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The so-called Hamming distance measures the difference between two binary strings A and B. In simplified form, it measures the number of changes in A to get B. This type of distance is very useful in classical computing in applications such as error correction. It is also advantageous in quantum computing, being for example widely used in quantum machine learning. Since current quantum computers have limited resources, this type of distance is particularly attractive because it can be computed using fewer qubits and operations than other distances such as Euclidean or Manhattan distances. In this paper, two circuits for calculating Hamming distances using exclusively Clifford+T gates are presented. The aim of both circuits is to reduce the quantum cost and number of T gates needed to compute the Hamming distance. The T gate is more expensive than the other gates, so this reduction will have a significant impact on the total cost of the circuits. Furthermore, the proposed circuits are implemented using only Clifford+T gates. The circuits implemented exclusively with this group of gates are compatible with proven error detection and correction codes. [ABSTRACT FROM AUTHOR]

As the rapidly progressing applications of uncertainty theories, the need for modifications to some of their existing mathematical tools or creating new tools to deal correctly with them in various environments is also exposed. Hesitant fuzzy numbers (HFNs), as a particular case of fuzzy numbers, are not an exception to this rule. Considering the necessity of determining the distance between given HFNs in many of their practical applications, this article shows that the existing methods either do not provide correct results or are not able to meet the needs of users. This paper aims to present new methods for distance measures of hesitant fuzzy numbers. To do them, three prevalent distance measures, i.e., the generalized distance measure, the Hamming distance measure, and the Euclidean distance measure, will be optimized into three distinct trinal categories. With the approach of reducing error propagation via reducing some unnecessary mathematical computations, new distance measures on HFNs will be introduced, first. The middle is the modification of the first category, which is more suitable when the given HFNs are equal-distance by the previous formula. Also, as the third category, the weighted form of these distance measures has been proposed, to be used where the real and membership parts of HFNs are not of equal importance. As an application of these, a TOPSIS-based technique for solving multi-attribute group decision-making problems with HFNs has been proposed. A numerical example will be implemented to describe the presented method. Finally, along with the validation of the proposed method, its numerical comparison with some other existing methods will be discussed in detail. [ABSTRACT FROM AUTHOR]

The energy consumption of mechanical products in China is enormous, and the energy utilization rate is low, which is increasingly receiving people's attention. Conducting product design for energy optimization is of great significance for improving energy utilization efficiency. The scheme design of a product is the key to achieving innovation in product design, and the evaluation and decision-making of the design scheme directly affect the results of the later stage of the design. Therefore, correctly evaluating and making reliable decisions on product design schemes that are oriented towards fuzzy decision optimization is an important aspect of product innovation conceptual design. The product modeling design quality evaluation is a multiple attribute group decision making (MAGDM) problems. Recently, the Combined Compromise Solution (CoCoSo) method and information entropy method has been employed to cope with MAGDM issues. The interval neutrosophic sets (INSs) are employed as a tool for portraying uncertain information during the product modeling design quality evaluation. In this paper, the CoCoSo method is designed for MAGDM under INSs. Then, the interval neutrosophic numbers CoCoSo (INN-CoCoSo) method based on the Hamming distance and Euclidean distance is built for MAGDM. The information Entropy method is employed to produce the weight information based on the Hamming distance and Euclidean distance under INNSs. Finally, a practical numerical example for product modeling design quality evaluation is supplied to show the INN-CoCoSo method. The main contributions of this paper are constructed: (1) This paper builds the novel MAGDM based on CoCoSo model under INSs; (2) The information Entropy method is employed to produce the weight information based on the Hamming distance and Euclidean distance under INNSs; (3) The new MAGDM method is proposed for product modeling design quality evaluation based on INN-CoCoSo. [ABSTRACT FROM AUTHOR]

