1-Soliton solution of the generalized KdV equation with generalized evolution

This paper obtains the 1-soliton solution of three variants of the generalized KdV equation with generalized evolution. The solitary wave ansatz is used to carry out the integration of such equation. The parameter domain is also identified in the process. A couple of conserved quantities are also calculated for each of these variants. The numerical simulations are also carried out.     

A generalization of generalized Fibonacci and generalized Pell numbers

This paper is concerned with developing a new class of generalized numbers. The main advantage of this class is that it generalizes the two classes of generalized Fibonacci numbers and generalized Pell numbers. Some new identities involving these generalized numbers are obtained. In addition, the two well-known identities of Sury and Marques which are recently developed are deduced as special cases. Moreover, some other interesting identities involving the celebrated Fibonacci, Lucas, Pell and Pell–Lucas numbers are also deduced.     

Axiomatic characterizations of generalized values

In the framework of cooperative game theory, the concept of generalized value, which is an extension of that of value, has been recently proposed to measure the overall influence of coalitions in games. Axiomatizations of two classes of generalized values, namely probabilistic generalized values and generalized semivalues, which extend probabilistic values and semivalues, respectively, are first proposed. The axioms we utilize are based on natural extensions of axioms involved in the axiomatizations of values. In the second half of the paper, special instances of generalized semivalues are also axiomatized.     

Finite extensions of generalized Bessel sequences to generalized frames

The objective of this paper is to investigate the question of modifying a given generalized Bessel sequence to yield a generalized frame or a tight generalized frame by finite extension. Some necessary and sufficient conditions for the finite extensions of generalized Bessel sequences to generalized frames or tight generalized frames are provided, and every result is illustrated by the corresponding example.     

Bivariate generalized exponential distribution

Recently it has been observed that the generalized exponential distribution can be used quite effectively to analyze lifetime data in one dimension. The main aim of this paper is to define a bivariate generalized exponential distribution so that the marginals have generalized exponential distributions. It is observed that the joint probability density function, the joint cumulative distribution function and the joint survival distribution function can be expressed in compact forms. Several properties of this distribution have been discussed. We suggest to use the EM algorithm to compute the maximum likelihood estimators of the unknown parameters and also obtain the observed and expected Fisher information matrices. One data set has been re-analyzed and it is observed that the bivariate generalized exponential distribution provides a better fit than the bivariate exponential distribution.     

New identities involving generalized Fibonacci and generalized Lucas numbers

This paper presents two new identities involving generalized Fibonacci and generalized Lucas numbers. One of these identities generalize the two well-known identities of Sury and Marques which are recently developed. Some other interesting identities involving the famous numbers of Fibonacci, Lucas, Pell and Pell-Lucas numbers are also deduced as special cases of the two derived identities. Performing some mathematical operations on the introduced identities yield some other new identities involving generalized Fibonacci and generalized Lucas numbers.     

Topological 1-soliton solution of the generalized KdV equation with generalized evolution

In this paper, the topological 1-soliton solution to the generalized Korteweg-de Vries equation with generalized evolution will be obtained. The solitary wave ansatz methods will be applied to obtain the solutions. In the process, the constraints relations between the soliton parameters will also be determined.     

SOME COMPARISONS BETWEEN GENERALIZED ORDER STATISTICS

Some stochastic comparisons of generalized order statistics under the right spread order,the location independent riskier order and the total time transform order are investigated in this paper.The underlying distributions and parameters on which generalized order statistics are based are also surveyed to obtain the conditions for increasing the expectations of spacings between the first two generalized order statistics and between the last two generalized order statistics.     

1-Soliton solution of the generalized Burgers equation with generalized evolution

The generalized Burgers equation with generalized evolution is studied in this paper. The soliton ansatz is used to carry out the integration of this generalized Burgers equation. This study is then extended to 1+2 dimensions, even with full nonlinearity.     

Left and right generalized Drazin invertible operators

In this paper, we define and study the left and the right generalized Drazin inverse of bounded operators in a Banach space. We show that the left (resp. the right) generalized Drazin inverse is a sum of a left invertible (resp. a right invertible) operator and a quasi-nilpotent one. In particular, we define the left and the right generalized Drazin spectra of a bounded operator and also show that these sets are compact in the complex plane and invariant under additive commuting quasi-nilpotent perturbations. Furthermore, we prove that a bounded operator is left generalized Drazin invertible if and only if its adjoint is right generalized Drazin invertible. An equivalent definition of the pseudo-Fredholm operators in terms of the left generalized Drazin invertible operators is also given. Our obtained results are used to investigate some relationships between the left and right generalized Drazin spectra and other spectra founded in Fredholm theory.     

Multivariate generalized S-estimators

In this paper we introduce generalized S-estimators for the multivariate regression model. This class of estimators combines high robustness and high efficiency. They are defined by minimizing the determinant of a robust estimator of the scatter matrix of differences of residuals. In the special case of a multivariate location model, the generalized S-estimator has the important independence property, and can be used for high breakdown estimation in independent component analysis. Robustness properties of the estimators are investigated by deriving their breakdown point and the influence function. We also study the efficiency of the estimators, both asymptotically and at finite samples. To obtain inference for the regression parameters, we discuss the fast and robust bootstrap for multivariate generalized S-estimators. The method is illustrated on a real data example.     

