向量优化中基于拟内部的弱C(\varepsilon)-有效解的标量化

首先获得了co-radiant集的一些拟内部性质. 进而在邻近C(\varepsilon)-次似凸性假设条件下, 建立了相应的择一性定理, 并给出了基于拟内部的集值向量优化问题弱C(\varepsilon)-有效解的线性标量化结果.     

Quasi interiors,lagrange multipliers,andL p spectral estimation with lattice bounds

Lagrange multipliers useful in characterizations of solutions to spectral estimation problems are proved to exist in the absence of Slater's condition provided a new constraint involving the quasi-relative interior holds. We also discuss the quasi interior and its relation to other generalizations of the interior of a convex set and relationships between various constraint qualifications. Finally, we characterize solutions to theL ^p spectral estimation problem with the added constraint that the feasible vectors lie in a measurable strip [, ].The authors wish to thank Jonathan M. Borwein and Adrian S. Lewis for many enlightening discussions and useful suggestions. The duality approach to the general problem inL ^p was suggested by J. M. Borwein.     

向量优化中改进集的一些拓扑性质

主要研究改进集的一些拓扑运算性质。首先在改进集条件下给出了拓扑向量空间中两个非空集之和的拓扑内部的一些运算性质。进一步,利用改进集获得了 Flores-Baz′an 和 Hern′andez 提出的假定 B 的一个加强形式。此外,给出了一些例子对主要结果进行了解释。     

A Method of Representing Rough Sets System Determined by Quasi Orders

In this paper, we study about the ordered structure of rough sets determined by a quasi order. A characterization theorem for rough sets of an approximation space (U, R) based on a quasi order R is given in Nagarajan and Umadevi (2010). Then using the characterization of rough sets determined by a quasi order, its rough sets system is represented by a new construction. This construction is generalized and abstracted into a new method of constructing Kleene based algebraic structures from dually isomorphic distributive lattices. Then by using different varieties of distributive lattices, we obtain various Kleene based algebraic structures. By this construction, we give various algebraic structures to the rough sets system determined by a quasi order R.     

Quasiopen and <Emphasis Type="Italic">p</Emphasis>-Path Open Sets,and Characterizations of Quasicontinuity

In this paper we give various characterizations of quasiopen sets and quasicontinuous functions on metric spaces. For complete metric spaces equipped with a doubling measure supporting a p-Poincaré inequality we show that quasiopen and p-path open sets coincide. Under the same assumptions we show that all Newton-Sobolev functions on quasiopen sets are quasicontinuous.     

Jensen inequalities for <Emphasis Type="Italic">P</Emphasis>-class functions

In this paper, we first give some characterizations for P-class functions. Then giving a Hermite–Hadamard type inequality for P-class functions, we prove equivalency of some significant metrics in normed linear spaces. We also obtain an operator version of the Jensen inequality for P-class functions. Introducing operator (mid) P-class functions, we present some characterizations for such functions.     

On Bipolar Anti Fuzzy h-ideals in Hemi-rings

In this work, we present notions of bipolar anti fuzzy $h$-ideals and bipolar anti fuzzy interior $h$-ideals in hemi-rings. Investigating some of their properties, we characterize hemi-rings by means of positive anti $\beta$-cut and negative anti $\alpha$-cut. Meanwhile, some results of homomorphisms, anti images and anti pre-images are given to show the rationality of the definitions introduced in the present paper. Also, we define an equivalence relation on bipolar anti fuzzy $h$-ideals. In particular, we investigate translations, extensions and multiplications of bipolar anti fuzzy $h$-ideals. Finally, we present characterizations of h-hemi-regular and h-semi-simple hemi-rings in terms of bipolar anti fuzzy $h$-ideals.     

Rigidity of ‐quasi Einstein manifolds

This paper deals with the study on ‐quasi Einstein manifolds. First, we give some characterizations of an ‐quasi Einstein manifold admitting closed conformal or parallel vector field. Then, we obtain some rigidity conditions for this class of manifolds. We prove that an ‐quasi Einstein manifold with a closed conformal vector field has a warped product structure of the form , where I is a real interval, is an ‐dimensional Riemannian manifold and q is a smooth function on I . Finally, a non‐trivial example of an ‐quasi Einstein manifold verifying our results in terms of the potential function is presented.     

On some dynamical aspects of NIP theories

We investigate some dynamical features of the actions of automorphisms in the context of model theory. We interpret a few notions such as compact systems, entropy and symbolic representations from the theory of dynamical systems in the realm of model theory. In this direction, we settle a number of characterizations of NIP theories in terms of dynamics of automorphisms and invariant measures. For example, it is shown that the property of NIP corresponds to the compactness property of some associated systems and also to the zero entropy property of automorphisms. These results give a correspondence between some notions of tameness in model theory and ergodic theory. Moreover, we study the concept of symbolic representation and consider it in some well known mathematical objects such as the circle group, Bohr sets, Sturmian sequences, the structure \((\mathbb {Z},+,U)\), and random graphs with a model theoretic point of view in mind. We establish certain characterizations for stability theoretic dividing lines, such as independence property, order property and strict order property in terms of associated symbolic representations. At the end, we propose some applications of symbolic representations and these characterizations by giving a proof for a classical theorem by Shelah and also introducing some invariants associated to the types and elements of models.     

