Solving the k-Independent Sets Problem of Graphs by Gröbner Bases

The aim of this paper is to given an algebraic computational method for finding maximal independent sets as well as the independent number of an arbitrary finite graph of n vertices G by strengthening the problem of finding maximal independent sets of G to the problem of finding k-independent sets in G for. It is shown that the existence of k-independent sets in G is equivalent to the existence of solutions of a system of multivariate polynomial equations. It follows that the problem of finding k-independent sets can be realized by using Gröbner bases of polynomial ideals. Since the number of k-independent sets is finite, the triangular equations composed by Gröbner bases are easier to be solved. Consequently, the maximal independent sets and the independent number of G are obtained after solving at most n such equations. Finally, the numerical example is presented to illustrate the effectiveness of this algebraic computational method.

Cosmology-Independent Photon Mass Limits from Localized Fast Radio Bursts by Using Artificial Neural Networks

A hypothetical photon mass m_(γ) can produce a frequency-dependent vacuum dispersion of light, which leads to an additional time delay between photons with different frequencies when they propagate through a fixed distance. The dispersion measure and redshift measurements of fast radio bursts(FRBs) have been widely used to constrain the rest mass of the photon. However, all current studies analyzed the effect of the frequency-dependent dispersion for massive photons in the standard ΛCDM cosmological context. In order to alleviate the circularity problem induced by the presumption of a specific cosmological model based on the fundamental postulate of the masslessness of photons, here we employ a new model-independent smoothing technique, artificial neural network(ANN), to reconstruct the Hubble parameter H(z) function from 34 cosmic-chronometer measurements.By combining observations of 32 well-localized FRBs and the H(z) function reconstructed by ANN, we obtain an upper limit of m_(γ) ≤ 3.5 × 10^(-51)kg, or equivalently m_(γ) ≤ 2.0 × 10^(-15)eV/c^(2)(m_(γ) ≤ 6.5 × 10^(-51)kg, or equivalently m_(γ) ≤ 3.6 × 10^(-15)eV/c_(2)) at the 1σ(2σ) confidence level. This is the first cosmology-independent photon mass limit derived from extragalactic sources.

A Constrained Local Neighborhood Approach for Efficient Markov Blanket Discovery in Undirected Independent Graphs

When learning the structure of a Bayesian network,the search space expands significantly as the network size and the number of nodes increase,leading to a noticeable decrease in algorithm efficiency.Traditional constraint-based methods typically rely on the results of conditional independence tests.However,excessive reliance on these test results can lead to a series of problems,including increased computational complexity and inaccurate results,especially when dealing with large-scale networks where performance bottlenecks are particularly evident.To overcome these challenges,we propose a Markov blanket discovery algorithm based on constrained local neighborhoods for constructing undirected independence graphs.This method uses the Markov blanket discovery algorithm to refine the constraints in the initial search space,sets an appropriate constraint radius,thereby reducing the initial computational cost of the algorithm and effectively narrowing the initial solution range.Specifically,the method first determines the local neighborhood space to limit the search range,thereby reducing the number of possible graph structures that need to be considered.This process not only improves the accuracy of the search space constraints but also significantly reduces the number of conditional independence tests.By performing conditional independence tests within the local neighborhood of each node,the method avoids comprehensive tests across the entire network,greatly reducing computational complexity.At the same time,the setting of the constraint radius further improves computational efficiency while ensuring accuracy.Compared to other algorithms,this method can quickly and efficiently construct undirected independence graphs while maintaining high accuracy.Experimental simulation results show that,this method has significant advantages in obtaining the structure of undirected independence graphs,not only maintaining an accuracy of over 96%but also reducing the number of conditional independence tests by at least 50%.This significant performance improvement is due to the effective constraint on the search space and the fine control of computational costs.

