Sharpness of the Satisfiability Threshold for Non-Uniform Random k-SAT

Power Law ◽

General Model ◽

Step Function ◽

Propositional Satisfiability ◽

Wide Range ◽

Random Instances ◽

Variable Probability

We study a more general model to generate random instances of Propositional Satisfiability (SAT) with n Boolean variables, m clauses, and exactly k variables per clause. Additionally, our model is given an arbitrary probability distribution (p_1, ..., p_n) on the variable occurrences. Therefore, we call it non-uniform random k-SAT. The number m of randomly drawn clauses at which random formulas go from asymptotically almost surely (a.a.s.) satisfiable to a.a.s. unsatisfiable is called the satisfiability threshold. Such a threshold is called sharp if it approaches a step function as n increases. We identify conditions on the variable probability distribution (p_1, ..., p_n) under which the satisfiability threshold is sharp if its position is already known asymptotically. This result generalizes Friedgut’s sharpness result from uniform to non-uniform random k -SAT and implies sharpness for thresholds of a wide range of random k -SAT models with heterogeneous probability distributions, for example such models where the variable probabilities follow a power-law.

Conditional Mixture Model for Correlated Neuronal Spikes

Neural Computation ◽

10.1162/neco.2010.04-08-766 ◽

2010 ◽

Vol 22 (7) ◽

pp. 1718-1736 ◽

Cited By ~ 2

Author(s):

Shun-ichi Amari

Keyword(s):

Mixture Models ◽

Mixture Model ◽

Computational Neuroscience ◽

General Model ◽

Real Data ◽

Spike Trains ◽

Cell Assemblies ◽

Competitive Model

Analysis of correlated spike trains is a hot topic of research in computational neuroscience. A general model of probability distributions for spikes includes too many parameters to be of use in analyzing real data. Instead, we need a simple but powerful generative model for correlated spikes. We developed a class of conditional mixture models that includes a number of existing models and analyzed its capabilities and limitations. We apply the model to dynamical aspects of neuron pools. When Hebbian cell assemblies coexist in a pool of neurons, the condition is specified by these assemblies such that the probability distribution of spikes is a mixture of those of the component assemblies. The probabilities of activation of the Hebbian assemblies change dynamically. We used this model as a basis for a competitive model governing the states of assemblies.

A New Limit Theorem for Quantum Walk in Terms of Quantum Bernoulli Noises

Entropy ◽

10.3390/e22040486 ◽

2020 ◽

Vol 22 (4) ◽

pp. 486

Author(s):

Caishi Wang ◽

Suling Ren ◽

Yuling Tang

Keyword(s):

Limit Theorem ◽

Quantum Walk ◽

Initial State ◽

Wide Range ◽

Gauss Type ◽

Limit Probability

In this paper, we consider limit probability distributions of the quantum walk recently introduced by Wang and Ye (C.S. Wang and X.J. Ye, Quantum walk in terms of quantum Bernoulli noises, Quantum Inf. Process. 15 (2016), no. 5, 1897–1908). We first establish several technical theorems, which themselves are also interesting. Then, by using these theorems, we prove that, for a wide range of choices of the initial state, the above-mentioned quantum walk has a limit probability distribution of standard Gauss type, which actually gives a new limit theorem for the walk.

Sigma-Pi Structure with Bernoulli Random Variables: Power-Law Bounds for Probability Distributions and Growth Models with Interdependent Entities

Entropy ◽

10.3390/e23020241 ◽

2021 ◽

Vol 23 (2) ◽

pp. 241

Author(s):

Arthur Matsuo Yamashita Rios de Sousa ◽

Hideki Takayasu ◽

Didier Sornette ◽

Misako Takayasu

Keyword(s):

