Corresponding Prime Number Distribution Equation

The conjecture of twin prime numbers is a mathematical problem. Proving the twin prime conjecture using traditional modern number theory is extremely profound and complex. We propose an elementary research method for corresponding prime number, proved that the conjecture of twin prime numbers and obtain the corresponding prime distribution equation. According to the distribution rate of corresponding prime numbers, the distribution pattern of twin prime numbers was proved the distribution rate theorem. This is the distribution rate of prime numbers corresponding to composite numbers, which approaches the distribution rate of prime numbers corresponding to integers. Based on the corresponding prime distribution equation, obtain the twin prime inequality function. Then, the formula for calculating twin prime numbers was discussed. There is also the Hardy Littlewood conjecture. This provides a practical and feasible approach for studying the distribution of twin prime numbers.

An Algorithm for Generating Random Numbers with Normal Distribution

A new algorithm is suggested based on the central limit theorem for generating pseudo-random numbers with a specified normal or Gaussian probability density function. The suggested algorithm is very simple but highly accurate, with an efficiency that falls between those of the Box-Muller and von Neumann rejection methods.

Number concentration and size distributions of submicron particles in Jinan urban area:Characteristics in summer and winter

The aerosol number concentration and size distribution were measured with the newly developed Wide-range Particle Spectrometer in summer and winter of 2006 at the urban site of Jinan City. Here reported the characteristics of fine particles of the different observation seasons. Relative high number concentrations for the particles in the diameter range of 10-500 nm were observed in both seasons. It was found that the dominant number distributed in particle diameter smaller than 100 nm and the percentage over the number concentration of all air particles is much higher than what has been measured in other urban sites over the world. The number mean diameter in summer was much smaller than in winter, strongly suggesting the different origin of ultrafine particles in different seasons. That is, particles in ultrafine mode mainly came from nucleation and new particle formation in summer while from traffic emission in winter. The diurnal variation also supported this point. Number concentration in the diameter range of 10-200 um got their peak values at noontime, well correlated with the mixing ratio of SO2 and the intensity of solar radiation in summer. While in winter, those in the same diameter range showed the main peaks during the traffic hours happened in the morning and evening.

Seasonal Variations of Number Size Distributions and Mass Concentrations of Atmospheric Particles in Beijing

Particle number and mass concentrations were measured in Beijing during the winter and summer periods in 2003, together with some other parameters including black carbon (BC) and meteorological conditions. Particle mass concentrations exhibited low seasonality, and the ratio of PM2.5/PM10 in winter was higher than that in summer. Particle number size distribution (PSD) was characterized by four modes and exhibited low seasonality. BC was well correlated with the number and mass concentrations of accumulation and coarse particles, indicating these size particles are related to anthropogenic activities. Particle mass and number concentrations (except ultra-fine and nucleation particles) followed well the trends of BC concentration for the majority of the day, indicating that most particles were associated with primary emissions. The diurnal number distributions of accumulation and coarse mode particles were characterized by two peaks.

Effects of atomic number Z on the energy distribution of hot electrons generated by femtosecond laser interaction with metallic targets

The effects of atomic number Z on the energy distribution of hot electrons generated by the interaction of 60fs, 130mJ, 800nm, and 7×10^17W/cm^2 laser pulses with metallic targets have been studied experimentally. The results show that the number and the effective temperature of hot electrons increase with the atomic number Z of metallic targets, and the temperature of hot electrons are in the range of 190-230keV, which is consistent with a scaling law of hot electrons temperature.