In this article, we discuss a few fresh approaches to the spherical vague normal set (SVNS) approach to multiple attribute decision‐making (MADM) problems. A new generalization of the vague set (VS) and the spherical interval valued fuzzy set (SIVFS) is the spherical vague set (SVS). The spherical vague number (SVN) concepts consolidate normal fuzzy number (NFN) and we defined the spherical vague normal number (SVNN) and some of its intriguing fundamental operations. The purpose of this article is to discuss a novel idea of spherical vague normal weighted averaging (SVNWA), spherical vague normal weighted geometric (SVNWG), generalized spherical vague normal weighted averaging (GSVNWA) and generalized spherical vague normal weighted geometric (GSVNWG) operators. We talked about a flowchart with an algorithm that uses this operators and the MADM approach. With the help of a numerical example, we interact the extended euclidean and hamming distance measures. In this communication, it is also important to elaborate on some key SVN approach characteristics based on various algebraic operations, such as idempotency, boundedness, commutativity, and monotonicity. They are quicker to find the best option, more straightforward and practical. Think about five farmers. The four factors that are taken into account for each of the five farmers are climate, water, soil, disease, and flood, and their corresponding weights are displayed. We want to select the best option from a large number of choices by comparing expert assessments with the criteria. As a result, the conclusions of the defined models are more precise and closely related to Θ. We contrast some of the current models with the ones that have been proposed in order to demonstrate the dependability and utility of the models under investigation. The study's findings are also intriguing and fascinating. [ABSTRACT FROM AUTHOR]

Combinative distance-based assessment (CODAS) is a multi-criteria decision-making (MCDM) method that is based on the Euclidean and Hamming distances of alternatives from the average scores of attributes. Spherical fuzzy sets, as the recent extensions of ordinary fuzzy sets, were developed based on Pythagorean and neutrosophic sets and enable decision-makers to express their membership, non-membership, and hesitancy degrees independently and in a larger domain than most other fuzzy extensions. This paper proposes a new interval-valued spherical fuzzy CODAS method and provides extra space for catching the vagueness in the nature of the problem. The feasibility and practicality of the proposed model are illustrated with an application for evaluating the reopening readiness of academic units for campus education in the era of COVID-19. Three decision-makers from a higher education institution evaluate four academic units with respect to five strategic criteria and prioritize them according to their readiness levels for the campus type of education. Sensitivity and comparative analyses, theoretical and practical contributions, limitations, and future research avenues are also presented in the study. [ABSTRACT FROM AUTHOR]

In this paper, we present novel approaches to multi-attribute decision-making problems using q -rung normal fuzzy numbers. The q -rung normal fuzzy sets are an important way to express uncertain information, and they are superior to the intuitionistic fuzzy sets, Pythagorean fuzzy sets and normal Pythagorean fuzzy sets. A combination of the q -rung fuzzy number and the normal fuzzy number becomes q -rung normal fuzzy numbers. The new averaging and geometric operations of q -rung normal fuzzy numbers are studied using the general aggregation function. There are both commutative, associative, idempotent and boundedness compatible using q -rung normal fuzzy numbers. There are four new aggregating operators presented such as q -rung normal fuzzy interaction weighted averaging, q -rung normal fuzzy interaction weighted geometric, q -rung generalized normal fuzzy interaction weighted averaging and q -rung generalized normal fuzzy interaction weighted geometric. Usually, the aggregation functions are taken to be the Euclidean distance and Hamming distance. The existence of these operators enables the development of algorithms that solve multi-attribute decision-making problems. We demonstrate the application of enhanced Euclidean and Hamming distances to problems arising in real-world situations. An important component of a robot sensor is computer science and machine tool technology. There are four factors that can be used to evaluate robotics systems: resolution, sensitivity, error and accuracy. In order to determine the best alternative, expert opinions should be compared to the criteria. The accuracy of these models is better and they are closer to q. Finally, we used some practical examples to illustrate the validity and superiority of the proposed method by comparing with other existing methods. [ABSTRACT FROM AUTHOR]

