Tuning mechanical behaviors of highly entangled hydrogels with the random distribution of mobile entanglements

Highly entangled hydrogels exhibit excellent mechanical properties,including high toughness,high stretchability,and low hysteresis.By considering the evolution of randomly distributed entanglements within the polymer network upon mechanical stretches,we develop a constitutive theory to describe the large stretch behaviors of these hydrogels.In the theory,we utilize a representative volume element(RVE)in the shape of a cube,within which there exists an averaged chain segment along each edge and a mobile entanglement at each corner.By employing an explicit method,we decouple the elasticity of the hydrogels from the sliding motion of their entanglements,and derive the stress-stretch relations for these hydrogels.The present theoretical analysis is in agreement with experiment,and highlights the significant influence of the entanglement distribution within the hydrogels on their elasticity.We also implement the present developed constitutive theory into a commercial finite element software,and the subsequent simulations demonstrate that the exact distribution of entanglements strongly affects the mechanical behaviors of the structures of these hydrogels.Overall,the present theory provides valuable insights into the deformation mechanism of highly entangled hydrogels,and can aid in the design of these hydrogels with enhanced performance.

Magnetized magma intrusions being sources of a weak and a strong lunar magnetic anomaly revealed by 3D distribution of magnetization

In this work,we aim to investigate the origin of the magnetic carriers in the lunar crust and the intensity of the ancient dynamo field.The magnetization and depth range of magnetic carriers are studied under a weak and a strong magnetic anomaly in Mare Tranquillitatis and in Oceanus Procellarum,respectively,where the surface ages are 3.6 and 3.3 billion years.A sophisticated three-dimensional amplitude inversion software program from a geophysical survey is used to reconstruct the distributions of magnetization in the lunar crust.Because no globally measured surface magnetic field exists for the Moon,a crustal magnetic anomaly model with a grid resolution of 0.2°is used.The depth range of the magnetic source is fixed by the boundary identified by a relative criterion,which is 20%of the recovered maximum magnetization.The central burial depths of the magnetic carriers are approximately 15 km and 25 km under Reiner Gamma and Mare Tranquillitatis,respectively.The volumes of the two magnetic sources are at scales of 104 and 105 km3,respectively.The aforementioned differences may imply a hotter crust under Reiner Gamma than Mare Tranquillitatis by 3.3 billion years.The results support the view that the magma intrusions magnetized by an ancient magnetic field could be the origin of magnetic anomalies under Reiner Gamma and Mare Tranquillitatis.Compared with previous works,the maximum magnetization of 3 A/m under Reiner Gamma supports the intensity of the field being several microteslas.

One-sided empirical Bayes test for location parameter in Gamma distribution

In this paper, we devote to constructing the one-sided empirical Bayes(EB) test for the location parameter in the Gamma distribution by nonparametric method. Under some mild conditions, we prove that the EB test is asymptotically optimal with the rate of the order O(n^(-δs/(2s+1))), where 1/2 ≤ δ < 1 and s > 1 is a given natural number. An example is also given to illustrate that the conditions of the main theorems are easily satisfied.

The Riemannian Structure of the Three-Parameter Gamma Distribution

In this paper, we will utilize the results already known in differential geometry and provide an intuitive understanding of the Gamma Distribution. This approach leads to the definition of new concepts to provide new results of statistical importance. These new results could explain Chen [1-3] experienced difficulty when he attempts to simulate the sampling distribution and power function of Cox’s [4,5] test statistics of separate families of hypotheses. It may also help simplify and clarify some known statistical proofs or results. These results may be of particular interest to mathematical physicists. In general, it has been shown that the parameter space is not of constant curvature. In addition, we calculated some invariant quantities, such as Sectional curvature, Ricci curvature, mean curvature and scalar curvature.

