Complexity of Injective Homomorphisms to Small Tournaments, and of Injective Oriented Colourings

Several possible definitions of local injectivity for a homomorphism of an oriented graph G to an oriented graph H are considered. In each case, we determine the complexity of deciding whether there exists such a homomorphism when G is given and H is a fixed tournament on three or fewer vertices. Each possible definition leads to a locally-injective oriented colouring problem. A dichotomy theorem is proved in each case.

EXPLICIT DESCRIPTION OF A CLASS OF INDECOMPOSABLE INJECTIVE MODULES

Let R be a commutative Noetherian ring and p be a prime ideal of R such that the ideal pRp is principal and ht(p)≠0. In this note, the anthors describe the explicit structure of the injective envelope of the R-module R/p.

围长至少为5的平面图的injective染色

通过构造一个(Δ+3)-临界图G,运用权转移的方法证明了该图G不存在.同时,用反证法证明了:对于围长至少为5的平面图G,若Δ(G)≥30,则χi(G)≤Δ+3.这个结论改进了现有的一个结果.

围长至少为5的平面图的injective列表染色

研究了围长至少为5的平面图的injective列表染色问题.通过分析极小反例的结构性质并利用权转移方法,证明了围长至少为5且最大度至少为12的平面图G的injective列表色数不超过Δ(G)+4.此结果进一步拓展了平面图关于injective色数的Luar猜想成立的充分条件.

Injective Envelopes of a Hilbert C~*Module

As in homology, the notion of injectivity is introduced in the category whose objects are Hilbert C * module over a C * algebra and whose morphism are bounded module operators. The definition of injective envelopes of an extension of a Hilbert C * modules over a C * algebra is introduced, and is characterized in terms of the injectivity and essence. It is shown that every Hilbert C * module has a unique (up to H isometrics) injective envelope if it exists. It is also shown that an extension of a Hilbert C * module is an injective envelope if and only if it is an injective and essential extension. Moreover, every Hilbert C * module over a W * algebra has a unique (up to H isometrics) injective envelope and the injective envelope of a Hilbert C * module H is maximal essential extension of H .

A Generalization of Gorenstein Injective and Flat Modules

In this article, we introduce and study the concept of n-Gorenstein injective （resp., n-Gorenstein flat） modules as a nontrivial generalization of Gorenstein injective （resp., Gorenstein flat） modules. We investigate the properties of these modules in various ways. For example, we show that the class of n-Gorenstein injective （resp., n -Gorenstein flat） modules is closed under direct sums and direct products for n ≥ 2. To this end, we first introduce and study the notions of n-injective modules and n-flat modules.

On Pseudo Small Injective Modules

The definition of principally pseudo injectivity motivates us to generalize tae notion of injectivity, noted SP pseudo injectivity. The aim of this paper is to investigate characterizations and properties of SP pseudo injective modules. Various results are devel- oped, many extending known results. As applications, we give some characterizations on Noetherian rings, QI rings, quasi-Frobenius rings.

On the Injective Equitable Domination of Graphs

A dominating set D in a graph G is called an injective equitable dominating set (Inj-equitable dominating set) if for every , there exists such that u is adjacent to v and . The minimum cardinality of such a dominating set is denoted by and is called the Inj-equitable domination number of G. In this paper, we introduce the injective equitable domination of a graph and study its relation with other domination parameters. The minimal injective equitable dominating set, the injective equitable independence number , and the injective equitable domatic number are defined.

n-strongly Gorenstein Projective and Injective and Flat Modules

In this paper, we study some properties of n-strongly Gorenstein projective,injective and flat modules, and discuss some connections between n-strongly Gorenstein injective, projective and flat modules. Some applications are given.

Notes on Strongly n-Gorenstein Projective, Injective and Flat Modules

In this paper, we mainly investigate some properties of strongly n-Gorenstein projective, injective and flat modules under the extension of rings, which mainly including excellent extensions, morita equivalences, polynomial extensions and localizations.

INJECTIVE MAPS ON PRIMITIVE SEQUENCES OVER Z/(p^e)

Let Z/（p^e） be the integer residue ring modulo p^e with p an odd prime and integer e ≥ 3. For a sequence a over Z/（p^e）, there is a unique p-adic decomposition a- = a-0 ＋a-1 .p ＋… ＋ a-e-l .p^e-1 where each a-i can be regarded as a sequence over Z/（p）, 0 ≤ i ≤ e - 1. Let f（x） be a primitive polynomial over Z/（p^e） and G＇（f（x）,p^e） the set of all primitive sequences generated by f（x） over Z/（p^e）. For μ（x） ∈ Z/（p）[x] with deg（μ（x）） ≥ 2 and gad（1 ＋ deg（μ（x））,p- 1） = 1, setφe-1 （x0, x1,… , xe-1） = xe-1. [μ（xe-2） ＋ ηe-3（x0, X1,…, xe-3）] ＋ ηe-2（x0, X1,…, xe-2） which is a function of e variables over Z/（p）. Then the compressing mapφe-1 ： G＇（f（x）,p^e） → （Z/（p））^∞ ,a-→φe-1（a-0,a-1, … ,a-e-1） is injective. That is, for a-,b-∈ G＇（f（x）,p^e）, a- = b- if and only if φe-1 （a-0,a-1, … ,a-e-1） = φe-1（b-0, b-1,… ,b-e-1）. As for the case of e = 2, similar result is also given. Furthermore, if functions φe-1 and ψe-1 over Z/（p） are both of the above form and satisfy φe-1（a-0,a-1,…,a-e-1）=ψe-1（b-0, b-1,… ,b-e-1） for a-,b-∈G＇（f（x）,p^e）, the relations between a- and b-, φe-1 and ψe-1 are discussed

