Symmetric Stable Processes in Cones

We study exponents of integrability of the first exit time from generalized cones for conditioned rotation invariant stable Lévy processes. Along the way, we introduce the spherical fractional Laplacian and derive some of its spectral properties.     

Number of Complete N-ary Subtrees on Galton-Watson Family Trees

We associate with a Bienaymé-Galton-Watson branching process a family tree rooted at the ancestor. For a positive integer , define a complete -ary tree to be the family tree of a deterministic branching process with offspring generating function . We study the random variables and counting the number of disjoint complete -ary subtrees, rooted at the ancestor, and having height and , respectively. Dekking (1991) and Pakes and Dekking (1991) find recursive relations for and involving the offspring probability generation function (pgf) and its derivatives. We extend their results determining the probability distributions of and . It turns out that they can be expressed in terms of the offspring pgf, its derivatives, and the above probabilities. We show how the general results simplify in case of fractional linear, geometric, Poisson, and one-or-many offspring laws.      

First-order and second-order ε-directional derivatives of a marginal function in convex programming with linear inequality constraints

Hiriart-Urruty gave formulas of the first-order and second-order -directional derivatives of a marginal function for a convex programming problem with linear equality constraints, that is, the image of a function under linear mapping (Ref. 1). In this paper, we extend his results to a problem with linear inequality constraints. The formula of the first-order derivative is given with the help of a duality theorem. A lower estimate for the second-order -directional derivative is given.The author wishes to thank Professor N. Furukawa and Dr. H. Kawasaki for their helpful comments and encouragements. He is also indebted to one referee for pointing out the proof of Proposition 3.1.     

A Generalized Double Crossproduct and Drinfeld Double

In this paper we construct a new and more complicated algebra construction of two algebras B and H, a generalized double crossproduct B H. The left generalized smash product, the right generalized smash product, Majids double crossproduct, especially, the smash product, the Drinfeld Double D(H) and Doi-Takeuchi algebra B H are all special cases as our algebra structure. Next, we analyze conditions under which this new algebra B H is a Hopf algebra termed a generalized double crossproduct of Hopf algebra, and describe a coquasitriangular structure over the generalized double crossproduct Hopf algebra B H. Finally, what we do is to construct a new braided monoidal category ^J_JMod^Q_Q obtained from the structure of the generalized double crossproduct, and establish a kind of new quantum Yang-Baxter operators.AMS Subject Classification (1991): 16W20, 16D90, 16S40, 16W30     

Spaces over a Category and Assembly Maps in Isomorphism Conjectures in K‐ and L‐Theory

We give a unified approach to the Isomorphism Conjecture of Farrell and Jones on the algebraicKandLtheory of integral group rings and to the Baum–Connes Conjecture on the topologicalKtheory of reducedC*algebras of groups. The approach is through spectra over the orbit category of a discrete groupG.We give several points of view on the assembly map for a family of subgroups and characterize such assembly maps by a universal property generalizing the results of Weiss and Williams to the equivariant setting. The main tools are spaces and spectra over a category and their associated generalized homology and cohomology theories, and homotopy limits.     

Simulation optimization of buffer allocations in production lines with unreliable machines

We use a recent simulationbased optimization method, sample path optimization, to find optimal buffer allocations in tandem production lines where machines are subject to random breakdowns and repairs, and the product is fluidtype. We explore some of the functional properties of throughput of such systems and exploit these properties to prove the almost sure convergence of our optimization technique, under a regularity condition on the steady state. Utilizing a generalized semiMarkov process (GSMP) representation of the system, we derive recursive expressions to compute onesided directional derivatives of throughput, from a single simulation run. Finally, we give computational results for lines with up to 50 machines. We also compare results for smaller lines with the results from a more conventional method, stochastic approximation, whenever applicable. In these numerical studies, our method performed quite well on problems that are considered difficult by current computational standards.     

