Finiteness Conditions Research Articles

Scrutinization of finite time stability of fractional impulsive neutral model with disturbance

Finite time stability practically examines the trajectories of a system which converge to equilibrium state in a short period of time. This notion requires predefined bounds on system parameters and bounded time interval. Considering the idea that many practical system often operates over time interval being finite rather than infinite, we explore the finite time stability concept of damped fractional system with neutral conditions and impulsive effects. The desired bounds for the stability of the system is derived by implementing Gronwall’s inequality conditions. Further, the finite time stability conditions of the proposed fractional linear model is extended to nonlinearity term with disturbance. Finally, numerical simulations are given to show the effectiveness of the derived results.

Exploiting neighborhood interference with low-order interactions under unit randomized design

Abstract Network interference, where the outcome of an individual is affected by the treatment assignment of those in their social network, is pervasive in real-world settings. However, it poses a challenge to estimating causal effects. We consider the task of estimating the total treatment effect (TTE), or the difference between the average outcomes of the population when everyone is treated versus when no one is, under network interference. Under a Bernoulli randomized design, we provide an unbiased estimator for the TTE when network interference effects are constrained to low-order interactions among neighbors of an individual. We make no assumptions on the graph other than bounded degree, allowing for well-connected networks that may not be easily clustered. We derive a bound on the variance of our estimator and show in simulated experiments that it performs well compared with standard estimators for the TTE. We also derive a minimax lower bound on the mean squared error of our estimator, which suggests that the difficulty of estimation can be characterized by the degree of interactions in the potential outcomes model. We also prove that our estimator is asymptotically normal under boundedness conditions on the network degree and potential outcomes model. Central to our contribution is a new framework for balancing model flexibility and statistical complexity as captured by this low-order interactions structure.

Multi-attribute group decision-making based on q-rung orthopair fuzzy Aczel–Alsina power aggregation operators

This article aims to develop the Aczel–Alsina operational laws based on power aggregation operators (PAOs) for the q-Rung orthopair fuzzy set (FS) (q-ROFS) framework. Changing the parameter can alter these sets according to the degree of fluctuation, providing a range of options. The decision-making sciences use “q-ROFS” to describe the family of the most distinct fuzzy information thoughts. In response to the degree of variation, the q-ROFSs can increasingly adjust the information region by fluctuating the restriction q≥1, making numerous options. But on the other hand, PAOs have the advantage of vanishing the influence of awkward data from the final results. To take advantages of POAs, based on Aczel- Alsina (AA) operational laws, q-Rung orthopair fuzzy (q-ROF) AA power-weighted averaging (q-ROFAAPWA) and q-ROF AA power-weighted geometric (q-ROFAAPWG) operators are stated. Also, it studied that the proposed AOs fulfilled the conditions of boundedness, monotonicity, and idempotency. Based on these newly constructed AOs, a technique for dealing the multi-attribute group decision-making (MAGDM) problems is suggested. Numerous research, correlations with another modern approach and a numerical example of selecting stock market companies have been used to show the suggested system’s applicability. The sensitivity analysis of the developed method is examined. A comparative study with other prevailing methods is also providing for superiority analysis. Finally, we demonstrated that in the results and discussion section, the proposed AOs in the q-ROPFS framework are more reliable in aggregating information than intuitionistic fuzzy (IF) set (IFS) and Pythagorean FS (PyFS) frameworks.

Exponential stability and boundedness of nonlinear perturbed systems by unbounded perturbation terms

We study the exponential stability and boundedness problem for perturbed nonlinear time-varying systems using Lyapunov Functions with indefinite derivatives. As the limiting function for the perturbation term, we use different forms and give stability and boundedness conditions in terms of the coefficients in these bounds. Contrary to most of the available conditions, we allow the coefficients to be unbounded. But instead, we put forward a condition that requires a series produced by coefficients to be limited and exponentially decaying. We perform our results on Linear time-varying systems and generalize many of the available results.

Convergence of solutions to nonlinear nonconvex optimal control problems

In this paper, we consider nonlinear nonconvex optimal control problems and study convergence conditions of their solutions. To be more precise, we first introduce a generalized boundedness condition and discuss its relations with some typical existing ones in the literature. Next, combining this condition with the Gronwall Lemma, we investigate the boundedness property of solutions to state equations and the compactness of feasible sets of the reference problems. Then, based on these obtained results, convergence conditions in the sense of Painlevé-Kuratowski for such problems are formulated. Finally, at the end of the paper, applications to two practical situations, problems of fuel-optimal frictionless horizontal motion of a mass point and glucose models, are also presented.

