On the Mean Value of High-Powers of a Special Character Sum Modulo a Prime

In this paper,we use the elementary methods,the properties of Dirichlet character sums and the classical Gauss sums to study the estimation of the mean value of high-powers for a special character sum modulo a prime,and derive an exact computational formula.It can be conveniently programmed by the“Mathematica”software,by which we can get the exact results easily.

APPROXIMATION BY FABER-LAURENT RATIONAL FUNCTIONS IN THE MEAN OF FUNCTIONS OF CLASS LP(T) WITH 1

Let Γ be a regular curve and Lp (Γ), 1<p<+∞. be the class of all complex - valued functions f de-fined on Γ which are such that |f|p is mtegrabie in sense of Lebesgue. In this work, we define the k th p-Fabcrpolynomial F k,p (z),the kth p-Faber principle part F k.p (1/z) for Γ , and defined the nth p-Fcber- Laurent rational function Rn.p (f, z) and p- generalized modulus of continuity Ωp(f, t) of a function f of Lp(Γ) We inves-tigate some properties of Fk,p (z) and Fk.p (1/z). And then we prove a direct theorem characterizing the degree of approximation with respect to Ω (. , t) in the mean of functions of Lp(Γ) by the rational junctions Rn.p (. . z).

On the Mean Value Formula of Dedekind Sums

The main purpose of this paper is to use the mean value theorem of the Dirichlet L-function to study the distribution property of Dedekind sums, and to give a sharper mean value formula.

A note on error bars as a graphical representation of the variability of data in biomedical research:choosing between standard deviation and standard error of the mean

Standard deviation(SD)and standard error of the mean(SEM)have been applied widely as error bars in scientific plots.Unfortunately,there is no universally accepted principle addressing which of these 2 measures should be used.Here we seek to fill this gap by outlining the reasoning for choosing SEM over SD and hope to shed light on this unsettled disagreement among the biomedical community.The utility of SEM and SD as error bars is further discussed by examining the figures and plots published in 2 research articles on pancreatic disease.

The Mean Man's Party

The notorious (声名狼藉的) cheap skate (吝啬鬼) finally decided to have a party.Explanining to a friend how to find his apartment(公寓),he said,“Come up to 5M and ring the

Study on the Dynamics of an SIR Epidemic Model with Saturated Growth Rate

In this paper, we investigate the dynamic properties of an SIR epidemic model with saturated growth rate. Under the conditions of an arbitrary initial value, we prove that the system exists unique positive solution, and give the sufficient conditions caused by random environmental factors leading to the extinction of infectious diseases. Moreover, we verify the conditions for the persistence of infectious diseases in the mean sense. Finally, we provide the biology interpretation and some strategies to control the infectious diseases.

Placebo responses in patients with gastrointestinal disorders

Over the last several years there has been a growing interest in placebo, not only as an inert control in clinical trials, but also in the placebo effect as a group effect as well as a reaction in individual subjects. Methodological factors such as regression to the mean and natural history of the disease play a role in the evaluation of a possible placebo effect. In this report, we discuss several factors including PavIovian conditioning, beliefs outcome, expectations, and other factors as potential mediators of the placebo response. Placebo effects are common in gastrointestinal diseases and there seems to be no clear difference between placebo effects in functional gastrointestinal diseases （functional dyspepsia and irritable bowel syndrome） and organic gastrointestinal disease （duodenal ulcer and inflammatory bowel disease）.

Blind spectrum sensing based on the ratio of mean square to variance

Anderson-Darling （AD） sensing, characteristic function （CF） sensing and order statistic （OS） sensing are three common spectrum sensing （SS） methods based on goodness of fit （GOF） testing. However, AD and OS sensing needs the prior information of noise variance; CF and OS sensing have high computation complexity. To circumvent those difficulties, in this paper, the ratio of the mean square to variance （RM2V） of the samples, after deriving its probability density function （PDF）, is employed as a test statistic to detect the availability of the vacant spectrum in the cognitive radio （CR） system. Then a blind SS method based on RM2V is proposed, which is dubbed as RM2V sensing, and its exact theoretical threshold is obtained via the derived PDF of RM2V. The performance of RM2V sensing is evaluated by theoretical analysis and Monte Carlo simulations. Comparing with the conventional energy detection （ED）, AD, CF and OS sensing, RM2V sensing, with no need of noise variance, has advantages from the aspect of computation complexity and detection performance.

