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1,118
Experimental Issues in Coherent QuantumState Manipulation of Trapped Atomic Ions
, 1998
"... this paper, we investigate a subset of these topics which involve the coherent manipulation of quantum states of trapped atomic ions. The focus will be on a proposal to implement quantum logic and quantum computation using trapped ions [1]. However, we will also consider related work on the generati ..."
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Cited by 68 (11 self)
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this paper, we investigate a subset of these topics which involve the coherent manipulation of quantum states of trapped atomic ions. The focus will be on a proposal to implement quantum logic and quantum computation using trapped ions [1]. However, we will also consider related work on the generation of nonclassical states of motion and entangled states of trapped ions [239]. Many of these ideas have been summarized in a recent review [40].
Adaptive Nonlinear Approximations
, 1994
"... The problem of optimally approximating a function with a linear expansion over a redundant dictionary of waveforms is NPhard. The greedy matching pursuit algorithm and its orthogonalized variant produce suboptimal function expansions by iteratively choosing the dictionary waveforms which best matc ..."
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Cited by 66 (1 self)
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The problem of optimally approximating a function with a linear expansion over a redundant dictionary of waveforms is NPhard. The greedy matching pursuit algorithm and its orthogonalized variant produce suboptimal function expansions by iteratively choosing the dictionary waveforms which best match the function's structures. Matching pursuits provide a means of quickly computing compact, adaptive function approximations. Numerical experiments show that the approximation errors from matching pursuits initially decrease rapidly, but the asymptotic decay rate of the errors is slow. We explain this behavior by showing that matching pursuits are chaotic, ergodic maps. The statistical properties of the approximation errors of a pursuit can be obtained from the invariant measure of the pursuit. We characterize these measures using group symmetries of dictionaries and using a stochastic differential equation model. These invariant measures define a noise with respect to a given dictionary. ...
Efficient collocational approach for parametric uncertainty analysis
 Commun. Comput. Phys
, 2007
"... Abstract. A numerical algorithm for effective incorporation of parametric uncertainty into mathematical models is presented. The uncertain parameters are modeled as random variables, and the governing equations are treated as stochastic. The solutions, or quantities of interests, are expressed as co ..."
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Cited by 57 (4 self)
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Abstract. A numerical algorithm for effective incorporation of parametric uncertainty into mathematical models is presented. The uncertain parameters are modeled as random variables, and the governing equations are treated as stochastic. The solutions, or quantities of interests, are expressed as convergent series of orthogonal polynomial expansions in terms of the input random parameters. A highorder stochastic collocation method is employed to solve the solution statistics, and more importantly, to reconstruct the polynomial expansion. While retaining the high accuracy by polynomial expansion, the resulting “pseudospectral ” type algorithm is straightforward to implement as it requires only repetitive deterministic simulations. An estimate on error bounded is presented, along with numerical examples for problems with relatively complicated forms of governing equations. Key words: Collocation methods; pseudospectral methods; stochastic inputs; random differential equations; uncertainty quantification. 1
Molecular Modeling Of Proteins And Mathematical Prediction Of Protein Structure
 SIAM Review
, 1997
"... . This paper discusses the mathematical formulation of and solution attempts for the socalled protein folding problem. The static aspect is concerned with how to predict the folded (native, tertiary) structure of a protein, given its sequence of amino acids. The dynamic aspect asks about the possib ..."
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Cited by 55 (5 self)
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. This paper discusses the mathematical formulation of and solution attempts for the socalled protein folding problem. The static aspect is concerned with how to predict the folded (native, tertiary) structure of a protein, given its sequence of amino acids. The dynamic aspect asks about the possible pathways to folding and unfolding, including the stability of the folded protein. From a mathematical point of view, there are several main sides to the static problem:  the selection of an appropriate potential energy function;  the parameter identification by fitting to experimental data; and  the global optimization of the potential. The dynamic problem entails, in addition, the solution of (because of multiple time scales very stiff) ordinary or stochastic differential equations (molecular dynamics simulation), or (in case of constrained molecular dynamics) of differentialalgebraic equations. A theme connecting the static and dynamic aspect is the determination and formation of...
The Diffusion Limit of Transport Equations II: Chemotaxis Equations
"... this paper we use the diusionlimit expansion of transport equations developed earlier [23] to study the limiting equation under avariety of external biases imposed on the motion. When applied to chemotaxis or chemokinesis, these biases produce modication of the turning rate, the movement speed or t ..."
