Generalized Koszul properties for augmented algebras

Under certain conditions, a filtration on an augmented algebra A admits a related filtration on the Yoneda algebra E(A) := Ext_A(K, K). We show that there exists a bigraded algebra monomorphism from gr E(A) to E_Gr(gr A), where E_Gr(gr A) is the graded Yoneda algebra of gr A. This monomorphism can be applied in the case where A is connected graded to determine that A has the K_2 property recently introduced by Cassidy and Shelton.Comment: 14 page

A note on positive energy of topologically massive gravity

I review how "classical SUGRA" embeddability establishes positive energy E for D=3 topologically massive gravity (TMG), with or without a cosmological term, a procedure familiar from D=4 Einstein gravity (GR). It also provides explicit expressions for E. In contrast to GR, E is not manifestly positive, due to the peculiar two-term nature of TMG.Comment: 7 page

Note on group distance magic graphs G[C4]G[C_4]

A \emph{group distance magic labeling} or a \gr-distance magic labeling of a graph $G(V,E)$ with $|V | = n$ is an injection $f$ from $V$ to an Abelian group \gr of order $n$ such that the weight $w(x)=\sum_{y\in N_G(x)}f(y)$ of every vertex $x \in V$ is equal to the same element \mu \in \gr, called the magic constant. In this paper we will show that if $G$ is a graph of order $n=2^{p}(2k+1)$ for some natural numbers $p$, $k$ such that \deg(v)\equiv c \imod {2^{p+1}} for some constant $c$ for any $v\in V(G)$, then there exists an \gr-distance magic labeling for any Abelian group \gr for the graph $G[C_4]$. Moreover we prove that if \gr is an arbitrary Abelian group of order $4n$ such that \gr \cong \zet_2 \times\zet_2 \times \gA for some Abelian group \gA of order $n$, then exists a \gr-distance magic labeling for any graph $G[C_4]$

Seshadri constants and Grassmann bundles over curves

Let $X$ be a smooth complex projective curve, and let $E$ be a vector bundle on $X$ which is not semistable. For a suitably chosen integer $r$, let $\text{Gr}(E)$ be the Grassmann bundle over $X$ that parametrizes the quotients of the fibers of $E$ of dimension $r$. Assuming some numerical conditions on the Harder-Narasimhan filtration of $E$, we study Seshadri constants of ample line bundles on $\text{Gr}(E)$. In many cases, we give the precise value of Seshadri constant. Our results generalize various known results for ${\rm rank}(E)=2$.Comment: Final version; Annales Inst. Fourier (to appear

Energy absorption of composite materials

Results of a study on the energy absorption characteristics of selected composite material systems are presented and the results compared with aluminum. Composite compression tube specimens were fabricated with both tape and woven fabric prepreg using graphite/epoxy (Gr/E), Kevlar (TM)/epoxy (K/E) and glass/epoxy (Gl/E). Chamfering and notching one end of the composite tube specimen reduced the peak load at initial failure without altering the sustained crushing load, and prevented catastrophic failure. Static compression and vertical impact tests were performed on 128 tubes. The results varied significantly as a function of material type and ply orientation. In general, the Gr/E tubes absorbed more energy than the Gl/E or K/E tubes for the same ply orientation. The 0/ + or - 15 Gr/E tubes absorbed more energy than the aluminum tubes. Gr/E and Gl/E tubes failed in a brittle mode and had negligible post crushing integrity, whereas the K/E tubes failed in an accordian buckling mode similar to the aluminum tubes. The energy absorption and post crushing integrity of hybrid composite tubes were not significantly better than that of the single material tubes