DYNAMIC FOR A STOCHASTIC MULTI-GROUP AIDS MODEL WITH SATURATED INCIDENCE RATE

In this paper,a stochastic multi-group AIDS model with saturated incidence rate is studied.We prove that the system is persistent in the mean under some parametric restrictions.We also obtain the sufficient condition for the existence of the ergodic stationary distribution of the system by constructing a suitable Lyapunov function.Our results indicate that the existence of ergodic stationary distribution does not rely on the interior equilibrium of the corresponding deterministic system,which greatly improves upon previous results.

Spatially Periodic System with Infinite Globally Coupled Oscillators Driven by Temporal-Spatial Noise

In this paper, we study spatially periodic system with infinite globally coupled oscillators driven by temporal-spatial noise and subject to a constant force. The results show that the system exhibits the phenomena of the non-equilibrium phase transition, transport of particles, and the anomalous hysteresis cycle for the mean field and the probability current.

Asymptotic analysis of a nonlinear stochastic eco-epidemiological system with feedback control

This paper proposes a new stochastic eco-epidemiological model with nonlinear incidence rate and feedback controls.First,we prove that the stochastic system has a unique global positive solution.Second,by constructing a series of appropriate stochastic Lyapunov functions,the asymptotic behaviors around the equilibria of deterministic model are obtained,and we demonstrate that the stochastic system exists a stationary Markov process.Third,the conditions for persistence in the mean and extinction of the stochastic system are established.Finally,we carry out some numerical simulations with respect to different stochastic parameters to verify our analytical results.The obtained results indicate that the stochastic perturbations and feedback controls have crucial effects on the survivability of system.

Simple analysis on the embodiment of golden mean(Zhong Yong) thoughts in tuina

The concept of golden mean （Zhong Yong）, i.e. moderation, is the highest principle of moral cultivation in Confucianism. It entails an exclusive methodology in addressing politics, philosophy and everything in the universe. This concept has permeated into all aspects of traditional Chinese culture. As an ancient therapy, tuina is an important part of Chinese medicine and has its characteristic theory and methods. Just like Chinese medicine, tuina has long been greatly influenced by the concept of golden mean. To better promote the development of tuina, it＇s necessary to investigate the historical evolution of tuina and the concept of golden mean as well as the influence of ＇golden mean＇ idea on tuina manipulation, mechanism and treatment.

Stability and output feedback stabilization for systems with Markovian switching and impulse effects

We investigate the exponential stability in the mean square sense for the systems with Markovian switching and impulse effects.Based on the statistic property of the Markov process,a stability criterion is established.Then,by the parameterizations via a family of auxiliary matrices,the dynamical output feedback controller can be solved via an LMI approach,which makes the closed-loop system exponentially stable.A numerical example is given to demonstrate the method.

Analysis and simulation of a stochastic COVID-19 model with large-scale nucleic acid detection and isolation measures: A case study of the outbreak in Urumqi, China in August 2022

In this paper,a stochastic COVID-19 model with large-scale nucleic acid detection and isolation measures is proposed.Firstly,the existence and uniqueness of the global positive solution is obtained.Secondly,threshold criteria for the stochastic extinction and persistence in the mean with probability one are established.Moreover,a sufficient condition for the existence of unique ergodic stationary distribution for any positive solution is also established.Finally,numerical simulations are carried out in combination with real COVID-19 data from Urumqi,China and the theoretical results are verified.

