Involutions and fixed-point-free involutions arise naturally as representatives for certain Borel orbits in invertible matrices. Similarly, partial involutions and partial fixed-point-free involutions represent certain Borel orbits in matrices which are not necessarily invertible. Inclusion relations among Borel orbit closures induce a partial order on these discrete parameterizing sets. In this dissertation we investigate the associated order complex of these posets. In particular, we prove that the order complex of the Bruhat poset of Borel orbit closures is shellable in symmetric as well as skew-symmetric matrices. / acase@tulane.edu

We obtain a tight upper bound for the genus of a partition, and calculate the number of partitions of maximal genus. The generating series for genus zero and genus one rooted hypermonopoles is obtained in closed form by specializing the genus series for hypermaps. We discuss the connection between partitions and rooted hypermonopoles, and suggest how a generating series for genus one partitions may be obtained via the generating series for genus one rooted hypermonopoles. An involution on the poset of genus one partitions is constructed from the associated hypermonopole diagrams, showing that the poset is rank-symmetric. Also, a symmetric chain decomposition is constructed for the poset of genus one partitions, which consequently shows that it is strongly Sperner.

We obtain a tight upper bound for the genus of a partition, and calculate the number of partitions of maximal genus. The generating series for genus zero and genus one rooted hypermonopoles is obtained in closed form by specializing the genus series for hypermaps. We discuss the connection between partitions and rooted hypermonopoles, and suggest how a generating series for genus one partitions may be obtained via the generating series for genus one rooted hypermonopoles. An involution on the poset of genus one partitions is constructed from the associated hypermonopole diagrams, showing that the poset is rank-symmetric. Also, a symmetric chain decomposition is constructed for the poset of genus one partitions, which consequently shows that it is strongly Sperner.

La transmission de données multimédia sur les réseaux sécurisés a une croissance exponentielle grâce aux progrès scientifique dans les technologies de l'information et de la communication. La sécurité des données dans certaines applications comme le stockage sécurisé, l'authentification, la protection des droits d'auteurs, la communication militaire ou la visioconférence confidentielles, nécessitent de nouvelles stratégies en matière de transmission sécurisée. Deux techniques sont couramment utilisées pour la transmission sécurisée de données visuelles, à savoir : la cryptographie et la stéganographie. La cryptographie sécurise les données en utilisant des clés secrètes afin de rendre les données illisibles, la stéganographie, elle, vise à insérer des données cruciales dans des signaux porteurs anodins.De plus, pour la confiance mutuelle et les systèmes distribués, le partage sécurisé de ressources est souvent une garantie suffisante pour les applications de communication. L'objectif principal de cette thèse est de réaliser une protection des données visuelles, en particulier les images numériques, par le biais des techniques modernes de cryptographie. Dans ce contexte, deux objectifs de recherche ont été développés durant ces travaux de thèse.La première partie de notre travail se concentre sur la sécurité des images numériques dans un environnement partagé. Ensuite, la deuxième partie porte sur l'intégrité des données visuelles pendant une transmission sécurisée.Nous avons proposé un nouveau schéma de partage des images qui exploite les propriétés d'addition et de multiplication homomorphique de deux crypto systèmes à clé publique largement utilisés : les algorithmes RSA et Paillier. Dans les schémas traditionnels de partage sécurisé, le ``dealer'' partitionne le secret en parties et le distribue à chacun des autres acteurs. Ainsi, aucun des acteurs impliqués ne participe à la création du partage sécurisé, mais il est toujours possible que le ``dealer'' transmette des données malveillantes. Au contraire, l'approche proposée utilise le système de partage de secret d'une manière qui limite l'influence du ‘‘dealer'' sur le protocole en permettant à chaque acteur de participer.La deuxième partie de ces travaux de thèse met l'accent sur l'intégrité des données visuelles lors de la transmission. L'intégrité des données signifie que les données gardent leurs structures complètes au cours d'une opération numérique comme le stockage, le transfert ou la récupération. Le changement d'un seul bit de données cryptées peut avoir un impact catastrophique sur les données décryptées. Nous abordons le problème de correction d'erreurs dans les images cryptées en utilisant le chiffrement à clé symétrique AES (Advanced Encryptions Standard) suivant différents modes. Trois mesures sont proposées afin d'exploiter les statistiques locales des données visuelles et l'algorithme de chiffrement, dans l'objectif de corriger les erreurs efficacement. / Due to the advancements in the information and communication technologies, the transmission of multimedia data over secure or insecure communication channels has increased exponentially. The security of data in applications like safe storage, authentications, copyright protection,remote military image communication or confidential video-conferencing require new strategies for secure transmission. Two techniques are commonly used for the secure transmission of visual data, i.e. cryptography and steganography. Cryptography achieves security by using secret keysto make the data illegible while steganography aims to hide the data in some innocent carrier signal. For shared trust and distributed environment, secret sharing schemes provide sufficient security in various communication applications. The principal objective of this thesis is to achieveprotection of visual data especially images through modern cryptographic techniques. In this context, the focus of the work in perspective, is twofolded. The first part of our work focuses on the security of image data in shared environment while the second part focuses on the integrity ofimage data in the encrypted domain during transmission.We proposed a new sharing scheme for images which exploits the additive and multiplicative homomorphic properties of two well known public key cryptosystems, namely, the RSA and the Paillier. In traditional secret sharing schemes, the dealer partitions the secret into shares and distributethe shares to each of the player. Thus, none of the involved players participate in the creation of the shared secret and there is always a possibilitythat the dealer can cheat some player. On the contrary, the proposed approach employs the secret sharing scheme in a way that limits the influence of the dealer over the protocol by allowing each player to participate. The second part of our thesis emphasizes on the integrity of visual data during transmission. Data integrity means that the data have its complete structure during any operation like storage, transfer or retrieval. A single bit change in encrypted data can have catastrophic impact over the decrypted data. We address the problem of error correction in images encrypted using symmetric key cryptosystem of the Advanced Encryption Standard (AES) algorithm. Three methods are proposed to exploit the local statistics of the visual data and the encryption algorithm to successfully correct the errors.

