Many real-life processes are black-box problems, i.e., the internal workings are inaccessible or a closed-form mathematical expression of the likelihood function cannot be defined. For continuous random variables, likelihood-free inference problems can be solved via Approximate Bayesian Computation (ABC). However, an optimal alternative for discrete random variables is yet to be formulated. Here, we aim to fill this research gap. We propose an adjusted population-based MCMC ABC method by re-defining the standard ABC parameters to discrete ones and by introducing a novel Markov kernel that is inspired by differential evolution. We first assess the proposed Markov kernel on a likelihood-based inference problem, namely discovering the underlying diseases based on a QMR-DTnetwork and, subsequently, the entire method on three likelihood-free inference problems: (i) the QMR-DT network with the unknown likelihood function, (ii) the learning binary neural network, and (iii) neural architecture search. The obtained results indicate the high potential of the proposed framework and the superiority of the new Markov kernel. In various scientific domains, an accurate simulation model can be designed, yet formulating the corresponding likelihood function remains a challenge. In other words, there is a simulator of a process available that, when provided an input, returns an output, but the inner workings of the process are not analytically available ,2,3,4,5.Commonly, likelihood-free inference problems for continuous data are solved via a group of methods known under the term Approximate Bayesian Computation (ABC) ,7. The m biology ,9, and s biology , Sequent biology , and neu biology ,13. In t biology  for disc biology . Thus, tOur main focus is on optimal proposal distribution design as there is no “natural” notion of the search direction and scale for discrete data spaces. Hence, the presented solution is inspired by Differential Evolution (DE) , which hThe contribution of the present paper is as follows. First, we introduce an alternative version of the MCMC-ABC algorithm, namely a population-based MCMC-ABC method, that is applicable to likelihood-free inference tasks with discrete random variables. Second, we propose a novel Markov kernel for likelihood-based inference methods in a discrete state space. Third, we present the utility of the proposed approach on three binary problems.y, x and Let x while only a simulator is available, then it is considered as an Approximate Bayesian Computation (ABC) problem, meaning that In many practical applications, the likelihood function is unknown, but it is possible to obtain (approximate) samples from nference  or a blanference . If the Typically, a likelihood-free inference problem or an ABC problem is solved through sampling. One of the most well-known sampling methods is the Metropolis–Hastings algorithm , where tGiven a population of chains, a question of interest is what is the best proposal distribution for an efficient sampling convergence. One approach is parallel tempering. It introduces an additional temperature parameter and initializes each chain at a different temperature ,24. HoweReversibility is met, because the suggested proposal could be inverted to obtain Reference  successfAperiodicity is met, because the Markov chain follows a random walk.Irreducibility is solved by applying the noise.Furthermore, the created Markov chain is ergodic, as the following two conditions are met:Hence, the resulting Markov chain has a unique stationary distribution. The results presented by  indicateIn this paper, we focus on binary variables, because categorical variables could always be transformed to a binary representation. Hence, the most straightforward proposal for binary variables is the independent sampler that utilizes the product of Bernoulli:x is represented by a string of bits, xoroperator between two bits We propose to utilize the ideas outlined by , but in ollowing , we propction in . Next, wm q(δ|C):xnew=xi⊗δRemark 1.The proposal defined in  fulfillsProof. Reversibility is met, as  side of . Aperiod side of  is determut) operation:The missing property of  could bermulated  as in PrProposition 1.The proposal defined as a mixture fined by  and qxorfined by , is a prProof. mut proposal assures that there is a positive transition probability across the entire search space.    □Reversibility and aperiodicity were shown in Proposition 1. The irreducibility is met, because the mut+xor. First, it introduces another hyperparameter, However, we notice that there are two potential issues with the mixture proposal mut operation in .We refer to this new Markov kernel for discrete random variables as the discrete differential evolution Markov chain  . See theAlgorithm 1 Population-MCMC-ABC.1:Given 2:mut+xor or dde-mc.▹ Either , mut+xor3:Simulate 4:ifthen5:    6:    7:ifthen    8:        9:returnxA single step of the population-MCMC-ABC algorithm is presented in Algorithm 1. Notice that in Line 5, we take advantage of the symmetricity of all the proposal. Moreover, in the procedure, we skip an outer loop over all chains for clarity. Without loss of generality, we assume a simulator to be a probabilistic program denoted by dde-mc, on a problem when the likelihood is known.QMR-DT network (likelihood-based case): First, we validate the novel proposal, QMR-DT network (likelihood-free case): Second, we verify the performance of the presented proposal by modifying the first test-bed as a likelihood-free problem.Binarized Neural Network Learning: Third, we investigate the performance of the proposed approach on a high-dimensional problem, namely learning binary neural networks.Neural architecture search: Lastly, we consider a problem of Neural Network Architecture Search (NAS).In order to verify our proposed approach, we use four test-beds:https://github.com/IlzeAmandaA/ABCdiscrete (accessed on 5 March 2021).With each test-bed, we increase the complexity of the problem. Hence, the number of iterations chosen varies per experiment. The code of the methods and all experiments is available at the following link: x) and findings (y):y and l and finding i, i.e., the probability that the disease l alone could cause the finding i to have a positive outcome. For a complete inference, the prior l.The overall setup was designed as described by , i.e., wDT model . qil is  made by  that thedde-mc kernel to the mut proposal, the mut-xor proposal, the mut+crx proposal (see [ind-samp) as in  as in  with sam work by , it was x and the most probable values found by the population-MCMC with different proposals. The number of diseases was set to In this experiment, we used the error that is defined as the average Hamming distance between the real values of dde-mc and mut+xor, are superior to kernels stemming from genetic algorithms or random search, i.e., mut+crx, mut, and ind-samp , while the time complexity of ind-samp is O(1).DE-inspired proposals, ind-samp . In partdde-mc, mut+xor, and ind-samp as a baseline. mut+crx and mut were not selected due to to their very slow convergence with high-dimensional problems.Based on the obtained results, the subsequent experiments were carried out only with In this test-bed, the QMR-DT network is redefined as a simulator model, i.e., the likelihood is assumed to be intractable. The Hamming distance is selected as the distance metric, but due to its equivocal nature for high-dimensional data, the dimensionality of the problem is reduced. In particular, the number of diseases and observations  are decreased to 10 and 20, respectively, while the probabilities of the network are sampled from a beta distribution, dde-mc converged faster and to a better  optimum than mut+xor. However, this effect could be explained by a lower dimensionality of the problem compared to the first experiment. Second, utilizing the exponential distribution had a profound positive effect on the convergence rate of both dde-mc and mut+xor  with a single fully-connected hidden layer on the image dataset of ten handwritten digits (MNIST ). We usex inspired by the work of [D denotes the dimensionality of x. As a result, the prior distribution acts as a regularization term as it favors parameter settings with fewer active weights. The distribution is independent of the data y thus, the partition function For the Metropolis acceptance rule, we define a Boltzmann distribution over the prior distribution of the weights  work of :(14)p(x)The original dataset consists of 60,000 training examples and 10,000 test examples. For our experiment, we selected the digits 0 and 1; hence, the dataset size was reduced to 12,665 training and 2115 test examples. Different tolerance values were investigated to obtain the best convergence, ranging from 0.03 to 0.2, and each experiment was run for at least 200,000 iterations. All experiments were cross-evaluated five times. Lastly, we evaluated the performance by computing both the minimum test error obtained by the final population, as well as the test error obtained by using a Bayesian approach, i.e., we computed the true predictive distribution via majority voting by utilizing an ensemble of models. In particular, we selected the five last updated populations, resulting in 5 × 24 × 5 = 600 models per run, and we repeated this with different seeds 10 times.Because the classification error function in  is non-dmut+xor proposal converged the fastest towards the optimal solution in the search space  was better for dde-mc and mut+xor compared to ind-samp  . The cordde-mc, for a population-based MCMC-ABC algorithm. The obtained results suggest that the newly designed proposal is a promising and effective solution for intractable problems in a discrete space.In this paper, we note that there is a gap in the available methods for likelihood-free inference on discrete problems. We propose to utilize ideas known from evolutionary computing similarly to , in ordeind-samp. This is accounted for by the fact that for high dimensions, it takes more time for the entire population to converge.Furthermore, Markov kernels based on differential evolution are also effective to traverse a discrete search space. Nonetheless, great attention has to be paid to the choice of the tolerance threshold for the MCMC-ABC methods. In other words, if the tolerance is set too high, then the performance of the DE-based proposals drops to that of an independent sampler, i.e., the error of the Markov chain is high. For high-dimensional problems, the proposed kernel seems to be most promising; however, its population error becomes similar to that of Alternative ABC algorithms like SMC should be further investigated.In this work, we focused on calculating distances in the data space. However, utilizing summary statistics is almost an obvious direction for future work.As the whole algorithm is based on logical operators and the input variables are also binary, the algorithm could be encoded using only bits, thus saving considerable amounts of memory storage. Consequently, any matrix multiplication could be replaced by an XNORoperation followed by a sum, thus reducing the computation costs and possibly allowing implementing the algorithm on relatively simple devices. Therefore, a natural consequence of this work would be a direct hardware implementation of the proposed methods.In this paper, we outline a number of potential applications of the presented methodology and indicate that the obtained results are of great practical potential. From the optimization perspective, a discrete ABC gives an opportunity to solve a problem in a principled manner. This is extremely important for applications associated with deep learning, e.g., NAS ,35, neurIn conclusion, we would like to highlight that the present work offers new research directions:


