Mersenne Numbers, Recursive Generation of Natural Numbers, and Counting the Number of Prime Numbers

A simple recursive algorithm to generate the set of natural numbers, based on Mersenne numbers: MN = 2N – 1, is used to count the number of prime numbers within the precise Mersenne natural number intervals: [0;MN]. This permits the formulation of an extended twin prime conjecture. Moreover, it is found that the prime numbers subsets contained in Mersenne intervals have cardinalities strongly correlated with the corresponding Mersenne numbers.

Effects of Seed Viability and Number of Generations on Genetic Integrity of Soybean Germplasm Zhonghuang 18 by AFLP Markers

The seeds of a soybean cultivar Zhonghuang 18 were subjected to accelerated aging for 0(population G_0-1), 112(population G_0-2), 154(population G_0-3) and 196 d(population G_0-4), whose germination percentage was found to be 98.0%, 95.0%, 81.0%, and 79.0%, respectively. Then, the four populations were regenerated twice in the field. The first descendant populations were marked as G_1-1, G_1-2, G_1-3 and G_1-4, and the second were marked as G_2-1, G_2-2, G_2-3 and G_2-4, respectively. The genetic variation between the control population(G_0-1) and the experimental populations was analyzed using 12 AFLP primer combinations. The results showed that there was no significant difference in genetic similarity between the 11 experimental populations and the control population G_0-1. The genetic similarity between population G_2-4 and G_0-1 was still as high as 0.933 3, indicating that the F_2 generation of the population whose germination percentage was only 79.0% still had a high genetic similarity to the control population. The results of t-tests revealed that the populations G_1-1, G_2-1, G_1-2 and G_2-2 showed no significant difference from the control population G_0-1 in effective number of alleles per locus(Ae), genetic diversity index(H) and Shannon's diversity index(I), while these indices of populations G_0-3, G_0-4, G_1-3, G_1-4, G_2-3 and G_2-4 were significantly reduced. χ~2 tests indicated that the populations G_1-1 and G_2-1 showed little difference, and the populations G_0-2, G_0-3, G_0-4, G_1-2, G_1-3, G_1-4, G_2-2, G_2-3, and G_2-4 showed great difference in allele frequency distribution from the control population G_0-1, and the difference was greater when the seed viability was lower. Compared with the control population G_0-1, the number of rare alleles(Nr) of the populations G_0-2, G_1-1, G_2-1, G_1-2 and G_2-2 showed no significant difference, while that of the populations G_0-3, G_0-4, G_1-3, G_1-4, G_2-3 and G_2-4 declined obviously. These results revealed that compared with the control population, the genetic diversity and Nr for the descendant populations of the populations with 98.0% and 95.0% germination percentages did not change significantly, but declined greatly for the descendant populations of the populations with 81.0.% and 79.0% percentages. The results suggested that the decline in seed viability has a greater impact than the number of generations on genetic structure of soybean germplasm. So, it is suggested that soybean seed with an initial germination percentage of 98.0% should be regenerated before its germination percentage declines to 81.0%.

Development of New Method for Generating Prime Numbers

The article is devoted to actual problems of prime numbers. A theorem that allows generating a sequence of prime numbers is proposed. An algorithm for generating prime numbers has been developed. A comparison of the proposed theorem, with Wilson’s theorem is also provided.

Effect of Richardson number on entropy generation over backward facing step

The flow over a backward facing step （BFS） has been taken as a useful proto- type to investigate intrinsic mechanisms of separated flow with heat transfer. However, to date, the open literature on the effect of Richardson number on entropy generation over the BFS is absent yet, although the flow pattern and heat transfer characteristic both will receive significant influence caused by the variation of Richardson number in many prac- tical applications, such as in microelectromechanical systems and aerocrafts. The effect of Richardson number on entropy generation in the BFS flow is reported in this paper for the first time. The entropy generation analysis is conducted through numerically solving the entropy generation equation. The velocity and temperature, which are the inputs of the entropy generation equation, are evaluated by the lattice Boltzmann method. It is found that the distributions of local entropy generation number and Bejan number are significantly influenced by the variation of Richardson number. The total entropy gen- eration number is a monotonic decreasing function of Richardson number, whereas the average Bejan number is a monotonic increasing function of Richardson number.

Products of Odd Numbers or Prime Number Can Generate the Three Members’ Families of Fermat Last Theorem and the Theorem Is Valid for Summation of Squares of More Than Two Natural Numbers

Fermat’s last theorem, had the statement that there are no natural numbers A, B, and C such that An + Bn = Cn, in which n is a natural number greater than 2. We have shown that any product of two odd numbers can generate Fermat or Pythagoras triple (A, B, C) following n = 2 and also it is applicable A2 + B2 + C2 + D2 + so on =An2 where all are natural numbers.

