In the decision analysis, due to the fuzziness, complexity and uncertainty of human thinking and objective things, decision-makers often cannot give accurate values when making judgements on things by pairwise comparison. Herrera et al. [1,2] noted the concept of using linguistic information to express evaluation results. The linguistic judgement matrix has become an important information form for decision-makers to compare objects. However, it is also important whether the linguistic judgement matrix is consistent before the decision-maker gives the judgement result. At present, the research in this field has attracted the extensive attention of scholars and achieved some results [3,4,5,6,7,8]. The number of illogical objects is found by constructing the directed graph, and then the satisfactory consistency degree of linguistic judgement is calculated ; this problem is solved by deriving the compatibility of matrix and weight matrix .
Many achievements on the consistency of linguistic judgement matrix have been obtained [9,10,11,12,13,14,15,16]. From the current research results, few scholars directly use the definition to determine the satisfactory consistency of the linguistic judgement matrix. Most scholars use graph theory knowledge or define a preference relation matrix to determine the satisfactory consistency of the linguistic judgement matrix. Whether the diagonal element of the reachable matrix is 0 was used to determine whether the linguistic judgement matrix has satisfactory consistency . A method for determining the satisfactory consistency of the linguistic judgement matrix was established by using the concepts of the derived matrix and preference relation matrix . The linguistic judgement matrix and graph theory were linked together to solve the problem of satisfactory consistency of linguistic judgement matrix with no difference between two objects . Fan  used the concept of the induced matrix to transform the linguistic judgement matrix into a quantitative matrix to solve this problem. However in these methods, multiple matrices are required to be multiplied. Based on the former, this paper also uses the preference relation matrix to analyse the satisfactory consistency of the linguistic judgement matrix, but it is different. It proposes the concepts of the equivalent scheme, row preference value, column preference value, and standard 0-1 permutation matrix. This paper also presents a method to determine the satisfactory consistency of the linguistic judgement matrix. The method not only solves the problem of satisfactory consistency but is also simple and effective.
However, only judging whether the linguistic judgement matrix is consistent cannot meet the requirements of decision-makers. In recent years, some achievements have been made in other aspects of the linguistic judgement matrix [17,18,19,20,21,22,23,24,25,26]. Definitions of the consistency, acceptable incompleteness and complete consistency of the linguistic judgement matrix are given as well as a method for ranking alternatives . A new proof method is given in reference . It proves that if the judgement matrices given by each decision-maker were acceptable, then their weighted geometric average judgement matrices were also acceptable. For uncertain linguistic multi-attribute decision-making problems with incomplete weight coefficients, an optimisation model was established by transforming uncertain linguistic information into trapezoidal fuzzy numbers to obtain scheme weight coefficients . The definition and properties of binary intuitionistic fuzzy language preference information are given, and a ranking method based on binary intuitionistic fuzzy language preference information is proposed . Pang  compared the preference information of individuals and groups and measured the influence of the preference information of individuals and groups on decision-making from three evaluation indicators of consistency, closeness and monotonicity. Hsu  used the direct and indirect pairwise comparison rules to construct a multi-criteria incomplete language preference relationship model to solve the problem of increasing solutions and decreasing consistency.
This paper discusses the method to determine and adjust the satisfactory consistency of the linguistic judgement matrix with strict preference relation between two objects. In this paper, the satisfactory consistency of the linguistic judgement matrix can be modified by using whether the 0-1 permutation preference relation matrix of the linguistic judgement matrix is an upper triangular matrix, and then the three objects for forming a circular circle of judgement results are found by using the cyclic matrix. Using the size of the row preference value to modify the cyclic circle, the 0-1 type permutation preference relation matrix with satisfactory consistency is obtained. Finally the linguistic judgement matrix with satisfactory consistency is obtained.
Orderliness: when Inverse operation neg: neg( Maximisation operation: if Minimisation operation: if
Inverse operation neg: neg(
Maximisation operation: if
Minimisation operation: if
[11,12] A linguistic judgement matrix on the finite object set
In the linguistic judgement matrix, if
According to the meaning of the objects in the linguistic term set, we can know that the objects in the linguistic judgement matrix have the following properties:
If If If
A preference relation matrix of the linguistic judgement matrix on the finite object set
In the preference relation matrix
A linguistic judgement matrix
In the last section, the definition of the linguistic judgement matrix is given, and the definitions of the equivalent scheme, the row preference value and the column preference value, and the preference relation matrix are proposed. On the basis of comparison of each object, the method to determine the satisfactory consistency of the linguistic judgement matrix is given, and the ranking method is also given.