We present uncertainty relations based on Wigner–Yanase–Dyson skew information with quantum memory. Uncertainty inequalities both in product and summation forms are derived. It is shown that the lower bounds contain two terms: one characterizes the degree of compatibility of two measurements, and the other is the quantum correlation between the measured system and the quantum memory. Detailed examples are given for product, separable and entangled states. R and S, where R. Schrödinger gave a further improved uncertainty relation [d-dimensional Hilbert space H, and H. The number c is the largest overlap among all R and S. Berta et al. [R and S in the presence of quantum memory B:d is the dimension of the subsystem A. The term A and the quantum memory B. The bound of Berta et al. has been further improved [h, Φ) entropies [The uncertainty principle is an essential feature of quantum mechanics, characterizing the experimental measurement incompatibility of non-commuting quantum mechanical observables in the preparation of quantum states. Heisenberg first introduced variance-based uncertainty . Later, relation :VV:(1)Iα wR, S and For a quantum state ty holds :3)Uα , and theA and B satisfy Theorem 1 gives a product form of the uncertainty relation. Comparing the results (Equation ) withoutEquation , while iEquation , even ifTheorem 2.Let Proof of Theorem 2.By definition and taking into account the fact that ≥Iα , we haveEquation . By usinEquation . ☐Corresponding to the product form of the uncertainty relation, we can also derive the sum form of the uncertainty relation:We note that Equation  reduces eorem in .B. It is shown that the lower bound contains two terms: one is the quantum correlation From Theorems 1 and 2, we obtain uncertainty relations in the form of the product and sum of skew information, which are different from the uncertainty of , which orelation .Example 1.We consider the 2-qubit Werner state Equation  are giveandrespectively; see .Similarly, we can obtain the values of the left- and right-hand sides of Equation :andrespectively; see .Here we see explicitly that, just as for the Shannon entropy, Rényi entropy, Tsallis entropy, Equation , the casquations  and 7)   enExample 2.Consider a separable bipartite state, .We still choose Equation  are zeroEquation  is zero.Equation  are 12 aEquation  are zero.Example 3.For the Werner state .We have the values of the left- and right-hand sides of Equation , respectandSee .We can also obtain the values of the left- and right-hand sides of Equation :andrespectively; see .Moreover, when .We have investigated the uncertainty relations both in product and summation forms in terms of the Wigner–Yanase–Dyson skew information with quantum memory. It has been shown that the lower bounds contain two terms: one is the quantum correlation


An analysis of low birth weight (LBW) births in ten contaminated prefectures of Japan, 1995–2018, finds a statistically significant increase in the LBW proportion in 2012–2013, but no increase after 2013. In the rest of Japan (37 prefectures), no increase in LBW births was found after the Fukushima accident. The effect was greater in area 1  than in area 2 , and not statistically significant in area 3 .In their recent article, Scherb et al.  investighttps://www.r-project.org/); the option family = quasibinomial was applied to adjust the variances for overdispersion. This means that F-tests were used instead of chi-square tests.Since the numbers of live births and LBW births for all three areas were provided in Table 1 of [p-value 0.074. A fifth-degree polynomial for the temporal trend, however, fitted the data better  than a fourth-degree polynomial . Now, the odds ratio for the level shift decreased to OR = 1.009 , p-value 0.347. Similarly, the regressions of the data for the three sub-areas yielded non-significant level shifts when a fifth-degree polynomial was applied: area 1: OR = 1.039 , p-value 0.152; area 2: OR = 1.013 , p-value 0.244; area 3: OR = 1.004 , p-value 0.730). For all three data sets, a fifth-degree polynomial yielded a better fit than a fourth-degree polynomial.With a fourth-degree polynomial for the time trend as applied in , the estp-value 0.152. No notable level shift was detected in the control region .To increase the statistical power, I pooled the data from areas 1 and 2 and termed them the study region; area 3 was used as the control region. The odds ratio of the level shift in the study region was 1.019 , p-value 0.041). The deviance was 40.2 (df = 17), considerably smaller than deviance = 45.9 (df = 17) obtained with model 1. A similar regression for the control region with model 2 yielded OR = 0.998 . The trend of LBW proportions in the study region and the regression lines for the two competing models are shown in Fig. A plot of the data shows an increase in LBW-proportions in 2012–2013. Therefore an additional regression analysis was carried out with a model that allowed for a short term increase in 2012–2013 (model 2) instead of a level shift in 2012–2018 (model 1). The estimate of the odds ratio in 2012–2013 was 1.016 , p-value 0.018) and model 2 , p-value 0.002), but model 2 fitted the data better  than model 1 . As the data showed no overdispersion, a chi-square test was applied for testing the significance of the odds ratios. The test for a possible level shift in addition to the effect in 2012–2013 yielded OR = 1.00 . Thus, the significant result for the shift in LBW proportion obtained with model 1 is driven by the peak in 2012–2013.A considerable improvement of the fit was obtained when 2002 was chosen as the beginning of the study period. Logistic regression with a fourth-degree polynomial turned out to be the best choice for the long-term trend. The results were statistically significant for both regression model 1 , To conclude, my re-analysis of the data shows that there was an increase in LBW proportion in the first 2 years after Fukushima, but no increase after 2013. Thus, the claim by Scherb et al. that their result is evidence of a genetic radiation effect is challenged by the present analysis.


For n-coalescents, this has been explicitly shown for only a limited subclass of n-coalescents and exponentially growing populations. This article gives a more general construction of time-changed n-coalescents as limits of specific Cannings models with rather arbitrary time changes.Multiple-merger coalescents, e.g.  As a more realistic alternative, multiple-merger coalescents, especially n-coalescents, introduced by Pitman  limit of genealogies from samples of size n taken from Cannings models. The limit is reached as population size N goes to infinity and time is rescaled, see Möhle and Sagitov  timescale. Such changes are featured in coalescent simulators as ms  transformed. However, size changes too extreme can yield a non-bifurcating (multiple merger) genealogy, see Birkner et al.  from Eldon and Wakeley  with changed time scale have been recently discussed and applied as models of genealogies, see Spence et al. , where ancestors of different sets are different. For The terminology from Möhle  is used ee Möhle  for detaN Cannings models for each n-coalescent n-coalescents. For this, we need some assumptions. Most importantly, an asymptotically infinite sum needs to be controlled. For this, we introduce a concept of o-terms: A sequence Consider a sequence of fixed-N leading to a well-defined population size profile in coalescent time, i.e. Population size changes of order The first class of Cannings models used to construct time-changed n-coalescents are modified Moran models. In a modified Moran model, only a single individual has more than one offspring (and may have many offspring). Following Huillet and Möhle , Matuszewski et al. -n-coalescent. In the discrete model, additional individuals can be added in any way so that the resulting model is still a modified Moran model.Let ying Eq.  for \docn-coalescents. For instance, if one only allows certain population size changes, one can also use the modified Moran model used in Theorem The specific models used in each of the two theorems are not the only possibilities of modified Moran models with variable population sizes to converge to entclass1pt{miniman-coalescent. In the discrete model, additional individuals can be added in any way so that the resulting model is still a modified Moran model.Let ying Eq.  for \docFinally, for the classic Moran model, i.e. the modified Moran model with  via Eq.  for \docFor the standard Moran model and a population size profile d by Eq.  so that n-coalescents, N Cannings models introduced in Schweinsberg  lets each individual in any generation r produce a number of  offspring X with N offspring are chosen to form the next generation. If less than N offspring are produced, the missing next generation individuals are arbitrarily associated with parents. Here, this is done by randomly choosing a parent, which preserves exchangeability and makes the model a Cannings model. The genealogies of a sample of size n converge as n-coalescent, see Schweinsberg  with growth rate n-coalescent with exponential growth, waiting times for coalescence events follow a Gompertz distribution, e.g. see Slatkin and Hudson , Eq.  appearedρt), Eq. .t in the continuous time limit. The approach of this study was to build on existing Cannings models that converge for fixed population size to the n-coalescent and just change the population sizes gradually from generation to generation, which includes adjusting parent-offspring allocation between generations. This raises the question whether the used Cannings models and the adjustment of ancestral relationships have biological interpretations and are a reasonable model for at least some real populations.As for the Wright–Fisher model, genealogies of samples taken from (haploid) modified Moran and other Cannings models can be approximated by a time-change of their limit coalescent process, when the population sizes of the discrete models are fluctuating, but are always of the same order of size. As for models with fixed population size, time intervals of n-coalescent with N): On top of a standard Moran model choice of one parent M with two offspring and one individual in the parent generation with no offpring, there is a random probability X for each other individual in the parent generation to not have offspring in the next generation. X is drawn from M reproduce once, and M replaces itself and all non-reproducing individuals by its offspring. These models capture the concept of sweepstake reproduction . Both models lead to the same Dirac coalescent limit and have the same time rescaling order X, a fixed number of X, similar models also appear in Hartmann and Huillet . Ad. Ad5). Af is regularly varying \document1 in Eq.  yields14Now, we establish an easy-to-verify variant of Möhle , N satisfyondition , \documer for n-coalescent as the fixed-size model. Eq.  Markovian process. This has an implication for modelling of real populations: The genealogy of a sudden population expansion, happening at a specific generation, where a single genotype/individual is responsible for the population growth, is not given by a time-changed continuous-time n-coalescent.Depending on the magnitude of a population size increase, adding individuals as further offspring of the multiplying parent from the fixed-size modified Moran model can strongly increase coalescence probabilities. For instance, for a population expansion of size N modified Moran models.We recall some properties of fixed-(i)As (ii)n coalescent if If (iii)If (i) from Huillet and Möhle , Eq. 10 n-coalescent after a suitable time change.The following proposition provides criteria for genealogies in modified Moran models with fluctuating population sizes to converge to a N modified Moran model so that r. If Consider a fixed-and that  holds fohat Eqs.  and 13)N modifieBased on the fixed-size modified Moran model and n-coalescent limit for the fixed-N modified Moran model.Then, This is shown by applying Lemma s are Stirling numbers of the first kind. From Eq.  is a logarithm, we have, using d Tavare , we can ince Eq.  ensures n-coalescents.The integral representation of the time change is a deterministic version of the coalescent intensity from Kaj and Krone -coalescents for To prove the Corollaries n-coalescents are covered by Theorem From Eq. , it follentclass1pt{minimaentclass1pt{minimaN case, again  model satisfies B is the Beta function, see Schweinsberg : for a us, only  needs toentclass1pt{minimat, since  holds,\dentclass1pt{minima


