集值映射的一种锥凸性及标量化

在一种集合偏序关系下提出了集值映射的标量锥拟凸概念, 讨论了它与各种锥凸性的关系. 然后对恰当锥拟凸性得到了某种水平集意义下的刻画. 同时建立了集值映射的各种锥凸性通过实值单调增加凸函数表示的标量化复合法则. 最后给出了利用Gerstewitz泛函表示的对集值映射的锥拟凸性的标量化刻画.     

Using store level scanner data to improve category management decisions: Developing positioning maps

This paper provides evidence of the usefulness of aggregated point-of-sale scanner data to infer the positioning of competing brands, providing valuable information for category management and hence facilitating decision making. Specifically, the authors propose a methodology to study the internal market structure based on market share models with latent heterogeneity when only macro-level time series data (not individual choices) are available. The proposed approach assumes a multidimensional decomposition, latent in the preference structure that is implicit to these types of models. By empirically applying this approach, the authors (1) simultaneously identify both latent dimensions of competing brands and latent segments with different brand preferences, (2) explain the competitive positioning of brands without using disaggregated consumer panel data, and (3) achieve greater predictive performance. The findings offer insights to academics and practitioners interested in improving the practice of category management.     

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.     

Nonlinear Stability of Riemann Ellipsoids with Symmetric Configurations

Using modern differential geometric methods, we study the relative equilibria for Dirichlet’s model of a self-gravitating fluid mass having at least two equal axes. We show that the only relative equilibria of this type correspond to Riemann ellipsoids for which the angular velocity and vorticity are parallel to the same principal axis of the body configuration. The two solutions found are MacLaurin and transversal spheroids. The singular reduced energy-momentum method developed in Rodríguez-Olmos (Nonlinearity 19(4):853–877, 2006) is applied to study their nonlinear stability and instability. We found that the transversal spheroids are nonlinearly stable for all eccentricities while for the MacLaurin spheroids, we recover the classical results. Comparisons with other existing results and methods in the literature are also made.      

Conservative cross diffusions and pattern formation through relaxation

This paper is aimed at studying the formation of patches in a cross-diffusion system without reaction terms when the diffusion matrix can be negative but with positive self-diffusion. We prove existence results for small data and global a priori bounds in space–time Lebesgue spaces for a large class of ‘diffusion’ matrices. This result indicates that blow-up should occur on the gradient. One can tackle this issue using a relaxation system with global solutions and prove uniform a priori estimates. Our proofs are based on a duality argument à la M. Pierre which we extend to treat degeneracy and growth of the diffusion matrix.We also analyze the linearized instability of the relaxation system and a Turing type mechanism can occur. This gives the range of parameters and data for which instability may occur. Numerical simulations show that patterns arise indeed in this range and the solutions tend to exhibit patches with stiff gradients on bounded solutions, in accordance with the theory.     

A duality principle for groups

The duality principle for Gabor frames states that a Gabor sequence obtained by a time-frequency lattice is a frame for L²(Rd) if and only if the associated adjoint Gabor sequence is a Riesz sequence. We prove that this duality principle extends to any dual pairs of projective unitary representations of countable groups. We examine the existence problem of dual pairs and establish some connection with classification problems for II₁ factors. While in general such a pair may not exist for some groups, we show that such a dual pair always exists for every subrepresentation of the left regular unitary representation when G is an abelian infinite countable group or an amenable ICC group. For free groups with finitely many generators, the existence problem of such a dual pair is equivalent to the well-known problem about the classification of free group von Neumann algebras.     

The Mazur product map on Hardy-type sequence spaces

We study certain Hardy-type sequence spaces Hp and , 1?p?∞, which are analogues of ?^∞ and c₀, respectively. We show that the Mazur product is not onto for every p∈(1,∞) with q=p⁻¹(p−1). We present corollaries for spaces defined via weighted ?p seminorms and for c₀. The latter corollary provides a new solution of Mazur's Problem 8 in the Scottish Book.     

On Characterizing the Solution Sets of Pseudoinvex Extremum Problems

In this paper, we study the minimization of a pseudoinvex function over an invex subset and provide several new and simple characterizations of the solution set of pseudoinvex extremum problems. By means of the basic properties of pseudoinvex functions, the solution set of a pseudoinvex program is characterized, for instance, by the equality , for each feasible point x, where is in the solution set. Our study improves naturally and extends some previously known results in Mangasarian (Oper. Res. Lett. 7: 21–26, 1988) and Jeyakumar and Yang (J. Opt. Theory Appl. 87: 747–755, 1995). This research was partially supported by National Natural Science Foundation of China Grants No. 10771228 and 10831009.     

How to define the dual of a higher-dimensional linear system

The classical duality of linear systems is generalized to higher dimensions. Consequently, a new insight is given to the concepts of controllability and observability, and autonomy.     

Common fixed point results for noncommuting mappings without continuity in generalized metric spaces

Using the setting of a generalized metric space, a fixed point theorem is proved for one map, and several fixed point theorems are proved for two maps. These results generalize several well known comparable results in the literature.