Properties of functions generalized convex with respect to a WT-system

Continuity and convergence properties of functions, generalized convex with respect to a continuous weak Tchebysheff system, are investigated. It is shown that, under certain non-degeneracy assumptions on the weak Tchebysheff system, every function in its generalized convex cone is continuous, and pointwise convergent sequences of generalized convex functions are uniformly convergent on compact subsets of the domain. Further, it is proved that, with respect to a continuous Tchebysheff system, L_p-convergence to a continuous function, pointwise convergence and uniform convergence of a sequence of generalized convex functions are equivalent on compact subsets of the domain.     

Generalized Quadrangles with a Spread of Symmetry

We present a common construction for some known infinite classes of generalized quadrangles. Whether this construction yields other (unknown) generalized quadrangles is an open problem. The class of generalized quadrangles obtained this way is characterized in two different ways. On the one hand, they are exactly the generalized quadrangles having a spread of symmetry. On the other hand, they can be characterized in terms of the group of projectivities with respect to a spread. We explore some properties of these generalized quadrangles. All these results can be applied to the theory of the glued near hexagons, a class of near hexagons introduced by the author in De Bruyn (1998) On near hexagons and spreads of generalized quadrangles, preprint.     

Linearizability of the polynomial differential systems with a resonant singular point

Integrability and linearizability of polynomial differential systems are studied. The computation of generalized period constants is a way to find necessary conditions for linearizable systems for any rational resonance ratio. A method to compute generalized period constants is given. The algorithm is recursive and easy to realize with computer algebraic system. As the application, we discuss linearizable conditions for several Lotka-Volterra systems, and where this is the first time that the linearizability is considered for 3:−4 and 3:−5 resonances.     

Distributive Lattices with a Generalized Implication: Topological Duality

In this paper we introduce the notion of generalized implication for lattices, as a binary function ⇒ that maps every pair of elements of a lattice to an ideal. We prove that a bounded lattice A is distributive if and only if there exists a generalized implication ⇒ defined in A satisfying certain conditions, and we study the class of bounded distributive lattices A endowed with a generalized implication as a common abstraction of the notions of annihilator (Mandelker, Duke Math J 37:377–386, 1970), Quasi-modal algebras (Celani, Math Bohem 126:721–736, 2001), and weakly Heyting algebras (Celani and Jansana, Math Log Q 51:219–246, 2005). We introduce the suitable notions of morphisms in order to obtain a category, as well as the corresponding notion of congruence. We develop a Priestley style topological duality for the bounded distributive lattices with a generalized implication. This duality generalizes the duality given in Celani and Jansana (Math Log Q 51:219–246, 2005) for weakly Heyting algebras and the duality given in Celani (Math Bohem 126:721–736, 2001) for Quasi-modal algebras.     

Analytic solutions of a reducible strain gradient elasticity model for solid cylinder with a cavity and its application in zonal failure

In this work, the reducibility condition of the fourth-order equilibrium equation in the strain gradient elasticity (SGE) model for solid cylinder with a cavity is obtained. When the reducibility condition is satisfied, the analytic displacement, generalized radial stress, and generalized angular stress can be solved out, and according to the higher-order coefficients, internal length scale, and Lamé constants, the displacement, generalized radial stress, and generalized angular stress are classified into four types: (1) conventional elasticity solution, (2) quasiperiodic SGE solution, (3) monotonous SGE solution, and (4) non-real-number solution. Quasiperiodic generalized radial stress and generalized angular stress are used to explain the occurrence of zonal failure of surrounding rock of a circular roadway. Numerical analysis with MATLAB is applied to study the influence of loading on zonal failure of surrounding rock of a circular raodway.     

Behavior of singularities of a nonsteady wave field near a caustic in a medium with a variable velocity

The Cauchy problem for the wave equation is considered. A discontinuity in the form of a homogeneous generalized function of degree is given at the initial front. An asymptotic solution is found in the vicinity of the space-time caustic in the interaction of the discontinuity surface with the caustic. The solution is represented in the form of a certain combination of the generalized function with associated Legendre polynomials.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 156, pp. 98–108, 1986.     

Generalized Bellman-Hamilton-Jacobi optimality conditions for a control problem with a boundary condition

In this paper we study conditions for optimality of a deterministic control problem where the state of the system is required to stop at the boundary. Using the Clarke generalized gradient, we refine the classical verification theorem and show that it is not only sufficient but also necessary for optimality. It is also shown that the solution to the generalized Bellman-Jacobi-Hamilton equation involving the Clarke generalized gradient is unique among the class of regular functions.     