In this article, some properties of neutrosophic derivative and neutrosophic numbers have been presented. This properties have been used to develop the neutrosophic differential calculus. By considering different types of first- and second-order derivatives, different kind of systems of derivatives have been developed. This is the first time where a second-order neutrosophic boundary-value problem has been introduced with different types of first- and second-order derivatives. Some numerical examples have been examined to explain different systems of neutrosophic differential equation. In 1965, Lotfi A Zadeh  introducIn our literature review, we have seen that the neutrosophic precalculus and neutrosophic calculus were first studied by Smarandache , which iNeutrosophic integral calculus plays very important role in the field of neutrosophic calculus. Neutrosophic integral is the extension of fuzzy integral, which was first introduced by Smarandache . In the Before discussing about neutrosophic differential equation, we should know about fuzzy differential equation which may be modified or generalized for neutrosophic environment. The theory of fuzzy differential equation was first introduced by Kaleva , which hi et al.  proposedi et al.  studied i et al. –43. Now,In the recent time, researcher is also working on neutrosophic differential equation. Sumanthi et al.  proposedIn our literature review, we have been seen that there are few works have been done on neutrosophic differential equation. However, there are almost no work have been done on neutrosophic boundary-value problem and neutrosophic initial value problem. Therefore, there is a big scope and opportunity to work in these area. Since there is not much work which have been done, then we must develop the basic properties and results which are needed for the proper development of this topic. Now, a proper development of basic differential equation already have been done in crisp and fuzzy environment which motivates us to think about the similar types of development and modification in neutrosophic environment. In the future, this article may help the other researcher for the further development of this topic.To build a theory of neutrosophic differential equation, the second-order neutrosophic boundary-value problem is developed in this article.To present some properties of neutrosophic number.To prove the neutrosophic derivative  and genen, m)-types differentiability of neutrosophic-valued function.To present  system, where To define two-point neutrosophic boundary-value problem in different -system.To solve two-point neutrosophic boundary-value problem and investigate the solutions in different .When H is an interval, then the definition written as:Then, it can be written as: Let \docLet Proof of this proposition is trivial and it can be done using extension principle . \documeLet Since Let Moi et al.  found sof(x) at  Let \docNow, type-1 first-order derivative is denoted by f(x). Now, type-3 derivative if type-4 derivative if type-5 derivative if type-6 derivative if type-7 derivative if type-8 derivative if However, we will use only type-1 and type-2 derivative of f(x) in the rest of this article.In the similar way, we can define another types of derivative of Let I. If f is type-1 differentiable, then f is generalized neutrosophic differentiable function of type-1. Again, if f is type-2 differentiable function, then,f is generalized neutrosophic differentiable function of type-2. Therefore, Definition Since Let f is type-2 differentiable. If If This completes the proof. F, f(x) at It is said to be type-1 derivative if Let nth-order derivative of a neutrosophic-valued function.By this similar process, we can define the f(x) is said to be -type differentiable at f(x) at Let If If If If Let If If The proof the third and fourth part of the theorem is similar to the second and first part, respectively. f(x) is type-1 differentiable function and g(x) is type-2 differentiable function on I. Then, I and Let f is type-1 differentiable, then we have:g is type-2 differentiable, then there exists Since From Eq. Now, multiplying Eq. f is type-2 and g is type-1 differentiable. By similar process, we can show the same result when f and g are second-order generalized neutrosophic differentiable function on I, such that f is -type and g is -type differentiable function or f is -type and g is -type differentiable function or f is -type and g is -type differentiable function or f is -type and g is -type differentiable function on I. Then, I and:Let f is -type differentiable and g is -type differentiable. Then, by the above Theorem For the first case, g is type-1 differentiable, then f.g is type-1 differentiable and If g is type-2 differentiable, then f.g is type-2 differentiable and If Let Subcase 1: Let g is type-1 differentiable, then: Subcase 2: Let There are two subcases. Subcase 1: Let g is type-2 differentiable, then: Subcase 2: Let There are two subcases. This completes the proof. a and b are neutrosophic number, and Let us consider the second-order neutrosophic boundary-value problem as follows:y is said to be -solution of Eq. Let To find the solution of the neutrosophic boundary-value problem Therefore, there are four types of possible system of boundary-value problem. System :where Here, Since there is no derivative involve in the boundary conditions. Therefore, for all remaining system, the boundary conditions will be same as above. Only the equation will be change. System :Here,  System :Here,  System :Here, y be the -solution of the boundary-value problem Let F.where In this section, we shall discuss some test problems and their numerical results. The Tables are calculated using Wolfram Mathematica 9.0 and the figures have been drawn using MATLAB R2018a. The Matlab code for the figures has been given below.Matlab code :x=k; %Choose k, where plot(hold on.plot(hold off.xlabel(’x’); %For x-axis.ylabel(’y’); %For y-axis.Let us consider second-order neutrosophic boundary-value problem as follows: Systemy is a  solution for the boundary-value problem, then:If entclass1pt{minimay is  differentiable for y gives a  solution of the neutrosophic boundary-value problem on Then, the type-1 derivative of the solution is: SystemThe solution of the boundary-value problem for 2,2) system is:,2 systemy is type-2 differentiable. Then, type-2 derivative of y is:y is type-2 differentiable. Then, we have:y gives a  solution on For  Systemy is:y gives a  solution on The solution of the boundary-value problem for 1,2) system is: system i System :y is:y gives a  solution on The solution of the boundary-value problem for 2,1) system is:,1 systemTherefore, the boundary-value problem gives -solution on In Fig. entclass1pt{minimaentclass1pt{minimaentclass1pt{minimax\in 0,1/$$\end{doLet us consider second-order neutrosophic boundary-value problem as follows: SystemThe solution of the boundary-value problem for -system is: System-solution of the boundary-value problem is: System-solution for this boundary-value problem does not exist. Because, if -solution exists, then the solution can be written in the form as follows: SystemBy similar argument, we can show that -solution does not exist.In Figs. entclass1pt{minimaentclass1pt{minimaentclass1pt{minimaentclass1pt{minimaLet us consider second-order neutrosophic boundary-value problem as follows: Systemy, The solution of the boundary-value problem for -system is: systemThe solution for  system is:By this similar process, we can show that for  system and  system, From Fig. n, m) differentiability of neutrosophic-valued function, where In this article, mainly, we have focused on the development of neutrosophic differential equation. Some properties of neutrosophic number have been presented here. In Definition n, m) system, where x.Here, we have considered different types of derivatives in the form of different (mentclass2pt{minimmentclass2pt{minim


Patent data is an established source of information for both scientific research and corporate intelligence. Yet, most patent-based technology indicators fail to consider firm-level dynamics regarding their technological quality and technological activity. Accordingly, these indicators are unlikely to deliver an unbiased view on the current state of firm-level innovation and are thus incomplete tools for researchers and corporate intelligence practitioners. In this paper, we develop DynaPTI, an indicator that tackles this particular shortcoming of existing patent-based measures. Our proposed framework extends the literature by incorporating a dynamic component and is built upon an index-based comparison of firms. Furthermore, we use machine-learning techniques to enrich our indicator with textual information from patent texts. Together, these features allow our proposed framework to provide precise and up-to-date assessments about firm-level innovation activities. To present an exemplary implementation of the framework, we provide an empirical application to companies from the wind energy sector and compare our results to alternatives. Our corresponding findings suggest that our approach can generate valuable insights that are complementary to existing approaches, particularly regarding the identification of recently emerging, innovation-overperformers in a particular technological field. Heat waves, droughts or melting glaciers in different parts of the world underline the need for a rapid green transition to mitigate the implications and to approach the challenges of climate change. While this involves structural changes across all sectors of today's society, the invention of new green technologies and their adoption by businesses play a key role for achieving a sustainable green transformation  provides an indication of the patent's technological importance, since these citations depict the extent to which other patents are making use and build upon its underlying invention. Notwithstanding this very appealing feature of patent citations, they are also subject to several distorting factors. For example, the citation intensity of patents varies over time and across technological fields and patent offices , which allows to investigate firms' technological strength and innovative quality using a dynamic perspective. This dynamic perspective is the core of our proposed framework, as it thereby extends the prior literature that typically uses static indicators  and additionally represent their corresponding average values (2a and 2b). Moreover, the sub-measures are characterized by the following two properties:First, the four sub-measures are all index-based, which means that their calculation steps take the patents of other companies from the same index as a benchmark into account. This allows to automatically mitigate distorting factors such as a company's size, geographical location, or technological area  the firms forming a stock index, e.g., the Nasdaq-100 for US firms, where c =100, (2) the firms assigned to a specific technology field, (3) the firms headquartered in a selected country, (4) the firms listed in a fund or firms active in an ESG technology.i represents end of year i, e.g., if i refers to 2022, it represents the end of the year 2022, i-1 then represents the end of the year 2021 etc. This means that the duration from due time (i–x) to (i-x + n) is n years. At due time i, only active patent publications are accounted for, and the same holds true for i-1, i-2, and so on.To derive the index-based metrics, we consider a set of patents denoted by This definitions and assignment processes allow all subsequent investigations to be index-specific.Although several approaches are possible to depict a patent's technological quality, it is typically approximated by a patent's citations it received from patent offices. Our framework follows this approach but can also be flexibly adapted to other quality definitions. Hence, the following describes our concept to approximate patents' technological quality in very general terms, so it can be identically applied to any other kind of quality measure at the patent level.As a starting point, recall that existing patent quality indicators typically leverage information from patent citations. The idea behind this approach is simple: When an arbitrary patent A  is cited by a subsequently applied patent B filed at some patent office, its disclosed technology is somehow related to patent B. Often patent A is relevant to novelty for one or more claims in patent B which define the scope of the invention in patent B.However, using patent A's raw number of received citations as a proxy for