On the stability of Jensen’s functional equation on groups

In this paper we establish the stability of Jensen’s functional equation on some classes of groups. We prove that Jensen equation is stable on noncommutative groups such as metabelian groups and T (2, K), where K is an arbitrary commutative field with characteristic different from two. We also prove that any group A can be embedded into some group G such that the Jensen functional equation is stable on G.     

The Brauer group of an affine double plane associated to a hyperelliptic curve

We study the Brauer group of an affine double plane π:X→𝔸² defined by an equation of the form z² = f in two separate cases. In the first case, f is a product of n linear forms in k[x,y] and X is birational to a ruled surface ?¹×C, where C is rational if n is odd and hyperelliptic if n is even. In the second case, f = y²?p(x) is the equation of an affine hyperelliptic curve. For π as well as the unramified part of π, we compute the groups of divisor classes, the Brauer groups, the relative Brauer groups, and all of the terms in the sequences of Galois cohomology.     

Uniqueness of solutions to Schrödinger equations on complex semi-simple Lie groups

In this note we study the time-dependent Schrödinger equation on complex semi-simple Lie groups. We show that if the initial data is a bi-invariant function that has sufficient decay and the solution has sufficient decay at another fixed value of time, then the solution has to be identically zero for all time. We also derive Strichartz and decay estimates for the Schrödinger equation. Our methods also extend to the wave equation. On the Heisenberg group we show that the failure to obtain a parametrix for our Schrödinger equation is related to the fact that geodesics project to circles on the contact plane at the identity.     

A Note on the Stability of Geodesics on Diffeomorphism Groups with One-side Invariant Metric

In this note,we consider the stability of geodesics on volume-preserving diffeomorphism groups with one-side invariant metric.We showed that for non-Beltrami fields on a three-dimensional compact manifold,there does not exist Eulerian stable flow which is Lagrangian exponential unstable.We noticed that a stationary flow corresponding to the KdV equation can be Eulerian stable while the corresponding motion of the fluid is at most exponentially unstable.     

Complete symmetry group and nonlocal symmetries for some two-dimensional evolution equations

The complete symmetry group of an 1+1 evolution equation of maximal symmetry has been demonstrated to be represented by the six-dimensional Lie algebra of point symmetries sl(2,R)s_⊕W, where W is the three-dimensional Heisenberg-Weyl algebra. We construct a complete symmetry group of a 1+2 evolution equation ut=(Fy(u)ux) for some functions F using the point symmetries admitted by the equation. The 1+2 equation is not completely specifiable by point symmetries alone for some specific functions F. We make use of Ansätze already reported by Myeni and Leach [S.M. Myeni, P.G.L. Leach, Nonlocal symmetries and complete symmetry groups of evolution equations, J. Nonlinear Math. Phys. 13 (2006) 377-392] which provide a route to the determination of the required generic nonlocal symmetries necessary to supplement the point symmetries for the complete specification of these 1+2 evolution equations. Further we find that taking some suitable linear combination of Lie point symmetries helps to optimise the procedure of specifying the equation. A general result concerning the number of symmetries required to form a complete symmetry group of evolution is presented in the Conclusion.     

Torsion-free constructive nilpotent <Emphasis Type="Italic">R</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-groups

We consider a torsion-free nilpotent R _p -group, the p-rank of whose quotient by the commutant is equal to 1 and either the rank of the center by the commutant is infinite or the rank of the group by the commutant is finite. We prove that the group is constructivizable if and only if it is isomorphic to the central extension of some divisible torsion-free constructive abelian group by some torsion-free constructive abelian R _p -group with a computably enumerable basis and a computable system of commutators. We obtain similar criteria for groups of that type as well as divisible groups to be positively defined. We also obtain sufficient conditions for the constructivizability of positively defined groups.     

自同构群阶为4p1p2···pn 的有限群

设G 是有限群, m 是已知正整数, 求解方程|Aut(G)| = m 中的有限群G 是一个难度很大的问题. 此课题的系统研究始于上世纪70 年代末. 后来陆续有包括本文作者在内的研究者对于m 是一些特殊的数时方程的求解问题进行了研究, 目前已经取得了一系列的结果. 本文在过去研究的基础上研究|Aut(G)| = 4p₁p₂···p_n 的求解问题, 并求出了全部有限群, 推广了杜妮和李世荣的结果.     

线性方程组解的有关问题

首先讨论了两个齐次线性方程组有非零公共解的充分必要条件并给出了非零公共解的一般形式,然后讨论了两个线性方程组同解的一个充分必要条件和非齐次线性方程组的线性无关解向量的个数以及非齐次线性方程组通解的表达式,最后证明了非齐次线性方程组有解的一个充分必要条件.     

The two-dimensional eikonal equation

We study the two-dimensional eikonal equation ψ _x ² + ψ _y ² = 1/v ²(x, y). We carry out the group analysis of the equation, establish a connection between the group properties and geometric characteristics of the Riemannian space with the metric ds ² = [dx ² + dy ²]/v ²(x, y). We select the most important classes of equations and derive some conditions for reducibility of a given equation to an equation of one of those classes. We find a condition for two equations to be equivalent (the theorem of seven invariants). For the equations corresponding to Riemannian spaces of constant curvature, we obtain explicit formulas for the solutions describing the wave front for a point source and also the ray equations.     

Classes of abelian groups related to sequential convergence

We study classes of abelian groups related to sequential com¬pactness and its generalizations (completeness, coarseness and sequential pre-compactness) in convergence groups. In particular, we describe the algebraic structure of the abelian groups on which every coarse convergence is complete and we prove that: i) every abelian group admits a sequentially precompact convergence; ii) every algebraically compact abelian group admits a sequen¬tially compact convergence.