Invariants, Lie-Bäcklund operators and Bäcklund transformations

This thesis is mainly concerned with the application of groups of transformations to differential equations and in particular with the connection between the group structure of a given equation and the existence of exact solutions and conservation laws. In this respect the Lie-Bäcklund groups of tangent transformations, particular cases of which are the Lie tangent and the Lie point groups, are extensively used.

In Chapter I we first review the classical results of Lie, Bäcklund and Bianchi as well as the more recent ones due mainly to Ovsjannikov. We then concentrate on the Lie-Bäcklund groups (or more precisely on the corresponding Lie-Bäcklund operators), as introduced by Ibragimov and Anderson, and prove some lemmas about them which are useful for the following chapters. Finally we introduce the concept of a conditionally admissible operator (as opposed to an admissible one) and show how this can be used to generate exact solutions.

In Chapter II we establish the group nature of all separable solutions and conserved quantities in classical mechanics by analyzing the group structure of the Hamilton-Jacobi equation. It is shown that consideration of only Lie point groups is insufficient. For this purpose a special type of Lie-Bäcklund groups, those equivalent to Lie tangent groups, is used. It is also shown how these generalized groups induce Lie point groups on Hamilton's equations. The generalization of the above results to any first order equation, where the dependent variable does not appear explicitly, is obvious. In the second part of this chapter we investigate admissible operators (or equivalently constants of motion) of the Hamilton-Jacobi equation with polynornial dependence on the momenta. The form of the most general constant of motion linear, quadratic and cubic in the momenta is explicitly found. Emphasis is given to the quadratic case, where the particular case of a fixed (say zero) energy state is also considered; it is shown that in the latter case additional symmetries may appear. Finally, some potentials of physical interest admitting higher symmetries are considered. These include potentials due to two centers and limiting cases thereof. The most general two-center potential admitting a quadratic constant of motion is obtained, as well as the corresponding invariant. Also some new cubic invariants are found.

In Chapter III we first establish the group nature of all separable solutions of any linear, homogeneous equation. We then concentrate on the Schrodinger equation and look for an algorithm which generates a quantum invariant from a classical one. The problem of an isomorphism between functions in classical observables and quantum observables is studied concretely and constructively. For functions at most quadratic in the momenta an isomorphism is possible which agrees with Weyl' s transform and which takes invariants into invariants. It is not possible to extend the isomorphism indefinitely. The requirement that an invariant goes into an invariant may necessitate variants of Weyl' s transform. This is illustrated for the case of cubic invariants. Finally, the case of a specific value of energy is considered; in this case Weyl's transform does not yield an isomorphism even for the quadratic case. However, for this case a correspondence mapping a classical invariant to a quantum orie is explicitly found.

Chapters IV and V are concerned with the general group structure of evolution equations. In Chapter IV we establish a one to one correspondence between admissible Lie-Bäcklund operators of evolution equations (derivable from a variational principle) and conservation laws of these equations. This correspondence takes the form of a simple algorithm.

In Chapter V we first establish the group nature of all Bäcklund transformations (BT) by proving that any solution generated by a BT is invariant under the action of some conditionally admissible operator. We then use an algorithm based on invariance criteria to rederive many known BT and to derive some new ones. Finally, we propose a generalization of BT which, among other advantages, clarifies the connection between the wave-train solution and a BT in the sense that, a BT may be thought of as a variation of parameters of some. special case of the wave-train solution (usually the solitary wave one). Some open problems are indicated.

Most of the material of Chapters II and III is contained in [I], [II], [III] and [IV] and the first part of Chapter V in [V].

Ultra-compact silicon-on-insulator optical filter based on sidewall Bragg grating

Micro-cavity structure composed of silicon wire with 240nm square cross section and two symmetrical sidewall waveguide Bragg gratings is fabricated and studied for the operation under telecommunication wavelengths. Optical filter of quasi-TE mode was realized based on this cavity. In such micro-cavity, optical quality factor (Q) was measured up to 380 with a 4.8nm free spectral range (FSR) and 12dB fringe contrast (FC). The measured group index of silicon waveguide with only 240nm square cross section was between 3.80 and 5.43. It is the first time group delay of silicon wire waveguide with such small core dimension is studied. Larger group delay can be expected after optimizing the design parameters and the fabrication process.

Theoretical and Experimental Studies of an Ultra-compact Photonic Crystal Corner Mirror Based on Silicon-On-Insulator

An ultra-compact silicon-on-insulator based photonic crystal corner mirror is designed and optimized. A sample is then successfully fabricated with extra losses 1.1 +/- 0.4dB for transverse-electronic (M) polarization for wavelength range of 1510-1630nm.

