Differential calculus and integration of generalized functions over membranes

In this paper we continue the development of the differential calculus started in Aragona et al. (Monatsh. Math. 144:13–29, 2005). Guided by the so-called sharp topology and the interpretation of Colombeau generalized functions as point functions on generalized point sets, we introduce the notion of membranes and extend the definition of integrals, given in Aragona et al. (Monatsh. Math. 144:13–29, 2005), to integrals defined on membranes. We use this to prove a generalized version of the Cauchy formula and to obtain the Goursat Theorem for generalized holomorphic functions. A number of results from classical differential and integral calculus, like the inverse and implicit function theorems and Green’s theorem, are transferred to the generalized setting. Further, we indicate that solution formulas for transport and wave equations with generalized initial data can be obtained as well.     

On the existence of solutions to generalized quasi-equilibrium problems

In this paper, we apply new results on variational relation problems obtained by D. T. Luc (J Optim Theory Appl 138:65–76, 2008) to generalized quasi-equilibrium problems. Some sufficient conditions on the existence of its solutions of generalized quasi-equilibrium problems are shown. As special cases, we obtain several results on the existence of solutions of generalized Pareto and weak quasi-equilibrium problems concerning C-pseudomonotone multivalued mappings. We deduce also some results on the existence of solutions to generalized vector Pareto and weakly quasivariational inequality and vector Pareto quasi-optimization problems with multivalued mappings.     

The semilocal convergence of a generalization of Brent's and Brown's methods

In this paper, we discuss the semilocal convergence of Martínez's generalization of Brent's and Brown's methods. Through a careful investigation of the algorithm structure, we convert Martínez's generalized method into an approximate Newton method with a special error term. Based on such equivalent variation, we prove the semilocal convergence theorem of Martínez's generalized method. This is a complementary result to the convergence theory of Martínez's generalized method.     

The degree of convergence of entire functions and generalized type

Summary The generalized order and generalized type of an entire function have been considered in this paper, using arbitrary growth functions. In place of the usual Taylor series expansion, polynomial series expansion having polynomial coefficients of an entire function have been considered and formula for generalized type obtained in terms of the polynomial coefficients. In the end, a result characterizing the set of entire functions of positive generalized order and finite type in terms of their degree of convergence on general sets has been obtained.     

Definability of second order generalized quantifiers

We study second order generalized quantifiers on finite structures. One starting point of this research has been the notion of definability of Lindström quantifiers. We formulate an analogous notion for second order generalized quantifiers and study definability of second order generalized quantifiers in terms of Lindström quantifiers.     

Polynomial extensions of generalized quasi-Baer rings

In this paper, we consider the behavior of polynomial rings over generalized quasi-Baer rings and show that the generalized quasi-Baer condition on a ring R is preserved by many polynomial extensions.     

Generalized Jordan derivations on Lie ideals associate with Hochschild 2-cocycles of rings

Let R be a 2-torsion free ring and L be a Lie ideal of R. In this paper we initiate the study of generalized derivations on Lie ideals associate with Hochschild 2-cocycles and prove that every generalized derivation associate with a Hochschild 2-cocycle on L is a generalized derivation on L under certain conditions.     

On choices of formulations of computing the generalized singular value decomposition of a large matrix pair

For the computation of the generalized singular value decomposition (GSVD) of a large matrix pair (A,B) of full column rank, the GSVD is commonly formulated as two mathematically equivalent generalized eigenvalue problems, so that a generalized eigensolver can be applied to one of them and the desired GSVD components are then recovered from the computed generalized eigenpairs. Our concern in this paper is, in finite precision arithmetic, which generalized eigenvalue formulation is numerically preferable to compute the desired GSVD components more accurately. We make a detailed perturbation analysis on the two formulations and show how to make a suitable choice between them. Numerical experiments illustrate the results obtained.

     

广义(ω)性质的一个注记

通过定义新的谱集,研究了Weyl定理的一个变形—广义(w)性质,给出了Banach空间上有界线性算子满足广义(w)性质的充要条件.同时,利用所得的主要结论,我们研究了广义(w)性质的摄动.     

矩阵值Caratheodory函数广义块Pick矩阵的秩不变性

Lasarow^[1]推导出矩阵值Carath\'{e}odory函数的第一、第二型广义块Pick矩阵及其变型的秩不变性. 这些矩阵由同一个Carath\'{e}odory函数的值与它的直到某阶的导数值确定. 利用文献[2]中提出的块Toeplitz向量方法, 该文断言,这些块矩阵的秩分别相关并重合于具有秩不变性的块Toeplitz矩阵的秩, 从而改进了这两类广义块Pick矩阵的秩不变性结论的证明.     

The existence of generalized mix functions

To enrich the message space of a cipher and guarantee security, Ristenpart and Rogaway defined mix functions on two sets of equal size. To mix inputs from two sets of different sizes, Stinson generalized the definition of mix functions (called generalized mix functions), and established an existence result for generalized mix functions with 10 undetermined pairs of input sizes. In this paper, we complete the solution to the existence problem for generalized mix functions.      

