Symmetric Interval Research Articles

A Markovian Gauss inequality for asymmetric deviations from the mode of symmetric unimodal distributions

For a random variable with a unimodal distribution and finite second moment Gauß (1823) proved a sharp bound on the probability of the random variable to be outside a symmetric interval around its mode. An alternative proof for it under the assumption of a finite r r -th absolute moment is given ( r ≥ 1 r\geq 1 ), based on the Khintchine representation of unimodal random variables. A special instance of the resulting Narumi–Gauß inequality is the one with finite first absolute moment, which might be called a Markovian Gauß inequality. For symmetric unimodal distributions with finite second moment Semenikhin (2019) generalized the Gauß inequality to arbitrary intervals. For the class of symmetric unimodal distributions with finite first absolute moment we construct a Markovian version of it. Related inequalities of Volkov (1969) and Sellke and Sellke (1997) will be discussed as well.

A Modified Bubnov-Galerkin Method for Solving Boundary Value Problems with Linear Ordinary Differential Equations

Introduction. The paper considers the solution of boundary value problems on an interval for linear ordinary differential equations, in which the coefficients and the right-hand side are continuous functions. The conditions for the orthogonality of the residual equation to the coordinate functions are supplemented by a system of linearly independent boundary conditions. The number of coordinate functions m must exceed the order n of the differential equation. Materials and Methods. To numerically solve the boundary value problem, a system of linearly independent coordinate functions is proposed on a symmetric interval [−1,1], where each function has a unit Chebyshev’s norm. A modified Petrov-Galerkin method is applied, incorporating linearly independent boundary conditions from the original problem into the system of linear algebraic equations. An integral quadrature formula with twelfth-order error is used to compute the scalar product of two functions. Results. A criterion for the existence and uniqueness of a solution to the boundary value problem is obtained, provided that n linearly independent solutions of the homogeneous differential equation are known. Formulas are derived for the matrix coefficients and the coefficients of the right-hand side in the system of linear algebraic equations for the vector expansion of the solution in terms of the coordinate function system. These formulas are obtained for second- and third-order linear differential equations. The modified Bubnov-Galerkin method is formulated for differential equations of arbitrary order. Discussion and Conclusions. The derived formulas for the generalized Bubnov-Galerkin method may be useful for solving boundary value problems involving linear ordinary differential equations. Three boundary value problems with second- and third-order differential equations are numerically solved, with the uniform norm of the residual not exceeding 10–11.

Metric ultraproducts of groups — Simplicity, perfectness and torsion

We characterise the simplicity of metric ultraproducts of a family of metric groups. We also present several new examples of simple groups, such as metric ultraproducts of finite and infinite symmetric groups, linear groups, and interval exchange transformation groups. Using similar methods, we also examine concepts such as genericity, perfectness, and torsion.

Increasing Gambler’s Ruin duration and Brownian motion exit times

. In Gambler’s Ruin when both players start with the same amount of money, we show the playing time stochastically increases when the games are made more fair. We give two different arguments for this fact that extend results from Peköz and Ross. We then use this to show that the exit time from a symmetric interval for Brownian motion with drift stochastically increases as the drift moves closer to zero; this result is not easily obtainable from the available explicit formulas for the density.

Bounds on eigenvalues of real symmetric interval matrices for αBB method in global optimization

In this paper, we investigate bounds on eigenvalues of real symmetric interval matrices. We present a method that computes bounds on eigenvalues of real symmetric interval matrices. It outperforms many methods developed in the literature and produces as sharp as possible bounds on eigenvalues of real symmetric interval matrices. The aim is to apply the proposed method to compute lower bounds on eigenvalues of a symmetric interval hessian matrix of a nonconvex function in the ?BB method and use them to produce a tighter underestimator that improves the ?BB algorithm for solving global optimization problems. In the end, we illustrate by example, the comparison of various approaches of bounding eigenvalues of real symmetric interval matrices. Moreover, a set of test problems found in the literature are solved efficiently and the performances of the proposed method are compared with those of other methods.

