We propose a new framework for understanding block Krylov subspace methods, which hinges on a matrix-valued inner product. We can recast the ``classical" block Krylov methods, such as O'Leary's block conjugate gradients, global methods, and loop-interchange methods, within this framework. Leveraging the generality of the framework, we develop an efficient restart procedure and error bounds for the shifted block full orthogonalization method (Sh-BFOM(m)). Regarding BFOM as the prototypical block Krylov subspace method, we propose another formalism, which we call modified BFOM, and show that block GMRES and the new block Radau-Lanczos method can be regarded as modified BFOM. In analogy to Sh-BFOM(m), we develop an efficient restart procedure for shifted BGMRES with restarts (Sh-BGMRES(m)), as well as error bounds. Using this framework and shifted block Krylov methods with restarts as a foundation, we formulate block Krylov subspace methods with restarts for matrix functions acting on multiple vectors f(A)B. We obtain convergence bounds for \bfomfom (BFOM for Functions Of Matrices) and block harmonic methods (i.e., BGMRES-like methods) for matrix functions. With various numerical examples, we illustrate our theoretical results on Sh-BFOM and Sh-BGMRES. We also analyze the matrix polynomials associated to the residuals of these methods. Through a variety of real-life applications, we demonstrate the robustness and versatility of B(FOM)^2 and block harmonic methods for matrix functions. A particularly interesting example is the tensor t-function, our proposed definition for the function of a tensor in the tensor t-product formalism. Despite the lack of convergence theory, we also show that the block Radau-Lanczos modification can reduce the number of cycles required to converge for both linear systems and matrix functions. / Mathematics

In this work we study properties of the higher-dimensional generally rotating black hole space-time so-called Kerr-NUT-(A)dS and the related spaces with the same explicit and hidden symetries as the Kerr-NUT-(A)dS spacetime. First, we search commuta- tivity conditions for classical (charged) observables and their operator analogues, then we investigate a fulfilment of these conditions in the metioned spaces. We calculate the curvature of these spaces and solve the charged Hamilton-Jacobi and Klein-Gordon equations by the separation of the variables for an electromagnetic field, which pre- serves integrability of motion of a charged particle and mutual commutativity of the corresponding operators.

The objects studied in this thesis are families of cycles on schemes. A space — the Chow variety — parameterizing effective equidimensional cycles was constructed by Chow and van der Waerden in the first half of the twentieth century. Even though cycles are simple objects, the Chow variety is a rather intractable object. In particular, a good functorial description of this space is missing. Consequently, descriptions of the corresponding families and the infinitesimal structure are incomplete. Moreover, the Chow variety is not intrinsic but has the unpleasant property that it depends on a given projective embedding. A main objective of this thesis is to construct a closely related space which has a good functorial description. This is partly accomplished in the last paper. The first three papers are concerned with families of zero-cycles. In the first paper, a functor parameterizing zero-cycles is defined and it is shown that this functor is represented by a scheme — the scheme of divided powers. This scheme is closely related to the symmetric product. In fact, the scheme of divided powers and the symmetric product coincide in many situations. In the second paper, several aspects of the scheme of divided powers are discussed. In particular, a universal family is constructed. A different description of the families as multi-morphisms is also given. Finally, the set of k-points of the scheme of divided powers is described. Somewhat surprisingly, cycles with certain rational coefficients are included in this description in positive characteristic. The third paper explains the relation between the Hilbert scheme, the Chow scheme, the symmetric product and the scheme of divided powers. It is shown that the last three schemes coincide as topological spaces and that all four schemes are isomorphic outside the degeneracy locus. The last paper gives a definition of families of cycles of arbitrary dimension and a corresponding Chow functor. In characteristic zero, this functor agrees with the functors of Barlet, Guerra, Kollár and Suslin-Voevodsky when these are defined. There is also a monomorphism from Angéniol's functor to the Chow functor which is an isomorphism in many instances. It is also confirmed that the morphism from the Hilbert functor to the Chow functor is an isomorphism over the locus parameterizing normal subschemes and a local immersion over the locus parameterizing reduced subschemes — at least in characteristic zero. / QC 20100908

This master's thesis addresses numerical methods of computing the typical ranks of tensors over the real numbers and explores some properties of tensors over finite fields. We present three numerical methods to compute typical tensor rank. Two of these have already been published and can be used to calculate the lowest typical ranks of tensors and an approximate percentage of how many tensors have the lowest typical ranks (for some tensor formats), respectively. The third method was developed by the authors with the intent to be able to discern if there is more than one typical rank. Some results from the method are presented but are inconclusive. In the area of tensors over nite filds some new results are shown, namely that there are eight GLq(2) GLq(2) GLq(2)-orbits of 2 2 2 tensors over any finite field and that some tensors over Fq have lower rank when considered as tensors over Fq2 . Furthermore, it is shown that some symmetric tensors over F2 do not have a symmetric rank and that there are tensors over some other finite fields which have a larger symmetric rank than rank.

