Symmetric Interval Research Articles

Disagreement of Hierarchical Opinion Dynamics with Changing Antagonisms

For opinion dynamics in social networks with antagonisms, distributed processes lead to challenging difficulties to characterize them, especially under changing network topologies. There may appear polarization, or neutralization, or fluctuation due to the simultaneous existence of cooperative and antagonistic interactions. In this paper, a leader-follower hierarchical framework is first established for opinion dynamics under the representation of signed digraphs and then a class of disagreement behaviors is identified by introducing a new rule to distinguish leaders and followers. It is shown that for opinion dynamics under switching signed digraphs, the agents in the leader layer always converge, whereas the agents in the follower layer generally may not converge but fluctuate within a symmetric interval spanned by the converged opinions of all leaders. Furthermore, the convergence of leaders is divided into clusters, and for each cluster, the opinions of leaders are ensured to polarize or neutralize, which are correspondingly tied to the structural balance or unbalance of the associated switching signed digraph with the cluster. Illustrative examples are presented to demonstrate the disagreement behaviors observed for opinion dynamics.

A Robust Optimization Approach to the Multiple Allocation p-Center Facility Location Problem

In this study, we apply a robust optimization approach to a p-center facility location problem under uncertainty. Based on a symmetric interval and a multiple allocation strategy, we use three types of uncertainty sets to formulate the robust problem: box uncertainty, ellipsoidal uncertainty, and cardinality-constrained uncertainty. The equivalent robust counterpart models can be solved to optimality using Gurobi. Comprehensive numerical experiments have been conducted by comparing the performance of the different robust models, which illustrate the pattern of robust solutions, and allocating a demand node to multiple facilities can reduce the price of robustness, and reveal that alternative models of uncertainty can provide robust solutions with different conservativeness.

Symmetric Triangular Interval Type-2 Intuitionistic Fuzzy Sets with Their Applications in Multi Criteria Decision Making

Type-2 intuitionistic fuzzy set (T2IFS) is a powerful and important extension of the classical fuzzy set, intuitionistic fuzzy set to measure the vagueness and uncertainty. In a practical decision-making process, there always occurs an inter-relationship among the multi-input arguments. To deal with this point, the motivation of the present paper is to develop some new interval type-2 (IT2) intuitionistic fuzzy aggregation operators which can consider the multi interaction between the input argument. To achieve it, we define a symmetric triangular interval T2IFS (TIT2IFS), its operations, Hamy mean (HM) operator to aggregate the preference of the symmetric TIT2IFS and then shows its applicability through a multi-criteria decision making (MCDM). Several enviable properties and particular cases together with following different parameter values of this operator are calculated in detail. At last a numerical illustration is to given to exemplify the practicability of the proposed technique and a comparative analysis is analyzed in detail.

Weakly Monotone Fock Space and Monotone Convolution of the Wigner Law

We study the distribution (w.r.t. the vacuum state) of family of partial sums Sm of position operators on weakly monotone Fock space. We show that any single operator has the Wigner law, and an arbitrary family of them (with the index set linearly ordered) is a collection of monotone independent random variables. It turns out that our problem equivalently consists in finding the m-fold monotone convolution of the semicircle law. For m = 2 we compute the explicit distribution. For any m > 2 we give the moments of the measure, and show it is absolutely continuous and compactly supported on a symmetric interval whose endpoints can be found by a recurrence relation.

Estimation and prediction of Marshall–Olkin extended exponential distribution under progressively type-II censored data

ABSTRACTIn this paper, we consider Marshall–Olkin extended exponential (MOEE) distribution which is capable of modelling various shapes of failure rates and aging criteria. The purpose of this paper is three fold. First, we derive the maximum likelihood estimators of the unknown parameters and the observed the Fisher information matrix from progressively type-II censored data. Next, the Bayes estimates are evaluated by applying Lindley’s approximation method and Markov Chain Monte Carlo method under the squared error loss function. We have performed a simulation study in order to compare the proposed Bayes estimators with the maximum likelihood estimators. We also compute 95% asymptotic confidence interval and symmetric credible interval along with the coverage probability. Third, we consider one-sample and two-sample prediction problems based on the observed sample and provide appropriate predictive intervals under classical as well as Bayesian framework. Finally, we analyse a real data set to illustrate the results derived.

