Arithmetic Operations of Generalized Trapezoidal Picture Fuzzy Numbers by Vertex Method

In this article, we define the arithmetic operations of generalized trapezoidal picture fuzzy numbers by vertex method which is assembled on a combination of the (α, γ, β)-cut concept and standard interval analysis. Various related properties are explored. Finally, some computations of picture fuzzy functions over generalized picture fuzzy variables are illustrated by using our proposed technique.

A Comparison of Arithmetic Operations for Dynamic Process Optimization Approach

A comparison of arithmetic operations of two dynamic process optimization approaches called quasi-sequential approach and reduced Sequential Quadratic Programming(rSQP)simultaneous approach with respect to equality constrained optimization problems is presented.Through the detail comparison of arithmetic operations,it is concluded that the average iteration number within differential algebraic equations(DAEs)integration of quasi-sequential approach could be regarded as a criterion.One formula is given to calculate the threshold value of average iteration number.If the average iteration number is less than the threshold value,quasi-sequential approach takes advantage of rSQP simultaneous approach which is more suitable contrarily.Two optimal control problems are given to demonstrate the usage of threshold value.For optimal control problems whose objective is to stay near desired operating point,the iteration number is usually small.Therefore,quasi-sequential approach seems more suitable for such problems.

Interval arithmetic operations for uncertainty analysis with correlated interval variables

A new interval arithmetic method is proposed to solve interval functions with correlated intervals through which the overestimation problem existing in interval analysis could be significantly alleviated. The correlation between interval parameters is defined by the multidimensional parallelepiped model which is convenient to describe the correlative and independent interval variables in a unified framework. The original interval variables with correlation are transformed into the standard space without correlation,and then the relationship between the original variables and the standard interval variables is obtained. The expressions of four basic interval arithmetic operations, namely addition, subtraction, multiplication, and division, are given in the standard space. Finally, several numerical examples and a two-step bar are used to demonstrate the effectiveness of the proposed method.

The Arithmetic of Natural Language:Toward a typology of numeral systems

Numeral systems in natural languages show astonishing variety,though with very strong unifying tendencies that are increasing as many indigenous numeral systems disappear through language contact and globalization.Most numeral systems make use of a base,typically 10,less commonly 20,followed by a wide range of other possibilities.Higher numerals are formed from primitive lower numerals by applying the processes of addition and multiplication,in many languages also exponentiation;sometimes,however,numerals are formed from a higher numeral,using subtraction or division.Numerous complexities and idiosyncrasies are discussed,as are numeral systems that fall outside this general characterization,such as restricted numeral systems with no internal arithmetic structure,and some New Guinea extended body-part counting systems.

NEW DEMODULATION TECHNOLOGY OF OPTIMIZING ENERGY OPERATOR AND APPLICATION TO GEAR FAULT DIAGNOSIS

Although many methods have been applied to diagnose the gear thult currently, the sensitivity of them is not very good. In order to make the diagnosis methods have more excellent integrated ability in such aspects as precision, sensitivity, reliability and compact algorithm, and so on, and enlightened by the energy operator separation algorithm （EOSA）, a new demodulation method which is optimizing energy operator separation algorithm （OEOSA） is presented. In the algorithm, the non-linear differential operator is utilized to its differential equation： Choosing the unit impulse response length of filter and fixing the weighting coefficient for inportant points. The method has been applied in diagnosing tooth broden and fatiguing crack of gear faults successfully. It provides demodulation analysis of machine signal with a new approach.

Bernoulli Numbers ＆ Vandermonde Matrices

Matrix structuring is a very beautiful way to place Bernoulli numbers, by which a new view to the numbers is opened. Natural Numbers are mathematics seeds and Natural Number System （NNS） breeds the whole world mathematically.

Lanchester equation for cognitive domain using hesitant fuzzy linguistic terms sets

Intelligent wars can take place not only in the physical domain and information domain but also in the cognitive domain.The cognitive domain will become the key domain to win in the future intelligent war.A Lanchester equation considering cognitive domain is proposed to fit the development tendency intelligent wars in this paper.One party is considered to obtain the exponential enhancement advantage on combat forces in combat if it can gain an advantage in the cognitive domain over the other party according to the systemic advantage function.The operational effectiveness of the cognitive domain in war is considered to consist of a series of indicators.Hesitant fuzzy sets and linguistic term sets are powerful tools when evaluating indicators,hence the indicators are scored by experts using hesitant fuzzy linguistic terms sets here.A unique hesitant fuzzy hybrid arithmetical averaging operator is used to aggregate the evaluation.

