Gaussian interval quadrature formula

Summary. We prove the existence of a Gaussian quadrature formula for Tchebycheff systems, based on integrals over non-overlapping subintervals of arbitrary fixed lengths and the uniqueness of this formula in the case the subintervals have equal lengths. Received July 6, 1999 / Published online August 24, 2000     

On the Complexity of Isolating Real Roots and Computing with Certainty the Topological Degree

In this contribution the isolation of real roots and the computation of the topological degree in two dimensions are considered and their complexity is analyzed. In particular, we apply Stenger's degree computational method by splitting properly the boundary of the given region to obtain a sequence of subintervals along the boundary that forms a sufficient refinement. To this end, we properly approximate the function using univariate polynomials. Then we isolate each one of the zeros of these polynomials on the boundary of the given region in various subintervals so that these subintervals form a sufficiently refined boundary.     

The inverse spectral problem for a Stieltjes eight-shaped string

We consider a spectral problem generated by the Stieltjes string equation on a metric figure-of-eight graph and the corresponding inverse problem which is stated as follows. Values of the point masses located on one of the loops and the lengths of subintervals on it are given together with the spectrum of the spectral problem on the whole graph, the total length of the second loop and the length of the first subinterval on it, and a certain constant. Values of the masses and the lengths of subintervals on the second loop are to be found. Conditions sufficient for such a problem to be solvable are given in the implicit form. An algorithm for recovering the masses and the subintervals on the second loop is proposed.     

Inverse problems for Sturm–Liouville operators on a compact equilateral graph by partial nodal data

Partial inverse nodal problems for Sturm–Liouville operators on a compact equilateral star graph are investigated in this paper. Uniqueness theorems from partial twin‐dense nodal subsets in interior subintervals or arbitrary interior subintervals having the central vertex are proved. In particular, we posed and solved a new type partial inverse nodal problems for the Sturm–Liouville operator on the compact equilateral star graph.     

New criteria for synchronization stability of continuous complex dynamical networks with non-delayed and delayed coupling

The synchronization stability problem of general complex dynamical networks with non-delayed and delayed coupling is investigated based on a piecewise analysis method, the variation interval of the time delay is firstly divided into several subintervals, by checking the variation of derivative of a Lyapunov functional in every subinterval, several new delay-dependent synchronization stability conditions are derived in the form of linear matrix inequalities, which are easy to be verified by the LMI toolbox. Some numerical examples show that, when the number of the divided subintervals increases, the corresponding criteria can provide much less conservative results.     

带松弛因子的Schwarz交替方法

§1.引言 Schwarz交替方法的收敛速度,依赖于子区域重迭部分的大小,重迭部分越大,收敛越快。然而重迭部分增大,必将引起计算量的增大,因此,在重迭部分不变的情况下,如何改善Schwarz交替过程的收敛速度,已成为人们感兴趣的问题。关于Schwarz算法收敛速度的讨论,许多文章都是对具体类型的微分方程展开的。     

关于Henstock积分的Harnack扩张的一个注记

本文定义了（ＲＬ）条件，利用（ＲＬ）条件给出了等式成立的充分必要条件．     

Splitting intervals

Given an interval we choose a point in the interval and one of the two subintervals thus obtained in a symmetric random manner. The process is repeated ad infinitum. We study the limiting distribution of the midpoint of the interval.     

An enhanced subinterval analysis method for uncertain structural problems

This paper proposes an enhanced subinterval analysis method to predict the bounds of structural response with interval parameters, which could deal with problems with relatively large uncertainties of the parameters. The intervals are first divided into several subintervals, and two expansion routes are then constructed based on the sensitivity analysis. Two subinterval sets are selected according to the expansion points on the routes, and the first order Taylor expansion method is then adopted to complete the subinterval analysis. Based on the selected subinterval sets, the upper and lower bounds of the structural response are further obtained by employing the interval union operation. An adaptive convergence approach is presented to determine the appropriate number of subintervals. Four numerical examples are investigated to demonstrate the validity of the proposed method.     

Digitization of nonautonomous control systems

Methods of the theory of nonautonomous differential equations are used to study the extent to which the properties of local null controllability and local feedback stabilizability are preserved when a control system with time-varying coefficients is digitized, e.g., approximated by piecewise autonomous systems on small time subintervals.     

EXISTENCE OF POSITIVE SOLUTIONS TO(n-1,1)CONJUGATE BOUNDARY VALUE PROBLEMS

1IntroductionInarecentpaper[fi,P.W.EloeandJ.Hendersonstudiedtheexistenceofpositivesolutionsofthefollowingnonlinear(n--l,l)conjugateboundaryvalueproblem'TheworkwassupportedbyNSFofHeilonaiiangandNNSFofChinawheref:[0,co)-[0,co),a:[0,l]-[0,co)arecontinuousand…     

Digitization of nonautonomous control systems

Methods of the theory of nonautonomous differential equations are used to study the extent to which the properties of local null controllability and local feedback stabilizability are preserved when a control system with time-varying coefficients is digitized, e.g., approximated by piecewise autonomous systems on small time subintervals.     