its technological quality can be problematic, since received citations can vary greatly due to several reasons. On the one hand, the patent office's examiner may have made a misjudgment, or the quoted text passages from patent A do not relate to the core of the invention of patent A. In addition, examiners may be biased and prefer to cite patents from their own geographic area and possibly from specific companies. It is also conceivable that once cited patents appear in later citation analyses, they are cited more frequently than patents that have not yet been cited, but which describe the same or at least very similar technical content. Patent A can also be too young to appear in a citation, although it is technologically located in a highly competitive and highly valued technology-environment.technology micro cluster (tmc) comes into play: instead of simply counting the number of citations of patent A and calculating a quality score based on that number, we derive a tmc formed by y patents describing the technology that is closest to the one codified in patent A. We then use this tmc's average technological quality and adjust it for the patents that are assigned to the company of patent A. In our case, the quality of the tmc is derived based on the citation frequency of the patents in the tmc, which is then assigned to patent A . However, as stated above, alternative quality definitions are possible as well. We identify the closest patents to patent A and define its tmc based on machine learning process, because machine learning approaches are particularly well suited to calculate the textual similarity between patent texts  based on the set of all patents described by N, iα, N, i-1α, N, i-2α… N, i-yα, plus patent A itself, adjusted by the patents publications of firm fNf, α, i, Nf, α, i-1, Nf, α, i-2… Nf, α, i-y for the index α and the period i, i-1, i-2 …i-y yielding the set Ntmc (patent A):Following this logic, we calculate the tmc for patent A  is then defined as the mean value of the quality values (indicted by a bar over the q) of the Ntmb(patent A) patents:maxM, iα, M, i-1α, M, i-2α… M, i-nα that forms the index alpha does not necessarily have to match the set of patents N, iα, N, i-1α, N, i-2α… N, i-yα that is used to calculate the tmc. However, it is advisable to select patents for which a quality can be derived. For example, if the citation frequency is the method of choice, the selected patents N, iα, N, i-1α, N, i-2α… N, i-yα should be of a certain age to have received any citations at all.At this point it should be noted that the set of patents Having outlined our chosen approach to approximate a patent's technological quality, we can now describe in more detail how we operationalize our 4 measures that define the DynaPTI indicator.Mf, α, i is the number of patents published in year i corresponding to firm f in the index α. The mean value of the technological quality q of these patents yields Qf, α, i = mean(q). Correspondingly, the technological quality of firm f assigned to the patents Mf, α, i-1, Mf, α, i-2… Mf, α, i-n is given by Qf, α, i-1, Qf, α, i-2… Qf, α, i-n. To calculate the technological quality change rate (sub-measure 1a), we first calculate the growth, i.e., the slope β = β of the linear regression line fitting the values Qf, α, i, Qf, α, i-1, Qf, α, i-2… Qf, α, i-n. The slope of the regression line is calculated based on the least squares method according towith the arithmetic mean value of the qualityand the arithmetic mean value of the due date yearsQf, α, i, n) represents the absolute growth and hence the absolute change of the technological quality of firm f assigned to the index α with the patent portfolios Mf, α, i, Mf, α, i-1, Mf, α, i-2… Mf, α, i-n. To calculate the relative growth γ of the technological quality, we divide the absolute growth by the arithmetic mean value according toβ, the mean value of the relative growth of the technological quality of all firms forming the index α, is calculated according toTo relate firm dQ,α,i,n is given byThe standard deviation f in the time period i, i-1, i-2…i-n according to the standard scoreWe can then use these calculations to derive an index-based change rate for the technological quality of firm f assigned to patents Mf, α, i, Mf, α, i-1, … Mf, α, i-n is given by Mf,α, i, Mf,α, i-1…Mf,α, i-n itself: The number of patents published in the corresponding year. To calculate the change rate of the activity (sub-measure 1b), we first calculate again the growth, i.e., slope β = β of the linear regression line fitting the values Mf,α, i, Mf,α, i-1, Mf,α, i-2…Mf, α, i-n. The slope of the regression line is calculated based on the least squares method according toThe technological activity change rate related to the index α is calculated in the same way as the technological quality change rate. The technological activity of firm with the activity's arithmetic mean valueMf, α, i, n) represents the absolute growth and the absolute change of the activity of firm f assigned to the index α with patents Mf,α, i, Mf,α, i-1…Mf,α, i-n.and Mf, α, i, n) is calculated by dividing the absolute growth by the arithmetic mean value according toThe information value of absolute changes must be considered as rather limited since this sub-measure can be biased in favor of firms with higher patenting activity. Hence, it is expected that firms with large patent portfolios tend to have larger absolute growth rates when referring to the change of the absolute number of active patent families. To account for this distorting size effect, the activity's relative growth γ, the mean value of the relative growth of the technological activity of all firms forming the index α according toTo relate the relative growth regarding the technological activity determined by the patent portfolio dM,α,i,n is given byThe corresponding standard deviation f in the time duration i, i-1, i-2…i-n is then given byEquivalent to the calculations for the technological quality, the index-based change rate of the technological activity for firm f forming the index α is calculated according toTo consider the mean value of the technological quality (sub-measure 2a) in a consistent manner with the change rates of the quality and activity, the mean value of the quality (equation 4) over the time duration i, i-1, i-2…i-n for all firms The standard deviation f in the time period i, i-1, i-2…i-n according to the standard scoreThe formulas above yield the index-based mean value of the technological quality of firm i, i-1, i-2…i-m for all firms f is calculated according toThe average value of the technological activity (sub-measure 2b) is derived equivalently to the corresponding calculations for the technological quality. The mean value of the technological activity (equation 11) over the time duration And the standard deviation f in the time period i, i-1, i-2…i-m according to the standard scoreThe formulas above yield the index-based mean value of the technological activity of firm m instead of n years like the other sub-measures. Of course, m can be set to n, but in practice the current technological activity  of a company is often considered instead of a longer time period, so in many cases m<n.Note that the mean value of the technological activity is given here for f, α, i, n, m, is calculated by simply adding the four sub-measures up. That is, the index-based change rate of firm f 's technological quality, δQ, f, α, i, n, the index-based change rate of its technological activity δM, f, α, i, n, the index-based mean value of its technological quality, The final value of DynaPTI, δf, α, i, n, mis 0:Due to the method of standard score, the mean value of δf can achieve a high indicator value if it has (i) a high technological quality (ii) a high technological activity (iii) an increasing technological quality, (iv) an increasing technological activity. Firms with an indicator value, δf, α, i,n,m > 0, are above average according to the DynaPTI measure, whilst firms with δf, α, i, n, m < 0 are below average. We next provide an illustration of this framework to demonstrate its implementation and main benefits.Firm In what follows, we demonstrate an application of the DynaPTI framework to present how it can be implemented and operationalized. We perform this application exercise for firms that are active inventors in over the recent past the technology field of wind energy. Although the primary goal of this application is to present an example implementation and not an assessment of a technology field, this selection to firms active in wind energy allows us to highlight how our proposed indicator can be used for corporate intelligence in a technology field that is highly important for the green transformation.In order to do this, the first step is to define an index α of firms from the technology field of wind energy and to extract their corresponding patents for a given time window. We identify such firms and their patents based on data and a definition of wind energy patents provided by EconSight AG.y in equations (1) and (2) is set to y = 5. The DynaPTI for innovators in wind energy is then derived according to the specifications in the previous section and calculated using a Python script. The corresponding results can be seen in We then set the following restrictions and use the following parameters to calibrate the framework: First, to keep our analysis concise, we require that all patents we use for any kind of calculation must be assigned to EconSight's wind energy technology field. Second, we want to focus on relevant players in this area. To ensure this, we restrict our selection of patents to those from firms that have published at least 5 patents in the time period 2020–2022, and have at least 1% of their patents in the field of wind energy. Third, we aim to focus on the very recent past to have an up-to date evaluation. Hence, we restrict the selection of patents to wind energy patents published in the 3 years' time period 2020–2022 for the calculation of our 4 sub-indicators. Instead, the set of patents for calculating the tmc is formed by all patents in the field of wind energy, which were published in the time period 2015–2018, and is not restricted to any set of firms. Again, with regard to the tmc, the number of patents to calculate the parameter Global X Wind Energy ETF (if they are included in the fund).In According to our results, the two companies Siemens Energy AG and Vestas Wind Systems AS show particularly high DynaPTI values. Both companies have a very high sub-measure 2b and thus a particularly high patent activity in the technology field wind energy. Compared to the other companies in the index, Vestas Wind Systems AS also has the highest sub-measure 1a and thus the highest dynamic in the development of patent quality.Another interesting case is Vinci SA, which has a particularly high sub-measure 1b. This indicates that its patent activity has increased significantly in the last 3 years compared to the other companies. Moreover, the company Holcim Ltd has a particularly high sub-measure 2a and thus a particularly high average patent quality compared to the other companies. This means that the companies Holcim Ltd and Vinci SA are also rated very highly in the DynaPTI even though they only have a relatively small proportion of their overall patents in the wind energy sector (see column F).Turning to a comparison with alternative measures, let us first focus on column (G), where we have included the Patent Asset Index (PAI) scores from PatentSight, which is an established patent-based indicator for corporate intelligence that although lacks a dynamic component.Global X Wind Energy ETF if they are present in the fund. The two top-performing companies according to DynaPTI are also included in the ETF and have relatively high portfolio weights. However, several green innovation over-performers, such as Vinci, ThyssenKrupp or Holcim are not reflected in this ETF. This may be due to their relatively low specialization in wind energy. Yet, despite their relatively low overall specialization our dynamic patent technology indicator suggests that they may be important innovators in wind energy and could make significant new contributions to this technology field. Thus, our approach can possibly offer investors a data-driven framework to