Support Varieties for Selfinjective Algebras

Support varieties for any finite dimensional algebra over a field were introduced in (Proc. London Math. Soc. 88 (3) (2004) 705–732) using graded subalgebras of the Hochschild cohomology ring. We mainly study these varieties for selfinjective algebras under appropriate finite generation hypotheses. Then many of the standard results from the theory of support varieties for finite groups generalize to this situation. In particular, the complexity of the module equals the dimension of its corresponding variety, all closed homogeneous varieties occur as the variety of some module, the variety of an indecomposable module is connected, the variety of periodic modules are lines and for symmetric algebras a generalization of Webbs theorem is true. As a corollary of a more general result we show that Webbs theorem generalizes to finite dimensional cocommutative Hopf algebras.Received November 2003Mathematics Subject Classifications (2000) Primary: 16E40, 16G10, 16P10, 16P20; Secondary: 16G70.     

Ideals in a perfect closure, linear growth of primary decompositions, and tight closure

This paper is concerned with tight closure in a commutative Noetherian ring of prime characteristic , and is motivated by an argument of K. E. Smith and I. Swanson that shows that, if the sequence of Frobenius powers of a proper ideal of has linear growth of primary decompositions, then tight closure (of ) `commutes with localization at the powers of a single element'. It is shown in this paper that, provided has a weak test element, linear growth of primary decompositions for other sequences of ideals of that approximate, in a certain sense, the sequence of Frobenius powers of would not only be just as good in this context, but, in the presence of a certain additional finiteness property, would actually imply that tight closure (of ) commutes with localization at an arbitrary multiplicatively closed subset of .

Work of M. Katzman on the localization problem for tight closure raised the question as to whether the union of the associated primes of the tight closures of the Frobenius powers of has only finitely many maximal members. This paper develops, through a careful analysis of the ideal theory of the perfect closure of , strategies for showing that tight closure (of a specified ideal of ) commutes with localization at an arbitrary multiplicatively closed subset of and for showing that the union of the associated primes of the tight closures of the Frobenius powers of is actually a finite set. Several applications of the strategies are presented; in most of them it was already known that tight closure commutes with localization, but the resulting affirmative answers to Katzman's question in the various situations considered are believed to be new.

     

Random Euclidean embeddings in spaces of bounded volume ratio

Let $\text{(}\text{R}{}^{\mathrm{N}}\text{,6·6)}$ be the space $\text{R}{}^{\mathrm{N}}$ equipped with a norm 6·6 whose unit ball has a bounded volume ratio with respect to the Euclidean unit ball. Let Γ be any random N×n matrix with N>n, whose entries are independent random variables satisfying some moment assumptions. We show that with high probability Γ is a good isomorphism from the n-dimensional Euclidean space $\text{(}\text{R}{}^{\mathrm{n}}\text{,|·|)}$ onto its image in $\text{(}\text{R}{}^{\mathrm{N}}\text{,6·6)}$: there exist α,β>0 such that for all $\text{x∈}\text{R}{}^{\mathrm{n}}$, $\text{α}\text{N}\text{|x|?6Γx6?β}\text{N}\text{|x|}$. This solves a conjecture of Schechtman on random embeddings of ?₂ⁿ into ?₁^N. To cite this article: A. Litvak et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Integrated fluidic adaptive zoom lens

An integrated fluidic adaptive zoom lens is demonstrated for what is believed to be the first time. A zoom lens was fabricated using an UV lithographic-galvanic-like process involving soft lithography and wafer bonding. The zooming capability of such a lens was achieved by varying the focal length instead of the distance between the lenses. A zoom ratio of greater than 2 was obtained for devices that are 8 mm thick and have a 20-mm lens diameter. Including the 30-mm image distance, the total physical length of the fluidic zoom lens was less than 43 mm. More-compact systems with a higher zoom ratio can be obtained by reduction of the aperture size.     

An analytical and experimental comparison of optimal actuator and error sensor location for vibration attenuation

Feedforward active control of the flexural waves in a single and L-shaped plate has been analytically and experimentally investigated. The plates are simply supported along two parallel edges, and free at the other two ends. Point forces were used to generate the primary and secondary plate excitations. The plate flexural displacement is described by a combination of a travelling wave solution and a modal expansion. The flexural wave coefficients were determined using the boundary conditions, continuity equations at the driving force locations, and continuity equations at the corner junction for the L-shaped plate. The control actuator and error sensor are optimally located in order to achieve the best control performance.     

A study of fused silica micro/nano patterning by focused-ion-beam

A dual-beam scanning electron microscopy (SEM)/focused-ion-beam (FIB) system was used to pattern fused silica substrates coated with a 15 nm thin Cr layer. The dimensions of fabricated features together with their surface morphology and profiles were investigated by SEM and atomic force microscopy (AFM). The study demonstrated that with the increase of the ion beam fluence the sputtering rate of the fused silica decreased non-linearly. Also, it was found that initially the sputtering rate increased with the increase of the beam current, after reaching a maximum value, it started decreasing when further beam current increment was performed. Compared with unprocessed areas, the surface finish of the features fabricated by FIB exhibited a significant improvement, and the ion fluence influence on the surface roughness of trenches with low aspect ratios could be considered as negligible. Using a fine beam probe, nano-gratings in the form of grooves with a width down to 54 nm and an aspect ratio higher than three were fabricated. The study showed that FIB machining could be an alternative technology to e-beam lithography for producing fused silica templates for UV nanoimprinting.     

Patterning of amorphous and polycrystalline Ni78B14Si8 with a focused-ion-beam

The machining response of amorphous and crystalline Ni₇₈B₁₄Si₈ was investigated when structuring substrates using focused-ion-beam (FIB) milling. In particular, the sputtering yield as a function of the scan speed, and the effects of ion fluence and scan speed on the milled depth were studied. The ion fluence dependent evolution of the cross-sectional profiles of trenches was examined by atomic force microscopy (AFM). When milling amorphous Ni₇₈B₁₄Si₈, it was found that the sputtering yield first decreased with increasing the beam scan speed, then kept constant within the scan speed range, up to 710 nm/s, investigated in this work; it was also found that the milled depth was almost proportional to the ion beam fluence. The patterning of polycrystalline Ni₇₈B₁₄Si₈ resulted in anisotropic milling-rates due to the varying orientation of the grains in the material. The analysis of the profile evolution in both materials indicated that the surface finish of trenches was scan speed, ion beam fluence and scan strategy dependent. The study demonstrated that direct patterning by FIB could be used for producing masters in amorphous Ni-based alloys for injection moulding and hot embossing.     

Functional MRI of working memory and selective attention in vibrotactile frequency discrimination

Background  

Focal lesions of the frontal, parietal and temporal lobe may interfere with tactile working memory and attention. To characterise the neural correlates of intact vibrotactile working memory and attention, functional MRI was conducted in 12 healthy young adults. Participants performed a forced-choice vibrotactile frequency discrimination task, comparing a cue stimulus of fixed frequency to their right thumb with a probe stimulus of identical or higher frequency. To investigate working memory, the time interval between the 2 stimuli was pseudo-randomized (either 2 or 8 s). To investigate selective attention, a distractor stimulus was occasionally presented contralaterally, simultaneous to the probe.