Variational-Like Inequalities with Generalized Monotone Mappings in Banach Spaces

In this paper, we introduce two classes of variational-like inequalities with generalized monotone mappings in Banach spaces. Using the KKM technique, we obtain the existence of solutions for variational-like inequalities with relaxed – monotone mappings in reflexive Banach spaces. We present also the solvability of variational-like inequalities with relaxed – semimonotone mappings in arbitrary Banach spaces by means of the Kakutani-Fan-Glicksberg fixed-point theorem. The results presented in this paper extend and improve the corresponding results of Refs. 1–6.     

On invexity-type nonlinear programming problems

In this paper, we propose a new class of nonlinear programing, called SFJ-invex programming. The optimality characterization shows that a problem is SFJ-invex if and only if a Fritz John point together with its multiplier, is a Fritz John saddle point of the problem. Under any constraint qualification assumption, a problem is SFJ-invex if and only if a Kuhn-Tucker point together with its multiplier is a Kuhn-Tucker saddle point of the problem. Furthermore, a generalization of the SFJ-invex, class is developed; the applications to (h, )-convex programming, particularly geometric programming, and to generalized fractional programming provide a relaxation in constraint qualification for differentiable problems to get saddle-point type optimality criteria.The author wishes to thank the referee for helpful comments.     

Topological influence of the presence of fully-λ-bases in locally convex spaces

This article reveals the topological impact of fully--bases in locally convex spaces where carries either the traditional normal topology or the fairly generalized-topology of Ruckle. It has been established that the generalized nuclearity of plays a significant role in influencing the topology of the space. Further, the equivalence of normal topology and the topology arising out of the fully--base ( being equipped with normal topology or-topology) has been investigated.We acknowledge with thanks the suggestions of the referee.     

Efficient Estimation of the Canonical Dependence Function

The canonical dependence function (z), z [0,1], is introduced and studied in detail for distributions, which belong to the -neighborhood of a bivariate generalized Pareto distribution. We establish local asymptotic normality (LAN) of the loglikelihood function of a 2×2 table sorting of n i.i.d. observations and derive efficient estimators of (z) from the Hájek-LeCam Convolution Theorem. These results extend results by Falk and Reiss (2003) for the canonical dependence parameter (1/2) to arbitrary z (0,1).     

Interpolation by Rational Functions with Nodes on the Unit Circle

From the Erds–Turán theorem, it is known that if f is a continuous function on and L _n (f, z) denotes the unique Laurent polynomial interpolating f at the (2 n + 1)th roots of unity, then Several years later, Walsh and Sharma produced similar result but taking into consideration a function analytic in and continuous on and making use of algebraic interpolating polynomials in the roots of unity.In this paper, the above results will be generalized in two directions. On the one hand, more general rational functions than polynomials or Laurent polynomials will be used as interpolants and, on the other hand, the interpolation points will be zeros of certain para-orthogonal functions with respect to a given measure on .     

A Note on Ultrametric Matrices

It is proved in this paper that special generalized ultrametric and special matrices are, in a sense, extremal matrices in the boundary of the set of generalized ultrametric and matrices, respectively. Moreover, we present a new class of inverse M-matrices which generalizes the class of matrices.     

Implementation and Test of Auction Methods for Solving Generalized Network Flow Problems with Separable Convex Cost

We describe the implementation and testing of two methods, based on the auction approach, for solving the problem of minimizing a separable convex cost subject to generalized network flow conservation constraints. The first method is the -relaxation method of Ref. 1; the second is an extension of the auction sequential/shortest path algorithm for ordinary network flow to generalized network flow. We report test results on a large set of randomly generated problems with varying topology, arc gains, and cost function. Comparison with the commercial code CPLEX on linear/quadratic cost problems and with the public-domain code PPRN on nonlinear cost ordinary network problems are also made. The test results show that the auction sequential/shortest path algorithm is generally fastest for solving quadratic cost problems and mixed linear/nonlinear cost problems with arc gain range near 1. The -relaxation method is generally fastest for solving nonlinear cost ordinary network problems and mixed linear/nonlinear cost problems with arc gain range away from 1. CPLEX is generally fastest for solving linear cost and mixed linear/quadratic cost problems with arc gain range near 1.     