Enhanced entropy generation and heat transfer characteristics of magnetic nano-encapsulated phase change materials in latent heat thermal energy storage systems

The objective of the current study is to investigate the importance of entropy generation and thermal radiation on the patterns of velocity,isentropic lines,and temperature contours within a thermal energy storage device filled with magnetic nanoencapsulated phase change materials(NEPCMs).The versatile finite element method(FEM)is implemented to numerically solve the governing equations.The effects of various parameters,including the viscosity parameter,ranging from 1 to 3,the thermal conductivity parameter,ranging from 1 to 3,the Rayleigh parameter,ranging from 102 to 3×10^(2),the radiation number,ranging from 0.1 to 0.5,the fusion temperature,ranging from 1.0 to 1.2,the volume fraction of NEPCMs,ranging from 2%to 6%,the Stefan number,ranging from 1 to 5,the magnetic number,ranging from 0.1 to 0.5,and the irreversibility parameter,ranging from 0.1 to 0.5,are examined in detail on the temperature contours,isentropic lines,heat capacity ratio,and velocity fields.Furthermore,the heat transfer rates at both the cold and hot walls are analyzed,and the findings are presented graphically.The results indicate that the time taken by the NEPCMs to transition from solid to liquid is prolonged inside the chamber region as the fusion temperatureθf increases.Additionally,the contours of the heat capacity ratio Cr decrease with the increase in the Stefan number Ste.

Effect of Spanwise Flexibility on Propulsion Performance of a Flapping Hydrofoil at Low Reynolds Number

Spanwise flexibility is a key factor influencing propulsion performance of pectoral foils. Performances of bionic fish with oscillating pectoral foils can be enhanced by properly selecting the spanwise flexibility. The influence law of spanwise flexibility on thrust generation and propulsion efficiency of a rectangular hydro-foil is discussed. Series foils constructed by the two-component silicon rubber are developed. NACA0015 shape of chordwise cross-section is employed. The foils are strengthened by fin rays of different rigidity to realize variant spanwise rigidity and almost the same chordwise flexibility. Experiments on a towing platform developed are carried out at low Reynolds numbers of 10 000, 15 000, and 20 000 and Strouhal numbers from 0.1 to 1. The following experimental results are achieved: (1) The average forward thrust increases with the St number increased; (2) Certain degree of spanwise flexibility is beneficial to the forward thrust generation, but the thrust gap is not large for the fins of different spanwise rigidity; (3) The fin of the maximal spanwise flexibility owns the highest propulsion efficiency; (4) Effect of the Reynolds number on the propulsion efficiency is significant. The experimental results can be utilized as a reference in deciding the spanwise flexibility of bionic pectoral fins in designing of robotic fish prototype propelled by flapping-wing.

Heat transfer and entropy generation analysis of non-Newtonian fluid flow through vertical microchannel with convective boundary condition

The entropy generation and heat transfer characteristics of magnetohydrodynamic (MHD) third-grade fluid flow through a vertical porous microchannel with a convective boundary condition are analyzed. Entropy generation due to flow of MHD non-Newtonian third-grade fluid within a microchannel and temperature-dependent viscosity is studied using the entropy generation rate and Vogel's model. The equations describing flow and heat transport along with boundary conditions are first made dimensionless using proper non-dimensional transformations and then solved numerically via the finite element method (FEM). An appropriate comparison is made with the previously published results in the literature as a limiting case of the considered problem. The comparison confirms excellent agreement. The effects of the Grashof number, the Hartmann number, the Biot number, the exponential space-and thermal-dependent heat source (ESHS/THS) parameters, and the viscous dissipation parameter on the temperature and velocity are studied and presented graphically. The entropy generation and the Bejan number are also calculated. From the comprehensive parametric study, it is recognized that the production of entropy can be improved with convective heating and viscous dissipation aspects. It is also found that the ESHS aspect dominates the THS aspect.

Effects of Thermophoresis on Unsteady MHD Free Convective Heat and Mass Transfer along an Inclined Porous Plate with Heat Generation in Presence of Magnetic Field

An analysis of Thermophoresis effect on unsteady magneto-hydrodynamic free convection flow over an inclined porous plate with time dependent suction in presence of magnetic field with heat generation has been considered by employing Nachtsheim-Swigert shooting iteration technique along with sixth order Runge-Kutta integration scheme. Resulting non-dimensional velocity, temperature and concentration profiles are then presented graphically for different values of the parameters entering into the problem. Finally, the effects of the pertinent parameters on the skin-friction coefficient, the rate of heat transfer (Nusselt number) and wall deposition flux (Stanton number), which are of physical interest, are exhibited in tabular form.

