It is known that the ring of invariants of any two-row group is Cohen-Macaulay. This result inspired the conjecture that the ring of invariants of any two-row group is a complete intersection. In this thesis, we study this conjecture in the case where the ground field is the prime field $\mathbb{F}_p$. We prove that all Abelian reflection two-row $p$-groups have complete intersection invariant rings. We show that all two-row groups with \textit{non-normal} Sylow $p$-subgroups have polynomial invariant rings. We also show that reflection two-row groups with \textit{normal} reflection Sylow $p$-subgroups have polynomial invariant rings. As an interesting application of a theorem of Nakajima about hypersurface invariant rings, we rework a classical result which says that the invariant rings of subgroups of $\text{SL}(2,\,p)$ are all hypersurfaces. In addition, we obtain a result that characterizes Nakajima $p$-groups in characteristic $p$, namely, if the invariant ring is generated by norms, then the group is a Nakajima $p$-group. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2009-09-29 15:08:40.705

Cette thèse s’articule autour des espaces de modules de représentations de carquois arbitraires, c’est-à-dire possédant d’éventuelles boucles. Nous obtenons trois types de résultats. Le premier concerne la base canonique de Lusztig, dont la définition est étendue à notre cadre, notamment en introduisant une algèbre de Hopf généralisant les groupes quantiques usuels (i.e. associés aux algèbres de Kac-Moody symétriques). On démontre au passage une conjecture faite par Lusztig en 1993, portant sur la catégorie de faisceaux pervers qu’il définit sur les variétés de représentations de carquois.Le second type de résultats, également inspiré par le travail de Lusztig, concerne la base semi- canonique et la variété Lagrangienne nilpotent de Lusztig. Pour un carquois arbitraire, on définit des sous-variétés de représentations semi-nilpotentes Λ(α), et nous montrons qu’elles sont Lagrangiennes. La démonstration repose sur l’existence de fibrations affines partielles entre diverses composantes de Λ(α), contrôlées par une combinatoire précise. Nous définissons une algèbre de convolution de fonctions constructibles sur ⊔Λ(α), et montrons qu’elle possède une base formée de fonctions quasi- caractéristiques des composantes irréductibles des Λ(α). La structure combinatoire qui se dégage ici est analogue à celle obtenue sur les faisceaux pervers de Lusztig, et fait apparaître des opérateurs plus généraux que ceux décrits par les cristaux de Kashiwara.Le troisième thème considéré est celui des variétés carquois de Nakajima, dont l’étude géomé- trique menée ici permet, conjointement avec ce qui est fait précédemment, de donner une définition de cristaux de Kashiwara généralisés. On définit à nouveau des sous-variétés Lagrangiennes, ainsi qu’un produit tensoriel sur leurs composantes irréductibles, comme fait dans le cas classique par Nakajima. / This thesis is about the moduli spaces of representations of arbitrary quivers, i.e. possibly carrying loops. We obtain three types of results. The first one deals with the Lusztig canonical basis, whose definition is here extended to our framework, thanks in particular to the definition of a Hopf algebra generalizing the usual quantum groups (i.e. associated to symmetric Kac-Moody algebras). We also prove a conjecture raised by Lusztig in 1993, which concerns the category of perverse sheaves he defines on varieties of representations of quivers.The second type of results, also inspired by the work of Lusztig, concerns the semicanonical basis. For an arbitrary quiver, we define subvarieties of seminilpotent representations Λ(α), and we show that they are Lagrangian. The proof relies on the existence of partial affine fibrations between some irreducible components of Λ(α), controled by a precise combinatorial structure. We define a convolution algebra of constructible functions on ⊔Λ(α), and show it is equipped with a basis of quasi-characteristic functions of the irreducible components of the Λ(α). The combinatorial structure arising from this construction is analogous to the one obtained on Lusztig perverse sheaves, and yields operators more general than the ones described by Kashiwara crystals.The third considered topic is the one of Nakajima quiver varieties, whose geometric study in this thesis allows, along with the previous (also geometric) work, to define generalized Kashiwara crystals. We define, again, Lagrangian subvarieties, and a tensor product of their irreducible components, as done by Nakajima on the classical case.

