A minimal design of order 11 on the 3-sphere

We prove that the 120 vertices of the regular four-dimensional polyhedron with the Schläfli symbol {3, 3, 5} form a minimal spheric design of order 11 onS ³. It is shown that the system of equal charges at these vertices provides the minimal potential energy among all systems of 120 equal charges onS ³.     

On finiteness of multiplication modules

Our main aim in this note, is a further generalization of a result due to D. D. Anderson, i.e., it is shown that if R is a commutative ring, and M a multiplication R-module, such that every prime ideal minimal over Ann (M) is finitely generated, then M contains only a finite number of minimal prime submodules. This immediately yields that if P is a projective ideal of R, such that every prime ideal minimal over Ann (P) is finitely generated, then P is finitely generated. Furthermore, it is established that if M is a multiplication R-module in which every minimal prime submodule is finitely generated, then R contains only a finite number of prime ideals minimal over Ann (M).      

Minimal configuration bicyclic graphs

The nullity η(G) of a graph G is the multiplicity of zero as an eigenvalue of the adjacency matrix of G. If η(G)?=?1, then the core of G is the subgraph induced by the vertices associated with the nonzero entries of the kernel eigenvector. The set of vertices which are not in the core is the periphery of G. A graph G with nullity one is minimal configuration if no two vertices in the periphery are adjacent and deletion of any vertex in the periphery increases the nullity. An ∞-graph ∞(p,?l,?q) is a graph obtained by joining two vertex-disjoint cycles C _p and C _q by a path of length l?≥?0. Let ?* be the class of bicyclic graphs with an ∞-graph as an induced subgraph. In this article, we characterize the graphs in ?* which are minimal configurations.     

一类具有任意次数的参数多项式极小曲面

极小曲面是在几何造型设计中有着重要应用的一类特殊曲面.本文从几何造型的视角提出一类次数任意的参数多项式极小曲面.所提出的极小曲面具有显式的参数表示,并具有一些重要的几何性质,如对称性、包含直线和自交性.根据几何性质,本文将该参数多项式极小曲面划分为4类:n=4k+1,n=4k+2,n=4k+3,n=4+4,其中n是极小曲面的次数,k是正整数.本文给出与之相对应的共轭极小曲面的显式参数形式,并实现其等距变形.     

A constructive proof by elementary transformations of Roth's removal theorems in the theory of matrix equations

W. E. Roth gave necessary and sufficient conditions for the existence of solution(s) of certain types of linear matrix equations. Proofs were based on invariant factors and were long and complicated. Other shorter but non-constructive proofs have since been provided by later authors. We present here very brief constructive proofs based on the simplest of mathematical techniques, namely row- and column-reduction of a matrix.     

Complete nonorientable minimal surfaces in a ball of

The existence of complete minimal surfaces in a ball was proved by N. Nadirashvili in 1996. However, the construction of such surfaces with nontrivial topology remained open. In 2002, the authors showed examples of complete orientable minimal surfaces with arbitrary genus and one end. In this paper we construct complete bounded nonorientable minimal surfaces in with arbitrary finite topology. The method we present here can also be used to construct orientable complete minimal surfaces with arbitrary genus and number of ends.

     

The optimal upper bound of the number of generalized Euler configurations

Consider a generalized 3-body problem. The attraction force between any two bodies is proportional to the two masses and the b-th power of the mutual distance. Albouy and Fu have obtained the optimal upper bound of the number of generalized Euler configurations for the cases b 1 and b = 2, 3. This paper obtains the optimal upper bound for the remaining real values of b in a systematic way.     

A symmetry of sphere map implies its chaos*

A well-known example, given by Shub, shows that for any |d| ≥ 2 there is a self-map of the sphere Sⁿ, n ≥ 2, of degree d for which the set of non-wandering points consists of two points. It is natural to ask which additional assumptions guarantee an infinite number of periodic points of such a map. In this paper we show that if a continuous map f : Sⁿ → Sⁿ commutes with a free homeomorphism g : Sⁿ → Sⁿ of a finite order, then f has infinitely many minimal periods, and consequently infinitely many periodic points. In other words the assumption of the symmetry of f originates a kind of chaos. We also give an estimate of the number of periodic points. *Research supported by KBN grant nr 2 P03A 045 22.     

Coloring of trees with minimum sum of colors

The chromatic sum of a graph is the smallest sum of colors among all proper colorings with natural numbers. The strength is the minimum number of colors needed to achieve the chromatic sum. We construct for each positive integer k a tree with strength k that has maximum degree only 2k − 2. The result is best possible. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 354–358, 1999     

Characterization of standard embeddings between complex Grassmannians by means of varieties of minimal rational tangents

