Lifting of Modular Forms

The existence and construction of vector-valued modular forms (vvmf) for any arbitrary Fuchsian group $\mathrm{G}$, for any representation $\rho:\mathrm{G} \longrightarrow \mathrm{GL}_{d}(\mathbb{C})$ of finite image can be established by lifting scalar-valued modular forms of the finite index subgroup $Ker(\rho)$ of $\mathrm{G}$. In this article vvmf are explicitly constructed for any admissible multiplier (representation) $\rho$, see Section 3 for the definition of admissible multiplier. In other words, the following question has been partially answered: For which representations $\rho$ of a given $\mathrm{G}$, is there a vvmf with at least one nonzero component ?Comment: 15 page

Meth in Allegan County -- Spreading to West Michigan?

According to the Office of National Drug Policy, White House, about 1.3 million people used meth in the year 2002. Moreover, in 2001, 607,000 people used methamphetamine. The University of Arkansas reports that businesses in Benton County were losing an estimated $21 million annually because of meth, mainly due to absenteeism and lost productivity. Methamphetamine is a profoundly addictive drug that seriously affects health, families, businesses, social services, and the environment. Why is meth use on the increase

On Orthogonal Hypergeometric Groups of Degree Five

A computation shows that there are 77 (up to scalar shifts) possible pairs of integer coefficient polynomials of degree five, having roots of unity as their roots, and satisfying the conditions of Beukers and Heckman [1], so that the Zariski closures of the associated monodromy groups are either finite or the orthogonal groups of non-degenerate and non-positive quadratic forms. Following the criterion of Beukers and Heckman [1], it is easy to check that only 4 of these pairs correspond to finite monodromy groups and only 17 pairs correspond to monodromy groups, for which, the Zariski closures have real rank one. There are remaining 56 pairs, for which, the Zariski closures of the associated monodromy groups have real rank two. It follows from Venkataramana [16] that 11 of these 56 pairs correspond to arithmetic monodromy groups and the arithmeticity of 2 other cases follows from Singh [11]. In this article, we show that 23 of the remaining 43 rank two cases correspond to arithmetic groups.Comment: 33 pages (To appear in Transactions of the American Mathematical Society

Preattentive texture discrimination with early vision mechanisms

We present a model of human preattentive texture perception. This model consists of three stages: (1) convolution of the image with a bank of even-symmetric linear filters followed by half-wave rectification to give a set of responses modeling outputs of V1 simple cells, (2) inhibition, localized in space, within and among the neural-response profiles that results in the suppression of weak responses when there are strong responses at the same or nearby locations, and (3) texture-boundary detection by using wide odd-symmetric mechanisms. Our model can predict the salience of texture boundaries in any arbitrary gray-scale image. A computer implementation of this model has been tested on many of the classic stimuli from psychophysical literature. Quantitative predictions of the degree of discriminability of different texture pairs match well with experimental measurements of discriminability in human observers

A computational model of texture segmentation

An algorithm for finding texture boundaries in images is developed on the basis of a computational model of human texture perception. The model consists of three stages: (1) the image is convolved with a bank of even-symmetric linear filters followed by half-wave rectification to give a set of responses; (2) inhibition, localized in space, within and among the neural response profiles results in the suppression of weak responses when there are strong responses at the same or nearby locations; and (3) texture boundaries are detected using peaks in the gradients of the inhibited response profiles. The model is precisely specified, equally applicable to grey-scale and binary textures, and is motivated by detailed comparison with psychophysics and physiology. It makes predictions about the degree of discriminability of different texture pairs which match very well with experimental measurements of discriminability in human observers. From a machine-vision point of view, the scheme is a high-quality texture-edge detector which works equally on images of artificial and natural scenes. The algorithm makes the use of simple local and parallel operations, which makes it potentially real-time

A network for multiscale image segmentation

Detecting edges of objects in their images is a basic problem in computational vision. The scale-space technique introduced by Witkin [11] provides means of using local and global reasoning in locating edges. This approach has a major drawback: it is difficult to obtain accurately the locations of the 'semantically meaningful' edges. We have refined the definition of scale-space, and introduced a class of algorithms for implementing it based on using anisotropic diffusion [9]. The algorithms involves simple, local operations replicated over the image making parallel hardware implementation feasible. In this paper we present the major ideas behind the use of scale space, and anisotropic diffusion for edge detection, we show that anisotropic diffusion can enhance edges, we suggest a network implementation of anisotropic diffusion, and provide design criteria for obtaining networks performing scale space, and edge detection. The results of a software implementation are shown