Detecting a Regularity in the Generation and Utilization of Primes in the Multiplicative Number Theory

If Goldbach’s conjecture is true, then for each prime number p there is at least one pair of primes symmetric with respect to p and whose sum is 2p. In the multiplicative number theory, covering the positive integers with primes, during the prime factorization, may be viewed as being the outcome of a parallel system which functions properly if and only if Euler’s formula of the product of the reciprocals of the primes is true. An exact formula for the number of primes less than or equal to an arbitrary bound is given. This formula may be implemented using Wolfram’s computer package Mathematica.

LIMITING BEHAVIOR OF RECURSIVE M-ESTIMATORS IN MULTIVARIATE LINEAR REGRESSION MODELS AND THEIR ASYMPTOTIC EFFICIENCIES

Recursive algorithms are very useful for computing M-estimators of regression coefficients and scatter parameters. In this article, it is shown that for a nondecreasing ul （t）, under some mild conditions the recursive M-estimators of regression coefficients and scatter parameters are strongly consistent and the recursive M-estimator of the regression coefficients is also asymptotically normal distributed. Furthermore, optimal recursive M-estimators, asymptotic efficiencies of recursive M-estimators and asymptotic relative efficiencies between recursive M-estimators of regression coefficients are studied.

A Hidden Stability in the Random Behaviour of the Consecutive Prime Numbers

The paper contains a geometric interpretation of Andrica’s conjecture about the gap between the square roots of the consecutive primes and brings empirical evidence that the random fluctuations of the gap between the quare roots of the consecutive primes seem to stabilize around the mean gap.

The Acute Effects of Aerobic Dance Exercise with and without Face Mask Use on Attention, Perceived Exertion and Mood States

The present study aimed to determine the effect of wearing a face mask during aerobic dance exercise on cognitive function,more specifically on attention,as well as on perceived exertion and mood states.Thirteen healthy college students(9 males and 4 females:mean age=17.5 years,height=1.72 m,weight=71.00 kg)volunteered to participate in this study.They were randomized to perform aerobic dance exercise while wearing a cloth face mask or no mask or a control condition(sitting on a comfortable chair and reading information about the health benefits of aerobic dance exercise)on three separate occasions(with at least one week of interval).Rate of perceived exertion(RPE),the Brunel Mood Scale(BRUMS)and d2 Attention assessment were assessed before and immediately after each condition.The results demonstrated higher concentration performance for the aerobic dance exercise without face mask than the control condition(p=0.05).Post RPE and BRUMS fatigue subscale values were significantly higher in the aerobic dance exercise with face mask as compared to the aerobic dance exercise without face mask and control condition(all,p<0.05).BRUMS vigor subscale value significantly differed across conditions(F=113.84,p<0.001,ES=0.86)and was significantly higher in the aerobic dance exercise group without face mask as compared to the aerobic dance exercise with face mask and the control conditions(both,p<0.001).This study suggests that face mask use during aerobic dance exercise with moderate intensity did not affect attention.Practitioners,students and athletes should avoid wearing face mask while practicing physical activity or aerobic dance exercise with moderate intensity to improve its acute effect on cognitive function.

Understanding the impact of HIV on mpox transmission in the MSM population: A mathematical modeling study

The recent mpox outbreak(in 2022e2023)has different clinical and epidemiological features compared with previous outbreaks of the disease.During this outbreak,sexual contact was believed to be the primary transmission route of the disease.In addition,the community of men having sex with men(MSM)was disproportionately affected by the outbreak.This population is also disproportionately affected by HIV infection.Given that both diseases can be transmitted sexually,the endemicity of HIV,and the high sexual behavior associated with the MSM community,it is essential to understand the effect of the two diseases spreading simultaneously in an MSM population.Particularly,we aim to understand the potential effects of HIV on an mpox outbreak in the MSM population.We develop a mechanistic mathematical model of HIV and mpox co-infection.Our model incorporates the dynamics of both diseases and considers HIV treatment with antiretroviral therapy(ART).In addition,we consider a potential scenario where HIV infection increases susceptibility to mpox,and investigate the potential impact of this mechanism on mpox dynamics.Our analysis shows that HIV can facilitate the spread of mpox in an MSM population,and that HIV treatment with ART may not be sufficient to control the spread of mpox in the population.However,we showed that a moderate use of condoms or reduction in sexual contact in the population combined with ART is beneficial in controlling mpox transmission.Based on our analysis,it is evident that effective control of HIV,specifically through substantial ART use,moderate condom compliance,and reduction in sexual contact,is imperative for curtailing the transmission of mpox in an MSM population and mitigating the compounding impact of these intertwined epidemics.

