On π-regularity of General Rings

A general ring means an associative ring with or without identity.An idempotent e in a general ring I is called left (right) semicentral if for every x ∈ I,xe=exe (ex=exe).And I is called semiabelian if every idempotent in I is left or right semicentral.It is proved that a semiabelian general ring I is π-regular if and only if the set N (I) of nilpotent elements in I is an ideal of I and I /N (I) is regular.It follows that if I is a semiabelian general ring and K is an ideal of I,then I is π-regular if and only if both K and I /K are π-regular.Based on this we prove that every semiabelian GVNL-ring is an SGVNL-ring.These generalize several known results on the relevant subject.Furthermore we give a characterization of a semiabelian GVNL-ring.

MAXIMAL L^(1)-REGULARITY OF GENERATORS FOR BOUNDED ANALYTIC SEMIGROUPS IN BANACH SPACES

In this paper,we prove that the generator of any bounded analytic semigroup in(θ,1)-type real interpolation of its domain and underlying Banach space has maximal L^(1)-regularity,using a duality argument combined with the result of maximal continuous regularity.As an application,we consider maximal L^(1)-regularity of the Dirichlet-Laplacian and the Stokes operator in inhomogeneous B_(q),^(s),1-type Besov spaces on domains of R^(n),n≥2.

The Cordiality on the Union of 3-regular Connected Graph and Cycle

Diab proved the following graphs are Cordial;Pm K1,n if and only if(m,n) =(1,2);Cm K1,n;Pm Kn;Cm Kn for all m and n except m ≡ 2(mod 4).In this paper,we proved the Cordiality on the union of 3-regular connected graph K3 and cycle Cm.First we have the Lemma 2,if uv ∈ E(G),G is Cordial,we add 4 vertices x,y,z,w in sequence to the edge uv,obtain a new graph denoted by G*,then G* is still Cordial,by this lemma,we consider four cases on the union of 3-regular connected graph R3,and for every case we distinguish four subcases on the cycle Cm.

COUNTING ROOTED NEAR-4-REGULAR EULERIAN MAPSON SOME SURFACES

In this article the rooted planar near-4-regular Eulerian trails are enum erated and an explicit form ula for such m aps is presented. Further, the rooted near-4-regular Eulerian m aps on the torus are counted in an exact w ay.

Extensions of strongly π-regular general rings

The concept of the strongly π-regular general ring （with or without unity） is introduced and some extensions of strongly π-regular general rings are considered. Two equivalent characterizations on strongly π- regular general rings are provided. It is shown that I is strongly π-regular if and only if, for each x ∈I, x^n =x^n＋1y = zx^n＋1 for n ≥ 1 and y, z ∈ I if and only if every element of I is strongly π-regular. It is also proved that every upper triangular matrix general ring over a strongly π-regular general ring is strongly π-regular and the trivial extension of the strongly π-regular general ring is strongly clean.

A POLYNOMIAL ALGORITHM FOR FINDING THEMINIMUM FEEDBACK VERTEX SET OF A3-REGULAR SIMPLE GRAPH

A subset of the vertex set of a graph is a feedback vertex set of the graph if the resulting graph is a forest after removed the vertex subset from the graph. A polynomial algorithm for finding a minimum feedback vertex set of a 3-regular simple graph is provided.

Counting Rooted Nearly 2-regular Planar Maps

The number of rooted nearly 2-regular maps with the valency of root-vertex, the number of non-rooted vertices and the valency of root-face as three parameters is obtained. Furthermore, the explicit expressions of the special cases including loopless nearly 2-regular maps and simple nearly 2-regular maps in terms of the above three parameters are derived.

Cover a 3-regular Claw-free Graph by Induced Matchings

The induced matching cover number of a graph G without isolated vertices, denoted by imc(G),is the minimum integer k such that G has k induced matchings {M1,M2,···,Mk}such that,V(M1)∪V(M2)∪···∪V(Mk)covers V(G).This paper shows that,if G is a 3-regular claw-free graph,then imc(G)∈{2,3}.

cs＊-regular Families and Metrization Theorems

In this paper, we proved that a sequential space has a cs＊-regular cs＊-network （or a cs＊-regular weak base） is metrizable, which generalized related results in [6], [12] and [14].