Evaluation of Industrial IoT Service Providers with TOPSIS Based on Circular Intuitionistic Fuzzy Sets

Industrial Internet of Things(IIoT)service providers have become increasingly important in the manufacturing industry due to their ability to gather and process vast amounts of data from connected devices,enabling manufacturers to improve operational efficiency,reduce costs,and enhance product quality.These platforms provide manufacturers with real-time visibility into their production processes and supply chains,allowing them to optimize operations and make informed decisions.In addition,IIoT service providers can help manufacturers create new revenue streams through the development of innovative products and services and enable them to leverage the benefits of emerging technologies such as Artificial Intelligence(AI)and machine learning.Overall,the implementation of IIoT platforms in the manufacturing industry is crucial for companies seeking to remain competitive and meet the ever-increasing demands of customers in the digital age.In this study,the evaluation criteria to be considered in the selection of IIoT service provider in small andmedium-sized(SME)manufacturing enterprises will be determined and IIoT service providers alternatives will be evaluated using the technique for order preference by similarity to an ideal solution(TOPSIS)method based on circular intuitionistic fuzzy sets.Based on the assessments conducted in accordance with the literature review and expert consultations,a set of 8 selection criteria has been established.These criteria encompass industry expertise,customer support,flexibility and scalability,security,cost-effectiveness,reliability,data analytics,as well as compatibility and usability.Upon evaluating these criteria,it was observed that the security criterion holds the highest significance,succeeded by cost-effectiveness,data analytics,flexibility and scalability,reliability,and customer support criteria,in descending order of importance.Following the evaluation of seven distinct alternatives against these criteria,it was deduced that the A6 alternative,a German service provider,emerged as the most favorable option.The identical issue was addressed utilizing sensitivity analysis alongside various multi-criteria decision-making(MCDM)methods,and after comprehensive evaluation,the outcomes were assessed.Spearman’s correlation coefficient was computed to ascertain the association between the rankings derived from solving the problem using diverse MCDM methods.

Fuzzy Multi-Criteria Decision Support System for the Best Anti-Aging Treatment Selection Process through Normal Wiggly Hesitant Fuzzy Sets

This socialized environment among educated and developed people causes themto focusmore on their appearance and health,which turns them towards medical-related treatments,leading us to discuss anti-aging treatment methods for each age group,particularly for urban people who are interested in this.Some anti-aging therapies are used to address the alterations brought on by aging in human life without the need for surgery or negative effects.Five anti-aging therapies such as microdermabrasion or dermabrasion,laser resurfacing anti-aging skin treatments,chemical peels,dermal fillers for aged skin,and botox injections are considered in this study.Based on the criteria of safety risk,investment cost,customer happiness,and side effects,the optimal alternative is picked.As a result,a NormalWiggly Hesitant Pythagorean Fuzzy Set(NWHPFS)is constructed and used in Multi-Criteria Decision-Making(MCDM)using traditional wavy mathematical approaches.The entropy approach is utilized to determine weight values,and the Normal Wiggly Hesitant Pythagorean-VlseKriterijumska Optimizacija I Kompromisno Resenje(NWHPF-VIKOR)method is utilized to rank alternatives using MCDM methodologies.Sensitivity analysis and comparative analysis were performed to ensure the robustness and reliability of the proposed method.The smart final choice will undoubtedly assist Decision Makers(DM)in making the right judgments,and the MCDM approach will undoubtedly assist individuals in understanding the medicine.

Origin of the Dashuigou independent tellurium deposit at Qinghai–Xizang Plateau: constraints from the light stable isotopes C, O, and H

By studying the light isotopic compositions of carbon,oxygen,and hydrogen,combined with previous research results on the ore-forming source of the deposit,the authors try to uncover its metallogenic origin.The δ^(18)O and δ^(13)C isotope signatures of dolomite samples vary between 10.2 and 13.0‰,and between−7.2 and−5.2‰,respectively,implying that the carbon derives from the upper mantle.δD and δ^(18) O of quartz,biotite,and muscovite from diff erent ore veins of the deposit vary between−82 and−59‰,and between 11.6 and 12.4‰,respectively,implying that the metallogenic solutions are mainly magmatic.According to the relevant research results of many isotope geologists,the fractionation degree of hydrogen isotopes increases as the depth to the Earth’s core increases,and the more diff erentiated the hydrogen isotopes are,the lower their values will be.In other words,mantle-derived solutions can have extremely low hydrogen isotope values.This means that the δD‰ value−134 of the pyrrhotite sample numbered SD-34 in this article may indicate mantle-derived oreforming fl uid of the deposit.The formation of the Dashuigou tellurium deposit occurred between 91.71 and 80.19 Ma.

