The calculation of expectations for classes of diffusion processes by Lie symmetry methods

This paper uses Lie symmetry methods to calculate certain expectations for a large class of It\^{o} diffusions. We show that if the problem has sufficient symmetry, then the problem of computing functionals of the form $E_x(e^{-\lambda X_t-\int_0^tg(X_s) ds})$ can be reduced to evaluating a single integral of known functions. Given a drift $f$ we determine the functions $g$ for which the corresponding functional can be calculated by symmetry. Conversely, given $g$, we can determine precisely those drifts $f$ for which the transition density and the functional may be computed by symmetry. Many examples are presented to illustrate the method.Comment: Published in at http://dx.doi.org/10.1214/08-AAP534 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Ancient fluvial processes in the equatorial highlands of Mars

Martian highland craters typically lack ejecta deposits, have no noticeable rim, and are flat floored. In addition, crater size frequency distribution curves show that highland craters have depleted populations less than 20 km in diameter. A variety of processes have been suggested to explain these observations including deposition of aeolian or volcanic materials up to the crater rim crests, thermal creep, terrain softening, and mass wasting. However, none of these processes adequately explains both the crater morphology and population distribution. In order to explain both the Martian highland crater morphology and population distribution, a fluvial process is proposed which is capable of removing the loose crater rim material. The resulting effect is to decrease the crater diameter, thereby causing the population curves to bendover. The eroded material is redistributed, burying or partially burying smaller diameter craters before complete erosion. This material may also be deposited into local topographic lows, creating the depositional basins observed. A fluvial process explains both sets of observations: crater morphology and crater population distribution curves

Geologic mapping of northern Lunae Planun, Mars

Lunae Planum is an elevated region east of the Tharsis rise, and ridged plains containing numerous Sacra Dorsa wrinkle ridges, cross-cutting Sacra Fossae grabens, and lobate scarps compose this Martian Plateau. Geologic mapping of the northern Lunae Planum region was undertaken to better understand to emplacement history of the ridge plains, the structural history of deformation, and the periods of fluvial processes that have modified the region. These investigations are important for several reasons: (1) the history of plains emplacement yields information valuable for understanding the evolution of Tharsis volcanism; (2) interpretation of structural deformation has implications on the lithology of the Martian crust; and (3) determining the history and fate of Martian volatiles is dependent upon knowing the periods of outflow activity. A discussion of the findings is presented

Lie Symmetry Methods for Multidimensional Linear, Parabolic PDES and Diffusions

In this paper we introduce methods based upon Lie symmetry analysis for the construction of explicit fundamental solutions of multidimensional parabolic PDEs. We give applications to the problem of finding transition probability densities for multidimensional diffusions and to representation theory.Lie symmetry groups; fundamental solutions; transition densities; representation theory

Lie Group Symmetries as Integral Transforms of Fundamental Solutions

We obtain fundamental solutions for PDEs of the form ut = x uxx +f(x)ux ??xru by showing that if the symmetry group of the PDE is nontrivial, it contains a standard integral transform of the fundamental solution. We show that in this case, the problem of finding a fundamental solution can be reduced to inverting a Laplace transform or some other classical transform.lie symmetry groups; fundamental solutions; transition densities; short range models; zero coupon bond pricing

Wrinkle ridges in the floor material of Kasei Valles, Mars: Nature and origin

Wrinkle ridges on Mars occur almost exclusively in smooth plains material referred to as ridged plains. One of the largest contiguous units of ridged plains occurs on Lunae Planum on the eastern flank of the Tharsis rise. The eastern, western, and northern margins of the ridged plains of Lunae Planum suffered extensive erosion in early Amazonian channel-forming events. The most dramatic example of erosion in early Amazonian plains is in Kasei Valles. The nature an origin of the wrinkle ridges in the floor material of Kasei Valles are discussed

Wideband microstrip patch antenna design for breast cancer tumour detection

A patch antenna is presented which has been designed to radiate into human breast tissue. The antenna is shown by means of simulation and practical measurement to possess a wide input bandwidth, stable radiation patterns and a good front-to-back ratio. Consideration is also given to its ability to radiate a pulse, and in this respect it is also found to be suitable for the proposed application