Infinite Interval Research Articles

Countable linear orders with disjoint infinite intervals are mutually orthogonal

Two linear orderings of a same set are perpendicular if the only self-mappings of this set that preserve them both are the identity and the constant mappings. Two linear orderings are orthogonal if they are isomorphic to two perpendicular linear orderings. We show that two countable linear orderings are orthogonal as soon as each one has two disjoint infinite intervals. From this and previously known results it follows in particular that each countably infinite linear ordering is orthogonal to itself.

Double phase problems with variable growth

We consider a class of double phase variational integrals driven by nonhomogeneous potentials. We study the associated Euler equation and we highlight the existence of two different Rayleigh quotients. One of them is in relationship with the existence of an infinite interval of eigenvalues while the second one is associated with the nonexistence of eigenvalues. The notion of eigenvalue is understood in the sense of pairs of nonlinear operators, as introduced by Fučík, Nečas, Souček, and Souček. The analysis developed in this paper extends the abstract framework corresponding to some standard cases associated to the p(x)-Laplace operator, the generalized mean curvature operator, or the capillarity differential operator with variable exponent. The results contained in this paper complement the pioneering contributions of Marcellini, Mingione et al. in the field of variational integrals with unbalanced growth.

A Lyapunov and Sacker–Sell spectral stability theory for one-step methods

Approximation theory for Lyapunov and Sacker–Sell spectra based upon QR techniques is used to analyze the stability of a one-step method solving a time-dependent (nonautonomous) linear ordinary differential equation (ODE) initial value problem in terms of the local error. Integral separation is used to characterize the conditioning of stability spectra calculations. The stability of the numerical solution by a one-step method of a nonautonomous linear ODE using real-valued, scalar, nonautonomous linear test equations is justified. This analysis is used to approximate exponential growth/decay rates on finite and infinite time intervals and establish global error bounds for one-step methods approximating uniformly, exponentially stable trajectories of nonautonomous and nonlinear ODEs. A time-dependent stiffness indicator and a one-step method that switches between explicit and implicit Runge–Kutta methods based upon time-dependent stiffness are developed based upon the theoretical results.

Fuzzy Laplace transform based on the Henstock integral and its applications in discontinuous fuzzy systems

The existing results on the fuzzy Laplace transform and their applications were based on Zaheh's decomposition theorem and were formally characterized by the integrals of real-valued functions directly. That is, the existence of the fuzzy Laplace transform in essence has not been solved. In this article, the fuzzy Laplace transform is incorporated into the framework of the Henstock integral and proposed by use of fuzzy Henstock integrals on infinite intervals. In addition, as a theoretical basis, the existence and the basic properties of the fuzzy Laplace transform are investigated, the convolution of fuzzy-valued functions and real-valued functions is defined, and the convolution theorem of the fuzzy Laplace transform is given. Finally, discontinuous fuzzy initial value problems and two kinds of fuzzy Volterra integral equations are discussed with use of the fuzzy Laplace transform presented in this article.

Cradle-to-gate CO2e emissions vs. in situ CO2 sequestration of structural concrete elements

Life cycle assessment (LCA) is an emerging methodological tool used in building design and construction to quantify the environmental impacts of materials, components, and whole buildings. While calculation of embodied cradle-to-gate greenhouse gas (GHG) emissions is commonplace for ordinary portland cement (OPC) structural concrete elements in building-related LCAs, the carbon dioxide (CO2) sequestered by in situ carbonation of exposed OPC concrete is often neglected because the quantity of sequesterable CO2 is assumed trivial compared to the initial GHG emissions (kg CO2e) associated with their manufacture. Using a screening cradle-to-gate LCA and a previously developed and validated CO2 sequestration model for OPC concrete, this paper quantifies and compares estimates of the initial CO2e emissions to the CO2 sequestration potential of several OPC concrete elements at both finite (25 years) and infinite time intervals. The results demonstrate that, depending on cement type, compressive strength, structural geometry, and time, approximately 19%—a non-trivial sum—of initial CO2e emissions could be recoverable via CO2 sequestration for the concrete elements considered herein. Notably, however, concrete elements that sequester the most CO2 do not always result in the lowest net CO2e emissions.