Novelty of Different Distance Approach for Multi-Criteria Decision-Making Challenges Using q-Rung Vague Sets

In this article,multiple attribute decision-making problems are solved using the vague normal set(VNS).It is possible to generalize the vague set(VS)and q-rung fuzzy set(FS)into the q-rung vague set(VS).A log q-rung normal vague weighted averaging(log q-rung NVWA),a log q-rung normal vague weighted geometric(log q-rung NVWG),a log generalized q-rung normal vague weighted averaging(log Gq-rung NVWA),and a log generalized q-rungnormal vagueweightedgeometric(logGq-rungNVWG)operator are discussed in this article.Adescription is provided of the scoring function,accuracy function and operational laws of the log q-rung VS.The algorithms underlying these functions are also described.A numerical example is provided to extend the Euclidean distance and the Humming distance.Additionally,idempotency,boundedness,commutativity,and monotonicity of the log q-rung VS are examined as they facilitate recognizing the optimal alternative more quickly and help clarify conceptualization.We chose five anemia patients with four types of symptoms including seizures,emotional shock or hysteria,brain cause,and high fever,who had either retrograde amnesia,anterograde amnesia,transient global amnesia,post-traumatic amnesia,or infantile amnesia.Natural numbers q are used to express the results of the models.To demonstrate the effectiveness and accuracy of the models we are investigating,we compare several existing models with those that have been developed.

An Intelligent MCGDM Model in Green Suppliers Selection Using Interactional Aggregation Operators for Interval-Valued Pythagorean Fuzzy Soft Sets

Green supplier selection is an important debate in green supply chain management(GSCM),attracting global attention from scholars,especially companies and policymakers.Companies frequently search for new ideas and strategies to assist them in realizing sustainable development.Because of the speculative character of human opinions,supplier selection frequently includes unreliable data,and the interval-valued Pythagorean fuzzy soft set(IVPFSS)provides an exceptional capacity to cope with excessive fuzziness,inconsistency,and inexactness through the decision-making procedure.The main goal of this study is to come up with new operational laws for interval-valued Pythagorean fuzzy soft numbers(IVPFSNs)and create two interaction operators-the intervalvalued Pythagorean fuzzy soft interaction weighted average(IVPFSIWA)and the interval-valued Pythagorean fuzzy soft interaction weighted geometric(IVPFSIWG)operators,and analyze their properties.These operators are highly advantageous in addressing uncertain problems by considering membership and non-membership values within intervals,providing a superior solution to other methods.Moreover,specialist judgments were calculated by the MCGDM technique,supporting the use of interaction AOs to regulate the interdependence and fundamental partiality of green supplier assessment aspects.Lastly,a statistical clarification of the planned method for green supplier selection is presented.

On Multi-Granulation Rough Sets with Its Applications

Recently,much interest has been given tomulti-granulation rough sets (MGRS), and various types ofMGRSmodelshave been developed from different viewpoints. In this paper, we introduce two techniques for the classificationof MGRS. Firstly, we generate multi-topologies from multi-relations defined in the universe. Hence, a novelapproximation space is established by leveraging the underlying topological structure. The characteristics of thenewly proposed approximation space are discussed.We introduce an algorithmfor the reduction ofmulti-relations.Secondly, a new approach for the classification ofMGRS based on neighborhood concepts is introduced. Finally, areal-life application from medical records is introduced via our approach to the classification of MGRS.

