http://repositorio.ufc.br/handle/riufc/63 Sun, 14 Jun 2026 17:48:48 GMT 2026-06-14T17:48:48Z http://repositorio.ufc.br/handle/riufc/86473 Título: Relação entre VC Dimension e menor testemunha Autor(es): Moura, Rafael Fernandes Abstract: The VC-dimension is an important combinatorial measure that, in a sense, indicates the complexity of a family of binary functions. It was originally defined in 1971 by Vapnik and Chervonenkis (VC) and gave rise to the so-called “Vapnik-Chervonenkis Theory”. It is of fundamental importance in Statistical Learning Theory, being used, for example, to predict probabilistic upper bounds for classification tests. Another important concept, now in Computational Learning Theory, is the “Teaching Dimension” (TD). It is related to the notion of “witnesses” for a family of binary functions. More precisely, if H is a set of binary functions with domain X, we say that S ⊆ X is a witness for a certain h ∈ H if it is possible to identify h (among the elements of H) from its restriction to S; that is, if no other function in H has the same restriction as h on S. Defining TDmin(H,h) as the size of the smallest witness for h, the Minimal Teaching Dimension of H, denoted by TDmin(H), is equal to the minimum value of TDmin(H,h) as h varies. The RTD (“Recursive Teaching Dimension”) conjecture relates the two above parameters, stating that the value of the Teaching Dimension of any family H, with finite domain, is less than or equal to the VC-dimension of H multiplied by an absolute constant. In this dissertation, we study the history and recent advances regarding this conjecture. Tipo: Dissertação Thu, 01 Jan 2026 00:00:00 GMT http://repositorio.ufc.br/handle/riufc/86473 2026-01-01T00:00:00Z http://repositorio.ufc.br/handle/riufc/86081 Título: Problema de Erdös-Rothschild em certos padrões de coloração de grafos conexos Autor(es): Alencar, George Lucas Diniz Abstract: A pattern ˆF is a graph with a precoloring of its edges. If G is a graph, then an edge-coloring of G avoids ˆF if G does not contain a subgraph isomorphic to F (considering also a color-preserving isomorphism). In this dissertation, we mention some known results, such as: (i) If ˆF is the pattern of a complete graph, then for any positive integer n, one of the n-vertex graphs that maximizes the number of colorings avoiding ˆF is a complete multipartite graph. (ii) If ˆF is the pattern of a monochromatic complete graph with k+1 vertices and r = 2 or 3, then for any sufficiently large positive integer n, the graph that maximizes the number of r-colorings avoiding ˆF is the Turán graph with n vertices and k parts. (iii) If ˆF is a monochromatic star with t +1 vertices, then for any graph G with n vertices, the number of r-colorings of G that avoid ˆF is at most Ä (r(t − 1))!(t − 1)! r ä n2 . The contributions of this work are: (iv) If ˆF is the pattern of a complete graph with at least 700 vertices and at most 5 colors, then for all sufficiently large r, the n-vertex graph that maximizes the number of r-colorings avoiding ˆF is not the Turán graph with n vertices and k parts. (v) Letting ˆF be a monochromatic star with t +1 vertices, there exists a graph G with n vertices such that the number of r-colorings of G that avoid ˆF is at least f (r) n(t − 1)2 , where f (r) ∼2πr Ä re2 är. Tipo: Dissertação Wed, 01 Jan 2025 00:00:00 GMT http://repositorio.ufc.br/handle/riufc/86081 2025-01-01T00:00:00Z http://repositorio.ufc.br/handle/riufc/85015 Título: Uma prova simples do Teorema de Calabi-Bernstein sobre superfícies tipo-espaço máximas Autor(es): Alves, Laiane Miranda Abstract: The Calabi-Bernstein Theorem, which states that the only entire solutions to the maximal surface equation are affine functions, is a very well known result in Lorentzian Geometry, collecting many different proofs along the years since its original, one due to E. Calabi. In this work, we study a direct and simple proof of this theorem given by A. Romero, aiming to be easily understood by beginning researchers. As such, this proof only requires basic Riemannian and Semi-Riemannian Geometry preliminaries, which have been included in the text, in addition to Liouville Theorem on holomorphic functions of one complex variable. Tipo: Dissertação Wed, 01 Jan 2025 00:00:00 GMT http://repositorio.ufc.br/handle/riufc/85015 2025-01-01T00:00:00Z http://repositorio.ufc.br/handle/riufc/84448 Título: Variação Lp de operadores maximais do tipo convolução Autor(es): Silva, Antônio Valderlanio Ribeiro da Abstract: Based on the work of (Carneiro; Svaiter, 2013), we will show in this dissertation that operators from a subclass M of Maximal Operator of Convolution Type, to which belong the Gauss Maximal Operator and the Poisson Maximal Operator, are bounded on the Sobolev spaces, 1,? ( R3 ) and have bounded !? variation holding: ∥∇ "i 5 ∥ !? ( R3 ) ≤ ∥∇ 5 ∥ !? ( R3 ) for "i ∈ M when ? ≥ 1 and 3 = 1 or when ? = 2 or ? = ∞ and 3 > 1 . We will also show similar results for when 5 is a discrete function and when it is a bounded pointwise variation function. Tipo: Dissertação Wed, 01 Jan 2020 00:00:00 GMT http://repositorio.ufc.br/handle/riufc/84448 2020-01-01T00:00:00Z