Lebesgue Integral Research Articles

Jensen and Ostrowski type inequalities for general Lebesgue integral with applications

Some inequalities related to Jensen and Ostrowski inequalities for general Lebesgue integral are obtained. Applications for f-divergence measure are provided as well.

The fractional Sturm–Liouville problem—Numerical approximation and application in fractional diffusion

The numerical method of solving the fractional eigenvalue problem is derived in the case when the fractional Sturm–Liouville equation is subjected to the mixed boundary conditions. This non-integer order differential equation is discretized to the scheme with the symmetric matrix representing the action of the numerically expressed composition of the left and the right Caputo derivative. The numerical eigenvalues are thus real, and the eigenvectors associated to distinct eigenvalues are orthogonal in the respective finite-dimensional Hilbert space. The advantage of the proposed method is the formulation which allows us to construct the approximate eigenfunctions which form an orthonormal function system in the infinite-dimensional weighted Lebesgue integrable function space. The developed numerical method of calculation of the eigenvalues and eigenfunctions is then applied in construction of the approximate solution to the 1D space-time fractional diffusion problem in a bounded domain.

A kernel estimate method for characteristic function-based uncertainty importance measure

In this paper, we propose a fast computation method based on a kernel function for the characteristic function-based moment-independent uncertainty importance measure θi. We first point out that the possible computational complexity problems that exist in the estimation of θi. Since the convergence rate of a double-loop Monte Carlo (MC) simulation is O(N−1/4), the first possible problem is the use of double-loop MC simulation. And because the norm of the difference between the unconditional and conditional characteristic function of model output in θi is a Lebesgue integral over the infinite interval, another possible problem is the computation of this norm. Then a kernel function is introduced to avoid the use of double-loop MC simulation, and a longer enough bounded interval is selected to instead of the infinite interval to calculate the norm. According to these improvements, a kind of fast computational methods is introduced for θi, and during the whole process, all θi can be obtained by using a single quasi-MC sequence. From the comparison of numerical error analysis, it can be shown that the proposed method is an effective and helpful approach for computing the characteristic function-based moment-independent importance index θi.

Theory of inclusion-exclusion integral

We propose an integral with respect to a nonadditive monotone measure. This integral is a generalization of the Lebesgue integral and also the Choquet integral. It has appropriate properties, and can be extensively applied to real data analysis.

Calculation of Lebesgue integrals by using uniformly distributed sequences

A certain modified version of Kolmogorov’s strong law of large numbers is used for an extension of the result of C. Baxa and J. Schoißengeier (2002) to a maximal set of uniformly distributed sequences in (0,1) which strictly contains the set of all sequences having the form ({αn})n∈N for some irrational number α and having the full ℓ1∞-measure, where ℓ1∞ denotes the infinite power of the linear Lebesgue measure ℓ1 in (0,1).

A unified approach to the monotone convergence theorem for nonlinear integrals

We present a unified approach to the monotone convergence theorem for nonlinear integrals such as the Choquet, the Šipoš, the Sugeno, and the Shilkret integral. A nonlinear integral may be viewed as a nonlinear functional defined on a set of pairs of a nonadditive measure and a measurable function. We thus formulate our general type of monotone convergence theorem for such a functional. The key tool is a perturbation of functional that manages not only the monotonicity of the functional but also the small change of the functional value arising as a result of adding small amounts to a measure and a function in the domain of the functional. Our approach is also applicable to the Lebesgue integral when a nonadditive measure is σ-additive.

A unifying approach to risk-measure-based optimal reinsurance problems with practical constraints

The design of optimal reinsurance treaties in the presence of multifarious practical constraints is a substantive but underdeveloped topic in modern risk management. To examine the influence of these constraints on the contract design systematically, this article formulates a generic constrained reinsurance problem where the objective and constraint functions take the form of Lebesgue integrals whose integrands involve the unit-valued derivative of the ceded loss function to be chosen. Such a formulation provides a unifying framework to tackle a wide body of existing and novel distortion-risk-measure-based optimal reinsurance problems with constraints that reflect diverse practical considerations. Prominent examples include insurers’ budgetary, regulatory and reinsurers’ participation constraints. An elementary and intuitive solution scheme based on an extension of the cost–benefit technique in Cheung and Lo [Cheung, K.C. & Lo, A. (2015, in press). Characterizations of optimal reinsurance treaties: a cost–benefit approach Scandinavian Actuarial Journal. doi:10.1080/03461238.2015.1054303.] is proposed and illuminated by analytically identifying the optimal risk-sharing schemes in several concrete optimal reinsurance models of practical interest. Particular emphasis is placed on the economic implications of the above constraints in terms of stimulating or curtailing the demand for reinsurance, and how these constraints serve to reconcile the possibly conflicting objectives of different parties.