Power Law ◽

Extreme Values ◽

Growth Models ◽

Random Variables ◽

Random Coefficients ◽

Bernoulli Random Variables ◽

Product Term ◽

Simple Growth

The Sigma-Pi structure investigated in this work consists of the sum of products of an increasing number of identically distributed random variables. It appears in stochastic processes with random coefficients and also in models of growth of entities such as business firms and cities. We study the Sigma-Pi structure with Bernoulli random variables and find that its probability distribution is always bounded from below by a power-law function regardless of whether the random variables are mutually independent or duplicated. In particular, we investigate the case in which the asymptotic probability distribution has always upper and lower power-law bounds with the same tail-index, which depends on the parameters of the distribution of the random variables. We illustrate the Sigma-Pi structure in the context of a simple growth model with successively born entities growing according to a stochastic proportional growth law, taking both Bernoulli, confirming the theoretical results, and half-normal random variables, for which the numerical results can be rationalized using insights from the Bernoulli case. We analyze the interdependence among entities represented by the product terms within the Sigma-Pi structure, the possible presence of memory in growth factors, and the contribution of each product term to the whole Sigma-Pi structure. We highlight the influence of the degree of interdependence among entities in the number of terms that effectively contribute to the total sum of sizes, reaching the limiting case of a single term dominating extreme values of the Sigma-Pi structure when all entities grow independently.

Probability Density Functions

Bayesian Statistics for Beginners ◽

10.1093/oso/9780198841296.003.0009 ◽

2019 ◽

pp. 108-130

Author(s):

Therese M. Donovan ◽

Ruth M. Mickey

Keyword(s):

Bayesian Inference ◽

Probability Density ◽

Random Variable ◽

Probability Density Functions ◽

Density Functions ◽

Normal Distributions ◽

Likelihood Profile ◽

Variable Probability

This chapter builds on probability distributions. Its focus is on general concepts associated with probability density functions (pdf’s), which are distributions associated with continuous random variables. The continuous uniform and normal distributions are highlighted as examples of pdf’s. These and other pdf’s can be used to specify prior distributions, likelihoods, and/or posterior distributions in Bayesian inference. Although this chapter specifically focuses on the continuous uniform and normal distributions, the concepts discussed in this chapter will apply to other continuous probability distributions. By the end of the chapter, the reader should be able to define and use the following terms for a continuous random variable: random variable, probability distribution, parameter, probability density, likelihood, and likelihood profile.

INTENSITY PROBABILITY DISTRIBUTION OF SPIKING-MODE OSCILLATIONS IN LASERS: POWER-LAW UNIVERSALITY

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127494000174 ◽

1994 ◽

Vol 04 (01) ◽

pp. 225-230

Author(s):

FUMIHIKO ISHIYAMA ◽

KENJU OTSUKA

Keyword(s):

Power Law ◽

Delayed Feedback ◽

Output Intensity ◽

External Perturbations ◽

Loss Modulation

We theoretically investigated the probability distributions of the output intensity of lasers with external perturbations. These perturbations were loss modulation and incoherent delayed feedback. We found that the intensity probability distributions usually obey a distinct power-law in spiking-mode oscillations both in periodic and chaotic regimes.

Linear electrical properties of striated muscle fibres observed with intracellular electrodes

Proceedings of the Royal Society of London Series B Biological Sciences ◽

10.1098/rspb.1964.0030 ◽

1964 ◽

Vol 160 (978) ◽

pp. 69-123 ◽

Cited By ~ 135

Keyword(s):

Electrical Properties ◽

Striated Muscle ◽

Surface Membrane ◽

Fibre Surface ◽

Step Function ◽

Muscle Fibres ◽

Appreciable Effect ◽

Current Application ◽

Current Path ◽

Wide Range

The linear electrical properties of muscle fibres have been examined using intracellular electrodes for a. c. measurements and analyzing observations on the basis of cable theory. The measurements have covered the frequency range 1 c/s to 10 kc/s. Comparison of the theory for the circular cylindrical fibre with that for the ideal, one-dimensional cable indicates that, under the conditions of the experiments, no serious error would be introduced in the analysis by the geometrical idealization. The impedance locus for frog sartorius and crayfish limb muscle fibres deviates over a wide range of frequencies from that expected for a simple model in which the current path between the inside and the outside of the fibre consists only of a resistance and a capacitance in parallel. A good fit of the experimental results on frog fibres is obtained if the inside-outside admittance is considered to contain, in addition to the parallel elements R m = 3100 Ωcm 2 and C m = 2.6 μF/cm 2 , another path composed of a resistance R e = 330 Ωcm 2 in series with a capacitance C e = 4.1 μF/cm 2 , all referred to unit area of fibre surface. The impedance behaviour of crayfish fibres can be described by a similar model, the corresponding values being R m = 680 Ωcm 2 , C m = 3.9 μF/cm 2 , R e = 35 Ωcm 2 , C e = 17 μF/cm 2 . The response of frog fibres to a step-function current (with the points of voltage recording and current application close together) has been analyzed in terms of the above two-time constant model, and it is shown that neglecting the series resistance would have an appreciable effect on the agreement between theory and experiment only at times less than the halftime of rise of the response. The elements R m and C m are presumed to represent properties of the surface membrane of the fibre. R e and C e are thought to arise not at the surface, but to be indicative of a separate current path from the myoplasm through an intracellular system of channels to the exterior. In the case of crayfish fibres, it is possible that R e (when referred to unit volume) would be a measure of the resistivity of the interior of the channels, and C e the capacitance across the walls of the channels. In the case of frog fibres, it is suggested that the elements R e , C e arise from the properties of adjacent membranes of the triads in the sarcoplasmic reticulum . The possibility is considered that the potential difference across the capacitance C e may control the initiation of contraction.

