Parametric generalized set-valued variational inclusions and resolvent equations

In this paper, we develop the sensitivity analysis for generalized set-valued variational inclusions and generalized resolvent equations. We establish the equivalence between the parametric generalized set-valued variational inclusions and parametric generalized resolvent equations, by using the resolvent operator technique without assuming the differentiability of the given data.     

Generalized p-values and generalized confidence regions for the multivariate Behrens-Fisher problem and MANOVA

For two multivariate normal populations with unequal covariance matrices, a procedure is developed for testing the equality of the mean vectors based on the concept of generalized p-values. The generalized p-values we have developed are functions of the sufficient statistics. The computation of the generalized p-values is discussed and illustrated with an example. Numerical results show that one of our generalized p-value test has a type I error probability not exceeding the nominal level. A formula involving only a finite number of chi-square random variables is provided for computing this generalized p-value. The formula is useful in a Bayesian solution as well. The problem of constructing a confidence region for the difference between the mean vectors is also addressed using the concept of generalized confidence regions. Finally, using the generalized p-value approach, a solution is developed for the heteroscedastic MANOVA problem.     

Generalized G-KKM Theorems in Generalized Convex Spaces and Their Applications

Some new generalized G-KKM and generalized S-KKM theorems are proved under the noncompact setting of generalized convex spaces. As applications, some new minimax inequalities, saddle point theorems, a coincidence theorem, and a fixed point theorem are given in generalized convex spaces. These theorems improve and generalize many important known results in recent literature.     

Existence and uniqueness of v-asymptotic expansions and Colombeau's generalized numbers

We define a type of generalized asymptotic series called v-asymptotic. We show that every function with moderate growth at infinity has a v-asymptotic expansion. We also describe the set of v-asymptotic series, where a given function with moderate growth has a unique v-asymptotic expansion. As an application to random matrix theory we calculate the coefficients and establish the uniqueness of the v-asymptotic expansion of an integral with a large parameter. As another application (with significance in the non-linear theory of generalized functions) we show that every Colombeau's generalized number has a v-asymptotic expansion. A similar result follows for Colombeau's generalized functions, in particular, for all Schwartz distributions.     

Reduction of Courant algebroids and generalized complex structures

We present a theory of reduction for Courant algebroids as well as Dirac structures, generalized complex, and generalized Kähler structures which interpolates between holomorphic reduction of complex manifolds and symplectic reduction. The enhanced symmetry group of a Courant algebroid leads us to define extended actions and a generalized notion of moment map. Key examples of generalized Kähler reduced spaces include new explicit bi-Hermitian metrics on CP².     

Cantorian-fractal space-time and particles in a generalized magnetic field

In the fractal space-time theory the generalized precession, the generalized Stern–Gerlach experiment, the generalized Zeeman effect, the generalized Landau's levels and the generalized fractional quantum Hall effect are analyzed. The Cantorian structure of space-time implies the supplemental decreasing rate of the Ecliptic tilt, the “global” quantization in units of 36 km s⁻¹ and the filling factor ν=1/2n+1.     

Generalized convexity and inequalities involving special functions

In this paper, the authors show a relation between the generalized convexity and super- (sub-)multiplicative property, and discuss some generalized convexity and inequalities involving the Gaussian hypergeometric function, the generalized η-distortion function and the generalized Grötzsch function μa(r).     

One-point extensions of generalized hexagons and octagons

In this note, we prove the uniqueness of the one-point extension S of a generalized hexagon of order 2 and prove the non-existence of such an extension S of any other finite generalized hexagon of classical order, different from the one of order 2, and of the known finite generalized octagons provided the following property holds: for any three points x, y and z of S, the graph-theoretic distance from y to z in the derived generalized hexagon S_x is the same as the distance from x to z in S_y.     

Arakawa-Kaneko L-functions and generalized poly-Bernoulli polynomials

We introduce and investigate generalized poly-Bernoulli numbers and polynomials. We state and prove several properties satisfied by these polynomials. The generalized poly-Bernoulli numbers are algebraic numbers. We introduce and study the Arakawa-Kaneko L-functions. The non-positive integer values of the complex variable s of these L-functions are expressed rationally in terms of generalized poly-Bernoulli numbers and polynomials. Furthermore, we prove difference and Raabe?s type formulae for these L-functions.     

Lattice paths and generalized cluster complexes

In this paper we propose a variant of the generalized Schröder paths and generalized Delannoy paths by giving a restriction on the positions of certain steps. This generalization turns out to be reasonable, as attested by the connection with the faces of generalized cluster complexes of types A and B. As a result, we derive Krattenthaler's F-triangles for these two types by a combinatorial approach in terms of lattice paths.     

Line graph of combinations of generalized Bethe trees: Eigenvalues and energy

We characterize the eigenvalues and energy of the line graph L(G) whenever G is (i) a generalized Bethe tree, (ii) a Bethe tree, (iii) a tree defined by generalized Bethe trees attached to a path, (iv) a tree defined by generalized Bethe trees having a common root, (v) a graph defined by copies of a generalized Bethe tree attached to a cycle and (vi) a graph defined by copies of a star attached to a cycle; in this case, explicit formulas for the eigenvalues and energy of L(G) are derived.     