English

In this paper, we prove and discuss averaging results for ordinary differential equations perturbed by a small parameter. The conditions we assume on the right-hand sides of the equations under which our averaging results are stated are more general than those considered in the literature. Indeed, often it is assumed that the right-hand sides of the equations are uniformly bounded and a Lipschitz condition is imposed on them. Sometimes this last condition is relaxed to the uniform continuity in the second variable uniformly with respect to the first one. In our results, we assume only that the right-hand sides of the equations are bounded by some locally Lebesgue integrable functions with the property that their indefinite integrals satisfy a Lipschitz-type condition. Also, we consider that they are only continuous in the second variable uniformly with respect to the first one.

Determination of Jumps in Terms of Linear Operators

A theorem of Lukacs [J. Reine Angew. Math., 150, 107–112 (1920)] states that the partial sums of conjugate Fourier series of a periodic Lebesgue integrable function f diverge with a logarithmic rate at the points of discontinuity of f of the first kind. Moricz [Acta Math. Hung., 98, 259–262 (2003)] proved a similar theorem for the rectangular partial sums of double conjugate trigonometric Fourier series. We consider analogs of the Moricz theorem for the generalized Cesaro means and positive linear means. In the present paper we prove a similar theorem in terms of linear operators satisfying certain conditions.

Structured populations with distributed recruitment: from PDE to delay formulation

In this work, first, we consider a physiologically structured population model with a distributed recruitment process. That is, our model allows newly recruited individuals to enter the population at all possible individual states, in principle. The model can be naturally formulated as a first‐order partial integro‐differential equation, and it has been studied extensively. In particular, it is well posed on the biologically relevant state space of Lebesgue integrable functions. We also formulate a delayed integral equation (renewal equation) for the distributed birth rate of the population. We aim to illustrate the connection between the partial integro‐differential and the delayed integral equation formulation of the model utilising a recent spectral theoretic result. In particular, we consider the equivalence of the steady state problems in the two different formulations, which then lead us to characterise irreducibility of the semigroup governing the linear partial integro‐differential equation. Furthermore, using the method of characteristics, we investigate the connection between the time‐dependent problems. In particular, we prove that any (non‐negative) solution of the delayed integral equation determines a (non‐negative) solution of the partial differential equation and vice versa. The results obtained for the particular distributed states at birth model then lead us to present some very general results, which establish the equivalence between a general class of partial differential and delay equation, modelling physiologically structured populations. Copyright © 2016 John Wiley & Sons, Ltd.

The Universal Approximation Capabilities of Cylindrical Approximate Identity Neural Networks

Universal approximation capability of feedforward neural networks is one of the important theoretical concepts in artificial neural networks. In this study, a type of single-hidden-layer feedforward neural networks is presented. The networks is called feedforward cylindrical approximate identity neural networks. Then, universal approximation capabilities of the networks are investigated in two function spaces. Whereby the notions of cylindrical approximate identity and cylindrical convolution are introduced. The analyses are divided into two cases: In the first case, universal approximation capability of a single-hidden-layer feedforward cylindrical approximate identity neural networks to continuous bivariate functions on the infinite cylinder is investigated. In the latter case, universal approximation capability of the networks is extended to the pth-order Lebesgue integrable bivariate functions on the infinite cylinder.

Verification of the Lebesgue averaging method

This work is devoted to the study of the accuracy of the Lebesgue averaging method for the spectra of resonance radiation in solving the transport equation. The method is tested for the problem of the thermal radiation transfer in the Earth atmosphere. The results of numerical calculations in two versions of the Lebesgue averaging method are compared with the results of the direct pointwise (lineby-line) spectral calculations. In the first version, the classical Lebesgue integral is used, and the absorption coefficient is taken in it as the independent variable instead of the photons energy. In the second version, we apply the Lebesgue-Stieltjes integral with the measure of Lebesgue’s set as the independent variable. All calculations are performed by using one scheme of spatial and angular discretization of the transport equation. This ensures the purity of the computational experiment. The second variant of averaging by the use of Lebesgue-Stieltjes integration showed the highest accuracy (within 5%) with a significant decrease of the number of arithmetic operations (approximately by a factor of 103–104 ) in relation to the direct spectral calculations.