Yangian symmetric correlators are defined as eigenfunctions of monodromy operators. Examples and relations are given and an evaluation method of the symmetry condition is described.

In this thesis we study relativistic models of gravitating uids with heat ow and electric charge. Firstly, we derive the model of a charged shear-free spherically symmetric cosmological model with heat ow. The solution of the Einstein-Maxwell equations of the system is governed by the pressure isotropy condition. This condition is a highly nonlinear partial di erential equation. We analyse this master equation using Lie's group theoretic approach. The Lie symmetry generators that leave the equation invariant are found. We provide exact solutions to the gravitational potentials using the rst symmetry admitted by the equation. Our new exact solutions contain the earlier results of Msomi et al (2011) without charge. Using the second symmetry we are able to reduce the order of the master equation to a rst order highly nonlinear di erential equation. Secondly, we study a shear-free spherically symmetric cosmological model with heat ow in higher dimensions. We establish the Einstein eld equations and nd the governing pressure isotropy condition. We use an algorithm due to Deng (1989) to provide several new classes of solutions to the model. The four-dimensional case is contained in our general result. Solutions due to Bergmann (1981), Maiti (1982), Modak (1984) and Sanyal and Ray (1984) for the four-dimensional case are regained. We also establish a new class of solutions that contains the results of Deng (1989) from four dimensions. / Thesis (M.Sc.)-University of KwaZulu-Natal, Durban, 2011.

We give natural correspondences of odd-degree characters of the symmetric groups and some of their subgroups, which can be described easily by restriction of characters, degrees and multiplicities.