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Cited by 54 (9 self)
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this paper we use the diusionlimit expansion of transport equations developed earlier [23] to study the limiting equation under avariety of external biases imposed on the motion. When applied to chemotaxis or chemokinesis, these biases produce modication of the turning rate, the movement speed or the preferred direction of movement. Depending on the strength of the bias, it leads to anisotropic diusion, to a drift term in the ux or to both, in the parabolic limit. We show that the classical chemotaxis equation  whichwe call the PatlakKellerSegelAlt (PKSA) equation  only arises when the bias is suciently small. Using this general framework, we derive phenomenological models for chemotaxis of agellated bacteria, of slime molds and of myxobacteria. We also show that certain results derived earlier for onedimensional motion can easily be generalized to two or threedimensional motion as well. ## ############# The linear transport equation @ @t p(x; v; t)+v ##p(x; v; t)=#p(x; v; t)+ # # T(v;v # )p(x; v # ;t)dv # ; (1.1) in which p(x; v; t) represents the density of particles at spatial position x # IR # moving with velocity v # V # IR # at time t # 0, arises when the movement of biological organisms is modeled byavelocityjump process [38]. Here the turning rate may be space or velocitydependent, but in other contexts it may also depend on internal variables that evolve in space and time, in which case (1.1) must be generalized. The turning kernel or turn angle distribution T (v; v # ) gives the probabilityofavelocity jump from v # to v if a jump occurs: in general it may also be spacedependent or depend on internal variables. In the present formulation we assume that the `decision' to turn as reected in is not coupled to th...
Statistical mechanics of neocortical interactions: A scaling paradigm applied to electroencephalography
 PHYS. REV. A
, 1991
"... A series of papers has developed a statistical mechanics of neocortical interactions (SMNI), deriving aggregate behavior of experimentally observed columns of neurons from statistical electricalchemical properties of synaptic interactions. While not useful to yield insights at the single neuron lev ..."
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Cited by 51 (44 self)
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A series of papers has developed a statistical mechanics of neocortical interactions (SMNI), deriving aggregate behavior of experimentally observed columns of neurons from statistical electricalchemical properties of synaptic interactions. While not useful to yield insights at the single neuron level, SMNI has demonstrated its capability in describing largescale properties of shortterm memory and electroencephalographic (EEG) systematics. The necessity of including nonlinear and stochastic structures in this development has been stressed. In this paper, a more stringent test is placed on SMNI: The algebraic and numerical algorithms previously developed in this and similar systems are brought to bear to fit large sets of EEG and evoked potential data being collected to investigate genetic predispositions to alcoholism and to extract brain “signatures” of shortterm memory. Using the numerical algorithm of Very Fast Simulated ReAnnealing, it is demonstrated that SMNI can indeed fit this data within experimentally observed ranges of its underlying neuronalsynaptic parameters, and use the quantitative modeling results to examine physical neocortical mechanisms to discriminate between highrisk and lowrisk populations genetically predisposed to alcoholism. Since this first study is a control to span relatively long time epochs, similar to earlier attempts to establish such correlations, this discrimination is inconclusive because of other neuronal activity which can mask such effects. However, the SMNI model is shown to be consistent
Fast evaluation of fluctuations in biochemical networks with the linear noise approximation
 Genome Research
"... Biochemical networks in single cells can display large fluctuations in molecule numbers, making mesoscopic approaches necessary for correct system descriptions. We present a general method that allows rapid characterization of the stochastic properties of intracellular networks. The starting point i ..."
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Cited by 50 (2 self)
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Biochemical networks in single cells can display large fluctuations in molecule numbers, making mesoscopic approaches necessary for correct system descriptions. We present a general method that allows rapid characterization of the stochastic properties of intracellular networks. The starting point is a macroscopic description that identifies the system’s elementary reactions in terms of rate laws and stoichiometries. From this formulation follows directly the stationary solution of the linear noise approximation (LNA) of the Master equation for all the components in the network. The method complements bifurcation studies of the system’s parameter dependence by providing estimates of sizes, correlations, and time scales of stochastic fluctuations. We describe how the LNA can give precise system descriptions also near macroscopic instabilities by suitable variable changes and elimination of fast variables. [Supplemental material is available online at www.genome.org.] A key element in systems biology is the design of mathematical models that faithfully describe the dynamics of intracellular chemical networks. In general, chemical reactions in single cells occur far from thermodynamic equilibrium (Keizer 1987), and the molecule copy numbers can sometimes be very small (Guptasarama 1995). Both these properties make it mandatory to ana
A theory of interdecadal climate variability of the North Pacific ocean–atmosphere system
 J. Climate
, 1997
"... A linear coupled model for the atmosphere–upperocean system is proposed to highlight the mechanisms of decadal to interdecadal climate variability in the North Pacific. In this model, wind stress anomalies over the North Pacific are related to anomalies in the meridional temperature gradient of the ..."
