Propriedades do primeiro autovalor e da primeira autofunção do operador laplaciano em variedades riemannianas são estudadas. Para variedades em que se pode estimar o laplaciano de funções distância, estimativas explícitas para o primeiro autovalor do laplaciano em domínios duplamente conexos são obtidas. Então observamos que hipóteses sobre as curvaturas da variedade e do bordo do domínio permitem estimar o laplaciano da distância. Além disso, autofunções em domínios não compactos do espaço hiperbólico EI" são estudadas. Mostramos que donn'nios contidos em horobolas não admitem autofunções limitadas associadas ao autovalor A(HIn), mas se o fecho assintótico do domínio contém um aberto de (9ooIHIn, então ele admite uma autofunção positiva que se anula em dfí U dooQ. A existência e o perfil de autofunções de autovalor A(IHI") em EI", em IHIn\Sr(o), em horobolas, em hiperbolas e no complementar de horobolas são analisados. Para alguns desses domínios apresentamos uma expressão explícita para a autofunção que depende apenas da distância à fronteira. Finalmente, técnicas de simetrização de Schwarz são adaptadas para variedades permitindo-nos obter estimativas para normas de autofunções. Primeiro um argumento de comparação demonstra que variedades mais simétricas maximizam certas normas. Obtenios também uma estimativa diretamente da função isoperimétrica da variedade. / Some properties of the first eigenvalue A and the first eigenfunction of the Laplace operator in a Riemannian manifold are studied. Assuming a bound for the Laplacian of the distance function, exphcit estimates for the first eigenvalue of a doubly counected domain are presented. Then some assumptions on the curvatures of the manifold and its boundary are made in order to have an estimate for the Laplacian of the distance function. Furthermore eigenfunctions of non compact domains in the hyperbohc space EIn are studied. We prove that a domain contained in a horoball does not admit a bounded eigenfunction of eigenvalue A(lHIn), but if the closure of the domain contains an open set of then it admits a positive eigenfunction that vanishes on dQ U daoíl. The existence and the profile of eigenfunctions of eigenvalue A(E[n ) in H71, in H [ r i \ 5 r ( o ) , in horoballs, hiperballs and in the complement of a horoball are analysed. For some of these domains we present an explicit expression for the eigenfunction that depends only on the distance to the boundary. Finally Schwarz symmetrization techniques are adapted for manifolds implying in estimates for the norm of the eigenfunctions. First a comparison argument proves that highly symmetric manifolds maximize some norm and then an estimated obtained directly from the isoperimetric function of the manifold is presented.

We consider a quantum system whose unperturbed form consists of a self-adjoint Δ-operator on a 2-dimensional compact Riemannian manifold, which may or may not have a boundary. Then as a perturbation, we add a delta potential/point scatterer at some select point ρ. The perturbed self-adjoint operator is constructed rigorously by means of self-adjoint extension theory. We also consider a corresponding classical dynamical system on the cotangent/cosphere bundle, consisting of geodesic flow on the manifold, with specular reflection if there is a boundary. Chapter 2 describes the mathematics of the unperturbed and perturbed quantum systems, as well as outlining the classical dynamical system. Included in the discussion on the delta-perturbed quantum system is consideration concerning the strength of the delta potential. It is reckoned that the delta potential effectively has negative infinitesimal strength. Chapter 3 continues on with investigations from [KMW10], concerned with perturbed eigenfunctions that approximate to a linear combination of only two "surrounding" unperturbed eigenfunctions. In Thm. 4.4 of [KMW10], conditions are derived under which a sequence of perturbed eigenfunctions exhibits this behaviour in the limit. The approximating pair linear combinations belong to a class of quasimodes constructed within [KMW10]. The aim of Chapter 3 in this thesis is to improve on the result in [KMW10]. In Chapter 3, preliminary results are first derived constituting a broad consideration of the question of when a perturbed eigenfunction subsequence approaches linear combinations of only two surrounding unperturbed eigenfunctions. Afterwards, the central result of this Chapter, namely Thm. 3.4.1, is derived, which serves as an improved version of Thm. 4.4 in [KMW10]. The conditions of this theorem are shown to be weaker than those in [KMW10]. At the same time though, the conclusion does not require the approximating pair linear combinations to be quasimodes contained in the domain of the perturbed operator. Cor. 3.5.2 allows for a transparent comparison between the results of this Chapter and [KMW10]. Chapter 4 deals with the construction of non-singular rank-one perturbations for which the eigenvalues and eigenfunctions approximate those of the delta-perturbed operator. This is approached by means of direct analysis of the construction and formulae for the rank-one-perturbed eigenvalues and eigenfunctions, by comparison that of the delta-perturbed eigenvalues and eigenfunctions. Successful results are derived to this end, the central result being Thm. 4.4.19. This provides conditions on a sequence of non-singular rank-one perturbations, under which all eigenvalues and eigenbasis members within an interval converge to those of the delta-perturbed operator. Comparisons have also been drawn with previous literature such as [Zor80], [AK00] and [GN12]. These deal with rank-one perturbations approaching the delta potential within the setting of a whole Euclidean space Rⁿ, for example by strong resolvent convergence, and by limiting behaviour of generalised eigenfunctions associated with energies at every Eℓ(0,∞). Furthermore in Chapter 4, the suggestion from Chapter 2 that the delta potential has negative infinitessimal strength is further supported, due to the coefficients of the approximating rank-one perturbations being negative and tending to zero. This phenomenon is also in agreement with formulae from [Zor80], [AK00] and [GN12]. Chapter 5 first reviews the correspondence between certain classical dynamics and equidistribution in position space of almost all unperturbed quantum eigenfunctions, as demonstrated for example in [MR12]. Equidistribution in position space of almost all perturbed eigenfunctions, in the case of the 2D rectangular flat torus, is also reviewed. This result comes from [RU12], which is only stated in terms of the "new" perturbed eigenfunctions, which would only be a subset of the full perturbed eigenbasis. Nevertheless, in this Chapter it is explained how it follows that this position space equidistribution result also applies to a full-density subsequence of the full perturbed eigenbasis. Finally three methods of approach are discussed for attempting to derive this position space equidistribution result in the case of a more general delta-perturbed system whose classical dynamics satisfies the particular key property.

