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Decoherence, einselection, and the quantum origins of the classical
 REVIEWS OF MODERN PHYSICS 75, 715. AVAILABLE ONLINE AT HTTP://ARXIV.ORG/ABS/QUANTPH/0105127
, 2003
"... 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) ..."
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Cited by 46 (1 self)
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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 environmentinduced 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 "Schrödingercat 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
Extracting Oscillations: Neuronal Coincidence Detection with Noisy Periodic Spike Input
, 1998
"... How does a neuron vary its mean output firing rate if the input changes from random to oscillatory coherent but noisy activity? What are the critical parameters of the neuronal dynamics and input statistics? To answer these questions, we investigate the coincidencedetection properties of an integra ..."
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Cited by 19 (6 self)
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How does a neuron vary its mean output firing rate if the input changes from random to oscillatory coherent but noisy activity? What are the critical parameters of the neuronal dynamics and input statistics? To answer these questions, we investigate the coincidencedetection properties of an integrateandfire neuron. We derive an expression indicating how coincidence detection depends on neuronal parameters. Specifically, we show how coincidence detection depends on the shape of the postsynaptic response function, the number of synapses, and the input statistics, and 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.
Decoherence, Einselection and the Existential Interpretation (The Rough Guide)
 PHIL. TRANS. R. SOC. LOND. A
, 1998
"... The roles of decoherence and environmentinduced 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 ..."
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Cited by 19 (0 self)
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The roles of decoherence and environmentinduced 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.
Fluctuations of the phase boundary in the 2D Ising ferromagnet
 Commun. Math. Phys
, 1997
"... in the 2D Ising Ferromagnet ..."
Some Integral Geometry Tools to Estimate the Complexity of 3D Scenes
, 1997
"... 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 ..."
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Cited by 15 (4 self)
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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 Hausdorff Dimension of Graphs of Weierstrass Functions
 Trans. Amer. Math. Soc
, 1996
"... 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 rigo ..."
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Cited by 11 (0 self)
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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 Weierstrasslike 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...
A Hilbert space of Dirichlet series and systems of dilated functions
 in L2 (0,1
, 1997
"... 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 multiplie ..."
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Cited by 10 (2 self)
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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 infinitedimensional 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.
Measuring Generalization Performance in Coevolutionary Learning
"... Coevolutionary 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., inputoutput mappings ..."
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Cited by 9 (5 self)
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Coevolutionary 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., inputoutput 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 coevolutionary learning even though the notion of generalization is wellunderstood in machine learning. In this paper, we introduce a theoretical framework to address this research issue. We present the framework in terms of gameplaying 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 closedform 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 gameplaying, it is wellknown that one is more interested in the generalization
Relative states and the environment: Einselection, envariance, quantum darwinism, and the existential interpretation
, 2007
"... Starting with basic axioms of quantum theory we revisit “Relative State Interpretation ” set out 50 years ago by Hugh Everett III (1957a,b). His approach explains “collapse of the wavepacket ” by postulating that observer perceives the state of the “rest of the Universe ” relative to his own state, ..."
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Cited by 7 (0 self)
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Starting with basic axioms of quantum theory we revisit “Relative State Interpretation ” set out 50 years ago by Hugh Everett III (1957a,b). His approach explains “collapse of the wavepacket ” by postulating that observer perceives the state of the “rest of the Universe ” relative to his own state, or – to be more precise – relative to the state of his records. This allows quantum theory to be universally valid. However, while Everett explains perception of collapse, relative state approach raises three questions absent in Bohr’s Copenhagen Interpretation which relied on independent existence of an ab intio classical domain. One is now forced one to seek sets of preferred, effectively classical but ultimately quantum states that can define branches of the universal state vector, and allow observers to keep reliable records. Without such (i) preferred basis relative states are “too relative”, and the approach suffers from basis ambiguity. Moreover, universal validity of quantum theory raises the issue of the (ii) origin of probabilities, and of the Born’s rule pk = ψk  2 which is simply postulated in textbook discussions. Last not least, even preferred quantum states (defined e.g. by the einselection – environment induced superselection) – are still quantum. Therefore they cannot be found out by initially ignorant observers through direct measurement without getting disrupted. Yet, states of macroscopic object exist objectively and can be found out by anyone. So,
Edge disjoint Hamilton cycles in sparse random graphs of minimum degree at least k
 J. Graph Theory
"... 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 occu ..."
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Cited by 6 (2 self)
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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.