Symmetric Gaussian Distribution Research Articles

Numerical Solution to Poisson’s Equation for Estimating Electrostatic Properties Resulting from an Axially Symmetric Gaussian Charge Density Distribution: Charge in Free Space

Poisson’s equation frequently emerges in many fields, yet its exact solution is rarely feasible, making the numerical approach notably valuable. This study aims to provide a tutorial-level guide to numerically solving Poisson’s equation, focusing on estimating the electrostatic field and potential resulting from an axially symmetric Gaussian charge distribution. The Finite Difference Method is utilized to discretize the desired spatial domain into a grid of points and approximate the derivatives using finite difference approximations. The resulting system of linear equations is then tackled using the Successive Over-Relaxation technique. Our results suggest that the potential obtained from the direct integration of the distance-weighted charge density is a reasonable choice for Dirichlet boundary conditions. We examine a scenario involving a charge in free space; the numerical electrostatic potential is estimated to be within a tolerable error range compared to the exact solution.

Statistical features of the general population of patients with genital endometriosis

Abstract. It is relevant to study the incidence of genital endometriosis, for which radical treatment methods have not been found yet. This study should also include a statistical analysis of the general population of patients included in the research, followed by clinical trials. Object and methods. A statistical analysis of the general population of patients was carried out with the following values: N (120), mode (Mo), median (Me), simple arithmetic mean (M) and its standard deviation (M ± δ), variation indicators (R-range): dispersion (D), standard deviation (δ); coefficient of variation (V); quartiles (Q) and their range (IQR). We also determined the tkp value (according to the Student distribution table) with a probability of values of 0.95 and texp, as well as scattering indicators — minimum (min) and maximum (max), dispersion (S2), asymmetry (As), kurtosis (Ex), and others. To test the hypothesis about the type of distribution, we used the Pearson goodness-of-fit test. Additionally, the standard sampling error for the mean (E), the marginal sampling error (ε), and the confidence interval (CI) were estimated. Using the distribution table, we found the χ2 values; the random error of the lower (tL) and upper bound (tU), and the confidence interval of the standard deviation were calculated. Results and discussion. The following indicators were calculated: average and standard deviation (M ± δ); Mo and Me had an equal value. The sample range was calculated; the sample quartiles divided the given population into four equal parts, where Q2 (25 %) was equal to 30 years, Q1 (25 %) corresponded to 27 years, Q3 — to 34 years (25 %), and IQR corresponded to 7 years. Notably, 28 people were under 27 years old and over 34. To exclude anomalous distribution, outliers were determined using the Tukey method; outliers over 55 years turned out to be rigid. The value of the indicator V=18.89 % was determined; that was less than 30 %, which indicated the homogeneity of the population and weak variation. The standard sampling error for the mean (E), marginal sampling error (ε), and confidence interval for dispersion were calculated. The value of the χ2 indicator was estimated according to the distribution table. The random error of the dispersion of the lower (tL) and upper bound (tU) was determined. The significance score of the asymmetry index was calculated using the mean square error of the asymmetry coefficient (SAs). Conclusion. As it follows from statistical analysis, this general population had all the signs of a symmetrical normal distribution, therefore, the results obtained are trustworthy.

Tight Lower Bound on Differential Entropy for Mixed Gaussian Distributions

In this paper, a tight lower bound for the differential entropy of the Gaussian mixture model is presented. First, the probability model of mixed Gaussian distribution that is created by mixing both discrete and continuous random variables is investigated in order to represent symmetric bimodal Gaussian distribution using the hyperbolic cosine function, on which a tighter upper bound is set. Then, this tight upper bound is used to derive a tight lower bound for the differential entropy of the Gaussian mixture model introduced. The proposed lower bound allows to maintain its tightness over the entire range of the model's parameters and shows more tightness when compared with other bounds that lose their tightness over certain parameter ranges. The presented results are then extended to introduce a more general tight lower bound for asymmetric bimodal Gaussian distribution, in which the two modes have a symmetric mean but differ in terms of their weights.