O objetivo deste trabalho é explorar um modelo para medidas quânticas de duração finita baseado na equação de Lindblad, com a análise de um sistema de 2 níveis acoplado a um reservatório térmico que ocasiona decoerência. A interação entre o sistema e o dispositivo de medida é markoviana, justificando o uso da equação de Lindblad para obter a dinâmica do processo de medida. Para analisar a influência do ambiente/reservatório térmico não-markoviano, cuja definição não inclui o aparato de medida, foi utilizada a abordagem de Redfield para a interação entre o sistema e o ambiente. Na teoria híbrida aqui exposta, para efetuar o traço parcial dos graus de liberdade do ambiente foi desenvolvido um método analítico baseado na álgebra de super-operadores e no uso dos super-operadores de Nakajima-Zwanzig. Foi verificado que medidas de duração finita sobre o sistema aberto de 2 níveis podem proteger o estado inicial dos efeitos do ambiente, desde que o observável medido não comute com a interação. Quando o observável medido comuta com a interação sistema-ambiente, a medida de duração finita acelera a decoerência induzida pelo ambiente. A validade das previsões analíticas foi testada comparando os resultados com uma abordagem numérica exata. Quando o acoplamento entre o sistema e o aparato de medida excede a faixa de validade da aproximação analítica, o estado inicial ainda é protegido pela medida de duração finita, como indicam os cálculos numéricos exatos. / The aim of this work is to explore a model for finite-time measurement based on the Lindblad equation, with analysis of a system consisting of a 2-level system coupled to a thermal reservoir. We assume a Markovian measuring device and, therefore, use a Lindbladian description for the measurement dynamics. For studying the case of noise produced by a non-Markovian environment, whose definition does not include the measuring apparatus, we use the Redfield approach to the interaction between system and environment. In the present hybrid theory, to trace out the environmental degrees of freedom, we introduce an analytic method based on superoperator algebra and Nakajima-Zwanzig superoperators. We show that measurements of finite duration performed on an open two-state system can protect the initial state from a phase-noisy environment, provided the measured observable does not commute with the perturbing interaction. When the measured observable commutes with the environmental interaction, the finite-duration measurement accelerates the rate of decoherence induced by the phase noise. We have tested the validity of the analytical predictions against an exact numerical approach. When the coupling between the system and the measuring apparatus increases beyond the range of validity of the analytical approximation, the initial state is still protected by the finite-time measurement, according with the exact numerical calculations.

O objetivo deste trabalho é explorar um modelo para medidas quânticas de duração finita baseado na equação de Lindblad, com a análise de um sistema de 2 níveis acoplado a um reservatório térmico que ocasiona decoerência. A interação entre o sistema e o dispositivo de medida é markoviana, justificando o uso da equação de Lindblad para obter a dinâmica do processo de medida. Para analisar a influência do ambiente/reservatório térmico não-markoviano, cuja definição não inclui o aparato de medida, foi utilizada a abordagem de Redfield para a interação entre o sistema e o ambiente. Na teoria híbrida aqui exposta, para efetuar o traço parcial dos graus de liberdade do ambiente foi desenvolvido um método analítico baseado na álgebra de super-operadores e no uso dos super-operadores de Nakajima-Zwanzig. Foi verificado que medidas de duração finita sobre o sistema aberto de 2 níveis podem proteger o estado inicial dos efeitos do ambiente, desde que o observável medido não comute com a interação. Quando o observável medido comuta com a interação sistema-ambiente, a medida de duração finita acelera a decoerência induzida pelo ambiente. A validade das previsões analíticas foi testada comparando os resultados com uma abordagem numérica exata. Quando o acoplamento entre o sistema e o aparato de medida excede a faixa de validade da aproximação analítica, o estado inicial ainda é protegido pela medida de duração finita, como indicam os cálculos numéricos exatos. / The aim of this work is to explore a model for finite-time measurement based on the Lindblad equation, with analysis of a system consisting of a 2-level system coupled to a thermal reservoir. We assume a Markovian measuring device and, therefore, use a Lindbladian description for the measurement dynamics. For studying the case of noise produced by a non-Markovian environment, whose definition does not include the measuring apparatus, we use the Redfield approach to the interaction between system and environment. In the present hybrid theory, to trace out the environmental degrees of freedom, we introduce an analytic method based on superoperator algebra and Nakajima-Zwanzig superoperators. We show that measurements of finite duration performed on an open two-state system can protect the initial state from a phase-noisy environment, provided the measured observable does not commute with the perturbing interaction. When the measured observable commutes with the environmental interaction, the finite-duration measurement accelerates the rate of decoherence induced by the phase noise. We have tested the validity of the analytical predictions against an exact numerical approach. When the coupling between the system and the measuring apparatus increases beyond the range of validity of the analytical approximation, the initial state is still protected by the finite-time measurement, according with the exact numerical calculations.

The thesis aimed to identify fracture strain for multiaxial loading for laminate material of Al and LDPE that is mostly used in the food packing technology specially in liquid packaging industry. These materials are thin and flexible which make it difficult to handle and test it. The investigation was through applying new Nakajima test setup on such a material by designing a custom setup and 3D printing the Nakajima setup prototype to be able to use it in this study. DIC technique was used to get the results from the Nakajima test and the software used to apply the DIC technique is GOM correlate. As a validation for measurement from such a technique, a tensile test has been done and measured strains were compared with those from the specimen from the Nakajima that has smallest width as it was close to a uniaxial loading. One more validation technique was by creating a FE-model using (Abaqus software) for the biaxial loading and comparing it to the biaxial loading results that obtained from the DIC analysis.

