An algebraic approach to chaos

The aim of this paper is to study chaotic actions on objects other than metric spaces, e.g. locales and commutative rings. To do so, point-free versions of topologically transitivity and the density of periodic points are obtained for actions on a locale, and then generalized to a category which includes the desired objects of study.Presented at the European Colloquium of Category Theory, Tours, France, 25–31 July 1994.     

An alternative approach to solve the mixed AKNS equations

The algebraic–geometric solutions of the mixed AKNS equations are investigated through a finite-dimensional Lie–Poisson Hamiltonian system, which is generated by the nonlinearization of the adjoint equation related to the AKNS spectral problem. First, each mixed AKNS equation can be decomposed into two compatible Lie–Poisson Hamiltonian flows. Then the separated variables on the coadjoint orbit are introduced to study these Lie–Poisson Hamiltonian systems. Further, based on the Hamilton–Jacobi theory, the relationship between the action-angle coordinates and the Jacobi-inversion problem is established. In the end, using Riemann–Jacobi inversion, the algebraic–geometric solutions of the first three mixed AKNS equations are obtained.     

An operator approach to some graph enumeration problems

An operator approach to some graph enumeration problems is developed together with the formal procedures related to the enumeration. Both annihilation and creation operators are defined for vertices, edges and Euler characteristics of a graph. An application to forest enumeration leads to compact expressions exhibiting the duality between the operators.     

An economic approach to some classical theorems in optimization theory

This short article offers economically intuitive proofs of the Euler equation and the maximum principle based on one of the best known results in economics, namely that the marginal utility of one extra dollar spent on each consumption goods is the same for all the consumption goods as required by budget-constrained utility maximization.     

An abstract approach to evaporation models in rarefied gas dynamics

An efficient algorithm is presented for determining the unique solvability of certain one-dimensional stationary transport problems. The non-existence of stationary evaporation states with supersonic drift velocities for one and three dimensional BGK model is recovered.

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Sommario È presentato un algoritmo efficiente per determinare la solubilità univoca di alcuni probiemi unidimensionali di trasporto stazionario. È ripresa la non esistenza di stati stazionari di evaporazione con velocità di deriva supersoniche per modelli BGK a una e tre dimensioni.

     

An elementary approach to discrete models of dividend strategies

The paper studies a discrete counterpart of Gerber et al. (2006). The surplus of an insurance company (before dividends) is modeled as a time-homogeneous Markov chain with possible changes of size +1,0,−1,−2,−3,…. If a barrier strategy is applied for paying dividends, it is shown that the dividends-penalty identity holds. The identity expresses the expected present value of a penalty at ruin in terms of the expected discounted dividends until ruin and the expected present value of the penalty at ruin if no dividends are paid. For the problem of maximizing the difference between the expected discounted dividends until ruin and the expected present value of the penalty at ruin, barrier strategies play a prominent role. In some cases an optimal dividend barrier exists. The paper discusses in detail the special case where the distribution of the change in surplus does not depend on the current surplus (so that in the absence of dividends the surplus process has independent increments). A closed-form result for zero initial surplus is given, and it is shown how the relevant quantities can be calculated recursively. Finally, it is shown how optimal dividend strategies can be determined; typically, they are band strategies.     

Application of chaos degree to some dynamical systems

Chaos degree defined through two complexities in information dynamics is applied to some deterministic dynamical models. It is shown that this degree well describes the chaotic feature of the models.     

An extremal metric approach to some problems on extremal decomposition

The paper presents a simple extremal metric approach to problems on extremal decomposition in families of systems of nonoverlapping domains of different types. The approach is based on using general properties of trajectories of the associated quadratic differentials for simpler extremal decomposition problems. Bibliography: 8 titles. Dedicated to the 100th anniversary of G. M. Goluzin’s birthday __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 337, 2006, pp. 191–211.     

An approach to constructing models of multiphase elastic porous media

A novel approach to describing the behaviour of multiphase elastic porous media is proposed. The average values of the physical quantities needed to describe the motions of porous media are formulated using an integral relation. The validity of this relation is taken as the fundamental hypothesis. The integral definition of the average values enables integral relations to be devised for the average values from the integral laws of conservation of mass, momentum and energy and the increase in entropy. Along with the average values, the integral relations contain new variables that can be identified with generalized thermodynamic forces, which can be used to take into account the phase interaction in a porous medium. The integral relations are used to derive differential equations for the rate of entropy change and Gibbs relations for a porous medium as a basis for obtaining the constitutive relations. Relationships between the thermomechanical parameters of the model are established from the Gibbs relations under additional assumptions. The equation for the rate of entropy change can be used to establish relations between the generalized thermodynamic forces and fluxes. A complete system of differential equations in the defining parameters, which describes the motion of multiphase elastic porous media, is finally obtained.     

Finite-dimensional models of diffusion chaos

Some parabolic systems of the reaction-diffusion type exhibit the phenomenon of diffusion chaos. Specifically, when the diffusivities decrease proportionally, while the other parameters of a system remain fixed, the system exhibits a chaotic attractor whose dimension increases indefinitely. Various finite-dimensional models of diffusion chaos are considered that represent chains of coupled ordinary differential equations and similar chains of discrete mappings. A numerical analysis suggests that these chains with suitably chosen parameters exhibit chaotic attractors of arbitrarily high dimensions.     