Group Decision-Making Model of Renal Cancer Surgery Options Using Entropy Fuzzy Element Aczel-Alsina Weighted Aggregation Operators under the Environment of Fuzzy Multi-Sets

Since existing selection methods of surgical treatment schemes of renal cancer patients mainly depend on physicians’clinical experience and judgments,the surgical treatment options of renal cancer patients lack their scientifical and reasonable information expression and group decision-making model for renal cancer patients.Fuzzy multi-sets(FMSs)have a number of properties,which make them suitable for expressing the uncertain information of medical diagnoses and treatments in group decision-making(GDM)problems.To choose the most appropriate surgical treatment scheme for a patient with localized renal cell carcinoma(RCC)(T1 stage kidney tumor),this article needs to develop an effective GDM model based on the fuzzy multivalued evaluation information of the renal cancer patients.First,we propose a conversionmethod of transforming FMSs into entropy fuzzy sets(EFSs)based on the mean and Shannon entropy of a fuzzy sequence in FMS to reasonably simplify the information expression and operations of FMSs and define the score function of an entropy fuzzy element(EFE)for ranking EFEs.Second,we present the Aczel-Alsina t-norm and t-conorm operations of EFEs and the EFE Aczel-Alsina weighted arithmetic averaging(EFEAAWAA)and EFE Aczel-Alsina weighted geometric averaging(EFEAAWGA)operators.Third,we develop a multicriteria GDM model of renal cancer surgery options in the setting of FMSs.Finally,the proposed GDM model is applied to two clinical cases of renal cancer patients to choose the best surgical treatment scheme for a renal cancer patient in the setting of FMSs.The selected results of two clinical cases verify the efficiency and rationality of the proposed GDM model in the setting of FMSs.

More about Bernoulli Numbers

Bernoulli Numbers are coded with Deterministic Redundancy of Arithmetic Operations, adding and multiplying or exponent, in Natural Number System. And based on the redundancy, a process for obtaining the Bernoulli Numbers is elaborated.

Efficient quantum arithmetic operation circuits for quantum image processing

Efficient quantum circuits for arithmetic operations are vital for quantum algorithms.A fault-tolerant circuit is required for a robust quantum computing in the presence of noise.Quantum circuits based on Clifford+T gates are easily rendered faulttolerant.Therefore,reducing the T-depth and T-Count without increasing the qubit number represents vital optimization goals for quantum circuits.In this study,we propose the fault-tolerant implementations for TR and Peres gates with optimized T-depth and T-Count.Next,we design fault-tolerant circuits for quantum arithmetic operations using the TR and Peres gates.Then,we implement cyclic and complete translations of quantum images using quantum arithmetic operations,and the scalar matrix multiplication.Comparative analysis and simulation results reveal that the proposed arithmetic and image operations are efficient.For instance,cyclic translations of a quantum image produce 50%T-depth reduction relative to the previous best-known cyclic translation.

Concept of Yager operators with the picture fuzzy set environment and its application to emergency program selection

Purpose-The aim of this study as to find out an approach for emergency program selection.Design/methodology/approach-The authors have generated six aggregation operators(AOs),namely picture fuzzy Yager weighted average(PFYWA),picture fuzzy Yager ordered weighted average,picture fuzzy Yager hybrid weighted average,picture fuzzy Yager weighted geometric(PFYWG),picture fuzzy Yager ordered weighted geometric and picture fuzzy Yager hybrid weighted geometric aggregations operators.Findings-First of all,the authors defined the score and accuracy function for picture fuzzy set(FS),and some fundamental operational laws for picture FS using the Yager aggregation operation.After that,using the developed operational laws,developed some AOs,namely PFYWA,picture fuzzy Yager ordered weighted average,picture fuzzy Yager hybrid weighted average,PFYWG,picture fuzzy Yager ordered weighted geometric and picture fuzzy Yager hybrid weighted geometric aggregations operators,have been proposed along with their desirable properties.A decision-making(DM)approach based on these operators has also been presented.An illustrative example has been given for demonstrating the approach.Finally,discussed the comparison of the proposed method with the other existing methods and write the conclusion of the article.Originality/value-To find the best alternative for emergency program selection.

Orthopair indeterminate information expression,aggregations and multiattribute decision making method with indeterminate ranges

A neutrosophic number(NN)(d=μ+vI)can flexibly represent the indeterminate information corresponding to values/ranges of the indeterminacy I.Regarding the hybrid concept of intuitionistic fuzzy set(IFS)and NN,this study presents an orthopair indeterminate set(OIS),an orthopair indeterminate element weighted arithmetic averaging(OIEWAA)operator and an orthopair indeterminate element weighted geometric averaging(OIEWGA)operator to simplify and generalise the existing IFS and interval-valued IFS expressions and aggregation forms.Thus,a multiattribute decision making(DM)approach with indeterminate ranges of decision makers is developed based on the OIEWAA and OIEWGA operators and the score and accuracy functions of orthopair indeterminate elements in OIS setting.Finally,the proposed DM approach is applied to a multi-attribute DM example of manufacturing schemes(alternatives)in OIS setting to demonstrate the applicability and flexibility of the proposed DM approach in OIS setting.