We establish the existence of radial solutions to the p-Laplacian equation ∆p u + f(u)=0 in RN, where f behaves like |u|q-1 u when u is large and f(u) < 0 for small positive u. We show that for each nonnegative integer n, there is a localized solution u which has exactly n zeros. Also, we look for radial solutions of a superlinear Dirichlet problem in a ball. We show that for each nonnegative integer n, there is a solution u which has exactly n zeros. Here we give an alternate proof to that which was given by Castro and Kurepa.

Dans la première partie de cette thèse, nous introduisons une nouvelle généralisation de l'opérateur de Laplace en géométrie de Finsler. Cette opérateur est défini en intégrant le long des fibres les dérivées directionnelles secondes d'une fonction par rapport à une mesure d'angle que nous construisons. Nous obtenons un opérateur différentiel d'ordre ..., elliptique, symétrique, et qui admet une bonne théorie spectrale. Nous calculons des exemples explicites de spectres pour des métriques de Katok-Ziller. En courbure négative, nous prouvons, grâce à un théorème d'Ancona que la frontière de Martin est Hölder-homéomorphe à la frontière visuelle. Ceci nous permet de déduire l'existence et l'ergodicité des mesures harmoniques pour cet opérateur. Dans la seconde partie, nous étudions les flots d'Anosov en dimension ... dont l'espace des feuilles est homéomorphe à .... Lorsque la variété est hyperbolique, Thurston démontra que le feuilletage (in)stable induit un flot ''orthogonal'' au premier. Nous utilisons ce second flot pour étudier les classes d'isotopie d'orbites périodiques du flot d'Anosov, ainsi que l'existence de cylindres plongés. / In the first part of this dissertation, we give a new definition of a Laplace operator for Finsler metric as an average, with regard to an angle measure, of the second directional derivatives. This operator is elliptic, symmetric with respect to the Holmes-Thompson volume, and coincides with the usual Laplace--Beltrami operator when the Finsler metric is Riemannian. We compute explicit spectral data for some Katok-Ziller metrics. When the Finsler metric is negatively curved, we show, thanks to a result of Ancona that the Martin boundary is Hölder-homeomorphic to the visual boundary. This allow us to deduce the existence of harmonic measures and some ergodic preoperties. In the second part of this dissertation, we study Anosov flows in 3-manifolds, with leaf-spaces homeomorphic to .... When the manifold is hyperbolic, Thurston showed that the (un)stable foliations induces an "orthogonal" flow. We use this second flow to study isotopy class of periodic orbits of the Anosov flow and existence of embedded cylinders.

Laplacian differential operator -- Vibrations of plucked strings and Hollow cylinders.

