Multiple objective fractional programming involving semilocally type I-preinvex and related functions

Sufficient optimality conditions are obtained for a nonlinear multiple objective fractional programming problem involving η-semidifferentiable type I-preinvex and related functions. Furthermore, a general dual is formulated and duality results are proved under the assumptions of generalized semilocally type I-preinvex and related functions. Our result generalize the results of Preda [V. Preda, Optimality and duality in fractional multiple objective programming involving semilocally preinvex and related functions, J. Math. Anal. Appl. 288 (2003) 365–382] and Stancu-Minasian [I.M. Stancu-Minasian, Optimality and duality in fractional programming involving semilocally preinvex and related functions, J. Inform. Optim. Sci. 23 (2002) 185–201].     

Integrals and derivatives connected to the r-Lambert function

In this paper, we derive some formulas for higher order derivatives of r-Lambert functions. Moreover, an integration formula involving powers of r-Lambert function is obtained.     

Asymptotic Approximations for Radial Spheroidal Wavefunctions with Complex Size Parameter

Radial spheroidal wavefunctions are functions of four variables, usually denoted by m , n , x , and γ, the last of which is known as the size parameter. This parameter becomes complex when the problem of scattering of a sound pulse by a spheroid is treated using a Laplace transform with respect to time together with the method of separation of variables. Several asymptotic approximations, involving modified Bessel functions, are developed and analyzed.     

On some Turán-type inequalities for q-special functions

Intrinsic inequalities involving Turán-type inequalities for some q-special functions are established. A special interest is granted to q-Dunkl kernel. The results presented here would provide extensions of those given in the classical case.     

Convexity andq-gamma function

Inequalities and convexity properties known for the gamma function are extended to theq-gamma function, 0<q<1. Applying some classical inequalities for convex functions, we deduce monotonicity results for several functions involving theq-gamma function. Further applications to the probability theory are given.     

Some limit relations in the A1 tableau of Dunkl-Cherednik operators

In this paper we study some limit relations involving some q-special functions related with the A₁ (root system) tableau of Dunkl-Cherednik operators. Concretely we consider the limits involving the nonsymmetric q-ultraspherical polynomials (q-Rogers polynomials), ultraspherical polynomials (Gegenbauer polynomials), q-Hermite and Hermite polynomials.     

Generalized isotone optimization with applications to starshaped functions

This article considers a curve-fitting problem involving the minimization of the distance from a functionf to a convex cone of functions. A weighted uniform norm is considered as a measure of the distance. The domain of the functions is a partially ordered set, and the convex cone is defined by the isotonicity and nonnegativity conditions on functions. The problem has a linear programming formulation; however, explicit expressions for the optimal solutions have been obtained directly, thereby eliminating the necessity of using linear programming techniques. The results are applied to approximation by starshaped functions.     

Two-sided inequalities for the Struve and Lommel functions

Abstract

Mathematical inequalities and other results involving such widely- and extensively-studied special functions of mathematical physics and applied mathematics as (for example) the Bessel, Struve and Lommel functions as well as the associated hypergeometric functions are potentially useful in many seemingly diverse areas of applications, especially in situations in which these functions are involved in solutions of mathematical, physical and engineering problems which can be modeled by ordinary and partial di?erential equations. With this objective in view, our present investigation is motivated by some open problems involving inequalities for a number of particular forms of the hypergeometric function ₁F₂(a; b, c; z). Here, in this paper, we apply a novel approach to such problems and obtain presumably new two-sided inequalities for the Struve function, the associated Struve function and the modified Struve function by first investigating inequalities for the general hypergeometric function ₁F₂(a; b, c; z). We also briefly discuss the analogous new inequalities for the Lommel function under some conditions and constraints. Finally, as special cases of our main results, we deduce several inequalities for the modified Lommel function and the normalized Lommel function.     

Multiple solutions to a class of inclusion problem with the p(x)-Laplacian

In this article, we obtain the existence of at least two nontrivial solutions for a nonlinear elliptic problem involving p(x)-Laplacian type operator and nonsmooth potentials. Our approach is variational and it is based on the nonsmooth critical point theory for locally Lipschitz functions.     

A predictor-corrector method for a class of equilibrium problems

We consider two classes of generalized monotone functions and generalized skew symmetric functions. By applying the method of auxiliary-principle, we propose and study a predictor-corrector algorithm for solving mixed quasiequilibrium problems involving σ -pseudomonotone functions with respect to a ξ -skew symmetric bifunction, which are concepts introduced in this paper. We prove the convergence of the iterative sequence under generalized pseudomonotonicity and skew symmetry assumptions, improving some known results in the recent literature. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Schwinger Functions in Noncommutative Quantum Field Theory

It is shown that the n-point functions of scalar massive free fields on the noncommutative Minkowski space are distributions which are boundary values of analytic functions. Contrary to what one might expect, this construction does not provide a connection to the popular traditional Euclidean approach to noncommutative field theory (unless the time variable is assumed to commute). Instead, one finds Schwinger functions with twistings involving only momenta that are on the mass-shell. This explains why renormalization in the traditional Euclidean noncommutative framework crudely differs from renormalization in the Minkowskian regime.     

q-Extensions of certain Erdélyi type integrals

Recently we discovered several new Erdélyi type integrals. In the present paper, it is shown how the q-extensions of all those integrals involving and representing certain q-hypergeometric functions can be developed. The well-known special cases and applications of these q-integrals are also pointed out.     

