Neste trabalho construimos constelacoes de sinais no plano hiperbolico. Analisamos o desempenho de constelacoes PAM, PSK e QAM-circular no plano hipebolico em relacao a constelacoes equivalentes do plano euc1idiano. Para estabelecermos... more
Neste trabalho construimos constelacoes de sinais no plano hiperbolico. Analisamos o desempenho de constelacoes PAM, PSK e QAM-circular no plano hipebolico em relacao a constelacoes equivalentes do plano euc1idiano. Para estabelecermos estas constelacoes introduzimos diversos conceitos de geometria hiperbolica, sendo o principal deles, o conceito de tesselacao do plano. Para podermos fazer decisoes em relacao a escolha de quais tesselacoes fornecem constelacoes de interesse, obtivemos funcoes enumeradoras, que nos permitem contar o numero de pontos em subconjuntos finitos das tesselacoes. Para podermos calcular o desempenho de constelacoes de interesse, obtivemos uma funcao densidade de probabilidade gaussiana para o plano hiperbolico e apresentamos suas principais propriedades. Partindo do conceito de funcao de probabilidade gaussiana hiperbolica, caracterizamos o ruido de um canal gaussiano hiperbolico, utilizando as isometrias do plano hiperbolico Abstract
Research Interests: Physics and Humanities
Current work defines Schur representation of a bilinear operator $$T: H \times H \rightarrow H$$ , where H is a separable Hilbert space. Introducing the concepts of self-adjoint bilinear operators, ordered eigenvalues and eigenvectors, we... more
Current work defines Schur representation of a bilinear operator $$T: H \times H \rightarrow H$$ , where H is a separable Hilbert space. Introducing the concepts of self-adjoint bilinear operators, ordered eigenvalues and eigenvectors, we prove that if T is compact, self-adjoint, and its eigenvalues are ordered, then T has a Schur representation, thus obtaining a spectral theorem for T on real Hilbert spaces. We prove that the hypothesis of the existence of ordered eigenvalues is fundamental.
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ABSTRACT In this paper we present classes of topological quantum codes (TQC) derived from the embedding of complete bipartite graphs on the corresponding compact surfaces. Every code achieves minimum distance 3 and its encoding rate is... more
ABSTRACT In this paper we present classes of topological quantum codes (TQC) derived from the embedding of complete bipartite graphs on the corresponding compact surfaces. Every code achieves minimum distance 3 and its encoding rate is such that k/n → 1 as n → ∞.
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Current work builds new families of non-binary nonlinear error-correcting codes from Finite Upper Half-Plane and p a prime number. A fundamental domain is defined to a discrete group acting over Hq. We establish some concepts and... more
Current work builds new families of non-binary nonlinear error-correcting codes from Finite Upper Half-Plane and p a prime number. A fundamental domain is defined to a discrete group acting over Hq. We establish some concepts and results on Hq, such that the geometric properties allow us to get codification and decodification.
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This current work propose a technique to generate polygonal color codes in the hyperbolic geometry environment. The color codes were introduced by Bombin and Martin-Delgado in 2007, and the called triangular color codes have a higher... more
This current work propose a technique to generate polygonal color codes in the hyperbolic geometry environment. The color codes were introduced by Bombin and Martin-Delgado in 2007, and the called triangular color codes have a higher degree of interest because they allow the implementation of the Clifford group, but they encode only one qubit. In 2018 Soares e Silva extended the triangular codes to the polygonal codes, which encode more qubits. Using an approach through hyperbolic tessellations we show that it is possible to generate Hyperbolic Polygonal codes, which encode more than one qubit with the capacity to implement the entire Clifford group and also having a better coding rate than the previously mentioned codes, for the color codes on surfaces with boundary with minimum distance d = 3.
Research Interests: Mathematics and Tema
In this work, we show that an n-dimensional sublattice Λ′=mΛ of an n-dimensional lattice Λ induces a G=Zmn tessellation in the flat torus Tβ′=Rn/Λ′, where the group G is isomorphic to the lattice partition Λ/Λ′. As a consequence, we... more
In this work, we show that an n-dimensional sublattice Λ′=mΛ of an n-dimensional lattice Λ induces a G=Zmn tessellation in the flat torus Tβ′=Rn/Λ′, where the group G is isomorphic to the lattice partition Λ/Λ′. As a consequence, we obtain, via this technique, toric codes of parameters [[2m2,2,m]], [[3m3,3,m]] and [[6m4,6,m2]] from the lattices Z2, Z3 and Z4, respectively. In particular, for n=2, if Λ1 is either the lattice Z2 or a hexagonal lattice, through lattice partition, we obtain two equivalent ways to cover the fundamental cell P0′ of each hexagonal sublattice Λ′ of hexagonal lattices Λ, using either the fundamental cell P0 or the Voronoi cell V0. These partitions allow us to present new classes of toric codes with parameters [[3m2,2,m]] and color codes with parameters [[18m2,4,4m]] in the flat torus from families of hexagonal lattices in R2.