Generalized higher order Bernoulli number pairs and generalized Stirling number pairs

From a delta series f(t) and its compositional inverse g(t), Hsu defined the generalized Stirling number pair . In this paper, we further define from f(t) and g(t) the generalized higher order Bernoulli number pair . Making use of the Bell polynomials, the potential polynomials as well as the Lagrange inversion formula, we give some explicit expressions and recurrences of the generalized higher order Bernoulli numbers, present the relations between the generalized higher order Bernoulli numbers of both kinds and the corresponding generalized Stirling numbers of both kinds, and study the relations between any two generalized higher order Bernoulli numbers. Moreover, we apply the general results to some special number pairs and obtain series of combinatorial identities. It can be found that the introduction of generalized Bernoulli number pair and generalized Stirling number pair provides a unified approach to lots of sequences in mathematics, and as a consequence, many known results are special cases of ours.     

Optimality and duality for a multi-objective programming problem involving generalized d-type-I and related n-set functions

In this paper, we introduce several generalized convexity for a real-valued set function and establish optimality and duality results for a multi-objective programming problem involving generalized d-type-I and related n-set functions.     

Generalized Shalika models of p-adic SO4n and functoriality

Let F denote a p-adic local field of characteristic zero. In this paper, we investigate the structures of irreducible admissible representations of SO_4n (F) having nonzero generalized Shalika models and find relations between the generalized Shalika models and the local Arthur parameters, which support our conjectures on the local Arthur parametrization and the local Langlands functoriality in terms of the dual group associated to the spherical variety, which is attached to the generalized Shalika models.     

Generalized inverses and their application to applied probability problems

The main aim of this paper is to examine the applicability of generalized inverses to a wide variety of problems in applied probability where a Markov chain is present either directly or indirectly through some form of imbedding. By characterizing all generalized inverses of I—P, where P is the transition matrix of a finite irreducible discrete time Markov chain, we are able to obtain general procedures for finding stationary distributions, moments of the first passage time distributions, and asymptotic forms for the moments of the occupation-time random variables. It is shown that all known explicit methods for examining these problems can be expressed in this generalized inverse framework. More generally, in the context of a Markov renewal process setting the aforementioned problems are also examined using generalized inverses of I—P. As a special case, Markov chains in continuous time are considered, and we show that the generalized inverse technique can be applied directly to the infinitesimal generator of the process, instead of to I—P, where P is the transition matrix of the discrete time jump Markov chain.     

Sufficiency and duality in nondifferentiable multiobjective programming involving generalized type I functions

Recently, Hachimi and Aghezzaf defined generalized (F,α,ρ,d)-type I functions, a new class of functions that unifies several concepts of generalized type I functions. In this paper, the generalized (F,α,ρ,d)-type I functions are extended to nondifferentiable functions. By utilizing the new concepts, we obtain several sufficient optimality conditions and prove mixed type and Mond-Weir type duality results for the nondifferentiable multiobjective programming problem.     

Lambert's W, infinite divisibility and Poisson mixtures

The Lambert W function is shown to be the Laplace exponent of a positive infinitely divisible law (i.e. W is a Bernstein function) called the standard Lambert law. This law is a generalized gamma convolution. At least three Poisson mixture families are defined in terms of W. One of these is the generalized Poisson laws which are shown to be generalized negative-binomial convolutions. Mixing with positive stable laws yields further generalizations.     

Generalized Hankel matrices of Markov parameters and their applications to control problems

The concept of Hankel matrices of Markov parameters associated with two polynomials is generalized for matrices. The generalized Hankel matrices of Markov parameters are then used to develop methods for testing the relative primeness of two matrices A and B, for determining stability and inertia of a matrix, and for constructing a class of matrices C such that A + C has a desired spectrum. Neither the method of construction of the generalized Hankel matrices nor the methods developed using these matrices require explicit computation of the characteristic polynomial of A (or of B).     

New systems of generalized quasi-variational inclusions in FC-spaces and applications

In this paper, we study some new systems of generalized quasi-variational inclusion problems in FC-spaces without convexity structure.By applying an existence theorem of maximal elements of set-valued mappings due to the author, some new existence theorems of solutions for the systems of generalized quasi-variational inclusion problems are proved in noncompact FC-spaces. As applications, some existence results of solutions for the system of quasi-optimization problems and mathematical programs with the systems of generalized quasi-variational inclusion constraints are obtained in FC-spaces.     

Effective condition numbers and small sample statistical condition estimation for the generalized Sylvester equation

In this paper,we investigate the effective condition numbers for the generalized Sylvester equation(AX-YB,DX-YE)=(C,F),where A,D∈R m×m,B,E∈R n×n and C,F ∈ R m×n.We apply the small sample statistical method for the fast condition estimation of the generalized Sylvester equation,which requires O(m2n+mn2) flops,comparing with O(m3+n3) flops for the generalized Schur and generalized HessenbergSchur methods for solving the generalized Sylvester equation.Numerical examples illustrate the sharpness of our perturbation bounds.