Event-triggered finite-time [formula omitted] control for positive Markov jump systems with partly known transition probability

The article studies the event-triggered finite-time L1 control design for positive Markov jump systems(PMJSs) with partly known transition probability. Firstly, with regard to the unique positive characteristic, a novel 1-norm based event-triggering condition is proposed for PMJSs. Then, sufficient conditions of positivity and finite-time L1 boundedness are provided for the closed-loop system under the constructed even-triggered control law. Based on this, by applying the matrix decomposition technology to the event-triggered controller gain matrix, an effective mode-dependent event-triggered finite-time L1 control scheme in the solvable linear programming(LP) form is presented for PMJSs while preserving the positivity and finite-time L1 boundedness in closed-loop. Further, an effective LP-based event-triggered finite-time L1 control scheme is also obtained for PMJSs with completely known transition probability and completely unknown transition probability, respectively. Finally, two examples are presented to verify the validity of the obtained results.

Peridynamic analysis of curved elastic beams

Peridynamics is an extended continuum theory which operates with non-local deformation measures as well as long range internal force/moment interactions. The aim of this paper is to present and to analyze peridynamic governing equations for elastic curved beams. To this end the strain energy density is formulated as a function of two non-local deformation measures including the axial deformation and curvature. Applying the Lagrangian formalism, peridynamic equations of motion are derived. To analyze curved beams of finite length peridynamic boundary conditions are required. Examples of boundary conditions including simple support and clamped edge are presented by introducing a fictitious domain. Solutions to PD equations of motion are presented for various problems including static analysis of simply supported and clamped arc beams as well as modal analysis of a slender ring and a simply supported arc beam. To validate peridynamic formulations, solutions for displacements, natural frequencies and vibration modes are compared with those by the classical beam theory. A very good agreement between the non-local and the classical theories is observed for the case of the small horizon sizes which shows the capability of the derived equations of motion and proposed boundary conditions.

Analysis of nonlinear compartmental model using a reliable method

The goal of this work is to investigate nonlinear models and their complexity using techniques that are universal and have connections to historical and material aspects. Using the premise of a constant population that is uniformly mixed, a nonlinear compartmental model that depicts the movement between voter classes is taken into consideration. In the current work, we investigate the dynamical framework that supports the interactions between the three parties. It is discussed how rate change affects various metrics. The conditions for boundedness, stability, existence, and other dynamics are obtained. We derive the effects of generalizing the model in any order. The current study supports investigations into complex real-world issues and forecasts of necessary plans.

Role of Flow Inertia in Aggregate Restructuring and Breakage at Finite Reynolds Numbers.

Forces acting on aggregates depend on their properties, such as size and structure. Breakage rate, stable size, and structure of fractal aggregates in multiphase flows are strongly related to the imposed hydrodynamic forces. While these forces are prevalently viscous for finite Reynolds number conditions, flow inertia cannot be ignored, thereby requiring one to fully resolve the Navier-Stokes equations. To highlight the effect of flow inertia on aggregate evolution, numerical investigation of aggregate evolution in simple shear flow at the finite Reynolds number is conducted. The evolution of aggregates exposed to shear flow is tracked over time. Particle coupling with the flow is resolved with an immersed boundary method, and flow dynamics are solved using a lattice Boltzmann method. Particle dynamics are tracked by a discrete element method, accounting for interactions between primary particles composing the aggregates. Over the range of aggregate-scale Reynolds numbers tested, the breakage rate appears to be governed by the combined effect of momentum diffusion and the ratio of particle interaction forces to the hydrodynamic forces. For higher shear stresses, even when no stable size exists, breakage is not instantaneous because of momentum diffusion kinetics. Simulations with particle interaction forces scaled with the viscous drag, to isolate the effect of finite Reynolds hydrodynamics on aggregate evolution, show that flow inertia at such moderate aggregate Reynolds numbers has no impact on the morphology of nonbreaking aggregates but significantly favors breakage probability. This is a first-of-its-kind study that establishes the role of flow inertia in aggregate evolution. The findings present a novel perspective into breakage kinetics for systems in low but finite Reynolds number conditions.