Asymptotic optimality of nonparametric tests for restricted alternatives in some multivariate mixed models

For some mixed models (involving both stochastic and nonstochastic predictors), a general class of permutationally distribution-free rank tests for some restricted alternative problems is considered. The proposed tests are asymptotically optimal in the light of the restricted likelihood ratio tests. For an ordered alternative problem in a two-way analysis of covariance model, the proposed tests are asymptotically optimal.     

An alternative approach for chaos: A plane pendulum with oscillating torque

The shooting method is applied to obtain chaotic motions for a pendulum with a oscillatory torque excitation on its support. It shows that if the pendulum is placed at certain spots, the corresponding motion will become chaotic. It proves the coexistence of uncountably many non-periodic motions and countably many periodic motions of the pendulum.     

An elementary approach to the meromorphic continuation of some classical Dirichlet series

Here we obtain the meromorphic continuation of some classical Dirichlet series by means of elementary and simple translation formulae for these series. We are also able to determine the poles and the residues by this method. The motivation to our work originates from an idea of Ramanujan which he used to derive the meromorphic continuation of the Riemann zeta function.     

On chaos, transient chaos and ghosts in single population models with Allee effects

Density-dependent effects, both positive or negative, can have an important impact on the population dynamics of species by modifying their population per-capita growth rates. An important type of such density-dependent factors is given by the so-called Allee effects, widely studied in theoretical and field population biology. In this study, we analyze two discrete single population models with overcompensating density-dependence and Allee effects due to predator saturation and mating limitation using symbolic dynamics theory. We focus on the scenarios of persistence and bistability, in which the species dynamics can be chaotic. For the chaotic regimes, we compute the topological entropy as well as the Lyapunov exponent under ecological key parameters and different initial conditions. We also provide co-dimension two bifurcation diagrams for both systems computing the periods of the orbits, also characterizing the period-ordering routes toward the boundary crisis responsible for species extinction via transient chaos. Our results show that the topological entropy increases as we approach to the parametric regions involving transient chaos, being maximum when the full shift R(L)^∞ occurs, and the system enters into the essential extinction regime. Finally, we characterize analytically, using a complex variable approach, and numerically the inverse square-root scaling law arising in the vicinity of a saddle-node bifurcation responsible for the extinction scenario in the two studied models. The results are discussed in the context of species fragility under differential Allee effects.     

An evaluation approach to engineering design in QFD processes using fuzzy goal programming models

Quality function deployment (QFD) is a product development process used to achieve higher customer satisfaction: the engineering characteristics affecting the product performance are designed to match the customer requirements. From the viewpoint of QFDs designers, product design processes are performed in uncertain environments, and usually more than one goal must be taken into account. Therefore, when dealing with the fuzzy nature in QFD processes, fuzzy approaches are applied to formulate the relationships between customer requirements (CRs) and engineering design requirements (DRs), and among DRs. In addition to customer satisfaction, the cost and technical difficulty of DRs are also considered as the other two goals, and are evaluated in linguistic terms. Fuzzy goal programming models are proposed to determine the fulfillment levels of the DRs. Differing from existing fuzzy goal programming models, the coefficients in the proposed model are also fuzzy in order to expose the fuzziness of the linguistic information. Our model also considers business competition by specifying the minimum fulfillment levels of DRs and the preemptive priorities between goals. The proposed approach can attain the maximal sum of satisfaction degrees of all goals under each confidence degree. A numerical example is used to illustrate the applicability of the approach.     

An operator-based approach to the analysis of ruin-related quantities in jump diffusion risk models

Recent developments in ruin theory have seen the growing popularity of jump diffusion processes in modeling an insurer’s assets and liabilities. Despite the variations of technique, the analysis of ruin-related quantities mostly relies on solutions to certain differential equations. In this paper, we propose in the context of Lévy-type jump diffusion risk models a solution method to a general class of ruin-related quantities. Then we present a novel operator-based approach to solving a particular type of integro-differential equations. Explicit expressions for resolvent densities for jump diffusion processes killed on exit below zero are obtained as by-products of this work.     

An analytic approach to purely nonlocal Bellman equations arising in models of stochastic control

Given a bounded domain Ω⊂Rd and two integro-differential operators L¹, L² of the form we study the fully nonlinear Bellman equation

(0.1)     

Assuring consistency in mixed models

Information systems security defines three properties of information: confidentiality, integrity, and availability. These characteristics remain major concerns throughout the commercial and military industry. Ordinary users have taken these features as basis for their businesses. Furthermore, users may find it necessary to combine policies in order to protect their information in a suitable way. However, inconsistencies may arise as a result of implementing multiple secrecy and privacy models; and therefore, render these services unsecure. In this paper, we propose an approach to detect and report inconsistencies when choosing mixed models for integrity and security. It is based on specifying the policies in first order logic and applying formal analysis. We demonstrate the feasibility of our proposition by applying it to the Clark Wilson and role based access control models. We use the Alloy language and analyzer to formalize the mixed model and check for any inconsistencies.     

Dynamical approach to some problems in integral geometry

As is well known, the main problem in integral geometry is to reconstruct a function in a given domain , where its integrals over a family of subdomains in are known. Such a problem is interesting not only as an object of pure analysis, but also in connection with various applications in practical disciplines. The most remarkable example of such a connection is the Radon problem and tomography. In this paper we solve one of these problems when is a bounded domain in with a piecewise smooth boundary. Some intermediate results related to dynamical systems with two generators and to some functional-integral equations are new and interesting per se. As an application of the results obtained we briefly study a boundary problem for a general third order hyperbolic partial differential equation in a bounded domain with data on the whole boundary .