ON HENSTOCK-STIELTJES INTEGRALS OF INTERVAL-VALUED FUNCTIONS ON TIME SCALES

In this paper we introduce the interval-valued Henstock- Stieltjes integral on time scales and investigate some properties of these integrals. 1. Introduction and preliminaries The Henstock integral for real functions was first defined by Henstock (2) in 1963. The Henstock integral is more powerful and simpler than the Lebesgue, Wiener and Feynman integrals. The Henstock delta integral on time scales was introduced by Allan Peterson and Bevan Thompson (5) in 2006. In 2000, Congxin Wu and Zengtai Gong introduced the concept of the Henstock integral of interval-valued functions (6). In this paper we introduce the concept of the Henstock-Stieltjes delta integral of interval-valued functions on time scales and investigate some properties of the integral. A time scale T is a nonempty closed subset of real number R with the subspace topology inherited from the standard topology of R. For t ∈ T we define the forward jump operator σ(t) = inf{s ∈ T : s > t} where inf ϕ = sup T , while the backward jump operator ρ(t) = sup{s ∈ T : s t, we say that t is right-scattered, while if ρ(t) 0 on (a, b)T , δR(t) > 0 on (a, b)T , δL(a) ≥ 0, δR(b) ≥ 0 and δR(t) ≥ µ(t) for each t ∈ (a, b)T.

L2–L∞ filtering for stochastic systems driven by Poisson processes and Wiener processes

This paper investigates the L2–L∞ filtering problem for stochastic systems driven by Poisson processes and Wiener processes. Firstly, this paper presents an approach to transform the expectation of stochastic integral with respect to Poisson process into the expectation of Lebesgue integral by the martingale theory. Then, based on this, a filter is designed to guarantee that the filtering error system is mean-square asymptotically stable and its L2–L∞ performance satisfies a prescribed level. Finally, a simulation example is given to illustrate the effectiveness of the proposed filtering scheme.

A Nice Example of Lebesgue Integration

We explore the properties of an interesting new example of a function which is Lebesgue integrable but not Riemann integrable.

Some results concerning the summability of spliced sequences

Abstract: A spliced sequence is formed by combining all of the terms of two or more convergent sequences, in their original order, into a new spliced sequence. In this paper replacing convergent sequences by bounded sequences, we study the summability of spliced sequences and give some inequalities that provide us with approximation of the core of transformation of these sequences by a summability matrix. We also present some further results via the Lebesgue integral.

General Lebesgue integral inequalities of Jensen and Ostrowski type for differentiable functions whose derivatives in absolute value are h-convex and applications

Some inequalities related to Jensen and Ostrowski inequalities for general Lebesgue integral of differentiable functions whose derivatives in absolute value are h-convex are obtained. Applications for f-divergence measure are provided as well.

On nonlinear integral equations in the space $${{\rm \Phi}BV (I)}$$ Φ B V ( I )

We present the existence and uniqueness of global and local \({{\rm \Phi}}\)-bounded variation (\({{\rm \Phi}BV}\)) solutions as well as continuous \({{\rm \Phi}BV}\)-solutions of nonlinear Hammerstein and Volterra–Hammerstein integral equations formulated in terms of the Lebesgue integral.

An Effect-Theoretic Account of Lebesgue Integration

Effect algebras have been introduced in the 1990s in the study of the foundations of quantum mechanics, as part of a quantum-theoretic version of probability theory. This paper is part of that programme and gives a systematic account of Lebesgue integration for [0,1]-valued functions in terms of effect algebras and effect modules. The starting point is the 'indicator' function for a measurable subset. It gives a homomorphism from the effect algebra of measurable subsets to the effect module of [0,1]-valued measurable functions which preserves countable joins.It is shown that the indicator is free among these maps: any such homomorphism from the effect algebra of measurable subsets can be thought of as a generalised probability measure and can be extended uniquely to a homomorphism from the effect module of [0,1]-valued measurable functions which preserves joins of countable chains. The extension is the Lebesgue integral associated to this probability measure. The preservation of joins by it is the monotone convergence theorem.

Measurable Soft Sets

The soft set is a mapping from a parameter to the crisp subset of universe. Molodtsov introduced the concept of soft sets as a generalized tool for modeling complex systems involving uncertain or not clearly defined objects. In this paper the concept of measurable soft sets are introduced and their properties are discussed. The open problem of this paper is to develop measurable functions and lebesgue integral in soft set theory context.