On existence of minimizers for the Hardy–Sobolev–Maz’ya inequality

We show existence of minimizers for the Hardy–Sobolev–Maz’ya inequality in when either m > 2, n≥ 1 or m = 1, n≥ 3. The authors expresses their gratitude to the faculties of mathematics departments at Technion - Haifa Institute of Technology, at the University of Crete and at the University of Cyprus for their hospitality. A.T. acknowledges partial support by the RTN European network Fronts–Singularities, HPRN-CT-2002-00274. K.T acknowledges support as a Lady Davis Visiting Professor at Technion and partial support from University of Crete, University of Cyprus and Swedish Research Council.     

The Gabriel–Roiter Measure for Right Pure Semisimple Rings

We show how the Gabriel–Roiter measure, introduced by Ringel in (Bull Sci Math 129:726–748, 2005 and Contemp Math 406:105–135, 2006), applies to indecomposable modules of finite length over right pure semisimple rings, and in particular to the study of the open problem whether any right pure semisimple ring is of finite representation type. Dedicated to the memory of Andrey Vladimirovich Roiter. Professor A. V. Roiter has died on 26 July 2006 in Riga, Latvia. He was born in 1937.     

Bifurcations of Fine 3-point-loop in Higher Dimensional Space

By using the linear independent fundamental solutions of the linear variational equation along the heteroclinic loop to establish a suitable local coordinate system in some small tubular neighborhood of the heteroclinic loop, the Poincaré map is constructed to study the bifurcation problems of a fine 3–point loop in higher dimensional space. Under some transversal conditions and the non–twisted condition, the existence, coexistence and incoexistence of 2–point–loop, 1–homoclinic orbit, simple 1–periodic orbit and 2–fold 1–periodic orbit, and the number of 1–periodic orbits are studied. Moreover, the bifurcation surfaces and existence regions are given. Lastly, the above bifurcation results are applied to a planar system and an inside stability criterion is obtained. This work is supported by the National Natural Science Foundation of China (10371040), the Shanghai Priority Academic Disciplines and the Scientific Research Foundation of Linyi Teacher’s University     

On dynamics of fluids in astrophysics

The object of this article is to show the existence of weak solutions to the Navier–Stokes–Fourier Poisson system on (in general) unbounded domains. The topic is a natural continuation of the author’s results on the existence of weak solutions to the problem on Lipschitz domains and to the Oxenius system on unbounded domains. Technique of the proof is based on the tools developed in a series of works by Feireisl (Oxford lecture series in mathematics and its applications, 26, Oxford University Press, Oxford) and others during the recent years. The weak solution’s sensitivity to a change of the domain is discussed as well.     

Bakry–Emery Curvature Operator and Ricci Flow

In this paper, we introduce some techniques of Bakry–Emery curvature operator to Ricci flow and prove the evolution equation for the Bakry–Emery scalar curvature. As its application, we can easily derive the Perelman’s entropy functional monotonicity formula. We also discuss some gradient estimates of Ricci curvature and L ²– estimates of scalar curvature.Project partially supported by Yumiao Fund of Putian University.     

A nonconvex dissipative system and its applications (I)

In order to study the uniformly translating solution of some non-linear evolution equations such as the complex Ginzburg–Landau equation, this paper presents a qualitative analysis to a Duffing–van der Pol non-linear oscillator. Monotonic property of the bounded exact solution is established based on the construction of a convex domain. Under certain parametric choices, one first integral to the Duffing–van der Pol non-linear system is obtained by using the Lie symmetry analysis, which constitutes one of the bases for further work of obtaining uniformly translating solutions of the complex Ginzburg–Landau equation. Dedicated to Professor G. Strang on the occasion of his 70th birthday     

Specht modules and semisimplicity criteria for Brauer and Birman–Murakami–Wenzl algebras

A construction of bases for cell modules of the Birman–Murakami–Wenzl (or B–M–W) algebra B _n (q,r) by lifting bases for cell modules of B _n−1(q,r) is given. By iterating this procedure, we produce cellular bases for B–M–W algebras on which a large Abelian subalgebra, generated by elements which generalise the Jucys–Murphy elements from the representation theory of the Iwahori–Hecke algebra of the symmetric group, acts triangularly. The triangular action of this Abelian subalgebra is used to provide explicit criteria, in terms of the defining parameters q and r, for B–M–W algebras to be semisimple. The aforementioned constructions provide generalisations, to the algebras under consideration here, of certain results from the Specht module theory of the Iwahori–Hecke algebra of the symmetric group. Research supported by Japan Society for Promotion of Science.     

Decompositions of Besov–Morrey spaces and Triebel–Lizorkin–Morrey spaces

The aim of this paper is to define the Besov–Morrey spaces and the Triebel– Lizorkin–Morrey spaces and to present a decomposition of functions belonging to these spaces. Our results contain an answer to the conjecture proposed by Mazzucato. The first author is supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists. The second author is supported by Fūjyukai foundation and the 21st century COE program at Graduate School of Mathematical Sciences, the University of Tokyo.     

A construction of multiscale bases for Petrov–Galerkin methods for integral equations

In this paper, a construction of multiscale bases for Petrov–Galerkin methods for Fredholm integral equations of the second kind is proposed. The properties of multiscale bases are presented including additional order of vanishing moments, compact supports and stability. Communicated by A. Zhou Dedicated to Professor Charles A. Micchelli on the occasion of his sixtieth birthday with friendship and esteem Mathematics subject classifications (2000) 41A10, 65R20, 65D15. Min Huang: Supported in part by Professor Yuesheng Xu's support under the program of “One Hundred Distinguished Young Scientists” of the Chinese Academy of Sciences and by the Graduate Innovation Foundation of the Chinese Academy of Sciences.     

α-Transience and α-Recurrence of Right Processes

In this article, we mainly discuss some potential theory in the framework of right Markov processes. We introduce the concept of α-excessive function, α-recurrence and α-transience for right processes with α ≤ 0, and give a thorough investigation.