This thesis deals with cohomologies on the symplectic quotient of a Frölicher space which is locally diffeomorphic to a Euclidean Frölicher subspace of Rn of constant dimension equal to n. The symplectic reduction under consideration in this thesis is an extension of the Marsden-Weinstein quotient (also called, the reduced space) well-known from the finite-dimensional smooth manifold case. That is, starting with a proper and free action of a Frölicher-Lie-group on a locally Euclidean Frölicher space of finite constant dimension, we study the smooth structure and the topology induced on a small subspace of the orbit space. It is on this topological space that we will construct selected cohomologies such as : sheaf cohomology, Alexander-Spanier cohomology, singular cohomology, ~Cech cohomology and de Rham cohomology. Some natural questions that will be investigated are for instance: the impact of the symplectic structure on these di erent cohomologies; the cohomology that will give a good description of the topology on the objects of category of Frölicher spaces; the extension of the de Rham cohomology theorem in order to establish an isomorphism between the five cohomologies. Beside the algebraic, topological and geometric study of these new objects, the thesis contains a modern formalism of Hamiltonian mechanics on the reduced space under symplectic and Poisson structures. / Mathematical Sciences / D. Phil. (Mathematics)

Cette thèse est divisée en trois parties. Dans la première, on étudie des systèmes articulés (mécanismes formés de tiges rigides) dont l'espace ambiant n'est pas le plan, mais diverses variétés riemanniennes. On étudie la question de l'universalité des mécanismes : cette notion correspond à l'idée que toute courbe serait tracée par un sommet d'un mécanisme, et que toute variété différentiable serait l'espace de configuration d'un mécanisme. On étend les théorèmes d'universalité au plan de Minkowski, au plan hyperbolique et enfin à la sphère.Toute surface dans R^3 peut être aplatie selon l'axe des z, et la surface aplatie s'approche d'une table de billard dans R^2. Dans la seconde partie, on montre que, sous certaines hypothèses, le flot géodésique de la surface converge localement uniformément vers le flot de billard. De plus, si le billard est dispersif, les propriétés chaotiques du billard remontent au flot géodésique : on montre qu'il est alors Anosov. En appliquant ce résultat à la théorie des systèmes articulés, on obtient un nouvel exemple de systèmes articulé Anosov, comportant cinq tiges.Dans la troisième partie, on s'intéresse aux variétés munies de connexions localement métriques, c'est-à-dire de connexions qui sont localement des connexions de Levi-Civita de métriques riemanniennes ; on donne dans ce cadre un analogue du théorème de décomposition de De Rham, qui s'applique habituellement aux variétés riemanniennes. Dans le cas où une telle connexion préserve une structure conforme, on montre que cette décomposition comporte au plus deux facteurs ; de plus, lorsqu'il y a exactement deux facteurs, l'un des deux est l'espace euclidien R^q. La démonstration des résultats de cette partie passe par l'étude des feuilletages munis d'une structure de similitude transverse. Sur ces feuilletages, on montre un résultat de rigidité qui peut être vu indépendamment des autres: ils sont soit transversalement plats, soit transversalement riemanniens. / This thesis is divided into three parts. In the first part, we study linkages (mechanisms made of rigid rods) whose ambiant space is no longer the plane, but various Riemannian manifolds. We study the question of the universality of linkages: this notion corresponds to the idea that every curve would be traced out by a vertex of some linkage, and that any differentiable manifold would be the configuration space of some linkage. We extend universality theorems to the Minkowski plane, the hyperbolic plane, and finally the sphere.Any surface in R^3 can be flattened with respect to the z-axis, and the flattened surface gets close to a billiard table in R^2. In the second part, we show that, under some hypotheses, the geodesic flow of the surface converges locally uniformly to the billiard flow. Moreover, if the billiard is dispersing, the chaotic properties of the billiard also apply to the geodesic flow: we show that it is Anosov in this case. By applying this result to the theory of linkages, we obtain a new example of Anosov linkage, made of five rods.In the third part, we first consider manifolds with locally metric connections, that is, connections which are locally Levi-Civita connections of Riemannian metrics; we give in this framework an analog of De Rham's decomposition theorem, which usually applies to Riemannian manifolds. In the case such a connection also preserves a conformal structure, we show that this decomposition has at most two factors; moreover, when there are exactly two factors, one of them is the Euclidean space R^q. The proofs of the results of this part use foliations with transverse similarity structures. On these foliations, we give a rigidity theorem of independant interest: they are either transversally flat, or transversally Riemannian.

