Algebraic Properties of Powers of Edge Ideals of Vertex-Weighted Oriented Cycles

Let t be any positive integer and I(C_(n)) the edge ideal of a vertex-weighted oriented n-cycle graph C_(n).We provide explicit formulas for the regularity and depth of I(C_(n))^(t).In particular,we find that the regularity of I(C_(n))^(t)is a linear function;Ass(I(C_(n))^(t),the set of associated prime ideals of I(C_(n))^(t),equals Ass(I(C_(n))^(t);and the depth of I(C_(n))^(t)is a constant for all.We also give some examples to show that these results are related to the direction selection of edges and the weight of vertices.

On the Auslander-Reiten Quiver with OrientedCyclesof Representation-finite Algebras

Let A be a basic,connected finite dimensional algebra over an algebraically closed field with finite representation type, α(A) be the maximum of set consisting of numbers of indecomposable summands in the middle term of Auslander Reiten sequences in mod A. In this paper we show that if α(A)=1, then Γ A contains oriented cycles if and only if Γ A contains DT r periodic modules. When α(A)2 we give counterexamples to the assertion.

Graded Hochschild Cohomology of a Path Algebra with Oriented Cycles

The aim of this paper is to characterize the first graded Hochschild cohomology of a hereditary algebra whose Gabriel quiver is admitted to have oriented cycles. The interesting conclusion we have obtained shows that the standard basis of the first graded Hochschild cohomology depends on the genus of a quiver as a topological object. In this paper, we overcome the limitation of the classical Hochschild cohomology for hereditary algebra where the Gabriel quiver is assumed to be acyclic. As preparation, we first investigate the graded differential operators on a path algebra and the associated graded Lie algebra.

Resourceability on nuclear fuel cycle by transmutation approach

A resourceability on nuclear fuel cycle by transmutation of fission products in the spent fuel of nuclear reactors is discussed in this paper to investigate the feasibility of ＂creation and utilization＂ of Apr6s ORIENT from Adv.-ORIENT cycle, in which chemical ＂separation and utilization＂ of nuclear rare metals （platinum group metals, Mo, Tc, rare earth, etc.） has been proposed since FY2006. Apr6s ORIENT research program was newly initiated in FY2011 for nuclear transmutation of fission products into stable or short-lived highly-valuable elements. In the resourceability of rare earth metals from fission products, non-radioactive Nd and Dy can be created from Pr and Tb, respectively, by transmutation. Especially, the Dy creation has relatively high feasibility of about 10-20 %/y in creation rate. A proper moderation of neutrons in blanket of fast reactors may be required to provide a high creation rate of La from Ba. In light platinum group metals, non-radioactive Ru can be created from Tc by transmutation, of which creation rate is about 4-5 %/y in blanket of fast reactors. Pd created from Rh is almost non-radioactively depending on the isotope fraction of 107pd. Rh creation from Ru is not feasible under the neutron irradiation of typical nuclear reactors.

Hochschild cohomology of truncated quiver algebras

For a truncated quiver algebra over a field of an arbitrary characteristic, its Hochschild cohomology is calculated. Moreover, it is shown that its Hochschild cohomology algebra is finitedimensional if and only if its global dimension is finite if and only if its quiver has no oriented cycles.