Thorium distribution on the Moon:new insights from Chang’E-2 gamma-ray spectrometer

We present the thorium distribution on the lunar surface derived from observations by the Chang’E-2 gamma-ray spectrometer(CE-2 GRS). This new map shows a similar thorium distribution to previous observations. In combination with this new thorium map and impact cratering model, we investigate the origination of thorium on the Moon’s highlands, which was previously thought to be contributed from Imbrium ejecta. We found that the Imbrium ejecta has a small contribution(~20%–30%) to the thorium on the lunar highlands but most thorium is likely to be indigenous before the deposition of the Imbrium ejecta. This new thorium map also confirms that the eastern highlands have a relatively higher thorium concentration than the western highlands. We propose that the thin crust and large basins on the eastern highlands are responsible for this difference in thorium.

What Is the Difference between Gamma and Gaussian Distributions?

An inequality describing the difference between Gamma and Gaussian distributions is derived. The asymptotic bound is much better than by existing uniform bound from Berry-Esseen inequality.

A Note on Finding Geodesic Equation of Two Parameters Gamma Distribution

Engineers commonly use the gamma distribution to describe the life span or metal fatigue of a manufactured item. In this paper, we focus on finding a geodesic equation of the two parameters gamma distribution. To find this equation, we applied both the well-known Darboux Theorem and a pair of differential equations taken from Struik [1]. The solution proposed in this note could be used as a general solution of the geodesic equation of gamma distribution. It would be interesting if we compare our results with Lauritzen’s [2].

On Robustness of a Sequential Test for Scale Parameter of Gamma and Exponential Distributions

The main aim of the present paper is to study the robustness of the developed sequential probability ratio test (SPRT) for testing the hypothesis about scale parameter of gamma distribution with known shape parameter and exponential distribution with location parameter. The robustness of the SPRT for scale parameter of gamma distribution is studied when the shape parameter has undergone a change. The similar study is conducted for the scale parameter of exponential distribution when the location parameter has undergone a change. The expressions for operating characteristic and average sample number functions are derived. It is found in both the cases that the SPRT is robust only when there is a slight variation in the shape and location parameter in the respective distributions.

The Kolmogorov-Smirnov test for three redshift distributions of long gamma-ray bursts in the Swift Era

We investigate redshift distributions of three long burst samples, with the first sample containing 131 long bursts with observed redshifts, the second including 220 long bursts with pseudo-redshifts calculated by the variability-luminosity relation, and the third including 1194 long bursts with pseudo-redshifls calculated by the lag-luminosity relation, respectively. In the redshift range 0-1 the Kolmogorov-Smirnov probability of the observed redshift distribution and that of the variability-luminosity relation is large. In the redshift ranges 1-2, 2-3, 3-6.3 and 0-37, the Kolmogorov-Smirnov probabilities of the redshift distribution from lag-luminosity relation and the observed redshift distribution are also large. For the GRBs, which appear both in the two pseudo-redshift burst samples, the KS probability of the pseudo-redshift distribution from the lag-luminosity relation and the observed reshift distribution is 0.447, which is very large. Based on these results, some conclusions are drawn： i） the V-Liso relation might be more believable than the τ-Liso relation in low redshift ranges and the τ-Liso relation might be more real than the V-Liso relation in high redshift ranges; ii） if we do not consider the redshift ranges, the τ-Liso relation might be more physical and intrinsical than the V-Liso relation.

Investigations into Some Simple Expressions of the Gamma Function in Wind Power Theoretical Estimate by the Weibull Distribution

The Weibull distribution is a probability density function (PDF) which is widely used in the study of meteorological data. The statistical analysis of the wind speed v by using the Weibull distribution leads to the estimate of the mean wind speed , the variance of v around and the mean power density in the wind. The gamma function Γ is involved in those calculations, particularly Γ (1+1/k), Γ (1+2/k) and Γ (1+3/k). The paper reports the use of the Weibull PDF f(v) to estimate the gamma function. The study was performed by looking for the wind speeds related to the maximum values of f(v), v2 f(v) and v3 f(v). As a result, some approximate relationships were obtained for Γ (1+1/k), Γ (1+2/k) and Γ (1+3/k), that use some fitting polynomial functions. Very good agreements were found between the exact and the estimated values of Γ (1+n/k) that can be used for the estimation of the mean wind speed , the variance σ2 of the wind speed v;around the mean speed and the average wind power density.