In this dissertation we study positive solutions to a singular p-Laplacian elliptic boundary value problem on a bounded domain with smooth boundary when a positive parameter varies. Our main focus is the analysis of a challenging class of singular p-Laplacian problems. We establish the existence of a positive solution for all positive values of the parameter and the existence of at least two positive solutions for a certain explicit range of the parameter. In the Laplacian case, we also prove the uniqueness of the positive solution for large values of the parameter. We extend our existence and multiplicity results to classes of singular systems and to the case when a domain is an exterior domain. We prove our existence and multiplicity results by the method of sub and supersolutions and our uniqueness result by establishing apriori and boundary estimates. Such results are well known in the literature for the nonsingular case. In this study, we extend these results to the more difficult singular case.

SILVA, A. K. V. Estimativas gradiente para autofunções do V-Laplaciano e métricas m-quasi-Einstein generalizadas compactas com bordo. 2017. 40 f. Tese (Doutorado em Matemática) – Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2017. / Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-04-18T14:49:03Z No. of bitstreams: 1 2015_tese_akvsilva.pdf: 322426 bytes, checksum: d6abc5541598409191f635ad25e1f501 (MD5) / Rejected by Rocilda Sales (rocilda@ufc.br), reason: Bom dia Andrea, Por favor repasse esse e-mail para o aluno. Estou devolvendo o trabalho pois tem alguns itens que não estão normalizados. O ano que deve constar na capa, folha de rosto e ficha catalográfica é o da entrega. E na ficha catalográfica o nome do autor e o título do trabalho não é em caixa alta. Somente as iniciais e quando necessário. Atenciosamente, Rocilda Sales on 2017-04-19T11:01:58Z (GMT) / Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-04-19T13:11:59Z No. of bitstreams: 1 2015_tese_akvsilva.pdf: 322367 bytes, checksum: c9bd22b4cb5917adacde8be99093e8d8 (MD5) / Rejected by Rocilda Sales (rocilda@ufc.br), reason: Alterar o ano. on 2017-04-19T14:52:13Z (GMT) / Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-04-19T16:22:45Z No. of bitstreams: 1 2015_tese_akvsilva.pdf: 322367 bytes, checksum: 1523e0c8bd0590358cda3a542819aa62 (MD5) / Rejected by Rocilda Sales (rocilda@ufc.br), reason: Correção na folha de aprovação. on 2017-04-19T16:39:17Z (GMT) / Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-04-19T17:04:43Z No. of bitstreams: 1 2015_tese_akvsilva.pdf: 322112 bytes, checksum: e3087741f0e7bb8b966418fb10253c7b (MD5) / Approved for entry into archive by Rocilda Sales (rocilda@ufc.br) on 2017-04-24T11:14:14Z (GMT) No. of bitstreams: 1 2015_tese_akvsilva.pdf: 322112 bytes, checksum: e3087741f0e7bb8b966418fb10253c7b (MD5) / Made available in DSpace on 2017-04-24T11:14:14Z (GMT). No. of bitstreams: 1 2015_tese_akvsilva.pdf: 322112 bytes, checksum: e3087741f0e7bb8b966418fb10253c7b (MD5) Previous issue date: 2015-08-17 / The main of this work was to study properties of Riemannian when subjected to conditions on Bakry-Émery-Ricci tensor. Essentially we study two cases. In the first case, motivated by the work of Barros and Ribeiro Jr. (2014), He, Petersen and Wylie (2012) and Miao and Tam (2011), was introduced generalized m-quasi-Einstein metrics compact with boundary, where we get a result that classify these metrics; more specifically, assuming that gradient field of the exponential of potential function is a conformal vector field, we obtain that this must be a geodesic ball in a simply connected space form. That we get some results that implies when these are trivial metrics. In the second case, we work the Bakry-Émery-Ricci tensor bounded bellow, initially in a compact Riemannian, with or without boundary, and later on balls in complete Riemannian. With this study, we obtain gradient estimates for eigenfunctions of V-Laplacian operator, that generalize results of (Li, 2005) and (Li, 2015). Finally, as consequence theses results, we show an Harnack’s inequality. / Este trabalho tem como principal objetivo estudar propriedades de variedades Riemannianas quando submetidas a condições sobre tensores de Ricci-Bakry-Émery. Essencialmente estudamos dois casos. No primeiro caso, motivados pelos trabalhos de Barros e Ribeiro Jr (2014), He, Petersen e Wylie (2012) e por Miao e Tam (2011), introduzimos métricas m-quasi-Einstein generalizadas compactas com bordo, donde obtemos um resultado que garante uma classificação para estas métricas; mais precisamente, assumindo que o gradiente da exponencial da função potencial é um campo conforme, obtemos que aquela deve ser uma bola geodésica de uma forma espacial simplesmente conexa. Disso, obtemos alguns resultados em que garantimos quando estas métricas são triviais. No segundo caso, trabalhos o tensor de Ricci-Bakry-Émery limitado por baixo, inicialmente, em variedades Riemannianas compactas, com bordo ou sem bordo, e posteriormente, sobre bolas em variedades Riemannianas completas. Com esse estudo, obtivemos estimativas do gradiente para autofunções do operador V-Laplaciano, generalizando resultados de (Li, 2005) e (Li, 2015). Finalmente, como consequências desses resultados, exibimos uma desigualdade de Harnack.