An Extended Step-Wise Weight Assessment Ratio Analysis with Symmetric Interval Type-2 Fuzzy Sets for Determining the Subjective Weights of Criteria in Multi-Criteria Decision-Making Problems

Determination of subjective weights, which are based on the opinions and preferences of decision-makers, is one of the most important matters in the process of multi-criteria decision-making (MCDM). Step-wise Weight Assessment Ratio Analysis (SWARA) is an efficient method for obtaining the subjective weights of criteria in the MCDM problems. On the other hand, decision-makers may express their opinions with a degree of uncertainty. Using the symmetric interval type-2 fuzzy sets enables us to not only capture the uncertainty of information flexibly but also to perform computations simply. In this paper, we propose an extended SWARA method with symmetric interval type-2 fuzzy sets to determine the weights of criteria based on the opinions of a group of decision-makers. The weights determined by the proposed approach involve the uncertainty of decision-makers’ preferences and the symmetric form of the weights makes them more interpretable. To show the procedure of the proposed approach, it is used to determine the importance of intellectual capital dimensions and components in a company. The results show that the proposed approach is efficient in determining the subjective weights of criteria and capturing the uncertainty of information.

The Asymptotic Formula for the Eigenvalues of Sturm–Liouville Problems with One Turning Point of Negative Order

In the present paper, we consider a linear second-order differential equation of Sturm–Liouville type with one turning point of negative order, on a symmetric finite interval. We provide a constructing method to find the asymptotic formula for the eigenvalues of the boundary value problem consisting of the Sturm–Liouville equation with Dirichlet boundary condition.

Diagonally Implicit Multistep Block Method of Order Four for Solving Fuzzy Differential Equations Using Seikkala Derivatives

In this paper, the solution of fuzzy differential equations is approximated numerically using diagonally implicit multistep block method of order four. The multistep block method is well known as an efficient and accurate method for solving ordinary differential equations, hence in this paper the method will be used to solve the fuzzy initial value problems where the initial value is a symmetric triangular fuzzy interval. The triangular fuzzy number is not necessarily symmetric, however by imposing symmetry the definition of a triangular fuzzy number can be simplified. The symmetric triangular fuzzy interval is a triangular fuzzy interval that has same left and right width of membership function from the center. Due to this, the parametric form of symmetric triangular fuzzy number is simple and the performing arithmetic operations become easier. In order to interpret the fuzzy problems, Seikkala’s derivative approach is implemented. Characterization theorem is then used to translate the problems into a system of ordinary differential equations. The convergence of the introduced method is also proved. Numerical examples are given to investigate the performance of the proposed method. It is clearly shown in the results that the proposed method is comparable and reliable in solving fuzzy differential equations.

О росте коэффициентов в полиномах Бернштейна для стандартного модуля на симметричном отрезке

О росте коэффициентов в полиномах Бернштейна для стандартного модуля на симметричном отрезке

Exploiting the Persymmetric Property of Covariance Matrices for Knowledge-Aided Space-Time Adaptive Processing

In space–time adaptive processing (STAP) technique, the estimation of the interference-plus-noise covariance matrix is one of the critical points. Incorporating a priori knowledge into STAP architectures can reduce the effect of the heterogeneous environment and substantially improve the estimation accuracy of the covariance matrix. Besides the prior information, the persymmetric structure in radar systems with symmetric spaced linear array and constant pulse repetition interval can also be exploited to improve the STAP performance. In this paper, we present a new computationally adaptive knowledge-aided STAP method that requires fewer samples by utilizing the persymmetric structure of the covariance matrix. In addition, based on the covariance matrix estimation technology of the newly proposed knowledge-aided STAP method, two knowledge-aided persymmetric adaptive detectors in the nonhomogeneous environment are proposed as well. First, a two-step design procedure-based detector is proposed for the partially homogeneous model, which is called knowledge-aided persymmetric adaptive coherence estimator. Second, we improve the stochastic heterogeneous model and propose a new knowledge-aided persymmetric generalized likelihood ratio test for this model. Finally, simulation results confirm the effectiveness of the proposed methods.

