Origin of Sexy Prime Numbers, Origin of Cousin Prime Numbers, Equations from Supposedly Prime Numbers, Origin of the Mersenne Number, Origin of the Fermat Number

We have found through calculations that the differences between the closest supposed prime numbers other than 2 and 3 defined in the articles are: 2;4: and 6. For those whose difference is equal to 6, we showed their origin then we classified them into two categories according to their classes, we showed in which context two prime numbers which differ from 6 are called sexy and in what context they are said real sexy prime. For those whose difference is equal to 4, we showed their origin then we showed that two prime numbers which differ from 4, that is to say two cousin prime numbers, are successive. We made an observation on the supposed prime numbers then we established two pairs of equations from this observation and deduced the origin of the Mersenne number and that of the Fermat number.

Preliminary Identification of a Prime Number Other Than 2 and 3, the Origin of Twin Prime Numbers, the Structure of the Chain of Prime Numbers and the Set of Prime Numbers Less Than a Given Integer

The application of the Euclidean division theorem for the positive integers allowed us to establish a set which contains all the prime numbers and this set we called it set of supposedly prime numbers and we noted it Esp. We subsequently established from the previous set the set of non-prime numbers (the set of numbers belonging to this set and which are not prime) denoted Enp. We then extracted from the set of supposedly prime numbers the numbers which are not prime and the set of remaining number constitutes the set of prime numbers denoted Ep. We have deduced from the previous set, the set of prime numbers between two natural numbers. We have explained during our demonstrations the origin of the twin prime numbers and the structure of the chain of prime numbers.

On the Number of Primes in the Interval (x, 2x) by an Elementary Method

An elementary formula to know the number of primes in the interval (x, 2x) close to the exact figure for a fixed x is given here. A new elementary equation is derived (a relation between prime numbers and composite numbers distributed in the interval [1, 2x]). An elementary method to know the number of primes in a given magnitude is suitably placed in the form of a general formula, and we have proved it. The general formula is applied to the terms of the equation, and a tactical simplification of the terms gives rise to an expression whose verification envisages scope for its further studies.

The wheat sucrose synthase gene TaSus1 is a determinant of grain number per spike

Some haplotypes of the sucrose synthase gene TaSus1 are associated with thousand-grain weight(TGW)in wheat(Triticum aestivum L.).However,no mutations have been identified within the gene to test this association.The effects of TaSus1 on grain number per spike(GNS)also are largely unknown.Our previous genome-wide association study identified TaSus-A1 as a candidate gene controlling fertile spikelet number per spike(FSN).In the present study,we generated two independent mutants for the three TaSus1 homoeologs by CRISPR/Cas9-mediated genome editing.The triple mutants displayed lower FSN,GNS,grain number per spikelet(GNST),and TGW than wild-type plants.In 306 hexaploid wheat accessions,two single-nucleotide polymorphisms in TaSus-A1 contributed differently to GNS.Introgression of the two alleles into a wheat genetic background confirmed their effects.The alleles differed in geographical distribution among the accessions.

The Immense Impact of Reverse Edges on Large Hierarchical Networks

Hierarchical networks are frequently encountered in animal groups,gene networks,and artificial engineering systems such as multiple robots,unmanned vehicle systems,smart grids,wind farm networks,and so forth.The structure of a large directed hierarchical network is often strongly influenced by reverse edges from lower-to higher-level nodes,such as lagging birds’howl in a flock or the opinions of lowerlevel individuals feeding back to higher-level ones in a social group.This study reveals that,for most large-scale real hierarchical networks,the majority of the reverse edges do not affect the synchronization process of the entire network;the synchronization process is influenced only by a small part of these reverse edges along specific paths.More surprisingly,a single effective reverse edge can slow down the synchronization of a huge hierarchical network by over 60%.The effect of such edges depends not on the network size but only on the average in-degree of the involved subnetwork.The overwhelming majority of active reverse edges turn out to have some kind of“bunching”effect on the information flows of hierarchical networks,which slows down synchronization processes.This finding refines the current understanding of the role of reverse edges in many natural,social,and engineering hierarchical networks,which might be beneficial for precisely tuning the synchronization rhythms of these networks.Our study also proposes an effective way to attack a hierarchical network by adding a malicious reverse edge to it and provides some guidance for protecting a network by screening out the specific small proportion of vulnerable nodes.