identify companies that may be neglected due to their size or specialization. Conversely, it might also be a tool to detect potential greenwashing from companies pretending to innovate in an ESG-related field but not showing significant performance based on our framework.A second main advantage of our indicator can be seen from column (H), where we report companies' portfolio weights in the Patent data has become an established source of information for both scientific research and corporate intelligence. However, most patent-based indicators fail to consider firm-level dynamics regarding technological quality and technological activity. Especially in rather dynamic technological areas that are particularly relevant for the green transition, this limitation can distort company assessments that are based on existing patent-based measures. In this paper, we have developed a framework to build a novel indicator, which we call DynaPTI, that tackles this particular shortcoming. Its main strength is that it provides researchers and practitioners a tool that is more reliable for up-to-date assessments about firm-level innovation activities.Our proposed approach is complementary to prior work and is based on three main building blocks: First, it focuses on index-based comparisons to mitigate distorting effects such as the technological area. Second, it uses a transformer-based approach to leverage textual information contained in patents. This procedure allows to approximate the technological quality of firms' innovation pipelines and to correct for distorting factors related to the patent examination process. Third, the indicator consists of four different sub-measures that capture technological quality and activity, as well as their corresponding dynamics over a specific time window. We operationalize these sub-measures using patent citations and patent applications as approximations for the technological quality and activity of firms, respectively. Note however that our proposed indicator is relatively generic and can be flexibly specified. Researchers and practitioners could thus easily adapt its empirical ingredients as long as there is corresponding temporal data available. For example, one may use patent claims instead of patent citations as a technological quality proxy.Yet, the main advantage of our proposed indicator is that it incorporates a dynamic perspective. This seems particularly useful for investigating companies in technological areas that are characterized by rapid changes, which is the case, for example, for several technology domains that are highly relevant for a successful green transition. We demonstrate this by providing an application of our framework to the wind energy sector. In doing so, we assess the strength of companies' innovation pipelines that are active in this domain based on different measures, including our own. Taken together, the results from this application exercise highlight the following: DynaPTI may be viewed as a tool that can uncover recently emerging innovation-overperformers in a particular technological field and is less biased in favor of large incumbents compared to existing patent-based indicators.Patent Asset Index. This is because DynaPTI puts less to no emphasis on innovations from the relatively distant past compared to static patent-based innovation indicators , how strongly is the calculation of the tmc influenced by the selected machine learning model. These parameter choices may depend on the underlying technology or the selected index: the faster a technology develops, the shorter the period should be defined. The period can also depend on the context in which the relevant index calculations are carried out: longer periods may have to be chosen for political decisions, i.e., for decisions at the economic management level. Ultimately, one must also consider that for patents with a completely new technological approach, the tmc calculation may lead to nonsensical outcomes, since such a patent would not have any semantically close counterparts.Hence, future research in the area of patent-based technology indicators may primarily focus on evaluating our measure's performance and those of other dynamic approaches in alternative settings. Another direction for future research would be to examine the parameter space of the model and, if necessary, identify meaningful values depending on the scenario. Furthermore, since our proposed framework is rather generic, it would be interesting to specify and calibrate DynaPTI based on alternative parameters  and data , and subsequently evaluate corresponding results. In principle, DynaPTI can even be expanded beyond patent data. Due to its index-based calculation and the method of the statistical standard score, parameters can be omitted or added at will. For example, it would be possible to include financial key figures such as sales and profits. With this regard, DynaPTI could also be interpreted and tested as an extension tool for classical financial analyses. Taken together, we believe that all of these open issues are interesting and important avenues for future research to improve patent-based indicators for scientific research and corporate intelligence.michael.freunek@econsight.ch.The data analyzed in this study is subject to the following licenses/restrictions: data is proprietary. Requests to access these datasets should be directed to MF: conceptual idea, methodological approach and calculation, empirical estimation, and conclusions. MN: literature, motivation, methodological evaluation, empirical evaluation, and conclusions. Both authors contributed to the article and approved the submitted version.


The methodological design was quasi-experimental with a duration of 8 weeks. The data were collected three times: pretest, posttest, and follow-up. The results showed differences favoring the experimental group compared to the control group after the implementation of the intervention program in the following variables: team identification (p < 0.001), role clarity (p < 0.001), intra-team communication , group cohesion (p < 0.05), social conflict (p = 0.001), commitment to the team (p < 0.001), inside sacrifice (p < 0.001), transactive memory systems , collective efficacy (p = 0.02) and team performance (p = 0.001). Consequently, the application of team-building strategies incorporated into specific technical-tactical training tasks in football seems to improve group dynamics in sports teams.The study aimed to analyze the effects of an intervention program based on team building developed in technical-tactical training tasks in football. Specifically, it examined the benefits of the intervention in variables related to the conceptual model of team building: role clarity, team identification, intra-team communication, cohesion, intra-group conflict, commitment to the team, inside sacrifice, transactive memory systems, collective efficacy, and perceived performance. The participants were 51 young elite footballers divided into an experimental group ( Within 1.1.This conceptual framework has served as a reference to develop interventions focused on the construction of sports teams . These iSecondly, the professional in charge of carrying out the intervention can be a key figure to develop the established strategies optimally. Previous studies have chosen two methods: (a) a direct method, in which the sports psychology specialist works directly with the players, or (b) an indirect method, in which the sport psychology specialist works exclusively with the coach, who is the one who implements the intervention . ResearcThird, the type of intervention focused on variables of the environment, structure, or processes of the team can also determine the program’s efficacy. Overall, interventions focusing on goal setting  or teamwAnother relevant aspect of team training interventions is the consequences analyzed. In general, improvements have been studied in different measures of performance  and psyc1.2.Taking into account the existing scientific literature on the creation of interventions to improve group dynamics, this research seeks to develop an intervention program: (a) intervening on the inputs ; and team processes  of the model of Therefore, the main objective of the study was to analyze the effects of an intervention with team-building strategies integrated into training tasks with technical-tactical objectives within the team’s game model to improve group dynamics and team performance. Accordingly, the central hypothesis of the study is that the intervention program developed will favor role clarity, improve team identification, optimize intra-group communication, increase group cohesion and commitment to the team, reduce intra-group conflict, increase teammate inside sacrifice and coordination in game situations, increase the levels of collective efficacy, and improve perceived performance.2.2.1.To develop the research, we followed the ethical guidelines of the Regarding the intervention program, a group of experts in sports psychology and football training was in charge of analyzing, relating, and integrating the technical-tactical objectives of the team , with the psychosocial variables of the model of The team-building strategies were carried out in 32 football training tasks  related to the main objective of each session and associated with the team’s technical-tactical contents . The control and experimental groups both performed the same training sessions, except for the modifications proposed in the specific tasks in the experimental group.2.2.age =M 16.54, SD = 1.23; years of seniority in the club: years =M 7.87, SD = 1.31), and 24 players from the under-16 elite team in the control group . The intervention was carried out with four coaches  between 34 and 42 years old , with a coaching license and experience of at least 5 years in training categories. Sample selection was intentional by clusters, considering the geographical proximity of the teams and the working group’s possibilities to access and monitor the sample.In the present study, participants were 51 male training footballers belonging to two teams of different categories but the same professional football club of the Argentine first division. Specifically, there were 27 players from the under-17 elite team in the experimental group (2.3.2.3.1.strongly disagree) to 9 (strongly agree).To evaluate role clarity, we used the Spanish version validated by 2.3.2.completely disagree) to 5 (completely agree).Team identification was assessed with five items  used in previous research . Items a2.3.3.almost never) to 7 .Intra-team communication was analyzed with the Spanish version of the SECTS-2 . This sc2.3.4.strongly disagree) to 9 (strongly agree).The Spanish version of the Group Environment Questionnaire developed by 2.3.5.never) to 7 .The Spanish adaptation of the Intra-Group Conflict Scale elaborated by 2.3.6.strongly disagree) to 5 (strongly agree).To measure players’ perceptions of commitment to the team in the season, we used the Spanish version of the KUT Objective Scale KUT: . This sc2.3.7.totally disagree) to 9 .Players’ perceptions of inside sacrifice were measured with the Group Sacrifice Scale (GSS), originally designed by 2.3.8.strongly disagree) to 5 (strongly agree).The Scale of Transactive Memory Systems in Sport (TMMS-S) developed by 2.3.9.bad) to 5 (excellent).This variable was analyzed with the instrument developed by 2.3.10.poor) to 10 (excellent) was used to evaluate the responses. This way of analyzing team performance seems to be ecologically valid and reliable  was used to test group differences in the dependent variables before starting the study. When significant group differences were observed in the pretest measures, they were included as covariates. Mixed-design ANOVAs  were used to check the effect of the intervention. If this analysis yielded a significant interaction effect, the main effect of time was calculated for each condition. The changes were considered significant when the value of p was less than p < 0.05.The data were analyzed with the SPSS 25.0 statistical program. First, means and standard deviations were calculated. Second, the normality of the data was checked with the Shapiro–Wilk test. The data had a normal distribution, so parametric tests were used. Third, one-way