• A CODAS-based group decision model is designed for the T-spherical fuzzy context. • A T-spherical fuzzy Minkowski distance operator based on Heronian mean is proposed. • A novel ordered weighted Heronian Minkowski distance (TSFOWHMD) operator is advanced. • The TSFOWHMD operator is utilized to improve the CODAS method. • The applicability of the developed methodology is verified through a case study. T-spherical fuzzy sets (TSFSs) are more flexible and efficient tools to deal with ambiguous, uncertain and vague information in complex real-world decision-making problems than various extended intuitionistic fuzzy sets. This paper aims to develop a novel T-spherical fuzzy (TSF) Combinative Distance-Based ASsessment (CODAS) based on the Heronian Minkowski distance aggregation operator, this new method can capture interrelationship between input arguments. Some TSF weighted Heronian Minkowski distance (TSFWHMD) aggregation operators with generalization are developed based on Heronian mean and Minkowski-type distance, their properties are discussed as well as their families are analyzed. Furthermore, the TSF MAGDM methodology based on the improved CODAS is designed, where the Minkowski-type distance is used to define the TSF similarity for computing the expert weights and to construct the maximizing deviation method (MDM) for determining the attribute weights, respectively. The TSF ordered weighted Heronian Hamming distance (TSFOWHHD) and TSF ordered weighted Heronian Euclidean distance (TSFOWHED) operators derived from the TSFOWHMD operator are integrated into the CODAS method, which is an improved method for both measuring the deviation of the negative ideal solution from each alternative and capturing the correlation between attributes. Finally, the feasibility and practicality of developed methodology are illustrated with an example of CAE (Computer Aided Engineering) software selection for lithium-ion power battery (LiPB) design, sensitivity analysis and method comparisons are performed to elucidate the reliability and validity of the developed methodology. [ABSTRACT FROM AUTHOR]

This study discusses novel approaches for solving multi-attribute decision-making problems using logarithmic square root vague sets. Logarithmic square root vague sets can be considered as an extension of vague sets and square root fuzzy sets provides a more profitable method to express the uncertainties in the data to deal with decision making problems. We developed some novel logarithmic square root vague sets, which can overcome the weaknesses of algebraic operations and capture the relationship between various square root vague sets and vague sets. The purpose of this study is to examine several aggregation operators using logarithmic square root vague sets. The weighted logarithmic averaging distance and weighted logarithmic geometric distance operators are distance measure that is based on aggregating model. We discuss the concepts of logarithmic square root vague weighted averaging, logarithmic square root vague weighted geometric, generalized logarithmic square root vague weighted averaging and generalized logarithmic square root vague weighted geometric operators. We communicate that algebraic structures such as associative, distributive, idempotent, bounded, commutativity and monotonicity properties are satisfied by logarithmic square root vague numbers. We discusses some mathematical properties of logarithmic square root vague sets, as well as the Hamming distance and Euclidean distance measures. It measures the similarity of two strings of information by measuring the Hamming distance. An analysis of the proposed operators is presented, along with alternative formulations and families, as well as their main properties. The weighting vector is characterized by multiple classical measures and we propose alternative solutions for dealing with the logarithmic properties of the operators. As a result of studying the weighting vectors of the operators, we present generalizations of them. We examine vague sets with important properties using algebraic operations in more detail. Furthermore, we develop an algorithm to solve multi-attribute decision-making problems using these operators. We provide real-life examples to illustrate how enhanced Euclidean distances, Hamming distances and score values can be applied in real-world situations. To show the superiority and the validity of the proposed aggregation operations, we compared it with the existing method, and concluded from the comparison and sensitivity analysis that our proposed technique is more effective and reliable. Based on the expert assessments, we compared the criteria with the most appropriate options. [ABSTRACT FROM AUTHOR]

In this paper, considering the traditional geometric operation laws and Pythagorean fuzzy information, a varies of new distance measures of Pythagorean fuzzy set are proposed, such as generalized Pythagorean fuzzy ordered geometric distance (GPFOWGD) measures and generalized Pythagorean fuzzy hybrid weighted geometric distance (GPFHWGD) measures. Besides, some special issues including Hamming distance, Euclidean distance and Hausdorff distance of these raised geometric distance measures are investigated. To testify the valid of these new presented distance measures, a decision-making model is built and illustrated by a mathematical calculation example to evaluate the service quality of sports clubs using Pythagorean fuzzy information. The results display that the approach is uncomplicated, valid and simple to compute. The example illustrates that the method presented in the paper can be used to deal with problems of uncertainty such as dynamic multiple attribute decision making. [ABSTRACT FROM AUTHOR]

