A remark on conditioned invariance in the behavioral approach

In this paper a definition for the property of behavioral invariance is proposed with the purpose of generalizing the state space geometric approach to the behavioral setting. Based on this notion together with the well-known notion of behavioral observer, a definition of conditioned invariance is also presented. The results obtained for the characterization of the defined properties put into evidence some problems that, in our opinion, should deserve attention. This could serve as a starting point for a discussion on the foundations of an analogue of the geometric theory within the behavioral setting

Geometric engineering of (framed) BPS states

BPS quivers for N=2 SU(N) gauge theories are derived via geometric engineering from derived categories of toric Calabi-Yau threefolds. While the outcome is in agreement of previous low energy constructions, the geometric approach leads to several new results. An absence of walls conjecture is formulated for all values of N, relating the field theory BPS spectrum to large radius D-brane bound states. Supporting evidence is presented as explicit computations of BPS degeneracies in some examples. These computations also prove the existence of BPS states of arbitrarily high spin and infinitely many marginal stability walls at weak coupling. Moreover, framed quiver models for framed BPS states are naturally derived from this formalism, as well as a mathematical formulation of framed and unframed BPS degeneracies in terms of motivic and cohomological Donaldson-Thomas invariants. We verify the conjectured absence of BPS states with "exotic" SU(2)_R quantum numbers using motivic DT invariants. This application is based in particular on a complete recursive algorithm which determine the unframed BPS spectrum at any point on the Coulomb branch in terms of noncommutative Donaldson-Thomas invariants for framed quiver representations.Comment: 114 pages; v2:minor correction

Estimating the Equity Premium

To estimate the equity premium, it is helpful to use finance theory: not the old-fashioned theory that efficient markets imply a constant equity premium, but theory that restricts the time-series behavior of valuation ratios, and that links the cross-section of stock prices to the level of the equity premium. Under plausible conditions, valuation ratios such as the dividend-price ratio should not have trends or explosive behavior. This fact can be used to strengthen the evidence for predictability in stock returns. Steady-state valuation models are also useful predictors of stock returns given the high degree of persistence in valuation ratios and the difficulty of estimating free parameters in regression models for stock returns. A steady-state approach suggests that the world geometric average equity premium was almost 4% at the end of March 2007, implying a world arithmetic average equity premium somewhat above 5%. Both valuation ratios and the cross-section of stock prices imply that the equity premium fell considerably in the late 20th Century, but has risen modestly in the early years of the 21st Century.

Estimating the Equity Premium

Finance theory restricts the time-series behaviour of valuation ratios and links the cross-section of stock prices to the level of the equity premium. This can be used to strengthen the evidence for predictability in stock returns. Steady-state valuation models are useful predictors of stock returns, given the persistence in valuation ratios. A steady-state approach suggests that the world geometric average equity premium fell considerably in the late twentieth century, rose modestly in the early years of the twenty-first century, and was almost 4% at the end of March 2007.Economic

Analysis of EEG Data Using Complex Geometric Structurization

Electroencephalogram (EEG) is a common tool used to understand brain activities. The data are typically obtained by placing electrodes at the surface of the scalp and recording the oscillations of currents passing through the electrodes. These oscillations can sometimes lead to various interpretations, depending on, for example, the subject’s health condition, the experiment carried out, the sensitivity of the tools used, or human manipulations. The data obtained over time can be considered a time series. There is evidence in the literature that epilepsy EEG data may be chaotic. Either way, the Embedding Theory in dynamical systems suggests that time series from a complex system could be used to reconstruct its phase space under proper conditions. In this letter, we propose an analysis of epilepsy EEG time series data based on a novel approach dubbed complex geometric structurization. Complex geometric structurization stems from the construction of strange attractors using Embedding Theory from dynamical systems. The complex geometric structures are themselves obtained using a geometry tool, the α-shapes from shape analysis. Initial analyses show a proof of concept in that these complex structures capture the expected changes brain in lobes under consideration. Further, a deeper analysis suggests that these complex structures can be used as biomarkers for seizure changes