Improved quantum randomness amplification with finite number of untrusted devices based on a novel extractor

Quantum randomness amplification protocols have increasingly attracted attention tbr their tantastic ability to ampllI~, weak randomness to almost ideal randomness by utilizing quantum systems. Recently, a realistic noise-tolerant randomness amplification protocol using a finite number of untrusted devices was proposed. The protocol has the composable security against non-signalling eavesdroppers and could produce a single bit of randomness from weak randomness sources, which is certified by the violation of certain Bell inequalities. However, the protocol has a non-ignorable limitation on the min- entropy of independent sources. In this paper, we further develop the randomness amplification method and present a novel quantum randomness amplification protocol based on an explicit non-malleable two independent-source randomness extractor, which could remarkably reduce the above-mentioned specific limitation. Moreover, the composable security of our improved protocol is also proposed. Our results could significantly expand the application range for practical quantum randomness amplification, and provide a new insight on the practical design method for randomness extraction.

Effect of Impeller Solidity on the Generating Performance for Solar Power Generation

According to current solar power research,both the generating unit’s minimum start-up speed and power generation system’s minimum flow rate for operation decrease with the increase in the impeller solidity.Ideally,a high solidity should be achieved,as this translates more power for a solar power system in the start-up and shut-down cycles.However,increasing the number of blades does not increase the impeller solidity;therefore,there is an optimal number of blades needed to achieve the preferred solidity.This paper begins by selecting the blade airfoil and then performs a theoretical analysis based on the relationship between the blade number and chord length.Experiments are conducted to measure the starting and stopping wind speeds and power characteristics for different numbers of blades.The results show that a maximum impeller solidity of 0.2862 is achieved,as well as the minimum flow speed at the start-up,and the maintenance of the solar chimney power generation system is optimized when there are four blades.

Radiation Effects on Free Convection MHD Couette Flow Started Exponentially with Variable Wall Temperature in Presence of Heat Generation

Radiation effects on free convection MHD Couette flow started exponentially with variable wall temperature in the presence of heat generation have been studied. The governing equations are solved analytically using the Laplace transform technique. The variations of velocity and fluid temperature are presented graphically. It is observed that the velocity decreases with an increase in either magnetic parameter or radiation parameter or Prandtl number. It is also observed that the velocity increases with an increase in either heat generation parameter or Grashof number or accelerated parameter or time. An increase in either radiation parameter or Prandtl number leads to fall in the fluid temperature. It is seen that the fluid temperature increases with an increase in either heat generation parameter or time. Further, it is seen that the shear stress at the moving plate decreases with an increase in either magnetic parameter or radiation parameter while it increases with an increase in either heat generation parameter or Prandtl number. The rate of heat transfer increases with an increase in either Prandtl number or time whereas it decreases with an increase in heat generation parameter.

Conjugate Effects of Radiation and Joule Heating on Magnetohydrodynamic Free Convection Flow along a Sphere with Heat Generation

The conjugate effects of radiation and joule heating on magnetohydrodynamic (MHD) free convection flow along a sphere with heat generation have been investigated in this paper. The governing equations are transformed into dimensionless non-similar equations by using set of suitable transformations and solved numerically by the finite difference method along with Newton’s linearization approximation. Attention has been focused on the evaluation of shear stress in terms of local skin friction and rate of heat transfer in terms of local Nusselt number, velocity as well as temperature profiles. Numerical results have been shown graphically for some selected values of parameters set consisting of heat generation parameter Q, radiation parameter Rd, magnetic parameter M, joule heating parameter J and the Prandtl number Pr.

Infinite Number of Twin Primes

This work is devoted to the theory of prime numbers. Firstly it introduced the concept of matrix primes, which can help to generate a sequence of prime numbers. Then it proposed a number of theorems, which together with theorem of Dirichlet, Siegel and Euler allow to prove the infinity of twin primes.

Effects of Viscous Dissipation on MHD Natural Convection Flow along a Vertical Wavy Surface with Heat Generation

The effect of external magnetic field and internal heat generation or absorption on a steady two-dimensional natural convection flow of viscous incompressible fluid along a uniformly heated vertical wavy surface has been investigated. The governing boundary layer equations are first transformed into a non-dimensional form using suitable set of dimensionless variables. The transformed boundary layer equations are solved numerically using the implicit finite difference method, known as Keller-box scheme. Numerical results for velocity, temperature, skin friction, the rate of heat transfer are obtained for different values of the selected parameters, such as viscous dissipation parameter (Vd), heat generation parameter (Q), magnetic parameter (M) and presented graphically and discussed. Streamlines and isotherms are presented for selected values of heat generation parameter and explained.