A Novel Thermoelectric Generation Array Reconfiguration to Reduce Mismatch Power Loss Under Nonuniform Temperature Distribution

In practice, industrial exhaust emissions as well as emissions from automobiles, ships, biomass combustion, etc., can be potential application areas for thermoelectric generation (TEG). However, the structural design of heat exchange equipment is usually limited by the internal flow field, resulting in uneven temperature distribution on the heat exchange equipment’s surface. The resulting mismatch power loss is a major challenge for thermoelectric power generation. In this study, based on the characteristics of the surface temperature distribution of heat exchange equipment in the context of gas emissions, a static reconfiguration scheme is proposed for reconfiguring honeycomb (HC) arrays using the symmetric interval crossing (SIC) method. Based on a fixed interconnect array configuration, the solution requires only a change in the location of the modules and no change in the electrical connections, thus reducing mismatch losses while lowering manufacturing costs. Test experiments are conducted for 6 × 6 TEG arrays, mismatch losses are evaluated for four nonuniform temperature distribution cases, and the performance of seven different TEG array configurations is compared. The findings demonstrate that, in nonuniform temperature distribution scenarios, the SIC method can effectively reduce mismatch losses and has a greater output power than alternative array configurations.

Optimal symmetric interval solution of fuzzy relation inequality considering the stability in P2P educational information resources sharing system

The max-min fuzzy relation inequalities have been recently introduced for modeling the Peer-to-Peer (P2P) educational information resources sharing system. Each solution of the max-min system is indeed a feasible scheme in the corresponding P2P network system. For a given solution, the allowed fluctuation tolerant for all the components, reflects the stability of its corresponding feasible scheme. Considering the stability, we are interested in the maximum fluctuation for a given solution in this work. In order to differentiate the absolute fluctuation and the relative fluctuation, we define and investigate the MAF-based and the MRF-based optimal symmetric interval solutions, respectively. Two independent algorithms are developed for solving these optimal symmetric interval solutions. The numerical examples show that our proposed algorithms are valid and efficient.

On Some Class of Normal Differential Operators for First Order

In this work, we construct the minimal and maximal operators generated by linear differential-operator expression for first order in the Hilbert space of vector-functions on finite symmetric interval. Then, deficiency indices of the minimal operator will be calculated and the space of boundary values of this operator will be constructed. By using of Calkin-Gorbachuk method, the general representation of all normal extensions of the formally normal minimal operator in terms of boundary values will also be established. Moreover we explore the spectrum structure of these extensions.

Distribution of Eigenvalues and Upper Bounds of the Spread of Interval Matrices

The distribution of eigenvalues and the upper bounds for the spread of interval matrices are significant in various fields of mathematics and applied sciences, including linear algebra, numerical analysis, control theory, and combinatorial optimization. We present the distribution of eigenvalues within interval matrices and determine upper bounds for their spread using Geršgorin’s theorem. Specifically, through an equality for the variance of a discrete random variable, we derive upper bounds for the spread of symmetric interval matrices. Finally, we give three numerical examples to illustrate the effectiveness of our results.

An Improved Harris Hawks Optimization Algorithm and Its Application in Grid Map Path Planning.

Aimed at the problems of the Harris Hawks Optimization (HHO) algorithm, including the non-origin symmetric interval update position out-of-bounds rate, low search efficiency, slow convergence speed, and low precision, an Improved Harris Hawks Optimization (IHHO) algorithm is proposed. In this algorithm, a circle map was added to replace the pseudo-random initial population, and the population boundary number was reduced to improve the efficiency of the location update. By introducing a random-oriented strategy, the information exchange between populations was increased and the out-of-bounds position update was reduced. At the same time, the improved sine-trend search strategy was introduced to improve the search performance and reduce the out-of-bound rate. Then, a nonlinear jump strength combining escape energy and jump strength was proposed to improve the convergence accuracy of the algorithm. Finally, the simulation experiment was carried out on the test function and the path planning application of a 2D grid map. The results show that the Improved Harris Hawks Optimization algorithm is more competitive in solving accuracy, convergence speed, and non-origin symmetric interval search efficiency, and verifies the feasibility and effectiveness of the Improved Harris Hawks Optimization in the path planning of a grid map.

Measurement uncertainty interval in case of a known relationship between precision and mean.