A Distributional Representation of Gamma Function with Generalized Complex Domian

In this paper, we present a new representation of gamma function as a series of complex delta functions. We establish the convergence of this representation in the sense of distributions. It turns out that the gamma function can be defined over a space of complex test functions of slow growth denoted by Z. Some properties of gamma function are discussed by using the properties of delta function.

Content subscribing mechanism in P2P streaming based on gamma distribution prediction

P2P systems are categorized into tree-based and mesh-based systems according to their topologies. Mesh-based systems are considered more suitable for large-scale lnternet applications, but require optimization on latency issue. This paper proposes a content subscribing mechanism （CSM） to eliminate unnecessary time delays during data relaying. A node can send content data to its neighbors as soon as it receives the data segment. No additional time is taken during the interactive stages prior to data segment transmission of streaming content. CSM consists of three steps. First, every node records its historical segments latency, and adopts gamma distribution, which possesses powerful expression ability, to express latency statistics. Second, a node predicts subscribing success ratio of every neighbor by comparing the gamma distribution parameters of the node and its neighbors before selecting a neighbor node to subscribe a data segment. The above steps would not increase latency as they are executed before the data segments are ready at the neighbor nodes. Finally, the node, which was subscribed to, sends the subscribed data segment to the subscriber immediately when it has the data segment. Experiments show that CSM significantly reduces the content data transmission latency.

Designing Bayesian Two-Sided Group Chain Sampling Plan for Gamma Prior Distribution

Acceptance sampling is used to decide either the whole lot will be accepted or rejected,based on inspection of randomly sampled items from the same lot.As an alternative to traditional sampling plans,it is possible to use Baye-sian approaches using previous knowledge on process variation.This study pre-sents a Bayesian two-sided group chain sampling plan(BTSGChSP)by using various combinations of design parameters.In BTSGChSP,inspection is based on preceding as well as succeeding lots.Poisson function is used to derive the probability of lot acceptance based on defective and non-defective products.Gamma distribution is considered as a suitable prior for Poisson distribution.Four quality regions are found,namely:(i)quality decision region(QDR),(ii)probabil-istic quality region(PQR),(iii)limiting quality region(LQR)and(iv)indifference quality region(IQR).Producer’s risk and consumer’s risk are considered to esti-mate the quality regions,where acceptable quality level(AQL)is associated with producer’s risk and limiting quality level(LQL)is associated with consumer’s risk.Moreover,AQL and LQL are used in the selection of design parameters for BTSGChSP.The values based on all possible combinations of design parameters for BTSGChSP are presented and inflection points’values are found.Thefinding exposes that BTSGChSP is a better substitute for the existing plan for industrial practitioners.

Analytical Evaluation of Non-Elementary Integrals Involving Some Exponential, Hyperbolic and Trigonometric Elementary Functions and Derivation of New Probability Measures Generalizing the Gamma-Type and Gaussian-Type Distributions

The non-elementary integrals involving elementary exponential, hyperbolic and trigonometric functions, and where α,η and β are real or complex constants are evaluated in terms of the confluent hypergeometric function 1F1 and the hypergeometric function 1F2. The hyperbolic and Euler identities are used to derive some identities involving exponential, hyperbolic, trigonometric functions and the hypergeometric functions 1F1 and 1F2. Having evaluated, these non-elementary integrals, some new probability measures generalizing the gamma-type and Gaussian distributions are also obtained. The obtained generalized probability distributions may, for example, allow to perform better statistical tests than those already known (e.g. chi-square (x2) statistical tests and other statistical tests constructed based on the central limit theorem (CLT)), while avoiding the use of computational approximations (or methods) which are in general expensive and associated with numerical errors.

