Finite Probability Research Articles

Effect of multiplicative noise on least-squares parameter estimation with applications to the atomic force microscope

Measurement of the power spectral density of (stochastic) Brownian fluctuations of micro- and nano-devices is used frequently to gain insight into their mechanistic properties. Noise is always present in these measurements and can directly influence any parameter estimation obtained through a least-squares analysis. Importantly, measurements of the spectral density of stationary random signals, such as Brownian motion, inherently contain multiplicative noise. In this article, we theoretically analyze the impact of multiplicative noise on fit parameters extracted using a least-squares analysis. A general analysis is presented that is valid for any fit function with any number of fit parameters. This yields closed-form expressions for the expected value and variance in the fit parameters and provides a rigorous theoretical framework for a priori determination of the effect of measurement uncertainty. The theory is demonstrated and validated through Monte Carlo simulation of synthetic data and by comparison to power spectral density measurements of the Brownian fluctuations of an atomic force microscope cantilever - analytical formulas for the uncertainty in the fitted resonant frequency and quality factor are presented. The results of this study demonstrate that precise measurements of fit parameters in the presence of noise are inherently problematic - individual measurements of the power spectral density are capable of yielding fit parameters that are many standard deviations away from the mean, with finite probability. This is of direct relevance to a host of applications in measurement science, including those connected with the atomic force microscope.

On the sampling of step length in Monte Carlo simulation of trajectories with very small mean free path

In detailed (event-by-event) simulation of trajectories of particles (electrons, ions) in liquids and solids the step length sampling method is often based on a total mean free path λ=(n·σ)−1, where n is the number of scatterers per unit volume and σ the total cross section. Typically, the step length s between scattering events is then generated by means of a sampling formula s=−λln(1−R), where R a random number in the interval 0<R<1; the probability for the particle to travel a distanced s without interaction is then exp(−s/λ). It is here argued that this “conventional” sampling method, which implies that the average distance between successive events is equal to λ, is erroneous unless λ is much larger than the distance dnn=n−1/3 between nearest neighbour scatterers. An alternative sampling method, M (monolayer) sampling, is proposed, with a fixed step length D=dnn and a finite probability I=1−exp(−D/λ) of interaction at the end of each step. According to M sampling, conventional sampling exaggerates the number of events per unit path length unless λ⪢dnn. Consequently, quantities like stopping power are overestimated, while transport mean free path is underestimated. The true (corrected) mean free path λc between events is found to be λc=D/(1−exp(−D/λ). The correction is substantial when λ is comparable to or smaller than the distance between the scatterers, in practice for very low energy particles in liquids and solids. In the opposite limit λ⪢D, conventional and M sampling produce the same result. Present results further indicate that conventional sampling using the corrected total mean free path λc is a good approximation to M sampling.

Uncontrolled growth resulting from dedifferentiation in a skin cell proliferation model

By introducing a small backward dedifferentiation probability of postmitotic cells to progenitor cells in a recently proposed skin cell proliferation model, the homeostasis of the system can be disrupted resulting in uncontrolled growth. It is found that when the dedifferentiation probability exceeds a small critical value, the stable fixed point of the system vanishes leading to unlimited cell growth resembling scenarios in carcinogenesis. Explicit expression for the critical dedifferentiation probability and phase diagram are calculated analytically and the associated nonlinear dynamics is analyzed. In the presence of stochastic fluctuations, our model predicts that the escape rate from homeostatic growth to uncontrolled growth is greatly enhanced by a small but finite dedifferentiation probability. These results are verified by numerical solutions of the dynamical system and chemical Langevin equations.

Acceptance, Aggregation and Scoring Rules

As the ongoing literature on the paradoxes of the Lottery and the Preface reminds us, the nature of the relation between probability and rational acceptability remains far from settled. This article provides a novel perspective on the matter by exploiting a recently noted structural parallel with the problem of judgment aggregation. After offering a number of general desiderata on the relation between finite probability models and sets of accepted sentences in a Boolean sentential language, it is noted that a number of these constraints will be satisfied if and only if acceptable sentences are true under all valuations in a distinguished non-empty set W. Drawing inspiration from distance-based aggregation procedures, various scoring rule based membership conditions for W are discussed and a possible point of contact with ranking theory is considered. The paper closes with various suggestions for further research.