Construction of simultaneous confidence bands for a percentile hyper-plane with predictor variables constrained in an ellipsoidal region

This paper provides the method of constructing the simultaneous confidence band for a percentile hyper-plane with predictor variables constrained in an ellipsoidal region. The band is compared with that on a rectangular region. It is shown that the band on the ellipsoidal region has a big advantage. The proposed method allows to construct the Type I band and the Type II band, and also allows to constrain the predictor variables either in finite intervals or in infinite intervals. In order to evaluate the efficiency of the band, the average width criterion is introduced, under which the Type I band and the Type II band are compared. All stated above are illustrated by a real example of a leaching rate study.

On the inversion of some integral transforms by the use of the generalized oblate spheroidal wave functions

ABSTRACTIn this paper, we introduce a new set of functions, which have the property of the completeness over a finite and infinite intervals. This family of functions, denoted for simplicity GOSWFs, are a generalization of the oblate spheroidal wave functions. They generalize also the Jacobi polynomials in some sense. The GOSWFs are nothing but the eigenfunctions of the finite weighted bilateral Laplace transform We compute this functions by the use the Gaussian quadrature method. As an application, we use the GOSWFs to invert the finite bilateral Laplace transform as well as the finite Laplace transform and the Mellin transform. Finally, we provide the reader by some numerical examples that illustrate the theoretical results. We reconstruct signals without noise and others with noise. We have used three kinds of noisy input signals: Sinusoidal noise given by Here and are two real parameters. The second type of noise is a functional noise given by: The third kind of the signal noise is the random noise which is normally distributed with mean 0 and standard deviation

An efficient performance evaluation model for the resource clusters in cloud environment using continuous time Markov chain and Poisson process

Cloud computing enables us to share the jobs across various cloud data centers with the internet as a backbone in order to process the jobs. The selections of the data centers to process the jobs are based on the potential of the data centers, arrival rate of the jobs and better resource utilization. Performance modeling and evaluation of the cloud systems shows a perfect picture of probability of distribution of jobs among the datacenters, mean number of jobs in the datacenters and makespan of the cloud environment. In order to provide a best performance evaluation model which leads to a prosperous cloud environment the proposed approach calculates the arrival rate of jobs using Poisson process in which the arrival rate can be calculated for infinite time intervals and the mean number of jobs and also the resource utilization are modeled using continuous Markov chain. The problems related to the calculation of arrival rate and resource utilization model of the existing approaches are solved to enhance the job scheduling by minimizing the makespan of the cloud environment.

The growth of entire Dirichlet series in terms of generalized orders

Let be a continuous function which increases to on an infinite interval of the form . A necessary and sufficient condition is found on a sequence increasing to which ensures that for each Dirichlet series of the form , , which is absolutely convergent in the following relation holds: where and are the maximum modulus and maximum term of the series, respectively. Bibliography: 10 titles.

On one problem of viscoelastic fluid dynamics with memory on an infinite time interval

In the present paper we establish the existence of weak solutions of one boundary value problem for one model of a viscoelastic fluid with memory along the trajectories of the velocity field on an infinite time interval. We use solvability of related approximating initial-boundary value problems on finite time intervals and responding pass to the limit.

Global Solvability and Estimates of Solutions to the Cauchy Problem for the Retarded Functional Differential Equations That Are Used to Model Living Systems

We study the Cauchy problem for the retarded functional differential equations that model the dynamics of some living systems. We find certain conditions ensuring the existence, uniqueness, and nonnegativity of solutions on finite and infinite time intervals. We obtain upper bounds for solutions and prove the continuous dependence of solutions on the initial data on finite time intervals.

Interrelationship of Linear Nonstationary Evasion Problems with Many Evaders

Abstract The evasion problem is considered in which group of pursuers and group of evaders participate under the condition that the pursuers include participants whose capabilities do not yield place to the capabilities of evaders and participants with lesser capabilities. The goal of the group of pursuers is to “catch” all the evaders. The goal of the group of evaders is to prevent this from happening, i.e., to make it possible for at least one of the evaders to avoid an encounter. The pursuers and the evaders employ piecewise-program strategies. It is shown that if in a differential game avoidance of an encounter by at least one evader takes place over an infinite time interval, then upon the addition of “weak” pursuers avoidance will take place on any finite time interval.