For example: if
According to the example above_St can be concluded that the 0-1 permutation preference relation matrix of the preference relation matrix is not unique.
For example, the following matrices are the standard 0-1 permutation matrices:
First, we prove the necessity. According to definition 2.5, if Then,
From equation (2), it can be concluded that,
First, we prove the necessity. According to definition 2.5, if
From equation (2), it can be concluded that,
The corresponding 0-1 permutation preference relation matrix is
It can be seen from the above that when the linguistic judgement matrix has satisfactory consistency, its 0-1 permutation preference relation matrix is the standard 0-1 permutation matrix.
Second, we prove the sufficiency. Let
Suppose the object
From the above proof, it can be concluded that the 0-1 permutation preference relation matrix is an upper triangular matrix when Step 1: Write the corresponding preference relation matrix according to the linguistic judgement matrix; Step 2: Write the 0-1 permutation preference relation matrix according to the preference relation matrix; Step 3: Judge whether the 0-1 permutation preference relation matrix is the standard 0-1 permutation matrix; Step 4: If it is an upper triangular matrix, it has a strict preference relationship; otherwise, proceed to the next step; Step 5: If it is a standard 0-1 permutation matrix, Step 6: Judge the end.
Step 1: Write the corresponding preference relation matrix according to the linguistic judgement matrix;
Step 2: Write the 0-1 permutation preference relation matrix according to the preference relation matrix;
Step 3: Judge whether the 0-1 permutation preference relation matrix is the standard 0-1 permutation matrix;
Step 4: If it is an upper triangular matrix, it has a strict preference relationship; otherwise, proceed to the next step;
Step 5: If it is a standard 0-1 permutation matrix,
Step 6: Judge the end.
According to the above judgement method, we can judge whether a linguistic judgement matrix has satisfactory consistency or not. At the same time, we can also get a ranking method of objects. The specific steps are as follows:
Step 1: Judge whether the linguistic judgement matrix given by the decision-maker has satisfactory consistency; Step 2: If there is no satisfactory consistency, it will be returned to the decision-maker for adjustment until satisfactory consistency is obtained; Step 3: If there is satisfactory consistency, write the 0-1 permutation preference relation matrix of the linguistic judgement matrix, that is, the standard 0-1 permutation matrix; Step 4: Judge the advantages and disadvantages of the scheme according to the standard 0-1 type permutation matrix; Step 5: Get the sequence of advantages and disadvantages of the object.
Step 1: Judge whether the linguistic judgement matrix given by the decision-maker has satisfactory consistency;
Step 2: If there is no satisfactory consistency, it will be returned to the decision-maker for adjustment until satisfactory consistency is obtained;
Step 3: If there is satisfactory consistency, write the 0-1 permutation preference relation matrix of the linguistic judgement matrix, that is, the standard 0-1 permutation matrix;
Step 4: Judge the advantages and disadvantages of the scheme according to the standard 0-1 type permutation matrix;
Step 5: Get the sequence of advantages and disadvantages of the object.
Following Definition 3.2 and Theorem 3.1, the conclusion can be easily derived.
The secondary sub-formulas of the 0-1 permutation preference relation matrix of linguistic judgement matrix are called the cyclic circle matrix, if they have the following form:
The cyclic circle matrix is the secondary sub-formula of the 0-1 permutation preference relation matrix, which is not necessarily true in reverse. There is only one element of the cyclic circle matrix in the main diagonal of the 0-1 permutation preference relation matrix, and the comparison relationship between the objects and the original linguistic judgement matrix remains unchanged. The comparison result of the three objects constituting the cyclic circle matrix is cyclic, and the cyclic circle matrix composed of the three objects can be the above four forms.