analysis of variance (3.p = 0.133), ERP (p = 0.079), personal inside sacrifice (p = 0.485), or team performance (p = 0.265). On the contrary, However, there were significant pretest group differences in RR (p = 0.007), team identification (p < 0.001), distinctiveness (p < 0.001), positive conflict (p < 0.001), acceptance (p < 0.001), negative conflict (p < 0.001), IG-T (p = 0.035), ATG-T (p < 0.001), IG-S (p = 0.003), ATG-S (p < 0.001), social conflict (p < 0.001), task conflict (p < 0.001), commitment to the team (p = 0.001), teammate inside sacrifice (p = 0.003), specialization (p < 0.001), credibility (p < 0.001), coordination (p = 0.002), collective efficacy (p < 0.001), and individual performance (p = 0.007).Before staring the intervention, there were no significant group differences in SR&B (p < 0.001). Pairwise comparisons revealed that the control group decrease  from pretest to follow-up (p < 0.001) and from posttest to follow-up (p = 0.001). There was also a significant Condition x Time interaction for ERP (p < 0.001). Pairwise comparison showed that the control group decrease from pretest to follow-up (p < 0.001) and from posttest to follow-up (p = 0.001). There was a significant Condition x Time interaction for RR (p < 0.001). The experimental group showed an increase in this variable from pretest to follow-up (p = 0.009), whereas the control group showed an increase from pretest to posttest (p = 0.006) with a decrease from posttest to follow-up (p = 0.008). Finally, there was a significant Condition x Time interaction for team identification (p < 0.001). The experimental group showed an increase from pretest to posttest (p = 0.013) and from pretest to follow-up (p = 0.043), whereas the control group showed a decrease from pretest to posttest (p = 0.020) and follow-up (p < 0.001), and from posttest to follow-up (p < 0.001).p = 0.029). Pairwise comparisons showed an increase in this variable from pretest to follow-up (p = 0.004) and from posttest to follow-up (p = 0.001) in the control group. There was also a significant Condition x Time interaction for positive conflict (p < 0.001). The control group showed a decrease from pretest to posttest (p = 0.049), from pretest to follow-up (p = 0.004), and from posttest to follow-up (p < 0.001), whereas the experimental group showed an increase in positive conflict from pretest to posttest (p < 0.001) and from pretest to follow-up (p = 0.007). There was a significant Condition x Time interaction for acceptance (p < 0.001). Pairwise comparison showed decreases in the control group from pretest to follow-up (p = 0.004) and from posttest to follow-up (p < 0.001), whereas experimental group showed an increase from pretest to follow-up (p < 0.001), and from posttest to follow-up (p < 0.001). Finally, there was a significant Condition x Time interaction for negative conflict (p < 0.001). Pairwise comparison showed decreases in the experimental group from pretest to follow-up (p < 0.001) and from posttest to follow-up (p < 0.001).p = 0.001). Pairwise comparisons showed a significant increase from pretest to posttest (p = 0.001) but also a decrease from posttest to follow-up (p = <0.001) in the control group, whereas an increase from pretest to follow-up was observed in the experimental group (p = 0.005). Also, there was a significant Condition x Time interaction for ATG-T (p = 0.005). Pairwise comparisons showed significant decreases from pretest to posttest (p < 0.001), from pretest to follow-up (p = <0.001), and from posttest to follow-up (p < 0.001) in the control group. In contrast, the experimental group showed increases from pretest to posttest (p < 0.001), from pretest to follow-up (p < 0.001), and from posttest to follow-up (p < 0.001). Regarding social cohesion, there was a significant Condition x Time interaction for IG-S (p < 0.001). Pairwise comparisons showed decreases from pretest to follow-up (p < 0.001) and from posttest to follow-up (p < 0.001) in the control group. Likewise, there was a significant Condition x Time interaction for ATG-S (p = 0.038). Pairwise comparisons showed that the control group decreased this variable from pretest to follow-up (p < 0.001) and from posttest to follow-up (p < 0.001), whereas the experimental group showed an increase from pretest to posttest (p = 0.048), and also an increase from pretest to follow-up (p < 0.001). By contrast, there was no significant Condition x Time interaction for task conflict (p = 0.066), but there was a significant interaction for social conflict (p = 0.001). Pairwise comparisons showed that the control group increased this social conflict from pretest to follow-up (p < 0.001), whereas the experimental group showed decrease from pretest to posttest (p = 0.001) and also an increase from posttest to follow-up (p < 0.001).p < 0.001). Pairwise comparisons showed an increase from pretest to posttest (p < 0.001) and from pretest to follow-up (p = 0.015), but a decrease from posttest to follow-up (p < 0.001) in the control group, whereas the experimental group showed increases from pretest to follow-up (p < 0.001) and from posttest to follow-up (p < 0.001). Secondly, there was also a significant Condition x Time interaction for personal inside sacrifice (p < 0.001). Pairwise comparisons showed a decrease in this variable from pretest to posttest (p < 0.001) and from pretest to follow-up (p < 0.001) in the control group. Thirdly, there was a significant Condition x time Interaction for teammate inside sacrifice (p < 0.001). Pairwise comparison showed decreases in the control group from pretest to follow-up (p < 0.001) and from posttest to follow-up (p < 0.001), whereas increases were observed from pretest to follow-up (p < 0.001) and from posttest to follow-up (p < 0.001) in the experimental group.Analyzing commitment to the team and inside sacrifice, firstly, the results showed a significant Condition × Time interaction for commitment to the team (p = 0.008). Pairwise comparison showed decreases from pretest to follow-up (p < 0.001) and from posttest to follow-up (p < 0.001) in the control group, whereas increases were observed from pretest to posttest (p = 0.001), from pretest to follow-up (p < 0.001), and from posttest to follow-up (p = 0.016) in the experimental group. There was a significant Condition x Time interaction for credibility (p < 0.001). Pairwise comparison showed a decrease from pretest to follow-up (p < 0.01) and an increase from posttest to follow-up (p = <0.01) in the control group, whereas increases were observed from pretest to posttest (p = 0.001). By contrast, there was not a significant Condition x Time interaction for coordination factor (p = 0.535).In relation to the TMS factors, there was a significant Condition x Time interaction for specialization (p = 0.021). Pairwise comparison showed a decrease from posttest to follow-up (p = 0.310) in the control group, whereas increases were observed from pretest to posttest (p = 0.020) and from posttest to follow-up (p = 0.03) in the experimental group. On the contrary, there was no significant Condition x Time interaction for individual performance (p = 0.092), but there was a significant Condition x Time interaction for team performance (p = 0.001). Pairwise comparison showed a decrease from pretest to follow-up in the control group (p = 0.045), whereas increases were observed from pretest to posttest (p = 0.003) and from pretest to follow-up (p = 0.012) in the experimental group.Regarding collective efficacy and performance, there was a significant Condition x Time interaction for collective efficacy  and at follow-up . Previous correlational studies have revealed the importance of roles  and teamConcerning the team processes, the different communication factors presented a similar line of results, with increases in the positive dimensions in the experimental group, mainly at follow-up , and a decrease in the negative dimension . On the other hand, the control group showed a decrease in the positive variables and an increase in the negative dimension at follow-up. In this sense, intra-team communication had already been identified as a key process that facilitates a better performance of group members . TherefoConcerning the model’s outcomes, in the dimensions of group cohesion and team conflict, the experimental group showed an improvement over the control group in group cohesion but not in team conflict. Two relevant aspects can be extracted from the results: (a) on the one hand, tactical training tasks seem to improve both task and social dimensions in the team, which had not been achieved previously ; (b) on Continuing with the analysis of the remaining variables, after the intervention, commitment to the team and inside sacrifice presented group differences, with a positive trend in the experimental group compared with the control group. Given the previously found relationship between the two variables with team performance , this stConcerning the TMS, coordination improved in both groups without differences. The reason for this may be that, as the competition progresses, coordination improves, if only because of the training and matches played by the players. However, the dimensions of specialization and credibility did present differences, with a positive trend in the experimental group compared with the control group. In this sense, it has been proven that the creation and development of shared memory systems , specifiFinally, collective efficacy and team performance presented group differences, with a positive trend in the experimental group after the intervention program. Similar results were found by other intervention programs that integrated strategies with tactical training tasks , as coop4.1.Despite the contribution of the study, there are several limitations that must be taken into account for future work. Firstly, the study was conducted in a specific sport with players in training, so caution is recommended when applying some of the strategies in other collective sports or population groups. It would be interesting to analyze other age groups and other collective sports, which could develop team-building strategies associated with tactical training due to their similar characteristics. Secondly, it should be noted that many psychosocial variables linked to the intervention were analyzed, but there was no assessment of the technical-tactical components trained. Therefore, developing an analysis of behaviors related to technical-tactical aspects in training and competitions would be essential. Another limitation of this study is that it used quantitative assessments exclusively, and the involved agents did not assess the intervention program itself through a qualitative approach, which could provide relevant information to perfect the program development. Finally, it should be noted that the number of participants and teams was reduced, so it would be relevant to develop this training program with more sports teams to corroborate the positive results of this intervention.5.The main conclusion of the present study is that the intervention program with strategies based on team-building implemented in the experimental group’s tactical training tasks improved almost all analyzed variables compared with the control group . Therefore, this study provides new knowledge on the development of intervention programs based on team building, mainly the integration of strategies from the model of The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.The studies involving human participants were reviewed and approved by Bioethics Committee of the University of the first author (Protocol number: 239/2019). Written informed consent to participate in this study was provided by the participants’ legal guardian/next of kin.JT and TG-C designed and performed the study, acquired and interpreted the data, and drafted the first version of the manuscript. JD-G interpreted the data and improved the manuscript. ML-G and FL designed the study, interpreted the data, had overall responsibility for the study, and improved the manuscript. All authors contributed to the article and approved the submitted version.Financial support provided by the European Regional Development Fund (ERDF) and the Government of Extremadura (GR18KA20).The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.


