The diamond partial order for strong Rickart rings

The diamond partial order has been first introduced for matrices, and then discussed also in the general context of *-regular rings. We extend this notion to Rickart rings, and state various properties of the diamond order living on the so-called strong Rickart rings. In particular, it is compared with the weak space preorder and the star order; also existence of certain meets and joins under diamond order is discussed.     

The L* Partial Order on the Set of Group Matrices

In this paper,we use the L(o)wner partial order and the star partial order to introduce a new partial order (denoted by "L*") on the set of group matrices,and get some characteristics and properties of the new partial order.In particular,we prove that the L* partial order is a special kind of the core par-tial order and it is equivalent to the star partial order under some conditions.We also illustrate its difference from other partial orders with examples and find out under what conditions it is equivalent to other partial orders.     

有限集合上偏序关系的矩阵判别方法

要从定义出发去判断集合X上的二元关系R是否是偏序关系比较困难,从关系矩阵出发,给出了偏序关系的矩阵判别方法,并对此方法进行了证明.     

矩阵广义Schur补的复合矩阵的 Lwner偏序与奇异值

本文把矩阵广义 Ｓｃｈｕｒ补和复合矩阵结合起来,研究了一个ｍ×ｎ复矩阵的广义Ｓｃｈｕｒ补及其共轭转置之积的复合矩阵的Ｌｗｎｅｒ偏序,并给出相关复合矩阵的奇异值不等式;推广了近期的一些结果．     

范畴中态射集减序的刻划

本文研究了范畴中态射集减序的刻划问题，其主要结果是：（１）证得一般范畴情形减序的十四种刻划；（２）将Ｍｉｔｒａ、Ｂａｋｓａｌａｒｙ等人的利用（１）－逆、（１，２）－逆的包含关系及平行和来刻划复矩阵减序的结果推广到素性预加法范畴与带对合的正性预加法范畴中；（３）给出了１９８０年Ｈａｒｔｗｉｇ关于半群减序刻划的一个反例，因此否定了这个刻画．     

Linear Extensions of Additive Partial Orders

Addive partial orders arise naturally in theories of comparativeprobability and subset preferences. An additive partial order is a partialorder on the family of subsets of ann-element set that satisfies . This is reformulated as a subset P of {1,0,–1}ⁿ that excludes 0 and containsx+y whenever x,y P and x+y {1,0,–1}ⁿ. Additional conditions of positivity andcompleteness give rise to positive additive partial orders and additivelinear orders respectively. The paper investigates conditions under which anadditive partial order is included in, or extendable to, an additive linearorder. The additive dimension of an extendable additive partial order isdefined and computed for several classes of additive orders.     

A Characterization and the Intersection of the Maximal Compatible Extensions of a Partial Order

We present a characterization of the maximal compatible extensions of a given compatible partial order ≤  _r on a unary algebra (A,f ). These extensions can be constructed by using the compatible linear extensions of ≤  _r*, where (A*,f*) is the so called contracted quotient algebra of (A,f) and the compatible partial order ≤  _r* on (A*,f*) is naturally induced by ≤  _r . Using this characterization, we determine the intersection of the maximal compatible extensions of ≤  _r .      

New matrix partial order based on spectrally orthogonal matrix decomposition

We investigate partial orders on the set of complex square matrices and introduce a new order relation based on spectrally orthogonal matrix decompositions. We also establish the relation of this concept with the known orders.     

Canonical State Representations and Hilbert Functions of Multidimensional Systems

A basic and substantial theorem of one-dimensional systems theory, due to R. Kalman, says that an arbitrary input/output behavior with proper transfer matrix admits an observable state representation which, in particular, is a realization of the transfer matrix. The state equations have the characteristic property that any local, better temporal, state at time zero and any input give rise to a unique global state or trajectory of the system or, in other terms, that the global state is the unique solution of a suitable Cauchy problem. With an adaption of this state property to the multidimensional situation or rather its algebraic counter-part we prove that any behavior governed by a linear system of partial differential or difference equations with constant coefficients is isomorphic to a canonical state behavior which is constructed by means of Gröbner bases. In contrast to the one-dimensional situation, to J.C. Willems’ multidimensional state space models and and to J.F. Pommaret’s modified Spencer form the canonical state behavior is not necessarily a first order system. Further first order models are due E. Zerz. As a by-product of the state space construction we derive a new variant of the algorithms for the computation of the Hilbert function of finitely generated polynomial modules or behaviors. J.F. Pommaret, J. Wood and P. Rocha discussed the Hilbert polynomial in the systems theoretic context. The theorems of this paper are constructive and have been implemented in MAPLE in the two-dimensional case and demonstrated in a simple, but instructive example. A two-page example also gives the complete proof of Kalman’s one-dimensional theorem mentioned above. We believe that for this standard case the algorithms of the present paper compare well with their various competitors from the literature.     