First, we prove the sufficiency. If there is a cyclic circle matrix in the 0-1 permutation preference relation matrix of the linguistic judgement matrix, it shows that the comparison results of the three objects are a cycle circle. Therefore, the comparison results of the objects are not transitive, and the linguistic judgement matrix is not of satisfactory consistency. Second, we prove the necessity. According to theorem 1, the linguistic judgement matrix has strict preference relation between two objects, then its 0-1 permutation preference relation matrix
First, we prove the sufficiency. If there is a cyclic circle matrix in the 0-1 permutation preference relation matrix of the linguistic judgement matrix, it shows that the comparison results of the three objects are a cycle circle. Therefore, the comparison results of the objects are not transitive, and the linguistic judgement matrix is not of satisfactory consistency.
Second, we prove the necessity. According to theorem 1, the linguistic judgement matrix has strict preference relation between two objects, then its 0-1 permutation preference relation matrix
Discussing circular circle matrix of
Discussing the cyclic circle matrix of
It is discussed that there is an element of 1 in the lower semi-triangular matrix, and the corresponding element is 0 in the upper triangular matrix of the 0-1 permutation preference relation matrix. Since 1 of the lower triangular matrix and 0 of the corresponding upper semi-triangular matrix are the results of the same pair of scheme comparison, the 0 element in the upper triangular matrix is no longer discussed. Theorem 2 is proved.
From Theorem 2, we get a method to judge whether the linguistic judgement matrix has satisfactory consistency. At the same time, we also find out the circle composed of three objects. If we modify the circle formed, we can get the order of the objects. Now, we give the criteria of correcting the circulation circle:
If the row preference values of the three objects are not equal, the one with the smallest row preference value is regarded as the worst object among the three objects; If the row preference values of two objects are equal and less than that of the other object, the object corresponding to the row with element 1 in the lower triangular matrix is recorded as the worst object; If the row preference values of the three objects are equal, the sum of the subscripts of the linguistic phrases in the linguistic judgement matrix is compared, and the one with the smallest subscript sum is regarded as the worst object among the three objects; if the sum of subscripts is equal, according to the comparison results with the optimal object, the worst one is marked as the worst one.
If the row preference values of the three objects are not equal, the one with the smallest row preference value is regarded as the worst object among the three objects;
If the row preference values of two objects are equal and less than that of the other object, the object corresponding to the row with element 1 in the lower triangular matrix is recorded as the worst object;
If the row preference values of the three objects are equal, the sum of the subscripts of the linguistic phrases in the linguistic judgement matrix is compared, and the one with the smallest subscript sum is regarded as the worst object among the three objects; if the sum of subscripts is equal, according to the comparison results with the optimal object, the worst one is marked as the worst one.
After finding the worst object, if the cyclic circle is
The specific steps of judgement and correction are given as follows:
Step 1: Give the preference relation matrix of linguistic judgement matrix; Step 2: 0-1 permutation preference relation matrix of linguistic judgement matrix is given; Step 3: Judge whether the 0-1 permutation preference relation matrix is an upper triangular matrix; if the linguistic judgement matrix has satisfactory consistency, the judgement ends, otherwise proceed to the next step; Step 4: Find the cyclic matrix in the 0-1 permutation matrix, and get the objects of forming the cyclic matrix; Step 5: According to the principle of modified cycle, the first cycle is modified to obtain the modified 0-1 permutation preference relation matrix; Step 6: Rearrange the 0-1 permutation preference relation matrix and turn to step 3; Step 7: According to the principle of determining the comparison results, the modified linguistic judgement matrix is obtained; Step 8: Get the ranking of the objects; Step 9: Judge the end.
Step 1: Give the preference relation matrix of linguistic judgement matrix;
Step 2: 0-1 permutation preference relation matrix of linguistic judgement matrix is given;
Step 3: Judge whether the 0-1 permutation preference relation matrix is an upper triangular matrix; if the linguistic judgement matrix has satisfactory consistency, the judgement ends, otherwise proceed to the next step;
Step 4: Find the cyclic matrix in the 0-1 permutation matrix, and get the objects of forming the cyclic matrix;
Step 5: According to the principle of modified cycle, the first cycle is modified to obtain the modified 0-1 permutation preference relation matrix;
Step 6: Rearrange the 0-1 permutation preference relation matrix and turn to step 3;
Step 7: According to the principle of determining the comparison results, the modified linguistic judgement matrix is obtained;
Step 8: Get the ranking of the objects;
Step 9: Judge the end.