Commonly used minimizer orders select more k-mers than necessary and therefore provide limited improvement in runtime and memory usage of downstream analysis tasks. The recently introduced universal k-mer hitting sets produce minimizer orders with fewer selected k-mers. Generating compact universal k-mer hitting sets is currently infeasible for k > 13, and thus, they cannot help in the many applications that require minimizer orders for larger k. Here, we close the gap of efficient minimizer orders for large values of k by introducing decycling-set-based minimizer orders: new minimizer orders based on minimum decycling sets. We show that in practice these new minimizer orders select a number of k-mers comparable to that of minimizer orders based on universal k-mer hitting sets and can also scale to a larger k. Furthermore, we developed a method that computes the minimizers in a sequence on the fly without keeping the k-mers of a decycling set in memory. This enables the use of these minimizer orders for any value of k. We expect the new orders to improve the runtime and memory usage of algorithms and data structures in high-throughput DNA sequencing analysis.Minimizers are ubiquitously used in data structures and algorithms for efficient searching, mapping, and indexing of high-throughput DNA sequencing data. Minimizer schemes select a minimum  A key parameter for evaluating and comparing sketching schemes is density   any L-long sequence. In terms of a complete de Bruijn graph (dBG) of order k, a minimum UHS is a minimum set of nodes whose removal leaves no path of length L − k + 1 nodes in the graph. Heuristic algorithms for finding a compact UHS include DOCKS  . A UHS ide DOCKS  and PASHde DOCKS , both ofdBG size .k, and currently, compact UHSs are available only for k ≤ 13. As a result, UHS-based minimizer orders could not be used in many applications that require larger values of k, such as long-read mapping . Polar-set-based sequence-specific minimizers achieved very low density for k ≤ 25  for alphabet Σ. Moreover, the same order is used for all L and a given k, unlike DOCKS and PASHA, which need to compute a UHS for every combination of k and L. Furthermore, the longest remaining path in a complete dBG after removal of an MDS was recently shown to be bounded by O(k3)  , so for k, precomputing, storing, and querying it become infeasible for large k. We overcome this limitation by presenting an efficient method to test in O(k) time if a k-mer belongs to the MDS, which enables computing the minimizers in a sequence according to our MDS-based order on the fly. Thus, the minimizer orders that we defined provide, for the first time, universal orders with low density that easily scale to any value of k and to any desired window length.As the MDS size grows exponentially in We begin by providing definitions and theoretical background and describing relevant related work  iff o(x1) < o(x2).A de Bruijn graph (dBG) of order k is a directed graph in which every node is labeled with a distinct k-mer, and there is a directed edge from node a to b iff the (k − 1)-long suffix of a is the same as the (k − 1)-long prefix of b. The edge is labeled with the (k + 1)-long merge of the two labels. A complete dBG has a node for every possible k-mer and an edge for every possible (k + 1)-mer. Paths in a dBG of order k represent sequences, and a path of w nodes represents a sequence of w overlapping k-mers.A minimizer scheme is a function fk,w,o:Σw+k−1 → {0, …, w − 1}. Function f returns the position of the minimum k-mer under o in a given window of w overlapping k-mers . By convention, ties are broken by choosing the leftmost k-mer. The minimizers of a string S, denoted as L-long windows of S.A expected density of a minimizer scheme is the expected fraction of k-mer positions that will be selected as minimizers over an infinitely long random i.i.d. sequence. The particular density of a minimizer scheme on a specific sequence S  is the fraction of k-mer positions selected by the scheme on that sequence. The expected (particular) density factor is the expected (particular) density multiplied by (w + 1).The k → ∞ or w → ∞ and prove a general lower bound of The expected density factor of a random minimizer is two . Marçaisk, k-mer order h, we say that minimizer order ohΠ, is compatible with Π and h if for i = j, then Given an ordered partition of Σdecycling set of a graph G =  is a set of nodes whose removal results in an acyclic graph. Finding an MDS  in an arbitrary graph is NP-hard (k. Mykkeltveit (k). A pure cycle is a set of nodes corresponding to all the cyclic rotations of some k-mer  (x is positive (negative) if x) > 0 (<0). All positive (negative) rotations of a k-mer are consecutive, and either all rotations have x) = 0 or the two blocks of consecutive positive and negative rotations are separated by at most one rotation with x) = 0 on either side (x) = 0, any arbitrary k-mer from the cycle can be selected.To determine which of the cyclic rotations of a , T = 3) . We say her side . The MDShttps://www-cs-faculty.stanford.edu/~knuth/programs/unavoidable2.w). This implementation uses the FKM algorithm  = 0 for all k-mer rotations, the lexicographically smallest k-mer is included in the decycling set. An MDS consists of Θ(|Σ|k/k) k-mers.Mykkeltveit's algorithm has an efficient implementation owing to Knuth  k-mers such that any L-long string contains at least one k-mer from k-mer from w = L − k + 1 overlapping k-mers, and thus, the nodes represented by w-long paths in a complete dBG of order k |Σ|k+1L) for p iterations of node removal. Using DOCKS, small UHSs have been constructed only for k ≤ 11. Minimizer orders compatible with these UHSs were shown to have lower density than random orders  time and compared to determine if x is the first positive counterclockwise rotation. If x) = x′) = 0, then the algorithm determines whether x is a lexicographically smallest rotation in O(k) time.For large values of Proposition 1in time O(k).Algorithm 1:Input: k-mer xOutput: Membership in the MDS fori ∈  doci = sin(2πi/k)1: 2: x′ = xk−1x0x1..xk−23: 4: ifx) > 0 thenx is the first rotation with x) > 05: ifx′) ≤ 0 then return true6:  else ifx) = 0 then7: ifx′) = 0 thenx is the lexico. smallest rotation8:  i ← 09:    forj ∈  do10:  ifthen return false11:    iftheni = 012:    elsei ← i + 113:    ifthen return true14:    15: return falseProof.k-mer x is positive, negative, or nonpositive if x) > 0, < 0, or ≤0, respectively. Recall that a k-mer k-mers in the conjugacy class have x) = 0 and x is a lexicographically smallest rotation.The proof follows from the definition of k-mer x is the first positive counterclockwise rotation in its conjugacy class; that is, x has x) > 0, and the one-letter clockwise rotation of x, denoted x′, has x′) ≤ 0.For 1 above, line 6 returns true iff the input k-mer x, x′ have x) = x′) = 0 (lines 7–8), then all k-mers in that conjugacy class have zero embedding by x and returns false if there is a k-mer that is lexicographically smaller than x (line 11). Otherwise, it returns true if either all possible rotations are lexicographically larger than x  or x is a lexicographically smallest rotation but is identical to another rotation .For 2 above, note that if two consecutive rotations of a O(k) time. The loop beginning on line 10 can run for at most 2k times and performs constant time computations per iteration for a total running time of O(k).The embedding computations  take k-mer x in each conjugacy class rather than the first positive one. We refer to this set as the symmetric decycling set By symmetry, Mykkeltveit's construction can be used to create an MDS using the first counterclockwise negative Theorem 1 (remaining path partition).In any remaining path in a complete dBG after removing, all the positive nodes  precede all the nonpositive nodes.positive part, containing only positive k-mers, followed by a nonpositive part consisting of only nonpositive k-mers. The proof relies on two lemmas:In other words, a remaining path must consist of two distinct parts: a Lemma 1.The k-mers associated with all incoming neighbors of a node x in a dBG have the samex).Proof.y of x differ only in y0 and have embedding with All incoming neighbors Lemma 2.The pure cycles factor a complete dBG; namely, every k-mer belongs to exactly one of the pure cycles.Proof.k-mer is on some pure cycle corresponding to its rotations. Assume the contrary that k-mer x is on two distinct pure cycles, C1 and C2. Let y be the last common node in the path in x. Then, the edges out of y in the two cycles are distinct, contradicting the fact that both correspond to the cyclic rotation of y.Every Proof (Theorem 1).xi be the first nonpositive node in a remaining path x1, …, xt and assume the contrary that there exists a positive xj for j > i. W.l.o.g. assume xj is the first with that property in the path. Let C be the pure cycle that contains xj. C exists, and it is well defined by y be the node preceding xj in C. By xj−1) = y). Because y is nonpositive, xj should be in C, a contradiction.Let double decycling set consisting of k-mers.By a similar argument, in a remaining path after removing We define a partition-compatible minimizer order based on double decycling sets with k and L values. Miniception uses lexicographic order by default, which was shown to have worse performance than random order. To be fair, here we modified Miniception to use the same random hash as our implementations and ran it with the recommended parameters.We compared the performance of our new MDS-based minimizer orders to random orders and to orders based on DOCKS, PASHA, and Miniception , across k-mer order. We calculated particular density factors on 10 randomly selected 10 million–nt segments from Chromosome X of the CHM13 telomere-to-telomere human genome assembly  combination.UHSs for L, and the relative improvement of UHS- and MDS-based orders compared with random orders increases with k. Conversely, as L grows for fixed k, the density factors of the different methods tend to perform similarly to random because longer windows are more likely to contain multiple k-mers from the sets defining the order (UHS or MDS), and the k-mers within the set are ordered randomly. MDS-based orders have much lower standard errors than the others, likely because the decycling-set k-mers remain the same for all repeated runs regardless of the random seed.The MDS-based