Artinian level algebras of codimension 3

In this paper, we continue the study of which $h$-vectors $\H=(1,3,..., h_{d-1}, h_d, h_{d+1})$ can be the Hilbert function of a level algebra by investigating Artinian level algebras of codimension 3 with the condition $\beta_{2,d+2}(I^{\rm lex})=\beta_{1,d+1}(I^{\rm lex})$, where $I^{\rm lex}$ is the lex-segment ideal associated with an ideal $I$. Our approach is to adopt an homological method called {\it Cancellation Principle}: the minimal free resolution of $I$ is obtained from that of $I^{\rm lex}$ by canceling some adjacent terms of the same shift. We prove that when $\beta_{1,d+2}(I^{\rm lex})=\beta_{2,d+2}(I^{\rm lex})$, $R/I$ can be an Artinian level $k$-algebra only if either $h_{d-1}<h_d<h_{d+1}$ or $h_{d-1}=h_d=h_{d+1}=d+1$ holds. We also apply our results to show that for $\H=(1,3,..., h_{d-1}, h_d, h_{d+1})$, the Hilbert function of an Artinian algebra of codimension 3 with the condition $h_{d-1}=h_d<h_{d+1}$, (a) if $h_d\leq 3d+2$, then $h$-vector \H cannot be level, and (b) if $h_d\geq 3d+3$, then there is a level algebra with Hilbert function \H for some value of $h_{d+1}$.Comment: 15 page

Generic Initial Ideals And Graded Artinian Level Algebras Not Having The Weak-Lefschetz Property

We find a sufficient condition that \H is not level based on a reduction number. In particular, we prove that a graded Artinian algebra of codimension 3 with Hilbert function $\H=(h_0,h_1,..., h_{d-1}>h_d=h_{d+1})$ cannot be level if $h_d\le 2d+3$, and that there exists a level O-sequence of codimension 3 of type \H for $h_d \ge 2d+k$ for $k\ge 4$. Furthermore, we show that \H is not level if $\beta_{1,d+2}(I^{\rm lex})=\beta_{2,d+2}(I^{\rm lex})$, and also prove that any codimension 3 Artinian graded algebra $A=R/I$ cannot be level if \beta_{1,d+2}(\Gin(I))=\beta_{2,d+2}(\Gin(I)). In this case, the Hilbert function of $A$ does not have to satisfy the condition $h_{d-1}>h_d=h_{d+1}$. Moreover, we show that every codimension $n$ graded Artinian level algebra having the Weak-Lefschetz Property has the strictly unimodal Hilbert function having a growth condition on $(h_{d-1}-h_{d}) \le (n-1)(h_d-h_{d+1})$ for every $d > \theta$ where $$h_0...>h_{s-1}>h_s.$$ In particular, we find that if $A$ is of codimension 3, then $(h_{d-1}-h_{d}) < 2(h_d-h_{d+1})$ for every $\theta< d <s$ and $h_{s-1}\le 3 h_s$, and prove that if $A$ is a codimension 3 Artinian algebra with an $h$-vector $(1,3,h_2,...,h_s)$ such that h_{d-1}-h_d=2(h_d-h_{d+1})>0 \quad \text{and} \quad \soc(A)_{d-1}=0 for some $r_1(A)<d<s$, then $(I_{\le d+1})$ is $(d+1)$-regular and \dim_k\soc(A)_d=h_d-h_{d+1}.Comment: 25 page

Hydrothermal synthesis of α-MnO<inf>2</inf> and β-MnO <inf>2</inf> nanorods as high capacity cathode materials for sodium ion batteries

Two types of MnO2 polymorphs, α-MnO2 and β-MnO2 nanorods, have been synthesized by a hydrothermal method. Their crystallographic phases, morphologies, and crystal structures were characterized by XRD, FESEM and TEM analysis. Different exposed crystal planes have been identified by TEM. The electrochemical properties of α-MnO 2 and β-MnO2 nanorods as cathode materials in Na-ion batteries were evaluated by galvanostatic charge/discharge testing. Both α-MnO2 and β-MnO2 nanorods achieved high initial sodium ion storage capacities of 278 mA h g-1 and 298 mA h g-1, respectively. β-MnO2 nanorods exhibited a better electrochemical performance such as good rate capability and cyclability than that of α-MnO2 nanorods, which could be ascribed to a more compact tunnel structure of β-MnO2 nanorods. Furthermore, the one-dimensional architecture of nanorods could also contribute to facile sodium ion diffusion in the charge and discharge process. © The Royal Society of Chemistry 2013