Study of baryon number transport dynamics and strangeness conservation effects using Ω-hadron correlations

In nuclear collisions at RHIC energies, an excess of Ω hyperons over ■ is observed, indicating that Ω has a net baryon number despite s and s quarks being produced in pairs. The baryon number in Ω may have been transported from the incident nuclei and/or produced in the baryon-pair production of Ω with other types of anti-hyperons such as Ξ. To investigate these two scenarios, we propose to measure the correlations between Ω and K and between Ω and anti-hyperons. We use two versions, the default and string-melting, of a multiphase transport(AMPT) model to illustrate the method for measuring the correlation and to demonstrate the general shape of the correlation. We present the Ω-hadron correlations from simulated Au+Au collisions at ■ =7.7 and 14.6 Ge V and discuss the dependence on the collision energy and on the hadronization scheme in these two AMPT versions. These correlations can be used to explore the mechanism of baryon number transport and the effects of baryon number and strangeness conservation on nuclear collisions.

Impacts of Aggregation Methods and Trophospecies Number on the Structure and Function of Marine Food Webs

Aggregation of species with similar ecological properties is one of the effective methods to simplify food web researches.However,species aggregation will affect not only the complexity of modeling process but also the accuracy of models’outputs.Selection of aggregation methods and the number of trophospecies are the keys to study the simplification of food web.In this study,three aggregation methods,including taxonomic aggregation(TA),structural equivalence aggregation(SEA),and self-organizing maps(SOM),were analyzed and compared with the linear inverse model–Markov Chain Monte Carlo(LIM-MCMC)model.Impacts of aggregation methods and trophospecies number on food webs were evaluated based on the robustness and unitless of ecological net-work indices.Results showed that aggregation method of SEA performed better than the other two methods in estimating food web structure and function indices.The effects of aggregation methods were driven by the differences in species aggregation principles,which will alter food web structure and function through the redistribution of energy flow.According to the results of mean absolute percentage error(MAPE)which can be applied to evaluate the accuracy of the model,we found that MAPE in food web indices will increase with the reducing trophospecies number,and MAPE in food web function indices were smaller and more stable than those in food web structure indices.Therefore,trade-off between simplifying food webs and reflecting the status of ecosystem should be con-sidered in food web studies.These findings highlight the importance of aggregation methods and trophospecies number in the analy-sis of food web simplification.This study provided a framework to explore the extent to which food web models are affected by dif-ferent species aggregation,and will provide scientific basis for the construction of food webs.

Effect of Sweptback Angle of a Delta Wing on Surface Pressure Distribution at Supersonic Mach Numbers

The purpose of this work is to shed light on the effect of the pivot position on the surface pressure distribution over a 3D wing in different flight conditions.The study is intended to support the design and development of aerospace vehicles where stability analysis,performance optimization,and aircraft design are of primary importance.The following parameters are considered:Mach numbers(M)of 1.3,1.8,2.3,2.8,3.3,and 3.8,angle of incidence(θ)in the range from 5°to 25°,pivot position from h=0.2 to 1.The results of the CFD numerical simulations match available analytical data,thereby providing evidence for the reliability of the used approach.The findings provide valuable insights into the relationship between the surface pressure distribution,the Mach number and the angle of incidence.

