Infinite Interval Research Articles

On a Multipoint Boundary Value Problem for a Fractional Order Differential Inclusion on an Infinite Interval

We investigate the existence of solutions for the following multipoint boundary value problem of a fractional order differential inclusionD0+αut+Ft,ut,u′t∋0,0<t<+∞,u0=u′0=0,Dα-1u+∞-∑i=1m-2‍βiuξi=0, whereD0+αis the standard Riemann-Liouville fractional derivative,2<α<3,0<ξ1<ξ2<⋯<ξm-2<+∞, satisfies0<∑i=1m-2‍βiξiα-1<Γ(α), and F:[0,+∞)×ℝ×ℝ→𝒫(ℝ)is a set-valued map. Several results are obtained by using suitable fixed point theorems when the right hand side has convex or nonconvex values.

The Rational Third-Kind Chebyshev Pseudospectral Method for the Solution of the Thomas-Fermi Equation over Infinite Interval

We propose a pseudospectral method for solving the Thomas-Fermi equation which is a nonlinear ordinary differential equation on semi-infinite interval. This approach is based on the rational third-kind Chebyshev pseudospectral method that is indeed a combination of Tau and collocation methods. This method reduces the solution of this problem to the solution of a system of algebraic equations. Comparison with some numerical solutions shows that the present solution is highly accurate.

L^p solutions of infinite time interval BSDEs and the corresponding g-expectations and g-martingales

L^p solutions of infinite time interval BSDEs and the corresponding g-expectations and g-martingales

Exclusion of dominated species in open replicator systems

An open replicator model is considered. A method is proposed to extend the results obtained for closed replicator systems to open replicator systems. Conditions are obtained for the extinction of species on an infinite time interval depending on the replicator matrix.

A representation theorem for generators of BSDEs with finite or infinite time intervals and linear-growth generators

Under the most elementary conditions on stochastic differential equations and some milder conditions on backward stochastic differential equations with finite or infinite time intervals and linear-growth generators, a representation theorem of generators and a converse comparison theorem of solutions are established in this paper.

Towards irreversibility with a finite bath of oscillators

We investigate the routes by which a bath composed of a finite number of oscillators at zero temperature approaches the induction of dissipation when it nears the usual limit of dense spectrum spread in an infinite interval. It is shown that, when this limit is taken, different distributions of environment frequencies can lead to the same irreversible evolution. However, when we move away from it, the dynamics departs from irreversibility in qualitatively different manners.

Existence results for nonlinear boundary value problems with integral boundary conditions on an infinite interval

In this paper, by using fixed point theorems in a cone, the existence of one positive solution and three positive solutions for nonlinear boundary value problems with integral boundary conditions on an infinite interval are established.MSC:34B10, 39A10, 34B18, 45G10.

On a class of sixth order viscous Cahn-Hilliard type equations

An initial-boundary-value problem for a class of sixth order viscous Cahn-Hilliard type equations with a nonlinear diffusion is considered. The study is motivated by phase-field modelling of various spatial structures, for example arising in oil-water-surfactant mixtures and in modelling of crystal growth on atomic length, known as phase field crystal model. For such problem we prove the existence and uniqueness of a global in time regular solution. First the finite-time existence is proved by means of the Leray-Schauder fixed point theorem. Then, due to suitable estimates, the finite-time solution is extended step by step on the infinite time interval.

Hypernumbers and extrafunctions: extending the classical calculus

-1. Introduction: How mathematicians solve unsolvable problems.-2. Hypernumbers(Definitions and typology,Algebraic properties,Topological properties).-3. Extrafunctions(Definitions and typology, Algebraic properties, Topological properties).-4. How to differentiate any real function (Approximations, Hyperdifferentiation).-5. How to integrate any continuous real function (Partitions and covers, Hyperintegration over finite intervals, Hyperintegration over infinite intervals). -6. Conclusion: New opportunities.- Appendix.- References.