Convolution kernels based on thin-plate splines

Quasi-interpolation using radial basis functions has become a popular method for constructing approximations to continuous functions in many space dimensions. In this paper we discuss a procedure for generating kernels for quasi-interpolation, using functions which have series expansions involving terms liker logr. It is shown that such functions are suitable if and only if is a positive even integer and the spatial dimension is also even.     

New convolutions for complete and elementary symmetric functions

In this paper, we give new relationships between complete and elementary symmetric functions. These results can be used to discover and prove some identities involving r-Whitney numbers, Jacobi–Stirling numbers, Bernoulli numbers and other numbers that are specializations of complete and elementary symmetric functions.     

Certain Families of Elliptic Functions Defined by <Emphasis Type="Italic">q</Emphasis>-Series

We introduce certain families of elliptic functions involving the Weierstrass ℘-function via q-series systematically and investigate their analytic and algebraic properties. Especially, we give examples of non-linear algebraic ordinary differential equations satisfied by some such elliptic functions. 2000 Mathematics Subject Classification: Primary–11M36, 33E05     

On a class of nonlinear variational–hemivariational inequalities

A three critical points theorem for nondifferentiable functions is pointed out and an existence result of multiple solutions for a Neumann elliptic variational–hemivariational inequality involving the p-laplacian is established. As an application, a Neumann problem for elliptic equations with discontinuous nonlinearities is studied.     

Piecewise-Linear Approximations of Multidimensional Functions

We develop explicit, piecewise-linear formulations of functions f(x):ℝ ⁿ ↦ℝ, n≤3, that are defined on an orthogonal grid of vertex points. If mixed-integer linear optimization problems (MILPs) involving multidimensional piecewise-linear functions can be easily and efficiently solved to global optimality, then non-analytic functions can be used as an objective or constraint function for large optimization problems. Linear interpolation between fixed gridpoints can also be used to approximate generic, nonlinear functions, allowing us to approximately solve problems using mixed-integer linear optimization methods. Toward this end, we develop two different explicit formulations of piecewise-linear functions and discuss the consequences of integrating the formulations into an optimization problem.     

Generalized V-univexity type-I for multiobjective programming with n-set functions

In the last time important results in multiobjective programming involving type-I functions were obtained (Yuan et al. in: Konnov et al. (eds) Lecture notes in economics and mathematical systems, 2007; Mishra et al. An Univ Bucureşti Ser Mat, LII(2): 207–224, 2003). Following one of these ways, we study optimality conditions and generalized Mond-Weir duality for multiobjective programming involving n-set functions which satisfy appropriate generalized univexity V-type-I conditions. We introduce classes of functions called (ρ, ρ′)-V-univex type-I, (ρ, ρ′)-quasi V-univex type-I, (ρ, ρ′)-pseudo V-univex type-I, (ρ, ρ′)-quasi pseudo V-univex type-I, and (ρ, ρ′)-pseudo quasi V-univex type-I. Finally, a general frame for constructing functions of these classes is considered. This research was supported by Grant PN II code ID No. 112/01.10.2007, CEEX code 1/2006 No. 1531/2006, and CNCSIS A No. 105 GR/2006.     

Approximation capabilities of immersed finite element spaces for elasticity Interface problems

We construct and analyze a group of immersed finite element (IFE) spaces formed by linear, bilinear, and rotated Q₁ polynomials for solving planar elasticity equation involving interface. The shape functions in these IFE spaces are constructed through a group of approximate jump conditions such that the unisolvence of the bilinear and rotated Q₁ IFE shape functions are always guaranteed regardless of the Lamé parameters and the interface location. The boundedness property and a group of identities of the proposed IFE shape functions are established. A multi‐point Taylor expansion is utilized to show the optimal approximation capabilities for the proposed IFE spaces through the Lagrange type interpolation operators.     

Construction of Cauchy integrals for one class of nonanalytic functions of complex variables

For some values of characteristics of p-analytic and (p, q)-analytic functions, the values of kernels of Cauchy integrals are found in an explicit form involving elementary and special functions. The application of the Cauchy integral for deriving the Muskhelishvili equation in the Stokes problem on a slow axially symmetric flow of a viscous liquid is demonstrated. Bibliography: 9 titles. Translated fromObchyslyuval’na ta Prykladna Matematyka, No. 76, 1992, pp. 98–111.