Are cities greener because of innovation?Independent innovation policy and green economic efficiency

The national independent innovation demonstration zone(NIIDZ)is an independent innovation policy that plays a crucial role in implementing strategies.Given the importance of the NIIDZ,this study uses panel data of 278 prefecture-level cities in China from 2006 to 2020 and empirically examines the effect and internal mechanism of the NIIDZ on green economic efficiency(GEE)using the difference-in-difference model(DID).The results show that the NIIDZ effectively enhances the growth of GEE,and the results remain valid through several robustness tests,such as year-by-year propensity score matching.The transmission mechanism suggests that the NIIDZ indirectly drives GEE by accelerating scientific and technological investment,promoting talent concentration,and optimizing the industrial structure.Moreover,heterogeneity analysis reveals that the promotion effect of the NIIDZ on GEE is more prominent in the eastern region and high green development level areas.The study’s findings can serve as a reference for China to further utilize the policy effectiveness of the NIIDZ and accelerate the high-quality development of the green economy in the future.

All-electron basis sets for H to Xe specific for ZORA calculations:Applications in atoms and molecules

A segmented basis set of quadruple zeta valence quality plus polarization functions(QZP)for H through Xe was developed to be used in conjunction with the ZORA Hamiltonian.This set was augmented with diffuse functions to describe electrons farther away from the nuclei adequately.Using the ZORA-CCSD(T)/QZP-ZORA theoretical model,atomic ionization energies and bond lengths,harmonic vibrational frequencies,and atomization energies of some molecules were calculated.The addition of core-valence corrections has been shown to improve the agreement between theoretical and experimental results for molecular properties.For atomization energies,a similar observation emerges when considering spin-orbit couplings.With the augmented QZP-ZORA set,static mean dipole polarizabilities of a set of atoms were calculated and compared with previously published recommended and experimental values.Performance evaluations of the ZORA and Douglas–Kroll–Hess Hamiltonians were made for each property studied.

Pseudo-Semi-Overlap Functions-Based Fuzzy Rough Sets Applied to Image Edge Extraction

As an extension of overlap functions, pseudo-semi-overlap functions are a crucial class of aggregation functions. Therefore, (I, PSO)-fuzzy rough sets are introduced, utilizing pseudo-semi-overlap functions, and further extended for applications in image edge extraction. Firstly, a new clustering function, the pseudo-semi-overlap function, is introduced by eliminating the symmetry and right continuity present in the overlap function. The relaxed nature of this function enhances its applicability in image edge extraction. Secondly, the definitions of (I, PSO)-fuzzy rough sets are provided, using (I, PSO)-fuzzy rough sets, a pair of new fuzzy mathematical morphological operators (IPSOFMM operators) is proposed. Finally, by combining the fuzzy C-means algorithm and IPSOFMM operators, a novel image edge extraction algorithm (FCM-IPSO algorithm) is proposed and implemented. Compared to existing algorithms, the FCM-IPSO algorithm exhibits more image edges and a 73.81% decrease in the noise introduction rate. The outstanding performance of (I, PSO)-fuzzy rough sets in image edge extraction demonstrates their practical application value.

Study on Independent Supervision Function of Design Assurance System

The effective operation of a design assurance system cannot be achieved without the effective performance of the independent supervision function.As one of the core functions of the design assurance system,the purpose of the independent supervision function is to ensure that the system operates within the scope of procedures and manuals.At present,the function of independent supervision is a difficult and confusing issue for various original equipment manufacturers as well as suppliers,and there is an urgent requirement to put forward relevant requirements and form relevant methods.Based on the above mentioned objective,the basic requirements of the independent supervision function of design assurance system were studied,the problems and deficiencies in the organization,staffing,and methods existing in the current independent supervision function were analyzed,the improvement suggestions and measures for the performance of the independent supervision function from the aspects of the organization,staffing,procedures,and suppliers were put forward.The present work and conclusions provide guidance and direction for the effective operation of the design assurance system.

Analysis of the Role of Problem-Based Independent Learning Model in Teaching Cerebral Ischemic Stroke First Aid in Emergency Medicine

Objective:To analyze the effect of using a problem-based(PBL)independent learning model in teaching cerebral ischemic stroke(CIS)first aid in emergency medicine.Methods:90 interns in the emergency department of our hospital from May 2022 to May 2023 were selected for the study.They were divided into Group A(45,conventional teaching method)and Group B(45 cases,PBL independent learning model)by randomized numerical table method to compare the effects of the two groups.Results:The teaching effect indicators and student satisfaction scores in Group B were higher than those in Group A(P<0.05).Conclusion:The use of the PBL independent learning model in the teaching of CIS first aid can significantly improve the teaching effect and student satisfaction.