orders consistently perform similarly or better than UHS-based orders. As hypothesized, the double decycling-set order has lower density than the decycling-set order. Miniception performs better than random and worse than the UHS-based orders, as was previously shown . As expek-mer usage is not uniformly distributed in real genomic sequences. However, the overall shape of the density curves and the performance ranking among the methods remain the same.The particular density factors are presented in k greater than those that are feasible to run with DOCKS and PASHA. k = 20, 50, and 100 on random segments of the human X Chromosome. Average density factors over the repeated runs are shown without error bars for visual clarity. The same plots with error bars as well as the expected density factors for k = 20, 50, and 100 are in We compared the MDS-based orders to the random baseline order and to Miniception for values of k grows, the advantage of the MDS-based orders becomes even more pronounced, and the double decycling-set-based order improves more significantly over the decycling-set-based order. This is true in particular for smaller L, with the differences between the decycling-set-based and double decycling-set-based orders beginning to diminish as L > 2k. The advantage compared to Miniception is maintained for large values of k across most L values, and the MDS-based minimizer orders achieve density factors that are up to 20% lower than Miniception-based orders. Miniception-based orders achieve lower density for small L as MDS-based orders are similar to random in that regime because most windows do not contain a member of the MDS. In contrast, Miniception approaches its theoretical density factor of 1.67 while L ≤ 2k and converges to (but remains below) the performance of random for larger L. In most cases, the particular density factor is slightly higher than the expected density factor, but the overall shape of the performance curves and the relative performance of the methods are the same for both particular and expected density.As L increases, whereas the MDS-based orders converge to two. Interestingly, the shape of the density factor curves is similar to the lower bound with a minimum around L = 2k.There exists a consistent gap between the theoretical lower bound for forward schemes, which is not known to be tight for minimizers and converges to 1.5 as k < 20, the MDS is precomputed and stored as a boolean vector. Thus, k-mer set membership is determined with an O(1) lookup. The exponential growth of the MDS makes it impossible to precompute or store in memory for even moderately larger k. Instead, we compute k-mer set membership on the fly for every k-mer in O(k) time using k-mers are represented using two bits per nucleotide, our implementation in C++ can use CPU-supported 128-bit operations for k ≤ 63 . We used the GMP library (https://gmplib.org/) to support operations for k > 63. As a result, the process is an order of magnitude slower than CPU-supported operations. We report runtimes to compute minimizers of a random 10 million–nt sequence for all methods and three values of k representing the different regimes in We report runtime measurements under three regimes of our MDS-compatible minimizer scheme implementations. For k and L values. The lookup and comparison used by the precomputed decycling set and UHS methods are also fast but limited to relatively small k. k. Our implementation's runtime increases by one to two orders of magnitude for k > 63. However, even the slowest double decycling order processes 10 million nucleotides in less than half a minute for k = 100, achieving much lower density than a random order. Miniception is implemented in Python and has a consistent runtime across all values of k but is much slower than other methods in two of the three regimes and achieves higher density than MDS-based orders.We implemented the pseudorandom order by a simple XOR operation and a comparison of the resulting integers. Thus, the random order is extremely fast for all kk-mers in a boolean vector. In contrast, the memory overhead of the random and MDS-based orders computed on the fly is negligible, and the memory usage is dominated by the sequence being processed. For the lookup-based methods, memory grows exponentially with k, and for a 10 million–nt sequence, the memory needed to store the set begins to dominate the memory for the sequence starting at k = 14. The memory usage, measured as the maximum resident set size, for k = 19 reaches 33.6 GB and is prohibitively large for k = 20.The implementation of lookup-based methods for the precomputed MDS-based order and UHS-based orders represents all 4k. A major limitation of minimizer orders based on UHSs was the need to create and store the whole set in advance. Instead, we based our new order only on an MDS, avoiding the need to add k-mers to make the set universal. This bypasses the costliest steps in DOCKS and PASHA and generates minimizer orders that are even better in terms of their density factors. Furthermore, this approach enables calculating the minimizers in a sequence efficiently on the fly, without the need to store the set. We showed that based on Mykkeltveit's algorithm, we can determine in O(k) if a k-mer belongs to the MDS, and thus, MDS membership can be checked for all k-mers in a sequence. The resulting new orders are comparable or better in their density than UHS-based minimizer orders, thus achieving good performance while avoiding escalating runtime and memory usage with the increase of k.In this study, we developed a method to generate highly effective minimizer orders for any k and L ≤ 2k, the double decycling-set-based orders yielded much lower density than the decycling-set-based orders . In adde window .There are also open theoretical and technical questions arising from our study. It is necessary to explain why the densities of the MDS-based minimizer orders match the shape of the theoretical lower bound for the more general forward sequence sketching schemes. The gap observed between the MDS-based order and the lower bound should be studied, and hopefully, a tighter lower bound for minimizer schemes can be found. Performance analysis of our new minimizer orders in terms of other criteria, including maximum bin load  and consk > 63 to speed up runtimes for very large k. In addition, the implementations using precomputed UHS or MDS could use more efficient data structures to store the set, for example, Bloom filters  and as All code developed under this project is publicly available at GitHub (Supplement 1Supplement 2


H, its reciprocal degree distance is defined asH and H. The goal of this paper is to establish some sufficient conditions to judge that a graph to be ħ-hamiltonian, ħ-path-coverable or ħ-edge-hamiltonian by employing the reciprocal degree distance.Given a connected graph  We call the sequence degree sequence for the network topology H. Throughout this paper we can assume that H from the symbols in the following.In this paper, we are only concerned with simple graphs . We assume these graphs are undirected, connected and finite. Let join (resp. union) of two graphs p vertices.We use H is ħ-hamiltonian. Therefore 0-hamiltonian is similar to that of hamiltonian. A hamiltonian path of a graph H is nothing but a sequence of incident edges covering exactly all vertices in the graph H. At this moment, we call H traceable. In general, a graph H is ħ-path-coverable if we can cover the vertex set by no more than ħ paths which do not have any two identical vertices. These paths are called vertex-disjoint in graph theory. Especially, 1-path-coverable is similar to that of traceable. We call a connected graph H ħ-edge-hamiltonian (ħ is a positive integer) under the following condition: If we choose a group of vertex-disjoint paths which have less than or equal to ħ edges totally, the group must form a subgraph of some hamiltonian cycle of H. Standard graph theory terminology, see e.g., If the subgraph induced by Molecular structure descriptors are also referred to as topological indices. These indices form an important part of the theoretical and mathematical chemistry, which are used to describe the characteristics of chemical compounds H is connected, then we define the reciprocal degree distanceadditively weighted Harary index because this invariant is regarded as an additively weighted variant of the Harary index. The present name for If Denote In graph theory, it is often interesting and challenging to determine whether a graph has some kind of important graph property. For example, it is NP-complete to decide whether a given graph is hamiltonian ħ-hamiltonian, ħ-path-coverable and ħ-edge-hamiltonian by using the reciprocal degree distance are given respectively in this paper. The following theorems are verified.Motivated by Theorem 1.1Letbe a graph withvertices. Ifthen H is ħ-hamiltonian.Theorem 1.2Letbe a graph withvertices. Letandbe two integers.i) Foris even and, oris odd andor, ifthen H is ħ-path-coverable. Ifthen π enforces ħ-hamiltonian.(b) Ifthen π enforces ħ-edge-hamiltonian., there exists an integer x is an integer, we haveAssume to the contrary that Case 1.H is ħ-hamiltonian.Case 2.Case 1, and we are done for this case. □Proof of Theorem 1.2H is not ħ-path-coverable. Therefore according to k with x is an integer, we obtainAssume to the contrary that Case 1.Since Subcase 1.1.H is ħ-path-coverable.Subcase 1.2.H is ħ-path-coverable.Case 2.Since Subcase 2.1.Subcase 1.1, and we are done for this case.Subcase 2.2.H is ħ-path-coverable. □Proof of Theorem 1.3H is not ħ-edge-hamiltonian. Thus by b), there exists an integer x is an integer, we haveAssume to the contrary that Case 1.H is ħ-edge-hamiltonian.Since Case 2.Case 1, and we are done for this case. □Since 4k-connectivity and β-deficiency of graphs are mainly studied in ħ-hamiltonian, ħ-path-coverable or ħ-edge-hamiltonian by means of RDD. In this paper, these problems are completely solved. Compared with the previous research, we improved the proof method and assisted matlab to solve the problems.In the previous relevant research literature, Mingqiang An, Yinan Zhang: Conceived and designed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper. Kinkar Chandra Das, Yilun Shang: Performed the experiments; Analyzed and interpreted the data; Wrote the paperThe authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.