The characterization of a class of subspace pseudoframes with arbitrary real number translations

In this article, the notion of generalized multiresolution structure is introduced. The concept of subspace pseudoframes with arbitrary real number translations is proposed. A new method for constructing a generalized multiresolution structure in Paley–Wiener subspace of L²(R) is presented. A pyramid decomposition scheme is established based on such a generalized multiresolution structure. Finally, affine frames of space L²(R) with arbitrary real number translations are obtained by virtue of the subspace pseudoframes and the pyramid decomposition scheme. Relation to some physical theories such as quarks confinement is also investigated.     

Analysis of generalized projective synchronization for a chaotic gyroscope with a periodic gyroscope

In this paper, a simple nonlinear controller is applied to investigate the generalized projective synchronization for a controlled chaotic gyroscope with a periodic gyroscope dynamical system. The necessary and sufficient conditions for generalized projective synchronization are developed through the theory of discontinuous dynamical systems. The synchronization invariant domain from the synchronization conditions is presented. The parameter maps are explored for a better understanding of the synchronicity of two gyroscopes with different motions. Finally, the partial and full generalized projective synchronizations of two nonlinear coupled gyroscope systems are carried out to verify the effectiveness of the scheme.     

Wave propagation in a micropolar generalized thermoelastic body with stretch

In the present investigation, we discuss two different problems namely

1. Rayleigh-Lamb problem in micropolar generalized thermoelastic layer with stretch and
2. Rayleigh wave in a micropolar generalized thermoelastic half-space with stretch. The frequency and wave velocity equations for symmetric and anti-symmetric vibrations are obtained for the first problem. The frequency equation has also been derived for the second problem. The special cases of the above said problems of micropolar generalized thermoelasticity with stretch for Green-Lindsay and Lord-Shulman theory have been discussed in detail. Results of these analysis reduce to those without thermal and stretch effects

     

Estimation of a non-negative location parameter with unknown scale

For a vast array of general spherically symmetric location-scale models with a residual vector, we consider estimating the (univariate) location parameter when it is lower bounded. We provide conditions for estimators to dominate the benchmark minimax MRE estimator, and thus be minimax under scale invariant loss. These minimax estimators include the generalized Bayes estimator with respect to the truncation of the common non-informative prior onto the restricted parameter space for normal models under general convex symmetric loss, as well as non-normal models under scale invariant \(L^p\) loss with \(p>0\) . We cover many other situations when the loss is asymmetric, and where other generalized Bayes estimators, obtained with different powers of the scale parameter in the prior measure, are proven to be minimax. We rely on various novel representations, sharp sign change analyses, as well as capitalize on Kubokawa’s integral expression for risk difference technique. Several properties such as robustness of the generalized Bayes estimators under various loss functions are obtained.     

Second-order risk comparison of SLSE with GLSE and MLE in a regression with serial correlation

First, the second-order bias of the estimator of the autoregressive parameter based on the ordinary least squares residuals in a linear model with serial correlation is given. Second, the second-order expansion of the risk matrix of a generalized least squares estimator with the above estimated parameter is obtained. This expansion is the same as that based on a suitable estimator of the autoregressive parameter independent of the sample. Third, it is shown that the risk matrix of the generalized least squares estimator is asymptotically equivalent to that of the maximum likelihood estimator up to the second order. Last, a sufficient condition is given for the term due to the estimation of the autoregressive parameter in this expansion to vanish under Grenander's condition for the explanatory variates.     

A generalization of the role expectation conflict theory with a methodological analysis

Gross’, Mason's and McEachern's role conflict theory with two incompatible role expectations is generalized by letting the number of incompatible role expectations increase. It is proved that the generalized theory is consistent and that its four axioms are independent. Three methodological results are proven. They show that the generalized theory is, in principle, testable from the point of view of three different well‐defined testing situations. Finally, it is shown that the original theory is a special case of the generalization.     

