Degree Formula Research Articles

Interval-valued n-person cooperative games with satisfactory degree constraints

The aim of this study is to develop several nonlinear programming models for interval-valued cooperative games in which taking into account the decision makers’ risk attitudes. First, we investigate several existing used satisfactory degree comparison methods for ranking interval-valued fuzzy numbers, and point out by an example that the method proposed by Liu et al. (Soft Comput 22:2557–2565, 2018) is more efficient than the method proposed by Hong and Li (Oper Res 17:1–19, 2016). Second, by taking into account decision makers’ risk attitudes, several corresponding nonlinear programming models are constructed based on satisfactory degree formulas that were proposed by Liu et al. (2018). Third, an illustrative example in conjunction with comparative analyses are employed to demonstrate the validity and applicability of the proposed models. Finally, to further highlight the validity of the proposed method, we discuss the relationship of the satisfactory degree formulas between Hong and Li (2016)’s method and Xu and Da (J Syst Eng 18:67–70, 2003)’s method.

A Novel Comparison of Probabilistic Hesitant Fuzzy Elements in Multi-Criteria Decision Making

The probabilistic hesitant fuzzy element is a common tool to deal with multi-criteria decision-making problems when the decision makers are irresolute in providing their evaluations. The existing methods for ranking probabilistic hesitant fuzzy elements are limited and not reasonable in practical applications. The main purpose of this paper is to find a more precise and appropriate comparison method so that we can deal with multi-criteria decision-making problems more efficiently. We first propose a chart technique to analyze the structure of a probabilistic hesitant fuzzy element. After that, we propose a novel possibility degree formula to rank probabilistic hesitant fuzzy elements. Last but not least, we provide a useful process to solve the actual multi-criteria decision-making problems, and make a real case study which demonstrates that our method is feasible and reliable.

Research on Landing Risk Assessment for Carrier-Based Aircrafts by M-V Multiattribute Decision-Making

For increasing accuracy rate of landing risk assessment of carrier-based aircraft, this article is committed to research on landing risk assessment of carrier-based aircraft based on mean-variance dynamic multi-attribute decision making. Attribute values are provided in the form of triangle fuzzy number. Mean sort values is acquired by formula of possibility degree and order. For solving a problem of the same or similar orders, the improved method combines mean and variance, analyzing and judging, getting alternatives rank. Finally, the feasibility and availability of the improved method is proved by simulation results.

Diversity and degrees of freedom in regression ensembles

Abstract Ensemble methods are a cornerstone of modern machine learning. The performance of an ensemble depends crucially upon the level of diversity between its constituent learners. This paper establishes a connection between diversity and degrees of freedom (i.e. the capacity of the model), showing that diversity may be viewed as a form of inverse regularisation. This is achieved by focusing on a previously published algorithm Negative Correlation Learning (NCL), in which model diversity is explicitly encouraged through a diversity penalty term in the loss function. We provide an exact formula for the effective degrees of freedom in an NCL ensemble with fixed basis functions, showing that it is a continuous, convex and monotonically increasing function of the diversity parameter. We demonstrate a connection to Tikhonov regularisation and show that, with an appropriately chosen diversity parameter, an NCL ensemble can always outperform the unregularised ensemble in the presence of noise. We demonstrate the practical utility of our approach by deriving a method to efficiently tune the diversity parameter. Finally, we use a Monte-Carlo estimator to extend the connection between diversity and degrees of freedom to ensembles of deep neural networks.

Exploration and Practice of a New Formula for Calculating the Degree of Freedom

The current “structural mechanics” in the textbook, the more sub items in the formula of calculating degrees of freedom, is not conducive for the students to master the understanding and calculation of degree of freedom concept. After a few years of teaching, the new calculating formula of degree of freedom for the system is put forward, and the old formula is verified by two examples. The result shows that the new calculating formula and the current mainstream material formula calculation results are consistent, and the new formula is more helpful for students to understand the physical meaning of the calculation of degrees of freedom.

