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Tato diplomová práce má za cíl seznámit čtenáře se stále ještě relativně novou a neznámou oblastí matematiky, s geometrickou algebrou. Nejdříve jsou uvedeny základní definice a poté jsou studovány vlastnosti obecné geometrické algebry. Další velká část textu se věnuje Konformní geometrické algebře, která je v současnosti jedna z nejvíce zkoumaných a aplikovaných geometrických algeber. Jsou popsány její algebraické a geometrické vlastnosti, konkrétně schopnost reprezentovat určité geometrické objekty jako vektory. Taktéž umožňuje počítat jejich průniky a konformní transformace. Další část textu je zaměřena na aplikace Konformní geometrické algebry, nejdříve k popisu kinematiky robotické ruky a poté v binokulárním viděni.

This thesis focuses on robotic 3D vision for application in Bin Picking. The new method based on Conformal Geometric Algebra (CGA) is proposed and tested for identification of spheres in Pointclouds created with 3D scanner. The speed, precision and scalability of this method is compared to traditional descriptors based method. It is proved that CGA maintains the same precision as the traditional method in much shorter time. The CGA based approach seems promising for the use in the future of robotic 3D vision for identification and localization of spheres.

<div>We consider actions of the current Lie algebras $\gl_{n}[t]$ and $\gl_{k}[t]$ on the space $\mathfrak{P}_{kn}$ of polynomials in $kn$ anticommuting variables. The actions depend on parameters $\bar{z}=(z_{1}\lc z_{k})$ and $\bar{\alpha}=(\alpha_{1}\lc\alpha_{n})$, respectively.</div><div>We show that the images of the Bethe algebras $\mathcal{B}_{\bar{\alpha}}^{\langle n \rangle}\subset U(\gl_{n}[t])$ and $\mathcal{B}_{\bar{z}}^{\langle k \rangle}\subset U(\gl_{k}[t])$ under these actions coincide.</div><div></div><div>To prove the statement, we use the Bethe ansatz description of eigenvectors of the Bethe algebras via spaces of quasi-exponentials. We establish an explicit correspondence between the spaces of quasi-exponentials describing eigenvectors of $\mathcal{B}_{\bar{\alpha}}^{\langle n \rangle}$ and the spaces of quasi-exponentials describing eigenvectors of $\mathcal{B}_{\bar{z}}^{\langle k \rangle}$.</div><div></div><div>One particular aspect of the duality of the Bethe algebras is that the Gaudin Hamiltonians exchange with the Dynamical Hamiltonians. We study a similar relation between the trigonometric Gaudin and Dynamical Hamiltonians. In trigonometric Gaudin model, spaces of quasi-exponentials are replaced by spaces of quasi-polynomials. We establish an explicit correspondence between the spaces of quasi-polynomials describing eigenvectors of the trigonometric Gaudin Hamiltonians and the spaces of quasi-exponentials describing eigenvectors of the trigonometric Dynamical Hamiltonians.</div><div></div><div>We also establish the $(\gl_{k},\gl_{n})$-duality for the rational, trigonometric and difference versions of Knizhnik-Zamolodchikov and Dynamical equations.</div>

Corrosion protection is very significant for metals in modern society from the view of industrial development. This thesis work project involves a research study that is aimed to investigate the effect of corrosion on the mechanical strength and magnetic properties of four no treated or treated Somaloy® component samples (Somaloy®700 1P,  Somaloy®700 3P, Somaloy®700HR 5P and Somaloy®110i 5P) provided by Höganäs AB and laminated steel sheets by salt spray test. The coatings for protection are phosphoric acid coating, sodium silicate coating, DCA-Modified silicone conformal coating and water-borne single coat paint respectively. Then the protective properties are evaluated by Electrochemical Impedance Spectroscopy (EIS) in 0.1 mol/L NaCl solution after 7 days exposure.   From transverse rupture strength (TRS) and hysteresis loop measurements by salt spray test, although the bar samples are treated coating, the corrosion decreases the mechanical strength to a certain extent more or less over time. For the magnetic properties, the corrosive environments hardly influence the magnetic parameters of the no treated or four types of coatings treated Somaloy® components. But the all kinds of magnetic parameters for laminated ring samples have a great variation after salt spray test.   In the EIS measurements, for the no treated samples, the initial corrosion resistance is only several hundred ohms and decreases after 1 hour, 8 hours and 1day exposure, then increases to a certain extent with time due to the corrosion products formed on the surface. For the sodium silicate coating, the initial corrosion resistance is approximately several ten thousands ohms and decreases rapidly only after 1 day exposure to several hundred ohms due to the sodium silicate film dissolves in the electrolyte solution and has no effective protective property. Then the following corrosion process is almost same as the no treated samples. For the water-borne single coat paint, the initial corrosion resistance can reach to several Giga-ohms and decreases over time, but can still stay at  level, indicating that this coating has a very good and effective protective properties.   EIS experiments indicate that water-borne single coat paint has a more effective protection than sodium silicate coating and can apply a better corrosion protection for the Somaloy® components   Key words: corrosion protection, soft magnetic composites, phosphoric acid coating, sodium silicate coating, DCA-Modified silicone conformal coating , water-borne single coat paint, salt spray test, TRS, hysteresis loop, EIS

