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Yes / We develop a novel approach to estimate the n unknown constituent frequencies of a noiseless signal that comprises of unknown number, n, of sinusoids of unknown phases and unknown amplitudes. The new two way approach uses two constraints to accurately estimate the unknown frequencies of the sinusoidal components in a signal. The new approach serves as a verification test for the estimated unknown frequencies through the estimated count of the unknown number of frequencies. The Hankel matrix, of the time domain samples of the signal, is used as a basis for further analysis in the Pisarenko harmonic decomposition. The new constraints, the Existence Factor (EF) and the Component Factor (CF), have been introduced in the methodology based on the relationships between the components of the sinusoidal signal and the eigenspace of the Hankel matrix. The performance of the developed approach has been tested to correctly estimate any number of frequencies within a signal with or without a fixed unknown bias. The method has also been tested to accurately estimate the very closely spaced low frequencies. / Innovate UK

Many real-world systems can be represented as network-graphs. Some of the networks have an inherent community structure based on interactions. The problem of identifying this grouping structure given a graph is termed as community detection problem which has certain existing algorithms. This thesis contributes by providing specific improvements to various community detection algorithms such as spectral clustering and extreme point algorithm. One of the main contributions is proposing a new sub-sampling method to make existing spectral clustering method scalable by reducing the computational complexity. Also, we have implemented extreme points algorithm for a general multiple communities detection case along with a sub-sampling based version to reduce the computational complexity. We have also developed spectral clustering algorithm for popularity-adjusted block model (PABM) model based graphs to make the algorithm exact thus improving its accuracy. / Master of Science

We consider the spectrum of the Schrödinger operator on an octagonal lattice using the Floquet-Bloch transform of the Laplacian. We will first consider the spectrum of the Laplacian in detail and prove various properties thereof, including spectral-band limits and locations of singularities. In addition, we will prove that Schrödinger operators with 1-1 periodic potentials can open at most two gaps in the spectrum precisely at energies $pm1$, and that a third gap can open at 0 for 2-2 periodic potentials. We describe in detail the structure of these operators for higher periods, and motivate our expectations of their spectra. / Master of Science / In quantum physics, we would like the capability to model environments, such as magnetic fields, that interact with electrons or other quantum entities. The fields of graph theory and functional analysis within mathematics provide tools which relate well-understood mathematical concepts to these physical interactions. In this work, we use these tools to describe these environments using previously employed techniques in new ways.

Spectroradiometer data (350 – 2500 nm) were acquired in late summer 1999 over various forest sites in Appomattox Buckingham State Forest, Virginia, to assess the spectral differentiability among six major forestry tree species, loblolly pine (Pinus taeda), Virginia pine (Pinus virginiana), shortleaf pine (Pinus echinata), scarlet oak (Quercus coccinea), white oak (Quercus alba), and yellow poplar (Liriodendron tulipifera). Data were smoothed using both moving (9-point) and static (10 nm average) filters and curve shape was determined using first and second differences of resultant data sets. Stepwise discriminant analysis decreased the number of independent variables to those significant for spectral discrimination at -level of 0.0025. Canonical discriminant analysis and a normal discriminant analysis were performed on the data sets to test separability between and within taxonomic groups. The hardwood and pine groups were shown to be highly differentiable with a 100% cross-validation accuracy. The three pines were less differentiable, with cross-validation results varying from 61.64% to 84.25%, while spectral separability among the three hardwood species showed more promise, with classification accuracies ranging from 78.36% to 92.54%. The second difference of the 9-point weighted average filter was the most effective data set, with accuracies ranging from 84.25% to 100.00% for the separability tests. Overall, variables needed for spectral discrimination were well distributed across the 350 nm to 2500 nm spectral range, indicating the usefulness of the whole wavelength range for discriminating between taxonomic groups and among species. Derivative analysis was shown to be effective for between and within group spectral discrimination, given that the data were smoothed first. Given the caveat of the limited species diversity examined, results of this study indicate that leaf-on hyperspectral remotely sensed data will likely afford spectral discrimination between hardwoods and softwoods, while discrimination within taxonomic groups might be more problematic. / Master of Science

International Telemetering Conference Proceedings / October 22-25, 2001 / Riviera Hotel and Convention Center, Las Vegas, Nevada / Performances of a subclass of a new spectral efficient modulation scheme, designated as Feher Keying [1], or FK, is evaluated. The Power Spectral Density (PSD) and Bit Error Rate (BER) characteristics of FK are presented. FK has ultra high spectral efficiency and satisfies the frequency mask for WLAN defined in FCC part 15, and it has a simple structure for high bit rate implementation.

