On Essential Stable Sets of Solutions in Set Optimization Problems

This paper introduces the essential stability for set optimization problems. Some kinds of essential stable sets of weakly minimal and minimal solutions are shown. The graph of minimal solution mappings is not necessarily closed, which is different from weakly minimal solution mappings. The existence of minimum essential sets of minimal solutions is proved.     

The convergence of a modified smoothing-type algorithm for the symmetric cone complementarity problem

The symmetric cone complementarity problem (denoted by SCCP) is a broad class of optimization problems, which contains the semidefinite complementarity problem, the second-order cone complementarity problem, and the nonlinear complementarity problem. In this paper we first extend the smoothing function proposed by Huang et al. (Sci. China 44:1107–1114, 2001) for the nonlinear complementarity problem to the context of symmetric cones and show that it is coercive under suitable assumptions. Based on this smoothing function, a smoothing-type algorithm, which is a modified version of the Qi-Sun-Zhou method (Qi et al. in Math. Program. 87:1–35, 2000), is proposed for solving the SCCP. By using the theory of Euclidean Jordan algebras, we prove that the proposed algorithm is globally and locally quadratically convergent under suitable assumptions. Preliminary numerical results for some second-order cone complementarity problems are reported which indicate that the proposed algorithm is effective.     

关于Kenichiro的一个丢番图方程问题

研究了Kenichiro提出的轮换对称形式的丢番图方程,即方程ab bc ca=ya+b+c3的求解问题,利用素数整除的一些性质,证明了该方程仅有平凡解a=b=c以及非平凡解(a,b,c)=(k,k,4k),(k,4k,k),(4k,k,k)(k∈N),从而完全解决了这个方程.     

Unit groups of group algebras of some small groups

Let FG be a group algebra of a group G over a field F and U (FG) the unit group of FG. It is a classical question to determine the structure of the unit group of the group algebra of a finite group over a finite field. In this article, the structure of the unit group of the group algebra of the non-abelian group G with order 21 over any finite field of characteristic 3 is established. We also characterize the structure of the unit group of FA ₄ over any finite field of characteristic 3 and the structure of the unit group of FQ ₁₂ over any finite field of characteristic 2, where Q ₁₂ = 〈x, y; x ⁶ = 1, y ² = x ³, x ^y = x ^?1〉.     

On the Largest Graph-Lagrangian of 3-Graphs with Fixed Number of Edges

The Graph-Lagrangian of a hypergraph has been a useful tool in hypergraph extremal problems. In most applications, we need an upper bound for the Graph-Lagrangian of a hypergraph. Frankl and Füredi conjectured that the \({r}\) -graph with \(m\) edges formed by taking the first \(\textit{m}\) sets in the colex ordering of the collection of all subsets of \({\mathbb N}\) of size \({r}\) has the largest Graph-Lagrangian of all \(r\) -graphs with \(m\) edges. In this paper, we show that the largest Graph-Lagrangian of a class of left-compressed \(3\) -graphs with \(m\) edges is at most the Graph-Lagrangian of the \(\mathrm 3 \) -graph with \(m\) edges formed by taking the first \(m\) sets in the colex ordering of the collection of all subsets of \({\mathbb N}\) of size \({3}\) .     

Strong Convergence of a Projection-Type Method for Mixed Variational Inequalities in Hilbert Spaces

Abstract

In this article, a projection-type method for mixed variational inequalities is proposed in Hilbert spaces. The proposed method has the following nice features: (i) The algorithm is well defined whether the solution set of the problem is nonempty or not, under some mild assumptions; (ii) If the solution set is nonempty, then the sequence generated by the method is strongly convergent to the solution, which is closest to the initial point; (iii) The existence of the solutions to variational inequalities can be verified through the behavior of the generated sequence. The results presented in this article generalize and improve some known results.     

A Lower Bound on the Second Fiber Coefficient of the Fiber Cones

Let (R, 𝔪) be a Cohen–Macaulay local ring of dimension d > 0, I an 𝔪-primary ideal of R and K an ideal containing I. When depth G(I) ≥ d ? 1 and r(I | K) < ∞, we present a lower bound on the second fiber coefficient of the fiber cones, and also provide a characterization, in terms of f ₂(I, K), of the condition depth F _K (I) ≥ d ? 1.