A Multi-Attribute Decision-Making Model Using Interval-Valued Intuitionistic Fuzzy Numbers and Attribute Correlation

This article describes a decision analysis method with attribute correlation and attribute values of interval-valued intuitionistic fuzzy numbers (IVIFN). This considers the correlation between attributes and undetermined attribute weight information was put forward against the case where solution attribute values are interval-valued intuitionistic fuzzy numbers. To achieve a comprehensive consideration of the undetermined evaluation attribute weight information, experts' positive and negative ideal solutions against each attribute are taken as reference points, attribute weight is obtained via decision making trial and evaluation laboratory (DEMATEL) method according to the direct relation matrix provided by experts against attribute sets. On this basis, evaluation information is aggregated by using weighted arithmetic average operators. Then, the closeness degree formula of interval-valued intuitionistic fuzzy numbers relative to the maximum IVIFN is provided by combining the Euclidean distance of IVIFN, thus determining closeness degree and making an ordering of all solutions. Finally, the feasibility and validity of the method proposed in this study are verified via a case analysis of the best information system decision-making.

Projective duality and a Chern-Mather involution

We observe that linear relations among Chern-Mather classes of projective varieties are preserved by projective duality. We deduce the existence of an explicit involution on a part of the Chow group of projective space, encoding the effect of duality on Chern-Mather classes. Applications include Plücker formulae, constraints on self-dual varieties, generalizations to singular varieties of classical formulas for the degree of the dual and the dual defect, formulas for the Euclidean distance degree, and computations of Chern-Mather classes and local Euler obstructions for cones.

Thermodynamic relation between irreversibility of heat transfer and cycle performance based on trapezoidal model

Irreversible degree of heat transfer in organic Rankine cycle (ORC) is proposed in this paper. Based on a trapezoidal cycle (TPC) and its theoretical model, the model and mathematic formulas of irreversible degree of heat transfer is built. Thermodynamic relation between irreversible degree and cycle performance (thermal efficiency, net power output and exergy efficiency) is built and studied, which can be applied to the coupling optimization between cycle performance and heat transfer processes. Similar to net power output, there exists a kind of shift-curve of irreversible degree and the corresponding shift-temperature of heating fluid for working fluids, which indicates the shift of irreversible degree from having optimum condition with a minimum to monotonic decrease with heating fluid temperature. The range of irreversible degree and range of cycle performance according to irreversible degree are obtained respectively. Moreover, a model of working fluid with linear saturation line is proposed to study the general thermodynamic principles of TPC (or ORC) theoretically without the restriction of actual working fluids.

Construction of a Mathematical Model for Calibrating Test Task Parameters and the Knowledge Level Scale of University Students by Means of Testing

The relevance of this study is determined by the algorithm developed for test task selection. The purpose of this article is to develop this test task selection algorithm having single and multiple-choice answers with various blocks of reactions. The main approach to the study of the problem is the construction of a mathematical model to calibrate test task parameters and the university students’ scale of knowledge level. The study has proven insufficient use of the classical theory of testing in an objective assessment of students’ knowledge. The method for calibrating test task parameters is developed. Scales of testees’ readiness level are defined. The adequate size of test reliability coefficient is found. The rule for test task selection is formulated. A formula for the complexity degree of a test task is found. An algorithm for allocation of test task types with unambiguous and multiple-choice answers with various blocks of reactions is offered.

Free functions with symmetry

In 1936, Margarete C. Wolf showed that the ring of symmetric free polynomials in two or more variables is isomorphic to the ring of free polynomials in infinitely many variables. We show that Wolf’s theorem is a special case of a general theory of the ring of invariant free polynomials: every ring of invariant free polynomials is isomorphic to a free polynomial ring. Furthermore, we show that this isomorphism extends to the free functional calculus as a norm-preserving isomorphism of function spaces on a domain known as the row ball. We give explicit constructions of the ring of invariant free polynomials in terms of representation theory and develop a rudimentary theory of their structures. Specifically, we obtain a generating function for the number of basis elements of a given degree and explicit formulas for good bases in the abelian case.