Ce mémoire a pour but d'étudier les propriétés des solutions à l'équation aux valeurs propres de l'opérateur de Laplace sur le disque lorsque les valeurs propres tendent vers l'in ni. En particulier, on s'intéresse au taux de croissance des normes ponctuelle et L1. Soit D le disque unitaire et @D sa frontière (le cercle unitaire). On s'inté- resse aux solutions de l'équation aux valeurs propres f = f avec soit des conditions frontières de Dirichlet (fj@D = 0), soit des conditions frontières de Neumann ( @f @nj@D = 0 ; notons que sur le disque, la dérivée normale est simplement la dérivée par rapport à la variable radiale : @ @n = @ @r ). Les fonctions propres correspondantes sont données par : f (r; ) = fn;m(r; ) = Jn(kn;mr)(Acos(n ) + B sin(n )) (Dirichlet) fN (r; ) = fN n;m(r; ) = Jn(k0 n;mr)(Acos(n ) + B sin(n )) (Neumann) où Jn est la fonction de Bessel de premier type d'ordre n, kn;m est son m- ième zéro et k0 n;m est le m-ième zéro de sa dérivée (ici on dénote les fonctions propres pour le problème de Dirichlet par f et celles pour le problème de Neumann par fN). Dans ce cas, on obtient que le spectre SpD( ) du laplacien sur D, c'est-à-dire l'ensemble de ses valeurs propres, est donné par : SpD( ) = f : f = fg = fk2 n;m : n = 0; 1; 2; : : :m = 1; 2; : : :g (Dirichlet) SpN D( ) = f : fN = fNg = fk0 n;m 2 : n = 0; 1; 2; : : :m = 1; 2; : : :g (Neumann) En n, on impose que nos fonctions propres soient normalisées par rapport à la norme L2 sur D, c'est-à-dire : R D F2 da = 1 (à partir de maintenant on utilise F pour noter les fonctions propres normalisées et f pour les fonctions propres quelconques). Sous ces conditions, on s'intéresse à déterminer le taux de croissance de la norme L1 des fonctions propres normalisées, notée jjF jj1, selon . Il est vi important de mentionner que la norme L1 d'une fonction sur un domaine correspond au maximum de sa valeur absolue sur le domaine. Notons que dépend de deux paramètres, m et n et que la dépendance entre et la norme L1 dépendra du rapport entre leurs taux de croissance. L'étude du comportement de la norme L1 est étroitement liée à l'étude de l'ensemble E(D) qui est l'ensemble des points d'accumulation de log(jjF jj1)= log : Notre principal résultat sera de montrer que [7=36; 1=4] E(B2) [1=18; 1=4]: Le mémoire est organisé comme suit. L'introdution et les résultats principaux sont présentés au chapitre 1. Au chapitre 2, on rappelle quelques faits biens connus concernant les fonctions propres du laplacien sur le disque et sur les fonctions de Bessel. Au chapitre 3, on prouve des résultats concernant la croissance de la norme ponctuelle des fonctions propres. On montre notamment que, si m=n ! 0, alors pour tout point donné (r; ) du disque, la valeur de F (r; ) décroit exponentiellement lorsque ! 1. Au chapitre 4, on montre plusieurs résultats sur la croissance de la norme L1. Le probl ème avec conditions frontières de Neumann est discuté au chapitre 5 et on présente quelques résultats numériques au chapitre 6. Une brève discussion et un sommaire de notre travail se trouve au chapitre 7. / The goal of this master's thesis is to explore the properties of the solutions of the eigenvalue problem for the Laplace operator on a disk as the eigenvalues go to in nity. More speci cally, we study the growth rate of the pointwise and the L1 norms of the eigenfunctions. Let D be the unit disk and @D be its boundary (the unit circle). We study the solutions of the eigenvalue problem f = f with either Dirichlet boundary condition (fj@D = 0) or Neumann boundary condition ( @f @nj@D = 0; note that for the disk the normal derivative is simply the derivative with respect to the radial variable: @ @n = @ @r ). The corresponding eigenfunctions are given by: f (r; ) = fn;m(r; ) = Jn(kn;mr)(Acos(n ) + B sin(n )) (Dirichlet) fN (r; ) = fN n;m(r; ) = Jn(k0 n;mr)(Acos(n ) + B sin(n )) (Neumann) where Jn is the nth order Bessel function of the rst type, kn;m is its mth zero and k0 n;m is the mth zero of its derivative (here we denote the eigenfunctions for the Dirichlet problem by f and those for the Neumann problem by fN). The spectrum of the Laplacian on D, SpD( ), that is the set of its eigenvalues, is given by: SpD( ) = f : f = fg = fk2 n;m : n = 0; 1; 2; : : :m = 1; 2; : : :g (Dirichlet) SpN D( ) = f : fN = fNg = fk0 n;m 2 : n = 0; 1; 2; : : :m = 1; 2; : : :g (Neumann) Finally, we normalize the L2 norm of the eigenfunctions on D, namely: R D F2 da = 1 (here and further on we use the notation F for the normalized eigenfunctions and f for arbitrary eigenfunctions). Under these conditions, we study the growth rate of the L1 norm of the normalized eigenfunctions, jjF jj1, in relation to . It is important to mention that the L1 norm of a function on a given domain corresponds to the iv maximum of its absolute value on the domain. Note that depends on two parameters, m and n, and the relation between and the L1 norm depends on the regime at which m and n change as goes to in nity. Studying the behavior of the L1 norm is linked to the study of the set E(D) which is the set of accumulation points of log(jjF jj1)= log : One of our main results is that [7=36; 1=4] E(B2) [1=18; 1=4]: The thesis is organized as follows. Introduction and main results are presented in chapter 1. In chapter 2 we review some well-known facts regarding the eigenfunctions of the Laplacian on the disk and the properties of the Bessel functions. In chapter 3 we prove results on pointwise growth of eigenfunctions. In particular, we show that, if m=n ! 0, then, for any xed point (r; ) on D, the value of F (r; ) decreases exponentially as ! 1. In chapter 4 we study the growth of the L1 norm. Eigenfunctions of the Neumann problem are discussed in chapter 5. Some numerical results are presented in chapter 6. A discussion and a summary of our work could be found in chapter 7.