Equivariant algebraic K-theory, G-theory and derived completions

In the mid 1980s, while working on establishing completion theorems for equivariant Algebraic K-Theory similar to the well-known Atiyah-Segal completion theorem for equivariant topological K-theory, the late Robert Thomason found the strong finiteness conditions that are required in such theorems to be too restrictive. Then he made a conjecture on the existence of a completion theorem in the sense of Atiyah and Segal for equivariant algebraic G-theory, for actions of linear algebraic groups on schemes that holds without any of the strong finiteness conditions that are required in such theorems proven by him, and also appearing in the original Atiyah-Segal theorem. The main goal of the present paper is to provide a proof of this conjecture in as broad a context as possible, making use of the technique of derived completion, and to consider several of the applications.Our solution is broad enough to allow actions by all linear algebraic groups, irrespective of whether they are connected or not, and acting on any quasi-projective scheme of finite type over a field, irrespective of whether they are regular or projective. This allows us therefore to consider the equivariant algebraic G-Theory of large classes of varieties like all toric varieties (for the action of a torus) and all spherical varieties (for the action of a reductive group). Restricting to actions by split tori, we are also able to consider actions on algebraic spaces. Moreover, the restriction that the base scheme be a field is also not required often, but is put in mainly to simplify some of our exposition. These enable us to obtain a wide range of applications, some of which are briefly sketched and which we plan to explore in detail in the future. In fact, we discuss an extension of our results to equivariant homotopy K-theory along with various Riemann-Roch theorems in a sequel. A comparison of our results with previously known results, none of which made use of derived completions, shows that without the use of derived completions one can only obtain results which are indeed very restrictive.

Theoretical solutions for 2D mode-I crack-tip stress fields in power-law plastic materials based on the stress factor derived from the developed median-energy–density equivalence method

This study presents the energy density equivalence equation, demonstrating that the strain energy density of an arbitrary representative volume element (RVE) under a complex stress state equals to that under a uniaxial stress state when volume change is neglected. Based on the developed median-energy–density equivalence method, the analytical solution for equivalent stress of the RVE at the energy density equivalence point in power-law plastic materials is derived in a new method and the concept of stress factor is proposed. Additionally, novel special functions are introduced to accurately describe the angular distributions of stresses in this study. By combining the stress factor, integral theoretical solutions for mode-I crack tip stress fields in power-law plastic materials under finite plane conditions are proposed. Finally, parameterized studies are conducted on bending and tension specimens under plane-strain and plane-stress conditions. The results of stress distributions and contour lines predicted by the integral theoretical solutions are compared with the finite element analysis and HRR solutions for five mode-I plane specimens, providing verification of the solutions’ applicability.

A posteriori error approximation in discontinuous Galerkin method on polygonal meshes in elliptic problems

The paper presents a posteriori error approximation concept based on residuals in the two-dimensional discontinuous Galerkin (DG) method. The approach is relatively simple and effective in application, and it takes advantage of some unique properties of the DG method. The error function is constructed in an enriched approximation space, utilizing the hierarchical nature of the basis functions. Among many versions of the DG method, the most popular one is based on the interior penalty approach. However, in this paper a DG method with finite difference (DGFD) is utilized, where the continuity of the approximate solution is enforced by finite difference conditions applied on the mesh skeleton. In the DG methods arbitrarily shaped finite elements can be used, so in this paper the meshes with polygonal finite elements are considered, including quadrilateral and triangular elements. Some benchmark examples are presented, in which Poisson’s and linear elasticity problems are considered. The examples use various mesh densities and approximation orders to evaluate the errors. The error estimation maps, generated for the discussed tests, indicate a good correlation with the exact errors. In the last example, the error approximation concept is applied for an adaptive hp mesh refinement.

Evaluating the importance of Raman and Brillouin side scattering at ignition conditions

A complete linear model for Raman and Brillouin side scattering is analyzed with infinite and finite boundary conditions. Analytic formulas of absolute thresholds and convective gains for side scatterings are derived when infinite boundary is assumed. Finite beam width introduces finite boundary conditions to the side scattering, which could influence both absolute thresholds and convective gains. Using these formulas and models, we have evaluated the importance of stimulated Raman side scattering and stimulated Brillouin side scattering at ignition conditions, including both direct drive and indirect drive. It shows that side scattering could be a possible way of scattering other than backscattering in several ignition situations.

Compactness of Commutators for Riesz Potential on Local Morrey-type spaces

The paper considers Morrey-type local spaces from LMpθw The main work is the proof of the commutator compactness theorem for the Riesz potential [b,Iα] in local Morrey-type spaces from LMpθw1 to LMpθw2. We also give new sufficient conditions for the commutator to be bounded for the Riesz potential [b,Iα] in local Morrey-type spaces from LMpθw1 to LMpθw2. In the proof of the commutator compactness theorem for the Riesz potential, we essentially use the boundedness condition for the commutator for the Riesz potential [b,Iα] in local Morrey-type spaces LMpθw, and use the sufficient conditions from the theorem of precompactness of sets in local spaces of Morrey type LMpθw. In the course of proving the commutator compactness theorem for the Riesz potential, we prove lemmas for the commutator ball for the Riesz potential [b,Iα]. Similar results were obtained for global Morrey-type spaces GMpθw and for generalized Morrey spaces Mpw.

