Clique-transversal number of graphs whose clique-graphs are trees

Given a graph G, a subgraph C is called a clique of G if C is a complete subgraph of G maximal under inclusion and ｜C｜ ≥2. A clique-transversal set S of G is a set of vertices of G such that S meets all cliques of G. The clique-transversal number, denoted as τC（G）, is the minimum cardinality of a clique-transversal set in G. The clique-graph of G, denoted as K（G）, is the graph obtained by taking the cliques of G as vertices, and two vertices are adjacent if and only if the corresponding cliques in G have nonempty intersection. Let F be a class of graphs G such that F = {G｜ K（G） is a tree}. In this paper the graphs in F having independent clique-transversal sets are shown and thus τC（G）/｜G｜ ≤ 1/2 for all G ∈F.

Bounds on the clique-transversal number of regular graphs

A clique-transversal set D of a graph G is a set of vertices of G such that D meets all cliques of G. The clique-transversal number, denoted τ c (G), is the minimum cardinality of a clique-transversal set in G. In this paper we present the bounds on the clique-transversal number for regular graphs and characterize the extremal graphs achieving the lower bound. Also, we give the sharp bounds on the clique-transversal number for claw-free cubic graphs and we characterize the extremal graphs achieving the lower bound.

Genome-wide investigation to assess copy number variants in the Italian local chicken population

Background Copy number variants(CNV)hold significant functional and evolutionary importance.Numerous ongoing CNV studies aim to elucidate the etiology of human diseases and gain insights into the population structure of livestock.High-density chips have enabled the detection of CNV with increased resolution,leading to the identification of even small CNV.This study aimed to identify CNV in local Italian chicken breeds and investigate their distribution across the genome.Results Copy number variants were mainly distributed across the first six chromosomes and primarily associated with loss type CNV.The majority of CNV in the investigated breeds were of types 0 and 1,and the minimum length of CNV was significantly larger than that reported in previous studies.Interestingly,a high proportion of the length of chromosome 16 was covered by copy number variation regions(CNVR),with the major histocompatibility complex being the likely cause.Among the genes identified within CNVR,only those present in at least five animals across breeds(n=95)were discussed to reduce the focus on redundant CNV.Some of these genes have been associated to functional traits in chickens.Notably,several CNVR on different chromosomes harbor genes related to muscle development,tissue-specific biological processes,heat stress resistance,and immune response.Quantitative trait loci(QTL)were also analyzed to investigate potential overlapping with the identified CNVR:54 out of the 95 gene-containing regions overlapped with 428 QTL associated to body weight and size,carcass characteristics,egg production,egg components,fat deposition,and feed intake.Conclusions The genomic phenomena reported in this study that can cause changes in the distribution of CNV within the genome over time and the comparison of these differences in CNVR of the local chicken breeds could help in preserving these genetic resources.

The first factor affecting dryland winter wheat grain yield under various mulching measures: Spike number

Water is the key factor limiting dryland wheat grain yield.Mulching affects crop yield and yield components by affecting soil moisture.Further research is needed to determine the relationships between yield components and soil moisture with yield,and to identify the most important factor affecting grain yield under various mulching measures.A long-term 9-yearifeld experiment in the Loess Plateau of Northwest China was carried out with three treatments:no mulch (CK),plastic mulch (M_(P)) and straw mulch (M_(S)).Yield factors and soil moisture were measured,and the relationships between them were explored by correlation analysis,structural equation modeling and significance analysis.The results showed that compared with CK,the average grain yields of M_(P) and M_(S) increased by 13.0and 10.6%,respectively.The average annual grain yield of the M_(P) treatment was 134 kg ha^(–1) higher than the M_(S) treatment.There were no significant differences in yield components among the three treatments (P<0.05).Soil water storage of the M_(S) treatment was greater than the M_(P) treatment,although the differences were not statistically signifiant.Soil water storage during the summer fallow period (SWSSF) and soil water storage before sowing (SWSS) of M_(S) were significantly higher than in CK,which increased by 38.5 and 13.6%,respectively.The relationship between M_(P) and CK was not statistically significant for SWSSF,but the SWSS in M_(P) was significantly higher than in CK.In terms of soil water storage after harvest (SWSH) and water consumption in the growth period(ET),there were no signi?cant differences among the three treatments.Based on the three analysis methods,we found that spike number and ET were positively correlated with grain yield.However,the relative importance of spike number to yield was the greatest in the M_(P )and M_(S) treatments,while that of ET was the greatest in CK.Suifcient SWSSF could indirectly increase spike number and ET in the three treatments.Based on these results,mulch can improve yield and soil water storage.The most important factor affecting the grain yield of dryland wheat was spike number under mulching,and ET with CK.These findings may help us to understand the main factors influencing dryland wheat grain yield under mulching conditions compared to CK.