In 1993,Tsal proved that a proper holomorphic mapping f:Ω→Ω' from an irreducible bounded symmetric domainΩof rank≥2 into a bounded symmetric domainΩ' is necessarily totally geodesic provided that r':=rank(Ω')≤rank(Ω):= r,proving a conjecture of the author's motivated by Hermitian metric rigidity.As a first step in the proof,Tsai showed that df preserves almost everywhere the set of tangent vectors of rank 1.Identifying bounded symmetric domains as open subsets of their compact duals by means of the Borel embedding,this means that the germ of f at a general point preserves the varieties of minimal rational tangents(VMRTs). In another completely different direction Hwang-Mok established with very few exceptions the Cartan- Fubini extension priniciple for germs of local biholomorphisms between Fano manifolds of Picard num- ber 1,showing that the germ of map extends to a global biholomorphism provided that it preserves VMRTs.We propose to isolate the problem of characterization of special holomorphic embeddings between Fano manifolds of Picard number 1,especially in the case of classical manifolds such as ratio- nal homogeneous spaces of Picard number 1,by a non-equidimensional analogue of the Cartan-Fubini extension principle.As an illustration we show along this line that standard embeddings between com- plex Grassmann manifolds of rank≤2 can be characterized by the VMRT-preserving property and a non-degeneracy condition,giving a new proof of a result of Neretin's which on the one hand paves the way for far-reaching generalizations to the context of rational homogeneous spaces and more generally Fano manifolds of Picard number 1,on the other hand should be applicable to the study of proper holomorphic mappings between bounded domains carrying some form of geometric structures.     

Minimal Surfaces Obtained by Ribaucour Transformations

We consider Ribaucour transformations between minimal surfaces and we relate such transformations to generating planar embedded ends. Applying Ribaucour transformations to Enneper's surface and to the catenoid, we obtain new families of complete, minimal surfaces, of genus zero, immersed in R ³, with infinitely many embedded planar ends or with any finite number of such ends. Moreover, each surface has one or two nonplanar ends. A particular family is obtained from the catenoid, for each pair (n,m), nm, such that n m0 is an irreducible rational number. For any such pair, we get a 1-parameter family of finite total curvature, complete minimal surfaces with n+2 ends, n embedded planar ends and two nonplanar ends of geometric index m, whose total curvature is –4(n+m). The analytic interpretation of a Ribaucour transformation as a Bäcklund type transformation and a superposition formula for the nonlinear differential equation = e^-2 is included.     

Further analysis of minimum residual iterations

The convergence behaviour of a number of algorithms based on minimizing residual norms over Krylov subspaces is not well understood. Residual or error bounds currently available are either too loose or depend on unknown constants that can be very large. In this paper we take another look at traditional as well as alternative ways of obtaining upper bounds on residual norms. In particular, we derive inequalities that utilize Chebyshev polynomials and compare them with standard inequalities. Copyright © 2000 John Wiley & Sons, Ltd.     

网络最小割计算的一点注记

给出了具有单一源和单一汇网络割的数目,完善了一些文献中的不完整之处.     

Subdifferential formulas for a class of non-convex infimal convolutions

In this paper, we provide a number of subdifferential formulas for a class of non-convex infimal convolutions in normed spaces. The formulas obtained unify several results on subdifferentials of the distance function and the minimal time function. In particular, we generalize the results obtained recently by Zhang et al.     

Parsimonious Least Norm Approximation

A theoretically justifiable fast finite successive linear approximation algorithm is proposed for obtaining a parsimonious solutionto a corrupted linear system Ax=b+p, where the corruption p is due to noise or error in measurement. The proposedlinear-programming-based algorithm finds a solution x by parametrically minimizing the number of nonzeroelements in x and the error Ax-b-p₁.Numerical tests on a signal-processing-based exampleindicate that the proposed method is comparable to a method that parametrically minimizesthe 1-norm of the solution x and the error Ax-b-p₁, and that both methods are superior, byorders of magnitude, to solutions obtained by least squares as well by combinatorially choosing an optimal solution with a specific number of nonzero elements.     

Realization of Configurations and the Löwner Ellipsoid

It is proved that any subset of an (m-1)-dimensional sphere of volume larger than l(m+ 1) of the volume of the entire sphere contains l+ 1 points forming a regular l-dimensional simplex. As a corollary, it is obtained that, if the exterior of a given m-dimensional filled ellipsoid contains no more than the 1/(m+ 1) fraction of some sphere, then the volume of the ellipsoid is no less than the volume of the corresponding ball. The existence of a pair of points a given spherical distance apart in a set of positive measure is examined.     

On p-nilpotency and minimal subgroups of finite groups

We call a subgroup H of a finite group G c-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K ≤ core(H). In this paper it is proved that a finite group G is p-nilpotent if G is S4-free and every minimal subgroup of P n GN is c-supplemented in NG(P), and when p = 2 P is quaternion-free, where p is the smallest prime number dividing the order of G, P a Sylow p-subgroup of G. As some applications of this result, some known results are generalized.     

单位球面中极小子流形的C∞紧性

本文研究了单位球面中极小子流形的C∞紧性,并得到两个紧性定理.作为应用,我们证明了存在正数δ(n),如果单位球面中极小子流形的第2基本形式的长度平方小于;2/3+δ(n),则它必须是全测地的或微分同胚于Veronese曲面.     

可图序列偏序集的极小元

ｎ项非增非负整数序列是可图的,若是某个阶简单图的度序列．所有项和为２ｍ、迹为ｆ的ｎ项可图序列的集合Ｇ_ｎ,ｍ,ｆ在优超关系下是一个偏序集．本文刻划了偏序集Ｇ_ｎ,ｍ,ｆ的极小元,并确定各种可图序列偏序集中极小元的个数．