Master of Science / Biomedical Sciences / Gregory F. Grauer / Chronic kidney disease (CKD) is a potentially life-threatening disease that reportedly affects 10% of dogs and 30% of cats over the age of 15. There is no cure available for CKD, but medical management is available for patients with this disease. Research has focused on earlier detection of CKD with the goal of instituting medical management and monitoring as early in the disease course as possible. Symmetric dimethylarginine (SDMA) has recently emerged as a novel renal excretory biomarker that may aid in early detection of CKD in cats and dogs. SDMA is non-protein bound and is freely filtered by the glomerulus, is not secreted or reabsorbed, and has greater than 90% excretion by the kidneys, making it a potential target for measurement of glomerular filtration rate (GFR). Previous studies have demonstrated a close parallel between SDMA and serum creatinine (sCr), which is the currently favored serum biomarker for assessment of GFR. Research has also demonstrated a correlation between SDMA and GFR. Serum concentrations of SDMA increase above normal when GFR is decreased by 25-40%; much earlier than the 75% decrease in GFR typically required for sCr to increase above its reference interval. The studies reported here demonstrate a potential use for the SDMA:sCr ratio as a predictor of volume responsive azotemia. Furthermore, longitudinal assessment of older dogs and cats for early detection of CKD showed that SDMA was a more sensitive indicator of CKD than sCr. The evaluation of SDMA reported in this thesis presents a novel perspective on SDMA and its use clinically.

PT-symmetric Hamiltonians proposed by Bender and Boettcher can have real energy spectra. As an extension of the Hermitian Hamiltonian, PT-symmetric systems have attracted a great interest in recent years. Understanding the underlying mathematical structure of these theories sheds insight on outstanding problems of physics. These problems include the nature of Higgs particles, the properties of dark matter, the matter-antimatter asymmetry in the universe, and neutrino oscillations. Furthermore, PT-phase transition has been observed in lasers, optical waveguides, microwave cavities, superconducting wires and circuits. The objective of this thesis is to extend the iterative method of solving Schrodinger equation used for an harmonic oscillator systems to Hamiltonians with PT-symmetric potentials. An important aspect of this approach is the high accuracy of eigenvalues and the fast convergence. Our method is a combination of Hill determinant method [8] and the power series expansion. eigenvalues and the fast convergence. One can transform the Schrodinger equation into a secular equation by using a trial wave function. A recursion structure can be obtained using the secular equation, which leads to accurate eigenvalues. Energy values approach to exact ones when the number of iterations is increased. We obtained eigenvalues for a set of PT-symmetric Hamiltonians.

Student Number : 9803192Y - MSc(Eng) Dissertation - School of Mechanical, Industrial and Aeronautical Engineering - Faculty of Engineering and the Built Environment / A computational °uid dynamics (CFD) study was carried out to improve the aero- dynamic performance of an agile high angle of attack missile. The normal force generated by the missile strakes had to be increased at the low angles of attack and the large side forces, experienced at high angles of attack due to the formation of steady asymmetric vortices had to be eliminated using strakelets on the missile nose. The ¯rst objective was achieved by increasing the missile strake span from 0:06D to 0:13D. The larger strake span increased the e®ective diameter of the missile body and prevented °ow reattachment to the body, a problem that was experienced when the strake span was 0:06D. Due to °ow separating further away from the body, strong vortices formed on the missile strakes, resulting in an increase in the normal force generated by the missile strakes at low angles of attack. The second objective was two-fold. Prior to analysing the e®ect of the strakelets on a steady asymmetric °ow¯eld, the steady asymmetric °ow¯eld had to ¯rst be created. This was achieved by placing a permanent, geometric perturbation on the missile nose. The size of the perturbation used in the study, which was determined by an iterative process, did not force °ow separation at low angles of attack and resulted in a steady asym- metric °ow¯eld that was representative of that on a blunt-ogive body. The e®ect of changing the span of the strakelets and the axial position of the strakelets were then investigated. It was found that the strakelets with a span of 0:09D, placed 1D from the nose tip eliminated the side forces by forcing vortex symmetry. Increasing or decreasing the span of the strakelet, positioned 1D from the nose tip or placing the strakelets with a span of 0:09D closer or further away from the nose tip did not eliminate the steady vortex asymmetry.