Uniformly Minimum-Variance Unbiased Estimator (UMVUE) for the Gamma Cumulative Distribution Function with Known and Integer Scale Parameter

Uniformly minimum-variance unbiased estimator (UMVUE) for the gamma cumulative distribution function with known and integer scale parameter. This paper applies Rao-Blackwell and Lehmann-Scheffeé Theorems to deduce the uniformly minimum-variance unbiased estimator (UMVUE) for the gamma cumulative distribution function with known and integer scale parameters. The paper closes with an example comparing the empirical distribution function with the UMVUE estimates.

Gamma-Generalized Inverse Gaussian Class of Distributions

Gamma distribution nests exponential, chi-squared and Erlang distributions;while generalized Inverse Gaussian distribution nests quite a number of distributions. The aim of this paper is to construct a gamma mixture using Generalized inverse Gaussian mixing distribution. The rth moment of the mixture is obtained via the rth moment of the mixing distribution. Special cases and limiting cases of the mixture are deduced.

Does the Redshift Distribution of Swift Long GRBs Trace the Star-Formation Rate?

Gamma-ray bursts (GRBs) are extremely powerful explosions that have been traditionally classified into two categories: long bursts (LGRBs) with an observed duration T90 > 2 s, and short bursts (SGRBs) with an observed duration T90 T90 is the time interval during which 90% of the fluence is detected. LGRBs are believed to emanate from the core-collapse of massive stars, while SGRBs are believed to result from the merging of two compact objects, like two neutron stars. Because LGRBs are produced by the violent death of massive stars, we expect that their redshift distribution should trace the star-formation rate (SFR). The purpose of our study is to investigate the extent to which the redshift distribution of LGRBs follows and reflects the SFR. We use a sample of 370 LGRBs taken from the Swift catalog, and we investigate different models for the LGRB redshift distribution. We also carry out Monte Carlo simulations to check the consistency of our results. Our results indicate that the SFR can describe the LGRB redshift distribution well for high redshift bursts, but it needs an evolution term to fit the distribution well at low redshift.

Reconstruction of Probability Density Function for Gamma Distribution

The probability distributions of small sample data are difficult to determine,while a large proportion of samples occur in the early failure period,so it is particularly important to make full use of these data in the statistical analysis.Based on gamma distribution,four methods of probability density function(PDF)reconstruction with early failure data are proposed,and then the mean time between failures(MTBF)evaluation expressions are concluded from the reconstructed PDFs.Both theory analysis and an example show that method 2 is the best evaluation method in dealing with early-failure-small-sample data.The reconstruction methods of PDF also have certain guiding significance for other distribution types.

Earthquake Magnitude Probability and Gamma Distribution

The earthquake magnitude probability distribution is one of the underlying input data for certain earthquake analyses, such as probabilistic seismic hazard analysis. Nowadays, the method proposed by McGuire and Arabasz (1990) is commonly used for obtaining the (simulated) earthquake magnitude probability distributions. However, based on the observed earthquake data in 5 regions (Taiwan, Japan, California, Turkey, and Greece), the model did not fit the observation well. Instead, all of the case studies show that using the newly proposed gamma distribution can improve the simulation significantly compared to the conventional method.

Determination of Gamma Angular Distribution from the Shape of Spectral Line for the First Excited State of Carbon Nucleus

An experiment investigating gamma emission in hadron therapy was performed at Cyclotron Centre Bronowice (CCB), Cracow, Poland, using two different phantom materials—carbon and poly(methyl methacrylate) PMMA. The measurements were carried out at 70 MeV proton beam energy and the gamma quanta were registered with the use of HP Ge detector with scintillation anti-Compton shielding. Although the primary aim was to establish a solid experimental data base for future applications in prompt gamma imaging, the data have also been analyzed with regards to the position and shape of the spectral line stemming from deexcitation of the carbon excited state 4.44 MeV. Measurements potentially useful to determine the cross section were performed only at 90° laboratory polar angle. However, benefiting from the very good energy resolution it turned out possible to extract information on angular distribution of the C* (4.44 MeV) deexcitation by analyzing the associated line shape. This paper presents the scheme of model calculations assuming the whole process can be divided into two stages: excitation of carbon nuclei by impinging protons and deexcitation of the C* (4.44 MeV) state.