NEIGHBORHOOD UNION OF INDEPENDENT SETS AND HAMILTONICITY OF CLAW-FREE GRAPHS

Let G be a graph,for any u∈V(G),let N(u) denote the neighborhood of u and d(u)=|N(u)| be the degree of u.For any UV(G),let N(U)=∪_~u∈U N(u), and d(U)=|N(U)|.A graph G is called claw-free if it has no induced subgraph isomorphic to K_~1,3 .One of the fundamental results concerning cycles in claw-free graphs is due to Tian Feng,et al.: Let G be a 2-connected claw-free graph of order n,and d(u)+d(v)+d(w)≥n-2 for every independent vertex set {u,v,w} of G, then G is Hamiltonian. It is proved that,for any three positive integers s,t and w,such that if G is a (s+t+w-1)-connected claw-free graph of order n,and d(S)+d(T)+d(W)>n-(s+t+w) for every three disjoint independent vertex sets S,T,W with |S|=s,|T|=t,|W|=w,and S∪T∪W is also independent,then G is Hamiltonian.Other related results are obtained too.

The Number of Maximal Independent Sets in Quasi-Tree Graphs and Quasi-Forest Graphs

A maximal independent set is an independent set that is not a proper subset of any other independent set. A connected graph (respectively, graph) G with vertex set V(G) is called a quasi-tree graph (respectively, quasi-forest graph), if there exists a vertex x &isin;V(G) such that G &minus;x?is a tree (respectively, forest). In this paper, we survey on the large numbers of maximal independent sets among all trees, forests, quasi-trees and quasi-forests. In addition, we further look into the problem of determining the third largest number of maximal independent sets among all quasi-trees and quasi-forests. Extremal graphs achieving these values are also given.

An Alternative Proof of the Largest Number of Maximal Independent Sets in Connected Graphs Having at Most Two Cycles

G. C. Ying, Y. Y. Meng, B. E. Sagan, and V. R. Vatter [1] found the maximum number of maximal independent sets in connected graphs which contain at most two cycles. In this paper, we give an alternative proof to determine the largest number of maximal independent sets among all connected graphs of order n ≥ 12, which contain at most two cycles. We also characterize the extremal graph achieving this maximum value.

Aggregation Operators for Decision Making Based on q-Rung Orthopair Fuzzy Hypersoft Sets:An Application in Real Estate Project

In this paper,a decision-making problem with a q-rung orthopair fuzzy hypersoft environment is developed,and two operators of ordered weighted average and induced ordered weighted average are developed.Several fundamental features are also derived.The induced ordered weighted average operator is essential in a q-ROFH environment as the induced ordered aggregation operators are special cases of the existing aggregation operators that already exist in q-ROFH environments.The main function of these operators is to help decision-makers gain a complete understanding of uncertain facts.The proposed aggregation operator is applied to a decision-making problem,with the aim of selecting the most promising real estate project for investment.

A NOTE ON THE JULIA SETS OF ENTIRE SOLUTIONS TO DELAY DIFFERENTIAL EQUATIONS

Let f be an entire solution of the Tumura-Clunie type non-linear delay differential equation.We mainly investigate the dynamical properties of Julia sets of f,and the lower bound estimates of the measure of related limiting directions is verified.

Quantum Algorithm for Approximating Maximum Independent Sets

We present a quantum algorithm for approximating maximum independent sets of a graph based on quantum non-Abelian adiabatic mixing in the sub-Hilbert space of degenerate ground states,which generates quantum annealing in a secondary Hamiltonian.For both sparse and dense random graphs G,numerical simulation suggests that our algorithm on average finds an independent set of size close to the maximum size α(G) in low polynomial time.The best classical algorithms,by contrast,produce independent sets of size about half of α(G)in polynomial time.