Non-smooth Curve Research Articles

The circular maximal operator on Heisenberg radial functions

Lebesgue space estimates are obtained for the circular maximal function on the Heisenberg group $\mathbb{H}^1$ restricted to a class of Heisenberg radial functions. Under this assumption, the problem reduces to studying a maximal operator on the Euclidean plane. This operator has a number of interesting features: it is associated to a non-smooth curve distribution and, furthermore, fails both the usual rotational curvature and cinematic curvature conditions.

Experimental Study on Drag Reduction Characteristics of Bionic Earthworm Self-Lubrication Surface.

In the present study, a coupling bionic method is used to study the drag reduction characteristics of corrugated surface with lubrication. In order to test the drag reduction features, bionic specimen was prepared inspired by earthworm surface and lubrication. Based on the reverse engineering method, nonsmooth curve of earthworm surface was extracted and the bionic corrugated sample was designed, and the position of lubrication hole was established by experimental testing. The lubricating drag reduction performance, the influence of normal pressure, the forward velocity, and the flow rate of lubricating fluid on the forward resistance of the bionic specimens were analyzed through a single factor test by using the self-developed test equipment. The model between the forward resistance and the three factors was established through the ternary quadratic regression test. The results show that the drag reduction effect is obvious, the drag reduction rate is 22.65% to 34.89%, and the forward resistance decreases with the increase of the forward velocity, increases with the increase of the normal pressure, and decreases first and then becomes stable with the increase of flow rate of lubricating fluid. There are secondary effects on forward resistance by the three factors, and the influencing order is as follows: normal pressure>flow rate of lubricating fluid>forward velocity.

Practical backstepping control for underactuated ship path following associated with disturbances

Robust tracking control of underactuated surface vessels is an important issue in marine control engineering all along. This study intends to enhance the robustness of the motion control system which associates with unknown dynamics and time-varying disturbances. A practical adaptive backstepping control scheme is developed to keep a surface ship along the desired course that generated by a virtual ship. This control scheme can guarantee the error dynamic system to be globally uniformly asymptotically stable by the Lyapunov stability theorem. Furthermore, this algorithm can be implemented without the exact information of the non-linear hydrodynamics and random ocean disturbances. The employed pre-filter can provide a smooth transition of the desired course angle and its derivatives that based on the line-of-sight guidance method when the desired path is a non-smooth curve. The proposed controller can steer the underactuated vessel and follow the reference course accurately whether the desired course is smooth or non-smooth. In addition, this method can effectively reduce the fluctuation of the control moment when the desired path is a non-smooth curve. Two simulation cases are drawn to illustrate the validity of the proposed control strategy.

The Cauchy Singular Integral on Non-Smooth Curve

We consider the Cauchy singular integral with Holder’s density on non-smooth curves: various versions of its definition, conditions for its existence, boundedness and so on.

Fractal Characteristics and Prediction of Backsilting Quantity in Yangtze Estuary Deepwater Channel

Fractal interpolation has been an important method applied to engineering in recent years. It can not only be used to fit smooth curve and stationary data but also show its unique superiorities in the fatting of non-smooth curve and non-stationary data. Through analyzing such characteristic values as average value, standard deviations, skewness and kurtosis of measured backsilting quantities in the Yangtze Estuary 12.5 m Deepwater Channel during 2011–2017, the fractal interpolation method can be used to study the backsilting quantity distribution with time. According to the fractal interpolation made on the channel backsilting quantities from January 2011 to December 2017, there was a good corresponding relationship between the annual (monthly) siltation quantities and the vertical scaling factor. On this basis, a calculation formula for prediction of the backsilting quantity in the Yangtze Estuary Deepwater Channel was constructed. With the relationship between the predicted annual backsilting quantities and the vertical scaling factor, the monthly backsilting quantities can be obtained. Thus, it provides a new method for estimating the backsilting quantity of the Yangtze Estuary Deepwater Channel.