A class of outer generalized inverses and a note on Drazin’s finiteness conditions in rings and semigroups

We introduce a class of outer generalized inverses in rings and semigroups, which affords a characterization of Greville’s spectral -inverses of complex matrices. Moreover, it is proved that Drazin’s intermediate finiteness and weak finiteness conditions for semigroups are the same notion.

Convergence and exponential stability of modified truncated Milstein method for stochastic differential equations

In this paper, we develop a new explicit scheme called modified truncated Milstein method which is motivated by truncated Milstein method proposed by Guo et al. (2018) and modified truncated Euler–Maruyama method introduced by Lan and Xia (2018). We obtain the strong convergence of the scheme under local boundedness and Khasminskii-type conditions, which are relatively weaker than the existing results, and we prove that the convergence rate could be arbitrarily close to 1 under given conditions. Moreover, pth moment exponential stability of the scheme is also obtained Three numerical experiments are offered to support our conclusions.

Plane-Wave Diffraction from Resistive-Filled Circular Hole in Infinite Resistive Plane: An Analytically Regularizing Approach

The study of the electromagnetic diffraction from penetrable screens with apertures and/or inhomogeneities is of great relevance today due to the huge number of modern applications in which they are involved. In this paper, the analysis of the plane wave scattering from a resistive-filled circular hole in a resistive plane is addressed. The uniquely solvable boundary value problem for the Maxwell equations, obtained via imposing generalized boundary conditions, power boundedness condition, and Silver–Muller radiation condition, is equivalently formulated in terms of an infinite set of singular dual integral equations in the vector Hankel transform domain. The Helmholtz–Galerkin technique allows for the discretization and, simultaneously, analytical regularization of the obtained integral equations. Fast convergence is guaranteed by a suitable choice of the basis functions reconstructing the physical behavior of the fields at the discontinuity between the two involved media. Moreover, the full-wave nature of the proposed approach allows the direct assessment of near-field and far-field parameters.

Classification of Solutions to Conformally Invariant Systems with Mixed Order and Exponentially Increasing or Nonlocal Nonlinearity

In this paper, without any assumption on and under the extremely mild assumption at for some arbitrarily large, we prove classification of solutions to the following conformally invariant system with mixed order and exponentially increasing nonlinearity in where , and that satisfies the finite total curvature condition . In order to show the integral representation formula and the crucial asymptotic property for , we derive and use an inequality, which is itself of independent interest. When , the system is closely related to single conformally invariant equations and on , which have been quite extensively studied (cf. [H. Brezis and F. Merle, Comm. Partial Differential Equations, 16 (1991), pp. 1223–1253; D. Cao, Comm. Partial Differential Equations, 17 (1992), pp. 407–435; S.-Y. A. Chang and P. C. Yang, Math. Res. Lett., 4 (1997), pp. 91–102; W. Chen and C. Li, Duke Math. J., 63 (1991), pp. 615–622; W. Chen, C. Li, and Y. Li, Adv. Math., 308 (2017), pp. 404–437; W. Chen, Y. Li, and R. Zhang, J. Funct. Anal., 272 (2017), pp. 4131–4157], etc.). We also derive classification results for nonnegative solutions to a conformally invariant system with mixed order and Hartree type nonlocal nonlinearity in . Extensions to mixed order conformally invariant systems in with general dimensions are also included.

Separation Conditions for Iterated Function Systems with Overlaps on Riemannian Manifolds

We formulate the weak separation condition and the finite type condition for conformal iterated function systems on Riemannian manifolds with nonnegative Ricci curvature, and generalize the main theorems by Lau et al. (Monatsch Math 156:325–355, 2009). We also obtain a formula for the Hausdorff dimension of a self-similar set defined by an iterated function system satisfying the finite type condition, generalizing a corresponding result by Jin and Yau (Commun Anal Geom 13:821–843, 2005) and Lau and Ngai (Adv Math 208:647–671, 2007) on Euclidean spaces. Moreover, we obtain a formula for the Hausdorff dimension of a graph self-similar set generated by a graph-directed iterated function system satisfying the graph finite type condition, extending a result by Ngai et al. (Nonlinearity 23:2333–2350, 2010).

Stability of Cayley dynamic systems with impulsive effects

Linear dynamic systems with impulsive effects are considered. For such a system we define a new impulsive exponential matrix. Necessary and sufficient conditions for exponential stability and boundedness have been established. The fundamental tool is an impulsive exponential matrix for exponential stability.