Robust and sparse M-estimation of DOA

A robust and sparse Direction of Arrival (DOA) estimator is derived for array data that follows a Complex Elliptically Symmetric (CES) distribution with zero-mean and finite second-order moments. The derivation allows to choose the loss function and four loss functions are discussed in detail: the Gauss loss which is the Maximum-Likelihood (ML) loss for the circularly symmetric complex Gaussian distribution, the ML-loss for the complex multivariate t-distribution (MVT) with ν degrees of freedom, as well as Huber and Tyler loss functions. For Gauss loss, the method reduces to Sparse Bayesian Learning (SBL). The root mean square DOA error of the derived estimators is discussed for Gaussian, MVT, and ϵ-contaminated data. The robust SBL estimators perform well for all cases and nearly identical with classical SBL for Gaussian array data.

An asymmetric probability density function

A probability density or distribution function of turbulence has been thought to be symmetric due to the symmetry of the partial differential equations from the first principles. However, the experimental data have been shown otherwise by a so-called Taylor correlation function, and this is an unresolved issue. A recent study shows that this probability density function can be asymmetric analytically by introducing negative eigenvalues. Here, in this work, we show the mathematical basis for this asymmetry, although the partial differential equations follow the symmetry of the Lie groups. We also demonstrate a complete solution of partial differential equations, including the exponential terms and negative eigenvalues, which plays a vital part in transient phenomena. Our analysis shows that the asymmetry is from the partition of velocities of the same or opposing direction, not from the negative eigenvalues. Fundamentally, the loss of rotational symmetry is caused by the exponential terms for the transient solution, which we demonstrate by the derivation of the complete solution. The new correlation produces excellent agreement with the experimental data. The universality and limitations of the correlation function are discussed, and through the parameter study, the variations and statistical nature of the probability function are clarified. The asymmetry probability function should have wider applicability than the symmetric Gaussian distribution, which is the special case of the asymmetry probability function.

A novel application of statistical process control charts in financial market surveillance with the idea of profile monitoring.

The implementation of statistical techniques in on-line surveillance of financial markets has been frequently studied more recently. As a novel approach, statistical control charts which are famous tools for monitoring industrial processes, have been applied in various financial applications in the last three decades. The aim of this study is to propose a novel application of control charts called profile monitoring in the surveillance of the cryptocurrency markets. In this way, a new control chart is proposed to monitor the price variation of a pair of two most famous cryptocurrencies i.e., Bitcoin (BTC) and Ethereum (ETH). Parameter estimation, tuning and sensitivity analysis are conducted assuming that the random explanatory variable follows a symmetric normal distribution. The triggered signals from the proposed method are interpreted to convert the BTC and ETH at proper times to increase their total value. Hence, the proposed method could be considered a financial indicator so that its signal can lead to a tangible increase of the pair of assets. The performance of the proposed method is investigated through different parameter adjustments and compared with some common technical indicators under a real data set. The results show the acceptable and superior performance of the proposed method.

On Distributed Lossy Coding of Symmetrically Correlated Gaussian Sources

A distributed lossy compression network with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> encoders and a decoder is considered. Each encoder observes a source and sends a compressed version to the decoder. The decoder produces a joint reconstruction of target signals with the mean squared error distortion below a given threshold. It is assumed that the observed sources can be expressed as the sum of target signals and corruptive noises which are independently generated from two symmetric multivariate Gaussian distributions. The minimum compression rate of this network versus the distortion threshold is referred to as the rate-distortion function, for which an explicit lower bound is established by solving a minimization problem. Our lower bound matches the well-known Berger-Tung upper bound for some values of the distortion threshold. The asymptotic gap between the upper and lower bounds is characterized in the large <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> limit.

Turbulence in Two-Phase Flows with Macro-, Micro- and Nanoparticles: A Review

Turbulent flows are nonstationary in nature. Since the turbulent fluctuations of most flow parameters satisfy a symmetric Gaussian distribution, the turbulent characteristics have the property of symmetry in the statistical meaning. A widespread simplest model of turbulent flows is the model of “symmetric” turbulence, namely, homogeneous isotropic turbulence (HIT). The presence of particles with non-uniform distribution of their concentration in the turbulent flow, even under HIT conditions, can lead to redistribution of different components of fluctuation velocities of the carrier gas, i.e., to the appearance of asymmetry. The subject of the review is turbulent flows of gas with solid particles. Particular attention is paid to the problem of the back influence of particles on carrier gas characteristics (first of all, on the turbulent kinetic energy). A review of the results of experimental and computational-theoretical studies of the effect of the presence of the dispersed phase in the form of particles on the parameters of the turbulent flow of the carrier gas phase has been carried out. The main physical mechanisms and dimensionless criteria determining the direction and magnitude of the impact of particles of different inertia on the carrier gas phase turbulence energy are described and analyzed. The peculiarities of the influence of particles on the turbulence energy of the gas for different classes of flows: homogeneous isotropic turbulence, homogeneous shear flow, and wall turbulence in a pipe (channel) have been considered. For the near-wall flow in the pipe, it is shown that the turbulizing effect of extremely low-inertia particles of sub-micrometer size (nanoparticles) is replaced by the laminarizing effect of low-inertia particles of micrometer size (microparticles), and then again it is replaced by turbulizing due to additional generation of turbulence in the wakes of large particles of millimeter size (macroparticles). The review is intended to some extent to fill in the currently existing gap associated with the absence of dimensionless criteria (or complexes of physical parameters) responsible for the direction (attenuation or enhancement) of turbulence modification, and the value of this change. Possible directions for further researches are given in the conclusion of the review.

