Poisson Law Research Articles

On a class of explicit Cauchy–Stieltjes transforms related to monotone stable and free Poisson laws

We consider a class of probability measures $\mu_{s,r}^{\alpha}$ which have explicit Cauchy–Stieltjes transforms. This class includes a symmetric beta distribution, a free Poisson law and some beta distributions as special cases. Also, we identify $\mu_{s,2}^{\alpha}$ as a free compound Poisson law with Lévy measure a monotone $\alpha$-stable law. This implies the free infinite divisibility of $\mu_{s,2}^{\alpha}$. Moreover, when symmetric or positive, $\mu_{s,2}^{\alpha}$ has a representation as the free multiplication of a free Poisson law and a monotone $\alpha$-stable law. We also investigate the free infinite divisibility of $\mu_{s,r}^{\alpha}$ for $r\neq2$. Special cases include the beta distributions $B(1-\frac{1}{r},1+\frac{1}{r})$ which are freely infinitely divisible if and only if $1\leq r\leq2$.

On seeing the wood from the leaves and the role of voxel size in determining leaf area distribution of forests with terrestrial LiDAR

Terrestrial LiDAR scanners have been shown to hold great potential for estimating and mapping three dimensional (3-D) leaf area distribution in forested environments. This is made possible by the capacity of LiDAR scanners to record the 3-D position of every laser pulse intercepted by plant material. The laser pulses emitted by a LiDAR scanner can be regarded as light probes whose transmission and interception may be used to derive leaf area density at different spatial scales using the Beer–Lambert law or Warren Wilson's contact frequency method among others. Segmenting the canopy into cubic volumes –or voxels- provides a convenient means to compute light transmission statistics and describe the spatial distribution of foliage area in tree crowns. In this paper, we investigate the optimal voxel dimensions for estimating the spatial distribution of within crown leaf area density. We analyzed LiDAR measurements from two field sites, located in Mali and in California, with trees having different leaf sizes during periods with and without leaves.We found that there is a range of voxel sizes, which satisfy three important conditions. The first condition is related to clumping and requires voxels small enough to exclude large gaps between crowns and branches. The second condition requires a voxel size large enough for the conditions postulated by the Poisson law to be valid, i.e., a turbid medium with randomly positioned leaves. And, the third condition relates to the appropriate voxel size to pinpoint the location of those volumes within the canopy which were insufficiently sampled by the LiDAR instrument to derive reliable statistics (occlusion effects). Here, we show that these requirements are a function of leaf size, branching structure, and the predominance of occlusion effects. The results presented provide guiding principles for using voxel volumes in the retrieval of leaf area distributions from terrestrial LiDAR measurements.

Energetic consistency conditions for standard impacts

The paper presents a full account of Poisson's impact law in inequality form. Based on an entirely new setting for the decompression phase, impact laws for unilateral and bilateral geometric and kinematic constraints and various friction elements are introduced and equipped with an impact coefficient in the sense of Poisson's impulse ratio. Energetic consistency is proven for small and similar impact coefficients via the condition number of the Delassus operator. Energetic consistency is generally proven for frictionless systems, as well as for systems containing one single frictional contact. A counter-example is developed, which demonstrates a possible energy increase for Poisson impacts in the presence of more than one frictional contact.

Some Utilizations of Lambert W Function in Distribution Theory

We introduce a new class of positive infinitely divisible probability laws calling them 𝔏γ distributions. Their cumulant-generating functions (cgf) are expressed in terms of the principal branch of the Lambert W function. The probability density functions (pdfs) of 𝔏γ laws are bounded resembling pdf of a Lévy stable distribution. The exponential dispersion model constructed starting from an 𝔏γ distribution admits the inverse Gaussian approximation. The natural exponential family constructed starting from an 𝔏γ distribution constitutes the reciprocal of the natural exponential family generated by a spectrally negative stable law with α = 1. We derive new results on 𝔏γ laws and the related exponential dispersion models, including their convolution and scaling closure properties. We generate another exponential dispersion model starting from an exponentially compounded 𝔏γ law. This distribution emerges in the Poisson mixture representation of a generalized Poisson law. We extend the Poisson approximation for the scaled Neyman type A exponential dispersion model. We derive saddlepoint-type approximations for some of these exponential dispersion models. The role of the Lambert W function is emphasized.