Towards the Dependence on Parameters for the Solution of the Thermostatted Kinetic Framework

Axioms ◽

10.3390/axioms10020059 ◽

2021 ◽

Vol 10 (2) ◽

pp. 59

Author(s):

Bruno Carbonaro ◽

Marco Menale

Keyword(s):

Complex System ◽

Continuous Dependence ◽

Social Role ◽

Functional Dependence ◽

Transition Probability ◽

Classical Mechanics ◽

Human Society ◽

Wide Range ◽

Probability Densities

A complex system is a system involving particles whose pairwise interactions cannot be composed in the same way as in classical Mechanics, i.e., the result of interaction of each particle with all the remaining ones cannot be expressed as a sum of its interactions with each of them (we cannot even know the functional dependence of the total interaction on the single interactions). Moreover, in view of the wide range of its applications to biologic, social, and economic problems, the variables describing the state of the system (i.e., the states of all of its particles) are not always (only) the usual mechanical variables (position and velocity), but (also) many additional variables describing e.g., health, wealth, social condition, social rôle ⋯, and so on. Thus, in order to achieve a mathematical description of the problems of everyday’s life of any human society, either at a microscopic or at a macroscpoic scale, a new mathematical theory (or, more precisely, a scheme of mathematical models), called KTAP, has been devised, which provides an equation which is a generalized version of the Boltzmann equation, to describe in terms of probability distributions the evolution of a non-mechanical complex system. In connection with applications, the classical problems about existence, uniqueness, continuous dependence, and stability of its solutions turn out to be particularly relevant. As far as we are aware, however, the problem of continuous dependence and stability of solutions with respect to perturbations of the parameters expressing the interaction rates of particles and the transition probability densities (see Section The Basic Equations has not been tackled yet). Accordingly, the present paper aims to give some initial results concerning these two basic problems. In particular, Theorem 2 reveals to be stable with respect to small perturbations of parameters, and, as far as instability of solutions with respect to perturbations of parameters is concerned, Theorem 3 shows that solutions are unstable with respect to “large” perturbations of interaction rates; these hints are illustrated by numerical simulations that point out how much solutions corresponding to different values of parameters stay away from each other as t→+∞.

Probability-Based Prediction of Activity in Multiple Arm Muscles: Implications for Functional Electrical Stimulation

Journal of Neurophysiology ◽

10.1152/jn.00956.2007 ◽

2008 ◽

Vol 100 (1) ◽

pp. 482-494 ◽

Cited By ~ 9

Author(s):

Chad V. Anderson ◽

Andrew J. Fuglevand

Keyword(s):

Electrical Stimulation ◽

Functional Electrical Stimulation ◽

Muscle Activity ◽

Probabilistic Methods ◽

Muscle Stimulation ◽

Implanted Electrodes ◽

Motor Behaviors ◽

Wide Range ◽

Arm Muscles

Functional electrical stimulation (FES) involves artificial activation of muscles with implanted electrodes to restore motor function in paralyzed individuals. The range of motor behaviors that can be generated by FES, however, is limited to a small set of preprogrammed movements such as hand grasp and release. A broader range of movements has not been implemented because of the substantial difficulty associated with identifying the patterns of muscle stimulation needed to elicit specified movements. To overcome this limitation in controlling FES systems, we used probabilistic methods to estimate the levels of muscle activity in the human arm during a wide range of free movements based on kinematic information of the upper limb. Conditional probability distributions were generated based on hand kinematics and associated surface electromyographic (EMG) signals from 12 arm muscles recorded during a training task involving random movements of the arm in one subject. These distributions were then used to predict in four other subjects the patterns of muscle activity associated with eight different movement tasks. On average, about 40% of the variance in the actual EMG signals could be accounted for in the predicted EMG signals. These results suggest that probabilistic methods ultimately might be used to predict the patterns of muscle stimulation needed to produce a wide array of desired movements in paralyzed individuals with FES.

