The strength of the Weak Lefschetz Property

We study a number of conditions on the Hilbert function of a level artinian algebra which imply the Weak Lefschetz Property (WLP). Possibly the most important open case is whether a codimension 3 SI-sequence forces the WLP for level algebras. In other words, does every codimension 3 Gorenstein algebra have the WLP? We give some new partial answers to this old question: we prove an affirmative answer when the initial degree is 2, or when the Hilbert function is relatively small. Then we give a complete answer to the question of what is the largest socle degree forcing the WLP.Comment: A few minor corrections; to appear in the Illinois J. Mat

On the degree two entry of a Gorenstein hh-vector and a conjecture of Stanley

In this note we establish a (non-trivial) lower bound on the degree two entry $h_2$ of a Gorenstein $h$-vector of any given socle degree $e$ and any codimension $r$. In particular, when $e=4$, that is for Gorenstein $h$-vectors of the form $h=(1,r,h_2,r,1)$, our lower bound allows us to prove a conjecture of Stanley on the order of magnitude of the minimum value, say $f(r)$, that $h_2$ may assume. In fact, we show that $$\lim_{r\to \infty} {f(r)\over r^{2/3}}= 6^{2/3}.$$ In general, we wonder whether our lower bound is sharp for all integers $e\geq 4$ and $r\geq 2$.Comment: A few minor changes. To appear in Proc. of the AM

Bounds and asymptotic minimal growth for Gorenstein Hilbert functions

We determine new bounds on the entries of Gorenstein Hilbert functions, both in any fixed codimension and asymptotically. Our first main theorem is a lower bound for the degree $i+1$ entry of a Gorenstein $h$-vector, in terms of its entry in degree $i$. This result carries interesting applications concerning unimodality: indeed, an important consequence is that, given $r$ and $i$, all Gorenstein $h$-vectors of codimension $r$ and socle degree $e\geq e_0=e_0(r,i)$ (this function being explicitly computed) are unimodal up to degree $i+1$. This immediately gives a new proof of a theorem of Stanley that all Gorenstein $h$-vectors in codimension three are unimodal. Our second main theorem is an asymptotic formula for the least value that the $i$-th entry of a Gorenstein $h$-vector may assume, in terms of codimension, $r$, and socle degree, $e$. This theorem broadly generalizes a recent result of ours, where we proved a conjecture of Stanley predicting that asymptotic value in the specific case $e=4$ and $i=2$, as well as a result of Kleinschmidt which concerned the logarithmic asymptotic behavior in degree $i= \lfloor \frac{e}{2} \rfloor$.Comment: Several minor changes; to appear in J. Algebr

A characterization of Gorenstein Hilbert functions in codimension four with small initial degree

The main goal of this paper is to characterize the Hilbert functions of all (artinian) codimension 4 Gorenstein algebras that have at least two independent relations of degree four. This includes all codimension 4 Gorenstein algebras whose initial relation is of degree at most 3. Our result shows that those Hilbert functions are exactly the so-called {\em SI-sequences} starting with (1,4,h_2,h_3,...), where $h_4 \leq 33$. In particular, these Hilbert functions are all unimodal. We also establish a more general unimodality result, which relies on the values of the Hilbert function not being too big, but is independent of the initial degree.Comment: A few changes. Final version, to appear in Math. Res. Let

Championing and promoting innovation in UK megaprojects

In this article we explore the role of innovation champions and the ways innovation is championed and promoted in five selected UK megaprojects. To date, this research area is surprisingly under-explored in the megaproject management literature. The study is based on thirty interviews with innovation champions to capture their own perceptions and narratives on the ways innovation is stimulated and promoted in megaprojects, combined with relevant textual materials. Innovation was commonly defined as a step change or best practice that creates value that could be financial, environmental, societal etc. The data demonstrate the role of CEOs and their collaboration with innovation champions and academic partners in introducing innovation into the vocabulary of megaprojects, developing and formalising innovation strategy, and transferring experiences across megaprojects. The data show the ways different communication channels are used to promote innovations by champions such as forums, portals, platforms, campaigns, events etc

Pure O-sequences and matroid h-vectors

We study Stanley's long-standing conjecture that the h-vectors of matroid simplicial complexes are pure O-sequences. Our method consists of a new and more abstract approach, which shifts the focus from working on constructing suitable artinian level monomial ideals, as often done in the past, to the study of properties of pure O-sequences. We propose a conjecture on pure O-sequences and settle it in small socle degrees. This allows us to prove Stanley's conjecture for all matroids of rank 3. At the end of the paper, using our method, we discuss a first possible approach to Stanley's conjecture in full generality. Our technical work on pure O-sequences also uses very recent results of the third author and collaborators.Comment: Contains several changes/updates with respect to the previous version. In particular, a discussion of a possible approach to the general case is included at the end. 13 pages. To appear in the Annals of Combinatoric

