systemic problem structuring and other participative methods. However, when they present evidence to support these claims, it is usually based on single case studies of intervention. There have been very few attempts at evaluating across methods and across interventions undertaken by different people. This is because, in any local intervention, contextual factors, the skills of the researcher and the purposes being pursued by stakeholders affect the perceived success or failure of a method. The use of standard criteria for comparing methods is therefore made problematic by the need to consider what is unique in each intervention. So, is it possible to develop a single evaluation approach that can support both locally meaningful evaluations and longer-term comparisons between methods? This paper outlines a methodological framework for the evaluation of systemic problem structuring methods that seeks to do just this.     

Hierarchy,cities size distribution and Zipf's law

We show that a hierarchical cities structure can be generated by a self-organized process which grows with a bottom-up mechanism, and that the resulting distribution is power law. First we analytically prove that the power law distribution satisfies the balance between the offer of the city and the demand of its basin of attraction, and that the exponent in the Zipf's law corresponds to the multiplier linking the population of the central city to the population of its basin of attraction. Moreover, the corresponding hierarchical structure shows a variable spanning factor, and the population of the cities linked to the same city up in the hierarchy is variable as well. Second a stochastic dynamic spatial model is proposed, whose numerical results confirm the analytical findings. In this model, inhabitants minimize the transportation cost, so that the greater the importance of this cost, the more stable is the system in its microscopic aspect. After a comparison with the existent methods for the generation of a power law distribution, conclusions are drawn on the connection of hierarchical structure, and power law distribution, with the functioning of the system of cities.     

Properties of on-line social systems

We study properties of five different social systems: (i) internet society of friends consisting of over 10⁶ people, (ii) social network consisting of 3 × 10⁴ individuals, who interact in a large virtual world of Massive Multiplayer Online Role Playing Games (MMORPGs), (iii) over 10⁶ users of music community website, (iv) over 5 × 10⁶ users of gamers community server and (v) over 0.25 × 10⁶ users of books admirer website. Individuals included in large social network form an Internet community and organize themselves in groups of different sizes. The destiny of those systems, as well as the method of creating of new connections, are different, however we found that the properties of these networks are very similar. We have found that the network components size distribution follow the power-law scaling form. In all five systems we have found interesting scaling laws concerning human dynamics. Our research has shown how long people are interested in a single task, how much time they devote to it and how fast they are making friends. It is surprising that the time evolution of an individual connectivity is very similar in each system.