Evaluation of the WDM6 scheme in the simulation of number concentrations and drop size distributions of warm-rain hydrometeors: comparisons with the observations and other schemes

The number concentrations and drop size distributions(DSDs)of warm-rain hydrometeors play an important role in the simulation of microphysical processes.To evaluate the performance of the WDM6 scheme,which predicts the cloud number concentration(Nc)explicitly in aspects of warm-rain hydrometeors number concentrations and DSDs,the simulation of the WDM6 scheme is compared with airborne observations of a flight trial,as well as with the simulations of the Thompson scheme and Morrison scheme.Results show that the WDM6 scheme produces smaller(larger)cloud(rain)number concentrations and wider cloud DSDs compared to the observations,with the largest biases at upper levels of stratiform cloud(SC).The Thompson scheme and the Morrison scheme,both of which set the Nc as a constant,compare better to the observations than the WDM6 scheme in aspects of Nc and DSD.Sensitivity tests of the initial cloud condensation nuclei(CCN)number concentration(CCN0)of the WDM6 scheme show that a better choice of the initial CCN0 may improve the simulation of convective cloud but helps little in terms of SC.The simulation of rain number concentration and DSD is not sensitive to the CCN0 in the WDM6 scheme.

On Prime Numbers between kn and (k + 1) n

In this paper along with the previous studies on analyzing the binomial coefficients, we will complete the proof of a theorem. The theorem states that for two positive integers n and k, when n ≥ k - 1, there always exists at least a prime number p such that kn p ≤ (k +1)n. The Bertrand-Chebyshev’s theorem is a special case of this theorem when k = 1. In the field of prime number distribution, just as the prime number theorem provides the approximate number of prime numbers relative to natural numbers, while the new theory indicates that prime numbers exist in the specific intervals between natural numbers, that is, the new theorem provides the approximate positions of prime numbers among natural numbers.

The effects of cold region meteorology and specific environment on the number of hospital admissions for chronic kidney disease:An investigate with a distributed lag nonlinear model

Objective:To explore the effects of daily mean temperature(°C),average daily air pressure(hPa),humidity(%),wind speed(m/s),particulate matter(PM)2.5(μg/m3)and PM10(μg/m3)on the admission rate of chronic kidney disease(CKD)patients admitted to the Second Affiliated Hospital of Harbin Medical University in Harbin and to identify the indexes and lag days that impose the most critical influence.Methods:The R language Distributed Lag Nonlinear Model(DLNM),Excel,and SPSS were used to analyze the disease and meteorological data of Harbin from 01 January 2010 to 31 December 2019 according to the inclusion and exclusion criteria.Results:Meteorological factors and air pollution influence the number of hospitalizations of CKD to vary degrees in cold regions,and differ in persistence or delay.Non-optimal temperature increases the risk of admission of CKD,high temperature increases the risk of obstructive kidney disease,and low temperature increases the risk of other major types of chronic kidney disease.The greater the temperature difference is,the higher its contribution is to the risk.The non-optimal wind speed and non-optimal atmospheric pressure are associated with increased hospital admissions.PM2.5 concentrations above 40μg/m3 have a negative impact on the results.Conclusion:Cold region meteorology and specific environment do have an impact on the number of hospital admissions for chronic kidney disease,and we can apply DLMN to describe the analysis.

MEASUREMENTS OF PARTICLE NUMBER SIZE DISTRIBUTIONS AND NEW PARTICLE FORMATION EVENTS DURING WINTER IN THE PEARL RIVER DELTA REGION,CHINA

Particle number size distribution(PNSD) between 10 nm and 20 μm were measured in the Pearl River Delta(PRD) region in winter 2011.The average particle number concentration of the nucleation mode(10-20 nm),Aitken mode(20-100 nm),accumulation mode(100 nm-1μm) and coarse mode(1-20 μm) particles were 1 552,7 470,4 012,and 19 cm-3,respectively.The volume concentration of accumulation mode particles with peak at 300 nm accounted for over 70% of the total volume concentration.Diurnal variations and dependencies on meteorological parameters of PNSD were investigated.The diurnal variation of nucleation mode particles was mainly influenced by new particle formation events,while the diurnal variation of Aitken mode particles correlated to the traffic emission and the growth process of nucleation mode particles.When the PRD region was controlled by a cold high pressure,conditions of low relative humidity,high wind speed and strong radiation are favorable for the occurrence of new particle formation(NPF) events.The frequency of occurrence of NPF events was 21.3% during the whole measurement period.Parameters describing NPF events,including growth rate(GR) and source rate of condensable vapor(Q),were slightly larger than those in previous literature.This suggests that intense photochemical and biological activities may be the source of condensable vapor for particle growth,even during winter in the PRD.