ɛ-duality theorem of nondifferentiable nonconvex multiobjective programming

Necessary Kuhn-Tucker conditions up to precision without constraint qualification for -Pareto optimality of multiobjective programming are derived. This article suggests the establishment of a Wolfe-type -duality theorem for nondifferentiable, nonconvex, multiobjective minimization problems. The -vector Lagrangian and the generalized -saddle point for Pareto optimality are studied.     

A Property of the Ansatz of Hirota's Method for Quasilinear Parabolic Equations

By using the recently discovered new invariant properties of the ansatz of R. Hirota's method, we prove that the classes of linear fractional solutions to some nonlinear equations are closed. This allows us to construct new solutions for a chosen class of dissipative equations. This algorithm is similar to the method of dressing the solutions of integrable equations. The equations thus obtained imply a compatibility condition and are known as a nonlinear Lax pair with variable coefficients. So we propose a method for constructing such pairs. To construct solutions of a more complicated form, we propose to use the property of zero denominators and factorized brackets, which has been discovered experimentally. The expressions thus constructed are said to be quasi-invariant. They allow us to find true relations between the functions contained in the ansatz, to correct the ansatz, and to construct a solution. We present some examples of new solutions constructed following this approach. Such solutions can be used for majorizing in comparison theorems and for modeling phase processes and process in neurocomputers. A program for computing solutions by methods of computer algebra is written. These techniques supplement the classical methods for constructing solutions by using their group properties.     

On a Positivity Property of the Riemann ξ‐Function

We consider a quantitative version of the positivity condition of the Riemann function. J. C. Lagarias obtained that for >10. The main aim of this paper is to prove the equality for 1.     

Asymptotic Normality of Kernel Density Estimators under Dependence

In this paper, we study the kernel methods for density estimation of stationary samples under generalized conditions, which unify both the linear and -mixing processes discussed in the literature and also adapt to the non-linear or/and non--mixing processes. Under general, mild conditions, the kernel density estimators are shown to be asymptotically normal. Some specific theorems are derived within various contexts, and their applications and relationship with the relevant references are considered. It is interesting that the conditions on the bandwidth may be very simple, even in the generalized context. The stationary sequences discussed cover a large number of (linear or nonlinear) time series and econometric models (such as the ARMA processes with ARCH errors).     

Asymptotic Behaviour of λ-Convex Sets in the Hyperbolic Plane

It is known that the limit Area/Length for a sequence of convex sets expanding over the whole hyperbolic plane is less than or equal 1, and exactly 1 when the sets considered are convex with respect to horocycles. We consider geodesics and horocycles as particular cases of curves of constant geodesic curvature with 0 1 and we study the above limit Area/Length as a function of the parameter .     

A New Approach to Multiobjective Programming with a Modified Objective Function

In this paper, optimality for multiobjective programming problems having invex objective and constraint functions (with respect to the same function ) is considered. An equivalent vector programming problem is constructed by a modification of the objective function. Furthermore, an -Lagrange function is introduced for a constructed multiobjective problem and modified saddle point results are presented.     

A Full Characterization of Multipliers for the Strong ϱ-Integral in the Euclidean Space

We study a generalization of the classical Henstock-Kurzweil integral, known as the strong -integral, introduced by Jarník and Kurzweil. Let be the space of all strongly -integrable functions on a multidimensional compact interval E, equipped with the Alexiewicz norm We show that each element in the dual space of can be represented as a strong -integral. Consequently, we prove that fg is strongly -integrable on E for each strongly -integrable function f if and only if g is almost everywhere equal to a function of bounded variation (in the sense of Hardy-Krause) on E.