Compressed sensing has been proposed as a computationally efficient method to estimate the finite-dimensional signals. The idea is to develop an undersampling operator that can sample the large but finite-dimensional sparse signals with a rate much below the required Nyquist rate. In other words, considering the sparsity level of the signal, the compressed sensing samples the signal with a rate proportional to the amount of information hidden in the signal. In this dissertation, first, we employ compressed sensing for physical layer signal processing of directional millimeter-wave communication. Second, we go through the theoretical aspect of compressed sensing by running a comprehensive theoretical analysis of compressed sensing to address two main unsolved problems, (1) continuous-extension compressed sensing in locally convex space and (2) computing the optimum subspace and its dimension using the idea of equivalent topologies using Köthe sequence. In the first part of this thesis, we employ compressed sensing to address various problems in directional millimeter-wave communication. In particular, we are focusing on stochastic characteristics of the underlying channel to characterize, detect, estimate, and track angular parameters of doubly directional millimeter-wave communication. For this purpose, we employ compressed sensing in combination with other stochastic methods such as Correlation Matrix Distance (CMD), spectral overlap, autoregressive process, and Fuzzy entropy to (1) study the (non) stationary behavior of the channel and (2) estimate and track channel parameters. This class of applications is finite-dimensional signals. Compressed sensing demonstrates great capability in sampling finite-dimensional signals. Nevertheless, it does not show the same performance sampling the semi-infinite and infinite-dimensional signals. The second part of the thesis is more theoretical works on compressed sensing toward application. In chapter 4, we leverage the group Fourier theory and the stochastical nature of the directional communication to introduce families of the linear and quadratic family of displacement operators that track the join-distribution signals by mapping the old coordinates to the predicted new coordinates. We have shown that the continuous linear time-variant millimeter-wave channel can be represented as the product of channel Wigner distribution and doubly directional channel. We notice that the localization operators in the given model are non-associative structures. The structure of the linear and quadratic localization operator considering group and quasi-group are studied thoroughly. In the last two chapters, we propose continuous compressed sensing to address infinite-dimensional signals and apply the developed methods to a variety of applications. In chapter 5, we extend Hilbert-Schmidt integral operator to the Compressed Sensing Hilbert-Schmidt integral operator through the Kolmogorov conditional extension theorem. Two solutions for the Compressed Sensing Hilbert Schmidt integral operator have been proposed, (1) through Mercer's theorem and (2) through Green's theorem. We call the solution space the Compressed Sensing Karhunen-Loéve Expansion (CS-KLE) because of its deep relation to the conventional Karhunen-Loéve Expansion (KLE). The closed relation between CS-KLE and KLE is studied in the Hilbert space, with some additional structures inherited from the Banach space. We examine CS-KLE through a variety of finite-dimensional and infinite-dimensional compressible vector spaces. Chapter 6 proposes a theoretical framework to study the uniform convergence of a compressible vector space by formulating the compressed sensing in locally convex Hausdorff space, also known as Fréchet space. We examine the existence of an optimum subspace comprehensively and propose a method to compute the optimum subspace of both finite-dimensional and infinite-dimensional compressible topological vector spaces. To the author's best knowledge, we are the first group that proposes continuous compressed sensing that does not require any information about the local infinite-dimensional fluctuations of the signal.

The diploma thesis is elaboration of project documentation of the object Housing for seniors in Brno - Dolní Heršpice. The site is located un the area with the planned development of the city. The object is detached, three-story with a partial basement. The building is divided into five parts. The building is based on concrete foundation pads and strips. The structural construction is a combined column and wall system in technological design as monolithic concrete. The building is designed as a low energy building, with contact insulation (ETICS). The whole building is designed with forced air exchange. An extensive and intensive walkable flat roof is designed. The aim is to create small, barrier-free, community housing with affordable services and support, with a possible life expectancy. Services for the elderly with reduced self-sufficiency and with mild health and mental dysfunctions are considered. The building is equipped with space for accommodation with a capacity of 48 beds, space for eating with a separate kitchen, space for leisure and physical activities, space for basic health and social care and space for administrative and technical facilities. The construction is divided into several buildings, the subject of the thesis is primarily the solution SO.01 - Housing for seniors

Thesis on "Planetarium with coffee" is processed in the form of project documentation for construction. The proposed facility is designed as a two-storeyed without basement with the main ground plan dimensions of 52,8 x 27,47 m and is located near the village Lošov close to Olomouc. The building is operationally divided into a part intended for public entrance and a part for operation and maintenance. Loadbearing structural system is reinforced concrete monolithic skeleton based on monolithic reinforced concrete foundation footings. Ceiling structure consists of reinforced concrete, monolithic, locally supported, cross reinforced floor slab. Vertical external walls are made of ceramic blocks with contact thermal insulation system. Part of the front rounded facade is designed as glass fronted. The building is covered by a warm flat roof. Roof cladding consists of vegetation layers, walkable terrace or non-walkable structure of roof cladding. The planetarium itself is covered by a reinforced concrete dome with a thermal insulation layer of plates of foamed glass. Fillings of doors and windows are made of aluminum profiles with isolating triple glazing. The building meets the requirements for barrier-free use of buildings.

This diploma thesis deals with the design and elaboration process of a project documentation of medical center in city Hlinsko. New building is situated in the northern part of the city in the area determined for constructions of public infrastructure. It´s a four-floor object with partial basement and slant mansard and flat vegetative roof.Object is based on foundation strips and footings made from reinforced concrete.The bearing and internal walls are designed from KALKSANDSTEIN lime sand blocks.The beams and collums are used in the placed designed with open disposition.The ceiling in 1.PP is designed as reinforced concrete for securing fire safety.The rest od ceiling constructions is designed from prestressed concrete panels SPIROLL,with exception in 4.NP,where the ceiling constructions are made from solid roof system YTONG KOMFORT,which is layed on frame system made of steel and reinforced concrete.The peripheral walls of 1.PP are made from lost formwork, which is insulated with extruded polystyrene.The walls in floors above ground are insulated with combination of external thermal insulation system (ETICS) and ventilated facade with CEMBRIT SOLID and PATINA facade cladding panels. The building is functionally divided into several part.Cafe, pharmacy, ambulance and medical center, each with it´s own entrance.Pharmacy is equipped with shop, medicine mixer and storage.Ambulance is equipped with garage slots, background for employees, room of crisis preparedness and operating center.Medical centrum is equipped with emergency, RTG, operating theater, rehabilitations and other types of examination rooms.Each examination room is equipped with waiting room.For more comfort the center has a background for employees.The whole object is designed with forced air exchange. Designed object is barrier-free.In front of the building is situated parking slots for cars, motorbikes and ambulances.Total number of slots is 74 places, which 5 is designed for disabled people.