The manner in which states of some quantum systems become effectively classical is of great significance for the foundations of quantum physics, as well as for problems of practical interest such as quantum engineering. In the past two decades it has become increasingly clear that many (perhaps all) of the symptoms of classicality can be induced in quantum systems by their environments. Thus decoherence is caused by the interaction in which the environment in effect monitors certain observables of the system, destroying coherence between the pointer states corresponding to their eigenvalues. This leads to environment-induced superselection or einselection, a quantum process associated with selective loss of information. Einselected pointer states are stable. They can retain correlations with the rest of the universe in spite of the environment. Einselection enforces classicality by imposing an effective ban on the vast majority of the Hilbert space, eliminating especially the flagrantly nonlocal ‘‘Schrödinger-cat states.’ ’ The classical structure of phase space emerges from the quantum Hilbert space in the appropriate macroscopic limit. Combination of einselection with dynamics leads to the idealizations of a point and of a classical trajectory. In measurements, einselection replaces quantum entanglement between the apparatus and the measured system with the classical correlation. Only the preferred pointer observable of the apparatus can store information

How does a neuron vary its mean output firing rate if the input changes from random to coherent activity? What are the critical parameters of the neuronal dynamics and input statistics? To answer these questions, we investigate the coincidence detection properties of an integrate-and-fire neuron. We derive an expression indicating how coincidence detection depends on neuronal parameters. Specifically, (i) we show how coincidence detection depends on the shape of the postsynaptic response function, the number of synapses, and the input statistics, and (ii) we demonstrate that there is an optimal threshold. Our considerations can be used to predict from neuronal parameters whether and to what extent a neuron can act as a coincidence detector and thus can convert a temporal code into a rate code. Physik-Department der TU Munchen (T35), D-85747 Garching bei Munchen, Germany y Swiss Federal Institute of Technology, Center of Neuromimetic Systems, EPFL-DI, CH-1015 Lausanne, Switz...

Many problems in computer graphics deal with complex 3D scenes where visibility, proximity, collision detection queries have to be answered. Due to the complexity of these queries and the one of the models they are applied to, data structures most often based on hierarchical decompositions have been proposed to solve them. As a result of these involved algorithms/data structures, most of the analysis have been carried out in the worst case and fail to report good average case performances in a vast majority of cases. The goal of this work is therefore to investigate geometric probability tools to characterize average case properties of standard scenes such as architectural scenes, natural models, etc under some standard visibility and proximity requests. In the first part we recall some fundamentals of integral geometry and discuss the classical assumption of measures invariant under the group of motions in the context of non uniform models. In the second one we present simple generali...

The roles of decoherence and environment-induced superselection in the emergence of the classical from the quantum substrate are described. The stability of correlations between the einselected quantum pointer states and the environment allows them to exist almost as objectively as classical states were once thought to exist: there are ways of finding out what is the pointer state of the system which uses redundancy of its correlations with the environment, and which leave einselected states essentially unperturbed. This relatively objective existence of certain quantum states facilitates operational definition of probabilities in the quantum setting. Moreover, once there are states that ‘exist ’ and can be ‘found out’, a ‘collapse ’ in the traditional sense is no longer necessary—in effect, it has already happened. The role of the preferred states in the processing and storage of information is emphasized. The existential interpretation based on the relatively objective existence of stable correlations between the einselected states of observers ’ memory and in the outside universe is formulated and discussed.

in the 2D Ising Ferromagnet

The Weierstrass nowhere differentiable function, and functions constructed from similar infinite series, have been studied often as examples of functions whose graph is a fractal. Though there is a simple formula for the Hausdorff dimension of the graph which is widely accepted, it has not been rigorously proved to hold. We prove that if arbitrary phases are included in each term of the summation for the Weierstrass function, the Hausdorff dimension of the graph of the function has the conjectured value for almost every sequence of phases. The argument extends to a much wider class of Weierstrass-like functions. AMS 1991 Mathematics Subject Classification. Primary: 28A80, 26A30, 28A78; Secondary: 58F12. 1 Introduction Perhaps the most famous example of a continuous but nowhere differentiable function is that of Weierstrass, w(x) = 1 X k=0 a k cos(2ßb k x) -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 1: Graph of w(x) with a = 0:5 and b = 3. w...

Co-evolutionary learning involves a training process where training samples are instances of solutions that interact strategically to guide the evolutionary (learning) process. One main research issue is with the generalization performance, i.e., the search for solutions (e.g., input-output mappings) that best predict the required output for any new input that has not been seen during the evolutionary process. However, there is currently no such framework for determining the generalization performance in co-evolutionary learning even though the notion of generalization is well-understood in machine learning. In this paper, we introduce a theoretical framework to address this research issue. We present the framework in terms of game-playing although our results are more general. Here, a strategy’s generalization performance is its average performance against all test strategies. Given that the true value may not be determined by solving analytically a closed-form formula and is computationally prohibitive, we propose an estimation procedure that computes the average performance against a small sample of random test strategies instead. We perform a mathematical analysis to provide a statistical claim on the accuracy of our estimation procedure, which can be further improved by performing a second estimation on the variance of the random variable. For game-playing, it is well-known that one is more interested in the generalization

Let Gn,m,k denote the space of simple graphs with n vertices, m edges and minimum degree at least k, each graph G being equiprobable. Let G have property Ak if G contains ⌊(k − 1)/2 ⌋ edge disjoint Hamilton cycles, and, if k is even, a further edge disjoint matching of size ⌊n/2⌋. For k ≥ 3, Ak occurs in Gn,m,k with probability tending to 1 as n → ∞, when 2m ≥ ckn for some suitable constant ck.

Abstract. For a function ϕ in L 2 (0, 1), extended to the whole real line as an odd periodic function of period 2, we ask when the collection of dilates ϕ(nx), n = 1, 2, 3,..., constitutes a Riesz basis or a complete sequence in L 2 (0, 1). The problem translates into a question concerning multipliers and cyclic vectors in the Hilbert space H of Dirichlet series f(s) = ∑ n ann −s, where the coefficients an are square summable. It proves useful to model H as the H 2 space of the infinite-dimensional polydisk, or, which is the same, the H 2 space of the character space, where a character is a multiplicative homomorphism from the positive integers to the unit circle. For given f in H and characters χ, fχ(s) = ∑ n anχ(n)n −s is a vertical limit function of f. We study certain probabilistic properties of these vertical limit functions. 1.

Let X0 = 0, X1, X2,... be an aperiodic random walk generated by a sequence ξ1, ξ2,... of i.i.d. integer-valued random variables with common distribution p (·) having zero mean and finite variance. For an N-step trajectory X = (X0, X1,..., XN) and a monotone convex function V: R + → R + with V (0) = 0, define V(X) = PN−1 j=1 V ` |Xj | ´. Further, let I a,b N,+ be the set of all non-negative paths X compatible with the boundary conditions X0 = a, XN = b. We discuss asymptotic properties of X ∈ w.r.t. the probability distribution I a,b N,+ P a,b N,+,λ (X) = ` Z a,b ´ n −1 N,+,λ exp −λ V(X) o N−1 Y p (Xi+1 − Xi) as N → ∞ and λ → 0, Z a,b