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Relação entre VC Dimension e menor testemunha

Thu, 01 Jan 2026 00:00:00 GMT

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

Lema de Hopf-Oleinik não homogêneo para operadores totalmente não lineares com termo hamiltoniano

Wed, 01 Jan 2025 00:00:00 GMT

Título: Lema de Hopf-Oleinik não homogêneo para operadores totalmente não lineares com termo hamiltoniano Autor(es): Silva, Antônio Valderlanio Ribeiro da Abstract: This work establishes a quantitative version of the inhomogeneous Hopf-Oleinik Lemma for supersolutions of fully nonlinear elliptic equations with Hamiltonian terms, and applies this result to obtain optimal regularity estimates for l-semiconvex supersolutions. Speciﬁcally, we analyze fully nonlinear equations of the form P_,Λ,W,`,< ( 2D, D, G) = 5 (G), where P_,Λ,W,`,< represents a class of Pucci-type extremal operators with gradient dependence and unbounded coeﬃcients. Our results generalize classical estimates for fully nonlinear PDEs, highlighting the construction of barriers and the strong version of the Harnack inequality, as well as providing fundamental tools and ideas to deal with gradient-dependent structures under diﬀerent growth regimes. It is worth noting that the Hopf-Oleinik Lemma established here remains valid even in the presence of unbounded nonlinearities, signiﬁcantly strengthening previous results in this direction. Tipo: Tese

Problema de Erdös-Rothschild em certos padrões de coloração de grafos conexos

Wed, 01 Jan 2025 00:00:00 GMT

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

ρ-Einstein solitons and critical metrics of the volume functional on complete manifolds

Thu, 01 Jan 2026 00:00:00 GMT

Título: ρ-Einstein solitons and critical metrics of the volume functional on complete manifolds Autor(es): Coimbra, Caio Adler Scalser Abstract: The purpose of this work is to investigate analytic and geometric properties of ρ-Einstein solitons and V -static metrics on complete Riemanninan manifolds. In the ﬁrst part, we study geometric and analytical features of complete non-compact ρ-Einstein solitons, which are self-similar solutions of the Ricci–Bourguignon ﬂow. We study the spectrum of the drifted Laplacian operator for complete gradient shrinking ρ-Einstein solitons. Moreover, similar to classical results due to Calabi–Yau and Bishop for complete Riemannian manifolds with nonnegative Ricci curvature, we prove new volume growth estimates for geodesic balls of complete noncompact ρ-Einstein solitons. In particular, the rigidity case is discussed. In addition, we establish weighted volume growth estimates for geodesic balls of such manifolds. For the second part, we investigate critical metrics of the volume functional (V -static metrics) on complete manifolds without boundary. We prove that every critical metric of the volume functional on a connected complete manifold with parallel Ricci tensor is isometric to one of the standard models. Moreover, we show that a Bach-ﬂat critical metric of the volume functional on a complete, simply connected manifold with proper potential function is isometric to one of the following: the standard sphere Sn, Euclidean space Rn, hyperbolic space Hn, or a warped product R×ϕ Σc, where Σc is a regular level set of the potential function. In particular, we establish classiﬁcation results in dimensions three and four under weaker assumptions on the Bach tensor. Tipo: Tese