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Video thumbnails are essential to represent the content and summary of a video. This thesis proposed a thumbnail extraction approach for fashion videos based on the presence of clothing items, sharpness, and contrast. Furthermore, this thesis investigated how the proposed thumbnail selection method performed concerning user engagement. Other research has been done on user engagement; however, the impact of clothing item presence has yet to be investigated. Firstly, a YOLOv7 model was trained on a fashion dataset to identify clothing items. The proposed selection method used the model to extract labels to determine what frames contain the maximum number of clothing items. The selected frames were filtered based on a contrast threshold, and the sharpest frame was kept as the proposed thumbnail from the remaining frames. The contrast was measured by calculating the standard deviation of the pixels in each frame. The sharpness was measured with the Laplacian operator. The user engagement was investigated by surveying 119 participants on thumbnail preference. The participants were presented with three frames, the thumbnail extracted with the proposed method, and two control frames: the middle frame of the video and a frame where the YOLOv7 model had only identified one object. The results show that the proposed thumbnail selection method performs well, receiving 59.75% of the total votes, compared to a middle frame and a single-item frame that received 17.46% and 22.79% of the votes, respectively. The results indicate that the proposed parameters for the thumbnail extraction could lead to higher user engagement. / Video-miniatyrbilder är en essentiell del av att presentera och sammanfatta videoinnehåll. Den här uppsatsen föreslår en miniatyrbilds extraktionsmetod för modevideos baserat på klädesplagg, skärpa och kontrast. Denna uppsats utvärderade hur den föreslagna metoden presterar i relation till användarengagemang. Tidigare forskning har utvärderat användarengagemang på miniatyrbilder, dock inte kopplat till närvaro av klädesplagg. Först tränades en YOLOv7 modell på ett modedataset för att identifiera klädesplagg. Den föreslagna metoden använde modellen för att extrahera etiketter för att fastställa vilka bilder som inkluderade flest klädesplagg. De utvalda bilderna filtrerades med en kontrast-tröskel, och den skarpaste bilden av de resterande bilderna behölls som en föreslagen miniatyrbild. Kontrasten mättes med standardavvikelsen mellan pixlar i varje bild. Skärpan mättes med Laplaceoperatorn. Användarengagemanget undersöktes med en enkät genomförd av 119 deltagare för att identifiera vilken miniatyrbild som föredrogs. Deltagarna blev presenterade med tre bilder, en extraherad med den föreslagna metoden och två kontrollbilder: mittenbilden från videon och en bild där YOLOv7 modellen endast identifierat ett objekt. Resultaten visar att den föreslagna metoden presterar bra, den fick 59,75% av rösterna, jämfört med mittenbilden och bilden med ett objekt, som fick respektive 17.46% och 22.79%. Resultaten indikerar att den föreslagna parametrarna kan bidra till ökat användarengagamang i modefilmer.

Neste trabalho estudamos a equação de ondas do tipo p-Laplaciano \'u IND. tt\' - \'DELTA\' IND.p u + \'(- \'DELTA\' POT. alpha\' u IND. t\' = \' [u] POT.q - 2 u, definida num domínio limitado limitado do \'R POT. n\', com 2 \' > ou = \' p < q e 0 < \' alpha\' < 1. Utilizando o método de Faedo-Galerkin provamos a existência de soluções fracas globais para dados iniciais pequenos. Para essas soluções estudamos também o decaimento polinomial da energia associada. A questão da não existência de soluções globais é considerada para o caso em que a energia inicial do sistema é negativa / In this work we study the p-Laplacian wave equation \'u IND. tt\' - \' DELTA\' IND p u + \'(- \'DELTA\' POT. \'alpha\' \' u IND. t\' = \'[u] POT. q - 2 u, defined in a bounded domain of \'R POT n\', with 2 \'> or =\' p < q and 0 < \' alpha\' < 1. By using the Faedo-Galerkin method we prove the existence of weak global solutions for small initial data. We also study the polynomial decay of the associate energy. The blow-up of solutions in finite time is considered for negative initial energy