Quantum phase distribution and the number-phase Wigner function of the generalized squeezed vacuum states associated with solvable quantum systems

The quantum phase properties of the generalized squeezed vacuum states associated with solvable quantum systems are studied by using the Pegg-Barnett formalism.Then,two nonclassical features,i.e.,squeezing in the number and phase operators,as well as the number-phase Wigner function of the generalized squeezed states are investigated.Due to some actual physical situations,the present approach is applied to two classes of generalized squeezed states：solvable quantum systems with discrete spectra and nonlinear squeezed states with particular nonlinear functions.Finally,the time evolution of the nonclassical properties of the considered systems has been numerically investigated.

Asymptotic Harmonic Behavior in the Prime Number Distribution

We consider on x > 0, where the sum is over all primes p. If Φ is bounded on x > 0, then the Riemann hypothesis is true or there are infinitely many zeros . The first 21 zeros give rise to asymptotic harmonic behavior in Φ(x) defined by the prime numbers up to one trillion.

Influence of air masses on particle number concentration and size distribution at Mt. Waliguan, Qinghai Province, China

Particle size distribution of 12-500 nm was measured at Mt. Waliguan, China Global Atmosphere Watch Baseline Observatory, from Aug. in 2005 to May in 2007.72-hr back-trajectories at 100-m arrival height above ground level for the same period were calculated at 6：00, 12：00, and 21：00 （Beijing Time） for each day using the Hybrid Single-Particle Lagrangian Integrated Trajectory （HYSPLIT-4） model developed by NOAA/ARL. It was found that air mass sources significantly impact particle number concentration and size distribution at Mt. Waliguan. Cluster analysis of back-trajectories show that higher Aitken mode particle number concentration was observed when air masses came from or passed by the northeastern section of Mt. Waliguan, with short trajectory length. High number concentration of nucleation mode was associated with air masses from clean regions, with long trajectory length.

Detecting the possibility of a type of photon number splitting attack in decoy-state quantum key distribution

The existing decoy-state quantum key distribution(QKD)beating photon-number-splitting(PNS)attack provides a more accurate method to estimate the secure key rate,while it still considers that only single-photon pulses can generate secure keys in any case.However,multiphoton pulses can also generate secure keys if we can detect the possibility of PNS attack in the channel.The ultimate goal of this line of research is to confirm the absence of all types of PNS attacks.In particular,the PNS attack mentioned and detected in this paper is only the weaker version of PNS attack which significantly changes the observed values of the legitimate users.In this paper,under the null hypothesis of no weaker version of PNS attack,we first determine whether there is an attack or not by retrieving the missing information of the existing decoy-state protocols,extract a Cauchy distribution statistic,and further provide a detection method and the type I error probability.If the result is judged to be an attack,we can use the existing decoy-state method and the GLLP formula to estimate the secure key rate.Otherwise,the pulses with the same basis received including both single-photon pulses and multiphoton pulses,can be used to generate the keys and we give the secure key rate in this case.Finally,the associated experiments we performed(i.e.,the significance level is 5%)show the correctness of our method.

On the Distribution Property of LehmerDH Number

This paper is to study the distribution property of Lehmer DH number by use the estimation of general Kloostermann sum and the estimation of trigonometric sum.

Distribution and number change of the Littorinid (Gastropoda) on intertidal hard bed in the Jiulong Estuary, Fujian Province

-Among four species of Littorinid, Littorina scabra, L. brevicula, Nodiliitorina millegrana and N. pyramidalis pyramidalis, on the hard intertidal zone in the Jiulong Estuary, Fujian, L. scabra has the most extensive distribution both vertically and horizontally, and the highest anual average density. The uppermost limit of vertical distribution of N. pyramidalis pyramidalis is higher than the other three Littorinid. Desiccation is a main factor affecting the uppermost limit of vertical distribution of the Littorinid. The increasing wave action will raise the uppermost limit vertical distribution of Littorinid. With seasonal changes, the densities of the Littorinid also change in different tidal zonations. Difference in salinity is the main factor affecting the horizontal distribution of the Littorinid in the estuary.