Interaction of a hollow cylinder of finite length and a plate with a cylindrical cavity with a rigid insert

Two problems of the interaction of a hollow circular cylinder with load-free ends and an unbounded plate with a cylindrical cavity and a symmetrically imbedded rigid insert are considered. Homogeneous solutions are found and the generalized orthogonality of these solutions is used when the modified boundary conditions are satisfied. As a result, we have a system of two integral equations in functions of the displacements of the outer and inner surfaces of the hollow cylinder. These functions are sought in the form of sums of a trigonometric series and a power function with a root singularity. The ill-posed infinite systems of linear algebraic equations obtained are regularized by the introduction of small positive parameters. Since the elements of the matrices of the systems as well as the contact stresses are defined by poorly converging numerical and functional series, an efficient method for calculating of the remainders of the above-mentioned series is developed. Formulae are found for the contact pressure distribution function and the integral characteristic. Examples of the calculation of the interaction of the cylinder and the plate with an insert are given.The method of solving contact problems described here has been used earlier1, 2 and the generalized orthogonality of the solutions found for bodies of finite dimensions, that is, for a rectangle and cylinders of finite length, is its basis. Problems for hollow cylinders with a band ² and an insert reduce to a system of two integral equations, and the problem for a rectangle¹ reduces to one integral equation. Solving these integral equations, ill-posed systems of linear algebraic equations are obtained which are subject to regularization³.     

Generalized second-order directional derivatives and optimization with C1,1 functions

In this paper, a new generalized second-order directional derivative and a set-valued generalized Hessian are introudced for C¹,¹ functions in real Banach spaces. It is shown that this set-valued generalized Hessian is single-valued at a point if and only if the function is twice weakly Gãteaux differentiable at the point and that the generalized second-order directional derivative is upper semi-continuous under a regularity condition. Various generalized calculus rules are also given for C^1,1 functions. The generalized second-order directional derivative is applied to derive second-order necessary optirnality conditions for mathematical programming problems.     

On the derivation of fractional diffusion equation with an absorbent term and a linear external force

Recently, the generalized fractional reaction–diffusion equation subject to an external linear force field has been proposed to describe the transport processes in disordered systems. The solution of this generalized model can be formally expressed in closed form through the Fox function. For the sack of completeness, we dedicate this work to construct a neatly derivation of the generalized fractional reaction–diffusion equation. Remarkably, such derivation could in general offer some novel and inspiring inspection to the phenomena of anomalous transport. For instance, there is a strong evidence that the fractional calculus offers some physical insight into the origin of fractional dynamics for a systems which exhibit multiple trapping.     

On a Generalized Finite Intersection Property

We continue the study of the right finite intersection property under a weaker condition on annihilators, introducing the concept of generalized right finite intersection property (simply, generalized right FIP). We observe the structure of rings with the generalized right FIP and examine the generalized right FIP for various kinds of basic extensions of rings with the property. We show that the generalized right FIP does not go up to polynomial rings, and that the 2-by-2 full matrix ring over a domain has the generalized right FIP. In the process, we also obtain an equivalent condition for which a nonzero polynomial, over the ring of integers modulo n ≥ 2, is a non-zero-divisor.     

The Igusa Zeta Function Associated with a Composite Power Function on the Space of Rectangular Matrices

On the space of real rectangular n × m matrices, we introduce a composite power function and study the zeta integral associated with it. We describe the properties of the Igusa zeta function on the basis of the properties of a generalized composite power function and establish a functional relation for the zeta integral. As a result, the Fourier transform of a generalized composite power function is found in explicit form.     

On generalized Petersen graphs labeled with a condition at distance two

An L(2,1)-labeling of graph G is an integer labeling of the vertices in V(G) such that adjacent vertices receive labels which differ by at least two, and vertices which are distance two apart receive labels which differ by at least one. The λ-number of G is the minimum span taken over all L(2,1)-labelings of G. In this paper, we consider the λ-numbers of generalized Petersen graphs. By introducing the notion of a matched sum of graphs, we show that the λ-number of every generalized Petersen graph is bounded from above by 9. We then show that this bound can be improved to 8 for all generalized Petersen graphs with vertex order >12, and, with the exception of the Petersen graph itself, improved to 7 otherwise.     

The asymptotics of the return map of a singular point with fixed Newton diagram

We exhibit a large class of nondegenerate singular points in which necessary and sufficient conditions are given for monodromy. We compute the generalized first Lyapunov value, which is expressed in terms of the Newton diagram of the singular point. The computational algorithm proposed is based on writing the return map as the composition of transition mappings constructed using the diagram. The nonvanishing of the generalized first Lyapunov value is a sufficient condition for the existence of a focus. Bibliography: 8 titles.Translated fromTrudy Seminara imeni I. G. Petrovskogo, No. 15, pp. 156–177, 1991.