Multicenter Experience on Eversion Versus Conventional Carotid Endarterectomy in Symptomatic Carotid Artery Stenosis: Observations from the Stent-Protected Angioplasty versus Carotid Endarterectomy (SPACE-1) Trial

Conclusion: Patients with cervical carotid artery stenosis undergoing conventional carotid endarterectomy (C-CEA) have better periprocedural outcomes than those undergoing eversion CEA (E-CEA). However, E-CEA may be more efficacious for long-term prevention of ipsilateral stroke. Summary: There are two basic techniques for CEA: C-CEA is performed through a longitudinal arteriotomy and is generally closed with a patch, and E-CEA is performed with an oblique transection of the internal carotid artery off the common carotid artery with reimplantation of the endarterectomized internal carotid artery onto the common carotid artery. A Cochrane review in 2009 concluded that there was no evidence that one technique was superior to the other with respect to periprocedural stroke, death, or restenosis (Cao P et al, Cochrane Database Syst Rev 2002;1:CD001921). In this report, the authors present a post hoc retrospective analysis of the surgical arm of the Stent Protected Angioplasty versus Carotid Endarterectomy trial (SPACE-1). All patients in the SPACE-1 trial were symptomatic, and choice of CEA technique in the surgical arm was based on surgeon preference. There were 601 eligible patients randomized into the surgical arm of the SPACE-1 trial. After excluding patients who withdrew consent before treatment, who did not undergo per-protocol treatment, who experienced primary outcome before treatment, or who had CEA without patch closure or prosthetic bypass graft interposition, there were 516 patients suitable for analysis. Of these, 310 (60.1%) underwent C-CEA and 206 (39.9%) underwent E-CEA. The primary end point was ipsilateral stroke or death ≤30 days after surgery. Secondary end points included perioperative adverse events and the 2-year risk of restenosis, stroke, and death. There were no restrictions with respect to type of anesthesia, neurologic monitoring, heparin dose, type of patch to close the arteriotomy, or indications for shunting. The C-CEA and E-CEA groups were similar in demographics and baseline clinical variables. Shunt frequency was higher in the C-CEA group (65% vs.17%; P < .0001). Ipsilateral stroke or death ≤30 days were significantly greater with E-CEA than C-CEA (9% vs 3%; P = .005). There were no statistically significant differences in perioperative secondary outcome events between the two groups with the exception of a higher risk of intraoperative ipsilateral stroke in the E-CEA group (4% vs 0.3%; P = .0035). The 2-year risk of ipsilateral stroke after 30 days was higher in the C-CEA group than in the E-CEA group (2.9% vs 0%; P = .017). Including perioperative events at 2 years, there were no differences between groups in ipsilateral stroke, any stroke, disabling stroke, death, or any stroke or death. Comment: The study is interesting in that it suggests that the choice of CEA technique may influence periprocedural events but not long-term ipsilateral stroke or overall death. The limitations of this study are obvious: it used a nonrandomized post hoc analysis, with no information about why one CEA technique was chosen over the other, and lacked a blinded outcome assessment. The authors also note that an infinite hazard ratio confidence interval for ipsilateral stroke >30 days and the wide odds ratio confidence intervals for death rates indicate that, statistically, a substantial technique-dependent effect has not truly been ruled out by this study. Nevertheless, this was an independently monitored, multicenter study and therefore may have more generally applicable and accurate data than a single-center study.

Discrete Sturm comparison theorems on finite and infinite intervals

The Sturm comparison theorem for second-order Sturm–Liouville difference equations on infinite intervals is established and discussed.

The Colonial Origins of Comparative Development: An Empirical Investigation: Comment

Acemoglu, Johnson, and Robinson's (2001) seminal article argues property-rights institutions powerfully affect national income, using estimated mortality rates of early European settlers to instrument capital expropriation risk. However, 36 of the 64 countries in the sample are assigned mortality rates from other countries, often based on mistaken or conflicting evidence. Also, incomparable mortality rates from populations of laborers, bishops, and soldiers—often on campaign—are combined in a manner that favors the hypothesis. When these data issues are controlled for, the relationship between mortality and expropriation risk lacks robustness, and instrumental-variable estimates become unreliable, often with infinite confidence intervals. (JEL D02, E23, F54, I12, N40, O43, P14)

Uniform Asymptotics for the Finite-Time Ruin Probability of a Time-Dependent Risk Model with Pairwise Quasiasymptotically Independent Claims

We consider a generalized time-dependent risk model with constant interest force, where the claim sizes are of pairwise quasiasymptotical independence structure, and the claim size and its interclaim time satisfy a dependence structure defined by a conditional tail probability of the claim size given the interclaim time before the claim occurs. As the claim-size distribution belongs to the dominated variation class, we establish some weakly asymptotic formulae for the tail probability of discounted aggregate claims and the finite-time ruin probability, which hold uniformly for all times in a relevant infinite interval.