A semi-empirical method for predicting hydrological drought magnitudes in the Canadian prairies

A hydrological drought magnitude (M T ) expressed in standardized terms is predicted on annual, monthly and weekly time scales for a sampling period of T years in streamflow data from the Canadian prairies. The drought episodes are considered to follow the Poisson law of probability and, when coupled with the gamma probability distribution function (pdf) of drought magnitude (M) in the extreme number theorem, culminate in a relationship capable of evaluating the expected value, E(M T ). The parameters of the underlying pdf of M are determined based on the assumption that the drought intensity follows a truncated normal pdf. The E(M T ) can be evaluated using only standard deviation (σ), lag-1 autocorrelation (ρ) of the standardized hydrological index (SHI) sequence, and a weighting parameter Φ (ranging from 0 to 1) to account for the extreme drought duration (L T ), as well as the mean drought duration (Lm ), in a characteristic drought length (Lc ). The SHI is treated as standard normal variate, equivalent to the commonly-used standardized precipitation index. A closed-form relationship can be used for the estimation of first-order conditional probabilities, which can also be estimated from historical streamflow records. For all rivers, at the annual time scale, the value of Φ was found equal to 0.5, but it tends to vary (in the range 0 to 1) from river to river at monthly and weekly time scales. However, for a particular river, the Φ value was nearly constant at monthly and weekly time scales. The proposed method estimates E(M T ) satisfactorily comparable to the observed counterpart. At the annual time scale, the assumption of a normal pdf for drought magnitude tends to yield results in close proximity to that of a gamma pdf. The M T , when transformed into deficit-volume, can form a basis for designing water storage facilities and for planning water management strategies during drought periods. Editor D. Koutsoyiannis; Associate editor C. Onof Citation Sharma, T.C. and Panu, U.S., 2013. A semi-empirical method for predicting hydrological drought magnitudes in the Canadian prairies. Hydrological Sciences Journal, 58 (3), 549–569.

Reliability and performance optimization of pipelined real-time systems

We consider pipelined real-time systems that consist of a chain of tasks executing on a distributed platform. The processing of the tasks is pipelined: each processor executes only one interval of consecutive tasks. We are interested in minimizing both the input–output latency and the period of application mapping. For dependability reasons, we are also interested in maximizing the reliability of the system. We therefore assign several processors to each interval of tasks, so as to increase the reliability of the system. Both processors and communication links are unreliable and subject to transient failures. We assume that the arrival of the failures follows a constant parameter Poisson law, and that the failures are statistically independent events. We study several variants of this multiprocessor mapping problem, with several hypotheses on the target platform (homogeneous/heterogeneous speeds and/or failure rates). We provide NP-hardness complexity results, and optimal mapping algorithms for polynomial problem instances. Efficient heuristics are presented to solve the general case, and experimental results are provided.

On the existence of hydrogen atoms in higher dimensional Euclidean spaces

The question of whether hydrogen atoms can exist or not in spaces with a number of dimensions greater than 3 is revisited, considering higher dimensional Euclidean spaces. Previous results which lead to different answers to this question are briefly reviewed. The scenario where not only the kinematical term of Schr\"odinger equation is generalized to a D-dimensional space but also the electric charge conservation law (expressed here by the Poisson law) should actually remains valid is assumed. In this case, the potential energy in the Schr\"odinger equation goes like 1/r^{D-2}. The lowest quantum mechanical bound states and the corresponding wave functions are determined by applying the Numerov numerical method to solve Schr\"odinger's eigenvalue equation. States for different angular momentum quantum number (l = 0; 1) and dimensionality (5 \leq D \leq 10) are considered. One is lead to the result that hydrogen atoms in higher dimensions could actually exist. For the same range of the dimensionality D, the energy eigenvalues and wave functions are determined for l = 1. The most probable distance between the electron and the nucleus are then computed as a function of D showing the possibility of tiny bound states.