A Synthetic Comprehension of the Way of Zhong in Early Confucian Philosophy

Zhong 中 is a very important philosophical concept in early Confucianism. Both the received ancient Confucian classics and the newly discovered ancient bamboo manuscripts tell us that adhering to the principle of zhong was an important charge that had been transmitted and inherited by early ancient Chinese political leaders from generation to generation. Confucius and his followers adopted the concept of zhong and further developed it into a sophisticated doctrine, which is usually called zhongdao ~_ （the Way ofzhong） or zhongyong ~lgCgi. Being a polysemous word, zhong has several different but philosophically related meanings. However, for a long time, people usually understood zhong in the sense of only one of its meanings, and zhongdao or zhongyong has been commonly interpreted as ＂the doctrine of the mean.＂ My argument in this paper is that a synthetic interpretation, which includes all the semantic meanings of zhong is necessary in order to acquire a deep and well-rounded comprehension of the philosophical significance of the Way of zhong. The Way of zhong features a dialectal view of the relationship between heaven and human beings, mind and materials, subjective desire and the available objective conditions, self and others, centrality and diversity, and so on. The Way of zhong has become a widely applied philosophical methodology in Confucianism, as well as a political principle and a kind of personal moral merit in early Confucian doctrines. Today, it still has relevance in contemporary Chinese social and cultural contexts.

The stationary distribution and stochastic persistence for a class of disease models:Case study of malarial

This paper presents a nonlinear family of stochastic SEIRS models for diseases such as malaria in a highly random environment with noises from the disease transmission and natural death rates,and also from the random delays of the incubation and immunity periods.Improved analytical methods and local martingale characterizations are applied to find conditions for the disease to persist near an endemic steady state,and also for the disease to remain permanently in the system over time.Moreover,the ergodic stationary distribution for the stochastic process describing the disease dynamics is defined,and the statistical characteristics of the distribution are given mumerically.The results of this study show that the disease will persist and become permanent in the system,regardless of(1)whether the noises are from the discase transmission rate and/or from the natural death rates or(2)whether the delays in the system are constant or random for individuals in the system.Furthermore,it is shown that"weak"noise is associated with the existence of an endemic stationary distribution for the disease,while"strong"noise is associated with extinction of the population over time.Numerical simulation examples for Plasnodiurr vitar malaria are given.

Stochastic dynamics of the transmission of Dengue fever virus bet ween mosquit oes aiul humans

Considering the impact of environmental white noise on the quantity and behavior ofvector of disease,a stochastic differential model describing the transmission of Denguefever between mosquitoes and humans,in this paper,is proposed.By using Lyapunovmethods and Ito's formula,we first prove the existence and uniqueness of a globalpositive solution for this model.Further,some sufficient conditions for the extinction andpersistence in the mean of this stochastic model are obtained by using the techniquesof a series of stochastic inequalities.In addition,we also discuss the existence of aunique stationary distribution which leads to the stochastic persistence of this disease.Finally,several numerical simulations are carried to illustrate the main results of thiscontribution.

Variational Principles for Asymptotic Variance of General Markov Processes

A variational formula for the asymptotic variance of general Markov processes is obtained.As application,we get an upper bound of the mean exit time of reversible Markov processes,and some comparison theorems between the reversible and non-reversible diffusion processes.

PERSISTENCE OF POPULATIONS FORA PREDATOR-PREY SYSTEM IN A POLLUTED ENVIRONMENT

A predator-prey system in a polluted environment is studied in this paper.Surveying the transformation of toxicants from prey to predator and the effects of toxicants on functional response of predator comprehensively, the thresholds between the weak persistence in the mean and extinction of populations are established.

Dynamics of a stochastic three-species competitive model with Levy jumps

This paper is concerned with a three-species competitive model with both white noises and Levy noises. We first carry out the almost complete parameters analysis for the model and establish the critical value between persistence in the mean and extinction for each species. The sufficient criteria for stability in distribution of solutions are obtained. Finally, numerical simulations are carried out to verify the theoretical results.