Wavelet-enhanced bayesian classification of surface nanostructures observed in pentacene thin films

In recent years, the wavelet transform has proven to be an extremely powerful image-processing tool. Here, we have developed a wavelet-enhanced Bayesian methodology to analyze the AFM images of organic crystals, such as pentacene, using the Haar wavelet transform. This methodology segments and identifies structures in AFM data and extracts quantitative information such as grain size, perimeter, and orientation. This methodology relies on a Naïve Bayesian classification system, which determines whether each data point corresponds to a peak or a valley with a finite statistical probability based on analysis of a synthetic teaching image. In this work, we apply this technique to AFM images of organic crystal films deposited on three different substrates.

Markov branching in the vertex splitting model

We study a special case of the vertex splitting model which is a recentmodel of randomly growing trees. For any finite maximum vertex degreeD, we find a one parameter model, with parameter which has a so-called Markov branching property. When we find a two parameter model with an additional parameter which also has this feature. In the caseD = 3, the model bearsresemblance to Ford’s α-model of phylogenetic trees and when it is similar to its generalization, theαγ-model.For α = 0, the model reduces to the well known model of preferential attachment.In the case α > 0, we prove convergence of the finite volume probability measures, generated by the growth rules,to a measure on infinite trees which is concentrated on the set of trees with a single spine. Weshow that the annealed Hausdorff dimension with respect to the infinite volume measure is1/α. Whenγ = 0 the model reduces to a model of growing caterpillar graphs inwhich case we prove that the Hausdorff dimension is almost surely1/α and that the spectral dimension is almost surely2/(1 + α). We comment briefly on the distribution of vertex degrees and correlations betweendegrees of neighbouring vertices.

Finite size emptiness formation probability of the XXZ spin chain at

In this paper, we compute the emptiness formation probability of a (twisted-) periodic XXZ spin chain of finite length at , thus proving the formulas conjectured by Razumov and Stroganov (2001 J. Phys. A: Math. Gen. 34 3185 and 5335–40). The result is obtained by exploiting the fact that the ground state of the inhomogeneous XXZ spin chain at satisfies a set of qKZ equations associated with .

SHARING OF D-DIMENSIONAL QUANTUM STATES

We propose a scheme for the deterministic sharing arbitrary qudit states among three distant parties and characterize the set of ideal quantum channels. We also show that the use of non-ideal quantum channels for quantum state sharing can be related to the problem of quantum state discrimination. This allows us to formulate a protocol which leads to perfect quantum state sharing with a finite success probability.

Virtual Processes and Quantum Tunnelling as Fictions

In this paper it is argued that virtual processes are dispensable fictions. The argument proceeds by a comparison with the phenomenon of quantum tunnelling. Building on an analysis of Levy-Leblond and Balibar, it is argued that, although the phenomenon known as quantum tunnelling certainly occurs and is at the basis of many paradigmatic quantum effects, the implied conceptualization of it as a free particle burrowing through a potential barrier is flawed. An alpha particle, for example, does not exist as a free particle inside a uranium nucleus and then “burrow through” the massive potential barrier of the repulsive Coulomb potential: rather, it can be interpreted as existing in a bound state which gives it a corresponding (absolutely tiny, but) finite probability of appearing on the other side of the barrier. If the part of the state function representing the transmission through the barrier is conceived as representing a particle trajectory, the particle will have imaginary momentum and negative kinetic energy. A similar analysis then applies to virtual processes. For example, if (as in Hawking’s conception of black hole radiation) one imagines a pair of particles created at the Schwarzschild radius, one of which drops into the black hole, at its creation that particle will have imaginary momentum and negative kinetic energy; so will the pion that is imagined as mediating the nuclear exchange force on the standard model. In each case, it is argued, the phenomenon can be understood in terms of a finite probability of transmission predicted by quantum theory, without appealing to particle trajectories. The idea that a particle “penetrates” a barrier that it does not have the energy to surmount, or that a pair of particles is “virtually” produced one on either side of the Schwarzschild radius, in defiance of energy conservation, should be discarded as unphysical.