SnO<inf>2</inf>@graphene nanocomposites as anode materials for Na-ion batteries with superior electrochemical performance

An in situ hydrothermal synthesis approach has been developed to prepare SnO2@graphene nanocomposites. The nanocomposites exhibited a high reversible sodium storage capacity of above 700 mA h g-1 and excellent cyclability for Na-ion batteries. In particular, they also demonstrated a good high rate capability for reversible sodium storage. © 2013 The Royal Society of Chemistry

Hydrothermal synthesis of I?-MnO2 and I?-MnO2 nanorods as high capacity cathode materials for sodium ion batteries

Two types of MnO2 polymorphs, I?-MnO2 and I?-MnO2 nanorods, have been synthesized by a hydrothermal method. Their crystallographic phases, morphologies, and crystal structures were characterized by XRD, FESEM and TEM analysis. Different exposed crystal planes have been identified by TEM. The electrochemical properties of I?-MnO2 and I?-MnO2 nanorods as cathode materials in Na-ion batteries were evaluated by galvanostatic charge/discharge testing. Both I?-MnO2 and I?-MnO2 nanorods achieved high initial sodium ion storage capacities of 278 mA h ga??1 and 298 mA h ga??1, respectively. I?-MnO2 nanorods exhibited a better electrochemical performance such as good rate capability and cyclability than that of I?-MnO2 nanorods, which could be ascribed to a more compact tunnel structure of I?-MnO2 nanorods. Furthermore, the one-dimensional architecture of nanorods could also contribute to facile sodium ion diffusion in the charge and discharge process

Octahedral tin dioxide nanocrystals as high capacity anode materials for Na-ion batteries

Single crystalline SnO2 nanocrystals (∼60 nm in size) with a uniform octahedral shape were synthesised using a hydrothermal method. Their phase and morphology were characterized by XRD and FESEM observation. TEM and HRTEM analyses identified that SnO2 octahedral nanocrystals grow along the [001] direction, consisting of dominantly exposed {221} high energy facets. When applied as anode materials for Na-ion batteries, SnO2 nanocrystals exhibited high reversible sodium storage capacity and excellent cyclability (432 mA h g-1 after 100 cycles). In particular, SnO 2 nanocrystals also demonstrated a good high rate performance. Ex situ TEM analysis revealed the reaction mechanism of SnO2 nanocrystals for reversible Na ion storage. It was found that Na ions first insert into SnO2 crystals at the high voltage plateau (from 3 V to ∼0.8 V), and that the exposed (1 × 1) tunnel-structure could facilitate the initial insertion of Na ions. Subsequently, Na ions react with SnO2 to form NaxSn alloys and Na2O in the low voltage range (from ∼0.8 V to 0.01 V). The superior cyclability of SnO 2 nanocrystals could be mainly ascribed to the reversible Na-Sn alloying and de-alloying reactions. Furthermore, the reduced Na2O "matrix" may help retard the aggregation of tin nanocrystals, leading to an enhanced electrochemical performance. This journal is © the Owner Societies 2013

The Gotzmann coefficients of Hilbert functions

AbstractIn this paper we investigate some algebraic and geometric consequences which arise from an extremal bound on the Hilbert function of the general hyperplane section of a variety (Green's Hyperplane Restriction Theorem). These geometric consequences improve some results in this direction first given by Green and extend others by Bigatti, Geramita, and Migliore.Other applications of our detailed investigation of how the Hilbert polynomial is written as a sum of binomials, are to conditions that must be satisfied by a polynomial if it is to be the Hilbert polynomial of a non-degenerate integral subscheme of Pn (a problem posed by R.P. Stanley). We also give some new restrictions on the Hilbert function of a zero-dimensional reduced scheme with the Uniform Position Property

Cosmic Ray Accelerators in the Large Magellanic Cloud

I point out a correlation between gamma-ray emissivity and the historical star formation rate in the Large Magellanic Cloud ~12.5 Myr ago. This correlation bolsters the view that CRs in the LMC are accelerated by conglomerations of supernova remnants: i.e. superbubbles and supergiant shells.Comment: Research Not