Trajectories of changes over twelve years in the health status of Canadians from late middle age

Aging in a given individual can be characterized by the number of deficits (symptoms, signs, laboratory abnormalities, disabilities) that they accumulate. The number of accumulated deficits, more than their nature, well characterizes health status in individuals — the proportion of deficits present in an individual to deficits considered is known as a frailty index. While on average deficits accumulate with age, individual trajectories in the number of deficits is highly dynamic. Transitions in the number of deficits over a fixed time interval can be represented by the Poisson law, with the Poisson mean dependent on the deficit numbers at baseline. Here we present an extension of the model to make possible predictions for any given time period. Using data from the Canadian National Population Health Survey of people aged 55 and over (n=4330), followed during 7cycles being the baseline and 6cycles of follow-up every 2years, we found that the transition in the number of deficits during any time period can be approximated using a time dependent Poisson distribution with the Poisson mean tending to decelerate over time, according to square-root-of-time kinetics characteristic for stochastic processes (e.g. diffusion, Brownian motion ) while the probability of death shows a pattern of time acceleration with a high degree of precision, “explaining” over 98% of variance. The model predicts a variety of changes in health status including the possibility of health improvement indicating the repair/remodeling abilities of the organism. The model is valuable for estimating how changes in health can influence mortality across the life course from late middle age.

Advanced theory of multiple exciton generation effect in quantum dots

The theoretical aspects of the effect of multiple exciton generation (MEG) in quantum dots (QDs) have been analysed in this work. The statistical theory of MEG in QDs based on Fermi’s approach is presented, taking into account the momentum conservation law. According to Fermi this approach should give the ultimate quantum efficiencies of multiple particle generation. The microscopic mechanism of this effect is based on the theory of electronic “shaking”. According to this approach, the wave function of “shaking” electrons can be selected as Plato’s functions with effective charges depending on the number of generated excitons. From the theory it is known increasing the number of excitons leads to enhancement of the Auger recombination of electrons which results in reduced quantum yields of excitons. The deviation of the averaged multiplicity of the MEG effect from the Poisson law of fluctuations has been investigated on the basis of synergetics approaches. In addition the role of interface electronic states of QDs and ligands has been considered by means of quantum mechanical approaches. The size optimisation of QDs has been performed to maximise the multiplicity of the MEG effect.

Poisson approximation for the number of visits to balls in non-uniformly hyperbolic dynamical systems

AbstractWe study the number of visits to balls Br(x), up to time t/μ(Br(x)), for a class of non-uniformly hyperbolic dynamical systems, where μ is the Sinai–Ruelle–Bowen measure. Outside a set of ‘bad’ centers x, we prove that this number is approximately Poissonnian with a controlled error term. In particular, when r→0, we get convergence to the Poisson law for a set of centers of μ-measure one. Our theorem applies for instance to the Hénon attractor and, more generally, to systems modelled by a Young tower whose return-time function has an exponential tail and with one-dimensional unstable manifolds. Along the way, we prove an abstract Poisson approximation result of independent interest.

Cumulants of the q-semicircular law, Tutte polynomials, and heaps

The q-semicircular law as introduced by Bożejko and Speicher interpolates between the Gaussian law and the semicircular law, and its moments have a combinatorial interpretation in terms of matchings and crossings. We prove that the cumulants of this law are, up to some factor, polynomials in q with nonnegative coefficients. This is done by showing that they are obtained by an enumeration of connected matchings, weighted by the evaluation at (1,q) of a Tutte polynomial. The two particular cases q=0 and q=2 have also alternative proofs, related with the fact that these particular evaluation of the Tutte polynomials count some orientations on graphs. Our methods also give a combinatorial model for the cumulants of the free Poisson law. La loi q-semicirculaire introduite par Bożejko et Speicher interpole entre la loi gaussienne et la loi semi-circulaire, et ses moments ont une interprétation combinatoire en termes de couplages et croisements. Nous prouvons que les cumulants de cette loi sont, à un facteur près, des polynômes en q à coefficients positifs. La méthode consiste à obtenir ces cumulants par une énumération de couplages connexes, pondérés par l’évaluation en (1,q) d'un polynôme de Tutte. Les cas particuliers q=0 et q=2 ont une preuve alternative, reliè au fait que des évaluations particulières du polynôme de Tutte comptent certaines orientations de graphes. Nos méthodes donnent aussi un modèle combinatoire aux cumulants de la loi de Poisson libre.

