Bounds for the fundamental solutions of elliptic and parabolic equations

It is shown that any elloptic or parabolic operator in nondivergence form with measurable coefficients has a global fundamental solution verifying certain pointwise bounds.     

Studies on defect formation in femtosecond laser-irradiated PMMA and PDMS

We report here our results on the spectroscopic and elemental analysis of femtosecond (fs) laser-modified regions in polymers of polymethylmethacrylate (PMMA) and polydimethylsiloxane (PDMS) in the context of defect formation and emission in the visible region. Different physical and chemical models are used to explain the changes in modified regions. We found that the emission intensity, recorded from the fs-modified regions of polymers, decreased over time to a constant value. We also demonstrate that these materials are suitable for the preparation of the microstructures en route for light guiding applications. The fs laser-irradiated regions exhibited paramagnetic behavior as was confirmed from electron spin resonance studies through the formation of peroxide-type free radicals. Raman mapping was performed in the modified regions which consisted of defects and found that the modulations in intensity are predominant in the central portion of the structure compared to edges. Elemental analysis has been performed in the modified regions using field emission scanning electron microscope instrument and energy-dispersive X-ray absorption spectroscopy to estimate the percentage contents of individual elements which resulted in defect formation such as paramagnetic and optical centers.     

Strong Convergence Theorems for the Generalized Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in the Intermediate Sense in Hilbert Spaces

Abstract

The aim of this paper is to introduce the generalized viscosity implicit rules of one asymptotically nonexpansive mapping in the intermediate sense in Hilbert spaces. We obtain some strong convergence theorems under certain assumptions imposed on the parameters. We also give a numerical example to support our main results. The results obtained in this paper improve and extend many recent ones in this culture.     

Multi-objective optimal design of feedback controls for dynamical systems with hybrid simple cell mapping algorithm

This paper presents a study of multi-objective optimal design of full state feedback controls. The goal of the design is to minimize several conflicting performance objective functions at the same time. The simple cell mapping method with a hybrid algorithm is used to find the multi-objective optimal design solutions. The multi-objective optimal design comes in a set of gains representing various compromises of the control system. Examples of regulation and tracking controls are presented to validate the control design.     

A screw dislocation near a non-elliptical piezoelectric inhomogeneity with internal uniform electroelastic field

Conformal mapping and analytic continuation are employed to prove the existence of an internal uniform electroelastic field inside a non-elliptical piezoelectric inhomogeneity interacting with a screw dislocation. We focus specifically on the case when the piezoelectric matrix surrounding the inhomogeneity is subjected to uniform remote anti-plane mechanical and in-plane electrical loading and a constraint is imposed between the remote loading and the screw dislocation. The constraint can be expressed in a relatively simple decoupled form by utilizing orthogonality relationships between two corresponding eigenvectors. The internal uniform electroelastic field is found to be independent of the presence of the screw dislocation; moreover, it can be expressed in decoupled form.     

Generalized matching theorems for closed coverings of convex sets

In this paper we use fixed point and coincidence theorems due to Park [8] to give matching theorems concerning closed coverings of nonempty convex sets in a real topological vector space. Our new results extend previously given ones due to Ky Fan [2], [3], Shih [10], Shih and Tan [11], and Park [7].     

Dynamic stability of a class of fractional-order nonlinear systems via fixed point theory

In this paper, the problem of the uniform stability for a class of fuzzy fractional-order genetic regulatory networks with random discrete delays, distributed delays, and parameter uncertainties is studied. Although there is a portion of literature on using fixed point theorems to study the stability of fractional neural networks, most of them required the fractional order to be in $\left(\frac{1}{2},1\right)$. However, the case of the fractional-order belonging to $(0,\frac{1}{2})$ has not been discussed. To solve it, this work proposes a novel idea of using fixed point theory to study the stability of fuzzy (0,1) order neural networks, the problem of the uniqueness of the solution of the considered genetic regulatory networks is resolved, and a novel sufficient condition to guarantee the uniform stability of above genetic regulatory networks is also derived. Eventually, an example is given to demonstrate that the obtained result is effective.     

Characterizations of ( m,n )-Jordan Derivations and ( m,n )-Jordan Derivable Mappings on Some Algebras

Let R be a ring, M be a R-bimodule and m, n be two fixed nonnegative integers with m + n = 0. An additive mapping δ from R into M is called an(m, n)-Jordan derivation if(m +n)δ(A~2) = 2 mAδ(A) + 2nδ(A)A for every A in R. In this paper, we prove that every(m, n)-Jordan derivation with m = n from a C*-algebra into its Banach bimodule is zero. An additive mappingδ from R into M is called a(m, n)-Jordan derivable mapping at W in R if(m + n)δ(AB + BA) =2mδ(A)B + 2 mδ(B)A + 2 nAδ(B) + 2 nBδ(A) for each A and B in R with AB = BA = W. We prove that if M is a unital A-bimodule with a left(right) separating set generated algebraically by all idempotents in A, then every(m, n)-Jordan derivable mapping at zero from A into M is identical with zero. We also show that if A and B are two unital algebras, M is a faithful unital(A, B)-bimodule and U = [A M N B] is a generalized matrix algebra, then every(m, n)-Jordan derivable mapping at zero from U into itself is equal to zero.     

Fixed points of multivalued quasi-nonexpansive mappings using a faster iterative process

In this article, we prove some strong and weak convergence theorems for quasi-nonexpansive multivalued mappings in Banach spaces. The iterative process used is independent of Ishikawa iterative process and converges faster. Some examples are provided to validate our results. Our results extend and unify some results in the contemporary literature.     

若干高维可积和不可积模型的严格解

可积和不可积模型可以描述自然科学中的诸多现象, 寻找高维非线性模型的严格解已成为可积系统的一个重要研究内容. 结合达布变换法和多线性分离变量法, 可以得到多个(2+1)维非线性模型包含任意函数的严格解, 通过选取不同的任意函数, 构造这些非线性模型新的相互激发模式. 进一步推广了形变映射理论, 建立了变系数 场和sine-Gordon以及双sine-Gordon场的形变映射关系, 从而得到高维不可积模型包含任意函数的新严格解. 对任意函数的不同选择, 构造了sine-Gordon和双sine-Gordon可积模型丰富的局域解和周期解, 如多solitoff解及其周期波推广、周期形变的蛇形孤波解以及变模的拟周期解等.