Symmetric bilinear forms and vertices in characteristic 2

Let $G$ be a finite group and let $k$ be an algebraically closed field of characteristic $2$ and let $M$ be an indecomposable $kG$-module which affords a non-degenerate $G$-invariant symmetric bilinear form. We introduce the symmetric vertices of $M$. Each of these is a $2$-subgroup of $G$ which contains a Green vertex of $M$ with index at most $2$. If $M$ is irreducible then its symmetric vertices are determined up to $G$-conjugacy. If $B$ is the real $2$-block of $G$ containing $M$, we show that each symmetric vertex of $M$ is contained in an extended defect group of $B$. Moreover, we characterise the extended defect groups in terms of symmetric vertices. In order to prove these results, we develop the theory of involutary $G$-algebras. This allows us to translate questions about symmetric $kG$-modules into questions about projective modules of quadratic type.Comment: Changes from v2: erroneous Lemma 2.3 (on lifting idempotents) corrected. Consequent minor changes made to the rest of the paper. Table of contents remove

Book Review: Nama Japa: Prayer of the Name in the Hindu and Christian Traditions

A review of Nama Japa: Prayer of the Name in the Hindu and Christian Traditions by Vandana Mataji

Spacecraft studies of Phobos and Mars

Utilizing the Termoskan data set of the Phobos '88 mission we have recognized a new feature on Mars: Ejecta blanket Distinct In the THermal infrared (EDITH). Virtually all of the more than one hundred of these features discovered in the Termoskan data are located on the plains near Valles Manneris. EDITH's have a startlingly clear dependence upon terrains of Hesperian age, implying a spatial or temporal dependence on Hesperian terrains. Almost no thermally distinct ejecta blankets are associated with any of the thousands of craters within the data set that occur on the older Noachian units. EDITH's also do not appear on the portions of the younger Tharsis Amazonian units seen in the data. The Hesperian terrain dependence cannot be explained by either atmospheric or impactor variations; Noachian and Hesperian terrains must have experienced identical atmospheric and impactor conditions during Hesperian times. Thermally distinct eject a blankets therefore reflect target material differences and/or secondary modification processes. A further discussion of EIDTH's is presented

Geometry and relative age of large patterned fractures in southern Acidalia Planitia, Mars

Extraordinary patterned fractures occur in Elysium, Utopia, and Acidalia Planitiae of Mars. Theories as to their origin range from permafrost patterning to tectonic fracturing. Recent studies indicate the patterned fractures may be a surface reflection of rough, buried terrane in which fracturing occurs as a result of differential compaction. The patterned terrane in the southern portion of the Acidalia region (MC4-SE, MC4-SC) is being studied to determine its age relative to other terranes and events, and to constrain models for the origin of the fractures. Fracturing in this area occurs between 5 and 25 degrees west longitude and 37 and 50 degrees north latitude. Many fracture patterns are roughly polygonal, with polygons ranging in size from approximately 11 to 32 km in diameter. Fracture traces range from the limit of resolution (10's of meters) to 35 km in length

Travelling waves in a tissue interaction model for skin pattern formation

Tissue interaction plays a major role in many morphogenetic processes, particularly those associated with skin organ primordia. We examine travelling wave solutions in a tissue interaction model for skin pattern formation which is firmly based on the known biology. From a phase space analysis we conjecture the existence of travelling waves with specific wave speeds. Subsequently, analytical approximations to the wave profiles are derived using perturbation methods. We then show numerically that such travelling wave solutions do exist and that they are in good agreement with our analytical results. Finally, the biological implications of our analysis are discussed

Ten Micron Photometry of 25 Stars from B8 To M7

A photometer employing a liquid hydrogen-cooled mercury-doped germanium photoconductor whose spectral response is limited to the 8 - 14 µ region by a low pass interference filter and a BaF_2 window coupled with the cell's threshold wavelength has been placed at the east arm Cassegrain focus of the 200 inch Hale telescope. Twenty-five stars have been measured. The earliest star for which two measurements have been obtained is the B8Ia star β Orionis. The latest star is the M7e star X Cygni. The brightest star, L37 X 10^(-14) watts/cm^2, is α Orionis. The carbon star DS Peg was also measured. In a two-color diagram formed with B and V there is an intrinsic increase in dispersion going to later type stars and a systematic trend away from the blackbody relation. The ratios of the stellar fluxes to those expected from blackbodies at the published stellar effective-temperatures and angular diameters are not far from one. A systematic trend exhibited may not be real because of the assumptions involved in inteferometric diameter determinations. DS Peg does not appear overly peculiar in the two-color plots, but X,Cygni falls on the opposite side of the blackbody curve ("blue excess") compared with most of the late type stars. The fluxes presented here have not been corrected for presently uncertain telescope transmission losses which may be important

Anomalous bulk behaviour in the free parafermion Z(N)Z(N) spin chain

We demonstrate using direct numerical diagonalization and extrapolation methods that boundary conditions have a profound effect on the bulk properties of a simple $Z(N)$ model for $N \ge 3$ for which the model hamiltonian is non-hermitian. For $N=2$ the model reduces to the well known quantum Ising model in a transverse field. For open boundary conditions the $Z(N)$ model is known to be solved exactly in terms of free parafermions. Once the ends of the open chain are connected by considering the model on a ring, the bulk properties, including the ground-state energy per site, are seen to differ dramatically with increasing $N$. Other properties, such as the leading finite-size corrections to the ground-state energy, the mass gap exponent and the specific heat exponent, are also seen to be dependent on the boundary conditions. We speculate that this anomalous bulk behaviour is a topological effect.Comment: 8 pages, 8 figures, minor change