On a universal best choice algorithm for partially ordered sets

For the only known universal best choice algorithm for partially ordered sets with known cardinality and unknown order (proposed by J. Preater) we improve the estimation of the lower bound of its chance of success from the hitherto known constant 1/8 to 1/4. We also show that this result is the best possible for this algorithm, i.e., the 1/4 bound cannot be further improved. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

富足半群上的自然偏序

本文研究富足半群上的自然偏序，得到Green关系和自然偏序之间的联系，确定了富足半群何时关于自然偏序具有单边(双边)相容，另外，也研究了富足半群的本原元。     

右逆半群上的偏序关系

It is well known that there exists the smallest inverse semigroup congruence on an orthodox semigroup. We denote by Y the smallest inverse semigroup congruence on an orthodox semigroup. Let S be a fight inverse semigroup. We construct partial orders on S by some kind of its subsemigroups and uncover that partial orders on S have close contact with partial orders on S/Y.     

On a weighted core inverse in a ring with involution

We define and investigate the e-core inverse and f-dual core inverse in a ring with involution as extensions of the core and dual core inverse, respectively. Using these definitions, we present and characterize the e-core partial order and the f-dual core partial order. We describe the sets of all elements a Rickart ?-ring which are below a given element under the e-core and f-dual core partial order. New characterizations of weighted-EP elements are given too.     

具有逆断面的局部逆半群

Let S be a locally inverse semigroup with an inverse transversal S°. In this paper, we construct an amenable partial order on S by an R-cone. Conversely, every amenable partial order on S can be constructed in this way. We give some properties of a locally inverse semigroup with a Clifford transversal. In particular, if S is a locally inverse semigroup with a Clifford transversal, then there is an order-preserving bijection from the set of all amenable partial orders on S to the set of all R-cones of S.     

Fractional‐order orthogonal Bernstein polynomials for numerical solution of nonlinear fractional partial Volterra integro‐differential equations

In this paper, a new two‐dimensional fractional polynomials based on the orthonormal Bernstein polynomials has been introduced to provide an approximate solution of nonlinear fractional partial Volterra integro‐differential equations. For this aim, the fractional‐order orthogonal Bernstein polynomials (FOBPs) are constructed, and its operational matrices of integration, fractional‐order integration, and derivative in the Caputo sense and product operational matrix are derived. These operational matrices are utilized to reduce the under study problem to a nonlinear system of algebraic equations. Using the approximation of FOBPs, the convergence analysis and error estimate associated to the proposed problem have been investigated. Finally, several examples are included to clarify the validity, efficiency, and applicability of the proposed technique via FOBPs approximation.     

On a Linear Partial Differential Equation of the Higher Order in Two Variables with Initial Condition Whose Coefficients are Real-valued Simple Step Functions

By using the method developed in the paper [Georg. Inter. J. Sci. Tech., Volume 3, Issue 1 (2011), 107-129], it is obtained a representation in an explicit form of the weak solution of a linear partial differential equation of the higher order in two variables with initial condition whose coefficients are real-valued simple step functions.     

SP.n矩阵的分解

J.J．Johnson在1974年给出了P．n．P矩阵的谱性质具有一负特征值的充分条件及P．n矩阵的两个增性质．许多国内同仁也于近几年从分析偏负阵的Schur补入手，从特殊到一般，得到了一系列偏负阵的判定方法．本文在偏负阵判定方法的基础上，专门对偏负阵的分解进行论述，得出了对称偏负阵能进行三角分解和正交分解的充要条件．