Analysis of pseudo-random number generators in QMC-SSE method

In the quantum Monte Carlo(QMC)method,the pseudo-random number generator(PRNG)plays a crucial role in determining the computation time.However,the hidden structure of the PRNG may lead to serious issues such as the breakdown of the Markov process.Here,we systematically analyze the performance of different PRNGs on the widely used QMC method known as the stochastic series expansion(SSE)algorithm.To quantitatively compare them,we introduce a quantity called QMC efficiency that can effectively reflect the efficiency of the algorithms.After testing several representative observables of the Heisenberg model in one and two dimensions,we recommend the linear congruential generator as the best choice of PRNG.Our work not only helps improve the performance of the SSE method but also sheds light on the other Markov-chain-based numerical algorithms.

Discovery of controllable high Chern number quantum anomalous Hall state in tetragonal lattice FeSIn

Quantum anomalous Hall(QAH) insulators have excellent properties driven by fancy topological physics, but their practical application is greatly hindered by the observed temperature of liquid nitrogen, and the QAH insulator with high Chern number is conducive to spintronic devices with lower energy consumption. Here, we find that monolayer Fe SIn is a good candidate for realizing the QAH phase;it exhibits a high magnetic transition temperature of 221 K and tunable C = ±2 with respect to magnetization orientation in the y–z plane. After the application of biaxial strain, the magnetic axis shifts from the x–y plane to the z direction, and the effect of the high C and ferromagnetic ground state on the stress is robust. Also, the effect of correlation U on C has been examined. These properties are rooted in the large size of the Fe atom that contributes to ferromagnetic kinetic exchange with neighboring Fe atoms. These findings demonstrate monolayer Fe SIn to be a major template for probing novel QAH devices at higher temperatures.

Distribution and Relation of Primes with Tesla Numbers 3, 6, and 9

This work presents a different approach to twin primes, an approach from the perspective of the Tesla numbers and gives a refresh and new observation of twin primes that could lead us to an answer to the Twin Prime Conjecture problem. We expose a peculiar relation between twin primes and the generation of prime numbers with Tesla numbers. Tesla numbers seem to be present in so many domains like time, vibration and frequency [1], and the space between twin primes is not the exception. Let us say that twin primes are more than just prime numbers plus 2 or minus 2, and Tesla numbers are more involved with twin primes than we think, and hopefully, this approach give us a better understanding of the distribution of the twin pairs.

A New Proof for Congruent Number’s Problem via Pythagorician Divisors

Considering Pythagorician divisors theory which leads to a new parameterization, for Pythagorician triplets ( a,b,c )∈ ℕ 3∗ , we give a new proof of the well-known problem of these particular squareless numbers n∈ ℕ ∗ , called congruent numbers, characterized by the fact that there exists a right-angled triangle with rational sides: ( A α ) 2 + ( B β ) 2 = ( C γ ) 2 , such that its area Δ= 1 2 A α B β =n;or in an equivalent way, to that of the existence of numbers U 2 , V 2 , W 2 ∈ ℚ 2∗ that are in an arithmetic progression of reason n;Problem equivalent to the existence of: ( a,b,c )∈ ℕ 3∗ prime in pairs, and f∈ ℕ ∗ , such that: ( a−b 2f ) 2 , ( c 2f ) 2 , ( a+b 2f ) 2 are in an arithmetic progression of reason n;And this problem is also equivalent to that of the existence of a non-trivial primitive integer right-angled triangle: a 2 + b 2 = c 2 , such that its area Δ= 1 2 ab=n f 2 , where f∈ ℕ ∗ , and this last equation can be written as follows, when using Pythagorician divisors: (1) Δ= 1 2 ab= 2 S−1 d e ¯ ( d+ 2 S−1 e ¯ )( d+ 2 S e ¯ )=n f 2;Where ( d, e ¯ )∈ ( 2ℕ+1 ) 2 such that gcd( d, e ¯ )=1 and S∈ ℕ ∗ , where 2 S−1 , d, e ¯ , d+ 2 S−1 e ¯ , d+ 2 S e ¯ , are pairwise prime quantities (these parameters are coming from Pythagorician divisors). When n=1 , it is the case of the famous impossible problem of the integer right-angled triangle area to be a square, solved by Fermat at his time, by his famous method of infinite descent. We propose in this article a new direct proof for the numbers n=1 (resp. n=2 ) to be non-congruent numbers, based on an particular induction method of resolution of Equation (1) (note that this method is efficient too for general case of prime numbers n=p≡a ( ( mod8 ) , gcd( a,8 )=1 ). To prove it, we use a classical proof by induction on k , that shows the non-solvability property of any of the following systems ( t=0 , corresponding to case n=1 (resp. t=1 , corresponding to case n=2 )): ( Ξ t,k ){ X 2 + 2 t ( 2 k Y ) 2 = Z 2 X 2 + 2 t+1 ( 2 k Y ) 2 = T 2 , where k∈ℕ;and solutions ( X,Y,Z,T )=( D k , E k , f k , f ′ k )∈ ( 2ℕ+1 ) 4 , are given in pairwise prime numbers.2020-Mathematics Subject Classification 11A05-11A07-11A41-11A51-11D09-11D25-11D41-11D72-11D79-11E25 .