Determining an optimal value for the convergence control parameter in the HAM

In the applications, the homotopy analysis method (HAM) is an often used method to determine an analytical approximate solution of lower-dimensional nonlinear ordinary differential equations. This approximation consists of an infinite series which depends on an auxiliary real parameter h. This parameter must be adjusted such that the series converges towards the exact solution of the given problem. In this paper we propose a computational approach, which is based on the residual of the truncated series, to determine an optimal value hopt or an optimal region for h. Using the numerical computing environment MATLAB, we describe several possibilities how this approach can be realised. Finally, by means of three examples (an IVP, a two-point BVP, as well as a BVP on an infinite interval) we show how this mathematically sophisticated strategy can be applied and we present the optimal parameter hopt for each example.

The unique positive solution for fractional integro-differential equations on infinite intervals

The unique positive solution for fractional integro-differential equations on infinite intervals

Recovering Linear Operators and Lagrange Function Minimality Condition

This article concerns the recovery of the operators by noisy information in the case that their norms are defined by integrals over infinite intervals. We study the conditions under which the dual extremal problem (often nonconvex) can be solved using the Lagrange function minimality condition.

Positive solutions for nonlinear $\phi$-Laplacian third-order impulsive differential equations on infinite intervals

Positive solutions for nonlinear $\phi$-Laplacian third-order impulsive differential equations on infinite intervals

Homogeneous Lyapunov Functions: From Converse Design to Numerical Implementation

The problem of the synthesis of a homogeneous Lyapunov function for an asymptotically stable homogeneous system is studied. First, for systems with nonnegative degree of homogeneity, several expressions of homogeneous Lyapunov functions are derived, which depend explicitly on the supremum or the integral (over finite or infinite intervals of time) of the system solutions. Second, a numeric procedure is proposed, which ensures the construction of a homogeneous Lyapunov function. The analytical results are illustrated by simulations.

Mechanical response and simulation for constitutive equations with distributed order derivatives

Mechanical response and simulation for constitutive equation with distributed order derivatives were considered. We investigated the creep compliance, creep recovery, relaxation modulus, stress–strain behavior under harmonic deformation for each case of two constitutive equations. We express these responses and results as easily computable forms and simulate them by using MATHEMATICA 8. The results involve the exponential integral function, convergent improper integrals on the infinite interval [Formula: see text] and the numerical integral method for the convolution integral. For both equations, stress responses to harmonic deformation display hysteresis phenomena and energy dissipation. The two constitutive equations characterize viscoelastic models of fluid-like and solid-like, respectively.

Numerical integration of oscillatory Airy integrals with singularities on an infinite interval

This work is devoted to the quadrature rules and asymptotic expansions for two classes of highly oscillatory Airy integrals on an infinite interval. We first derive two important asymptotic expansions in inverse powers of the frequency ω . Then, based on structure characteristics of the two asymptotic expansions in inverse powers of the frequency ω , both the so-called Filon-type method and the more efficient Clenshaw–Curtis–Filon-type method are introduced and analyzed. The required moments in the former can be explicitly expressed by the Meijer G-functions. The latter can be implemented in O ( N log N ) operations, based on fast Fourier transform (FFT) and fast computation of the modified moments. Here, we can construct two useful recurrence relations for computing the required modified moments accurately, with the help of the Airy’s equation and some properties of the Chebyshev polynomials. Particularly, we also provide their error analyses in inverse powers of the frequency ω . Furthermore, the presented error analysis shows the advantageous property that the accuracy improves greatly as ω increases. Numerical examples are provided to illustrate the efficiency and accuracy of the proposed methods.

Existence and Ulam’s Stability for Conformable Fractional Differential Equations with Constant Coefficients

In this article, we develop a standard idea in seeking the solution of linear ODEs to derive the representation of solutions to conformable fractional linear differential equations with constant coefficients by adopting the variation of constants method. In addition, we present the existence of solutions to conformable fractional nonlinear differential equations with constant coefficients under mild conditions on the nonlinear term. Also, we transfer the concepts of Ulam’s stability for ODEs to this type of equation and give the Ulam–Hyers and Ulam–Hyers–Rassias stability results on finite time and infinite time intervals.