This thesis deals with credit risk modeling and related mathematical issues. In particular we study first-passage models for credit risk, where obligors default upon first passage of a ``credit quality" process to zero. The first passage problem for correlated Brownian motion is a mathematical structure which arises quite naturally in such models, in particular the seminal multivariate Black-Cox model. In general this problem is analytically intractable, however in two dimensions analytic results are available. In addition to correcting mistakes in several published formulae, we derive an exact simulation scheme for sampling the passage times. Our algorithm exploits several interesting properties of planar Brownian motion and conformal local martingales. The main contribution of this thesis is the development of a novel multivariate framework for credit risk. We allow for both stochastic trend and volatility in credit qualities, with dependence introduced by letting these quantities be driven by systematic factors common to all obligors. Exploiting a conditional independence structure we are able to express the proportion of defaults in an asymptotically large portfolio as a path functional of the systematic factors. The functional in question returns crossing probabilities of time-changed Brownian motion to continuous barriers, and is typically not available in closed form. As such the distribution of portfolio losses is in general analytically intractable. As such we devise a scheme for simulating approximate losses and demonstrate almost sure convergence of this approximation. We show that the model calibrates well, across both tranches and maturities, to market quotes for CDX index tranches. In particular we are able to calibrate to data from 2006, as well as more recent ``distressed" data from 2008.

This thesis deals with credit risk modeling and related mathematical issues. In particular we study first-passage models for credit risk, where obligors default upon first passage of a ``credit quality" process to zero. The first passage problem for correlated Brownian motion is a mathematical structure which arises quite naturally in such models, in particular the seminal multivariate Black-Cox model. In general this problem is analytically intractable, however in two dimensions analytic results are available. In addition to correcting mistakes in several published formulae, we derive an exact simulation scheme for sampling the passage times. Our algorithm exploits several interesting properties of planar Brownian motion and conformal local martingales. The main contribution of this thesis is the development of a novel multivariate framework for credit risk. We allow for both stochastic trend and volatility in credit qualities, with dependence introduced by letting these quantities be driven by systematic factors common to all obligors. Exploiting a conditional independence structure we are able to express the proportion of defaults in an asymptotically large portfolio as a path functional of the systematic factors. The functional in question returns crossing probabilities of time-changed Brownian motion to continuous barriers, and is typically not available in closed form. As such the distribution of portfolio losses is in general analytically intractable. As such we devise a scheme for simulating approximate losses and demonstrate almost sure convergence of this approximation. We show that the model calibrates well, across both tranches and maturities, to market quotes for CDX index tranches. In particular we are able to calibrate to data from 2006, as well as more recent ``distressed" data from 2008.

<p>In general relativity, the Riemannian Penrose inequality (RPI) provides a lower bound for the ADM mass of an asymptotically flat manifold of nonnegative scalar curvature in terms of the area of the outermost minimal surface, if one exists. In physical terms, an equivalent statement is that the total mass of an asymptotically flat spacetime admitting a time-symmetric spacelike slice is at least the mass of any black holes that are present, assuming nonnegative energy density. The main goal of this thesis is to deduce geometric lower bounds for the ADM mass of manifolds to which neither the RPI nor the famous positive mass theorem (PMT) apply. This is the case, for instance, for manifolds that contain metric singularities or have boundary components that are not minimal surfaces.</p> <p>The fundamental technique is the use of conformal deformations of a given Riemannian metric to arrive at a new Riemannian manifold to which either the PMT or RPI applies. Along the way we are led to consider the geometry of certain types non-smooth metrics. We prove a result regarding the local structure of area-minimizing hypersurfaces with respect such metrics using geometric measure theory.</p> <p>One application is to the theory of ``zero area singularities,'' a type of singularity that generalizes the degenerate behavior of the Schwarzschild metric of negative mass. Another application deals with constructing and understanding some new invariants of the harmonic conformal class of an asymptotically flat metric.</p> / Dissertation

Conformal arrays possess certain desirable characteristics for deployment on unmanned aerial vehicles and other payload-limited platforms: aerodynamic design, minimal payload weight, increased field of view, and ease of integration with diverse sensor functions. However, the conformal arrays nonplanar geometry causes high adaptive losses in conventional space-time adaptive processing (STAP) algorithms. In this thesis, we develop a conformal array signal model and apply it to evaluate the performance of conventional STAP algorithms on simulated ground clutter data. We find that array-induced clutter nonstationarity leads to high adaptive losses, which greatly burden detection performance. To improve adaptive performance, we investigate the application of existing equivalent-linear-array transformations and develop novel deterministic and adaptive angle-Doppler compensation techniques, which align nonstationary clutter returns. Through the application of these techniques, we are able to nearly fully mitigate the nonstationary behavior yielding performance similar to that of a conventional planar array. Finally, we investigate the impact of array errors on the performance of conformal arrays, and propose several array calibration techniques as ameliorating solutions.