Estimation of the geometrical measurement error at the stage of stereoscopic system design

The article is dedicated to the error estimation method for stereoscopic systems measuring three-dimensional coordinates and geometric parameters of objects. This method is required for stereoscopic system design to optimize the parameters of the image acquisition system and the data processing algorithms. The technique should be suitable for different mathematical models of image acquisition systems and allow to access the measurement uncertainty with a known uncertainty in determining the coordinates of the corresponding points on the images and the uncertainty of the calibration parameters. We analyzed known methods by comparing their results with the Monte Carlo simulation for the pinhole and the ray tracing models. It is shown that the method using the unscented transformation provides better accuracy and versatility than the linearization method. Using the example of measuring the length of a segment, it is demonstrated that the use of a symmetric confidence interval constructed from the mean and variance can lead to an inaccurate estimation of the error in measuring geometric parameters. We propose a method for calculating confidence intervals based on a combination of unscented transformation and interval analysis and confirm its effectiveness by the computer simulation. The analysis is applicable to the design of both passive stereoscopic devices and active triangulation systems as well as improving their software.

Robust-fuzzy model for supplier selection under uncertainty: An application to the automobile industry

This paper proposes an innovative robust-fuzzy method for multi-objective, multi-period supplier selection problem under multiple uncertainties. This approach integrates robust optimization and fuzzy programming. Uncertain parameters are modeled as random variables that take value within a symmetrical interval. However, due to the complexity or ambiguity of some real world problems and specially the nature of some of the available input data, the length of interval is also highly uncertain. This ambiguity motivated us to present a new approach, which can be applicable to multiple uncertainties conditions. Thus, in our approach the half-length of these intervals is also represented by fuzzy membership function. We develop a model and a solution approach to select suppliers by considering risk. The proposed method is applied to a real case of supplier selection in automobile industry under uncertainty and ambiguity conditions. To verify the proposed model, we evaluated the results by simulation technique and compared values of objective function under different scenarios.

Probabilistic Scatter Band with Error Distribution for Fatigue Life Comparisons

This paper presents a new probabilistic evaluation technique of fatigue life based on the adaptation of control chart concept incorporated in a scatter band. Conventionally, scatter band is a tool for visual evaluation of fatigue life accuracy. Such band is also used as an evaluation criterion for the estimation methods of fatigue strain–life equation. However, the current scatter band technique provides no probabilistic interpretation of the deviation that occurred during comparison. This technique is only used as a comparison tool considering whether the data points fall within or outside the predefined accuracy band. Therefore, a new framework of probabilistic scatter band is introduced in this study. The symmetrical tolerance interval with normal distribution is used to characterise the probabilistic field with upper and lower tolerance limit according to two standard deviations or a 95% tolerance limit. Using this technique, the comparison between fatigue life prediction and experimental data is provided in regions according to conservatism. The probability of error and rejection index values according to the statistical distribution of fatigue life is also presented. The proposed technique is applied in fatigue life evaluation between strain–life models and experimental data available in literature. The new technique is practically informative in probabilistic interpretation of fatigue life results with scatter and uncertainty factors included.

Interval type-2 fuzzy Hamy mean operators and their application in multiple criteria decision making

Interval type-2 fuzzy set (IT2FS) is a widely used tool for handling vagueness and imprecision in real-world problems. To aggregate the interval type-2 fuzzy information, some useful aggregation operators have been developed such as IT2-OWA, IT2-FWA, and IT2-Frank operators. In particular, some aggregation operators which can capture and reflect and interrelationships among aggregated arguments are intensively studied, among which Hamy mean (HM) is an important alternative. However, the traditional HM does not focus on fusion of fuzzy information and the weighted vector of the aggregated arguments. In this study, we extend the HM to accommodate interval type-2 fuzzy environment and show its application to multiple criteria decision making (MADM). In order to simplify the computing, we first define the concept of symmetric triangular interval type-2 fuzzy set and propose some operational laws. Motivated by the idea of HM operator, we develop the symmetric triangular interval type-2 fuzzy Hamy mean (STIT2FHM) operator for aggregating the interval type-2 fuzzy information. Some useful properties such as monotonicity, boundedness, and idempotency are studied in detail. Furthermore, we discuss some special cases with respect to different parameter values of the STIT2FHM operator. For the situations where the input arguments have different importance, we further develop the weighted symmetric triangular interval type-2 fuzzy Hamy mean (WSTIT2FHM) operator to aggregate interval type-2 fuzzy information. Based on it, an approach to MCDM problems with symmetric triangular interval type-2 fuzzy information is developed. Finally, a real-world example concerning tourism recommender system is provided to illustrate the practicality and effectiveness of the proposed method.