The idea of an interval complex neutrosophic soft set (I-CNSS) emerges from the interval neutrosophic soft set (I-NSS) by the extension of its three membership functions (T, I, F) from real space to complex space (unit disc) to better handle uncertainties, vagueness, indeterminacy, and imprecision of information in the periodic nature. The novelty of I-CNSS lies in its more significant range of activity compared to CNS. Measures of similarity and distance are important tools that can be used to solve many problems in real life. Hence, this paper presents some interval complex neutrosophic soft similarities based on Hamming and Euclidean distances of I-CNSSs to deal with real-life problems that include uncertain information such as decision-making issues and medical diagnosis stats. Firstly, this paper reviews the definition of an interval complex neutrosophic soft set. Secondly, we defined distance Hamming measures and distance Euclidean measures on I-CNSSs. Next, the axiomatic definition of similarity measures based on Hamming and Euclidean distances of I-CNSSs is proposed. Moreover, a numerical example is given and relations between these measures are introduced and verified. Meanwhile, some applications are given to show how similarity can be used to help the user in making decisions and making medical diagnoses. Finally, a comparison of some current approaches is used to back up this study. [ABSTRACT FROM AUTHOR]

Generalized hesitant fuzzy numbers (GHFNs) are able to directly manage situations in which we may encounter a finite set of known values with a finite set of degrees of doubt as quantitative approximations of an uncertain situation/quantification of a linguistic expression. They are new extensions of hesitant fuzzy sets, which have been considered in this paper. In fact, in this paper, GHFNs will be utilized to model the uncertainty of the assessment values of options against criteria in multi-attribute decision making (MADM) problems. It means that all of the elements of decision matrix are GHFNs. Then, the technique for order of preference by similarity to ideal solution (TOPSIS) method, as a very successful method in solving MADM problems, will be updated to be used with GHFNs. To this end, the distance between GHFNs must be defined to obtain the distances between given alternatives from each of two subjective alternatives (positive/negative ideal solutions). Thus, three existing famous distance measures, i.e., general distance ( d g ), Hamming distance ( d h ), and Euclidean distance ( d e ) measures, have been updated for GHFNs firstly. Then, the new TOPSIS method will be proposed based on GHFNs. Finally, the numerical examples have been appointed to illustrate the proposed method, analyze comparatively and validate it. [ABSTRACT FROM AUTHOR]

In this paper, various E-learning Apps have been compared and analysed by two distance measures and formulate their comparison with limited number of students and factors with the scope of their extension. Hamming Distance and Euclidean Distance are two distance measures used in this paper. Moreover, the variables considered for this purpose follows the Intuitionistic Fuzzy Set (IFS) theory. [ABSTRACT FROM AUTHOR]

Mining clusters from data is an important endeavor in many applications. The k‐means method is a popular, efficient, and distribution‐free approach for clustering numerical‐valued data, but does not apply for categorical‐valued observations. The k‐modes method addresses this lacuna by replacing the Euclidean with the Hamming distance and the means with the modes in the k‐means objective function. We provide a novel, computationally efficient implementation of k‐modes, called Optimal Transfer Quick Transfer (OTQT). We prove that OTQT finds updates to improve the objective function that are undetectable to existing k‐modes algorithms. Although slightly slower per iteration due to algorithmic complexity, OTQT is always more accurate and almost always faster (and only barely slower on some datasets) to the final optimum. Thus, we recommend OTQT as the preferred, default algorithm for k‐modes optimization. [ABSTRACT FROM AUTHOR]

