Generalized z-Distributions and Related Stochastic Processes

The class of generalized z–distributions is defined and their properties are investigated. Ornstein–Uhlenbeck–type and self–similar generalized z–processes are constructed and described. Esscher transforms of the generalized z–processes and the mixed generalized z–processes are characterized. Finally, construction and some properties of generalized z–diffusions are also discussed.     

Conservation laws for Camassa–Holm equation, Dullin–Gottwald–Holm equation and generalized Dullin–Gottwald–Holm equation

In this paper we construct the conservation laws for the Camassa–Holm equation, the Dullin–Gottwald–Holm equation (DGH) and the generalized Dullin–Gottwald–Holm equation (generalized DGH). The variational derivative approach is used to derive the conservation laws. Only first order multipliers are considered. Two multipliers are obtained for the Camassa–Holm equation. For the DGH and generalized DGH equations the variational derivative approach yields two multipliers; thus two conserved vectors are obtained.     

An Expansion Formula for the Askey–Wilson Function

The Askey–Wilson function transform is a q-analogue of the Jacobi function transform with kernel given by an explicit non-polynomial eigenfunction of the Askey–Wilson second order q-difference operator. The kernel is called the Askey–Wilson function. In this paper an explicit expansion formula for the Askey–Wilson function in terms of Askey–Wilson polynomials is proven. With this expansion formula at hand, the image under the Askey–Wilson function transform of an Askey–Wilson polynomial multiplied by an analogue of the Gaussian is computed explicitly. As a special case of these formulas a q-analogue (in one variable) of the Macdonald–Mehta integral is obtained, for which also two alternative, direct proofs are presented.     

Polynomial Fibonacci–Hessenberg matrices

A Fibonacci–Hessenberg matrix with Fibonacci polynomial determinant is referred to as a polynomial Fibonacci–Hessenberg matrix. Several classes of polynomial Fibonacci–Hessenberg matrices are introduced. The notion of two-dimensional Fibonacci polynomial array is introduced and three classes of polynomial Fibonacci–Hessenberg matrices satisfying this property are given.     

A Clifford–Finslerian physical unification and fractal dynamics

A Clifford–Finslerian physical unification is proposed based on Clifford–Finslerian mathematical structures and three physical principles. In the Clifford–Finslerian mathematical structure, spontaneous symmetry breaking is automatically embedded in fractal branches. With the action principle, connection principle and computation principle, physics can be unified, in which the Riemman–Einstein system and Euclid–Newton system are naturally included when quaternion are reduced to complex and real phases.     

Dynamic Robust Output Min–Max Control for Discrete Uncertain Systems

Min–max control is a robust control, which guarantees stability in the presence of matched uncertainties. The basic min–max control is a static state feedback law. Recently, the applicability conditions of discrete static min–max control through the output have been derived. In this paper, the results for output static min–max control are further extended to a class of output dynamic min–max controllers, and a general parametrization of all such controllers is derived. The dynamic output min–max control is shown to exist in many circumstances under which the output static min–max control does not exist, and usually allows for broader bounds on uncertainties. Another family of robust output min–max controllers, constructed from an asymptotic observer which is insensitive to uncertainties and a state min–max control, is derived. The latter is shown to be a particular case of the dynamic min–max control when the nominal system has no zeros at the origin. In the case where the insensitive observer exists, it is shown that the observer-controller has the same stability properties as those of the full state feedback min–max control.     

On a Generalized Class of Minimax Inequalities

Some results on a generalized class of minimax inequalities based on the r–I–G–H-KKM mapping theorems in a G–H-space setting are presented. The r–I–G–H-KKM mappings represent a new class of KKM mappings in G–H-spaces as well as in the interval spaces.     

Generalizations of the Nikodym boundedness and Vitali–Hahn–Saks theorems

Generalizations of the Nikodym boundedness and Vitali–Hahn–Saks theorems for scalar-valued measures on rings of sets that are in general not σ-rings are presented. As a consequence, the rings of subsets of N with density zero and uniform density zero are shown to have the Nikodym property. In addition, vector measure generalizations of the Vitali–Hahn–Saks theorem are given.     

Apparatus for fatigue testing glass-reinforced plastic specimens in symmetrical tension-compression at acoustic frequencies

The characteristics of an electromechanical apparatus for fatigue testing glass-reinforced plastic specimens in symmetrical tension-compression at loads up to 10–40 kN and acoustic frequencies on the interval 300–5000 Hz are described.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 3, pp. 525–528, May–June, 1972.     

Multiparticle Direct Correlations

We study the properties of multiparticle direct correlations in detail and construct a system of successive approximations of the generating functional of direct correlation functions. Moreover, in contrast to the Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy, this approach does not require approximating higher correlations. Our approximations determine the corresponding approximations of the thermodynamic potential. The irreducible parts of multiparticle direct correlations are constructed explicitly.     

Genericity of Well-Posedness, Perturbations and Smooth Variational Principles

We show that three important topics in nonlinear analysis and optimization are intimately related: the theory of perturbations, the notion of well-posedness and variational principles in the sense of Ekeland, Borwein–Preiss and Deville–Godefroy–Zizler. The concept of genericity and the new notion of flexible perturbation play a key role in these connections. This notion enables one to consider topologies on spaces of functions which have been introduced recently. A link between the Asplund and Ekeland–Lebourg methods and the Palais–Smale condition, another important topic in nonlinear analysis, is pointed out.     