FOURIER ANALYSIS AND THE KINETIC THEORY OF GASES

In the so-called Bernoulli model of the kinetic theory of gases, where (1) the particles are dimensionless points, (2) they are contained in a cube container, (3) no attractive or exterior force is acting on them, (4) there is no collision between the particles, (5) the collisions against the walls of the container are according to the law of elastic collision, we deduce from Newtonian mechanics two basic probabilistic limit laws about particle-counting. The first one is a global central limit theorem, and the second one is a local Poisson law. The key fact is that we can prove the “realistic limit”: first the time interval is fixed and the number of particles tends to infinity (where the density particle/volume remains a fixed constant), and then, in the second step, the time tends to infinity.

Statistics of quantum beats in the time dependence of the neutron scattering

The numerical calculations of the delay time distributions for the neutron scattering have been carried out. The scattering of the short wave packet of neutrons with the broad energy interval 500–800 keV by 58Ni nuclei has been studied. The decay curves and average times of the compound scattering were found as for the forward scattering as for the specific spin-parity states. The time oscillations of the decay curves were identified as quantum beats that were caused by the interference of the excited resonance set. The statistical analysis of the frequency spectra of the decay curves oscillations has been carried out. The behavior of the examined statistical observables turns out to be close to the Poisson law.

The monotone cumulants

In the present paper we define the notion of generalized cumulants which gives a universal framework for commutative, free, Boolean, and especially, monotone probability theories. The uniqueness of generalized cumulants holds for each independence, and hence, generalized cumulants are equal to the usual cumulants in the commutative, free and Boolean cases. The way we define (generalized) cumulants needs neither partition lattices nor generating functions and then will give a new viewpoint to cumulants. We define ``monotone cumulants'' in the sense of generalized cumulants and we obtain quite simple proofs of central limit theorem and Poisson's law of small numbers in monotone probability theory. Moreover, we clarify a combinatorial structure of moment-cumulant formula with the use of ``monotone partitions''.

Rapid Titration of Measles and Other Viruses: Optimization with Determination of Replication Cycle Length

BackgroundMeasles virus (MV) is a member of the Paramyxoviridae family and an important human pathogen causing strong immunosuppression in affected individuals and a considerable number of deaths worldwide. Currently, measles is a re-emerging disease in developed countries. MV is usually quantified in infectious units as determined by limiting dilution and counting of plaque forming unit either directly (PFU method) or indirectly from random distribution in microwells (TCID50 method). Both methods are time-consuming (up to several days), cumbersome and, in the case of the PFU assay, possibly operator dependent.Methods/FindingsA rapid, optimized, accurate, and reliable technique for titration of measles virus was developed based on the detection of virus infected cells by flow cytometry, single round of infection and titer calculation according to the Poisson's law. The kinetics follow up of the number of infected cells after infection with serial dilutions of a virus allowed estimation of the duration of the replication cycle, and consequently, the optimal infection time. The assay was set up to quantify measles virus, vesicular stomatitis virus (VSV), and human immunodeficiency virus type 1 (HIV-1) using antibody labeling of viral glycoprotein, virus encoded fluorescent reporter protein and an inducible fluorescent-reporter cell line, respectively.ConclusionOverall, performing the assay takes only 24–30 hours for MV strains, 12 hours for VSV, and 52 hours for HIV-1. The step-by-step procedure we have set up can be, in principle, applicable to accurately quantify any virus including lentiviral vectors, provided that a virus encoded gene product can be detected by flow cytometry.

CONDITIONALLY MONOTONE INDEPENDENCE I: INDEPENDENCE, ADDITIVE CONVOLUTIONS AND RELATED CONVOLUTIONS

We define a product of algebraic probability spaces equipped with two states. This product is called a conditionally monotone product. This product is a new example of independence in noncommutative probability theory and unifies the monotone and Boolean products, and moreover, the orthogonal product. Then we define the associated cumulants and calculate the limit distributions in central limit theorem and Poisson's law of small numbers. We also prove a combinatorial moment-cumulant formula using monotone partitions. We investigate some other topics such as infinite divisibility for the additive convolution and deformations of the monotone convolution. We define cumulants for a general convolution to analyze the deformed convolutions.