In this paper, we give a method to judge whether the linguistic judgement matrix with the not equivalent object have satisfactory consistency. Then, for the linguistic judgement matrix with strict preference relation of pairwise objects, we give a method to determine the satisfactory consistency of the linguistic judgement matrix. We use whether the 0-1 permutation preference relation matrix is an upper triangular matrix to determine the satisfactory consistency of the linguistic judgement matrix. This method is simple and intuitive. It does not need to establish models and complex operations but only needs to make simple transformation of the matrix. Then, three objects of forming a cyclic circle are found by using the cyclic matrix, and the modified linguistic judgement matrix is obtained by giving the correction principle. However, the modified method is only applicable to the matrix with strict preference relationship, and there are some defects in the matrix with an equivalent scheme. Further research is needed.
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of Listed Companies Based on Fisher Linear Discriminant Analysis High Simulation Reconstruction of Crowd Animation Based on Optical Flow Constraint Equation Construction of Intelligent Search Engine for Big Data Multimedia Resource Subjects Based on Partial Least Squares Structural Equation 3D Animation Simulation of Computer Fractal and Fractal Technology Combined with Diamond-Square Algorithm Analysis of the Teaching Quality of Physical Education Class by Using the Method of Gradient Difference The Summation of Series Based on the Laplace Transformation Method in Mathematics Teaching Optimal Solution of the Fractional Differential Equation to Solve the Bending Performance Test of Corroded Reinforced Concrete Beams under Prestressed Fatigue Load Animation VR scene mosaic modeling based on generalized Laplacian equation Radial Basis Function Neural Network in Vibration Control of Civil Engineering Structure Optimal Model Combination of Cross-border E-commerce Platform Operation Based on Fractional Differential Equations The influence of accounting computer information processing technology on enterprise internal control under panel data simultaneous equation Research on Stability of Time-delay Force Feedback Teleoperation System Based on Scattering Matrix BIM Building HVAC Energy Saving Technology Based on Fractional Differential Equation Construction of comprehensive evaluation index system of water-saving irrigation project integrating penman Montei the quation Human Resource Management Model of Large Companies Based on Mathematical Statistics Equations Data Forecasting of Air-Conditioning Load in Large Shopping Malls Based on Multiple Nonlinear Regression Analysis of technical statistical indexes of college tennis players under the win-lose regression function equation Automatic extraction and discrimination of vocal main melody based on quadratic wave equation Analysis of wireless English multimedia communication based on spatial state model equation 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on vectorized image processing technology BIM Engineering Management Oriented to Curve Equation Model Relationship Between Physical Activity Sedentary Behavior and Body Mass Index in Children and Adolescents Based on FIT Modelo Green supply chain innovation management strategy based on the combination of low carbon economy and e-commerce with big data technology Leakage control of urban water supply network and mathematical analysis and location of leakage points based on machine learning Analysis of higher education management strategy based on entropy and dissipative structure theory Grid data asset relationship and intelligent classification integrating knowledge graph and Internet of Things Prediction of corporate financial distress based on digital signal processing and multiple regression analysis Mathematical Method to Construct the Linear Programming of Football Training Multimedia sensor image detection based on constrained underdetermined equation The Size of Children's 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and character intelligence analysis based on semantic ontology and Poisson equation Analysis of digital humanistic knowledge production based on natural language processing Design of language assisted learning model and online learning system under the background of artificial intelligence Study on the influence of adolescent smoking on physical training vital capacity in eastern coastal areas Application of machine learning in stock selection Comparative analysis of CR of ideological and political education in different regions based on improved fuzzy clustering The Construction of Judgment Model for Cultivating Intelligent Accounting Professionals in Universities Based on Logarithmic Periodic Power Law Model Action of Aut( G) on the set of maximal subgroups of p-groups Digital Regeneration and Database Construction of Hunan Embroidery Needlework under the Perspective of Artificial Intelligence Research on the practice of innovation and entrepreneurship education in universities under the environment of big data Application of digital technology in environmental art design Research on loyalty prediction of e-commerce customer based on data mining Research on the evaluation system of tourism management classroom teaching based on multimedia big data analysis The application of ARCS motivation model in college oral English teaching Research on Innovation of Translation