A Note on the Girth of 3-Regular Hamiltonian Graph

It is well-known that the Petersen graph is nonhamiltonian.A very short proof for this result was presented in[2]due to D.B.West.In this note,by extending the proof technique in[2],we briefly show that the girth of every 3-regular hamiltonian graph on n≥10 vertices is at most(n+4)/3.

I-semi-π-regular Rings

Let R be a ring and I an ideal of R. A ring R is called I-semi-π--regular if R/I is π-regular and idempotents of R can be strongly lifted modulo I. Characterizations of I-semi-π-regular rings are given and relations between semi-π-regular rings and semiregular rings are explored.

A Note on the π- regularity of Rings

In this paper,we investigate unitπ regularity and strongly π regularity. We show that every corner ring eRe inherits the unit π regularity from a ring R. We also generalize some important characterizations of abelian regular rings to π regular rings and describe strongly π regular rings by virtue of group members.

Notes on Noncommutative VNL-rings and GVNL-rings

In this note, a counterexample is given to show that a noncommutative VNL-ring need not be an SVNL-ring, answering an open question of Chen and Tong （Glasgow Math. J., 48（1）（2006）） negatively. Moreover, some new results about VNL-rings and GVNL-ringsare also given.

ε_0-Regularity for Mean Curvature Flow from Surface to Flat Riemannian Manifold

In this paper we prove an so-regularity theorem for mean curvature flow from surface to a flat Riemannian manifold. More precisely, we prove that if the initial energy ∫∑0 ｜A｜^2 ≤ ε0 and the initial area u0（∑0） is not large, then along the mean curvature flow, we have ∫∑t ｜A｜^2 ≤ ε0. As an application, we obtain the long time existence and convergence result of the mean curvature flow.

A Fixed Point Method for Convex Systems

We present a new fixed point technique to solve a system of convex equations in several variables. Our approach is based on two powerful algorithmic ideas: operator-splitting and steepest descent direction. The quadratic convergence of the proposed approach is established under some reasonable conditions. Preliminary numerical results are also reported.

When Exchange Rings are Von Neumann Regular

We study when exchange rings are von Neumann regular. An exchange ring R with primitive factors Artinian is von Neumann regular, if the Jacobson radical of any indecomposable homomorphic image of R is T-nilpotent, and if any indecomposable homomorphic image of R is semiprime. Every indecomposable semiprimitive factor ring of R is regular, if R is an exchange ring such that every left primitive factor ring of R is a ring of index at most n and if R has nil-property.

A Note on the Prime Spectrums of Character Rings of Finite Groups

Let G be a finite group with order g and S be a subring of the algebraic number field which contains the integral extension over Z generated by a g-th primitive root co of unity, and R（G） be the character ring of G. The prime spectrum of the commutative ring S×Z R（G） iv denoted by Spec（S×Z R（G）） and set π={p｜p is a rational prime number such that p^-1 S}. We prove that when G is a regroup, a π＇-group, or a finite Abelian group, the number of the connetted components of Spec（ S×Z R （G） ） coincides with the number of the π-regular classes in G,

REGULARITY FOR CERTAIN QUASILINEAR ELLIPTIC SYSTEMS OF DIVERGENCE STRUCTURE

The regularity of the gradient of Holder continuous solutions of quasi-linear elliptic systems of the form -Dj(aij(x,u,Du)Diuk)=-Difki=gk is investigated. Partial regularity and ε-regularity are shown to hold under the structural assumption -Dj(aij(x,u,Du))=hi∈L∞

Quantile Regression Based on Laplacian Manifold Regularizer with the Data Sparsity in <i>l</i>1 Spaces

In this paper, we consider the regularized learning schemes based on l1-regularizer and pinball loss in a data dependent hypothesis space. The target is the error analysis for the quantile regression learning. There is no regularized condition with the kernel function, excepting continuity and boundness. The graph-based semi-supervised algorithm leads to an extra error term called manifold error. Part of new error bounds and convergence rates are exactly derived with the techniques consisting of l1-empirical covering number and boundness decomposition.

On Green’s Relations, 2~0-regularity and Quasi-ideals in Γ-semigroups

In this paper, we introduce the definition of （m, n）0-regularity in Г-semigroups. we in- vestigate and characterize the 20-regular class of F-semigroups using Green＇s relations. Extending and generalizing the Croisot＇s Theory of Decomposition for F-semigroups, we introduce and study the absorbent and regular absorbent Г-semigroups. We approach this problem by examining quasi-ideals using Green＇s relations.