Conductive Heat Transfer in Materials under Intense Heat Flows

The paper presents the solution of the spatial transient problem of the impact of a moving heat flux source induced by the laser radiation on the surface of a half-space using the superposition principle and the method of transient functions. The hyperbolic equation of transient thermal conductivity accounting for the relaxation time is used to model the laser heating process. It is assumed that the heat flux is distributed symmetrically with respect to the center of the heating spot. The combined numerical and analytical algorithm has been developed and implemented, which allows one to determine the temperature distribution both on the surface and on the depth of the half-space. In this case, the principle of superposition is used with the use of a special symmetric Gaussian distribution to describe the model of a source of high-intensity heat flux. The use of such a symmetric distribution made it possible to calculate the integrals over the spatial variables analytically. The results of the work could be used to estimate the contribution of the conductive component in the overall heat transfer of materials exposed to intense heat flows (laser surface treatment, laser additive technologies, streamlining and heating of materials by high-enthalpy gases, etc.).

Strategic Entry with Correlated Private Information

This article studies how the correlation of players’ private information affects their strategic be haviour. We introduce information correlation into a static two-player strategic substitutes entry game. The degree of information correlation is measured by the correlation coefficient of a symmetric joint normal distribution, which is used to model players’ prior distribution. It is found that a cutoff strategy cannot be used for all values of correlation coefficient to solve the game, and there exists a threshold correlation coefficient value to differentiate the unique-equilibrium and the multiple (three)-equilibria situations, given other parameters. Finally, by comparative statics analysis of symmetric equilibrium strategies, we find that increasing the payoff of entry encourages players to adopt a lower entry threshold, while increasing the information correlation or jointly increasing the variances of prior distribution increases the positive entry threshold and lowers the negative entry threshold. JEL Codes: C72, D21, D82, L13

Expression for uncertainty intervals handling skewness when the relative standard uncertainty is independent of the measurand level

Uncertainty intervals for many measurement results are typically reported as symmetric intervals around the measured value. However, at large standard uncertainties (> approx. 15 %–20 %), it is necessary to consider asymmetry of the uncertainty intervals. Here, an expression for calculating uncertainty intervals handling asymmetry when the relative standard uncertainty is independent of the measurand level is presented. The expression is based on implementation of a power transformation ({x}^{B}) for transformation of measurement results in order to achieve results that have a symmetric and approximate normal distribution. Uncertainty intervals are then calculated in the transformed space and back-transformed to the original space. The transformation includes a parameter, B, that needs to be optimized, and this can be based on real results, modelling of results, or on judgement. Two important reference points are B equal to 1 that corresponds to an approximate normal distribution of the original measurement results, and B approaching 0 that corresponds to an approximate log-normal distribution of the original measurement results. Comparisons are made with uncertainty intervals calculated using other expressions where it is assumed that measurement results have a normal distribution or a log-normal distribution. Implementation of the approach is demonstrated with several examples from chemical analysis.

The Application of Mixed Smoothing Spline and Fourier Series Model in Nonparametric Regression

In daily life, mixed data patterns are often found, namely, those that change at a certain sub-interval or that follow a repeating pattern in a certain trend. To handle this kind of data, a mixed estimator of a Smoothing Spline and a Fourier Series has been developed. This paper describes a simulation study of the estimator in nonparametric regression and its implementation in the case of poor households. The minimum Generalized Cross Validation (GCV) was used in order to select the best model. The simulation study used generation data with a Uniform distribution and a random error with a symmetrical Normal distribution. The result of the simulation study shows that the larger the sample size n, the better the mixed estimator as a model of nonparametric regression for all variances. The smaller the variance, the better the model for all combinations of samples n. Very poor households are characterized predominantly in their consumption of carbohydrates compared to that of fat and protein. The results of this study suggest that the distribution of assistance to poor households is not the same, because in certain groups there are poor households that consume higher carbohydrates, and some households may consume higher fats.

