Academic literature on the topic 'Beltrami's representation'

One of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data. We consider the problem of constructing a representation for data lying on a low-dimensional manifold embedded in a high-dimensional space. Drawing on the correspondence between the graph Laplacian, the Laplace Beltrami operator on the manifold, and the connections to the heat equation, we propose a geometrically motivated algorithm for representing the high-dimensional data. The algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality-preserving properties and a natural connection to clustering. Some potential applications and illustrative examples are discussed.

For some rank 1 non-linear σ models we prove that a necessary and sufficient condition of multiplicative renormalizability for composite fields is that they should be eigenfunctions of the coset Laplace-Beltrami operator. These eigenfunctions span the irreducible representation space of the isometry group and may be finite- or infinite-dimensional. The zero mode of the Laplace-Beltrami operator plays a particular role since it is not renormalized at all.

Beltrami equations L¯t(g)=μ(·,t)Lt(g) on S3 (where Lt, |t|<1, are the Rossi operators i.e., Lt spans the globally nonembeddable CR structure H(t) on S3 discovered by H. Rossi) are derived such that to describe quasiconformal mappings f:S3→N⊂C2 from the Rossi sphere S3,H(t). Using the Greiner–Kohn–Stein solution to the Lewy equation and the Bargmann representations of the Heisenberg group, we solve the Beltrami equations for Sobolev-type solutions gt such that gt−v∈WF1,2S3,θ with v∈CR∞S3,H(0).

Plusieurs auteurs ont utilisé les champs de contraintes pour résoudre l'équation d’équilibre de la mécanique des milieux continus. Airy (1863) a résolu le cas bidimensionnel, Maxwell (1870) et Morera (1892) ont étudié le cas tridimensionnel. Les solutions obtenues sont des cas particuliers de celles de Beltrami (1892). Gurtin a donné un exemple de solutions ne satisfaisant pas la représentation S = CurlCurlA de Beltrami, ce qui signifie que la représentation précédente est incomplète. De plus, il a montré que si l’ouvert est régulier, alors elle est complète dans l’espace des champs réguliers de contraintes auto-équilibrés.Dans cette thèse intitulée ”Caractérisations de champs de matrices, potentiels matrices et applications aux opérateurs traces”, on s’intéresse à diverses caractérisations de champs de vecteurs, de champs de matrices et spécialement au résultat de Gurtin dans le cas où l’ouvert et les champs de contraintes ne sont pas réguliers.Cette thèse est décomposée en cinq chapitres. Le premier chapitre expose la problématique de recherche traitée dans cette thèse. Il présente également l’origine du sujet de recherche. Dans le deuxième chapitre, on étudie l’opérateur et en particulier l’existence de potentiels vecteurs dans différents cadres fonctionnels.Dans les chapitres 3 et 4, on va montrer quelques versions de la complétude de la représentation de Beltrami et en déduire des décompositions de Helmholtz pour les champs de matrices.Le dernier chapitre est consacré à l’étude de l’image de différents opérateurs traces de fonctions W 2,p (Ω), W 3,p (Ω) lorsque Ω est un ouvert borné de R 2 lipschitzien. L’ingrédient essentiel est donné par la fonction d’Airy ou par la représentation de Beltrami
Many authors have used stress fields to solve the equilibrium equation of continuum me- chanics. Airy (1863) solved the two-dimensional case, Maxwell (1870) and Morera (1892) solved the three-dimensional case. The above solutions are special cases of those of Beltrami (1892). Gurtin gave an example of solutions that do not have Beltrami’s S = CurlCurlA representation. He showed that if the domain Ω is regular, then this representation is complete in the class of regular stress fields which are self-equilibrated.My thesis title is ”Characterizations of matrix fields, potential matrices and applications to trace operators”. In this work, we are interested by showing many characterizations ofvector fields, of matrix fields and especially by generalizing the result of Gurtin in the case when the open set and the stress fields are not regular.This thesis consists of five chapters. The first chapter presents the research problem ad- dressed in this thesis. It also presents the origin of the subject of research.In the second chapter, we study the operator . In particular, the existence of potential vectors in different functional frameworks.In Chapters 3 and 4, we will show some versions of Beltrami’s completeness and we deduce some Helmholtz decomopsitions for symmetric matrix fields.The last chapter is devoted to the study of the image of different trace operators of functions W 2,p (Ω), W 3,p (Ω) when Ω is a bounded open of R 2 with Lipschitz boundary. The essential ingredient is given by the Airy’s function or by the Beltrami representation

La tesi considera la varietà di formule espressive e di destinazioni funzionali della fotografia in rapporto alla costruzione di spazi e architetture nella Milano del primo cinquantennio dell’unità d’Italia. Strumento di lavoro per architetti e ingegneri, la fotografia delle principali operazioni edilizie eseguite nella città lombarda a partire dagli anni sessanta dell’Ottocento rappresenta per la nuova classe dirigente un fondamentale strumento di legittimazione del proprio operato amministrativo e di illustrazione dei progressi compiuti dalla città sulla via della modernizzazione. Attraverso un itinerario storico della ‘fotografia di cantiere’, costruito anche grazie al confronto con casi italiani e internazionali, lo studio presenta una puntuale analisi del fenomeno in ambito milanese. Tra i progetti affrontati, il piano di riforma del centro cittadino (1865-1878) di Giuseppe Mengoni e i restauri del Castello Sforzesco (1893-1907) di Luca Beltrami, oltre a una serie di interventi minori, documentati da alcuni dei principali fotografi attivi in città e dai redattori della pubblicistica illustrata, tipologia editoriale che tra Otto e Novecento si serve con sempre maggiore consapevolezza del mezzo fotografico. L’evoluzione del linguaggio fotografico viene dunque indagata nel suo contributo alla costruzione di un’iconografia nazionale della modernità, fenomeno che vede la città di Milano porsi in prima fila nel contesto italiano.
The study investigates the variety of expressive formulas and functional destinations of the photography production concerning the architectural renovation occurred in the city of Milan between 1861 and 1911. Design tool for architects and engineers, construction photography reveals itself as a fundamental tool of political legitimation for the new ruling class and illustrates the progress made by the city in terms of modernization. Through a historical reconstruction of the genre of construction photography, which also considers Italian and international cases, the study presents a detailed analysis of the Milanese phenomenon. Among the projects, the reform plan of the city center (1865-1878) developed by Giuseppe Mengoni and the restoration of the Sforza Castle (1893-1907) led by Luca Beltrami , as well as a number of minor operations. The inquiry considers the production by some of the main photographers working in the city and the iconographic apparatus of the most important illustrated press of the period, which during the late nineteenth century uses with increasing awareness the narrative qualities of the photographic medium. The evolution of the photographic language is therefore inquired with regard to his contribution to the construction of a national iconography of modernity , a phenomenon that sees the city of Milan in the front line.