Trustworthy Service Selection Integrating Cloud Model and Possibility Degree Ranking of Interval Numbers

It has been challenging to select suitable services from abundant candidates in cloud environment. Aiming to the characteristics of batch computing mode and stream computing mode, a novel trustworthy service selection approach is proposed integrating cloud model and interval numbers theory. To facilitate potential users to understand the quality of service, the trustworthiness of service is described with interval number using reverse cloud generator, and the services with poor performance are filtered out by employing deviation degree or proximity degree. Two formulas of possibility degree of interval numbers are designed to compare the trustworthiness values between cloud services by utilizing probability zone analysis and geometrical analysis respectively, and the ranking method for possibility degree of interval numbers is exploited to select the most trustworthy service. The experiments show that this approach is effective to improve the accuracy of service selection and select trustworthy service for potential users in cloud paradigm.

Homogeneity degree of cones

The homogeneity degree of a space X is the number of orbits for the action of the autohomeomorphisms group of X. We determine the homogeneity degree of the cone over a locally connected curve X not being a local dendrite in terms of that of X. Using the result of Pellicer-Covarrubias and Santiago-Santos, it gives us a formula for the homogeneity degree of the cone over any locally connected curve X.

Topological Characterization of Carbon Graphite and Crystal Cubic Carbon Structures.

Graph theory is used for modeling, designing, analysis and understanding chemical structures or chemical networks and their properties. The molecular graph is a graph consisting of atoms called vertices and the chemical bond between atoms called edges. In this article, we study the chemical graphs of carbon graphite and crystal structure of cubic carbon. Moreover, we compute and give closed formulas of degree based additive topological indices, namely hyper-Zagreb index, first multiple and second multiple Zagreb indices, and first and second Zagreb polynomials.

Two-layer weight large group decision-making method based on multi-granularity attributes

To overcome the information ambiguity inherent in multi-attribute large group decision-making problems, an interval 2-tuple linguistic large group decision-making method based on multi-granularity attributes is proposed. Considering the preferences of individual decision makers, the group is cluste red and classified to form an aggregate structure. The two-layer weight model is developed from the decision makers’ different contribution to the existing clusters’ and different contribution of different clusters. The possible degree formula for expansion is used to calculate the weighted value of the decision makers in all clusters. Fuzzy entropy is used to confirm the weighted value of each cluster, and the fuzzy relative entropy is then used to sort the schemes. Finally, an illustrative example indicates that the proposed model is feasible and effective for large group decision-making.

Innovation Analysis Approach to Design Parameters of High Speed Train Carriage and Their Intrinsic Complexity Relationships

In view of the problem that it’s difficult to accurately grasp the influence range and transmission path of the vehicle top design requirements on the underlying design parameters. Applying directed-weighted complex network to product parameter model is an important method that can clarify the relationships between product parameters and establish the top-down design of a product. The relationships of the product parameters of each node are calculated via a simple path searching algorithm, and the main design parameters are extracted by analysis and comparison. A uniform definition of the index formula for out-in degree can be provided based on the analysis of out-in-degree width and depth and control strength of train carriage body parameters. Vehicle gauge, axle load, crosswind and other parameters with higher values of the out-degree index are the most important boundary conditions; the most considerable performance indices are the parameters that have higher values of the out-in-degree index including torsional stiffness, maximum testing speed, service life of the vehicle, and so on; the main design parameters contain train carriage body weight, train weight per extended metre, train height and other parameters with higher values of the in-degree index. The network not only provides theoretical guidance for exploring the relationship of design parameters, but also further enriches the application of forward design method to high-speed trains.

