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Convexity of the Krein space tracial numerical range and Morse theory
Nakazato, Hiroshi; Bebiano, Natália; Providência, João daIn this paper we present a Krein space convexity theorem on the tracial-numerical range of a matrix. This theorem is the analogue of Westwick's theorem. The proof is an application of Morse theory.
Product of diagonal entries of the unitary orbit of a 3-by-3 normal matrix
Nakazato, Hiroshi; Bebiano, Natália; Providência, João daLet N be a 3×3 normal matrix. We investigate the sets where U(3) is the group of 3×3 unitary matrices and 1[less-than-or-equals, slant]k[less-than-or-equals, slant]3. Geometric properties of these sets are studied, namely, star-shapedness and simple connectedness are investigated. A method for the numerical estimation of is also provided for normal matrices of size 3. ; http://www.sciencedirect.com/science/a...
The J-numerical range of a J-Hermitian matrix and related inequalities
Nakazato, Hiroshi; Bebiano, Natália; Providência, João daRecently, indefinite versions of classical inequalities of Schur, Ky Fan and Rayleigh-Ritz on Hermitian matrices have been obtained for J-Hermitian matrices that are J-unitarily diagonalizable, J=Ir[circle plus operator](-Is),r,s>0. The inequalities were obtained in the context of the theory of numerical ranges of operators on indefinite inner product spaces. In this paper, the subject is revisited, relaxing th...
On the corners of certain determinantal ranges
Kovačec, Alexander; Bebiano, Natália; Providência, João daLet A be a complex n×n matrix and let SO(n) be the group of real orthogonal matrices of determinant one. Define [Delta](A)=det(AoQ):Q[set membership, variant]SO(n), where o denotes the Hadamard product of matrices. For a permutation [sigma] on 1,...,n, define It is shown that if the equation z[sigma]=det(AoQ) has in SO(n) only the obvious solutions (Q=([epsilon]i[delta][sigma]i,j), [epsilon]i=±1 such that [epsi...
The Elliptical and the Hyperbolical Range Theorems Revisited
Teixeira, Ricardo Emanuel Cunha; Bebiano, Natália; Providência, João daInternational Conference on Engineering and Mathematics (ENMA'2007), July 9-11, 2007 - Bilbao, Spain. ; A geometrical proof of the Hyperbolical Range Theorem, concerning the numerical range of linear operators on 2-dimensional Krein spaces, is given. The classical Elliptical Range Theorem, which is the correspondent result for Hilbert spaces, is also obtained using the same technique. Both proofs depend on the...
Exact solutions for the LMG model Hamiltonian based on the Bethe ansatz
Morita, Hiroyuki; Ohnishi, Hiromasa; Providência, João da; Nishiyama, SeiyaExact solutions for the Lipkin-Meshkov-Glick (LMG) model Hamiltonian are obtained by solving the Bethe ansatz equation (BAE) which is derived from the variation equation based on the Bethe ansatz. Unlike Pan and Draayer, we do not use bosonization and infinite-dimensional algebra techniques. Consequently there are no restrictions on parameters specifying strengths of the interactions included in the LMG Hamilto...
The numerical range of 2-dimensional Krein spaces operators
Nakazato, Hiroshi; Bebiano, Natália; Providência, João daThe tracial numerical range of operators on a 2-dimensional Krein space is investigated. Results in the vein of those obtained in the context of Hilbert spaces are stated
"McDonaldizar" a Matemática da geração Nintendo?
Providência, Natália Bebiano; Jr. Providência, João daNa sociedade global, em particular na sociedade portuguesa, o preconceito contra a matemática assume contornos merecedores de reflexão. Ciência morta, onde tudo está feito, ciência elitista ou imperialista, com exagerado peso selectivo nos curricula e no ingresso no ensino superior, são alguns epítetos que exornam a sua imagem. Existe uma subavaliação da matemática por part...
Long-wave fluctuations of the two-dimensional Heisenberg antiferromagnet
Belinicher, Victor; Providência, João daWe have reduced the Heisenberg model for the quantum antiferromagnet in the leading approximation over 1/s to the spin-rotator model which represents some sort of quantum lattice rotator model. We perform the long-wavelength reduction of this spin-rotator model. An intermediate scale [Lambda] was introduced and the short-wavelength antiferromagnetic spin fluctuations were integrated out. As a result, we obtaine...
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