Aircraft measurements of gravity waves in the upper troposphere and lower stratosphere during the START08 field experiment

Atmospheric Chemistry and Physics ◽

10.5194/acp-15-7667-2015 ◽

2015 ◽

Vol 15 (13) ◽

pp. 7667-7684 ◽

Cited By ~ 13

Author(s):

Fuqing Zhang ◽

Junhong Wei ◽

Meng Zhang ◽

K. P. Bowman ◽

L. L. Pan ◽

...

Keyword(s):

Power Law ◽

Gravity Waves ◽

Lower Stratosphere ◽

Potential Temperature ◽

Shear Instability ◽

Aircraft Measurements ◽

Upper Troposphere ◽

Airborne Measurements ◽

Wide Range

Abstract. This study analyzes in situ airborne measurements from the 2008 Stratosphere–Troposphere Analyses of Regional Transport (START08) experiment to characterize gravity waves in the extratropical upper troposphere and lower stratosphere (ExUTLS). The focus is on the second research flight (RF02), which took place on 21–22 April 2008. This was the first airborne mission dedicated to probing gravity waves associated with strong upper-tropospheric jet–front systems. Based on spectral and wavelet analyses of the in situ observations, along with a diagnosis of the polarization relationships, clear signals of mesoscale variations with wavelengths ~ 50–500 km are found in almost every segment of the 8 h flight, which took place mostly in the lower stratosphere. The aircraft sampled a wide range of background conditions including the region near the jet core, the jet exit and over the Rocky Mountains with clear evidence of vertically propagating gravity waves of along-track wavelength between 100 and 120 km. The power spectra of the horizontal velocity components and potential temperature for the scale approximately between ~ 8 and ~ 256 km display an approximate −5/3 power law in agreement with past studies on aircraft measurements, while the fluctuations roll over to a −3 power law for the scale approximately between ~ 0.5 and ~ 8 km (except when this part of the spectrum is activated, as recorded clearly by one of the flight segments). However, at least part of the high-frequency signals with sampled periods of ~ 20–~ 60 s and wavelengths of ~ 5–~ 15 km might be due to intrinsic observational errors in the aircraft measurements, even though the possibilities that these fluctuations may be due to other physical phenomena (e.g., nonlinear dynamics, shear instability and/or turbulence) cannot be completely ruled out.

Poisson mixtures

Natural Language Engineering ◽

10.1017/s1351324900000139 ◽

1995 ◽

Vol 1 (2) ◽

pp. 163-190 ◽

Cited By ~ 146

Author(s):

Kenneth W. Church ◽

William A. Gale

Keyword(s):

Negative Binomial ◽

Hidden Variables ◽

Heterogeneous Structure ◽

Text Compression ◽

Inverse Document Frequency ◽

Poisson Mixtures ◽

Document Frequency ◽

Wide Range ◽

Better Than

AbstractShannon (1948) showed that a wide range of practical problems can be reduced to the problem of estimating probability distributions of words and ngrams in text. It has become standard practice in text compression, speech recognition, information retrieval and many other applications of Shannon's theory to introduce a “bag-of-words” assumption. But obviously, word rates vary from genre to genre, author to author, topic to topic, document to document, section to section, and paragraph to paragraph. The proposed Poisson mixture captures much of this heterogeneous structure by allowing the Poisson parameter θ to vary over documents subject to a density function φ. φ is intended to capture dependencies on hidden variables such genre, author, topic, etc. (The Negative Binomial is a well-known special case where φ is a Г distribution.) Poisson mixtures fit the data better than standard Poissons, producing more accurate estimates of the variance over documents (σ2), entropy (H), inverse document frequency (IDF), and adaptation (Pr(x ≥ 2/x ≥ 1)).