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- On derivatives and norms of generalized matrix functions and respective symmetric powersPublication . Carvalho, Sonia; Freitas, Pedro J.In recent papers, the authors obtained formulas for directional derivatives of all orders, of the immanant and of the m-th xi-symmetric tensor power of an operator and a matrix, when xi is a character of the full symmetric group. The operator norm of these derivatives was also calculated. In this paper, similar results are established for generalized matrix functions and for every symmetric tensor power.
- The norm of the K- Th derivative of the X-symmetric power of an operatorPublication . Carvalho, Sonia; Freitas, Pedro JorgeIn this paper, the exact value for the norm of directional derivatives, of all orders, for symmetric tensor powers of operators on finite dimensional vector spaces is presented. Using this result, an upper bound for the norm of all directional derivatives of immanants is obtained.
- The k-th derivatives of the immanent and the chisymmetric power of an operatorPublication . Carvalho, Sonia; Freitas, Pedro J.In recent papers, formulas are obtained for directional derivatives, of all orders, of the determinant, the permanent, the m-th compound map and the m-th induced power map. This paper generalizes these results for immanants and for other symmetric powers of a matrix.
- The K-Th derivatives of the immanant and the X-symmetric power of an operatorPublication . Carvalho, Sonia; Freitas, Pedro JorgeIn recent papers, formulas are obtained for directional derivatives, of all orders, of the determinant, the permanent, the m-th compound map and the m-th induced power map. This paper generalizes these results for immanants and for other symmetric powers of a matrix.
- The norm of the k-the derivative of the Chisymmetric power of an operatorPublication . Carvalho, Sonia; Freitas, Pedro J.In this paper, the exact value for the norm of directional derivatives, of all orders, for symmetric tensor powers of operators on finite dimensional vector spaces is presented. Using this result, an upper bound for the norm of all directional derivatives of immanants is obtained. This work is inspired in results by R. Bhatia, J. Dias da Silva, P. Grover, and T. Jain.
- Information measures and synchronization in complete networksPublication . Rocha, J. Leonel; Carvalho, SoniaThe main purpose of this talk is to present information measures and synchronization of complete networks with local identical chaotic dynamical systems. The networks topologies are characterized by circulant matrices and the conditional Lyapunov exponents are explicitly determined. For different types of local dynamics, necessary and sufficient conditions for the occurrence of synchronization with or without the negativity of the conditional Lyapunov exponents are presented. Some properties of the mutual information rate and the Kolmogorov-Sinai entropy are established, depending on the topological entropy of the individual chaotic nodes and on the synchronization interval. Numerical studies are included.
- An alternative proof on higher order derivatives of a multilinear mapPublication . Carvalho, SoniaAs a generalization of the formulas proved by Bhatia, Grover and Jain (Derivatives of tensor powers and their norms. Electron J Linear Algebra. 2013;26:604-619), in recent papers (The kth derivative of the immannant and the chi-symmetric tensor power of an operator. Electron J Linear Algebra. 2014;27:Article 18, On derivatives and norms of generalized matrix functions and respective symmetric powers. Electron J Linear Algebra. 2015;30:Article 22) Carvalho and Freitas obtained formulas for directional derivatives, of all orders, for generalized matrix functions and for every symmetric tensor power associated with a character xi of a subgroup G of the symmetric group S-m. Throughout our work, we used some well-known formulas for the derivatives of all orders of a multilinear map, since the maps that we studied are all multilinear. In this paper, we intend to present an alternative proof of these formulas, using the multilinearity argument.