Fuzzy Multi-objective Optimization of EMT Based on the Minimum Average Weighted Deviation Algorithm

The optimal control of the vehicles with electro-mechanical transmission is a problem of multi-objective optimization. In this paper, based on the working characteristics of the engine, the battery pack and the motor/generators (MGs) as well as their interactions, a multi-objective optimization model with the optimal objective functions which include drivability, electric supply performance and fuel consumption is established. Using the relative membership degree formulas, the multi-objective optimization is transformed to fuzzy multi-objective optimization, and the problem of multi-objective optimization is converted to single-objective optimization with the corresponding weights of the objective functions gained by the minimum average weighted deviation algorithm. The forward model based on Matlab is implemented. Compared with the rule-based method, this method had enabled the drivability, the electric supply performance and the fuel consumption to improve by 29.8%, 38.1% and 11.8%, respectively.

Hesitant fuzzy linguistic multi-criteria decision making based on possibility theory

Hesitant fuzzy linguistic term sets (HFLTSs) have attracted a lot of attention recently due to their distinguished power and efficiency for dealing with the situations in which the decision makers may hesitate among several linguistic terms to assess alternatives. To extend the applicability of HFLTSs, this paper firstly analyses the existing comparison methods for HFLTSs, and then develops some new different possibility degree formulas which are convenience and easy of calculation. After that, two novel methods of hesitant fuzzy linguistic multi-criteria decision making are developed based on the priority method of fuzzy complementary preference relation and the idea of PROMETHEE, in which multiple steps may be integrated into one formula to simplify the computational process. Especially, the two proposed methods are designed to reflect the degree of one alternative to be superior or inferior to another one properly. Finally, an experiment is designed to test and compare the proposed methods and other existing ones. The results show that the proposed methods provide us with a useful way for decision making in fuzzy environments.

On possibility-degree formulae for ranking interval numbers

Since interval numbers are used to evaluate the opinions of decision makers and express the weights of alternatives in various decision making problems, it is requisite to give a feasible method to rank them. In the present paper, the two existing possibility degree formulae for ranking interval numbers are proved to be equivalent. A generalized possibility degree formula is proposed by considering the attitude of decision makers with a prescribed function. Some known possibility degree formulae for ranking interval numbers can be recovered by choosing a special function. The proposed method is applied to uniformly define the weak transitivity of interval multiplicative and additive reciprocal preference relations. Numerical examples are carried out to illustrate the new formula.

No counts, no variance: allowing for loss of degrees of freedom when assessing biological variability from RNA-seq data.

RNA sequencing (RNA-seq) is widely used to study gene expression changes associated with treatments or biological conditions. Many popular methods for detecting differential expression (DE) from RNA-seq data use generalized linear models (GLMs) fitted to the read counts across independent replicate samples for each gene. This article shows that the standard formula for the residual degrees of freedom (d.f.) in a linear model is overstated when the model contains fitted values that are exactly zero. Such fitted values occur whenever all the counts in a treatment group are zero as well as in more complex models such as those involving paired comparisons. This misspecification results in underestimation of the genewise variances and loss of type I error control. This article proposes a formula for the reduced residual d.f. that restores error control in simulated RNA-seq data and improves detection of DE genes in a real data analysis. The new approach is implemented in the quasi-likelihood framework of the edgeR software package. The results of this article also apply to RNA-seq analyses that apply linear models to log-transformed counts, such as those in the limma software package, and more generally to any count-based GLM where exactly zero fitted values are possible.

Radon transport model into a porous ground layer of finite capacity

The model of radon transfer is considered in a porous ground layer of finite power. With the help of the Laplace integral transformation, a numerical solution of this model is obtained which is based on the construction of a generalized quadrature formula of the highest degree of accuracy for the transition to the original - the function of solving this problem. The calculated curves are constructed and investigated depending on the diffusion and advection coefficients.The work was a mathematical model that describes the effect of the sliding attachment (stick-slip), taking into account hereditarity. This model can be regarded as a mechanical model of earthquake preparation. For such a model was proposed explicit finite- difference scheme, on which were built the waveform and phase trajectories hereditarity effect of stick-slip.