Finiteness Conditions Research Articles

Conditions for finiteness and bounds on moments of generalized order statistics

We present necessary and sufficient conditions for the finiteness of moments of a fixed order of generalized order statistics based on an arbitrary life baseline distribution with a finite expectation. The conditions depend on a relation between the moment order and the minimal parameter of the generalized order statistic and its multiplicity. Furthermore, under these conditions we determine sharp upper bounds on the moments of generalized order statistic expressed in the scale units being the respective powers of the mean of the baseline distribution.

The statistical complexity of early-stopped mirror descent

Abstract Recently there has been a surge of interest in understanding implicit regularization properties of iterative gradient-based optimization algorithms. In this paper, we study the statistical guarantees on the excess risk achieved by early-stopped unconstrained mirror descent algorithms applied to the unregularized empirical risk. We consider the set-up of learning linear models and kernel methods for strongly convex and Lipschitz loss functions while imposing only boundedness conditions on the unknown data-generating mechanism. By completing an inequality that characterizes convexity for the squared loss, we identify an intrinsic link between offset Rademacher complexities and potential-based convergence analysis of mirror descent methods. Our observation immediately yields excess risk guarantees for the path traced by the iterates of mirror descent in terms of offset complexities of certain function classes depending only on the choice of the mirror map, initialization point, step size and the number of iterations. We apply our theory to recover, in a clean and elegant manner via rather short proofs, some of the recent results in the implicit regularization literature while also showing how to improve upon them in some settings.

Notes on a paper of Pokhozhaev

We prove a second order identity for the Kirchhoff equation which yields, in particular, a simple and direct proof of Pokhozhaev’s second order conservation law when the nonlinearity has the special form . As applications, we give: an estimate of order ɛ −4 for the lifespan T ɛ of the solution of the Cauchy problem with initial data of size ɛ in Sobolev spaces when the nonlinearity is given by any C 2 function ; a necessary and sufficient condition for boundedness of a second order energy of the solutions.

Research on Practical Path Tracking Control of Autonomous Underwater Vehicle Based on Constructive Dynamic Gain Controller

In this study, the stochastic nonlinear system-based trajectory tracking control problem of an autonomous underwater vehicle (AUV) is studied. We investigate the time-varying gain adaptive control method to find possible approaches to reduce the excessive computational burden. Enhanced adaptive algorithms are devised by considering the dynamic characteristics of AUV motion. By transforming the original controller design problems into parameter selection problems and subsequently solving them using the functional time-varying observer technical theorem, we can achieve optimal control performance. The control system is shown to constrain system state error due to stochastic disturbances within arbitrarily small domains. A coordinate transformation is proposed for all system states to meet boundedness conditions. We show that the closed-loop stability is confirmed, the system is asymptotically probabilistically stable, and contraction limits given in the stability analysis may be used to certify the convergence of the AUV trajectory errors. A large number of simulation studies using an underwater vehicle model have proved the effectiveness and robustness of the proposed approach. A real-time, time-varying gain constructive control strategy is further developed for the hardware-in-the-loop simulation; the effectiveness of the controller design is confirmed by introducing the controller into the AUV actuator model.

Numerical simulation of ultrasonic impact treatment (UIT) assisted laser directed energy deposition (DED) CrCoNi medium entropy alloy process

In order to enhance the microstructure and mechanical properties of additively manufactured metal parts, the application of ultrasonic impact treatment (UIT) after the additive manufacturing process is introduced as a means to modulate and optimize the material's microstructure and properties. This study systematically investigates the effect of UIT on the distribution of residual stresses in CrCoNi medium entropy alloy (CrCoNi-MEA) through numerical simulation. A suitable finite element model and boundary conditions are established to simulate the UIT assisted laser DED process. The reliability of the finite element model is verified by XRD residual test results and EBSD observation results. The numerical simulation results shows that the specimen's surface exhibited predominantly compressive stress within a specific depth range, with the maximum compressive stress in the vertical direction. Furthermore, it is observed that UIT amplitude, frequency, and impact needle diameter also significantly influenced the residual stresses. By appropriately adjusting the UIT parameters, the magnitude and distribution of residual stresses could be further controlled and regulated. In addition, it is found that multiple UITs at different surface locations exhibited no significant mutual influence between each other.