Regularity of the solution and well-posedness of a mixed problem for an elliptic system with quadratic nonlinearity in gradients

A boundary value problem for an elliptic system of equations is studied that arises in the analysis of a new hydrodynamic model describing charge transport in a planar semiconductor MESFET (metal semiconductor field effect transistor). The problem has a number of features, specifically, the equations of the system involve squared components of the gradients of the unknown functions; the boundary conditions are of a mixed character, i.e., Dirichlet and Neumann conditions are set on different portions of the boundary; and the boundary of the domain is a nonsmooth curve, namely, a rectangle. Under a certain optimal condition, the C 1,α-regularity of a weakened solution of the problem is justified and its existence is proved, while its uniqueness is shown under additional constraints. The results are used to justify the stabilization method as applied to finding approximate stationary solutions of the hydrodynamic model.

Fractional differentiability of the non-smooth heat curve

There are many non-smooth objects in nature, such as coastline, rock fracture, cross section, whose differentiabilities cannot be described by ordinary calculus and methods in Euclidean geometry. The local fractional derivative is one of the potential tools to investigate the non-smooth problems. This study revisits the non-smooth curves generated from the fractional integrals and Cantor-like set. From the view of the fractional differentiable functions, the differentiabilities of the non-smooth curves are derived by using a binomial expansion.

Local-in-time well-posedness of a regularized mathematical model for silicon MESFET

We prove the local-in-time well-posedness of the initial boundary value problem for a system of quasilinear equations. This system is used for finding numerical stationary solutions of the hydrodynamical model of charge transport in the silicon MESFET (metal semiconductor field effect transistor). The initial boundary value problem has the following peculiarities: the quasilinear system is not a Cauchy-Kovalevskaya-type system; the boundary is a non-smooth curve and has angular points; nonlinearity of the problem is mainly connected with squares of gradients of the unknown functions. By using a special representation for the solution of a model problem we reduce the original problem to an integro-differential system. The local-in-time existence of a weakened generalized solution of this system is then proved by the fixed-point argument.

Computation of the separatrix of basins of attraction in a non-smooth dynamical system

This paper describes a new numerical method to compute the separatrix of the basins of attraction of coexisting attractors in a forced friction oscillator. Numerical results show that its intersection with a Poincaré section is a non-smooth curve.

A mechanism for dangerous border collision bifurcations

Abstract In this paper we investigate a mechanism causing dangerous border collision bifurcations which is numerically characterized by exhibiting a stable fixed point before and after the critical bifurcation point, but the unbounded behavior of orbits at the critical bifurcation point. In particular, we provide that at the critical bifurcation value μ0, the qualitative type of the fixed point without having Jacobian information is saddle, which can be induced by invariant manifolds of the periodic saddle orbit on the boundary of the basin of attractor at infinity at the parameter μ ≠ μ0. In addition, we show that invariant sets of such fixed point are nonsmooth curve, and also point out a dangerous situation of numerical simulation.

Airline market share and customer service quality: a reference-dependent model

Traditional models that explain the nature of the relationship between customer service quality and airline demand assume that the relationship can be approximated by using smooth or differentiable curves. Suzuki and Tyworth, however, recently argued that this assumption may not be valid, and that, if it is invalid, the model performance can be improved by using non-smooth functions to represent the relationship (Suzuki, Y., Tyworth, J.E., 1998. A theoretical framework for modeling sales-service relationships in the transportation industry. Transportation Research E 34 (2), 87–100). We use their framework to develop a model that represents the relationship between service quality and market share in the airline industry and then empirically compare its performance with conventional airline demand models. The results indicate that the relationship is characterized by a non-smooth curve and that our model provides a significantly better goodness of fit than other conventional demand models.

Numerical Integration of Vector Fields Over Curves with Zero Area

The boundary of a Jordan domain A may be a nonsmooth curve $\gamma$. If F is a smooth vector field defined near $\gamma$, then F is integrable over $\gamma$ provided $\gamma$ has two-dimensional Lebesgue measure zero and F is sufficiently smooth. When actually computing the integral ${\smallint _\gamma }F \bullet ds$, one might hope that ${\lim _{k \to \infty }}{\smallint _{{\gamma _k}}}F \bullet ds = {\smallint _\gamma }F \bullet ds$ for PL or smooth approximators ${\gamma _k}$ of $\gamma$. Several examples show that this is not the case. However, there are algorithms for choosing ${\gamma _k}$ so that ${\smallint _{{\gamma _k}}}F \bullet ds$ converges to ${\smallint _\gamma }F \bullet ds$ exponentially quickly.