A Number Theoretic Analysis of the Enthalpy, Enthalpy Energy Density, Thermodynamic Volume, and the Equation of State of a Modified White Hole, and the Implications to the Quantum Vacuum Spacetime, Matter Creation and the Planck Frequency

In this paper, we analyze the enthalpy, enthalpy energy density, thermodynamic volume, and the equation of state of a modified white hole. We obtain new possible mathematical connections with some sectors of Number Theory, Ramanujan Recurring Numbers, DN Constant and String Theory, that enable us to extract the quantum geometrical properties of these thermodynamic equations and the implication to the quantum vacuum spacetime geometry of our early universe as they act as the constraints to the nature of quantum gravity of the universe.

Flow and sound fields of scaled high-speed trains with different coach numbers running in long tunnel

Segregated incompressible large eddy simulation and acoustic perturbation equations were used to obtain the flow field and sound field of 1:25 scale trains with three,six and eight coaches in a long tunnel,and the aerodynamic results were verified by wind tunnel test with the same scale two-coach train model.Time-averaged drag coefficients of the head coach of three trains are similar,but at the tail coach of the multi-group trains it is much larger than that of the three-coach train.The eight-coach train presents the largest increment from the head coach to the tail coach in the standard deviation(STD)of aerodynamic force coefficients:0.0110 for drag coefficient(Cd),0.0198 for lift coefficient(Cl)and 0.0371 for side coef-ficient(Cs).Total sound pressure level at the bottom of multi-group trains presents a significant streamwise increase,which is different from the three-coach train.Tunnel walls affect the acoustic distribution at the bottom,only after the coach number reaches a certain value,and the streamwise increase in the sound pressure fluctuation of multi-group trains is strengthened by coach number.Fourier transform of the turbulent and sound pressures presents that coach number has little influence on the peak frequencies,but increases the sound pressure level values at the tail bogie cavities.Furthermore,different from the turbulent pressure,the first two sound pressure proper orthogonal decomposition(POD)modes in the bogie cavities contain 90%of the total energy,and the spatial distributions indicate that the acoustic distributions in the head and tail bogies are not related to coach number.

Binding Number and Fractional k-Factors of Graphs

In this paper, we consider the relationship between the binding number and the existence of fractional k-factors of graphs. The binding number of G is defined by Woodall as bind(G)=min{ | NG(X) || X |:∅≠X⊆V(G) }. It is proved that a graph G has a fractional 1-factor if bind(G)≥1and has a fractional k-factor if bind(G)≥k−1k. Furthermore, it is showed that both results are best possible in some sense.