Microlocal Analysis of Generalized Pullbacks of?Colombeau Functions

In distribution theory the pullback of a general distribution by a C ^∞-function is well-defined whenever the normal bundle of the C ^∞-function does not intersect the wave front set of the distribution. However, the Colombeau theory of generalized functions allows for a pullback by an arbitrary c-bounded generalized function. It has been shown in previous work that in the case of multiplication of Colombeau functions (which is a special case of a C ^∞ pullback), the generalized wave front set of the product satisfies the same inclusion relation as in the distributional case, if the factors have their wave front sets in favorable position. We prove a microlocal inclusion relation for the generalized pullback (by a c-bounded generalized map) of Colombeau functions. The proof of this result relies on a stationary phase theorem for generalized phase functions, which is given in the Appendix. Furthermore we study an example (due to Hurd and Sattinger), where the pullback function stems from the generalized characteristic flow of a partial differential equation.      

Generalized positive continuous additive functionals of multidimensional Brownian motion and their associated Revuz measures

We extend the notion of positive continuous additive functionals of multidimensional Brownian motions to generalized Wiener functionals in the setting of Malliavin calculus. We call such a functional a generalized PCAF. The associated Revuz measure and a characteristic of a generalized PCAF are also extended adequately. By making use of these tools a local time representation of generalized PCAFs is discussed. It is known that a Radon measure corresponds to a generalized Wiener functional through the occupation time formula. We also study a condition for this functional to be a generalized PCAF and the relation between the associated Revuz measure of the generalized PCAF corresponding to Radon measure and this Radon measure. Finally we discuss a criterion to determine the exact Meyer–Watanabe’s Sobolev space to which this corresponding functional belongs.     

Outer Generalized Inverses in Rings

ABSTRACT

In this article we formulate basic results on outer generalized inverses of elements in rings. We characterize elements which have the same idempotents related to their particular outer generalized inverses and investigate positive generalized inverses in C*-algebras.     

Generalized Watson transforms I: General theory

This paper introduces two main concepts, called a generalized Watson transform and a generalized skew-Watson transform, which extend the notion of a Watson transform from its classical setting in one variable to higher dimensional and noncommutative situations. Several construction theorems are proved which provide necessary and sufficient conditions for an operator on a Hilbert space to be a generalized Watson transform or a generalized skew-Watson transform. Later papers in this series will treat applications of the theory to infinite-dimensional representation theory and integral operators on higher dimensional spaces.

     

Interpolation of Herz Spaces and Applications

C. Herz introduced in [Hr] some new spaces to study properties of functions. An Interesting account, with many applications, of some particular cases of the generalized Herz spaces is given in [BS]. In this paper we first identify the duals of the generalized Herz spaces. Then, we characterize their intermediate spaces when the complex method of interpolation for families of spaces Is used. Applications are given that show the bounded ness of many operators on the generalized Herz spaces.     

Existence of Solutions for Generalized Vector Variational-like Inequalities

In this paper, we consider a generalized vector variational-like inequality problem (for short, GVVLIP), which includes generalized vector variational inequalities, vector variational inequalities and classical variational inequalities as special cases. The concepts of generalized C-pseudomonotone-like and generalized H-hemicontinuous-like operators are introduced. Some existence results for GVVLIP are obtained under the assumptions of generalized C-pseudomonotone-like property and generalized H-hemicontinuous-like property. These results appear to be new and interesting. New existence results of the classical variational inequality are also obtained. In this research, the first author was partially supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, China and the Dawn Program Foundation in Shanghai. The third author was partially supported by Grant NSC 94-2213-E-110-035.     

The differentiability of real functions on normed linear space using generalized subgradients

The modification of the Clarke generalized subdifferential due to Michel and Penot is a useful tool in determining differentiability properties for certain classes of real functions on a normed linear space. The Gâteaux differentiability of any real function can be deduced from the Gâteaux differentiability of the norm if the function has a directional derivative which attains a constant related to its generalized directional derivative. For any distance function on a space with uniformly Gâteaux differentiable norm, the Clarke and Michel-Penot generalized subdifferentials at points off the set reduce to the same object and this generates a continuity characterization for Gâteaux differentiability. However, on a Banach space with rotund dual, the Fréchet differentiability of a distance function implies that it is a convex function. A mean value theorem for the modified generalized subdifferential has implications for Gâteaux differentiability.     

On some geometric properties of the generalized CES production functions

In this paper we prove that the generalized CES production function has constant return to scale if and only if the corresponding hypersurface is developable. Moreover, we establish that this production function has decreasing/increasing return to scale if and only if the corresponding hypersurface has positive/negative Gaussian curvature. These results are a generalization of some recent results concerning the generalized Cobb-Douglas production functions [G.E. Vîlcu, A geometric perspective on the generalized Cobb-Douglas production functions, Appl. Math. Lett. 24 (2011) 777-783].     

On Several Types of Basic Constraint Qualifications via Coderivatives for Generalized Equations

In this paper, we study several types of basic constraint qualifications in terms of Clarke/Fréchet coderivatives for generalized equations. Several necessary and/or sufficient conditions are given to ensure these constraint qualifications. It is proved that basic constraint qualification and strong basic constraint qualification for convex generalized equations can be obtained by these constraint qualifications, and the existing results on constraint qualifications for the inequality system can be deduced from the given conditions in this paper. The main work of this paper is an extension of the study on constraint qualifications from inequality systems to generalized equations.