Decay measures on locally compact abelian topological groups

Source: PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS Volume: 131 Pages: 1257-1273 Part: Part 6 Published: 2001 Times Cited: 5 References: 23 Citation MapCitation Map beta Abstract: We show that the Banach space M of regular sigma-additive finite Borel complex-valued measures on a non-discrete locally compact Hausdorff topological Abelian group is the direct sum of two linear closed subspaces M-D and M-ND, where M-D is the set of measures mu is an element of M whose Fourier transform vanishes at infinity and M-ND is the set of measures mu is an element of M such that nu is not an element of MD for any nu is an element of M \ {0} absolutely continuous with respect to the variation \mu\. For any corresponding decomposition mu = mu(D) + mu(ND) (mu(D) is an element of M-D and mu(ND) is an element of M-ND) there exist a Borel set A = A(mu) such that mu(D) is the restriction of mu to A, therefore the measures mu(D) and mu(ND) are singular with respect to each other. The measures mu(D) and mu(ND) are real if mu is real and positive if mu is positive. In the case of singular continuous measures we have a refinement of Jordan's decomposition theorem. We provide series of examples of different behaviour of convolutions of measures from M-D and M-ND.

The rotational broadening of V395 Carinae. Implications on the compact object's mass

Context: The masses previously obtained for the X-ray binary 2S 0921-630 inferred a compact object that was either a high-mass neutron star or low-mass black-hole, but used a previously published value for the rotational broadening (v sin i) with large uncertainties. Aims: We aim to determine an accurate mass for the compact object through an improved measurement of the secondary star's projected equatorial rotational velocity. Methods: We have used UVES echelle spectroscopy to determine the v sin i of the secondary star (V395 Car) in the low-mass X-ray binary 2S 0921-630 by comparison to an artificially broadened spectral-type template star. In addition, we have also measured v sin i from a single high signal-to-noise ratio absorption line profile calculated using the method of Least-Squares Deconvolution (LSD). Results: We determine v sin i to lie between 31.3±0.5 km s-1 to 34.7±0.5 km s-1 (assuming zero and continuum limb darkening, respectively) in disagreement with previous results based on intermediate resolution spectroscopy obtained with the 3.6 m NTT. Using our revised v sin i value in combination with the secondary star's radial velocity gives a binary mass ratio of 0.281±0.034. Furthermore, assuming a binary inclination angle of 75° gives a compact object mass of 1.37±0.13 M_?. Conclusions: We find that using relatively low-resolution spectroscopy can result in systemic uncertainties in the measured v sin i values obtained using standard methods. We suggest the use of LSD as a secondary, reliable check of the results as LSD allows one to directly discern the shape of the absorption line profile. In the light of the new v sin i measurement, we have revised down the compact object's mass, such that it is now compatible with a canonical neutron star mass.

VLT FORS spectra of blue supergiants in the Local Group galaxy NGC 6822

Half hour exposures using the ESO VLT/FORS1 combination at Paranal in Chile have allowed us to obtain spectra for three B supergiants in the dwarf irregular galaxy NGC 6822. The spectra have been analysed using non-LTE techniques and temperatures, gravities, helium content and abundances have been obtained. Overall the metallicity of NGC 6822 is found to lie between that of the LMC and of the SMC, in agreement with previous observations of H II regions and in contrast to the earlier findings of Massey et al. (1995). The analysis of H-alpha yields estimates of the mass-loss rates and wind momenta. These results demonstrate that significantly longer exposures with the same instruments will allow us to perform quantitative spectroscopy of blue supergiants in galaxies far beyond the Local Group.

An isomorphism between group C*-algebras of ax + b-like groups

We give a necessary and sufficient condition for two ax+b-like groups to have isomorphic C*-algebras. In particular, we show that there are many non-isomorphic ax+b -like Lie groups having isomorphic group C*-algebras.

Thermolyzable polymer networks and star polymers containing a novel, compact, degradable acylal-based dimethacrylate cross-linker: Synthesis, characterization, and thermolysis

A compact, cleavable acylal dimethacrylate cross-linker, 1,1-ethylenediol dimethacrylate (EDDMA), was synthesized from the anhydrous iron(III) chloride-catalyzed reaction between methacrylic anhydride and acetaldehyde. The ability of EDDMA to act as cross-linker was demonstrated by using it for the preparation of one neat cross-linker network, four star polymers of methyl methacrylate (MMA), and four randomly cross-linked MMA polymer networks using group transfer polymerization (GTP). For comparison, the corresponding polymer structures based on the commercially available ethylene glycol dimethacrylate (EGDMA) cross-linker (isomer of EDDMA) were also prepared via GTR The number of arms of the EDDMA-based star polymers was lower than that of the corresponding EGDMA polymers, whereas the degrees of swelling in tetrahydrofuran of the EDDMA-based MMA networks were higher than those of their EGDMA-based counterparts. Although none of the EDDMA-containing polymers could be cleanly hydrolyzed under basic or acidic conditions, they could be thermolyzed at 200 degrees C within 1 day giving lower molecular weight products.

