This paper is devoted to consider the notions of complementary problem (CP) and set-valued complementary problem (SVCP). The set-valued complementary problem is compared with the classical single-valued complementary problem. Also, the solution set of the set-valued complementary problem is characterized. Our results illustrated by some examples. This paper is devoted to co...

Consider the affine variational inequality problem. It is shown that the distance to the solution set from a feasible point near the solution set can be bounded by the norm of a natural residual at that point. This bound is then used to prove linear convergence of a matrix splitting algorithm for solving the symmetric case of the problem. This latter result improves upon a recent result of Luo ...

We define and investigate HC-forcing invariant formulas of set theory, whose interpretations in the hereditarily countable sets are well behaved under forcing extensions. This leads naturally to a notion of cardinality ||Φ|| for sentences Φ of Lω1,ω, which counts the number of sentences of L∞,ω that, in some forcing extension, become a canonical Scott sentence of a model of Φ. We show this card...

This paper proposes an effective algorithm based on the level set method (LSM) to solve shape and topology optimization problems. Since the conventional LSM has several limitations, a binary level set method (BLSM) is used instead. In the BLSM, the level set function can only take 1 and -1 values at convergence. Thus, it is related to phase-field methods. We don’t need to solve the Hamilton-Jac...

Abstract. The aim of this paper is to investigate the application of a semiimplicit additive operator splitting scheme based binary level set method to source reconstruction problems. We reformulate the original model to be a new constrained optimization problem under the binary level set framework and solve it by the augmented Lagrangian method. Then we propose an efficient gradient-type algor...

In team environments, models are often shared and edited by multiple developers. To allow modularity and facilitate developer independence, we consider the problem of splitting a large monolithic model into sub-models. We propose an approach that assists users in incrementally discovering the set of desired sub-models. Our approach is supported by an automated tool that performs model splitting...

We define and investigate HC-forcing invariant formulas of set theory, whose interpretations in the hereditarily countable sets are well behaved under forcing extensions. This leads naturally to a notion of cardinality ||Φ|| for sentences Φ of Lω1,ω, which counts the number of sentences of L∞,ω that, in some forcing extension, become a canonical Scott sentence of a model of Φ. We show this card...

Let X,, X,, ,be i.i.d. random variables, which are uniformly distributed on [O,l]. Further let Z,(O) = [0, l] and let Z,(n) denote the k th largest interval generated by the points 0, X,, X,, . , X,_ 1, 1 (or equivalently, the interval corresponding to the kth largest spacing at the nth stage). This note studies the question for which classes of sequences k = k(n), will the interval Z,(,,(n) be...