Zabczyk Type Criteria for Asymptotic Behavior of Dynamical Systems and Applications

AbstractWe obtain new characterizations for nonuniform and uniform asymptotic behaviors of variational systems in infinite dimensional spaces, following the line of studies recently developed in Dragičević et al. (J Differ Equ 268:4786–4829, 2020, J Dyn Differ Equ 34:1107–1137, 2022, Appl Math Comput 414:1–22, 2022, J Math Anal Appl 515:1–37, 2022). First, we give complete descriptions for both nonuniform and uniform stability by means of some nonuniform conditions of convergence for series of suitable nonlinear trajectories. We apply our criteria to present a novel method of exploring the robustness of the nonuniform exponential stability under additive and multiplicative perturbations and we also deduce some consequences for the robustness of the uniform exponential stability. After that, we obtain characterizations for nonuniform and uniform instability in terms of certain nonuniform conditions imposed to the sums of some well-chosen series of nonlinear trajectories. We apply our results to provide a new technique of exploring the robustness of the nonuniform exponential instability under additive and multiplicative perturbations and discuss the consequences for uniform exponential instability. Hence, for both stability and instability we generalize the previous Zabczyk type results and extend their applicability. Next, as another class of applications, we provide nonuniform criteria of Rolewicz type for stability and instability of skew-product semiflows. Thus, we extend the Zabczyk–Rolewicz type methods in two directions: by giving local conditions for global behaviors and by employing specific nonuniform boundedness conditions. Our criteria can be applied to broad classes of discrete and continuous variational dynamical systems.

Finite element modeling of assembling rivet-fastened rectangular hollow flange beams in bending

The assembling rivet-fastened rectangular hollow flange beam (ARHFB) is a new type of cold-formed steel beam. The ARHFB section has two rectangular hollow flanges formed by U-shaped and C-shaped components that connected by self-locking rivets. A finite element model is developed to simulate 16 four-point bending experiments. Details of the modeling including finite element meshing, loading and boundary conditions, contact and analysis methods are presented. The accuracy of the developed finite element model is verified by comparing the ultimate moment capacity, applied load versus deflection curves, bending moment versus deflection curves, strain versus deflection curves and failure modes obtained from the tests and the finite element analysis. Based on the validated finite element model, a parametric analysis is carried out to discuss the effects of rivet spacing, section depth, flange thickness, and web thickness on the bending performance of ARHFB. The moment capacities of these sections obtained from the parametric study are compared with predictions from Australia and New Zealand, Chinese design standards for cold-rolled sections and the DSM (Direct Strength Method). It is found that GB50018 can conservatively predict the bearing capacity of ARHFB with a large error, while AS / NZS 4600 and DSM (Direct Strength Method) cannot provide reliable prediction results.

BIFURCATION ANALYSIS OF A MODIFIED LESLIE–GOWER PREDATOR–PREY MODEL WITH HUNTING COOPERATION AND FAVORABLE ADDITIONAL FOOD FOR PREDATOR

In this paper, we consider a modified Lasslie–Gower-type predator–prey model with the effect of hunting cooperation and favorable additional food for predator. We establish the conditions of positivity, boundedness, and permanence of solutions of the proposed model. Along with the trivial, predator free, prey free equilibrium points the system contains at most two coexistence equilibrium points. The system experiences the transcritical, saddle-node, Hopf, cusp, Bautin, and Bogdanov–Takens bifurcation depending on the model parameters. All the theoretical analyses are verified using numerical simulations. It is numerically established that the cooperation and extra food have high impact on the model dynamics.