Chronic Lunar Dust Exposure on Rat Cornea: Evaluation by Gene Expression Profiling

Lunar dust is capable of entering habitats and vehicle compartments by sticking to spacesuits or other objects that are transferred into the spacecraft from the lunar surface and has been reported to cause irritation upon exposure. During the Apollo missions, crewmembers reported irritation specifically to the skin and eyes after contamination of the lunar and service modules. It has since been hypothesized that ocular irritation and abrasion might occur as a result of such exposure, impairing crew vision. Recent work has shown that both ultrafine and unground lunar dust exhibited minimal irritancy of the ocular surface (i.e., cornea); however, the assessment of the severity of ocular damage resulting from contact of lunar dust particles to the cornea has focused only on macroscopic signs of mechanical irritancy and cytotoxicity. Given the chemical reactive properties of lunar dust, exposure of the cornea may contribute to detrimental effects at the molecular level including but not limited to oxidative damage. Additionally, low level chronic exposures may confound any results obtained in previous acute studies. We report here preliminary results from a tissue sharing effort using 10weekold Fischer 344 male rats chronically exposed to filtered air or jet milled lunar dust collected during Apollo 14 using a JaegerNYU noseonly chamber for a total of 120 hours (6 hours daily, 5 days a week) over a 4week period. RNA was isolated from corneas collected from rats at 1 day and 7 days after being exposed to concentrations of 0, 20, and 60 mg/m3 of lunar dust. Microarray analysis was performed using the Affymetrix GeneChip Rat Genome 230 2.0 Array with Affymetrix Expression Console and Transcriptome Analysis Console used for normalization and secondary analysis. An Ingenuity iReport"TM" was then generated for canonical pathway identification. The number of differentially expressed genes identified increases with dose compared to controls suggesting a more severe response to the lunar dust insult at higher levels. Pathways of interests that have been identified in all exposed samples include oxidative stress response, mitochondrial dysfunction, fibrosis, epithelial healing, TGF-Beta signaling, and extracellular matrix remodeling. Several biological processes related to cell migration, cellular proliferation, and eye development were also identified to be altered by exposure to lunar dust. Our preliminary results suggest that even a chronic insult of lunar dust as low as 20 mg/m(exp 3) elicits a molecular response in cornea tissue. Lunar dust on the surface of the moon would have the added properties of ionization and activation potentially leading to further damage to the cornea and greater sensitivity to any other environmental insult such as exposure to radiation. Additional studies are required to fully assess the risk of vision impairment and the mechanistic responses initiated in cornea exposed to lunar dust as well as the potential for longterm effects to astronaut healt

A novel experimental approach for the detection of the dynamic Casimir effect

The Casimir effect is a well-known macroscopic consequence of quantum vacuum fluctuations, but whereas the static effect (Casimir force) has long been observed experimentally, the dynamic Casimir effect is up to now undetected. From an experimental viewpoint a possible detection would imply the vibration of a mirror at gigahertz frequencies. Mechanical motions at such frequencies turn out to be technically unfeasible. Here we present a different experimental scheme where mechanical motions are avoided, and the results of laboratory tests showing that the scheme is practically feasible. We think that at present this approach gives the only possibility of detecting this phenomenon.Comment: Submitted to the Physical Review Letters. RevTeX. 4 pages, 2 figure

Large Eddy Simulations of sediment entrainment induced by a lock-exchange gravity current

Large Eddy simulations of lock-exchange gravity currents propagating over a mobile reach are presented. The numerical setting allows to investigate the sediment pick up induced by the currents and to study the underlying mechanisms leading to sediment entrainment for different Grashof numbers and grain sizes. First, the velocity field and the bed shear-stress distribution are investigated, along with turbulent structures formed in the flow, before the current reaches the mobile bed. Then, during the propagation of the current above the erodible section of the bed the contour plots of the entrained material are pre- sented as well as the time evolution of the areas covered by the current and by the sediment at this section. The numerical outcomes are compared with experimental data showing a very good agreement. Overall, the study confirms that sediment pick up is prevalent at the head of the current where the strongest turbulence occurs. Further, above the mobile reach of the bed, settling process seems to be of minor importance, with the entrained material being advected downstream by the current. Additionally, the study shows that, although shear stress is the main mechanism that sets particles in motion, turbu- lent bursts as well as vertical velocity fluctuations are also necessary to counteract the falling velocity of the particles and maintain them into suspension. Finally, the analysis of the stability conditions of the current shows that, from one side, sediment concentration gives a negligible contribution to the stability of the front of the current and from the other side, the stability conditions provided by the current do not allow sediments to move into the ambient fluid