The modeling and prediction of chaotic time series require proper reconstruction of the state space from the available data in order to successfully estimate invariant properties of the embedded attractor. Thus, one must choose appropriate time delay ${\tau }^{\ast }$ and embedding dimension p for phase space reconstruction. The value of ${\tau }^{\ast }$ can be estimated from the Mutual Information, but this method is rather cumbersome computationally. Additionally, some researchers have recommended that ${\tau }^{\ast }$ should be chosen to be dependent on the embedding dimension p by means of an appropriate value for the time delay ${\tau }_w=(p-1){\tau }^{\ast }$, which is the optimal time delay for independence of the time series. The C-C method, based on Correlation Integral, is a method simpler than Mutual Information and has been proposed to select optimally ${\tau }_w$ and ${\tau }^{\ast }$. In this paper, we suggest a simple method for estimating ${\tau }^{\ast }$ and ${\tau }_w$ based on symbolic analysis and symbolic entropy. As in the C-C method, ${\tau }^{\ast }$ is estimated as the first local optimal time delay and ${\tau }_w$ as the time delay for independence of the time series. The method is applied to several chaotic time series that are the base of comparison for several techniques. The numerical simulations for these systems verify that the proposed symbolic-based method is useful for practitioners and, according to the studied models, has a better performance than the C-C method for the choice of the time delay and embedding dimension. In addition, the method is applied to EEG data in order to study and compare some dynamic characteristics of brain activity under epileptic episodes

Let R be a commutative ring, G a group, and RG its group ring. Let ϕ: RG → RG denote the R-linear extension of an involution ϕ defined on G. An element x in RG is said to be symmetric if ϕ (x) = x. A characterization is given of when the symmetric elements (RG)_ϕ of RG form a ring. For many domains R it is also shown that (RG)ϕ is a ring if and only if the symmetric units form a group. The results obtained extend earlier work of Bovdi ( 2001 Bovdi , V. ( 2001 ). On symmetric units in group algebras . Comm. Algebra 29 ( 12 ): 5411 – 5422 . [CSA] [Taylor & Francis Online], [Web of Science ®] [Google Scholar] ), Bovdi et al. ( 1996 Bovdi , V. , Kovács , L. G. , Sehgal , S. K. (1996). Symmetric units in modular group algebras. Comm. Algebra 24(3):803–808. [CSA] [Taylor & Francis Online], [Web of Science ®] [Google Scholar] ), Bovdi and Parmenter ( 1997 Bovdi , V. , Parmenter , M. M. ( 1997 ). Symmetric units in integral group rings . Publ. Math. Debrecen 50 ( 3–4 ): 369 – 372 . [CSA] [Google Scholar] ), Broche Cristo ( 2003 Broche Cristo , O. ( 2003 ). A Commutatividade dos Elementos Simétricos e Anti-simétricos en Anéis de Grupo, Ph.D. thesis, Univ. Sao Paulo . [Google Scholar] , to appear), Giambruno and Sehgal ( 1993 Giambruno , A. , Sehgal , S. K. ( 1993 ). Lie nilpotence of group rings . Comm. Algebra 21 : 4253 – 4261 . [CSA] [Taylor & Francis Online], [Web of Science ®] [Google Scholar] ), and Lee ( 1999 Lee , G. T. ( 1999 ). Group rings whose symmetric elements are Lie nilpotent . Proc. Amer. Math. Soc. 127 ( 11 ): 3153 – 3159 . [CROSSREF] [CSA] [Google Scholar] ), who dealt with the case that ϕ is the involution * mapping g ∈ G onto g⁻¹.

This paper, based on the concept of symbolic correlation integral, introduces a set of symbolic recurrence plots and associated invariant measures, which are independent of the distance parameter, serving as a tool for quantifying the dynamic structure. These new measures allow the study of transient behavior, coexistence of attractors, bifurcations, and structural change. The final user does not have to choose the free distance parameter. An empirical application to electrocardiography data illustrates the use of symbolic recurrence measures.

The etiology of complex diseases is due to the combination of genetic and environmental factors, usually many of them, and each with a small effect. The identification of these small-effect contributing factors is still a demanding task. Clearly, there is a need for more powerful tests of genetic association, and especially for the identification of rare effects