Beauville surfaces and finite simple groups

A Beauville surface is a rigid complex surface of the form (C1 x C2)/G, where C1 and C2 are non-singular, projective, higher genus curves, and G is a finite group acting freely on the product. Bauer, Catanese, and Grunewald conjectured that every finite simple group G, with the exception of A5, gives rise to such a surface. We prove that this is so for almost all finite simple groups (i.e., with at most finitely many exceptions). The proof makes use of the structure theory of finite simple groups, probability theory, and character estimates.

Minimal martingale measure on a finite probability space

Minimal martingale measure on a finite probability space

Continuous-time random walks that alter environmental transport properties

We consider continuous-time random walks (CTRWs) in which the walkers have a finite probability to alter the waiting-time and/or step-length transport properties of their environment, resulting in possibly transient anomalous diffusion. We refer to these CTRWs as transmogrifying continuous-time random walks (TCTRWs) to emphasize that they change the form of the transport properties of their environment, and in a possibly strange way. The particular case in which the CTRW waiting-time density has a finite probability to be permanently altered at a given site, following a visitation by a walker, is considered in detail. Master equations for the probability density function of transmogrifying random walkers are derived, and results are compared with Monte Carlo simulations. An interesting finding is that TCTRWs can generate transient subdiffusion or transient superdiffusion without invoking truncated or tempered power law densities for either the waiting times or the step lengths. The transient subdiffusion or transient superdiffusion arises in TCTRWs with Gaussian step-length densities and exponential waiting-time densities when the altered average waiting time is greater than or less than, respectively, the original average waiting time.

Ruin probability analysis in geometric inhomogeneous claims case

The discrete time risk model with inhomogeneous claims is analyzed. The finite time ruin probability expression is obtained for the case when claims are distributed by geometric distribution with changing parameters. Some quantitave examples are also given.

Nonlocal unambiguous discrimination among N nonorthogonal qudit states lying in a higher-dimensional Hilbert space

We give a strategy for nonlocal unambiguous discrimination (UD) among N linearly independent nonorthogonal qudit states lying in a higher-dimensional Hilbert space. The procedure we use is a nonlocal positive operator valued measurement (POVM) in a direct sum space. This scheme is designed for obtaining the conclusive nonlocal measurement results with a finite probability of success. We construct a quantum network for realizing the nonlocal UD with a set of two-level remote rotations, and thus provide a feasible physical means to realize the nonlocal UD.

Anderson Localization or Nonlinear Waves: A Matter of Probability

In linear disordered systems Anderson localization makes any wave packet stay localized for all times. Its fate in nonlinear disordered systems (localization versus propagation) is under intense theoretical debate and experimental study. We resolve this dispute showing that, unlike in the common hypotheses, the answer is probabilistic rather than exclusive. At any small but finite nonlinearity (energy) value there is a finite probability for Anderson localization to break up and propagating nonlinear waves to take over. It increases with nonlinearity (energy) and reaches unity at a certain threshold, determined by the initial wave packet size. Moreover, the spreading probability stays finite also in the limit of infinite packet size at fixed total energy. These results generalize to higher dimensions as well.

Frequency Diversity in Mode-Division Multiplexing Systems

In the regime of strong mode coupling, the modal gains and losses and the modal group delays of a multimode fiber are known to have well-defined statistical properties. In mode-division multiplexing, mode-dependent gains and losses are known to cause fluctuations in the channel capacity, so that the capacity at finite outage probability can be substantially lower than the average capacity. Mode-dependent gains and losses, when frequency-dependent, have a coherence bandwidth that is inversely proportional to the modal group delay spread. When mode-division-multiplexed signals occupy a bandwidth far larger than the coherence bandwidth, the mode-dependent gains and losses are averaged over frequency, causing the outage capacity to approach the average capacity. The difference between the average and outage capacities is found to be inversely proportional to the square-root of a diversity order that is given approximately by the ratio of the signal bandwidth to the coherence bandwidth.

