Some Properties of Sums of Independent Random Sets

ＳｏｍｅＰｒｏｐｅｒｔｉｅｓｏｆＳｕｍｓｏｆＩｎｄｅｐｅｎｄｅｎｔＲａｎｄｏｍＳｅｔｓ＊）ＷａｎｇＲｏｎｇｍｉｎｇ（汪荣明）（ＤｅｐａｒｔｍｅｎｔｏｆＳｔａｔｉｓｔｉｃｓ，ＥａｓｔＣｈｉｎａＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ，Ｓｈａｎｇｈａｉ，２０００...     

The Strong Law of Large Numbers for ■-Mixing Random Variables

In this paper we present some results for the general strong laws of large numbers of ■-mixing random variables by a maximal inequality of Utev and Peligrad.These results extend and improve the related known works in the literature.     

The φ-Group for n Submodules Systems as a Generalization of the K-Theory

In this paper,the authors construct a φ-group for n submodules,which generalizes the classical K-theory and gives more information than the classical ones.This theory is related to the classification theory for indecomposable systems of n subspaces.     

The Hilbert Problem for the First Order Elliptic Complex Equation With Discontinuous Coefficients

本文研究一阶带间断系数的椭圆型复方程的Ｈｉｌｂｅｒｔ边值问题，在一定条件下给出了上述问题解的存在性、可解条件以及解的积分表示式．     

The Discontinuous Riemann–Hilbert Problem for Elliptic Complex Equations of First Order

In this article, we first transform the general uniformly elliptic systems of first order equations with certain conditions into the complex equations, and propose the discontinuous Riemann- Hilbert problem and its modified well-posedness for the complex equations. Then we give a priori estimates of solutions of the modified discontinuous Riemann-Hilbert problem for the complex equations and verify its solvability. Finally the solvability results of the original discontinuous Riemann-Hilbert boundary value problem can be derived. The discontinuous boundary value problem possesses many applications in mechanics and physics etc.     

The Third Order Estimate of Mixed Finite Element for the Navier－Stokes Problem

ＴｈｅＴｈｉｒｄＯｒｄｅｒＥｓｔｉｍａｔｅｏｆ Ｍｉｘｅｄ ＦｉｎｉｔｅＥｌｅｍｅｎｔ ｆｏｒｔｈｅ Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ ＰｒｏｂｌｅｍＬｕｏＺｈｅｎｄｏｎｇ（罗振东）（ＵｎｉｖｅｒｓｉｔｙｏｆＳｃｉｅｎｃｅａｎｄＴｅｃｈｎｏｌｏｇｙｏｆＣｈｉｎ?..     

On The Order of Weyl—Heisenburg Wavelet Approximation

The order of a concrete Weyl-Heisenburing wavelet approximation is given.This order isshown to be optimal.     

Exact Controllability for the Fourth Order Schr¨odinger Equation

The boundary controllability of the fourth order Schr(o)dinger equation in a bounded domain is studied.By means of an L2-Neumann boundary control,the authors prove that the solution is exactly controll...     

The Order of Hypersubstitutions of Type （2, 2）

Hypersubstitutions are mappings which map operation symbols to terms of the corresponding arities. They were introduced as a way of making precise the concept of a hyperidentity and generalizations to M-hyperidentities. A variety in which every identity is satisfied as a hyperidentity is called solid. If every identity is an M-hyperidentity for a subset M of the set of all hypersubstitutions, the variety is called M-solid. There is a Galois connection between monoids of hypersubstitutions and sublattices of the lattice of all varieties of algebras of a given type. Therefore, it is interesting and useful to know how semigroup or monoid properties of monoids of hypersubstitutions transfer under this Galois connection to properties of the corresponding lattices of M-solid varieties. In this paper, we study the order of each hypersubstitution of type （2, 2）, i.e., the order of the cyclic subsemigroup generated by that hypersubstitution of the monoid of all hypersubstitutions of type （2, 2）. The main result is that the order is 1, 2, 3, 4 or infinite.     

On the Weak＊ Drop Property for Polar of Closed Bounded Convex Sets

We define and study the weak* drop property for the polar of a closed bounded convex set in a Banach space which is both a generalization of the weak* drop property for dual norm in a Banach space and a characterization of the sub-differentialmapping x→эp(x) from S(X) into 2^S(X*) that is norm upper semi-continuous and norm compact-valued.     

Splittability for Partially Ordered Sets

A topological space X is said to be splittable over a class P of spaces if for every AX there exists continuous f:XYP such that f(A)f(XA) is empty. A class P of topological spaces is said to be a splittability class if the spaces splittable over P are precisely the members of P. We extend the notion of splittability to partially ordered sets and consider splittability over some elementary posets. We identify precisely which subsets of a poset can be split along over an n-point chain. Using these results it is shown that the union of two splittability classes need not be a splittability class and a necessary condition for P to be a splittability class is given.     