Background: Measurement uncertainty is typically expressed in terms of a symmetric interval y±U, where y denotes the measurement result and U the expanded uncertainty. However, in the case of heteroscedasticity, symmetric uncertainty intervals can be misleading. In this paper, a different approach for the calculation of uncertainty intervals is introduced. Methods: This approach is applicable when a validation study has been conducted with samples with known concentrations. In a first step, test results are obtained at the different known concentration levels. Then, on the basis of precision estimates, a prediction range is calculated. The measurement uncertainty for a given test result can then be obtained by projecting the intersection of the test result with the limits of the prediction range back onto the axis of the known values, now interpreted as representing the measurand. Results: It will be shown how, under certain circumstances, asymmetric uncertainty intervals arise quite naturally and lead to more reliable uncertainty intervals. Conclusions: This article establishes a conceptual framework in which measurement uncertainty can be derived from precision whenever the relationship between the latter and concentration has been characterized. This approach is applicable for different types of distributions. Closed expressions for the limits of the uncertainty interval are provided for the simple case of normally distributed test results and constant relative standard deviation.

Design of control algorithms for false information dissemination in smart city social media based on symmetric triangle interval

Design of control algorithms for false information dissemination in smart city social media based on symmetric triangle interval

A Fast Algorithm for the Eigenvalue Bounds of a Class of Symmetric Tridiagonal Interval Matrices

The eigenvalue bounds of interval matrices are often required in some mechanical and engineering fields. In this paper, we improve the theoretical results presented in a previous paper “A property of eigenvalue bounds for a class of symmetric tridiagonal interval matrices” and provide a fast algorithm to find the upper and lower bounds of the interval eigenvalues of a class of symmetric tridiagonal interval matrices.

On eigenvalues of real symmetric interval matrices: Sharp bounds and disjointness

In this paper, the eigenvalue problem of real symmetric interval matrices is studied. First, in the case of $2 \times 2$ real symmetric interval matrices, all the four endpoints of the two eigenvalue intervals are determined. These are not necessarily eigenvalues of vertex matrices, but it is shown that such a real symmetric interval matrix can be constructed from the original one. Then, necessary and sufficient conditions are provided for the disjointness of eigenvalue intervals. In the general $n\times n$ case, due to Hertz, a set of special vertex matrices determines the maximal eigenvalue and a similar statement holds for the minimal one. In a special case, namely if the right endpoints of the off-diagonal intervals are not smaller than the absolute value of the left ones, he concluded the vertex matrix of the right endpoints provides the maximal eigenvalue. Generalizing it, it is shown that in the case of real symmetric interval matrices with special sign pattern, a single vertex matrix determines one of the extremal bounds.

The shortest confidence interval for the ratio of quantiles of the Dagum distribution

Jȩdrzejczak et al. (REVSTAT-Statistical Journal 19(1), 87–97, 2021) constructed a confidence interval for a ratio of quantiles coming from the Dagum distribution, which is frequently applied as a theoretical model in numerous income distribution analyses. The proposed interval is symmetric with respect to the ratio of sample quantiles, which result may be unsatisfactory in many practical applications. The search for a confidence interval with a smaller length resulted in the derivation of the shortest interval with the ends being asymmetric relative to the ratio of sample quantiles. In the paper, the existence of the shortest confidence interval is shown and the method of obtaining such an interval is presented. The results of the calculations show a reduction in the length of the proposed confidence interval by several percent compared to the symmetrical confidence interval.

Spin it as you like: The (lack of a) measurement of the spin tilt distribution with LIGO-Virgo-KAGRA binary black holes

Context. The growing set of gravitational-wave sources is being used to measure the properties of the underlying astrophysical populations of compact objects, black holes, and neutron stars. Most of the detected systems are black hole binaries. While much has been learned about black holes by analyzing the latest LIGO-Virgo-KAGRA (LVK) catalog, GWTC-3, a measurement of the astrophysical distribution of the black hole spin orientations remains elusive. This is usually probed by measuring the cosine of the tilt angle (cosτ) between each black hole spin and the orbital angular momentum, with cosτ = +1 being perfect alignment. Aims. The LVK Collaboration has modeled the cosτ distribution as a mixture of an isotropic component and a Gaussian component with mean fixed at +1 and width measured from the data. We want to verify if the data require the existence of such a peak at cosτ = +1. Methods. We used various alternative models for the astrophysical tilt distribution and measured their parameters using the LVK GWTC-3 catalog. Results. We find that (a) augmenting the LVK model, such that the mean μ of the Gaussian is not fixed at +1, returns results that strongly depend on priors. If we allow μ > +1, then the resulting astrophysical cosτ distribution peaks at +1 and looks linear, rather than Gaussian. If we constrain −1 ≤ μ ≤ +1, the Gaussian component peaks at μ = 0.48−0.99+0.46 (median and 90% symmetric credible interval). Two other two-component mixture models yield cosτ distributions that either have a broad peak centered at 0.19−0.18+0.22 or a plateau that spans the range [ − 0.5, +1], without a clear peak at +1. (b) All of the models we considered agree as to there being no excess of black hole tilts at around −1. (c) While yielding quite different posteriors, the models considered in this work have Bayesian evidences that are the same within error bars. Conclusions. We conclude that the current dataset is not sufficiently informative to draw any model-independent conclusions on the astrophysical distribution of spin tilts, except that there is no excess of spins with negatively aligned tilts.