Understanding the Variability of Z-R Relationships Caused by Natural Variations in Raindrop Size Distributions(DSD):Implication of Drop Size and Number

In the issue of rainfall estimation by radar through the necessary relationship between radar reflectivity Z and rain rate R (Z-R), the main limitation is attributed to the variability of this relationship. Indeed, several pre-vious studies have shown the great variability of this relationship in space and time, from a rainfall event to another and even within a single rainfall event. Recent studies have shown that the variability of raindrop size distributions and thereby Z-R relationships is therefore, more the result of complex dynamic, thermody-namic and microphysical processes within rainfall systems than a convective/stratiform classification of the ground rainfall signature. The raindrop number and size at ground being the resultant of various processes mentioned above, a suitable approach would be to analyze their variability in relation to that of Z-R relation-ship. In this study, we investigated the total raindrop concentration number NT and the median volume di-ameter D0 used in numerous studies, and have shown that the combination of these two ‘observed’ parame-ters appears to be an interesting approach to better understand the variability of the Z-R relationships in the rainfall events, without assuming a certain analytical raindrop size distribution model (exponential, gamma, or log-normal). The present study is based on the analysis of disdrometer data collected at different seasons and places in Africa, and aims to show the degree of the raindrop size and number implication in regard to the Z-R relationships variability.

Distribution and number change of the Littorinidae (Gastropoda) on intertidal hard bed in the Jiulong River Estuary

There are 4 species of Littorinidae on the intertidal hard bed in the Jiulong River Estu-ary. They are Littorina scabra, L. brevicula, Nodilittorina pyramidalis pyramidalis and N.millegrana. Of these species, L. scabra is a dominant species. According to the results of in-vestigation, we find distribution patterns of the Littorinidae on intertidal hard bed in theestuary as follows: (1)There is no Littorinid species at Haicheng Station. (2)There is only Lit-

Oscillation behaviour in the photon-number distribution of squeezed coherent states

From the normally ordered form of the density operator of a squeezed coherent state（SCS）,we directly derive the compact expression of the SCS＇s photon-number distribution（PND）.Besides the known oscillation characteristics,we find that the PND is a periodic function with a period of π and extremely sensitive to phase.If the squeezing is strong enough,and the compound phase which is relevant to the complex squeezing and displacement parameters are assigned appropriate values,different oscillation behaviours in PND for even and odd photon numbers appear,respectively.

Passive decoy state SARG04 quantum-key-distribution with practical photon-number resolving detectors

SARG04 protocol has its advantages in defending photon number splitting attack, benefited from two-photon pulses part. In this paper, we present a passive decoy state SARG04 scheme combining with practical photon number resolving （PNR） detectors. Two kinds of practical detectors, transition-edge sensor and time-multiplexing detector, are taken into consideration. Theoretical analysis shows that both of them are compatible with the passive decoy state SARG04. Compared with the original SARG04, two detectors can boost the key generation rate and maximal secure distance obviously. Meanwhile, the result shows that quantum efficiency and dark count of the detector influence the maximal distance slightly, which indicates the prospect of implementation in real quantum key distribution system with imperfect practical PNS detectors.

Photon-number distribution of two-mode squeezed thermal states by entangled state representation

Using the entangled state representation, we convert a two-mode squeezed number state to a Hermite polynomial excited squeezed vacuum state. We first analytically derive the photon number distribution of the two-mode squeezed thermal states. It is found that it is a Jacobi polynomial; a remarkable result. This result can be directly applied to obtaining the photon number distribution of non-Gaussian states generated by subtracting from （adding to） two-mode squeezed thermal states.