Infinite time interval RBSDEs with non-Lipschitz coefficients

This paper aims at solving infinite horizon reflected backward stochastic differential equations (RBSDEs) under weak assumptions. We obtain a more general existence and uniqueness theorem about the solutions to RBSDEs with non-Lipschitz generators.

Some new Gronwall-Bellman type nonlinear dynamic inequalities containing integration on infinite intervals on time scales

Abstract In this paper, some new Gronwall-Bellman-type nonlinear dynamic inequalities containing integration on infinite intervals on time scales are established, which provides new bounds on unknown functions and can be used as a handy tool in the qualitative analysis of solutions of certain dynamic equations on time scales. MSC:26E70, 26D15, 26D10.

Global solvability of a problem governing the motion of two incompressible capillary fluids in a container

We deal with the motion of two incompressible fluids in a container one of which is inside another, taking into account the surface tension. We prove that this problem is uniquely solvable in an infinite time interval provided that the initial velocity of the fluids is small and the initial configuration of the inner fluid is close to a ball. Moreover, we show that the velocity decays exponentially at infinity with respect to time and that the interface between the fluids tends to a sphere of a certain radius. The proof is based on an exponential estimate of a generalized energy and on a local existence theorem for the problem in anisotropic Holder spaces. We follow the scheme developed by one of the authors for proving the global solvability of a problem governing the motion of one incompressible capillary fluid bounded by a free surface. Bibliography: 18 titles.

A New Extension Method in Diagnosis of Transformer Overheating Fault

For the transformer overheating fault diagnosis based on extenics, this paper proposes the extension diagnosis method that uses the matter element having all the characteristics and piecewise linearization correlation function. The method ensures the consistency of each overheating fault characteristics and overcomes the limitation of elementary correlation function in the infinite interval. The simulation result with 20 groups of overheating fault of which type is known shows that the proposed method overcomes the improved three-ratio method’s defects that multiple overheating fault can’t be diagnosed and the diagnosis precision isn’t high. Better diagnosis effect is obtained.

A generalization of the Wiener rational basis functions on infinite intervals, Part II — Numerical investigation

In Part I we introduced the generalized Wiener rational basis functions, and here in Part II we continue our investigation with numerical experiments. Wiener’s generalized basis can utilize the fast Fourier transform for integer values of the decay parameter s; we outline two algorithms for doing so. In addition, the issue of Galerkin representations for polynomial nonlinearities of expansions is addressed.The Wiener basis set is compared against domain truncation methods (Fourier and Chebyshev polynomials), Hermite functions, Sinc interpolations, and mapped Chebyshev expansions, and we show that for both exponentially and algebraically decaying functions, the Wiener approximation is as good as or superior to these alternatives. In addition, we carry out preliminary investigations regarding tuning of the decay parameter s. Numerical simulations of Korteweg–de Vries type equations show the effectiveness of the Wiener expansion. We also explore the practical use of the Wiener basis functions on the semi-infinite interval, which is compared against Laguerre function methods and other Jacobi polynomial mappings.

Study of a special model of resource allocation over an infinite interval of time

A special model of resource allocation over an infinite interval of time is studied. Using the Pontryagin maximum principle, an extreme solution is constructed whose optimality is proven with the help of a theorem on sufficient conditions, in the form of constructions of Pontryagin’s maximum principle. A concrete example in which the classical maximum principle is inapplicable is considered.

H 2-stabilization of the Isothermal Euler equations: a Lyapunov function approach

The authors consider the problem of boundary feedback stabilization of the 1D Euler gas dynamics locally around stationary states and prove the exponential stability with respect to the H 2-norm. To this end, an explicit Lyapunov function as a weighted and squared H 2-norm of a small perturbation of the stationary solution is constructed. The authors show that by a suitable choice of the boundary feedback conditions, the H 2-exponential stability of the stationary solution follows. Due to this fact, the system is stabilized over an infinite time interval. Furthermore, exponential estimates for the C 1-norm are derived.