By Edward Cline
Believe that $R$ is a finite dimensional algebra and $T$ is a correct $R$-module. permit $A = \textnormal{ End}_R(T)$ be the endomorphism algebra of $T$. This memoir provides a scientific learn of the relationships among the illustration theories of $R$ and $A$, specially these related to real or power constructions on $A$ which "stratify" its homological algebra. The unique motivation comes from the speculation of Schur algebras and the symmetric team, Lie conception, and the illustration idea of finite dimensional algebras and finite teams. The publication synthesizes universal positive aspects of a few of the above components, and provides a few new instructions. incorporated are an summary "Specht/Weyl module" correspondence, a brand new idea of stratified algebras, and a deformation idea for them. The method reconceptualizes many of the modular illustration thought of symmetric teams concerning Specht modules and locations that idea in a broader context. eventually, the authors formulate a few conjectures regarding the idea of stratified algebras and finite Coexeter teams, aiming towards figuring out the modular illustration idea of finite teams of Lie kind in all features.
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Adachi et al. calculated the eigenvalues of the Jacobian Matrix of the CNN and concluded that by using simple chaos control methods, the controlled CNN cannot be stabilized to a stored pattern, and a new control strategy must be developed for the CNN [36]. For the associative memory dynamics, Nakamura et al. [37], Kushibe et al. [38], He et al. [39]–[45], and Shrimali et al. [46]–[47] have proposed some chaos control methods for chaotic neural networks and investigated the controlled dynamics. In this chapter, we focus on several chaos control methods concerned with associative memory and the relation between the stable output and the initial state in the CNN.
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N, (1) where f (·, a) is a one-dimensional unimodal map with a control parameter a. xn is the value of the state variable at time n, and k represents an additional parameter for constant feedback. The chaotic dynamics can be converted to stable periodic one by tuning k externally [Gueron (1998)]. In what we propose here, we suppose that the dynamical system (see Eq. (1)) possesses a region in the parameter k, where chaotic behaviors are observed with a great number of intermingled periodic windows.