Basis-free expressions for derivatives of a subclass of nonsymmetric isotropic tensor functions

The present paper generalizes the method for solving the derivatives of symmetric isotropic tensor-valued functions proposed by Dui and Chen （2004） to a subclass of nonsymmetric tensor functions satisfying the commutative condition. This subclass of tensor functions is more general than those investigated by the existing methods. In the case of three distinct eigenvalues, the commutativity makes it possible to introduce two scalar functions, which will be used to construct the general nonsymmetric tensor functions and their derivatives. In the cases of repeated eigenvalues, the results are acquired by taking limits.

Complex Maxwell’s Equations

Maxwell’s equations in electromagnetism can be categorized into three dis-tinct groups based on the electromagnetic source when employing quaterni-ons. Each group represents a self-contained system in which Maxwell’s equations are applied and validated concurrently, in contrast to the previous approach that did not account for this. It has been noted that the formulation of these Maxwell equations ultimately results in the formulation of Max-well’s equations utilizing the scalar function.

Continuity of Solution Mappings for Parametric Set Optimization Problems under Partial Order Relations

This paper mainly investigates the semicontinuity of solution mappings for set optimization problems under a partial order set relation instead of upper and lower set less order relations. To this end, we propose two types of monotonicity definition for the set-valued mapping introduced by two nonlinear scalarization functions which are presented by these partial order relations. Then, we give some sufficient conditions for the semicontinuity and closedness of solution mappings for parametric set optimization problems. The results presented in this paper are new and extend the main results given by some authors in the literature.

The Existence and Uniqueness of the Solution for Perio dical Boundary Value Problems of 2k th Order Differential Equations with Resonance

In this paper, a new set of sufficient conditions related to an initial value problem and global homeomorphism is obtained in discussing the existence and uniqueness of 2π-periodic solution for 2kth order differential equations with resonance. The key role is played by nonnegative auxiliary scalar coercive function. The result of this paper generalizes some existed theorems.

Derivatives on the isotropic tensor functions

The derivative of the isotropic tensor function plays an important part in continuum mechanics and computational mechanics, and also it is still an opening problem. By means of a scalar response function f ( Λ i , I 1 , I2) and solving a tensor equation, this problem is well studied. A compact explicit expression for the derivative of the isotropic tensor function is presented, which is valid for both distinct and repeated eigenvalue cases. Throughout the analysis, the formulation holds for general isotropic tensor functions without need to solve eigenvector problems or determine coefficients. On the theoretical side, a very simple solution of a tensor equation is obtained. As an application to continuum mechanics, a base-free expression for the Hill’s strain rate is given, which is more compact than the existent results. Finally, with an example we compute the derivative of an exponent tensor function. And the efficiency of the present formulations is demonstrated.

Approximate Weak Minimal Solutions of Set-Valued Optimization Problems

This paper deals with approximate weak minimal solutions of set-valued optimization problems under vector and set optimality criteria.The relationships between various concepts of approximate weak minimal solutions are investigated.Some topological properties and existence theorems of these solutions are given.It is shown that for set-valued optimization problems with upper(outer)cone-semicontinuous objective values or closed objective maps the approximate weak minimal and strictly approximate lower weak minimal solution sets are closed.By using the polar cone and two scalarization processes,some necessary and sufficient optimality conditions in the sense of vector and set criteria are provided.

On Ekeland's Variational Principle for Set-Valued Mappings

In this paper, we derive a general vector Ekeland variational principle for set-valued mappings, which has a dosed relation to εk^0 -efficient points of set-valued optimization problems. The main result presented in this paper is a generalization of the corresponding result in [3].