Dans le domaine de la synthèse d'image, la simulation de l'apparence visuelle des matériaux nécessite, la résolution rigoureuse de l'équation du transport de la lumière. Cela implique d'incorporer dans les modèles tous les éléments pouvant avoir une influence sur la luminance spectrale énergétique reçue par l'œil humain. La caractérisation des propriétés de réflectance des matériaux, encore sujette à de nombreuses recherches, est très évoluée. Cependant, l'utilisation de cartes d'environnement, pour simuler leurs comportements visuels restent essentiellement trichromatiques. Caractériser la lumière naturelle avec précision, est une interrogation ancienne et il n'existe pas aujourd'hui de cartes d'environnement comportant à la fois les informations de luminance spectrale énergétique et de polarisations correspondant à des ciels réels. Il nous est donc apparu nécessaire, de proposer à la communauté de l'informatique graphique des environnements lumineux complets exploitables dans un moteur de rendu adapté en conséquence.Dans ce travail, nous exploitons des résultats issus d'autres domaines scientifiques tels que la météorologie, la climatologie..., pour proposer un modèle de ciel clair, c'est-à-dire sans nuage.Toutes les situations réelles ne pouvant pas être abordées par cette méthode, nous développons et caractérisons un dispositif de capture d'environnement lumineux incorporant à la fois, la gamme dynamique de l'éclairage, la répartition spectrale et les états de polarisation.Nous proposons, dans le but de standardiser les échanges, un format de données utilisable dans un moteur de rendu spectral, exploitant le formalisme de "Stokes - Mueller". / In the field of computer graphics, the simulation of the visual appearance of materials requires, a rigorous solving of the light transport equation. This implies to incorporate into models all elements that can influence the spectral received by human eyes. The characterization of the reflectance properties of materials, still subject to many researches is very advanced. However, the uses of environment maps, to simulate their visual behaviors remain essentially trichromaticity. Characterize the natural light with precision, is an old question. Today, there are no environment maps, including both spectral radiance and polarization informations, corresponding to a real sky. It was therefore necessary for us to design and propose to the computer graphics community a full of bright environments exploitable in a rendering engine adapted accordingly. In this work, we use the results of other scientific fields as meteorology, climatology..., to propose a new model of clear sky. As all actual situations are not addressed by this method, we develop and characterize an environment capturing device both incorporating the light dynamic range, the spectral distribution and the polarization states.