Phase segregation and transport in a two-species multi-lane system

We present a two-channel driven lattice gas model with oppositely directed species movingon two parallel lanes with lane switching processes. We study the correlated lane switchingmechanism for particles so that switching may occur with finite probability only whenoppositely directed species meet on the same channel. The system is analyzed for a closedring with conserved total particle number. For asymmetric particle exchange between thelanes, the system exhibits a unique polarization phenomenon with segregation of oppositelydirected species between the two lanes. The polarization phenomenon can be understood asa consequence of the existence of an absorbing steady state. For symmetric exchangerate of particles between the lanes, the system remains unpolarized, with equalparticle density on both the lanes in the thermodynamic limit of large system size.We study the system using a combination of a mean field (MF) analysis andMonte Carlo simulations. The nature of phase segregation that we see for this systemis distinct from driven particle systems which are in contact with the particlereservoir. The features observed for this minimal model will have ramifications forbiofilament based intracellular transport, wherein cellular cargoes, e.g. organelles andvesicles, are transported by oppositely directed particles on multiple filamenttracks.

Quarkonium dissociation in quark-gluon plasma via ionization in a magnetic field

We study the impact of magnetic fields generated in relativistic heavy ion collisions on the decay probability of quarkonium produced in the central rapidity region. The quark and anti-quark components are subject to mutually orthogonal electric and magnetic fields in the quarkonium comoving frame. In the presence of an electric field, quarkonium has finite dissociation probability. We use the WKB approximation to derive the dissociation probability. We found that quarkonium dissociation energy, i.e. the binding energy at which dissociation probability is of order unity, increases with the magnetic field strength. It also increases with quarkonium momentum in the laboratory frame due to Lorentz boost of electric field in the comoving frame. As a consequence, J/Psi in plasma dissociates at lower temperature then it would be in the absence of a magnetic field. We argue that J/Psi's produced in heavy-ion collisions at LHC with P_T>9GeV would dissociate even in vacuum. In plasma, J/Psi dissociation in magnetic field is much stronger due to decrease of its binding energy with temperature. We discuss the phenomenological implications of our results.

Ruin problems for an autoregressive risk model with dependent rates of interest

In this paper, we consider a discrete insurance risk model in which the claims, the premiums and the rates of interest are assumed to have dependent autoregressive structures (AR(1)). We derive recursive and integral equations for expected discounted penalty function. By these equations, we obtain generalized Lundberg inequality for the infinite time severity of ruin and hence for the infinite time ruin probability, consider asymptotic formula for the finite time ruin probability when loss distributions have regularly varying tails, and study some probability properties of the duration of ruin.

Finite time non-ruin probability for Erlang claim inter-arrivals and continuous inter-dependent claim amounts

A closed form expression, in terms of some functions which we call exponential Appell polynomials, for the probability of non-ruin of an insurance company, in a finite time interval is derived, assuming independent, non-identically Erlang distributed claim inter-arrival times, , any continuous joint distribution of the claim amounts and any non-negative, non-decreasing real function, representing its premium income. In the special case when it is shown that our main result yields a formula for the probability of non-ruin expressed in terms of the classical Appell polynomials. We give another special case of our non-ruin probability formula for , i.e. when the inter-arrival times are non-identically exponentially distributed, and also show that it coincides with the formula for Poisson claim arrivals, given in [18], when . The main result is extended further to a risk model in which inter-arrival times are dependent random variables, obtained by randomizing the Erlang shape and/or rate parameters. We also give some useful auxiliary results that characterize and express explicitly (and recurrently) the exponential Appell polynomials that appear in our finite time non-ruin probability formulae.