Two-Stage Fuzzy Interactive Multi-Objective Approach under Interval Type-2 Fuzzy Environment with Application to the Remanufacture of Old Clothes

In this study, a two-stage approach is introduced to obtain a more interactive and flexible solution to deal with the multi-objective programming under interval type-2 fuzzy environment. In the first stage, the fuzzy multi-objective chance-constrained programming with regular symmetric triangular interval type-2 fuzzy set parameters is proposed and transferred into its crisp equivalent form. In the second stage, we use the fuzzy interactive approach to address the crisp multi-objective programming obtained in the first stage by introducing the trade-off rate, which helps the decision maker react via updating their reference member values to obtain a satisfactory solution. Finally, taking a remanufacture of old clothes problem as an example, the comparison of experimental results obtained using a non-interactive method and interactive method shows that the proposed approach is conducive to obtaining satisfactory solutions effectively and efficiently, which broadens the application scope of the multi-objective programming with regular symmetric triangular interval type-2 fuzzy set parameters for sustainable manufacturing.

Controller robust placement with dynamic traffic in software-defined networking

For dynamic traffic of switches in the software-defined networking, the existing methods mostly adopt the switch migration to balance the controller with an uneven load. However, these researches have overlooked that the frequent migration of switches will cause instability in the control plane and increase the network delay. To reduce switch migration, this paper proposes a robust controller placement model that considers dynamic traffic while placing the controllers. Firstly, the bounded symmetric interval is used to describe the dynamic traffic in the network, and the magnitude of the traffic change is represented by the variation parameter σ. Secondly, the control parameter Γ is introduced to control the conservativeness of the solution and construct a multi-controller robust placement model. Finally, the model is transformed into a linear programming problem based on the strong duality theory, and we use the tabu search algorithm to implement an efficient solution. The simulation results show that different controller placement strategies can be obtained by changing the control parameter. When the service level is low, the cost increase can greatly improve the service level; however, when the service level exceeds 90%, the 10% increase in placement cost can only improve the service level by 1%.

On estimating the boundaries of a uniform distribution under additive measurement errors

This paper presents the method-of-moments (MM) and the maximum likelihood (ML) estimators for the width of a uniform random variable U when measured with additive errors. We study situations where the support is either the symmetric interval [−a, a] or asymmetric interval [a, b]. Both MM and ML estimators for the boundaries are discussed in two cases, namely, when the noise level σ2 is known and when it is unknown. While the MM estimators have a closed form, the ML estimators need an iterative algorithm to solve a system of nonlinear equations. A reliable algorithm to compute the ML estimators is proposed. We also establish sufficient conditions for the existence of the MM estimators and their asymptotic normality. Numerical experiments on real and synthetic data are presented to validate our results.

Variance and Semi-Variances of Regular Interval Type-2 Fuzzy Variables

In this paper, we define the variance and semi-variances of regular interval type-2 fuzzy variables (RIT2-FVs) as well as derive a calculation formula of them based on the credibility distribution. Following the relationship between the variance and the semi-variances of the regular symmetric triangular interval type-2 fuzzy variables (RSTIT2-FVs), a special type of interval type-2 fuzzy variable is discovered and proved. Furthermore, for applying the two measures, we propose the operational law for the variance and semi-variances of the linear function of mutually independent RSTIT2-FVs. Some numerical examples are illustrated. The consequences of examples prove that the formulas we proposed can be effectively applied to the calculation of the variance of RSTIT2-FVs. The results indicate that they play a great role in the application of variance of type-2 fuzzy sets in various fields.