Affine Isomorphism for Partially Ordered Sets

Let A and B be the adjacency matrices of graphs G₁ and G₂ (or the strict zeta matrices of posets P₁ and P₂). Associated with A and B is a particular affine space of matrices, denoted _A,B, such that G₁ is isomorphic to G₂ (resp., P₁ is isomorphic to P₂) if and only if there is a 0-1 matrix in _A,B. Solving this integer programming problem is (notoriously) of unknown complexity, and researchers have considered its relaxation; if there is a nonnegative member of _A,B, then one says that G₁ is fractionally isomorphic to G₂ (resp., P₁ is fractionally isomorphic to P₂.) Several combinatorial characterizations of fractional isomorphism for graphs are known.In this paper we note that fractional isomorphism is not an equivalence relation for posets and introduce a further relaxation by defining P₁ to be affinely isomorphic to P₂ if _A,B is nonempty. (Asking whether _A,B is nonempty is a natural first question preceding the question of whether or not _A,B has a binary member, i.e., whether the graphs or posets are isomorphic.) We prove that affine isomorphism is indeed an equivalence relation on posets and that two posets are affinely ismorphic if and only if the f-vectors of their order complexes are the same. One consequence of this is a proof of an affine version of the Poset Reconstruction Conjecture.     

Path-Connected Partially Ordered Sets

The graph of a partially ordered set (X, ?) has X as its set of vertices and (x,y) is an edge if and only if x covers y or y covers x. The poset is path-connected if its graph is connected. Two integer-valued metrics, distance and fence, are defined for path-connected posets. Together the values of these metrics determine a path-connected poset to within isomorphism and duality. The result holds for path-connected preordered sets where distance and fence are pseudometrics. The result fails for non-path-connected posets.     

P-Faithful Partially Ordered Sets

We prove a theorem that describes P-faithful partially ordered sets.     

Homomorphism-Homogeneous Partially Ordered Sets

A structure is called homogeneous if every isomorphism between finite substructures of the structure extends to an automorphism of the structure. Recently, P. J. Cameron and J. Nešetřil introduced a relaxed version of homogeneity: we say that a structure is homomorphism-homogeneous if every homomorphism between finite substructures of the structure extends to an endomorphism of the structure. In this paper we characterize homomorphism-homogeneous partially ordered sets (where a homomorphism between partially ordered sets A and B is a mapping f : A →B satisfying ). We show that there are five types of homomorphism-homogeneous partially ordered sets: partially ordered sets whose connected components are chains; trees; dual trees; partially ordered sets which split into a tree and a dual tree; and X ₅-dense locally bounded partially ordered sets. Supported by the Ministry od Science and Environmental Protection of the Republic of Serbia, Grant No. 144017.     

偏序集的关联环的同构问题

设Ｘ是局部有限偏序集（或拟序集），Ｒ是含１的结合环，Ⅰ（Ｘ，Ｒ）是Ｒ上Ｘ的关联环，关联环的同构问题是指：问题１：怎样的环，能使环同构Ⅰ（Ｘ，Ｒ）Ⅰ（Ｘ，Ｒ）推出偏序集之间的同构Ｘ芒Ｘ’？问题２：怎样的环或偏序集，能使环同构Ⅰ（Ｘ，Ｒ）Ⅰ（Ｘ，Ｓ）推出Ｒ　Ｓ？本文证明了对唯一幂等元环（非交换），问题１有正面回答；对问题２，我们证明了对交换不可分解环Ｒ、Ｓ，由环同构Ⅰ（Ｘ，Ｒ）Ⅰ（Ｘ，Ｒ）可得到Ｒ＝Ｓ，Ｘ＝Ｘ’。     

Chen-Fox-Lyndon Factorization for Words over Partially Ordered Sets

The Chen-Fox-Lyndon factorization theorem for words over totally ordered sets is a well-known and important theorem; its applications concern Chen iterated integrals and are used in control and filtering theories. We give a natural generalization for words over partially ordered sets, a question which came up in the context of generation theorems for (generalized) overlapping shuffle algebras. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 14, Algebra, 2004.     

Scaling Limits for Width Two Partially Ordered Sets: The Incomparability Window

We study the structure of a uniformly randomly chosen partial order of width 2 on n elements. We show that under the appropriate scaling, the number of incomparable elements converges to the height of a one dimensional Brownian excursion at a uniformly chosen random time in the interval [0, 1], which follows the Rayleigh distribution.     

Partially Well-Ordered Closed Sets of Permutations

It is known that the pattern containment order on permutations is not a partial well-order. Nevertheless, many naturally defined subsets of permutations are partially well-ordered, in which case they have a strong finite basis property. Several classes are proved to be partially well-ordered under pattern containment. Conversely, a number of new antichains are exhibited that give some insight as to where the boundary between partially well-ordered and not partially well-ordered classes lies.