Sets of multiplicity and closable multipliers on group algebras

We undertake a detailed study of the sets of multiplicity in a second countable locally compact group G and their operator versions. We establish a symbolic calculus for normal completely bounded maps from the space B(L-2(G)) of bounded linear operators on L-2 (G) into the von Neumann algebra VN(G) of G and use it to show that a closed subset E subset of G is a set of multiplicity if and only if the set E* = {(s,t) is an element of G x G : ts(-1) is an element of E} is a set of operator multiplicity. Analogous results are established for M-1-sets and M-0-sets. We show that the property of being a set of multiplicity is preserved under various operations, including taking direct products, and establish an Inverse Image Theorem for such sets. We characterise the sets of finite width that are also sets of operator multiplicity, and show that every compact operator supported on a set of finite width can be approximated by sums of rank one operators supported on the same set. We show that, if G satisfies a mild approximation condition, pointwise multiplication by a given measurable function psi : G -> C defines a closable multiplier on the reduced C*-algebra G(r)*(G) of G if and only if Schur multiplication by the function N(psi): G x G -> C, given by N(psi)(s, t) = psi(ts(-1)), is a closable operator when viewed as a densely defined linear map on the space of compact operators on L-2(G). Similar results are obtained for multipliers on VN(C).

Sets of p-multiplicity in locally compact groups

We initiate the study of sets of p-multiplicity in locally compact groups and their operator versions. We show that a closed subset E of a second countable locally compact group G is a set of p-multiplicity if and only if E∗={(s,t):ts−1∈E} is a set of operator p-multiplicity. We exhibit examples of sets of p-multiplicity, establish preservation properties for unions and direct products, and prove a p-version of the Stone–von Neumann Theorem.

A historical case study analysis of the establishment of charismatic leadership in a Protestant Reformation cultic group and its role in the recourse to violence

La recherche sur les questions touchant aux leaders de groupes sectaires et à la violence sectaire a mené à l’étude du rôle joué par l’autorité charismatique, tel que défini par Weber (1922) et repris par Dawson (2010). À ce sujet, d’éminents spécialistes des études sur les sectes sont d’avis qu’un vide important dans la recherche sur l’autorité charismatique dans le contexte de groupes sectaires et de nouveaux mouvements religieux reste à combler (ajouter les références ‘d’éminents spécialistes’). Ce mémoire vise à contribuer à l’étude cet aspect négligé, le rôle de l’autorité charismatique dans le recours è la violence dans les groupes sectaires, par une étude de cas historique d’un groupe de la Réformation protestante du XVIe siècle, le Royaume anabaptiste de Münster (AKA), sous l’influence d’un leader charismatique, Jan van Leiden. Cette recherche s’intéresse plus spécifiquement aux divers moyens utilisés par Jan van Leiden, pour asseoir son autorité charismatique et à ceux qui ont exercé une influence sur le recours à des actes de violence. L’étude de cas est basé sur le matériel provenant de deux comptes-rendus des faits relatés par des participants aux événements qui se sont déroulés à pendant le règne de Leiden à la tête du AKA. L’analyse du matériel recueilli a été réalisé à la lumière de trois concepts théoriques actuels concernant le comportement cultuel et le recours à la violence.. L’application de ces concepts théoriques a mené à l’identification de quatre principales stratégies utilisées par Jan van Leiden pour établir son autorité charismatique auprès de ses disciples, soit : 1) la menace du millénarisme, 2) l’exploitation d’une relation bilatérale parasitique avec ses disciples, 3) l’utilisation de l’extase religieuse et de la prophétie, 4) l’utilisation du désir de voir survenir des changements sociaux et religieux. En plus de ces quatre stratégies, trois autres dimensions ont été retenues comme signes que le recours à la violence dans le Royaume anabaptiste de Münster résultait de l’établissement de l’autorité charismatique de son leader, soit : 1) la violence liée au millénarisme, 2) la notion d’identité et de violence partagée, 3) des facteurs systémiques, physiques et culturels menant à la violence.