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This paper is devoted to the study of the stability of the compactness property of bilinear operators acting on the products of interpolated Banach spaces. We prove one-sided compactness results for bilinear operators on products of... more
This paper is devoted to the study of the stability of the compactness property of bilinear operators acting on the products of interpolated Banach spaces. We prove one-sided compactness results for bilinear operators on products of Banach spaces generated by abstract methods of interpolation, in the sense of Aronszajn and Gagliardo. To get these results, we prove a key one-sided bilinear interpolation theorem on compactness for bilinear operators on couples satisfying an extra approximation property. We give applications to general cases, including Peetre’s method and the general real interpolation methods.
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Current work presents a new approach to quantum color codes on compact surfaces with genus g\geq2 using the identification of these surfaces with hyperbolic polygons and hyperbolic tessellations. We show that this method may give rise to... more
Current work presents a new approach to quantum color codes on compact surfaces with genus g\geq2 using the identification of these surfaces with hyperbolic polygons and hyperbolic tessellations. We show that this method may give rise to color codes with a very good parameters and we present tables with several examples of these codes whose parameters had not been shown before. We also present a family of codes with minimum distance d=4 and the encoding rate asymptotically going to 1 while n\rightarrow\infty.
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Research Interests: Mathematics and Physics
Geometrically uniform codes are fundamental in communication systems, mainly for modulation. Typically, geometrically uniform codes are dependent on a given alphabet. The current work establishes the necessary and sufficient conditions... more
Geometrically uniform codes are fundamental in communication systems, mainly for modulation. Typically, geometrically uniform codes are dependent on a given alphabet. The current work establishes the necessary and sufficient conditions for obtaining a matched labeling between a group G and a signal set S. It introduces the concept of the G-isometric signal set, allowing for the establishment of equivalences between different types of signal sets. In particular, we obtain isometries between groups and geometrically uniform codes with a minimal generator. We also draw attention to the influence of the environment metric space, the group metric, and the matched mapping on the labeling of a signal set. The results are valid for all environment metric spaces. The alphabet emerges naturally from the relationship between the signal set S and the label group derived from its symmetry group, Γ(S).
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Current research builds labelings for geometrically uniform codes on the double torus through tiling groups. At least one labeling group was provided for all of the 11 regular tessellations on the double torus, derived from triangular... more
Current research builds labelings for geometrically uniform codes on the double torus through tiling groups. At least one labeling group was provided for all of the 11 regular tessellations on the double torus, derived from triangular Fuchsian groups, as well as extensions of these labeling groups to generate new codes. An important consequence is that such techniques can be used to label geometrically uniform codes on surfaces with greater genera. Furthermore, partitioning chains are constructed into geometrically uniform codes using soluble groups as labeling, which in some cases results in an Ungerboeck partitioning for the surface. As a result of these constructions, it is demonstrated that, as in Euclidean spaces, modulation and encoding can be combined in a single step in hyperbolic space.
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We study variants of s-numbers in the context of multilinear operators. The notion of an $$s^{(k)}$$ s ( k ) -scale of k-linear operators is defined. In particular, we shall deal with multilinear variants of the $$s^{(k)}$$ s ( k )... more
We study variants of s-numbers in the context of multilinear operators. The notion of an $$s^{(k)}$$ s ( k ) -scale of k-linear operators is defined. In particular, we shall deal with multilinear variants of the $$s^{(k)}$$ s ( k ) -scales of the approximation, Gelfand, Hilbert, Kolmogorov and Weyl numbers. We investigate whether the fundamental properties of important s-numbers of linear operators are inherited to the multilinear case. We prove relationships among some $$s^{(k)}$$ s ( k ) -numbers of k-linear operators with their corresponding classical Pietsch’s s-numbers of a generalized Banach dual operator, from the Banach dual of the range space to the space of k-linear forms, on the product of the domain spaces of a given k-linear operator.
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ABSTRACT In this paper we present six classes of topological quantum codes (TQC) on compactsurfaces with genus g ≥ 2. These codes are derived from self-dual, quasi self-dual anddenser tessellations associated with embeddings of self-dual... more
ABSTRACT In this paper we present six classes of topological quantum codes (TQC) on compactsurfaces with genus g ≥ 2. These codes are derived from self-dual, quasi self-dual anddenser tessellations associated with embeddings of self-dual complete graphs and com-plete bipartite graphs on the corresponding compact surfaces. The majority of the newclasses has the self-dual tessellations as their algebraic and geometric supporting math-ematical structures. Every code achieves minimum distance 3 and its encoding rate issuch that k/n → 1 as n → ∞, except for the one case where k/n → 1/3 as n → ∞.