Teaching and Translation Strategies for College Students in Multimedia Background A multi-channel convolutional neural network based on attention mechanism fusion for facial expression recognition Exploring the development of cultural and creative tourism under the perspective of ecological theory of tourism destination management Modeling Analysis of User Consumption Data of Internet Culture Industry in the Context of Emerging Technologies Reference: An algorithm for recognizing the main melody of orchestral music based on artificial intelligence of music melody contour Algebraic Equations in 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game review IoT-based interaction design of smart home products for elderly families Mathematical Calculus Modeling in Improving the Teaching Performance of Shot Put Model Construction of Influencing Factors of Physical Education Core Quality Cultivation Based on Multiple Linear Regression A multidimensional interpretation analysis of Marxism globalization theory based on distributed Internet Application of Nonlinear Differential Equation in Electric Automation Control System Research on the operation and management mechanism of cross-border e-commerce shared practice training base Strategy for financing mode optimization in international trade supply chain based on deep learning model Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Strategies for improving Chinese language proficiency based on artificial intelligence technology Early warning and detection of geological disasters based on intelligent genetic algorithm The success rate model of college students’ new entrepreneurship based on nonlinear differential equations Computational Algorithm to Solve Two–Body Problem Using Power Series in Geocentric System Research on the application of marketing strategy of national fitness exercise and dance events in the construction of sports culture based on big data technology Exploring the Application of Modern Fine Arts Artistic Modeling and Color in Modern Art Design Based on Information Fusion Technology Decisions of competing supply chain with altruistic retailer under risk aversion Strategies for constructing mathematical models of nonlinear systems based on multiple linear regression models Research on China’s tourism marketing model in the data perspective: based on ecological perspective Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations Research on human resource management performance evaluation based on BP algorithm The Marketing of Cross-border E-commerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory Application of Linear Partial Differential Equation Theory in Guiding Football Scientific Training Nonlinear Channel Estimation for Internet of Vehicles Some Necessary Conditions for Feedback Functions of de Bruijn Sequences The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function System Model of Shipping Enterprise Safety Culture Based on Dynamic Calculation Matrix Model An empirical research on economic growth from industrial structure optimisation in the Three Gorges Reservoir area The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Analysis and Research on Influencing Factors of Ideological and Political Education Teaching Effectiveness Based on Linear Equation Study of agricultural finance policy information extraction based on ELECTRA-BiLSTM-CRF Fractional Differential Equations in 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processing system based on ordinary differential equations Modeling of fractional differential equation in cloud computing image fusion algorithm Conventional Algorithms in Sports Training Based on Fractional Differential Equations Dynamic Nonlinear System Based on Complex System Theory in the Development of Vocational Education The Algorithm Accuracy of Mathematical Model to Improve the Transmission Speed of E-commerce Platform The Security of Database Network Model Based on Fractional Differential Equations Electric Vehicle Mechanical Transmission System Based on Fractional Differential Equations The Practice System of Physics and Electronics Courses in Higher Vocational Colleges Based on Fractional Differential Equations The Economic Model of Rural Supply and Demand Under the Data Analysis Function Based on Ordered Probit 3D Modeling System of Indoor Environment Art Landscape Design under Statistical Equation Electronic Information Security Model of Nonlinear Differential Equations The Optimization Model of College Students' Physical Exercise Motivation and Self-control Ability Based on the Mathematical Model of Probability Theory Health monitoring of Bridges based on multifractal theory Health status diagnosis of the bridges based on multi-fractal de-trend fluctuation analysis Application and risk assessment of the energy performance contracting model in energy conservation of public buildings Sensitivity analysis of design parameters of envelope enclosure performance in the dry-hot and dry-cold areas Criminal law imputation path for biometric information Research on composite dynamic disaster prevention and control system of mine earthquake and shock in thick and hard rock mines Research on urban energy Internet planning based on fuzzy comprehensive method Research on innovative strategies of college students’ English teaching under the background of artificial intelligence Electromagnetic interference prediction technology of new energy motor drive system Research on the intelligent path of college students’ network ideological and political