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.

The prediction of donor number and acceptor number of electrolyte solvent molecules based on machine learning

Electrolyte solvents have a critical impact on the design of high performance and safe batteries.Gutmann's donor number(DN) and acceptor number(AN) values are two important parameters to screen and design superior electrolyte solvents. However, it is more time-consuming and expensive to obtain DN and AN values through experimental measurements. Therefore, it is essential to develop a method to predict DN and AN values. This paper presented the prediction models for DN and AN based on molecular structure descriptors of solvents, using four machine learning algorithms such as Cat Boost(Categorical Boosting), GBRT(Gradient Boosting Regression Tree), RF(Random Forest) and RR(Ridge Regression).The results showed that the DN and AN prediction models based on Cat Boost algorithm possesses satisfactory prediction ability, with R^(2) values of the testing set are 0.860 and 0.96. Moreover, the study analyzed the molecular structure parameters that impact DN and AN. The results indicated that TDB02m(3D Topological distance based descriptors-lag 2 weighted by mass) had a significant effect on DN, while HATS1s(leverage-weighted autocorrelation of lag 1/weighted by I-state) plays an important role in AN. The work provided an efficient approach for accurately predicting DN and AN values, which is useful for screening and designing electrolyte solvents.

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.

A novel Effective Panicle Number per Plant 4 haplotype enhances grain yield by coordinating panicle number and grain number in rice

Increasing effective panicle number per plant(EPN)is one approach to increase yield potential in rice.However,molecular mechanisms underlying EPN remain unclear.In this study,we integrated mapbased cloning and genome-wide association analysis to identify the EPN4 gene,which is allelic to NARROW LEAF1(NAL1).Overexpression lines containing the Teqing allele(TQ)of EPN4 had significantly increased EPN.NIL-EPN4^(TQ) in japonica(geng)cultivar Lemont(LT)exhibited significantly improved EPN but decreased grain number and flag leaf size relative to LT.Haplotype analysis indicated that accessions with EPN4-1 had medium EPN,medium grain number,and medium grain weight,but had the highest grain yield among seven haplotypes,indicating that EPN4-1 is an elite haplotype of EPN4 for positive coordination of the three components of grain yield.Furthermore,accessions carrying the combination of EPN4-1 and haplotype GNP1-6 of GNP1 for grain number per panicle showed higher grain yield than those with other allele combinations.Therefore,pyramiding of EPN4-1 and GNP1-6 could be a preferred approach to obtain high yield potential in breeding.

How Prime Numbers Are Interconnected and Built with Two Equations: Addition and Subtraction Rules the Function (6µ)

Are all prime numbers linked by four simple functions? Can we predict when a prime will appear in a sequence of primes? If we classify primes into two groups, Group 1 for all primes that appear before ζ (such that , for instance 5, ), an even number divisible by 3 and 2, and Group 2 for all primes that are after ζ (such that , for instance 7), then we find a simple function: for each prime in each group, , where n is any natural number. If we start a sequence of primes with 5 for Group 1 and 7 for Group 2, we can attribute a μ value for each prime. The μ value can be attributed to every prime greater than 7. Thus for Group 1, and . Using this formula, all the primes appear for , where μ is any natural number.

基于决策者偏好构造的概率语言型Z-number多属性决策方法

由于决策环境的日益复杂,专家评价信息往往会包含专家偏好,因此有必要对专家偏好进行识别与处理。针对这一情况,提出了基于决策者偏好构造的概率语言型Z-number并应用于多属性决策问题中。首先,针对专家偏好未知的情况,提出了一种基于修正的平均得分距离,由专家评价信息确定专家偏好。然后,依据得到的专家偏好,利用GOWPA算子与最大熵生成概率语言型Z-number。同时,获得专家评价信息的可信度,由专家可信度聚合专家信息并采用TODIM方法对方案进行排序。最后,在国家大力提倡节能减排的背景下,将所提方法应用于国家绿色矿山试点企业的选取上,验证了所提方法的有效性和可行性。

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.

Lychrel Numbers in Base 10: A Probabilistic Approach

For decades, Lychrel numbers have been studied on many bases. Their existence has been proven in base 2, 11 or 17. This paper presents a probabilistic proof of the existence of Lychrel number in base 10 and provides some properties which enable a mathematical extraction of new Lychrel numbers from existing ones. This probabilistic approach has the advantage of being extendable to other bases. The results show that palindromes can also be Lychrel numbers.

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.

