Harmonic Analysis On Heisenberg Nilmanifolds

In these lectures we plan to present a survey of certain aspects of harmonic analysis on a Heisenberg nilmanifold Gammakslash}H-n. Using Weil-Brezin-Zak transform we obtain an explicit decomposition of L-2 (Gammakslash}H-n) into irreducible subspaces invariant under the right regular representation of the Heisenberg group. We then study the Segal-Bargmann transform associated to the Laplacian on a nilmanifold and characterise the image of L-2 (GammakslashH-n) in terms of twisted Bergman and Hermite Bergman spaces.

Variations in the Levels of Chloroplast tRNAs and Aminoacyl-tRNA Synthetases in Senescing Leaves of Phaseolus vulgaris

The relative amounts of chloroplast tRNAs(Leu), tRNA(Glu), tRNA(Phe), tRNAs(Thr), and tRNA(Tyr) and of chloroplastic and cytoplasmic aminoacyl-tRNA synthetases were compared in green leaves, yellowing senescing leaves, and N(6)-benzyladenine-treated senescing leaves from bean (Phaseolus vulgaris, var Contender). Aminoacylation of the tRNAs using Escherichia coli aminoacyl-tRNA synthetases indicated that in senescing leaves the relative amount of chloroplast tRNA(Phe) was significantly lower than in green leaves. Senescing leaves treated with N(6)-benzyladenine contained higher levels of this tRNA than untreated senescing leaves. No significant change in the relative amounts of chloroplast tRNAs(Leu), tRNAs(Thr), and tRNA(Tyr) was detected in green, yellow senescing, or N(6)-benzyladine-treated senescing leaves. Relative levels of chloroplast tRNAs were also estimated by hybridization of tRNAs to DNA blots of gene specific probes. These experiments confirmed the results obtained by aminoacylation and revealed in addition that the relative level of chloroplast tRNA(Glu) is higher in senescing leaves than in green leaves. Transcription run-on assays indicated that these changes in tRNA levels are likely to be due to a differential rate of degradation rather than to a differential rate of transcription of the tRNA genes. Chloroplastic and cytoplasmic leucyl-, phenylalanyl-, and tyrosyl-tRNA synthetase activities were greatly reduced in senescing leaves as compared to green leaves, whereas N(6)-benzyladenine-treated senescing leaves contained higher enzyme activities than untreated senescing leaves. These results suggest that during senescence, as well as during senescence-retardation by cytokinins, changes in enzyme activities, such as aminoacyl-tRNA synthetases, rather than reduced levels of tRNAs, affect the translational capacity of chloroplasts.

Solutions of cubic equations in quadratic fields

Let K be any quadratic field with O-K its ring of integers. We study the solutions of cubic equations, which represent elliptic curves defined over Q, in quadratic fields and prove some interesting results regarding the solutions by using elementary tools. As an application we consider the Diophantine equation r + s + t = rst = 1 in O-K. This Diophantine equation gives an elliptic curve defined over Q with finite Mordell-Weil group. Using our study of the solutions of cubic equations in quadratic fields we present a simple proof of the fact that except for the ring of integers of Q(i) and Q(root 2), this Diophantine equation is not solvable in the ring of integers of any other quadratic fields, which is already proved in [4].

On continuous timed automata with input-determined guards

We consider a general class of timed automata parameterized by a set of “input-determined” operators, in a continuous time setting. We show that for any such set of operators, we have a monadic second order logic characterization of the class of timed languages accepted by the corresponding class of automata. Further, we consider natural timed temporal logics based on these operators, and show that they are expressively equivalent to the first-order fragment of the corresponding MSO logics. As a corollary of these general results we obtain an expressive completeness result for the continuous version of MTL.

An estimate of the number of tropical tree species

The high species richness of tropical forests has long been recognized, yet there remains substantial uncertainty regarding the actual number of tropical tree species. Using a pantropical tree inventory database from closed canopy forests, consisting of 657,630 trees belonging to 11,371 species, we use a fitted value of Fisher's alpha and an approximate pantropical stem total to estimate the minimum number of tropical forest tree species to fall between similar to 40,000 and similar to 53,000, i.e., at the high end of previous estimates. Contrary to common assumption, the Indo-Pacific region was found to be as species-rich as the Neotropics, with both regions having a minimum of similar to 19,000-25,000 tree species. Continental Africa is relatively depauperate with a minimum of similar to 4,500-6,000 tree species. Very few species are shared among the African, American, and the Indo-Pacific regions. We provide a methodological framework for estimating species richness in trees that may help refine species richness estimates of tree-dependent taxa.

Remotely sensed debris thickness mapping of Bara Shigri Glacier, Indian Himalaya

Despite the important role of supraglacial debris in ablation, knowledge of debris thickness on Himalayan glaciers is sparse. A recently developed method based on reanalysis data and thermal band satellite imagery has proved to be potentially suitable for debris thickness estimation without the need for detailed field data. In this study, we further develop the method and discuss possibilities and limitations arising from its application to a glacier in the Himalaya with scarce in situ data. Surface temperature patterns are consistent for 13 scenes of Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) and Landsat 7 imagery and correlate well with incoming shortwave radiation and air temperature. We use an energy-balance approach to subtract these radiation or air temperature effects, in order to estimate debris thickness patterns as a function of surface temperature. Both incoming shortwave and longwave radiation are estimated with reasonable accuracy when applying parameterizations and reanalysis data. However, the model likely underestimates debris thickness, probably due to incorrect representation of vertical debris temperature profiles, the rate of heat storage and turbulent sensible heat flux. Moreover, the uncertainty of the result was found to increase significantly with thicker debris, a promising result since ablation is enhanced by thin debris of 1-2 cm.