Main problems in FPGA-based ring oscillator (RO) PUFs are that the RO frequencies are highly sensitive to operating conditions and other types of global variations. In addition, the RO frequencies are highly correlated by local variations. Therefore, in practice, conventional RO-PUF application schemes using Hamming distance normally require complex identification (ID) extraction algorithms and/or a large number of ROs to ensure a high level of uniqueness and reliability of the extracted IDs. In this work, we proposed a novel scheme of ID extraction based on Euclidean distances. Our proposed scheme stably and reliably generates the ID using a non-selective small number of ROs. Specifically, the generated IDs are mostly non-sensitive to global variables and operating conditions such as ambient temperature. In oppose to Hamming-based extraction, the close-frequencies ROs are normally removed to avoid flipping bits, our proposed scheme takes all those into account and simplifies the extraction process. Experiments on our available hardware have shown a very good level of reliability and uniqueness with ID collision rate is estimated less than 2 × 10 − 9 . • A unique ID extraction scheme using RO based on Euclidean distance. • IDs can be reliably generated utilizing a non-selective small number of ROs. • The ID extraction scheme is experimentally verified on FPGA devices. • The ID collision rate from the experiment is estimated to be less than 2-9. • The ID stability is confirmed under different operating conditions. [ABSTRACT FROM AUTHOR]

The similarity measure is used to tackle many issues that include indistinct as well as blurred information excluding is not in a position to deal with the general fuzziness along with obscurity of the problems that have various information. The main purpose of this research is to propose a multipolar interval-valued neutrosophic soft set (mPIVNSS) with operations and basic properties. We also develop Hamming distance and Euclidean distance by using mPIVNSS and numerical examples and use the developed distances to introduce similarity measures. By using the developed similarity measures a decision-making approach is presented for mPIVNSS. Finally, we used the developed decision-making approach for medical diagnosis. [ABSTRACT FROM AUTHOR]

In this study, considering the traditional geometric operation laws and Pythagorean fuzzy information, we propose a variety of new distance measures of Pythagorean fuzzy sets such as generalized Pythagorean fuzzy geometric distance (GPFGD) measures and generalized Pythagorean fuzzy weighted geometric distance (GPFWGD) measures. Besides, some special issues including Hamming distance, Euclidean distance, and Hausdorff distance of these raised geometric distance measures are investigated. To testify the valid of these new presented distance measures, we build a decision-making model illustrated by a mathematical calculation example to evaluate the sustainable development of new urbanization from the perspective of urban agglomeration using Pythagorean fuzzy information. [ABSTRACT FROM AUTHOR]

Image representation and approximate query is always a research challenge and is affected greatly by the dimension and size of images. Since hash-based methods and binary encodings in combination with other techniques, such as kernel tricks, a longer binary code and mapping vectors rotation, can maintain a linear query time and query accuracy, they have been used in this area broadly. This paper develops principal component analysis hashing (PCAH) and unequal length of binary coding to divide images into more categories, denoted as PCA-MD, to improve accuracy of the representation and lookup of images. This paper firstly proves that the eigenvector mapping is locality sensitive, which is the basis for more classes division. For the anisotropy of the eigenvectors, PCA-MD utilizes an unequal length of binary coding and fewer eigenvectors, rather than an equal code, to divide the images mapped on every eigenvector to more categories. Moreover, L1-norm distance is applied to measure the distances of images to avoid the enormous computation of Euclidean distance. Theoretical analysis and extensive experimental results demonstrate that the PCA-MD has a higher query performance and a slight longer run time than the state-of-the-art approaches based on the Hamming distance. This in turn verifies that PCAH is a locality sensitive hash and that partitioning into more categories rather than only two categories is reasonable. [ABSTRACT FROM AUTHOR]