Solution of Inverse Problem on Distributed Generation Using Complex-Valued Network Inversion

Solutions of inverse problems are required in various fields of science and engineering. The concept of network inversion has been studied as a neural-network-based solution to inverse problems. In general, inverse problems are not limited to a real-valued area. Recently, complex-valued neural networks have been actively studied in the field of neural networks. As an extension of network inversion to complex numbers, a complex-valued network inversion has been proposed. Moreover, inverse problems for estimating the parameters of distributed generation systems such as distributed energy plants or smart grids from observed electric circuit data have been studied in the field of natural energy. These emphasize the need to handle complex numbers in an alternating current （AC） circuit. In this paper, the authors propose an application of the complex-valued network inversion to the inverse estimation of a distributed generation. Further, the authors confirm the effectiveness of the complex-valued network inversion on the basis of simulation results.

Non Degeneration of Fibonacci Series, Pascal’s Elements and Hex Series

Generally Fibonacci series and Lucas series are the same, they converge to golden ratio. After I read Fibonacci series, I thought, is there or are there any series which converges to golden ratio. Because of that I explored the inter relations of Fibonacci series when I was intent on Fibonacci series in my difference parallelogram. In which, I found there is no degeneration on Fibonacci series. In my thought, Pascal triangle seemed like a lower triangular matrix, so I tried to find the inverse for that. In inverse form, there is no change against original form of Pascal elements matrix. One day I played with ring magnets, which forms hexagonal shapes. Number of rings which forms Hexagonal shape gives Hex series. In this paper, I give the general formula for generating various types of Fibonacci series and its non-degeneration, how Pascal elements maintain its identities and which shapes formed by hex numbers by difference and matrices.

400 Gb/s physical random number generation based on deformed square self-chaotic lasers

A circular-sided square microcavity laser etched a central hole has achieved chaos operation with a bandwidth of 20.8 GHz without external optical feedback or injection,in which the intensity probability distribution of a chaotic signal with a twopeak pattern was observed.Based on the self-chaotic microlaser,physical random numbers at 400 Gb/s were generated by extracting the four least significant bits without other complex post-processing methods.The solitary chaos laser and minimal post-processing have predicted a simpler and low-cost on-chip random number generator in the future.

Estimating effective reproduction number using generation time versus serial interval,with application to COVID-19 in the Greater Toronto Area,Canada

BACKGROUND.The effective reproduction number Re(t)is a critical measure of epidemic potential.Re(t)can be calculated in near real time using an incidence time series and the generation time distribution:the time between infection events in an infector-infectee pair.In calculating Re(t),the generation time distribution is often approximated by the serial interval distribution:the time between symptom onset in an infector-infectee pair.However,while generation time must be positive by definition,serial interval can be negative if transmission can occur before symptoms,such as in COVID-19,rendering such an approximation improper in some contexts.METHODS.We developed a method to infer the generation time distribution from parametric definitions of the serial interval and incubation period distributions.We then compared estimates of Re(t)for COVID-19 in the Greater Toronto Area of Canada using:negative-permitting versus non-negative serial interval distributions,versus the inferred generation time distribution.RESULTS.We estimated the generation time of COVID-19 to be Gamma-distributed with mean 3.99 and standard deviation 2.96 days.Relative to the generation time distribution,non-negative serial interval distribution caused overestimation of Re(t)due to larger mean,while negative-permitting serial interval distribution caused underestimation of Re(t)due to larger variance.IMPLICATIONS.Approximation of the generation time distribution of COVID-19 with non-negative or negative-permitting serial interval distributions when calculating Re(t)may result in over or underestimation of transmission potential,respectively.

Random-injection-based two-channel chaos with enhanced bandwidth and suppressed time-delay signature by mutually coupled lasers: Proposal and numerical analysis

Simultaneous bandwidth(BW) enhancement and time-delay signature(TDS) suppression of chaotic lasing over a wide range of parameters by mutually coupled semiconductor lasers(MCSLs) with random optical injection are proposed and numerically investigated. The influences of system parameters on TDS suppression(characterized by autocorrelation function(ACF) and permutation entropy(PE) around characteristic time) and chaos BW are investigated. The results show that, with the increasing bias current, the ranges of parameters(detuning and injection strength) for the larger BW(> 20 GHz) are broadened considerably, while the parameter range for optimized TDS(< 0.1) is not shrunk obviously.Under optimized parameters, the system can simultaneously achieve two chaos outputs with enhanced BW(> 20 GHz)and perfect TDS suppression. In addition, the system can generate two-channel high-speed truly physical random number sequences at 200 Gbits/s for each channel.