Mode III critical distance determination with optimized V-notched specimen under torsional fatigue and size effects on the inverse search probability distribution

An optimized V-notched specimen is proposed for the determination of mode III, or torsional, critical distance. Numerical procedures are provided for the inverse determination of this length, and then for a statistical analysis of the resulting skew-normal distribution, as obtained from (symmetric) normal distributions of the plain and notched specimen strengths. An experimental example is shown on steel 42CrMo4 + QT, with clear evidence of a significantly larger mode III critical distance than mode I. A parametric analysis is then presented to evaluate the effects of size of the specimen on the standard deviation of the inverse search critical distance probability distribution.

A robust Birnbaum–Saunders regression model based on asymmetric heavy-tailed distributions

Skew-normal/independent distributions provide an attractive class of asymmetric heavy-tailed distributions to the usual symmetric normal distribution. We use this class of distributions here to derive a robust generalization of sinh-normal distributions (Rieck in Statistical analysis for the Birnbaum–Saunders fatigue life distribution, 1989), we then propose robust nonlinear regression models, generalizing the Birnbaum–Saunders regression models proposed by Rieck and Nedelman (Technometrics 33:51–60, 1991) that have been studied extensively. The proposed regression models have a nice hierarchical representation that facilitates easy implementation of an EM algorithm for the maximum likelihood estimation of model parameters and provide a robust alternative to estimation of parameters. Simulation studies as well as applications to a real dataset are presented to illustrate the usefulness of the proposed model as well as all the inferential methods developed here.

Uncertainty in protein-ligand binding constants: asymmetric confidence intervals versus standard errors.

Equilibrium binding constants (Kb) between chemical compounds and target proteins or between interacting proteins provide a quantitative understanding of biological interaction mechanisms. Reported uncertainties of measured experimental parameters are critical for decision-making in many scientific areas, e.g., in lead compound discovery processes and in comparing computational predictions with experimental results. Uncertainties in measured Kb values are commonly represented by a symmetric normal distribution, often quoted in terms of the experimental value plus-minus the standard deviation. However, in general, the distributions of measured Kb (and equivalent Kd) values and the corresponding free energy change ΔGb are all asymmetric to varying degree. Here, using a simulation approach, we illustrate the effect of asymmetric Kb distributions within the realm of isothermal titration calorimetry (ITC) experiments. Further we illustrate the known, but perhaps not widely appreciated, fact that when distributions of any of Kb, Kd and ΔGb are transformed into each other, their degree of asymmetry is changed. Consequently, we recommend that a more accurate way of expressing the uncertainties of Kb, Kd, and ΔGb values is to consistently report 95% confidence intervals, in line with other authors' suggestions. The ways to obtain such error ranges are discussed in detail and exemplified for a binding reaction obtained by ITC.

Gravitational Measurements in Higher Dimensions

We attempt to study three significant tests of general relativity in higher dimensions, both in commutative and non-commutative spaces. In the context of non-commutative geometry, we will consider a solution of Einstein’s equation in higher dimensions, with a source given by a static, spherically symmetric Gaussian distribution of mass. The resulting metric would describe a regular or curvature singularity free black hole in higher dimensions. The metric should smoothly interpolate between Schwarzschild geometry at large distance, and de-Sitter spacetime at short distance. We will consider gravitational redshift, lensing, and time delay in each sector. It will be shown that, compared to the four-dimensional spacetime, there can be significant modifications due to the presence of extra dimensions and the non-commutative corrected black holes. Finally, we shall attempt to obtain a lower bound on the size of the extra dimensions and on the mass needed to form a black hole in different dimensions.

