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

High resolution thermal infrared mapping of Martian channels

Viking Infrared Thermal Mapper (IRTM) high resolution (2 to 5 km) data were compiled and compared to Viking Visual Imaging Subsystem (VIS) data and available 1:5M geologic maps for several Martian channels including Dao, Harmakhis, Mangala, Shalbatana, and Simud Valles in an effort to determine the surface characteristics and the processes active during and after the formation of these channels. Results show a dominance of aeolian processes active in and around the channels. These processes have left materials thick enough to mask any genuine channel deposits. Results also indicate that very comparable Martian channels and their surrounding terrain are blanketed by deposits which are homogeneous in their thermal inertia values. However, optimum IRTM data does not cover the entire Martian surface and because local deposits of high thermal inertia material may not be large enough in areal extent or may be in an unfavorable location on the planet, a high resolution data track may not always occur over these deposits. Therefore, aeolian processes may be even more active than the IRTM data tracts can always show

Breast cancer tumour detection using microwave radar techniques

A breast cancer detection technique using multi-static radar is proposed herein. Images of a breast tumour are produced using this technique, with backscatter data. A wideband antenna design suitable for a breast cancer detection system is also described. Practical measurements are performed using a network analyser and a pair of antennas that are used to simulate an array. These initial images demonstrate the successful detection of a tumour phantom immersed in a liquid phantom with similar dielectric properties as the breast tissues

Numerical analysis of microwave detection of breast tumours using synthetic focussing techniques

Microwave detection of breast tumours is a non-ionising and potentially low-cost and more certain alternative to X-ray mammography. Analogous to ground penetrating radar (GPR), microwaves are transmitted using an antenna array and the reflected signals, which contain reflections from tumours, are recorded. The work presented here employs a post reception synthetically focussed detection method developed for land mine detection (R. Benjamin et al., IEE Proc. Radar, Sonar and Nav., vol. 148, no.4, pp. 233-40, 2001); all elements of an antenna array transmit a broadband signal in turn, the elements sharing a field of view with the current transmit element then record the received signal. By predicting the path delay between transmit and receive antennas via any desired point in the breast, it is then possible to extract and time-align all signals from that point. Repeated for all points in the breast, this yields an image in which the distinct dielectric properties of malignant tissue are potentially visible. This contribution presents a theoretical evaluation of the breast imaging system using FDTD methods. The FDTD model realistically models a practical system incorporating wide band antenna elements. One major challenge in breast cancer detection using microwaves is the clutter arising from skin interface. Deeply located tumours can be detected using windowing techniques (R. Nilavalan et al., Electronics Letters, vol. 39, pp. 1787-1789, 2003); however tumours closer to the skin interface require additional consideration, as described herein

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