A New Proof on the Bipartite Turán Number of Bipartite Graphs

The bipartite Turán number of a graph H, denoted by ex(m,n;H), is the maximum number of edges in any bipartite graph G=(A,B;E(G))with | A |=mand | B |=nwhich does not contain H as a subgraph. Whenmin{ m,n }>2t, the problem of determining the value of ex(m,n;Km−t,n−t)has been solved by Balbuena et al. in 2007, whose proof focuses on the structural analysis of bipartite graphs. In this paper, we provide a new proof on the value of ex(m,n;Km−t,n−t)by virtue of algebra method with the tool of adjacency matrices of bipartite graphs, which is inspired by the method using { 0,1 }-matrices due to Zarankiewicz [Problem P 101. Colloquium Mathematicum, 2(1951), 301].

Proof of Concept for a Twin Prime Number Universe Using Set Theory

The Fine Structure Constant (eFSC) Model attempts to give a classical definition to a magical number that underlies much of quantum physics. The Fine Structure Constant (α) value equal to 137.03599206 represents a dimensionless constant that characterizes the strength of the electromagnetic (EM) interaction between subatomic charged particles. Python-generated property counts for the twin prime force F{139/137} show that the adjusted ratio gives a value of α = 137.036. This implies a mathematical framework underlying this constant is based on twin prime numbers and set theory. This study attempts to demonstrate a proof of concept that a hierarchy of fractional twin prime (αII) forces replicates the quantum nature of the universe and is aligned with the Standard Model of Particle Physics. An expanded eFSC Model demonstrates that twin prime forces and their property sets are mathematically viable substitutes for nuclear reactions, as demonstrated for the Beta-minus decay of neutrons into protons. Most significantly, the positive and negative prime numbers define these nuclear reactants and products as positive or negatively charged ions. Furthermore, the eFSC Model provides new insights regarding the hierarchy of EM forces underlying the quantum nature of the universe.

Dynamic Analysis of Customer Demand Based on Intuitionistic Fuzzy Number in Product Planning

To address the fuzziness and variability in determining customer demand importance,a dynamic analysis method based on intuitionistic fuzzy numbers is proposed.First,selected customers use intuitionistic fuzzy numbers to represent the importance of each demand.Then,the preference information is aggregated using customer weights and time period weights through the intuitionistic fuzzy ordered weighted average operator,yielding a dynamic vector of the subjective importance of the demand index.Finally,the feasibility of the proposed method is demonstrated through an application example of a vibrating sorting screen.

Relationship Between Gene-Phenotype and Clinical Manifestations of Chromosomal Copy Number Variations Indicated by Non-Invasive Prenatal Testing

Objective:To analyze the clinical value of non-invasive prenatal testing(NIPT)in detecting chromosomal copy number variations(CNVs)and to explore the relationship between gene expression and clinical manifestations of chromosomal copy number variations.Methods:3551 naturally conceived singleton pregnant women who underwent NIPT were included in this study.The NIPT revealed abnormalities other than sex chromosome abnormalities and trisomy 13,18,and 21.Pregnant women with chromosome copy number variations underwent genetic counseling and prenatal ultrasound examination.Interventional prenatal diagnosis and chromosome microarray analysis(CMA)were performed.The clinical phenotypes and pregnancy outcomes of different prenatal diagnoses were analyzed.Additionally,a follow-up was conducted by telephone to track fetal development after birth,at six months,and one year post-birth.Results:A total of 53 cases among 3551 cases showed chromosomal copy number variation.Interventional prenatal diagnosis was performed in 36 cases:27 cases were negative and 8 were consistent with the NIPT test results.This indicates that NIPT’s positive predictive value(PPV)in CNVs is 22.22%.Conclusion:NIPT has certain clinical significance in screening chromosome copy number variations and is expected to become a routine screening for chromosomal microdeletions and microduplications.However,further interventional prenatal diagnosis is still needed to identify fetal CNVs.

Maximum Number of Edges of (rK_t)-Free Graphs

With positive integers r,t and n,where n≥rt and t≥2,the maximum number of edges of a simple graph of order n is estimated,which does not contain r disjoint copies of K_r for r=2 and 3.