An asymptotics look at the generalized inference

This paper provides an asymptotics look at the generalized inference through showing connections between the generalized inference and two widely used asymptotic methods, the bootstrap and plug-in method. A generalized bootstrap method and a generalized plug-in method are introduced. The generalized bootstrap method can not only be used to prove asymptotic frequentist properties of existing generalized confidence regions through viewing fiducial generalized pivotal quantities as generalized bootstrap variables, but also yield new confidence regions for the situations where the generalized inference is unavailable. Some examples are presented to illustrate the method. In addition, the generalized F-test (Weerahandi, 1995 [26]) can be derived by the generalized plug-in method, then its asymptotic validity is obtained.     

Balanced generalized handcuffed designs

The definition of balanced generalized handcuffed designs (BHD) is of course more specific than that of the generalized handcuffed designs that we introduced in 1987.

In the first part of this paper, we present a particular property of a BHD, which is not necessarily that of a generalized handcuffed design.

Then, we provide the reader with a general procedure that enables one to obtain such designs, and is called a ‘difference method’. We also show how this difference method can be made more useful in the case where the set V on which a BHD is constructed is the residue classes of integers mod V.

The third part of this paper deals with the problem of the existence of a BHD; and a solution is given for a particular case. We assume that the method applied for solving this problem will allow for the constructing of many more theorems analogous to Theorem 3.     

Integral representations for the generalized Bedient polynomials and the generalized Cesàro polynomials

The main object of this paper is to investigate several general families of hypergeometric polynomials and their associated multiple integral representations. By suitably specializing our main results, the corresponding integral representations are deduced for such familiar classes of hypergeometric polynomials as (for example) the generalized Bedient polynomials and the generalized Cesàro polynomials. Each of the integral representations, which are derived in this paper, may be viewed also as a linearization relationship for the product of two different members of the associated family of hypergeometric polynomials.     

Extending the generalized tanh method to the generalized Hamiltonian equations: New soliton-like solutions

To certain nonlinear evolution equations, the tanh method has been generalized for constructing not only solitary-wave but also soliton-like solutions. In this paper, no loss of conciseness, we further extend the generalized tanh method with computerized symbolic computation to a pair of generalized Hamiltonian equations. A new family of soliton-like analytical solutions is obtained, of which the solitary waves and previously-claimed soliton-like solutions are shown to be the special cases.     

Efficient solutions in generalized linear vector optimization

This paper establishes several new facts on generalized polyhedral convex sets and shows how they can be used in vector optimization. Among other things, a scalarization formula for the efficient solution sets of generalized linear vector optimization problems is obtained. We also prove that the efficient solution set of a generalized linear vector optimization problem in a locally convex Hausdorff topological vector space is the union of finitely many generalized polyhedral convex sets and it is connected by line segments.     

Lipschitzian stability of fully parameterized generalized equations

This paper concerns the study of Lipschitzian stability of fully parameterized generalized equations in which both single-valued and set-valued functions depend on parameters. Various relationships between the Lipschitz-like and metric regularity properties of the solution mapping, the base mapping, or field mapping in the fully perturbed generalized equations are established by using the Dontchev–Hager Fixed Point Theorem. The implicit mapping theorem for metric regularity is also extended to fully parameterized generalized equations.     

The generalized Gompertz distribution

This paper deals with a new generalization of the exponential, Gompertz, and generalized exponential distributions. This distribution is called the generalized Gompertz distribution (GGD). The main advantage of this new distribution is that it has increasing or constant or decreasing or bathtub curve failure rate depending upon the shape parameter. This property makes GGD is very useful in survival analysis. Some statistical properties such as moments, mode, and quantiles are derived. The failure rate function is also derived. The maximum likelihood estimators of the parameters are derived using a simulations study. Real data set is used to determine whether the GGD is better than other well-known distributions in modeling lifetime data or not.     

On Hermitian generalized inverses and positive semidefinite generalized inverses

The main aim of this paper is to investigate the Hermitian and positive semidefinite generalized inverses of a square matrix. First, we present some conditions for the existence of Hermitian and positive semidefinite generalized inverses. Further, expressions of these generalized inverses are given. Finally, we give two numerical examples to demonstrate our results.     

Coxian representations of generalized Erlang distributions

This paper studies Coxian representations of generalized Erlang distributions. A nonlinear program is derived for computing the parameters of minimal Coxian representations of generalized Erlang distributions. The nonlinear program is also used to characterize the triangular order and the admissible region of generalized Erlang distributions. It is shown that the admissible region associated with a triangular order may not be convex. For generalized Erlang distributions of ME-order 3, a minimal Coxian representation is found explicitly. In addition, an algorithm is developed for computing a special type of ordered Coxian representations - the bivariate Coxian representation - for generalized Erlang distributions.