Optimal investment and reinsurance of an insurer with model uncertainty

We introduce a novel approach to optimal investment–reinsurance problems of an insurance company facing model uncertainty via a game theoretic approach. The insurance company invests in a capital market index whose dynamics follow a geometric Brownian motion. The risk process of the company is governed by either a compound Poisson process or its diffusion approximation. The company can also transfer a certain proportion of the insurance risk to a reinsurance company by purchasing reinsurance. The optimal investment–reinsurance problems with model uncertainty are formulated as two-player, zero-sum, stochastic differential games between the insurance company and the market. We provide verification theorems for the Hamilton–Jacobi–Bellman–Isaacs (HJBI) solutions to the optimal investment–reinsurance problems and derive closed-form solutions to the problems.     

Studying Baskakov–Durrmeyer operators and quasi-interpolants via special functions

We prove that the kernels of the Baskakov–Durrmeyer and the Szász–Mirakjan–Durrmeyer operators are completely monotonic functions. We establish a Bernstein type inequality for these operators and apply the results to the quasi-interpolants recently introduced by Abel. For the Baskakov–Durrmeyer quasi-interpolants, we give a representation as linear combinations of the original Baskakov–Durrmeyer operators and prove an estimate of Jackson–Favard type and a direct theorem in terms of an appropriate K-functional.     

Complexity of the min–max (regret) versions of min cut problems

This paper investigates the complexity of the min–max and min–max regret versions of the min s–t cut and min cut problems. Even if the underlying problems are closely related and both polynomial, the complexities of their min–max and min–max regret versions, for a constant number of scenarios, are quite contrasted since they are respectively strongly NP-hard and polynomial. However, for a non-constant number of scenarios, these versions become strongly NP-hard for both problems. In the interval scenario case, min–max versions are trivially polynomial. Moreover, for min–max regret versions, we obtain the same contrasted results as for a constant number of scenarios: min–max regret min s–t cut is strongly NP-hard whereas min–max regret min cut is polynomial.     

Harmonic Analysis on the SU(2) Dynamical Quantum Group

Dynamical quantum groups were recently introduced by Etingof and Varchenko as an algebraic framework for studying the dynamical Yang–Baxter equation, which is precisely the Yang–Baxter equation satisfied by 6j-symbols. We investigate one of the simplest examples, generalizing the standard SU(2) quantum group. The matrix elements for its corepresentations are identified with Askey–Wilson polynomials, and the Haar measure with the Askey–Wilson measure. The discrete orthogonality of the matrix elements yield the orthogonality of q-Racah polynomials (or quantum 6j-symbols). The Clebsch–Gordan coefficients for representations and corepresentations are also identified with q-Racah polynomials. This results in new algebraic proofs of the Biedenharn–Elliott identity satisfied by quantum 6j-symbols.     

Max-Plus Stochastic Processes

This paper is concerned with processes which are max-plus counterparts of Markov diffusion processes governed by Ito sense stochastic differential equations. Concepts of max-plus martingale and max-plus stochastic differential equation are introduced. The max-plus counterparts of backward and forward PDEs for Markov diffusions turn out to be first-order PDEs of Hamilton–Jacobi–Bellman type. Max-plus additive integrals and a max-plus additive dynamic programming principle are considered. This leads to variational inequalities of Hamilton–Jacobi–Bellman type.     

Polymer fractography and fracture kinetics

The fractographic method makes it possible to determine the test temperature and time at which anomalies appear in the temperature-time dependence of the strength of polymethyl methacrylate and polycaprolactam by finding the conditions of disappearance of specular zones from the fracture surfaces of these polymers. For PMMA these values are –40°C and 10^–2 sec, for PCL –120°C and 10^–7 sec, respectively.For communication 1 see [2].Ioffe Physicotechnical Institute, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, No. 2, pp. 232–237, March–April, 1971.     

Bäcklund autotransformation for the equationu xt=eu−e−2u

The system of differential relations that arises in connection with the Bullough-Dodd-Zhiber-Shabat equationu _xt=e^u–e^–2u is considered. The consistency of this system is established, and it is shown that the system realizes a Bäcklund autotransformation for the equationu _xt=e^u–e^–2u. The associated three-dimensional dynamical systems, which are compatible on a two-dimensional invariant submanifold, are investigated, and a construction of their general solution, which gives the explicit form of the three-parameter soliton for the equationu _xt=e^u–e^–2u, is proposed.Bashkir State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 1, pp. 146–159, April, 1993.     

An improvement of the numerical stability results for nonlinear neutral delay-integro-differential equations

This paper deals with stability of the extended Runge–Kutta methods for nonlinear neutral delay-integro-differential equations. The stability results in the reference [Y. Yu, L. Wen, S. Li, Nonlinear stability of Runge–Kutta methods for neutral delay integro-differential equations, Appl. Math. Comput. 191 (2007) 543–549] are improved. With this improvement, several new numerical stability criteria are obtained, it is proven that the extended Runge–Kutta methods are globally and asymptotically stable under the suitable conditions.     

The Criteria for Globally Stable Equilibrium in n-Dimensional Lotka–Volterra Systems

In this paper, a set of sufficient conditions are obtained for the existence of a globally asymptotically stable equilibrium point in various submodels of the classic n-dimensional Lotka–Volterra system. The submodels are the following systems: competition (cooperative or predator–prey) chain system and competition (cooperative or predator–prey) model between one and multispecies. The criteria in this paper are in explicit forms of the parameters and thus are easily verifiable.