A dynamic-programming algorithm for hierarchical discretization of continuous attributes

Discretization techniques can be used to reduce the number of values for a given continuous attribute, and a concept hierarchy can be used to define a discretization of a given continuous attribute. Traditional methods of building a concept hierarchy from a continuous attribute are usually based on the level-wise approach. Unfortunately, this approach suffers from three weaknesses: (1) it only seeks a local optimal solution instead of a global optimal, (2) it is usually subject to the constraint that each interval can only be partitioned into a fixed number of subintervals, and (3) the constructed tree may be unbalanced. In view of these weaknesses, this paper develops a new algorithm based on dynamic-programming strategy for constructing concept hierarchies from continuous attributes. The constructed trees have three merits: (1) they are global optimal trees, (2) each interval is partitioned into the most appropriate number of subintervals, and (3) the trees are balanced. Finally, we carry out an experimental study using real data to show its efficiency and effectiveness.     

An extension of hölder’s theorem on the gamma function

The Gamma function is differentially transcendental over certain differential domains of real-valued functions defined on fixed subintervals of (0, ∞). In memory of Lee A. Rubel Research supported by NSF Postdoctoral Fellowship No. DMS-9407549.     

一类中立型微分方程的区间振动准则

通过引人参数函数,对一类中立型时滞微分方程建立了一些区间振动准则,这些振动准则不同于已知依赖于整个[tο,∞）的性质的结果,而是仅依赖于[tο,∞）上的子区间列的性质。     

一种广义插值法

本文考虑一种广义插值问题，插值条件为小区间上的积分值，以弥补现有的插值方法在L2空间不再适用的不足，除了多项式插值外，还讨论了两种一次样条插值方法。     

The use of approximate factorization in stiff ODE solvers

We consider implicit integration methods for the numerical solution of stiff initial-value problems. In applying such methods, the implicit relations are usually solved by Newton iteration. However, it often happens that in subintervals of the integration interval the problem is nonstiff or mildly stiff with respect to the stepsize. In these nonstiff subintervals, we do not need the (expensive) Newton iteration process. This motivated us to look for an iteration process that converges in mildly stiff situations and is less costly than Newton iteration. The process we have in mind uses modified Newton iteration as the outer iteration process and a linear solver for solving the linear Newton systems as an inner iteration process. This linear solver is based on an approximate factorization of the Newton system matrix by splitting this matrix into its lower and upper triangular part. The purpose of this paper is to combine fixed point iteration, approximate factorization iteration and Newton iteration into one iteration process for use in initial-value problems where the degree of stiffness is changing during the integration.     

Robust stability results for nonlinear Markovian jump systems with mode‐dependent time‐varying delays and randomly occurring uncertainties

This article is concerned with the robust stability analysis for Markovian jump systems with mode‐dependent time‐varying delays and randomly occurring uncertainties. Sufficient delay‐dependent stability results are derived with the help of stability theory and linear matrix inequality technique using direct delay‐decomposition approach. Here, the delay interval is decomposed into two subintervals using the tuning parameter η such that , and the sufficient stability conditions are derived for each subintervals. Further, the parameter uncertainties are assumed to be occurring in a random manner. Numerical examples are given to validate the derived theoretical results. © 2015 Wiley Periodicals, Inc. Complexity 21: 50–60, 2016     

Subperiodic Dubiner distance,norming meshes and trigonometric polynomial optimization

We extend the notion of Dubiner distance from algebraic to trigonometric polynomials on subintervals of the period, and we obtain its explicit form by the Szeg? variant of Videnskii inequality. This allows to improve previous estimates for Chebyshev-like trigonometric norming meshes, and suggests a possible use of such meshes in the framework of multivariate polynomial optimization on regions defined by circular arcs.     

Reduction of overestimation in interval arithmetic simulation of biological wastewater treatment processes

A novel interval arithmetic simulation approach is introduced in order to evaluate the performance of biological wastewater treatment processes. Such processes are typically modeled as dynamical systems where the reaction kinetics appears as additive nonlinearity in state. In the calculation of guaranteed bounds of state variables uncertain parameters and uncertain initial conditions are considered. The recursive evaluation of such systems of nonlinear state equations yields overestimation of the state variables that is accumulating over the simulation time. To cope with this wrapping effect, innovative splitting and merging criteria based on a recursive uncertain linear transformation of the state variables are discussed. Additionally, re-approximation strategies for regions in the state space calculated by interval arithmetic techniques using disjoint subintervals improve the simulation quality significantly if these regions are described by several overlapping subintervals. This simulation approach is used to find a practical compromise between computational effort and simulation quality. It is pointed out how these splitting and merging algorithms can be combined with other methods that aim at the reduction of overestimation by applying consistency techniques. Simulation results are presented for a simplified reduced-order model of the reduction of organic matter in the activated sludge process of biological wastewater treatment.