Additive manufacturing can bring several advantages in tooling applications especially hot working tooling as high pressure die casting. Printing of conformal cooling channels can lead to improved cooling and faster solidification, which, in turn, can possibly result in better quality of the cast part. However, few studies on advantages of additive manufactured tools in high pressure die casting are published.The aim of this study was to investigate and quantify the effect of conformal cooling on microstructure and mechanical properties of high pressure die cast aluminum alloy. Two tools each consisting of two die inserts were produced with and without conformal channels using additive manufacturing. Both tools were used in die casting of aluminum alloy. Aluminum specimens were then characterized microstructurally in light optical microscope for secondary arm spacing measurements and subjected to tensile and hardness testing. Cooling behavior of different inserts was studied with a thermal camera and by monitoring the temperature change of cooling oil during casting. Surface roughness of die inserts was measured with profilometer before and after casting.Thermal imaging of temperature as a function of time and temperature change of oil during casting cycle indicated that conformal insert had faster cooling and lower temperature compared to conventional insert. However, thermal imaging of temperature after each shot in a certain point of time showed higher maximum and minimum temperature on conformal die surface but no significant difference in normalized temperature gradient compared to the conventional insert.The average secondary dendrite arm spacing values were fairly similar for samples from conventional and conformal inserts, while more specimens from conventional insert demonstrated coarser structure. Slower cooling in conventional insert could result in the coarser secondary dendrite arm spacing.Tensile strength and hardness testing revealed no significant difference in mechanical properties of the specimens cast in conventional and conformal die inserts. However, reduced deviations in hardness was observed for samples cast with conformal insert. This is in agreement with secondary dendrite arm spacing measurements indicating improved cooling with conformal insert.Surface roughness measurement showed small wear of the inserts. More castings are needed to observe a possible difference in wear between the conventional and conformal inserts.Small observed differences in cooling rate and secondary arm spacing did not result in evident difference in mechanical properties of the aluminum alloy but the variation in properties were reduced for samples cast with conformal cooling. Future work may include more accurate measurement of cooling behavior with a thermocouple printed into the die insert, casting of thicker specimen for porosity evaluation and fatigue testing and longer casting series to evaluate the influence of conformal cooling on tool wear.