An Extended VIKOR Method Based on q-Rung Orthopair Shadowed Set and Its Application to Multi-Attribute Decision Making

In the multi-attribute decision making (MADM) process, the attribute values are sometimes provided by experts or the public in the form of words. To model the linguistic evaluation more accurately, this paper proposes the q-rung orthopair shadowed set (q-ROSS) to represent attribute values and extends the VIKOR (VIsekriterijumska optimizacija i KOmpromisno Resenje) method to solve MADM problems in the q-ROSS context. First, we propose the q-ROSS to express evaluation information. Some basic operation rules and distance measures are investigated accordingly. When the amount of data is large, the left and right endpoints of the collected interval numbers will obey symmetric normal distribution. Secondly, based on the normal distribution assumption, the collected data intervals are mapped to shadowed sets through a data processing approach. Furthermore, we extend the VIKOR model to tackle the MADM problem where the evaluation values are expressed by q-rung orthopair shadowed numbers. A location selection problem verifies the practicability of our method, and the effectiveness and superiority of the presented approach are reflected through comparative analysis.

A spectroscopic Monte-Carlo model to simulate the response of pixelated CdTe based detectors

A fully spectroscopic Monte Carlo model has been developed to predict the spectroscopic performance of pixelated CdTe based detectors. The model incorporates photon attenuation by the photoelectric effect, Compton scattering and Rayleigh scattering. Charge transport equations are used to simulate the size of the electron cloud, approximated by a symmetrical two-dimensional Gaussian distribution, as it drifts to be read out at the detector anode. Direct comparisons are made between simulated data and experimentally acquired spectra from a 1 mm thick CdTe sensor coupled to the HEXITEC detector ASIC. The probability of an absorbed photon leading to charge sharing across pixels as a function of incoming photon energy is investigated. The charge cloud size was found to be dominated by cloud growth during drift for photon energies <100 keV. Furthermore, the portion of charge sharing events due to fluorescence from within the CdTe sensor is calculated—these events are distinguished from regular charge sharing events since their energy response differs. The model described is shown to give a good estimate of the total probability of charge sharing for energies up to 140 keV. CdTe sensor thickness, bias voltage, pixel size and electronic noise threshold can be adjusted to model a range of detector architectures.

Information Correlation in a 2 × 2 Game and an Extension of Purification Rationale

In this paper, we study a 2 × 2 Bayesian entry game with correlated private information. The distribution of private information is modelled by a symmetric joint normal distribution. Therefore, the correlation coefficient of the private information distribution reflects the degree of dependence of players' private information. Under such specification, players' private information can be correlated flexibly, which is not confined to the typical additive specification of private payoff shocks or private information by Carlson and van Damme (1993), where the private information is correlated due to the common payoff shock. In our game, if the private information is correlated, we find that given the variances of the private information, there exists a restriction on the degree of correlation of players' private information that allows the game can be solved by cutoff strategies. Specifically, given the variances of the private information, if players' private information in strategic substitutes (strategic complements) Bayesian games are positively (negatively) correlated, the range of correlation coefficient that allows the game can be solved by cutoff strategies is restricted so that if the correlation is out of the range, the game cannot be solved by cutoff strategies. Alternatively, given positive (negative) correlation of private information, the value of variances that allows a strategic substitutes (strategic complements) Bayesian games can be solved by cutoff strategies are restricted within certain range. If the value of variances fall out of the range, the Bayesian game cannot be solved by cutoff strategies. However, given negative (positive) correlation of players' private information in strategic substitutes (strategic complements) Bayesian games, in which the Bayesian games can always be solved by cutoff strategies, we prove that as the variances converge to zero, all pure strategy Bayesian Nash equilibria of the perturbed games converge to the respective Nash equilibria of the corresponding strategic substitutes (strategic complements) complete information games. Based on the result, we conclude that the purification rationale proposed by Harsanyi (1973) can be extended to games with dependent perturbation errors that follow a symmetric joint normal distribution if the correlation coefficient is positive for the strategic complements games or negative for the strategic substitutes games.

Stress fluctuations and shear thickening in dense granular suspensions

We experimentally investigate the rheology and stress fluctuations of granules densely suspended in silicone oil. We find that both thickening strength and stress fluctuations significantly weaken with oil viscosity $\eta_0$. Comparison of our rheological results to the Wyart-Cates model for describing different dynamic jamming states suggests a transition from frictional contacts to lubrication interactions as $\eta_0$ increases. To clarify the contribution from viscous interactions to the rheology, we systematically measure stress fluctuations in various flow states. Reduction of stress fluctuations with $\eta_0$ indicates that a strong lubrication layer greatly inhibits force correlations among particles. Measuring stress fluctuations in the strong shear thickening regime, we observe a crossover from asymmetric Gamma to symmetric Gaussian distributions and associated with it a decrease of lateral (radial) correlation length $\xi$ with increasing shear rate.