Quantum Isometry Groups of Duals of Free Powers of Cyclic Groups

We study the quantum isometry groups of the noncommutative Riemannian manifolds associated to discrete group duals. The basic representation theory problem is to compute the law of the main character of the relevant quantum group, and our main result here is as follows: for the group Z_s^{*n}, with s>4 and n>1, half of the character follows the compound free Poisson law with respect to the measure $\underline{\epsilon}$/2, where $\epsilon$ is the uniform measure on the s-th roots of unity, and $\epsilon\to\underline{\epsilon}$ is the canonical projection map from complex to real measures. We discuss as well a number of technical versions of this result, notably with the construction of a new quantum group, which appears as a representation-theoretic limit, at s equal to infinity.

Wave Speed in Excitable Random Networks with Spatially Constrained Connections

Very fast oscillations (VFO) in neocortex are widely observed before epileptic seizures, and there is growing evidence that they are caused by networks of pyramidal neurons connected by gap junctions between their axons. We are motivated by the spatio-temporal waves of activity recorded using electrocorticography (ECoG), and study the speed of activity propagation through a network of neurons axonally coupled by gap junctions. We simulate wave propagation by excitable cellular automata (CA) on random (Erdös-Rényi) networks of special type, with spatially constrained connections. From the cellular automaton model, we derive a mean field theory to predict wave propagation. The governing equation resolved by the Fisher-Kolmogorov PDE fails to describe wave speed. A new (hyperbolic) PDE is suggested, which provides adequate wave speed that saturates with network degree , in agreement with intuitive expectations and CA simulations. We further show that the maximum length of connection is a much better predictor of the wave speed than the mean length. When tested in networks with various degree distributions, wave speeds are found to strongly depend on the ratio of network moments rather than on mean degree , which is explained by general network theory. The wave speeds are strikingly similar in a diverse set of networks, including regular, Poisson, exponential and power law distributions, supporting our theory for various network topologies. Our results suggest practical predictions for networks of electrically coupled neurons, and our mean field method can be readily applied for a wide class of similar problems, such as spread of epidemics through spatial networks.

On the distribution of profile coordinates of equilibrium friction surface in wear-resistant powder coatings

The microgeometry of the equilibrium friction surface of wear-resistant gas-thermal coatings is discussed. To describe the steady-state profile of powder coatings during sliding friction, a theoretical probabilistic approach is used to statistically justify the hypothesis of the normal distribution of heights of the equilibrium friction surface of materials on the basis of the de Moivre-Laplace theorem. The limiting Poisson law and computer simulation of the powder environment with the probabilistic-geometric system is used to study the profile of the equilibrium friction surface of wear-resistant powder coatings. The findings obtained can be applied to build the support curve of the friction surface of wear-resistant powder coatings.

Quantum automorphisms of twisted group algebras and free hypergeometric laws

We prove that we have an isomorphism of type A a u t ( C σ [ G ] ) ≃ A a u t ( C [ G ] ) σ A_{aut}(\mathbb C_\sigma [G])\simeq A_{aut}(\mathbb C[G])^\sigma , for any finite group G G , and any 2 2 -cocycle σ \sigma on G G . In the particular case G = Z n 2 G=\mathbb Z_n^2 , this leads to a Haar measure-preserving identification between the subalgebra of A o ( n ) A_o(n) generated by the variables u i j 2 u_{ij}^2 and the subalgebra of A s ( n 2 ) A_s(n^2) generated by the variables X i j = ∑ a , b = 1 n p i a , j b X_{ij}=\sum _{a,b=1}^np_{ia,jb} . Since u i j u_{ij} is “free hyperspherical” and X i j X_{ij} is “free hypergeometric”, we obtain in this way a new free probability formula, which at n = ∞ n=\infty corresponds to the well-known relation between the semicircle law and the free Poisson law.