Numerical study of clathrin-mediated endocytosis of nanoparticles by cells under tension

In this study, a three-dimensional mathematical model was used to study the contribution of clathrins during the process of cellular uptake of spherical nanoparticles under different membrane tensions. The clathrin-coated pit (CCP) that forms around the inward budding of the cell membrane was modeled as a vesicle with bending rigidity. An optimization algorithm was proposed for minimizing the total energy of the system, which comprises the deforming nanoparticle, receptor-ligand bonds, cell membrane, and CCP, in which way, the profile of the system is acquired. The results showed that the CCP enable full wrapping of the nanoparticles at various membrane tensions. When the cell membrane tension increases, the total deformation energy also increases, but the ratio of CCP bending to the minimum value of the total energy of the system decreases. The results also showed that the diameter of the endocytic vesicles determined by the competition between the stretching of the cell membrane and confinement of the coated pits are much larger than the nanoparticles, which is quit different as the results in passive endocytosis that is not facilitated by the CCPs. The present results indicate that variations of tension on cell membranes constitutes a biophysical marker for understanding the size distribution of CCPs observed in experiments. The present results also suggest that the early abortion of endocytosis is related to that the receptor-ligand bonds cannot generate adequate force to wrap the nanoparticles into the cell membrane before the clathrins respond to support the endocytic vesicles. Correspondingly, late abortion may relate to the inability of CCPs to confine the nanoparticles until the occurrence of the necking stage of endocytosis.

A Numerical Study of Passive Receptor-Mediated Endocytosis of Nanoparticles:The Effect of Mechanical Properties

In this work,a three-dimensional axisymmetric model with nanoparticle,receptor-ligand bonds and cell membrane as a system was used to study the quasi-static receptor-mediated endocytosis process of spherical nanoparticles in drug delivery.The minimization of the system energy function was carried out numerically,and the deformations of nanoparticle,receptor-ligand bonds and cell membrane were predicted.Results show that passive endocytosis may fail due to the rupture of receptor-ligand bonds during the wrapping process,and the size and rigidity of nanoparticles affect the total deformation energy and the terminal wrapping stage.Our results suggest that,in addition to the energy requirement,the success of passive endocytosis also depends on the maximum strength of the receptor-ligand bonds.

Accuracy analysis of immersed boundary method using method of manufactured solutions

The immersed boundary method is an effective technique for modeling and simulating fluid-structure interactions especially in the area of biomechanics.This paper analyzes the accuracy of the immersed boundary method.The procedure contains two parts,i.e.,the code verification and the accuracy analysis.The code verification provides the confidence that the code used is free of mistakes,and the accuracy analysis gives the order of accuracy of the immersed boundary method.The method of manufactured solutions is taken as a means for both parts.In the first part,the numerical code employs a second-order discretization scheme,i.e.,it has second-order accuracy in theory.It matches the calculated order of accuracy obtained in the numerical calculation for all variables.This means that the code contains no mistake,which is a premise of the subsequent work.The second part introduces a jump in the manufactured solution for the pressure and adds the corresponding singular forcing terms in the momentum equations.By analyzing the discretization errors,the accuracy of the immersed boundary method is proven to be first order even though the discretization scheme is second order.It has been found that the coarser mesh may not be sensitive enough to capture the influence of the immersed boundary,and the refinement on the Lagrangian markers barely has any effect on the numerical calculation.

Weighted Gini-Simpson Quadratic Index of Biodiversity for Interdependent Species

The weighted Gini-Simpson quadratic index is the simplest measure of biodiversity which takes into account the relative abundance of species and some weights assigned to the species. These weights could be assigned based on factors such as the phylogenetic distance between species, or their relative conservation values, or even the species richness or vulnerability of the habitats where these species live. In the vast majority of cases where the biodiversity is measured the species are supposed to be independent, which means that the relative proportion of a pair of species is the product of the relative proportions of the component species making up the respective pair. In the first section of the paper, the main versions of the weighted Gini-Simpson index of biodiversity for the pairs and triads of independent species are presented. In the second section of the paper, the weighted Gini-Simpson quadratic index is calculated for the general case when the species are interdependent. In this instance, the weights reflect the conservation values of the species and the distribution pattern variability of the subsets of species in the respective habitat induced by the inter-dependence between species. The third section contains a numerical example.

Effect of regularized delta function on accuracy of immersed boundary method

The immersed boundary method is an effective technique for modeling and simulating fluid-structure interactions especially in the area of biomechanics. The effect of the regularized delta function on the accuracy is an important subject in the property study. A method of manufactured solutions is used in the research. The computational code is first verified to be mistake-free by using smooth manufactured solutions. Then, a jump in the manufactured solution for pressure is introduced to study the accuracy of the immersed boundary method. Four kinds of regularized delta functions are used to test the effect on the accuracy analysis. By analyzing the discretization errors, the accuracy of the immersed boundary method is proved to be first-order. The results show that the regularized delta function cannot improve the accuracy, but it can change the discretization errors in the entire computational domain.