Partitioning of Any Infinite Set with the Aid of Non-Surjective Injective Maps and the Study of a Remarkable Semigroup

In this article, we will present a particularly remarkable partitioning method of any infinite set with the aid of non-surjective injective maps. The non-surjective injective maps from an infinite set to itself constitute a semigroup for the law of composition bundled with certain properties allowing us to prove the existence of remarkable elements. Not to mention a compatible equivalence relation that allows transferring the said law to the quotient set, which can be provided with a lattice structure. Finally, we will present the concept of Co-injectivity and some of its properties.

SEEMINGLY INJECTIVE VON NEUMANN ALGEBRAS

We show that a QWEP von Neumann algebra has the weak* positive approximation property if and only if it is seemingly injective in the following sense: there is a factorization of the identity of Id_(M)= vu : M→uB(H)→vM with u normal, unital, positive and v completely contractive. As a corollary, if M has a separable predual, M is isomorphic(as a Banach space) to B(l2). For instance this applies(rather surprisingly) to the von Neumann algebra of any free group. Nevertheless, since B(H) fails the approximation property(due to Szankowski) there are M ’s(namely B(H)^(**) and certain finite examples defined using ultraproducts) that are not seemingly injective.Moreover, for M to be seemingly injective it suffices to have the above factorization of I dM through B(H) with u, v positive(and u still normal).

Practical construction of globally injective parameterizations with positional constraints

We propose a novel method to compute globally injective parameterizations with arbitrary positional constraints on disk topology meshes.Central to this method is the use of a scaffold mesh that reduces the globally injective constraint to a locally flipfree condition.Hence,given an initial parameterized mesh containing flipped triangles and satisfying the positional constraints,we only need to remove the flips of a overall mesh consisting of the parameterized mesh and the scaffold mesh while always meeting positional constraints.To successfully apply this idea,we develop two key techniques.Firstly,an initialization method is used to generate a valid scaffold mesh and mitigate difficulties in eliminating flips.Secondly,edgebased remeshing is used to optimize the regularity of the scaffold mesh containing flips,thereby improving practical robustness.Compared to state-of-the-art methods,our method is much more robust.We demonstrate the capability and feasibility of our method on a large number of complex meshes.

Injective Cogenerators,Cotilting Modules and Cosilting Modules

It is well known that the concept of cotilting modules generalizes injective cogenerators and in turn,the concept of cosilting modules generalizes cotilting modules.In this paper,we further investigate the close connections among injective cogenerators,cotilting modules and cosilting modules from the viewpoint of morphism categories.Some applications are also given.

Injective △+2 Coloring of Planar Graph Without Short Cycles

A coloring of graph G is an injective coloring if its restriction to the neighborhood of any vertex is injective, which means that any two vertices get different colors if they have a common neighbor. The injective chromatic number χi(G) of G is the least integer k such that G has an injective k-coloring. In this paper, we prove that(1) if G is a planar graph with girth g ≥ 6 and maximum degree △ ≥ 7, then χi(G) ≤ △ + 2;(2) if G is a planar graph with △ ≥ 24 and without 3,4,7-cycles, then χi(G) ≤ △ + 2.

A Note on AP-Injective Rings

The purpose of this paper is to study the following two questions on AP-injective rings: (1) R is a regular ring if and only if R is a left PP-ring and R is left AP-injective; (2) Let R be a right .AP-injective ring. Then R is self-injective if and only if R is weakly injective. Hence we get some new results of P-injective rings.

A Note on DG-Gorenstein Injective Complexes

The notion of DG-Gorenstein injective complexes is studied in this article.It is shown that a complex G is DG-Gorenstein injective if and only if G is exact with Z_(n)(G)Gorenstein injective in R-Mod for each n∈Zand any morphism f:E→G is null homotopic whenever E is a DG-injective complex.

围长至少为5的平面图的injective-染色

图G的injective k-染色是指映射c:V(G)→{1,2,…,k},使有公共邻点的两个顶点u,v满足c(u)≠c(v),用X_i(G)表示使G有一个injective k-染色的最小正整数k.对g(G)≥5的平面图G,若△(G)≥20,证明了X_i(G)≤△+3.

Some Remarks on Projective Generators and Injective Cogenerators

In this paper,we give a relationship between projective generators（resp.,injective cogenerators） in the category of R-modules and the counterparts in the category of complexes of R-modules.As a consequence,we get that complexes of W^--Gorenstein modules are actually W-Gorenstein complexes whenever W is a subcategory of R-modules satisfying W⊥W,where W^- is the subcategory of exact complexes with all cycles in W.We also study when all cycles of a W-Gorenstein complexes are W^--Gorenstein modules.