On Serial Noetherian Rings

It is well known that every serial Noetherian ring satisfies the restricted minimum condition. In particular, following Warfield (1975 Warfield , R. B. ( 1975 ). Serial rings and finitely presented modules . J. Algebra 37 : 187 – 222 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]), such a ring is a direct sum of an Artinian ring and hereditary prime rings. The aim of this note is to show that every serial ring having the restricted minimum condition is Noetherian.     

Palindromic Resolutions of Simple Modules

For a finite dimensional hereditary k-algebra H of Dynkin type, it is known that the trivial extension T(H) has the property that minimal projective resolutions of T(H)-modules are periodic. We examine conditions in which T(H)-modules have additional symmetry in their resolutions—namely, their resolutions are “palindromic.” A more general framework in which “palindromic” resolutions occur is also presented.     

Cross-intersecting sub-families of hereditary families

Families A₁,A₂,…,Ak of sets are said to be cross-intersecting if for any i and j in {1,2,…,k} with i≠j, any set in Ai intersects any set in Aj. For a finite set X, let X² denote the power set of X (the family of all subsets of X). A family H is said to be hereditary if all subsets of any set in H are in H; so H is hereditary if and only if it is a union of power sets. We conjecture that for any non-empty hereditary sub-family H≠{∅} of X² and any k?|X|+1, both the sum and the product of sizes of k cross-intersecting sub-families A₁,A₂,…,Ak (not necessarily distinct or non-empty) of H are maxima if A₁=A₂=?=Ak=S for some largest starSofH (a sub-family of H whose sets have a common element). We prove this for the case when H is compressed with respect to an element x of X, and for this purpose we establish new properties of the usual compression operation. As we will show, for the sum, the condition k?|X|+1 is sharp. However, for the product, we actually conjecture that the configuration A₁=A₂=?=Ak=S is optimal for any hereditary H and any k?2, and we prove this for a special case.     

On hereditary clique-Helly self-clique graphs

A graph is clique-Helly if any family of mutually intersecting (maximal) cliques has non-empty intersection, and it is hereditary clique-Helly (HCH) if its induced subgraphs are clique-Helly. The clique graph of a graph G is the intersection graph of its cliques, and G is self-clique if it is connected and isomorphic to its clique graph. We show that every HCH graph is an induced subgraph of a self-clique HCH graph, and give a characterization of self-clique HCH graphs in terms of their constructibility starting from certain digraphs with some forbidden subdigraphs. We also specialize this results to involutive HCH graphs, i.e. self-clique HCH graphs whose vertex-clique bipartite graph admits a part-switching involution.     

Hereditary unigraphs and Erdős–Gallai equalities

We give characterizations of the structure and degree sequence of hereditary unigraphs, those graphs for which every induced subgraph is the unique realization of its degree sequence. The class of hereditary unigraphs properly contains the threshold and matrogenic graphs, and the characterizations presented here naturally generalize those known for these other classes of graphs.     

两族可积的Hamilton方程

把屠规彰格式应用于等谱问题得到了两族可积的Ｈａｍｉｌｔｏｎ方程．得到后一族需要对屠格式进行变更，这为扩大屠格式的应用范围指出了一条途径．     

Cellularity of a New Class of Diagram Algebras

A diagram algebra in which the diagrams include n rows of n points is constructed. Moreover, we prove that the algebra is cellular and quasi-hereditary.     

A New Characterization of Hereditary Algebras

In this short article, we prove that a finite dimensional algebra is hereditary if and only if there is no loop in its ordinary quiver and every τ-tilting module is tilting.     

Weakly Stable Modules

ABSTRACT

A new notion which is called weakly stable module is introduced in this article. It is a nontrivial generalization of the modules with endomorphism rings having stable range one. We deduce that weakly stable projective modules have the cancellation property, and so any commutative hereditary ring has the cancellation property, i.e., if R is a commutative hereditary ring, then for any R-modules B and C, R ⊕ B ? R ⊕ C implies B ? C.     

Selective Hydrogenation of CO2 Dictated by Isomers in Au28(SR)20 Nanoclusters: Which One is Better?

It is a challenge to make clear how isomerism in a heterogeneous catalyst induces distinct differences in catalytic properties, as attainment of the structural isomerism in a conventional catalyst is difficult. By successfully identifying the isomerism in the atomically precise Au nanoclusters, an exciting opportunity for unravelling catalysis of isomeric catalysts is opened up. Herein, we report that the isomerism in the Au₂₈(SR)₂₀ nanoclusters with different surface atom arrangements can indeed render different catalytic behaviors in the selective hydrogenation of CO₂. We anticipate that our studies will serve as a starting point for fundamental investigations about how to control the catalytic activity and selectivity by the isomerism-induced catalysis.