T＊-extension of Lie Sup ertriple Systems

In this article, we study the Lie supertriple system （LSTS） T over a field K admitting a nondegenerate invariant supersymmetric bilinear form （call such a Tmetrisable）. We give the definition of T＊ω-extension of an LSTS T , prove a necessary and sufficient condition for a metrised LSTS （T ,Ф） to be isometric to a T＊-extension of some LSTS, and determine when two T＊-extensions of an LSTS are ＂same＂, i.e., they are equivalent or isometrically equivalent.

An Invariance for （2＋1）-Extension of Burgers Equation and Formulae to Obtain Solutions of KP Equation

In this article, we study the (2+1)-extension of Burgers equation and the KPequation. At first, based on a known Baecklund transformation and corresponding Lax pair, aninvariance which depends on two arbitrary functions for (2+1)-extension of Burgers equation isworked out. Given a known solution and using the invariance, we can find solutions of the(2+1)-extension of Burgers equation repeatedly. Secondly, we put forward an invariance of Burgersequation which cannot be directly obtained by constraining the invariance of the (2+1)-extension ofBurgers equation. Furthermore, we reveal that the invariance for finding the solutions of Burgersequation can help us find the solutions of KP equation. At last, based on the invariance of Burgersequation, the corresponding recursion formulae for finding solutions of KP equation are digged out.As the application of our theory, some examples have been put forward in this article and somesolutions of the (2+1)-extension of Burgers equation, Burgers equation and KP equation are obtained.

Class numbers of cyclic 2-extensions and Gross conjecture over Q

The Gross conjecture over Q was first claimed by Aoki in 1991.However,the original proof contains too many mistakes and false claims to be considered as a serious proof.This paper is an attempt to find a sound proof of the Gross conjecture under the outline of Aoki.We reduce the conjecture to an elementary conjecture concerning the class numbers of cyclic 2-extensions of Q.

Formulas to Solve the （2＋1）—Dimensional System：The （2＋1）—Extension of Classical Boussinesq System

We deal with the (2 + 1)-extension of classical Boussinesq system,which can reduce to several meaningful(1 + 1)-dimensional systems.By studying its Lax pair,we put forward invariances of Lax pair at first,then a recursionformula depending on an arbitrary function is derived.At last,some solutions of the (2 + 1)-extension of classicalBoussinesq system are digged out by using the formula.

On Two Classes of Extended 3-Lie Algebras

In this paper, based on the existing research results, we obtain the unary extension 3-Lie algebras by one-dimensional extension of the known Lie algebra L. For two known 3-Lie algebras H, M, the (μ, ρ, β)-extension of H through M is given, and the necessary and sufficient conditions for the (μ, ρ, β)-extension algebra of H through M being 3-Lie algebra are obtained, and the structural characteristics and properties of these two kinds of extended 3-Lie algebras are given.

The Rank of the 2-Class Group of Some Fields with LargeDegree

Let n≥3 be an integer and d an odd square-free integer.We compute the rank of the 2-class group of some fields of the form L_(n,d)=Q(ζ_(2^(n)),√d)when all the prime divisors of d are congruent to±3(mod 8)or 9(mod 16).

A converse of Hormander's L^(2)-estimate and new positivity notions for vector bundles

We study conditions of Hormander's L^(2)-estimate and the Ohsawa-Takegoshi extension theorem.Introducing a twisted version of the Hormander-type condition,we show a converse of Hormander's L^(2)-estimate under some regularity assumptions on an n-dimensional domain.This result is a partial generalization of the one-dimensional result obtained by Berndtsson(1998).We also define new positivity notions for vector bundles with singular Hermitian metrics by using these conditions.We investigate these positivity notions and compare them with classical positivity notions.