Exploring the finite-time dissipativity of Markovian jump delayed neural networks

In this paper, we study the finite-time dissipativity analysis of Markovian jump-delayed neural networks (MJDNNs). The goal is to establish less conservative results for extended dissipativity conditions for delayed MJDNNs. To achieve this, an appropriate Lyapunov-Krasovskii functional (LKF) with novel inequality like composite slack-matrix-based integral inequality (CSMBII). Next, the CSMBII and other sufficient conditions are employed to estimate the derivative of the constructed LKF. Using these techniques, a delay-dependent finite-time dissipativity condition is derived in terms of linear matrix inequalities (LMIs). These LMIs are used to formulate the finite dissipativity condition for the delayed MJNNs. The utility of the suggested approach is then confirmed by a number of interesting numerical examples, one of which has been confirmed by a real-world application of the benchmark problem that is associated with the designed MJDNNs. The illustrative simulation results conclusively demonstrate the superior performance and success of the developed CSMBII technique in this proposal, surpassing the limitations of existing techniques.

Linear and Quadratic Chabauty for Affine Hyperbolic Curves

Abstract We give sufficient conditions for finiteness of linear and quadratic refined Chabauty–Kim loci of affine hyperbolic curves. We achieve this by constructing depth $\leq 2$ quotients of the fundamental group, following a construction of Balakrishnan–Dogra in the projective case. We also apply Betts’ machinery of weight filtrations to give unconditional explicit upper bounds on the number of $S$-integral points when our conditions are satisfied.

A fractional mathematical model of heartwater transmission dynamics considering nymph and adult amblyomma ticks

Heartwater is a tick-borne illness that affects ruminants and is carried by the amblyomma ticks. The condition may sometimes be deadly. This paper studies the Caputo fractional version of the disease spread in domestic ruminants and amblyomma ticks. We obtain the positivity and boundedness condition through the Laplace transform. The stability state of the proposed model is obtained using the Ulam–Hyers and Ulam–Hyers–Rassias stability conditions. The heartwater-free equilibrium point is obtained. The fractional model is fitted to the heartwater incidence data from 2006 to 2019. The heartwater reproduction number, R0τ, from the parameter estimation is R0τ=1.9345 with a fitting fractional order, τ, of 0.6990. The residuals from the data fitting were randomly distributed, indicating that the proposed model could be used for further predictions. Furthermore, we observed the dynamic effect of varying the fractional order and noticed that changing the fractional order produces crisscrossed behaviour in infected compartments of domestic ruminants and infected adult amblyomma ticks. Finally, we showed the relative impact of varying the transmission rates and the infectivity potential of peractive, active, and recovered carrier ruminants on the overall dynamics of the disease. Thus, a reduction in the rate of transmission from nymph and adult amblyomma ticks to vaccinated ticks will increase the number of healthy domestic ruminants.

Classification of Consistent Systems of Handlebody Group Representations

Abstract The classifying spaces of handlebody groups form a modular operad. Algebras over the handlebody operad yield systems of representations of handlebody groups that are compatible with gluing. We prove that algebras over the modular operad of handlebodies with values in an arbitrary symmetric monoidal bicategory $\mathcal{M}$ (we introduce for these the name ansular functor) are equivalent to self-dual balanced braided algebras in $\mathcal{M}$. After specialization to a linear framework, this proves that consistent systems of handlebody group representations on finite-dimensional vector spaces are equivalent to ribbon Grothendieck-Verdier categories in the sense of Boyarchenko-Drinfeld. Additionally, it produces a concrete formula for the vector space assigned to an arbitrary handlebody in terms of a generalization of Lyubashenko’s coend. Our main result can be used to obtain an ansular functor from vertex operator algebras subject to mild finiteness conditions. This includes examples of vertex operator algebras whose representation category has a non-exact monoidal product.

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.