education based on big data mining technology Research on the application of PLC technology in electrical automation engineering Application of matrix multiplication in signal sensor image perception Analysing the variation of metadiscourse verb patterns in English academic papers from within and between disciplines Impact of COVID-19 policy on stock prices of listed property companies Research on the experimental principle of deep integration of LETS software and criminal procedure under the background of artificial intelligence VR-based computer maintenance practical training platform development design and application research Design and implementation of an integrated management system for discipline construction and postgraduate education in universities Research on innovative human capital for China’s economic development based on STI model In-depth analysis of the artistic expression of paper-cut elements in the design of boat space Knowledge graph construction and Internet of Things optimisation for power grid data knowledge extraction Modeling the pathway of breast cancer in the Middle East Construction and intelligent analysis of power grid physical data knowledge graph based on Internet of Things for power system Research on industrial Internet of Things and power grid technology application based on knowledge graph and data asset relationship model Research on the effects of financial literacy on rural household credit constraint Research on educational resource recommendation system based on MRLG Rec Visual analysis of the trend of mental health changes among Chinese college students in the context of the normalisation of the new crown epidemic A study on the phenomenon of anaphoric correction in college students’ English conversation Limit cycles of a generalised Mathieu differential system Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Application of data mining in basketball statistics The nonlinear effects of ageing on national savings rate – An Empirical Study based on threshold model Design of fitness walker for the elderly based on ergonomic SAPAD model Research on the innovation of collaborative education model of scientific research and teaching in colleges and universities based on the theory of process reengineering AtanK-A New SVM Kernel for Classification The innovative path of diversified vocal music education reform in colleges and universities under the advanced educational concepts of China and the West Mechanical behaviour of continuous girder bridge with corrugated steel webs constructed by RW Rhino application in ceramic packaging design teaching Study of a linear-physical-programming-based approach for web service selection under uncertain service quality Analysis and countermeasures of cultivating independent learning ability in colleges teaching English based on OBE theory Virtual Museum Scene Design Based on VRAR Realistic Interaction under PMC Artificial Intelligence Model A mathematical model of plasmid-carried antibiotic resistance transmission in two types of cells Projection of Early Warning Identification of Hazardous Sources of Gas Explosion Accidents in Coal Mines Based on NTM Deep Learning Network Research on innovative cultivation mode of English and American literature translation talents based on internet environment Burnout of front-line city administrative law-enforcing personnel in new urban development areas: An empirical research in China Analysis of quantitative management of online intelligent monitoring of tailing ponds based on the perspective of safety prevention and control Optimization of college physical education curriculum based on G-DINA Optimization of local ethnic music teaching transmission path based on logistic regression model RETRACTION NOTE Research on the mining of ideological and political knowledge elements in college courses based on the combination of LDA model and Apriori algorithm Research on non-linear visual matching model under inherent constraints of images Good congruences on weakly U-abundant semigroups Can policy coordination facilitate unimpeded trade? An empirical study on factors influencing smooth trade along the Belt and Road Research on the processing method of multi-source heterogeneous data in the intelligent agriculture cloud platform Internal control index and enterprise growth: An empirical study of Chinese listed-companies in the automobile manufacturing industry Research on design of customer portrait system for E-commerce Research on rule extraction method based on concept lattice of intuitionistic fuzzy language Fed-UserPro: A user profile construction method based on federated learning Design and evaluation of intelligent teaching system on basic movements in PE Research on identifying psychological health problems of college students by logistic regression model based on data mining Research on evaluation system of cross-border E-commerce platform based on the combined model A mathematical model of PCNN for image fusion with non-sampled contourlet transform Financial customer classification by combined model Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Retraction: Wei, Wei and Lu, Liu. “The influence of X fuzzy mathematics method in basketball tactics scoring” Applied Mathematics and Nonlinear Sciences, vol.7, no.1, 2022, pp.677-684 Retraction Retraction: Retraction of: Mundewadi, R. A. and S, Kumbinarasaiah. “Numerical Solution of Abel's Integral Equations using Hermite Wavelet” Applied Mathematics and Nonlinear Sciences, vol.4, no.2, 2019, pp.395-406