On some Density Theorems in Number Theory and Group Theory

Gowers, dans son article sur les matrices quasi-aléatoires, étudie la question, posée par Babai et Sos, de l'existence d'une constante $c>0$ telle que tout groupe fini possède un sous-ensemble sans produit de taille supérieure ou égale a $c|G|$. En prouvant que, pour tout nombre premier $p$ assez grand, le groupe $PSL_2(\mathbb{F}_p)$ (d'ordre noté $n$) ne posséde aucun sous-ensemble sans produit de taille $c n^{8/9}$, il y répond par la négative. Nous allons considérer le probléme dans le cas des groupes compacts finis, et plus particuliérement des groupes profinis $SL_k(\mathbb{Z}_p)$ et $Sp_{2k}(\mathbb{Z}_p)$. La premiére partie de cette thése est dédiée à l'obtention de bornes inférieures et supérieures exponentielles pour la mesure suprémale des ensembles sans produit. La preuve nécessite d'établir préalablement une borne inférieure sur la dimension des représentations non-triviales des groupes finis $SL_k(\mathbb{Z}/(p^n\mathbb{Z}))$ et $Sp_{2k}(\mathbb{Z}/(p^n\mathbb{Z}))$. Notre théoréme prolonge le travail de Landazuri et Seitz, qui considérent le degré minimal des représentations pour les groupes de Chevalley sur les corps finis, tout en offrant une preuve plus simple que la leur. La seconde partie de la thése à trait à la théorie algébrique des nombres. Un polynome monogéne $f$ est un polynome unitaire irréductible à coefficients entiers qui endengre un corps de nombres monogéne. Pour un nombre premier $q$ donné, nous allons montrer, en utilisant le théoréme de densité de Tchebotariov, que la densité des nombres premiers $p$ tels que $t^q -p$ soit monogéne est supérieure ou égale à $(q-1)/q$. Nous allons également démontrer que, quand $q=3$, la densité des nombres premiers $p$ tels que $\mathbb{Q}(\sqrt[3]{p})$ soit non monogéne est supérieure ou égale à $1/9$.

A study of the polycomb group complexes in the maintenance of heterochromatic genome stability and Alzheimer's disease.

La démence d'Alzheimer est une maladie neurodégénérative caractérisée par une perte progressive et irreversible des fonctions cognitives et des compétences intellectuelles. La maladie d’Alzheimer se présente sous deux formes: la forme familiale ou précoce (EOAD) qui représente 5% des cas et elle est liée à des mutations génétiques affectant le métabolisme des peptides amyloïde; et la forme tardive ou sporadique (LOAD) qui représente 95% des cas mais son étiologie est encore mal définie. Cependant, le vieillissement reste le principal facteur de risque pour développer LOAD. Les changements épigénétiques impliquant des modifications des histones jouent un rôle crucial dans les maladies neurodégénératives et le vieillissement lié à l'âge. Des données récentes ont décrit LOAD comme un désordre de l'épigénome et ont associé ce trouble à l'instabilité génomique. Les protéines Polycomb sont des modificateurs épigénétiques qui induisent le remodelage de la chromatine et la répression des gènes à l'hétérochromatine facultative. Nous rapportons que les souris hétérozygotes pour une protéine Polycomb développent avec l'âge un trouble neurologique ressemblant à LOAD caractérisé par l’altération des fonctions cognitives, la phosphorylation de la protéine tau, l'accumulation des peptides amyloïde, et le dysfonctionnement synaptique. Ce phénotype pathologique est précédé par la décondensation de l’hétérochromatine neuronale et l'activation de la réponse aux dommages à l'ADN. Parallèlement, une réduction d’expression de polycomb, malformations de l'hétérochromatine neuronale, et l'accumulation de dommages à l'ADN étaient également présents dans les cerveaux de patients LOAD. Remarquablement, les dommages de l'ADN ne sont pas distribués de façon aléatoire sur le génome mais sont enrichis au niveau des séquences répétitives. Les conclusions présentées dans cette thèse ont identifié des modifications épigénétiques spécifiques qui conduisent à une instabilité génomique aberrante menant à la formation de LOAD. Ces résultats vont aider au développement de nouveaux traitements qui peuvent potentiellement ralentir la neurodégénérescence.

Analytical Equations for Compact Dual Frequency Microstrip Antenna

Design equations are presented for calculating the resonance frequencies for a compact dual frequency arrow-shaped microstrip antenna. This provides a fast and simple way to predict the resonant frequencies of the antenna. The antenna is also analyzed using the IE3D simulation package. The theoretical predictions are found to be very close to the IE3D results and thus establish the validity of the design formulae

Slot coupled sqaure microstrip antenna for compact dual-frequency operation

A novel dual frequency design of a square microstrip antenna coaxially fed along the diagonal with a chip capacitor mounted at the bottom of the stub is introduced. This design provides enhanced area reduction and good cross-polarization levels. The antenna design can be used as a compact antenna system where limited size is a major requirement. The details of the antenna design and experimental results are presented