The Proof of Goldbach’s Conjecture on Prime Numbers

Goldbach’s Conjecture (“Every even positive integer strictly larger than 4 is the sum of two primes”) has remained unproven since 1742. This paper contains the proof that every positive composite integer n strictly larger than 3, is located at the middle of the distance between two primes, which implicitly proves Goldbach’s Conjecture for 2n as well.

Confidence Intervals for the Mean of Non-Normal Distribution: Transform or Not to Transform

In many areas of applied statistics, confidence intervals for the mean of the population are of interest. Confidence intervals are typically constructed as-suming normality although non-normally distributed data are a common occurrence in practice. Given a large enough sample size, confidence intervals for the mean can be constructed by applying the Central Limit Theorem or by the bootstrap method. Another commonly used method in practice is the back-transformation method, which takes on the following three steps. First, apply a transformation to the data such that the transformed data are normally distributed. Second, obtain confidence intervals for the transformed mean in the usual manner, which assumes normality. Third, apply the back- transformation to obtain confidence intervals for the mean of the original, non-transformed distribution. The parametric Wald method and a small sample likelihood-based third order method, which can address non-normality, are also reviewed in this paper. Our simulation results suggest that common approaches such as back-transformation give erroneous and misleading results even when the sample size is large. However, the likelihood-based third order method gives extremely accurate results even when the sample size is small.

Inference for the Normal Mean with Known Coefficient of Variation

Inference for the mean of a normal distribution with known coefficient of variation is of special theoretical interest be- cause the model belongs to the curved exponential family with a scalar parameter of interest and a two-dimensional minimal sufficient statistic. Therefore, standard inferential methods cannot be directly applied to this problem. It is also of practical interest because this problem arises naturally in many environmental and agriculture studies. In this paper we proposed a modified signed log likelihood ratio method to obtain inference for the normal mean with known coeffi- cient of variation. Simulation studies show the remarkable accuracy of the proposed method even for sample size as small as 2. Moreover, a new point estimator for the mean can be derived from the proposed method. Simulation studies show that new point estimator is more efficient than most of the existing estimators.

Theoretical Properties of Composite Likelihoods

The general functional form of composite likelihoods is derived by minimizing the Kullback-Leibler distance under structural constraints associated with low dimensional densities. Connections with the I-projection and the maximum entropy distributions are shown. Asymptotic properties of composite likelihood inference under the proposed information-theoretical framework are established.

Evaluation of Third-Order Method for the Tests of Variance Component in Linear Mixed Models

Mixed models provide a wide range of applications including hierarchical modeling and longitudinal studies. The tests of variance component in mixed models have long been a methodological challenge because of its boundary conditions. It is well documented in literature that the traditional first-order methods: likelihood ratio statistic, Wald statistic and score statistic, provide an excessively conservative approximation to the null distribution. However, the magnitude of the conservativeness has not been thoroughly explored. In this paper, we propose a likelihood-based third-order method to the mixed models for testing the null hypothesis of zero and non-zero variance component. The proposed method dramatically improved the accuracy of the tests. Extensive simulations were carried out to demonstrate the accuracy of the proposed method in comparison with the standard first-order methods. The results show the conservativeness of the first order methods and the accuracy of the proposed method in approximating the p-values and confidence intervals even when the sample size is small.

Comparing the transmission potential from sequence and surveillance data of 2009 North American influenza pandemic waves

Technological advancements in phylodynamic modeling coupled with the accessibility of real-time pathogen genetic data are increasingly important for understanding the infectious disease transmission dynamics.In this study,we compare the transmission potentials of North American influenza A(H1N1)pdm09 derived from sequence data to that derived from surveillance data.The impact of the choice of tree-priors,informative epidemiological priors,and evolutionary parameters on the transmission potential estimation is evaluated.North American Influenza A(H1N1)pdm09 hemagglutinin(HA)gene sequences are analyzed using the coalescent and birth-death tree prior models to estimate the basic reproduction number(R_(0)).Epidemiological priors gathered from published literature are used to simulate the birth-death skyline models.Path-sampling marginal likelihood estimation is conducted to assess model fit.A bibliographic search to gather surveillancebased R_(0)values were consistently lower(mean≤1.2)when estimated by coalescent models than by the birth-death models with informative priors on the duration of infectiousness(mean≥1.3 to≤2.88 days).The user-defined informative priors for use in the birth-death model shift the directionality of epidemiological and evolutionary parameters compared to non-informative estimates.While there was no certain impact of clock rate and tree height on the R_(0)estimation,an opposite relationship was observed between coalescent and birth-death tree priors.There was no significant difference(p=0.46)between the birth-death model and surveillance R0 estimates.This study concludes that treeprior methodological differences may have a substantial impact on the transmission potential estimation as well as the evolutionary parameters.The study also reports a consensus between the sequence-based R_(0)estimation and surveillanceased R_(0)stimates.Altogether,these outcomes shed light on the potential role of phylodynamic modeling to augment existing surveillance and epidemiological activities to better assess and respond to emerging infectious diseases.

