Decay rate of solutions of Navier-Stokes equations in 3-dimensional half space

In this article, we derive the L² spatial decay rate of a weak solution of the incompressible flow in the half space. When the half space (or some other fluid region with boundary) is concerned, pressure estimate is main obstacle since we do not have enough information of the pressure on the boundary. In this paper, we give an idea which does not require any pressure information on the boundary, and we hope that our idea could be applied to other problems such as exterior domain problem.     

Binary clustering

In many clustering systems (hierarchies, pyramids and more generally weak hierarchies) clusters are generated by two elements only.This paper is devoted to such clustering systems (called binary clustering systems). It provides some basic properties, links with (closed) weak hierarchies and some qualitative versions of bijection theorems that occur in Numerical Taxonomy. Moreover, a way to associate a binary clustering system to every clustering system is discussed.Finally, introducing the notion of weak ultrametrics, a bijection between indexed weak hierarchies and weak ultrametrics is obtained (the standard theorem involves closed weak hierarchies and quasi-ultrametrics).     

Realizing irreducible semigroups and real algebras of compact operators

Let B(X) be the algebra of bounded operators on a complex Banach space X. Viewing B(X) as an algebra over R, we study the structure of those irreducible subalgebras which contain nonzero compact operators. In particular, irreducible algebras of trace-class operators with real trace are characterized. This yields an extension of Brauer-type results on matrices to operators in infinite dimensions, answering the question: is an irreducible semigroup of compact operators with real spectra realizable, i.e., simultaneously similar to a semigroup whose matrices are real?     

Optimality conditions for weak and firm efficiency in set-valued optimization

This paper concerns the study of weak and firm local efficiency in constrained mathematical problems governed by set-valued mappings. We derive optimality conditions by means of the Bouligand derivative and by means of the Mordukhovich coderivative as well.     

Well-posedness of the ferrimagnetic equations

In this paper, we show the existence of global weak solutions of the ferrimagnetic equations on compact Riemannian manifold using the penalty method. We also show the uniqueness of the solution and its well-posedness by the energy estimates method in lower dimensions. In particular, when the space dimension is one, we can prove that the problem is globally well-posed.     

Conservation Laws and Potential Symmetries of Linear Parabolic Equations

We carry out an extensive investigation of conservation laws and potential symmetries for the class of linear (1+1)-dimensional second-order parabolic equations. The group classification of this class is revised by employing admissible transformations, the notion of normalized classes of differential equations and the adjoint variational principle. All possible potential conservation laws are described completely. They are in fact exhausted by local conservation laws. For any equation from the above class the characteristic space of local conservation laws is isomorphic to the solution set of the adjoint equation. Effective criteria for the existence of potential symmetries are proposed. Their proofs involve a rather intricate interplay between different representations of potential systems, the notion of a potential equation associated with a tuple of characteristics, prolongation of the equivalence group to the whole potential frame and application of multiple dual Darboux transformations. Based on the tools developed, a preliminary analysis of generalized potential symmetries is carried out and then applied to substantiate our construction of potential systems. The simplest potential symmetries of the linear heat equation, which are associated with single conservation laws, are classified with respect to its point symmetry group. Equations possessing infinite series of potential symmetry algebras are studied in detail.     

半单弱Hopf代数的作用

令H是半单弱Hopf代数, A是左H-模代数.我们证明了正则A-模的内射维数, A#H-模A的内射维数和正则A#H-模的内射维数三者是相等的. 而且,利用H在A上的不动点代数我们给出了A是Gorenstein代数的充要条件.     

Multidimensional Raman spectroscopic signature of sweat and its potential application to forensic body fluid identification

This proof-of-concept study demonstrated the potential of Raman microspectroscopy for nondestructive identification of traces of sweat for forensic purposes. Advanced statistical analysis of Raman spectra revealed that dry sweat was intrinsically heterogeneous, and its biochemical composition varies significantly with the donor. As a result, no single Raman spectrum could adequately represent sweat traces. Instead, a multidimensional spectroscopic signature of sweat was built that allowed for the presentation of any single experimental spectrum as a linear combination of two fluorescent backgrounds and three Raman spectral components dominated by the contribution from lactate, lactic acid, urea and single amino acids.