Supervised learning tasks like building a classifier, estimating the error rate of the predictors, are typically performed with labeled data. In most cases, obtaining labeled data is costly as it requires manual labeling. On the other hand, unlabeled data is available in abundance. In this thesis, we discuss methods to perform supervised learning tasks with no labeled data. We prove consistency of the proposed methods and demonstrate its applicability with synthetic and real world experiments. In some cases, small quantities of labeled data maybe easily available and supplemented with large quantities of unlabeled data (semi-supervised learning). We derive the asymptotic efficiency of generative models for semi-supervised learning and quantify the effect of labeled and unlabeled data on the quality of the estimate. Another independent track of the thesis is efficient computational methods for nonnegative tensor factorization (NTF). NTF provides the user with rich modeling capabilities but it comes with an added computational cost. We provide a fast algorithm for performing NTF using a modified active set method called block principle pivoting method and demonstrate its applicability to social network analysis and text mining.

In this work is analyzed a co-frame bundle first differential extension . Received local co-ordinates transformation law of the space, constructed this space linear connection and linear co-connection. In this work is proved basis space linear connection‘s object inducts objects linear connection and linear co-connection in space. Founded inducted connection curvature tensors.

Šiame darbe nagrinėjama speciali atraminių elementų erdvė – tiesinių elementų erdvė. Šios geometrijos bendrąją tiesinių ir afiniųjų siečių teoriją sukūrė V. Bliznikas. Jis parodė, [5] kaip tiesinės sieties geometrinis objektas indukuoja aukštesniųjų eilių afiniųjų, taip pat tenzorinių siečių objektams. V. Blizniko sukurtais tyrimo metodais dalinai naudojomės ir šiame darbe. Metrinių hiperplokštuminių elementų erdvė yra taip vadinamų normalizuotų erdvių atvejis. Normalizuotos erdvės tai tokios, kuriose apibrėžtos koks nors diferencialinis – geometrinis objektas, kurio invariantai ir sudaro normalizuotos erdvės geometrijos turinį. Tokiais objektais dažnai būna skaliarinė funkcija. (Finslerio ar Kartano erdvės), metrinis tenzorius (tiesinių ar hiperplokštuminių elementų erdvės), afiniosios sieties objektas (afiniosios sieties erdvės) ir pan. Šiame darbe nagrinėjamos metrinių tiesinių elementų erdvės, kurios yra normalizuojamos metrinio tenzoriaus pagalba. Be to, tas tenzorius turi specialią struktūrą (žr. [1]). Ta struktūra charakteringa tuo, kad visuomet tokios erdvės yra Landsbergo erdvių analogai. Darbe pavyko tokioms metrikoms sukonstruoti vidines beveik kompleksines ir beveik sandaugos struktūras, surasti jų integruojamumo sąlygą, kurios dėka metrikos specifika yra kitokia nei analogiškos sąlygos Finslerio erdvėse. Darbas sudarytas ir iš įžangos ir 8 paragrafų. Pirmajame paragrafe dėstomas įvadas į liestinių sluoksniuočių geometriją. Antrajame nagrinėjama šių erdvių... [toliau žr. visą tekstą] / The elements of metric space with a special form of metric are dealt with in the work. It is shown how in such spase linear and affine links are defined with the help of metric tenzor, the ogjects of curvature are founds the existence of the type of metric affine links is proved. It is proved that the metric tenzor induces two parametric almost complex and almost the structures of product, the integration criteria of these structures are found. Keywords: • differentiable manifold • tangent stratified; • linear and affine traceable; • integrated struktures; • structural tensors.

Die vorliegende Dissertation befasst sich mit der ramanspektroskopischen Charakterisierung von Galliumnitrid (GaN). Der Zusammenhang zwischen Waferkrümmung und mechanischer Restspannungen wird diskutiert. Mit Hilfe konfokaler Mikro-Ramanmessungen wurden Dotierprofile nachgewiesen sowie die Ladungsträgerkonzentration und -beweglichkeit ermittelt. Sämtliche Ramantensorelemente von wz-GaN wurden erstmals durch die Anwendung verschiedener Streugeometrien bestimmt. Eine neu entwickelte Vorwärtsstreuanordnung ermöglichte die Beobachtung von Phonon-Polaritonen. Es konnte gezeigt werden, dass von der theoretischen und experimentellen Betrachtung der Ramanstreuintensitäten dieser Elementaranregungen eindeutig das Vorzeichen der Faust-Henry-Koeffizienten von wz-GaN abgeleitet werden kann. Im Rahmen dieser Arbeit wurden alle Faust-Henry-Koeffizienten für GaN experimentell bestimmt. / This thesis focuses on special aspects of the Raman spectroscopical characterization of wurtzite gallium nitride (wz-GaN). The correlation between wafer curvature and residual stress is discussed. By means of confocal micro-Raman measurements doping profiles were detected as well as the density and mobility of free charge carriers were deduced. All Raman scattering cross sections of wz-GaN were determined the first time using different scattering configurations. A novel method for near-forward scattering was developed in order to observe phonon polaritons with pure symmetry. It is shown that the theoretical and experimental consideration of the Raman scattering efficiency of these elementary excitations allow for determining the sign of the Faust-Henry coefficients of wz-GaN unambiguously. The Faust-Henry coefficients of GaN were deduced from Raman scattering efficiencies of corresponding TO and LO phonons.

