A theory of concordance for non-spherical 3-knots

Consider a closed connected oriented 3-manifold embedded in the -sphere, which is called a -knot in this paper. For two such knots, we say that their Seifert forms are spin concordant, if they are algebraically concordant with respect to a diffeomorphism between the 3-manifolds which preserves their spin structures. Then we show that two simple fibered 3-knots are geometrically concordant if and only if they have spin concordant Seifert forms, provided that they have torsion free first homology groups. Some related results are also obtained.

     

Indexes and Special Discretization Methods for Linear partial Differential Algebraic Equations

Linear partial differential algebraic equations (PDAEs) of the form Au _t(t, x) + Bu _xx(t, x) + Cu(t, x) = f(t, x) are studied where at least one of the matrices A, B R ^n×n is singular. For these systems we introduce a uniform differential time index and a differential space index. We show that in contrast to problems with regular matrices A and B the initial conditions and/or boundary conditions for problems with singular matrices A and B have to fulfill certain consistency conditions. Furthermore, two numerical methods for solving PDAEs are considered. In two theorems it is shown that there is a strong dependence of the order of convergence on these indexes. We present examples for the calculation of the order of convergence and give results of numerical calculations for several aspects encountered in the numerical solution of PDAEs.     

一类非线性代数方程组的并行迭代算法

１．引言考虑非线性代数方程组这里,　　　　　　　　　　　　　　　　　　　　　　　　　　　　为连续的对角映射,二者的导函数均存在,但并不一定连续．这类非线性代数方程组具有丰富的实际背景．譬如,Ｓｔｅｆａｎ问题和许多弱非线性椭圆型偏微分方程,就可归结为（１．１）的数值求解问题．根据方程组（１．１）的特殊结构,并利用矩阵多重分裂思想,文ｔＺ］讨论了一类并行非线性Ｇａｕｓｓ－Ｓｅｉｄｅｌ型迭代算法．这类算法具有很好的数值性质和较高的并行效率·在此基础上,运用松弛加速技术,文［８］进一步研究了一类并行多分…     

Complex Zeros of Algebraic Polynomial with Non-Zero Mean Random Coefficients

In this paper we obtain a formula for the average density of the distribution of complex zeros of an algebraic polynomial with random coefficients. The coefficients are assumed independent identical normally distributed random variables with mean and variance ². The value of the average density for the case of =0 and ²=1 was obtained previously. Some limits of the distribution of the complex zeros are provided using the presented formula.     

Computation of generalized real radicals of polynomial ideals

For an ordered field (K,T) and an idealI of the polynomial ring , the construction of the generalized real radical ofI is investigated. When (K,T) satisfies some computational requirements, a method of computing is presented. Project supported by the National Natural Science Foundation of China (Grant No. 19661002) and the Climbing Project.     

Algebraic domain decomposition solver for linear elasticity

We generalize the overlapping Schwarz domain decomposition method to problems of linear elasticity. The convergence rate independent of the mesh size, coarse-space size, Korn's constant and essential boundary conditions is proved here. Abstract convergence bounds developed here can be used for an analysis of the method applied to singular perturbations of other elliptic problems.     

G. Martens′ Dimension Theorem for Curves of Odd Gonality

For a smooth projective irreducible algebraic curve C of odd gonality, the maximal possible dimension of the variety of special linear systems W _d ^r (C) is d-3r by a result of M. Coppens et al. Furthermore it is known that if the maximum dimension of W(C) for a curve C of odd gonality is attained then C is of very special type of curves by the recent progress made by G. Martens. The purpose of this paper is to chase an extension of the result of G. Martens; if dim W(C)=d-3r-1 for a curve C of odd gonality for some dg-4 and r1, then C must be either a smooth plane sextic, a pentagonal curve of bounded genus or a smooth plane octic.     

K1 of Twisted Rings of Polynomials

We prove that for an arbitrary endomorphism of a ring R the group K1(R[t]) splits into the direct sum of K1(R) and Ñil (r;). Moreover, for any such R and Ñil (R; ) is isomorphic to Ñil (R ; ) for some ring R with : R R – an isomorphism.     

On random algebraic polynomials

This paper provides asymptotic estimates for the expected number of real zeros and -level crossings of a random algebraic polynomial of the form , where are independent standard normal random variables and is a constant independent of . It is shown that these asymptotic estimates are much greater than those for algebraic polynomials of the form .

     

Unitarily Inequivalent Representations in Algebraic Quantum Theory

It has been maintained that the physical content of a model of a system is completely contained in the C∗-algebra of quasi-local observables that is associated with the system. The reason given for this is that the unitarily inequivalent representations of are physically equivalent. But, this view is dubious for at least two reasons. First, it is not clear why the physical content does not extend to the elements of the von Neumann algebras that are generated by representations of . It is shown here that although the unitarily inequivalent representations of are physically equivalent, the extended representations are not. Second, this view detracts from special global features of physical systems such as temperature and chemical potential by effectively relegating them to the status of fixed parameters. It is desirable to characterize such observables theoretically as elements of the algebra that is associated with a system rather than as parameters, and thereby give a uniform treatment to all observables. This can be accomplished by going to larger algebras. One such algebra is the universal enveloping von Neumann algebra, which is generated by the universal representation of ; another is the direct integral of factor representations that are associated with the set of values of the global features. Placing interpretive significance on the von Neumann algebras mentioned earlier sheds light on the significance of unitarily inequivalent representations of , and it serves to show the limitations of the notion of physical equivalence.