Factoriality of Certain Threefolds Complete Intersection in P 5 with Ordinary Double Points

Abstract

Let V ? P ⁵ be a reduced and irreducible threefold of degree s, complete intersection on a smooth hypersurface of degree t, with s > t ² ? t. In this paper, we prove that if the singular locus of V consists of δ < 3s/8t ordinary double points, then any projective surface contained in V is a complete intersection on V. In particular, V is Q-factorial.     

On the Equations De.ning Curves in a Polynomial Algebra

Let A be a commutative Noetherian ring of dimension n (n ≥ 3). Let I be a local complete intersection ideal in A[T] of height n. Suppose I/I ² is free A[T]/I-module of rank n and (A[T]/I) is torsion in K ₀(A[T]). It is proved in this article that I is a set theoretic complete intersection ideal in A[T] if one of the following conditions holds: (1) n ≥ 5, odd; (2) n is even, and A contains the field of rational numbers; (3) n = 3, and A contains the field of rational numbers.     

Remarks on Naimark's duality

We present an extension of a version of Naimark's dilation theorem which states that complete systems in a Hilbert space are projections of -linearly independent systems of elements of an ambient Hilbert space. This result is presented in the context of other known extensions of Naimark's theorem.

     

Curves on Generic Surfaces of High Degree Through a Complete Intersection in P 3

We consider general surfaces, S, of high degree containing a given complete intersection space curve, Y. We study integral curves in the subgroup of Pic(S) generated by Y and the plane section. We determine the cohomological invariants of these curves and classify the subcanonical ones. Then using these subcanonical curves we produce stable rank two vector bundles on P ³.     

论演化方程的结构

本文对演化方程进行了讨论，从而能够唯象地描述微观结构行为，如Ｍａｘｗｅｌｌ气体的部分可逆流动、恢复性、结构松弛以及从光散射和分子动力学得到的其它实验结果。讨论所得结果涉及Ｚａｒｅｍｂａ补充式（ａｎｓａｔｚ）的重新评价，可用来解决输运方程中具有显著非线性的问题。     

Remarks on the Linearization of Differential Equations

The Hamiltonian formalism and the theory of canonical transformationsare used in this paper, first of all, to show that, given anordinary non-linear differential equation it is always possiblein principle to find a variable transformation reducing it toa linear equation, or a system of linear equations. The proofgiven is not to be construed as a general practical method forfinding this transformation; it merely shows that such a transformationmust always exist. It is suggested that this may also hold true for partial differentialequations. The conjecture is made plausible, in two cases, bythe use of canonical transformation procedures for linearizingsimple non-linear partial differential equations—one beinga slight generalization of Burger's equation and the other anequation in three independent variables reminiscent of the Eulerequations for fluid flow.     

Some Remarks on Signs in Functional Equations

We discuss the computation of the sign in certain functional equations.     

关于齐次四元Monge-Ampère方程的一些注记北大核心CSCD

本文考虑了四元数空间Hn中齐次四元Monge-Ampère方程的狄利克雷问题解的正则性.首先,当区域是边界为C1,1的强拟凸域时,作者给出了解的Lipschitz估计.其次,考虑了四元MongeAmpère算子的收敛性.最后,讨论了齐次四元Monge-Ampère方程的粘性次解与F-次调和函数之间的关系.     

Remarks on the Notion of Order of Difference Equations

In the paper, the notion of order of a difference equation is introduced in such a way that this order is invariant with respect to the change of the independent variable. For the general case, a formula for the general solution of linear difference equation of k -th order is given. It is shown that, in contrast to differential equations, the dimension of the linear space of solutions of linear homogenous difference equation can be lowered if their domain of definition is restricted appropriately.     

Remarks on the Scattering Result of Nonlinear Schroedinger Equations

0 Introduction 　　In this paper, we give another proof of the scattering result of the following Cauchy problem in the energy space     

Remarks on the Global Existence for Incompressible Navier-Stokes Equations

In this article, the authors use the special structure of helicity for the threedimensional incompressible Navier-Stokes equations to construct a family of finite energysmooth solutions to the Navier-Stokes equations which critical norms can be arbitrarilylarge.     

Existence of Solutions to Nonlinear Schrödinger Equations Involving N-Laplacian and Potentials Vanishing at Infinity

We study the existence of solutions for the following class of nonlinear Schrödinger equations -Δ_Nu + V (x)u=K(x)f(u) in R^N where V and K are bounded and decaying potentials and the nonlinearity f(s) has exponential critical growth. The approaches used here are based on a version of the Trudinger-Moser inequality and a minimax theorem.     

Remarks on the Regularity Criteria of Three-Dimensional Navier-Stokes Equations in Margin Case

In the study of the regularity criteria for Leray weak solutions to threedimensional Navier-Stokes equations, two sufficient conditions such that the horizontal velocity u satisfies u∈L2（0,T;BMO（R3）） or u∈L^2/1＋r（0,T;B∞,∞（R3）） for 0 〈 r 〈 1 are considered.     

Remarks on the Regularity to 3-D Ideal Magnetohydrodynamic Equations

In this paper we are interested in the sufficient conditions which guarantee the regularityof solutions of 3-D ideal magnetohydrodynamic equations in the arbitrary time interval [0,T].Fivesufficient conditions are given.Our results are motivated by two main ideas:one is to control theaccumulation of vorticity alone;the other is to generalize the corresponding geometric conditions of3-D Euler equations to 3-D ideal magnetohydrodynamic equations.     

Remarks on the Regularity to 3-D Ideal Magnetohydrodynamic Equations

In this paper we are interested in the sufficient conditions which guarantee the regularity of solutions of 3-D ideal magnetohydrodynamic equations in the arbitrary time interval [0,T]. Five sufficient conditions are given. Our results are motivated by two main ideas: one is to control the accumulation of vorticity alone; the other is to generalize the corresponding geometric conditions of 3-D Euler equations to 3-D ideal magnetohydrodynamic equations.     

有关微分中值定理的一个推广

给出了微分中值定理的一个高次幂形式的推广结果.     

Two Remarks on Solutions of Gross-Pitaevskii Equations on Zhidkov Spaces

We consider the so-called Gross-Pitaevskii equations supplemented with non-standard boundary conditions. We prove two mathematical results concerned with the initial value problem for these equations in Zhidkov spaces.     

A note on complete intersections of height three

Let be a field of characteristic 0. If is a complete intersection generated by three homogeneous elements of degrees with , then the reduction of by a general linear form is minimally generated by three elements if and only if .