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Cited by 48 (1 self)
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A linear coupled model for the atmosphere–upperocean system is proposed to highlight the mechanisms of decadal to interdecadal climate variability in the North Pacific. In this model, wind stress anomalies over the North Pacific are related to anomalies in the meridional temperature gradient of the upper ocean. The latter depends upon air–sea thermodynamical feedbacks and meridional heat transport by upperocean currents. Slow adjustment of the oceanic gyre circulation to the change in wind stress is accomplished by the forced baroclinic oceanic Rossby waves, which carry out the meridional heat transport. Uncoupled ocean dynamic adjustment can produce a weak decadal to interdecadal peak in the power spectrum of the meridional transport under temporal white noise wind stress forcing with organized spatial structure. Coupled dynamics produce a basinscale interdecadal oscillatory mode. This mode arises from the dynamic coupling and the memory of the system, residing in the slow gyre circulation adjustment. Its stability is heavily controlled by the ocean thermal damping, and its period is about one and onehalf to three times the decadal ocean dynamic adjustment time. In the relevant parameter regime, this coupled mode produces a robust and pronounced interdecadal spectral peak in the upperocean temperature and the Sverdrup transport of the gyre circulation. The interdecadal oscillations reproduced in the simple model provide insights into main physical mechanisms of the North Pacific decadal–interdecadal variability observed in nature and simulated in coupled general circulation models. 1.
Measuring dynamical prediction utility using relative entropy
 J. Atmos. Sci
, 2002
"... A new parameter of dynamical system predictability is introduced that measures the potential utility of predictions. It is shown that this parameter satisfies a generalized second law of thermodynamics in that for Markov processes utility declines monotonically to zero at very long forecast times. E ..."
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Cited by 48 (5 self)
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A new parameter of dynamical system predictability is introduced that measures the potential utility of predictions. It is shown that this parameter satisfies a generalized second law of thermodynamics in that for Markov processes utility declines monotonically to zero at very long forecast times. Expressions for the new parameter in the case of Gaussian prediction ensembles are derived and a useful decomposition of utility into dispersion (roughly equivalent to ensemble spread) and signal components is introduced. Earlier measures of predictability have usually considered only the dispersion component of utility. A variety of simple dynamical systems with relevance to climate and weather prediction is introduced, and the behavior of their potential utility is analyzed in detail. For the climate systems examined here, the signal component is at least as important as the dispersion in determining the utility of a particular set of initial conditions. The simple ‘‘weather’ ’ system examined (the Lorenz system) exhibited different behavior with the dispersion being more important than the signal at short prediction lags. For longer lags there appeared no relation between utility and either signal or dispersion. On the other hand, there was a very strong relation at all lags between utility and the location of the initial conditions on the attractor. 1.
The dynamics of choice among multiple alternatives
 Journal of Mathematical Psychology
, 2006
"... We consider neurallybased models for decisionmaking in the presence of noisy incoming data. The twoalternative forcedchoice task has been extensively studied, and in that case it is known that mutuallyinhibited leaky integrators in which leakage and inhibition balance can closely approximate a ..."
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Cited by 47 (8 self)
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We consider neurallybased models for decisionmaking in the presence of noisy incoming data. The twoalternative forcedchoice task has been extensively studied, and in that case it is known that mutuallyinhibited leaky integrators in which leakage and inhibition balance can closely approximate a driftdiffusion process that is the continuum limit of the optimal sequential probability ratio test (SPRT). Here we study the performance of neural integrators in n ≥ 2 alternative choice tasks and relate them to a multihypothesis sequential probability ratio test (MSPRT) that is asymptotically optimal in the limit of vanishing error rates. While a simple race model can implement this ‘maxvsnext ’ MSPRT, it requires an additional computational layer, while absolute threshold crossing tests do not require such a layer. Race models with absolute thresholds perform relatively poorly, but we show that a balanced leaky accumulator model with an absolute crossing criterion can approximate a ‘maxvsave ’ test that is intermediate in performance between the absolute and maxvsnext tests. We consider free and fixed time response protocols, and show that the resulting mean reaction times under the former and decision times for fixed accuracy under the latter obey versions of Hick’s law in the low error rate range, and we interpret this in terms of information gained. Specifically, we derive relationships of the forms log(n − 1), log(n), or log(n + 1) depending on error rates, signaltonoise ratio, and the test itself. We focus on linearized models, but also consider nonlinear effects of neural activities (firing rates) that are bounded below and show how they modify Hick’s law. KEYWORDS: leaky accumulator, driftdiffusion model, neural network, Hick’s law, multihypothesis sequential test, sequential ratio test.