Dans la description à basse énergie de systèmes fortement corrélés, les champs de jauge peuvent émerger comme excitations collectives couplées à des quasiparticules fractionalisées. En particulier, certains aimants bidimensionnels dits frustrés sont décrits par un liquide de spin de Dirac comportant une symétrie de jauge U(1) compacte. La description infrarouge est donnée par une théorie conforme des champs, soit l'électrodynamique quantique en 2+1 dimensions avec 2N saveurs de fermions sans masse. Dans les aimants typiques, N=2 ou 4. L'aspect compact du champ de jauge implique également l'existence d'excitations topologiques, soit des instantons créés, dans ce contexte, par des opérateurs monopôles. Cette thèse porte sur les transitions de phase quantiques à partir d'un liquide de spin de Dirac et les propriétés des monopôles aux points critiques correspondants. Ces transitions sont induites en activant diverses interactions de type Gross-Neveu. Dans tous les cas à l'étude, la dimension d'échelle des monopôles est obtenue grâce à la correspondance état-opérateur et à un développement en 1/N. L'accent est d'abord mis sur une transition de confinement-déconfinement vers une phase antiferromagnétique décrite par la condensation d'un monopôle. Une levée de dégénérescence est observée au point critique alors que certaines dimensions d'échelle de monopôles sont réduites par rapport à leur valeur dans le liquide de spin de Dirac. Cette hiérarchie est caractérisée quantitativement en comparant les dimensions d'échelle dans des secteurs distincts du spin magnétique à l'ordre dominant en 1/N, puis qualitativement par une analyse en théorie des représentations. Des exposants critiques pour d'autres observables dans la théorie non compacte sont également obtenus. Enfin, deux transitions vers des liquides de spin topologiques, soit le liquide de spin chiral et le liquide de spin Z2, sont considérées. Les dimensions anormales des monopôles sont obtenues à l'ordre sous-dominant en 1/N. Ces résultats permettent de vérifier une dualité conjecturée avec un modèle bosonique et la valeur d'un coefficient universel pour les théories de jauge U(1) / In strongly correlated systems, gauge fields can emerge as collective excitations coupled to fractionalized quasiparticles. In particular, certain frustrated two-dimensional quantum magnets are described by a Dirac spin liquid which has a U(1) gauge symmetry. The infrared description is given by a conformal field theory, namely quantum electrodynamics in 2+1 dimensions with 2N flavours of massless fermions. In typical magnets, N=2 or 4. The compact aspect of the gauge field also implies the existence of topological excitations corresponding to instantons, which are created by monopole operators in this context. This thesis focuses on quantum phase transitions out of a Dirac spin liquid and the properties of monopoles at the corresponding critical points. These transitions are driven by activating various types of Gross-Neveu interactions. In all the cases studied, the scaling dimension of monopoles are obtained using the state-operator correspondence and a 1/N expansion. The confinement-deconfinement transition to an antiferromagnetic order produced by a monopole condensate is first studied. A degeneracy lifting is observed at the critical point, as certain monopoles have their scaling dimension reduced in comparison with the value in the Dirac spin liquid. This hierarchy is charactized quantitatively by comparing monopole scaling dimensions in distinct magnetic spin sector at leading-order in 1/N, and qualitatively by an analysis in representation theory. Critical exponents of various other operators are obtained in the non-compact model. Transitions to two topological spin liquids, namely a chiral spin liquid and a Z2 spin liquid, are also considered. Anomalous dimensions of monopoles are obtained at sub-leading order in 1/N. These results allow the verification of a conjectured duality with a bosonic model and the value of a universal coefficient in U(1) gauge theories.

[pt] Segundo a teoria desenvolvida por Kolmogorov, Arnold e Moser na década de sessenta, a grande maioria dos toros invariantes persistem após uma perturbação C3 de um Hamiltoniano integrável. Uma pergunta natural é se perturbações em topologias Ck, para k < 3, ainda preservam tais toros. Bangert mostrou que a situação é a oposta na topologia C1 : arbitrariamente próximo de uma métrica Riemanniana plana no toro existem métricas sem nenhum toro invariante. Ruggiero estendeu esses resultados para Lagrangeanos mecânicos no toro e mostrou que, no caso de métricas Riemannianas, esse fenômeno é C1 genérico. Neste trabalho, mostramos que, dado ǫ > 0, E 2 R e um Hamiltoniano de Tonelli reversível H : TT2 -> R, existe β E (0, 1) e uma ǫ perturbação H0 de H tal que H0 não possui gráficos contínuos invariantes. Para tal, construimos explicitamente uma métrica Finsler, sem nenhum campo contínuo de minimizantes, através de um estudo analítico do operador de Jacobi. / [en] According to the theory developed by Kolmogorov, Arnold and Moser in the sixties, the majority of invariant tori persists under a C3 perturbation of a integrable Hamiltonian. A natural question is if a perturbation in the Ck topology, k < 3, still preserves such tori. Bangert showed that, in the C1 topology, what happens is the opposite: there are metrics with no invariant torus arbitrarily close to any given Riemannian metric. Ruggiero extended these results to mechanical Lagrangians in the torus and showed that for Riemannian metrics this phenomenon is C1 generic. In this work, we show that, given e > 0, e 2 R and a reversible Tonelli Hamiltonian H : TT2 -> R, there exists β E (0, 1) and an ǫ perturbation H0 of H in the C1,β topology such that H0 has no continuous invariant graphs. The result is achieved by explicitly exhibiting a Finsler metric, without any continuous field of minimizers, constructed after an analytic study of the Jacobi operator.

<p>Holographic duality as a rigorous approach to quantum gravity claims that a quantum gravitational system is exactly equal to a quantum theory without gravity in lower spacetime dimensions living on the boundary of the quantum gravitational system. The duality maps key questions about the emergence of spacetime to questions on the non-gravitational boundary system that are accessible to us theoretically and experimentally. Recently, various aspects of quantum information theory on the boundary theory have been found to be dual to the geometric aspects of the bulk theory. In this thesis, we study the exact and approximate quantum error corrections (QEC) in a general quantum system (von Neumann algebras) focused on QFT and gravity. Moreover, we study entanglement theory in the presence of conserved charges in QFT and the multiparameter multistate generalization of quantum relative entropy.</p>

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