Cette thèse s'articule autour des espaces de modules de représentations de carquois arbitraires, c'est-à-dire possédant d'éventuelles boucles. Nous obtenons trois types de résultats. Le premier concerne la base canonique de Lusztig, dont la définition est étendue à notre cadre, notamment en introduisant une algèbre de Hopf généralisant les groupes quantiques usuels (i.e. associés aux algèbres de Kac-Moody symétriques). On démontre au passage une conjecture faite par Lusztig en 1993, portant sur la catégorie de faisceaux pervers qu'il définit sur les variétés de représentations de carquois.Le second type de résultats, également inspiré par le travail de Lusztig, concerne la base semi- canonique et la variété Lagrangienne nilpotent de Lusztig. Pour un carquois arbitraire, on définit des sous-variétés de représentations semi-nilpotentes Λ(α), et nous montrons qu'elles sont Lagrangiennes. La démonstration repose sur l'existence de fibrations affines partielles entre diverses composantes de Λ(α), contrôlées par une combinatoire précise. Nous définissons une algèbre de convolution de fonctions constructibles sur ⊔Λ(α), et montrons qu'elle possède une base formée de fonctions quasi- caractéristiques des composantes irréductibles des Λ(α). La structure combinatoire qui se dégage ici est analogue à celle obtenue sur les faisceaux pervers de Lusztig, et fait apparaître des opérateurs plus généraux que ceux décrits par les cristaux de Kashiwara.Le troisième thème considéré est celui des variétés carquois de Nakajima, dont l'étude géomé- trique menée ici permet, conjointement avec ce qui est fait précédemment, de donner une définition de cristaux de Kashiwara généralisés. On définit à nouveau des sous-variétés Lagrangiennes, ainsi qu'un produit tensoriel sur leurs composantes irréductibles, comme fait dans le cas classique par Nakajima.

This work aims at giving a systematic review of the numerical approaches and obtained results published in recent years. Focus is set on both the recent trends and achievements as well as challenges and open questions. Thesis critically reviews work done in previous thesis related to food packaging material and design, at Blekinge Institute of Technology with the collaboration of Tetra Pak liquid food packaging company. Here, 27 numbers of master’s thesis are critical review all the while using 4 PHD works for referencing.  Focus of these thesis works relates to the mechanics and its sub-branch fracture mechanics. Due to this all the theory related to mechanics and fracture mechanics, which is compulsory to know for understanding is defined initially. Main material which are under consideration throughout the work are PP, LDPE, PET-LDPE, and aluminum foil. As three materials are used in the liquid food packaging which are PE, paper board and aluminum. Uses of this material and there required physical properties are measured. Testing methods which are used to check whether the material meets the required parameters are tensile testing, tear testing, peel testing and shear test.  Also, some methods are used along with these tests to perform the test more precisely, to evaluate and compare result i.e., Nakajima test, Digital image correlation. DIC testing technique is used to compare the results obtained from simulation. It is also understood that DIC method provided the user with increased quality of obtained results.  Most of the topics of discussions have been reviewed for corresponding thesis, displaying the used ideas, theory, applied experimentation and realized conclusions. Using these above it was able to derive a conclusion based on the developed research questions and hypothesis. Defined problems in previous work are studied also solution as a future work is suggested to overcome these problems. Mainly the slippage issue while holding PE in clamps. Furthermore, various development is discussed which is done to introduce new material which is more possible being used in the future.

Le travail effectué dans cette thèse consiste à construire et adapter dans d'autres cadres des objets issus de la géométrie algébrique. Nous nous intéressons d'abord à la théorie des classes de Chern pour les faisceaux cohérents. Sur les variétés projectives, elle est complètement achevée dans les anneaux de Chow grâce à l'existence de résolutions globales localement libres et se ramène formellement à la théorie pour les fibrés. Un résultat de Voisin montre que ces résolutions n'existent pas toujours sur des variétés complexes compactes générales. Nous construisons ici par récurrence sur la dimension de la variété de base des classes de Chern en cohomologie de Deligne rationnelle pour les faisceaux analytiques cohérents en imposant la formule de Grothendieck-Riemann-Roch pour les immersions et en utilisant des méthodes de dévissage. Ces classes sont les seules à vérifier la formule de fonctorialité par pull-back, la formule de Whitney et la formule de Grothendieck-Riemann-Roch pour les immersions; elles coïncident donc avec les classes topologiques et les classes d'Atiyah. Elles vérifient aussi le théorème de Grothendieck-Riemann-Roch pour les morphismes projectifs. Notre second travail est l'étude des schémas de Hilbert ponctuels d'une variété symplectique ou presque complexe de dimension 4. Ils ont été construits par Voisin et généralisent les schémas de Hilbert connus pour les surfaces projectives. En utilisant les structures complexes relatives intégrables introduites dans la construction de Voisin, nous pouvons étendre au cas presque complexe ou symplectique la théorie classique. Nous calculons les nombres de Betti, nous construisons les opérateurs de Nakajima, nous étudions l'anneau de cohomologie et la classe de cobordisme de ces schémas de Hilbert, et nous prouvons dans ce contexte un cas particulier de la conjecture de la résolution crêpante de Ruan.