Remarks on small sets of reals

We show that the Dual Borel Conjecture implies that \boldsymbol\aleph_1 $"> and find some topological characterizations of perfectly meager and universally meager sets.

     

On a question by Corson about point-finite coverings

We answer in the affirmative the following question raised by H. H. Corson in 1961: “Is it possible to cover every Banach space X by bounded convex sets with non-empty interior in such a way that no point of X belongs to infinitely many of them?”Actually, we show the way to produce in every Banach space X a bounded convex tiling of order 2, i.e., a covering of X by bounded convex closed sets with non-empty interior (tiles) such that the interiors are pairwise disjoint and no point of X belongs to more than two tiles.     

On quasi $$\epsilon $$-solution for robust convex optimization problems

This paper devotes to the quasi \(\epsilon \)-solution (one sort of approximate solutions) for a robust convex optimization problem in the face of data uncertainty. Using robust optimization approach (worst-case approach), we establish approximate optimality theorem and approximate duality theorems in term of Wolfe type on quasi \(\epsilon \)-solution for the robust convex optimization problem. Moreover, some examples are given to illustrate the obtained results.     

Piecewise Linear Bases and Besov Spaces on Fractal Sets

For a class of closed sets F R ⁿ admitting a regular sequence of triangulations or generalized triangulations, the analogues on F of the Faber—Schauder and Franklin bases are discussed. The characterizations of the Besov spaces on F in the terms of coefficients of functions with respect to these bases are proved. As a consequence, analogous characterizations of the Besov spaces on some fractal domains (including the Sierpinski gasket and the von Koch curve) by coefficients of functions with respect to the wavelet bases constructed in [26] are obtained.     

Parametric AE-solution sets to the parametric linear systems with multiple right-hand sides and parametric matrix equation A(p)X = B(p)

In this paper, the parametric matrix equation A(p)X = B(p) whose elements are linear functions of uncertain parameters varying within intervals are considered. In this matrix equation A(p) and B(p) are known m-by-m and m-by-n matrices respectively, and X is the m-by-n unknown matrix. We discuss the so-called AE-solution sets for such systems and give some analytical characterizations for the AE-solution sets and a sufficient condition under which these solution sets are bounded. We then propose a modification of Krawczyk operator for parametric systems which causes reduction of the computational complexity of obtaining an outer estimation for the parametric united solution set, considerably. Then we give a generalization of the Bauer-Skeel and the Hansen-Bliek-Rohn bounds for enclosing the parametric united solution set which also enables us to reduce the computational complexity, significantly. Also some numerical approaches based on Gaussian elimination and Gauss-Seidel methods to find outer estimations for the parametric united solution set are given. Finally, some numerical experiments are given to illustrate the performance of the proposed methods.     

Littlewood-Paley Characterizations of Hajłasz-Sobolev and Triebel-Lizorkin Spaces via Averages on Balls

Let \(p\in (1,\infty )\) and \(q\in [1,\infty )\). In this article, the authors characterize the Triebel-Lizorkin space \({F}^{\alpha }_{p,q}(\mathbb {R}^{n})\) with smoothness order α ∈ (0, 2) via the Lusin-area function and the \(g_{\lambda }^{*}\)-function in terms of difference between f(x) and its ball average \(B_{t}f(x):=\frac 1{|B(x,t)|}{\int }_{B(x,t)}f(y)\,dy\) over the ball B(x, t) centered at \(x\in \mathbb {R}^{n}\) with radius t ∈ (0, 1). As an application, the authors obtain a series of characterizations of \(F^{\alpha }_{p,\infty }(\mathbb {R}^{n})\) via pointwise inequalities, involving ball averages, in spirit close to Haj?asz gradients, here some interesting phenomena naturally appear that, in the end-point case when α = 2, some of these pointwise inequalities characterize the Triebel-Lizorkin spaces \(F^{2}_{p,2}(\mathbb {R}^{n})\), while not \(F^{2}_{p,\infty }(\mathbb {R}^{n})\), and that some of other obtained pointwise characterizations are only known to hold true for \(F^{\alpha }_{p,\infty }(\mathbb {R}^{n})\) with \(p\in (1,\infty )\), α ∈ (0, 2) or α ∈ (n/p, 2). In particular, some new pointwise characterizations of Haj?asz-Sobolev spaces via ball averages are obtained. Since these new characterizations only use ball averages, they can be used as starting points for developing a theory of Triebel-Lizorkin spaces with smoothness orders not less than 1 on spaces of homogeneous type.     

Frames and non linear approximations in Hilbert spaces

A simple proof of the proposition, stated in ([2], p. 346), asserting that in Hilbert spaces a Riesz basis is greedy, is given. Also, greedy approximant for frames in Hilbert spaces is defined and it is shown that frames satisfy the quasi greedy and almost greedy conditions. Finally, we give the characterizations of approximation spaces A^s(Ψ), A_q^s(Ψ) by means of weak-l_p and Lorentz sequence spaces for frames.     

A characterization of the generalized quadrangle Q(4,q), q ODD

In [3], [ 4 ] we introduced the concept of (0,2)-set in generalized quadrangles, in order to obtain characterizations for P(S,()) and T ₂ ^* (O). Using these sets we are now able to formulate a characterization for Q(4,q), q odd, by assuming local conditions in an antiregular point x of a generalized quadrangle of order s.