Numerical stability in the calculation of eigenfrequencies using integral equations

We comment on a phenomenon of instability that appears while computing eigenfrequencies using the integral equation framework. More precisely, it is currently known that the real symmetric matrices are well, and sometimes the best, adapted to numerical treatment. However, we show that this is not the case, if we wish to determine with high accuracy the spectrum of elliptic, and other related operators, using integral representations.     

On the calculation of the largest eigenvalue of an integral equation

Two methods are given for obtaining computable bounds for the largest eigenvalue of a linear integral equation with a continuous, symmetric, and nonnegative kernel. Several numerical examples are presented.Sponsored by the Mathematics Research Center, United States Army, Madison, Wisconsin, under Contract No.: DA-31-9 24-ARO-D-462.     

Second kind integral equation formulation for the mode calculation of optical waveguides

We present a second kind integral equation (SKIE) formulation for calculating the electromagnetic modes of optical waveguides, where the unknowns are only on material interfaces. The resulting numerical algorithm can handle optical waveguides with a large number of inclusions of arbitrary irregular cross section. It is capable of finding the bound, leaky, and complex modes for optical fibers and waveguides including photonic crystal fibers (PCF), dielectric fibers and waveguides. Most importantly, the formulation is well conditioned even in the case of nonsmooth geometries. Our method is highly accurate and thus can be used to calculate the propagation loss of the electromagnetic modes accurately, which provides the photonics industry a reliable tool for the design of more compact and efficient photonic devices. We illustrate and validate the performance of our method through extensive numerical studies and by comparison with semi-analytical results and previously published results.     

The fuzzy integral

In this paper we define the fuzzy integral of a positive, measurable function, with respect to a fuzzy measure. We show that the monotone convergence theorem and Fatou's lemma are still true in this new setting. We study some of the properties of this integral, and show that it coincides with another fuzzy integral defined in the literature. Our main result is a convergence theorem, that is in a way stronger than the Lebesgue-dominated convergence theorem. This holds when the fuzzy measure is also assumed to be subadditive.     

The definite integral

<正>1.What does the definite integral mean?The definite integral of f(x)fromato b is defined the limit of the sum as n→∞.That is limn→∞∑n i=1f(ξi)·Δxi.We divide the interval[a,b]into n subintervals of equal widthΔx=(b-a)n.Let x0=a,x1,x2,…,xn=b be the endpoints of these subintervals and we chooseξiis any point in the ith subinterval,that is,xi-1≤ξi≤xi,then,the sum∑n f(ξi)·Δxiis called a Rie-     

The definite integral

1. What does the definite integral mean？
The definite integral off（x） from a to b is defined the limit of the sum as n→∞.     

The product integral

The basic concepts and properties of the product integral are discussed and various applications of it are examined.Translated fromItogi Nauki i Tekhniki, Ser. Problemy Geometrii, Vol. 22, 1990, pp. 167–215.     

The exponential integral distribution

The exponential integral distribution is introduced. The gamma distribution is a subclass of this distribution. The higher order exponential integrals are closely related to both the gamma and the beta function. The physical relevance of this new distribution is discussed.     

The Kontorovich-Lebedev integral transform

One considers the Kontorovich-Lebedev integral transform

The widest continuous integral

An approach to a definition of an integral, which differs from definitions of Lebesgue and Henstock-Kurzweil integrals, is considered. We use trigonometrical polynomials instead of simple functions. Let V be the space of all complex trigonometrical polynomials without the free term. The definition of a continuous integral on the space V is introduced. All continuous integrals are described in terms of norms on V. The existence of the widest continuous integral is proved, the explicit form of its norm is obtained and it is proved that this norm is equivalent to the Alexiewicz norm. It is shown that the widest continuous integral is wider than the Lebesgue integral. An analog of the fundamental theorem of calculus for the widest continuous integral is given.     

The weak McShane integral

We present a weaker version of the Fremlin generalized McShane integral (1995) for functions defined on a σ-finite outer regular quasi Radon measure space (S,Σ, T, µ) into a Banach space X and study its relation with the Pettis integral. In accordance with this new method of integration, the resulting integral can be expressed as a limit of McShane sums with respect to the weak topology. It is shown that a function f from S into X is weakly McShane integrable on each measurable subset of S if and only if it is Pettis and weakly McShane integrable on S. On the other hand, we prove that if an X-valued function is weakly McShane integrable on S, then it is Pettis integrable on each member of an increasing sequence (S _l ) _l?1 of measurable sets of finite measure with union S. For weakly sequentially complete spaces or for spaces that do not contain a copy of c ₀, a weakly McShane integrable function on S is always Pettis integrable. A class of functions that are weakly McShane integrable on S but not Pettis integrable is included.     

The set-indexed Ito integral

We construct an Ito-type stochastic integral where the integrator is a process indexed by a semilattice of compact subsets of a fixed topological spaceT and the integrands, which are indexed by the points inT, possess a natural form of predictability. The definition of the integral involves, among other things, an Ito-type isometry defined in terms of the set-indexed quadratic variation of the integrator. The martingale property and quadratic variation for the resulting integral process are derived. In addition, employing the notion of stopping set from Ivanoff and Merzbach (1995), we construct and study a set-indexed local integral. A novel and flexible notion of predictability for set-indexed processes is defined and characterized, permitting the integration of a set-indexed integrand against a set-indexed process. Research supported in part by the Israel Science Foundation (grant no.: 0321423). Research supported in part by a grant from the Natural, Sciences and Engineering Research Council of Canada.