For an elliptic curve E/K without potential complex multiplication we bound the index of the image of Gal K/K in GL2( Z), the representation being given by the action on the Tate modules of E at the various primes. The bound is explicit and only depends on [K : Q] and on the stable Faltings height of E. We also prove a result relating the structure of closed subgroups of GL2(Z ℓ ) to certain Lie algebras naturally attached to them.
Let A be the product of an abelian variety and a torus defined over a number field K. Fix some prime number ℓ. If α ∈ A(K) is a point of infinite order, we consider the set of primes p of K such that the reduction (α mod p) is well-defined and has order coprime to ℓ. This set admits a natural density. By refining the method of R. Jones and J. Rouse (2010), we can express the density as an ℓ-adic integral without requiring any assumption. We also prove that the density is always a rational number whose denominator (up to powers of ℓ) is uniformly bounded in a very strong sense. For elliptic curves, we describe a strategy for computing the density which covers every possible case.
Behavior-based robotics considers perception as a holistic process, strongly connected to behavioral needs of the robot. We present a bio-inspired framework for sensing-perception-action, applied to a roving robot in a random foraging task. Perception is here considered as a complex and emergent phenomenon where a huge amount of information coming from sensors is used to form an abstract and concise representation of the environment, useful to take a suitable action or sequence of actions. In this work a model for perceptual representation is formalized by means of RD-CNNs showing Turing patterns. They are used as attractive states for particular set of environmental conditions in order to associate, via a reinforcement learning, a proper action. Learning is also introduced at the afferent stage to shape the environment information according to the particular emerging pattern. The basins of attraction for the Turing patterns are so dynamically tuned by an unsupervised learning in order to form an internal, abstract and plastic representation of the environment, as recorded by the sensors.
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