A FUNDAMENTAL SOLUTION FOR THE LAPLACE OPERATOR ON THE QUATERNIONIC HEISENBERG GROUP

In this paper, the author studies the Laplace operator on the quaternionic Heisenberg group, construct a fundamental solution for it and use this solution to prove the Lp-boundedness and the weak (1-1) boundedness of certain singular convolution operators on the quaternionic Heisenberg group.     

鞅方法在一类破产问题中的应用

本文从鞅条件出发 ,推导出了总理赔过程分别为复合 Poisson过程与复合二项过程 ,利率强度波动为带跳的 Poisson过程情形下的调节方程 ,并由此得到了一些有趣的结果。     

Elliptic and parabolic problems in unbounded domains

We consider elliptic and parabolic problems in unbounded domains. We give general existence and regularity results in Besov spaces and semi‐explicit representation formulas via operator‐valued fundamental solutions which turn out to be a powerful tool to derive a series of qualitative results about the solutions. We give a sample of possible applications including asymptotic behavior in the large, singular perturbations, exact boundary conditions on artificial boundaries and validity of maximum principles. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Survey on Explicit Representation Formulae for Fundamental Solutions of Linear Partial Differential Operators

In the present article, we aim at treating the existence of fundamental solutions of linear partial differential operators with constant coefficients from the viewpoint of setting up explicit formulae yielding fundamental solutions.     

Infinite-dimensional linear groups with restrictions on subgroups that are not soluble <Emphasis Type="Italic">A</Emphasis><Subscript>3</Subscript>-groups

We are concerned with infinite-dimensional locally soluble linear groups of infinite central dimension that are not soluble A₃-groups and all of whose proper subgroups, which are not soluble A₃-groups, have finite central dimension. The structure of groups in this class is described. The case of infinite-dimensional locally nilpotent linear groups satisfying the specified conditions is treated separately. A similar problem is solved for infinite-dimensional locally soluble linear groups of infinite fundamental dimension that are not soluble A₃-groups and all of whose proper subgroups, which are not soluble A₃-groups, have finite fundamental dimension. __________ Translated from Algebra i Logika, Vol. 46, No. 5, pp. 548–559, September–October, 2007.     

Martin Boundaries of Elliptic Skew Products, Semismall Perturbations, and Fundamental Solutions of Parabolic Equations

We consider positive solutions of elliptic partial differential equations on non-compact domains of Riemannian manifolds. We explicitly determine Martin boundaries and Martin kernels for a class of elliptic equations in skew product form by exploiting and developing perturbation theory for elliptic equations and short/long-time estimates for fundamental solutions of parabolic equations.     

Certain condition on the second fundamental form of CR submanifolds of maximal CR dimension of complex Euclidean space

We treat m-dimensional real submanifolds M of complex space forms ̿M when the maximal holomorphic tangent subspace is (m−1)-dimensional. On these manifolds there exists an almost contact structure F which is naturally induced from the ambient space and in this paper we study the condition h(FX,Y)−h(X,FY) = g(FX,Y)η, η∊ T⊥(M), on the structure F and on the second fundamental form h of these submanifolds. Especially when the ambient space ̿M is a complex Euclidean space, we obtain a complete classification of submanifolds M which satisfy these conditions.Mathematics Subject Classifications (2000): 53C15, 53C40, 53B20.     

Hyperbolic spaces are of strictly negative type

We study finite metric spaces with elements picked from, and distances consistent with, ambient Riemannian manifolds. The concepts of negative type and strictly negative type are reviewed, and the conjecture that hyperbolic spaces are of strictly negative type is settled, in the affirmative. The technique of the proof is subsequently applied to show that every compact manifold of negative type must have trivial fundamental group, and to obtain a necessary criterion for product manifolds to be of negative type.

     

Topologically knotted Lagrangians in simply connected four-manifolds

Vidussi was the first to construct knotted Lagrangian tori in simply connected four-dimensional manifolds. Fintushel and Stern introduced a second way to detect such knotting. This note demonstrates that similar examples may be distinguished by the fundamental group of the exterior.