On the Clique-Transversal Number in(Claw,K_4 )-Free 4-Regular Graphs

A clique-transversal set D of a graph G is a set of vertices of G such that D meets all cliques of G.The clique-transversal number,denoted by τC（G）,is the minimum cardinality of a clique-transversal set in G.In this paper,we first present a lower bound on τC（G） and characterize the extremal graphs achieving the lower bound for a connected（claw,K4）-free 4-regular graph G.Furthermore,we show that for any 2-connected（claw,K4）-free 4-regular graph G of order n,its clique-transversal number equals to [n/3].

On Perron’s Formula and the Prime Numbers

The Riemann hypothesis is intimately connected to the counting functions for the primes. In particular, Perron’s explicit formula relates the prime counting function to fixed points of iterations of the explicit formula with particular relations involving the trivial and non-trivial roots of the Riemann Zeta function and the Primes. The aim of the paper is to demonstrate this relation at the fixed points of iterations of explicit formula, defined by functions of the form limT∈Ν→∞fT(zw)=zw,where, zwis a real number.

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.

Three- and Four-Dimensional Generalized Pythagorean Numbers

The Pythagorean triples (a, b | c) of planar geometry which satisfy the equation a2+b2=c2 with integers (a, b, c) are generalized to 3D-Pythagorean quadruples (a, b, c | d) of spatial geometry which satisfy the equation a2+b2+c2=d2 with integers (a, b, c, d). Rules for a parametrization of the numbers (a, b, c, d) are derived and a list of all possible nonequivalent cases without common divisors up to d2 is established. The 3D-Pythagorean quadruples are then generalized to 4D-Pythagorean quintuples (a, b, c, d | e) which satisfy the equation a2+b2+c2+d2=e2 and a parametrization is derived. Relations to the 4-square identity are discussed which leads also to the N-dimensional case. The initial 3D- and 4D-Pythagorean numbers are explicitly calculated up to d2, respectively, e2.

Complex Decision Modeling Framework with Fairly Operators and Quaternion Numbers under Intuitionistic Fuzzy Rough Context

The main goal of informal computing is to overcome the limitations of hypersensitivity to defects and uncertainty while maintaining a balance between high accuracy,accessibility,and cost-effectiveness.This paper investigates the potential applications of intuitionistic fuzzy sets(IFS)with rough sets in the context of sparse data.When it comes to capture uncertain information emanating fromboth upper and lower approximations,these intuitionistic fuzzy rough numbers(IFRNs)are superior to intuitionistic fuzzy sets and pythagorean fuzzy sets,respectively.We use rough sets in conjunction with IFSs to develop several fairly aggregation operators and analyze their underlying properties.We present numerous impartial laws that incorporate the idea of proportionate dispersion in order to ensure that the membership and non-membership activities of IFRNs are treated equally within these principles.These operations lead to the development of the intuitionistic fuzzy rough weighted fairly aggregation operator(IFRWFA)and intuitionistic fuzzy rough ordered weighted fairly aggregation operator(IFRFOWA).These operators successfully adjust to membership and non-membership categories with fairness and subtlety.We highlight the unique qualities of these suggested aggregation operators and investigate their use in the multiattribute decision-making field.We use the intuitionistic fuzzy rough environment’s architecture to create a novel strategy in situation involving several decision-makers and non-weighted data.Additionally,we developed a novel technique by combining the IFSs with quaternion numbers.We establish a unique connection between alternatives and qualities by using intuitionistic fuzzy quaternion numbers(IFQNs).With the help of this framework,we can simulate uncertainty in real-world situations and address a number of decision-making problems.Using the examples we have released,we offer a sophisticated and systematically constructed illustrative scenario that is intricately woven with the complexity ofmedical evaluation in order to thoroughly assess the relevance and efficacy of the suggested methodology.

Novel Rifle Number Recognition Based on Improved YOLO in Military Environment

Deep neural networks perform well in image recognition,object recognition,pattern analysis,and speech recog-nition.In military applications,deep neural networks can detect equipment and recognize objects.In military equipment,it is necessary to detect and recognize rifle management,which is an important piece of equipment,using deep neural networks.There have been no previous studies on the detection of real rifle numbers using real rifle image datasets.In this study,we propose a method for detecting and recognizing rifle numbers when rifle image data are insufficient.The proposed method was designed to improve the recognition rate of a specific dataset using data fusion and transfer learningmethods.In the proposed method,real rifle images and existing digit images are fusedas trainingdata,andthe final layer is transferredto theYolov5 algorithmmodel.The detectionand recognition performance of rifle numbers was improved and analyzed using rifle image and numerical datasets.We used actual rifle image data(K-2 rifle)and numeric image datasets,as an experimental environment.TensorFlow was used as the machine learning library.Experimental results show that the proposed method maintains 84.42% accuracy,73.54% precision,81.81% recall,and 77.46% F1-score in detecting and recognizing rifle numbers.The proposed method is effective in detecting rifle numbers.