• Proposing of new method for measuring the distance between pairs of clusters in the dataset. • The proposed distance accurately captures both the variability of the cluster centers as well as the variability of shapes and directions of their respective covariance matrices • The method has a number of advantages including simplicity, interpretability, and computational efficiency • Both the classical and the robust versions of the distances are provided • The distance is illustrated by several motivating examples that demonstrate the need of the new proposed distance and applied to both real and synthetic data • Proving that the Ward distance and the Euclidian distance are equivalent. Clustering methods are based on the computations of both the distances between every pair of the n observations in a multivariate dataset as well as the distances between every pair of clusters in the dataset. The clusters can have different locations and varying elliptical shapes and directions. Numerous methods have been proposed in the literature for computing both of these two types of distances. The contributions of this paper are two folds. First, we propose a new elliptical distance between pairs of clusters in a dataset with different cluster centers and elliptical shapes and directions, Second, we proved analytically that the Ward distance and the Euclidean distance are equivalent. We propose a new classical method for computing the distance between a pair of clusters in the dataset. It is the only distance that does not assume spherical clusters. The proposed classical distances could also be made robust by replacing estimates of location and scale by their respective robust estimators. The proposed distance has a number of advantages including simplicity, interpretability, computational efficiency as well as the ability to accurately capture both the variability of the cluster centers as well as the variability of shapes and directions of their respective covariance matrices. The method is also illustrated by several motivating examples that demonstrate the need of the new proposed distance. The superiority of the proposed method is also demonstrated by application to real-life as well as challenging synthetic data. [ABSTRACT FROM AUTHOR]

Evaluating and selecting suitable low-carbon suppliers is one of the most disputed topics in different industries. Some studies have more specifically investigated the role of sustainable low-carbon suppliers in the maritime sector. In general, sustainability is recognized as a vital philosophy for many industrial sectors since it enhances environmental protection and raises people's and organizations' awareness about social obligations. Therefore, choosing low-carbon suppliers based on the sustainability perspective is important for firms and organizations in order to effectively promote sustainable supplier chain management (SSCM). The present study develops a new fuzzy decision-making approach to rank and evaluate low-carbon, sustainable suppliers (LCSS) through the integration of the Combinative Distance-Based Assessment (CODAS) framework and intuitionistic fuzzy sets (IFSs), which results in a new approach called intuitionistic fuzzy CODAS (IF-CODAS). The paper is aimed at making a decision on the IFSs context by considering the decision makers' hesitancy based on both Hamming and Euclidean distances in accordance with the anti-ideal point. To evaluate the weights of the criteria, a new discrimination measure is developed. To show the practicality of the proposed approach, a case study for the assessment of low carbon sustainable suppliers is presented. To validate the developed approach, comparative research and sensitivity analysis were conducted in this study. The evaluation results showed that the fourth low-carbon sustainable supplier with a maximum assessment score degree (0.4747) using the proposed method was found the best option for selecting LCSS based on the sustainability perspective. Furthermore, the sensitivity analysis was made to observe the difference of alternative ranking when the importance values of criteria weights change. [Display omitted] • New intuitionistic fuzzy combinative distance-based assessment (IF-CODAS) framework is developed. • To determine criteria weights, new intuitionistic fuzzy divergence measure is proposed. • The proposed method is applied to address low carbon sustainable suppliers selection issues in the maritime industry. • Sensitivity analysis and comparative discussion are made to explore the accuracy and validity of proposed approach. [ABSTRACT FROM AUTHOR]

The existing hashing methods mainly handle either the feature based nearest-neighbor search or the category-level image retrieval, whereas a few efforts are devoted to instance retrieval problem. In this paper, we propose a binary multi-view fusion framework for directly recovering a latent Hamming subspace from the multi-view features for instance retrieval. More specifically, the multi-view subspace reconstruction and the binary quantization are integrated in a unified framework so as to minimize the discrepancy between the original multi-view high-dimensional Euclidean space and the resulting compact Hamming subspace. Besides, our method is essentially an unsupervised learning scheme without any labeled data involved, and thus can be used in the cases when the supervised information is unavailable or insufficient. Experiments on public benchmark and large-scale datasets reveal that our method achieves competitive retrieval performance comparable to the state-of-the-arts and has excellent scalability in large-scale scenario. [ABSTRACT FROM AUTHOR]