面向多源大数据云端处理的成本最小化方法

云计算为大数据处理提供了一种强大而高效的解决方案.在此模式下,数据管理者(data manager,简称DM)可以租用多个数据中心实时处理地理分散的数据.然而,由于数据产生的动态性以及资源价格的波动性,将数据迁移至哪些数据中心并提供合适的计算资源来处理它们,成为DM低成本处理多源数据的一大问题.首先,将以上问题转换成联合随机优化问题;然后,利用李雅普诺夫(Lyapunov)优化框架将原问题分解成两个独立的子问题进行求解;最后,基于求解结果设计在线算法.理论分析结果表明:所提算法可不断趋近线下最优解,并能够保证数据处理时延.在WorldCup98和Youtube数据集上的实验验证了理论分析结果的正确性以及该方法的优越性.

Time-consistent investment-reinsurance strategies towards joint interests of the insurer and the reinsurer under CEV models

The present paper studies time-consistent solutions to an investment-reinsurance problem under a mean-variance framework.The paper is distinguished from other literature by taking into account the interests of both an insurer and a reinsurer jointly.The claim process of the insurer is governed by a Brownian motion with a drift.A proportional reinsurance treaty is considered and the premium is calculated according to the expected value principle.Both the insurer and the reinsurer are assumed to invest in a risky asset,which is distinct for each other and driven by a constant elasticity of variance model.The optimal decision is formulated on a weighted sum of the insurer’s and the reinsurer’s surplus processes.Upon a verification theorem,which is established with a formal proof for a more general problem,explicit solutions are obtained for the proposed investment-reinsurance model.Moreover,numerous mathematical analysis and numerical examples are provided to demonstrate those derived results as well as the economic implications behind.

Asymptotic Inference for the Weak Stationary Double AR(1) Model

An AR(1) model with ARCH(1) error structure is known as the first-order double autoregressive (DAR(1)) model. In this paper, a conditional likelihood based method is proposed to obtain inference for the two scalar parameters of interest of the DAR(1) model. Theoretically, the proposed method has rate of convergence O(n-3/2). Applying the proposed method to a real-life data set shows that the results obtained by the proposed method can be quite different from the results obtained by the existing methods. Results from Monte Carlo simulation studies illustrate the supreme accuracy of the proposed method even when the sample size is small.

Assessing the mechanism of citywide test-trace-isolate Zero-COVID policy and exit strategy of COVID-19 pandemic

Background:Countries that aimed for eliminating the cases of COVID-19 with test-trace-isolate policy are found to have lower infections,deaths,and better economic performance,compared with those that opted for other mitigation strategies.However,the continuous evolution of new strains has raised the question of whether COVID-19 eradication is still possible given the limited public health response capacity and fatigue of the epidemic.We aim to investigate the mechanism of the Zero-COVID policy on outbreak containment,and to explore the possibility of eradication of Omicron transmission using the citywide test-trace-isolate(CTTI)strategy.Methods:We develop a compartmental model incorporating the CTTI Zero-COVID policy to understand how it contributes to the SARS-CoV-2 elimination.We employ our model to mimic the Delta outbreak in Fujian Province,China,from September 10 to October 9,2021,and the Omicron outbreak in Jilin Province,China for the period from March 1 to April 1,2022.Projections and sensitivity analyses were conducted using dynamical system and Latin Hypercube Sampling/Partial Rank Correlation Coefcient(PRCC).Results:Calibration results of the model estimate the Fujian Delta outbreak can end in 30(95%confdence interval CI:28–33)days,after 10(95%CI:9–11)rounds of citywide testing.The emerging Jilin Omicron outbreak may achieve zero COVID cases in 50(95%CI:41–57)days if supported with sufcient public health resources and population compliance,which shows the efectiveness of the CTTI Zero-COVID policy.Conclusions:The CTTI policy shows the capacity for the eradication of the Delta outbreaks and also the Omicron outbreaks.Nonetheless,the implementation of radical CTTI is challenging,which requires routine monitoring for early detection,adequate testing capacity,efcient contact tracing,and high isolation compliance,which constrain its benefts in regions with limited resources.Moreover,these challenges become even more acute in the face of more contagious variants with a high proportion of asymptomatic cases.Hence,in regions where CTTI is not possible,personal protection,public health control measures,and vaccination are indispensable for mitigating and exiting the COVID-19 pandemic.