No description available.

In this thesis, we prove that local genericity implies globally genericity for the quasi-split unitary group U3 for a quadratic extension of number fields E/F. We follow [Fli1992] and [GJR2001] closely, using the relative trace formula approach. Our main result is the existence of smooth transfer for the relative trace formulae in [GJR2001], which is circumvented there. The basic idea is to compute the Mellin transform of Shalika germ functions and show that they are equal in the unitary case and the general linear case.

The Fourier-Mellin transform was implemented on a digital computer and applied towards the recognition and differentiation of images of plant leaves regardless of translation, rotation or scale. Translated, rotated and scaled leaf images from seven species of plants were compared: avocado ( Persea americana), trembling aspen (Populus tremuloides), lamb's-quarter (Chenopodium album), linden (Tilla americana), silver maple (Acer saccharinum), plantain (Plantago major) and sumac leaflets (Rhus typhina ). The rate of recognition was high among translated and rotated leaf images for all plant species. The rates of recognition and differentiation were poor, however, among scaled leaf images and between leaves of different species. Improvements to increase the effectiveness of the algorithm are suggested.

No description available.

The paper contains the proof of the index formula for manifolds with conical points. For operators subject to an additional condition of spectral symmetry, the index is expressed as the sum of multiplicities of spectral points of the conormal symbol (indicial family) and the integral from the Atiyah-Singer form over the smooth part of the manifold. The obtained formula is illustrated by the example of the Euler operator on a two-dimensional manifold with conical singular point.

The structure of symplectic (canonical) transformations on manifolds with conical singularities is established. The operators associated with these transformations are defined in the weight spaces and their properties investigated.

Option is an asymmetric contract between two parties with future payoff derived from the price of underlying asset. Methods of pricing di erent types of options under more or less general assumptions have been extensively studied since the Nobel price winning works of Black and Scholes [1] and Merton [12] were published in 1973. A new way of pricing options with the use of Mellin transforms have been recently introduced by Panini and Srivastav [15] in 2004. This thesis offers a brief introduction to option pricing with Mellin transforms and a revision of some of the recent research in this field.

Magistro darbe yra gaunamas Z1(s) analizinis pratęsimas į pusplokštumę ir įverčiai. / In the master work, we prove that the function Z1(s) is analytically continuable to the halfplane and estimate.

Darbe pateikta funkcijų tyrimo apžvalga, svarbiausi žinomi rezultatai, suformuluota problema. Pagrindinė ribinė teorema įrodoma, taikant tikimybinius metodus, analizinių funkcijų savybes, aproksimavimo absoliučiai konvertuojančiu integralu principą. / The main limit theorem is proved using probabilistic methods, the analytical functions of the properties.

Diffuse optical tomography (DOT) is an imaging technique that utilizes near-infrared (NIR) light to probe biological tissue and ultimately recover the optical parameters of the tissue. Broadly, the process for image reconstruction in DOT involves three parts: (1) the detected measurements, (2) the modeling of the medium being imaged, and (3) the algorithm that incorporates (1) and (2) to finally estimate the optical properties of the medium. These processes have long been established in the DOT field but are also known to suffer drawbacks. The measurements themselves tend to be susceptible to experimental noise that could degrade reconstructed image quality. Furthermore, depending on the DOT configuration being utilized, the total number of measurements per capture can get very large and add additional computational burden to the reconstruction algorithms. DOT algorithms are reliant on accurate modeling of the medium, which includes solving a light propagation model and/or generating a so-called sensitivity matrix. This process tends to be complex and computationally intensive and, furthermore, does not take into account real system characteristics and fluctuations. Similarly, the inverse algorithms typically utilized in DOT also often take on a high computational volume and complexity, leading to long reconstruction times, and have limited accuracy depending on the measurements, forward model, and experimental system. The purpose of this dissertation is to address and develop computational methods, especially incorporating deep learning, to improve each of these components. First, I evaluated several time-domain data features involving the Mellin and Laplace transforms to incorporate measurements that were robust to noise and sensitive at depth for reconstruction. Furthermore, I developed a method to find the optimal values to use for different imaging depths and scenarios. Second, I developed a neural network that can directly learn the forward problem and sensitivity matrix for simulated and experimental measurements, which allows the computational forward model to adapt to the system's characteristics. Finally, I employed learning-based approaches based on the previous results to solve the inverse problem to recover the optical parameters in a high-speed manner. Each of these components were validated and tested with numerical simulations, phantom experiments, and a variety of in vivo data. Altogether, the results presented in this dissertation depict how these computational approaches lead to an improvement in DOT reconstruction quality, speed, and versatility. It is the ultimate hope that these methods, algorithms, and frameworks developed as a part of this dissertation can be directly used on future data to further validate the research presented here and to further validate DOT as a valuable imaging tool across many applications.