On p-adic lattices and Grassmannians

It is well-known that the coset spaces G(k((z)))/G(k[[z]]), for a reductive group G over a field k, carry the geometric structure of an inductive limit of projective k-schemes. This k-ind-scheme is known as the affine Grassmannian for G. From the point of view of number theory it would be interesting to obtain an analogous geometric interpretation of quotients of the form G(W(k)[1/p])/G(W(k)), where p is a rational prime, W denotes the ring scheme of p-typical Witt vectors, k is a perfect field of characteristic p and G is a reductive group scheme over W(k). The present paper is an attempt to describe which constructions carry over from the function field case to the p-adic case, more precisely to the situation of the p-adic affine Grassmannian for the special linear group G=SL_n. We start with a description of the R-valued points of the p-adic affine Grassmannian for SL_n in terms of lattices over W(R), where R is a perfect k-algebra. In order to obtain a link with geometry we further construct projective k-subvarieties of the multigraded Hilbert scheme which map equivariantly to the p-adic affine Grassmannian. The images of these morphisms play the role of Schubert varieties in the p-adic setting. Further, for any reduced k-algebra R these morphisms induce bijective maps between the sets of R-valued points of the respective open orbits in the multigraded Hilbert scheme and the corresponding Schubert cells of the p-adic affine Grassmannian for SL_n.Comment: 36 pages. This is a thorough revision, in the form accepted by Math. Zeitschrift, of the previously published preprint "On p-adic loop groups and Grassmannians

A Light, Transmission and Scanning Electron Microscope Study of Snuff-Treated Hamster Cheek Pouch Epithelium

The effects of smokeless tobacco (snuff) on hamster cheek mucosa were studied by light microscopy, transmission (TEM) and scanning electron microscopy (SEM). Two grams of commercially available smokeless tobacco were placed into the blind end of the right cheek pouch of each experimental animal, once a day and five days a week for 24 months. The control animals did not receive smokeless tobacco. After 24 months treatment with smokeless tobacco, hamster cheek mucosal epithelium lost its translucency and had become whitish in color. By light microscopy hyperorthokeratosis, prominent granular cell layers with increased keratohyalin granules and hyperplasia were seen. At the ultrastructural level, wider intercellular spaces filled with microvilli, numerous shorter desmosomes, many thin tonofilament bundles, increased number of mitochondria, membrane coating granules and keratohyalin granules were seen in snuff-treated epithelium. The changes in the surface of the epithelium as seen by SEM were the development of an irregular arrangement of the microridges and the disappearance of the normal honeycomb pattern. The microridges were irregular, widened and surrounded the irregular elongated pits. Some smooth areas without microridges and pits were also seen. The long-term histological, TEM and SEM changes induced by smokeless tobacco treatment of the epithelium are well correlated with each other and were similar to those reported in human leukoplakia without dyskeratosis. They imply changes of pathological response resulting from topically applied snuff

Synonymy and stratigraphic ranges of Belemnopsis in the Heterian and Ohauan Stages (Callovian-Tithonian), southwest Auckland, New Zealand.

Belemnopsis stevensi, Belemnopsis maccrawi, and Belemnopsis sp. A (Challinor 1979a) are synonymous; B. stevensi has priority. New belemnite material from Kawhia Harbour and Port Waikato, together with graphical study methods, indicates that many small fragmentary specimens associated with B. stevensi in the lower part of its stratigraphic range are probably the same taxon. B. stevensi has been found only in the Middle and Upper Heterian Stage (Lower Kimmeridgian) at Kawhia and only in the Lower Ohauan Stage (Upper Kimmeridgian) at Port Waikato. This apparently disjunct distribution is attributed to poor exposure in the relevant sections. Belemnopsis kiwiensis n.sp., Belemnopsis cf. sp. B, Belemnopsis sp. B, Belemnopsis sp. D, and Belemnopsis spp. are associated with B. stevensi near the lowest known point in its stratigraphic range. The distribution of stratigraphically useful belemnites within the Heterian and Ohauan Stages is: Conodicoelites spp. (Lower Heterian; correlated with Lower Callovian); Belemnopsis annae (Lower and Middle Heterian; Lower Callovian/Lower Kimmeridgian); Belemnopsis stevensi (Middle Heterian/Lower Ohauan; Kimmeridgian); Belemnopsis keari (Upper Heterian; Kimmeridgian); Belemnopsis trechmanni (Upper Ohauan; Upper Kimmeridgian/Middle Tithonian). The apparently extreme range of Belemnopsis annae remains unexplained. Klondyke Sandstone (new) is recognised as the basal member of Moewaka Formation (Port Waikato area)

Computing Hilbert Class Polynomials

We present and analyze two algorithms for computing the Hilbert class polynomial $H_D$ . The first is a p-adic lifting algorithm for inert primes p in the order of discriminant D < 0. The second is an improved Chinese remainder algorithm which uses the class group action on CM-curves over finite fields. Our run time analysis gives tighter bounds for the complexity of all known algorithms for computing $H_D$, and we show that all methods have comparable run times

Variational approximation for mixtures of linear mixed models

Mixtures of linear mixed models (MLMMs) are useful for clustering grouped data and can be estimated by likelihood maximization through the EM algorithm. The conventional approach to determining a suitable number of components is to compare different mixture models using penalized log-likelihood criteria such as BIC.We propose fitting MLMMs with variational methods which can perform parameter estimation and model selection simultaneously. A variational approximation is described where the variational lower bound and parameter updates are in closed form, allowing fast evaluation. A new variational greedy algorithm is developed for model selection and learning of the mixture components. This approach allows an automatic initialization of the algorithm and returns a plausible number of mixture components automatically. In cases of weak identifiability of certain model parameters, we use hierarchical centering to reparametrize the model and show empirically that there is a gain in efficiency by variational algorithms similar to that in MCMC algorithms. Related to this, we prove that the approximate rate of convergence of variational algorithms by Gaussian approximation is equal to that of the corresponding Gibbs sampler which suggests that reparametrizations can lead to improved convergence in variational algorithms as well.Comment: 36 pages, 5 figures, 2 tables, submitted to JCG

Cubic Curves, Finite Geometry and Cryptography

Some geometry on non-singular cubic curves, mainly over finite fields, is surveyed. Such a curve has 9,3,1 or 0 points of inflexion, and cubic curves are classified accordingly. The group structure and the possible numbers of rational points are also surveyed. A possible strengthening of the security of elliptic curve cryptography is proposed using a `shared secret' related to the group law. Cubic curves are also used in a new way to construct sets of points having various combinatorial and geometric properties that are of particular interest in finite Desarguesian planes.Comment: This is a version of our article to appear in Acta Applicandae Mathematicae. In this version, we have corrected a sentence in the third paragraph. The final publication is available at springerlink.com at http://www.springerlink.com/content/xh85647871215644