A stochastic limit approach to the SAT problem

We propose a new approach to solve an NP complete problem by means of stochastic limit.Comment: 8 page

Quantum Algorithm for SAT Problem and Quantum Mutual Entropy

It is von Neumann who opened the window for today's Information epoch. He defined quantum entropy including Shannon's information more than 20 years ahead of Shannon, and he introduced a concept what computation means mathematically. In this paper I will report two works that we have recently done, one of which is on quantum algorithum in generalized sense solving the SAT problem (one of NP complete problems) and another is on quantum mutual entropy properly describing quantum communication processes.Comment: 19 pages, Proceedings of the von Neumann Centennial Conference: Linear Operators and Foundations of Quantum Mechanics, Budapest, Hungary, 15-20 October, 200

Generic q-Markov semigroups and speed of convergence of q-algorithms

We study a special class of generic quantum Markov semigroups, on the algebra of all bounded operators on a Hubert space HS, arising in the stochastic limit of a generic system interacting with a boson Fock reservoir. This class depends on an orthonormal basis of HS. We obtain a new estimate for the trace distance of a state from a pure state and use this estimate to prove that, under the action of a semigroup of this class, states with finite support with respect to the given basis converge to equilibrium with a speed which is exponential, but with a polynomial correction which makes the convergence increasingly worse as the dimension of the support increases (Theorem 5.1). We interpret the semigroup as an algorithm, its initial state as input and, following Belavkin and Ohya,10 the dimension of the support of a state as a measure of complexity of the input. With this interpretation, the above results mean that the complexity of the input "slows down" the convergence of the algorithm. Even if the convergence is exponential and the slow down the polynomial, the constants involved may be such that the convergence times become unacceptable from a computational standpoint. This suggests that, in the absence of estimates of the constants involved, distinctions such as "exponentially fast" and "polynomially slow" may become meaningless from a constructive point of view. We also show that, for arbitray states, the speed of convergence to equilibrium is controlled by the rate of decoherence and the rate of purification (i.e. of concentration of the probability on a single pure state). We construct examples showing that the order of magnitude of these two decays can be quite differen

A study on morphological and dynamical properties of neuronal growth cones

In the developing nervous system and in the adult brain, neurons constantly need to solve mechanical problems. Neuronal growth cones are the main motile structures located at the tip of neurites and are composed of a lamellipodium from which thin filopodia emerge. They are responsible for extension of neurite processes and for transducing signals from extracellular cues to alter directionality, branching, and motility. They must decide how to explore the environment and in which direction to grow; they also need to establish appropriate contacts, to avoid obstacles and to determine how much force to exert. The complete understanding of the nervous system and its basic unit, the neuron, demands a quantification of the behavioral pattern and the morphological characteristics..