This thesis is devoted to several aspects of geometry and morphology in wetting problems and hard sphere packings. First, we propose a new method to simulate wetting and slip on nanostructured substrates: a phase field model associated with a dynamical density theory approach. We showed omniphobicity, meaning repellency, no matter the chemical properties of the liquid on monovalued surfaces, i.e. surfaces without overhangs, which is in contradiction with the macroscopic Cassie-Baxter-Wenzel theory, can produce so-called We checked systematically the impact of the surface parameters on omniphobic repellency, and we show that the key ingredient are line tensions, which emerge from needle shaped surface structures. Geometrical effects have also an important influence on glassy or jammed systems, for example amorphous hard sphere systems in infinite pressure limit. Such hard sphere packings got stuck in a so-called jammed phase, and we shall demonstrate that the local structure in such systems is universal, i.e. independent of the protocol of the generation. For this, robust order parameters - so-called Minkowski tensors - are developed, which overcome robustness deficiencies of widely used order parameters. This leads to a unifying picture of local order parameters, based on geometrical principles. Furthermore, we find with the Minkowski tensor analysis crystallization in jammed sphere packs at the random closed packing point

Dans les communications coopératives, deux ou plusieurs terminaux de transmissionsont combinés pour accroître la diversité et/ou la puissance des signaux arrivant à un récepteur. Récemment, l'analyse tensorielle s'est avérée une approche efficace pour l'estimation de canaux dans les systèmes coopératifs. Cependant, parmi les quelques travaux consacrés à cette tâche, l'utilisation de la décomposition tensorielle PARAFAC pour modéliser les signaux reçus ne permet pas l'estimation conjointe des symboles et des canaux de communication. Afin d'éviter l'utilisation de séquences de symboles pilotes, l'objectif de cette thèse est de fournir de nouvelles approches tensorielles, en termes de systèmes de transmission et de récepteurs semi-aveugles, pour des systèmes de communication MIMO avec relai mono-directionnels, à deux sauts. Deux systèmes de transmission sont proposés en utilisant un codage spatio-temporel du type Khatri-Rao et deux stratégies de traitement Amplify-and-Forward (AF) au relai. Pour ces systèmes, appelés PT2-AF et NP-AF, les signaux reçus au niveau de la destination satisfont respectivement des modèles tensoriels du type PARATUCK2 et nested PARAFAC. En exploitant les propriétés d'unicité de ces modèles tensoriels établies dans la thèse, plusieurs récepteurs semi-aveugles sont dérivés. Certains de ces récepteurs sont du type ALS, tandis que d'autres sont des solutions non itératives basées sur des factorisations de produits de Khatri-Rao. Des résultats de simulation sont présentés pour illustrer les performances des récepteurs proposés qui sont comparés à des estimateurs supervisés. / In cooperative communication systems, two or more transmitting terminals arecombined to increase the diversity and/or the power of the signals arriving at aparticular receiver. Recently, the so-called tensor analysis has been an efficient approach for channel estimation in systems with cooperative diversity. However, among the few works devoted to this task, the utilization of the PARAFAC tensor decomposition for modeling the received signals did not allow the development of techniques for joint symbol and channel estimation. Aiming to avoid the use of pilot-based sequences, the objective of this thesis is to provide new tensor-based strategies, including transmission systems and semi-blind receivers, for one-way two-hop relaying systems. Based on a Khatri-Rao space-time coding at the source and two different Amplify-and-Forward (AF) relaying strategies, two transmission systems are proposed. For these systems, named PT2-AF and NP-AF, the received signals at the destination node follow respectively a PARATUCK2 and a nested PARAFAC tensor model. Exploiting uniqueness properties of these tensor models which are established in the thesis, several semi-blind receivers are derived. Some of these receivers are of iterative form using an ALS algorithm, whereas some other ones are close-form solutions associated with Khatri-Rao factorizations. Some simulation results are finally presented to illustrate the performance of the proposed receivers which are compared to some state-of-the-art supervised techniques.