Many electromagnetic problems can be simulated by FDTD method. Mainly, we use orthogonal cartesian coordinate in normal situations when we deal with the electromagnetic problems. Because in most situations, the structures simulated are simply rectangular. But sometimes we may need to simulate the structures which are not rectangular like the sharps of arc and circle. For this kind of problems, the tranditional FDTD method no longer works, so the tranditional FDTD method must be modified to fit the simulation of irregular structures. Besides the FDTD method we mention above, we even combine it with non-uniform grid method in more applications. And the time to apply it is when the object simulated both has the rectangular and curved structures in the same time like the microstrip fed by the coaxial cable. The situations like that would be a good time to apply it.

No description available.

Spin-1/2 chains demonstrate some of the striking effects of interactions and quantum fluctuations in one-dimensional systems. The XXZ model has been used to study the unusual properties of anisotropic spin chains in an external magnetic field. The zero temperature phase diagram for this model exhibits a critical or quasi-long-range-ordered phase which is a realization of a Luttinger liquid. While many static properties of spin-1/2 chains have been explained by combinations of analytical techniques such as bosonization and Bethe ansatz, the standard approach fails in the calculation of some time-dependent correlation functions. I present a study of the longitudinal dynamical structure factor for the XXZ model in the critical regime. I show that an approximation for the line shape of the dynamical structure factor in the limit of small momentum transfer can be obtained by going beyond the Luttinger model and treating irrelevant operators associated with band curvature effects. This approach is able to describe the width of the on-shell peak and the high-frequency tail at finite magnetic field. Integrability is shown to affect the low-energy effective model at zero field, with consequences for the line shape. The power-law singularities at the thresholds of the particle-hole continuum are investigated using an analogy with the X-ray edge problem. Using methods of Bethe ansatz and conformal field theory, I compute the exact exponents for the edge singularities of the dynamical structure factor. The same methods are used to study the long-time asymptotic behavior of the spin self-correlation function, which is shown to be dominated by a high-energy excitation.

Intensity-modulated radiation therapy (IMRT) is an advanced mode of high- precision radiation therapy that utilizes computer-controlled x-ray accelerators to deliver precise radiation doses to malignant tumors. The radiation dose is designed to conform to the three-dimensional (3-D) shape of a tumor by modulating the intensity of the radiation beam to focus a higher radiation dose to the tumor while minimizing radiation exposure to surrounding normal tissue. One form of IMRT is known as tomotherapy. Tomotherapy achieves dose conformity to a tumor by modulating the intensity of a fan beam of radiation as the source revolves about a patient. Current available tomotherapy machines use x-ray linear accelerators (linacs) as a source of radiation. However, since linacs are technologically complex, the world- wide use of linac-based tomotherapy is limited. This thesis involves an investigation of Cobalt 60 (Co-60) based tomotherapy. The inherent simplicity of Co-60 has the potential to extend the availability of this technique to clinics throughout the world. The goal of this thesis is to generate two-dimensional (2-D) Co-60 tomotherapy con- formal dose distributions with a computer program and experimentally validate them on ¯lm using a ¯rst generation bench-top tomotherapy apparatus. The bench-top apparatus consists of a rotation-translation stage that can mimic a 2-D tomotherapy delivery by translating the phantom across a thin, "pencil- like" photon beam from various beam orientations. In this thesis, several random and clinical patterns are planned using an in-house inverse treatment planning system and are delivered on ¯lm using the tomotherapy technique. The delivered dose plans are compared with the simulated plans using the gamma dose comparison method. The results show a reasonably good agreement between the plans and the measurements, suggesting that Co-60 tomotherapy is indeed capable of providing state-of-the-art conformal dose delivery. / Thesis (Master, Physics, Engineering Physics and Astronomy) -- Queen's University, 2008-04-25 02:20:56.102 / Canadian Institutes of Health Research (CIHR) and the ORDCF’s Ontario Consortium for Image-guided Therapy and Surgery.

No description available.

2011 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE 2011), June 27-30, 2011, Grand Hyatt Taipei, Taipei, Taiwan