Spectral Stability for Classical Periodic Waves of the Ostrovsky and Short Pulse Models

AbstractWe consider the Ostrovsky and short pulse models in a symmetric spatial interval, subject to periodic boundary conditions. For the Ostrovsky case, we rederive the formulas for the classical periodic traveling waves, while for the short pulse model, we explicitly construct traveling waves in terms of Jacobi elliptic functions. In both cases, we show spectral stability, for all values of the parameters. This is achieved by studying the nonstandard eigenvalue problems in the form , where is a Hill operator.

Reflection positivity on real intervals

We study functions f : (a,b) ---> R on open intervals in R with respect to various kinds of positive and negative definiteness conditions. We say that f is positive definite if the kernel f((x + y)/2) is positive definite. We call f negative definite if, for every h > 0, the function e^{-hf} is positive definite. Our first main result is a L\'evy--Khintchine formula (an integral representation) for negative definite functions on arbitrary intervals. For (a,b) = (0,\infty) it generalizes classical results by Bernstein and Horn. On a symmetric interval (-a,a), we call f reflection positive if it is positive definite and, in addition, the kernel f((x - y)/2) is positive definite. We likewise define reflection negative functions and obtain a L\'evy--Khintchine formula for reflection negative functions on all of R. Finally, we obtain a characterization of germs of reflection negative functions on 0-neighborhoods in R.

Extremal eigenvalue intervals of symmetric tridiagonal interval matrices

SummaryComputing the extremal eigenvalue bounds of interval matrices is non‐deterministic polynomial‐time (NP)‐hard. We investigate bounds on real eigenvalues of real symmetric tridiagonal interval matrices and prove that for a given real symmetric tridiagonal interval matrices, we can achieve its exact range of the smallest and largest eigenvalues just by computing extremal eigenvalues of four symmetric tridiagonal matrices.

Model for multi-messages spreading over complex networks considering the relationship between messages

A novel messages spreading model is suggested in this paper. The model is a natural generalization of the SIS (susceptible-infective-susceptible) model, in which two relevant messages with same probability of acceptance may spread among nodes. One of the messages has a higher priority to be adopted than the other only in the sense that both messages act on the same node simultaneously. Node in the model is termed as supporter when it adopts either of messages. The transition probability allows that two kinds of supports may transform into each other with a certain rate, and it varies inversely with the associated levels which are discretely distributed in the symmetrical interval around original point. Results of numerical simulations show that individuals tend to believe the messages with a better consistency. If messages are conflicting with each other, the one with higher priority would be spread more and another would be ignored. Otherwise, the number of both supports remains at a uniformly higher level. Besides, in a network with lower connected degree, over a half of the individuals would keep neutral, and the message with lower priority becomes harder to diffuse than the prerogative one. This paper explores the propagation of multi-messages by considering their correlation degree, contributing to the understanding and predicting of the potential propagation trends.

Eigenvalue bounds for symmetric matrices with entries in one interval

In this paper, we consider the eigenvalue bounds for real symmetric matrices whose entries are in a given interval. Based on some matrix computation theories, a new algorithm and some theorems are presented which provide ways for solving the left problems in former papers and obtain satisfying results. Furthermore, the ideas are spread to general symmetric interval matrices. Finally, numerical examples are presented which show the validity of the proposed methods.

An interval-matrix branch-and-bound algorithm for bounding eigenvalues

We present and explore the behaviour of a branch-and-bound algorithm for calculating valid bounds on the kth largest eigenvalue of a symmetric interval matrix. Branching on the interval elements of the matrix takes place in conjunction with the application of Rohn's method (an interval extension of Weyl's theorem) in order to obtain valid outer bounds on the eigenvalues. Inner bounds are obtained with the use of two local search methods. The algorithm has the theoretical property that it provides bounds to any arbitrary precision (assuming infinite precision arithmetic) within finite time. In contrast with existing methods, bounds for each individual eigenvalue can be obtained even if its range overlaps with the ranges of other eigenvalues. Performance analysis is carried out through nine examples. In the first example, a comparison of the efficiency of the two local search methods is reported using 4000 randomly generated matrices. The eigenvalue bounding algorithm is then applied to five randomly generated matrices with overlapping eigenvalue ranges. Valid and sharp bounds are indeed identified given a sufficient number of iterations. Furthermore, most of the range reduction takes place in the first few steps of the algorithm so that significant benefits can be derived without full convergence. Finally, in the last three examples, the potential of the algorithm for use in algorithms to identify index-1 saddle points of nonlinear functions is demonstrated.