Continuous Functions Of Order Research Articles

An Enhanced Data Processing Framework for Mapping Tree Root Systems Using Ground Penetrating Radar

The preservation of natural assets is nowadays an essential commitment. In this regard, root systems are endangered by fungal diseases which can undermine the health and stability of trees. Within this framework, ground penetrating radar (GPR) is emerging as a reliable non-destructive method for root investigation. A coherent GPR-based root-detection framework is presented in this paper. The proposed methodology is a multi-stage data analysis system that is applied to semi-circular measurements collected around the investigated tree. In the first step, the raw data are processed by applying several standard and advanced signal processing techniques in order to reduce noise-related information. In the second stage, the presence of any discontinuity element within the survey area is investigated by analysing the signal reflectivity. Then, a tracking algorithm aimed at identifying patterns compatible with tree roots is implemented. Finally, the mass density of roots is estimated by means of continuous functions in order to achieve a more realistic representation of the root paths and to identify their length in a continuous and more realistic domain. The method was validated in a case study in London (UK), where the root system of a real tree was surveyed using GPR and a soil test pit was excavated for validation purposes. Results support the feasibility of the data processing framework implemented in this study.

The Relation between Hölder Continuous Function of Order α ∈ (0,1) and Function of Bounded Variation

The Cantor ternary function is the most famous example of a continuous function of bounded variation for which it satisfies the Holder continuous function of order α = log3 2, but does not satisfy for order α = 1. In this paper, based on previous work of Hölder continuous function of order α ∈ (0, 1) and using F α - calculus on fractal set F, we show the relation between the Hölder continuous function of order α ∈ (0, 1) and function of bounded variation. In particular, we give the necessary and sufficient condition for the variation of function satisfies Hölder condition or bi-Hölder condition.

Nonlinear approximation via compositions

Given a function dictionary D and an approximation budget N∈N, nonlinear approximation seeks the linear combination of the best N terms {Tn}1≤n≤N⊆D to approximate a given function f with the minimum approximation error εL,f≔min{gn}⊆R,{Tn}⊆D‖f(x)−∑n=1NgnTn(x)‖.Motivated by recent success of deep learning, we propose dictionaries with functions in a form of compositions, i.e., T(x)=T(L)∘T(L−1)∘⋯∘T(1)(x)for all T∈D, and implement T using ReLU feed-forward neural networks (FNNs) with L hidden layers. We further quantify the improvement of the best N-term approximation rate in terms of N when L is increased from 1 to 2 or 3 to show the power of compositions. In the case when L>3, our analysis shows that increasing L cannot improve the approximation rate in terms of N.In particular, for any function f on [0,1], regardless of its smoothness and even the continuity, if f can be approximated using a dictionary when L=1 with the best N-term approximation rate εL,f=O(N−η), we show that dictionaries with L=2 can improve the best N-term approximation rate to εL,f=O(N−2η). We also show that for Hölder continuous functions of order α on [0,1]d, the application of a dictionary with L=3 in nonlinear approximation can achieve an essentially tight best N-term approximation rate εL,f=O(N−2α∕d). Finally, we show that dictionaries consisting of wide FNNs with a few hidden layers are more attractive in terms of computational efficiency than dictionaries with narrow and very deep FNNs for approximating Hölder continuous functions if the number of computer cores is larger than N in parallel computing.

Global stability of a class of futile cycles

In this paper, we prove the global asymptotic stability of a class of mass action futile cycle networks which includes a model of processive multisite phosphorylation networks. The proof consists of two parts. In the first part, we prove that there is a unique equilibrium in every positive compatibility class. In the second part, we make use of a piecewise linear in rates Lyapunov function in order to prove the global asymptotic stability of the unique equilibrium corresponding to a given initial concentration vector. The main novelty of the paper is the use of a simple algebraic approach based on the intermediate value property of continuous functions in order to prove the uniqueness of equilibrium in every positive compatibility class.

Random regression models for the estimation of genetic and environmental covariance functions for growth traits in Santa Ines sheep.

Polynomial functions of different orders were used to model random effects associated with weight of Santa Ines sheep from birth to 196 days. Fixed effects included in the models were contemporary groups, age of ewe at lambing, and fourth-order Legendre polynomials for age to represent the average growth curve. In the random part, functions of different orders were included to model variances associated with direct additive and maternal genetic effects and with permanent environmental effects of the animal and mother. Residual variance was fitted by a sixth-order ordinary polynomial for age. The higher the order of the functions, the better the model fit the data. According to the Akaike information criterion and likelihood ratio test, a continuous function of order, five, five, seven, and three for direct additive genetic, maternal genetic, animal permanent environmental, and maternal permanent environmental effects (k = 5573), respectively, was sufficient to model changes in (co)variances with age. However, a more parsimonious model of order three, three, five, and three (k = 3353) was suggested based on Schwarz's Bayesian information criterion for the same effects. Since it was a more flexible model, model k = 5573 provided inconsistent genetic parameter estimates when compared to the biologically expected result. Predicted breeding values obtained with models k = 3353 and k = 5573 differed, especially at young ages. Model k = 3353 adequately fit changes in variances and covariances with time, and may be used to describe changes in variances with age in the Santa Ines sheep studied.

ESTIMATES FOR THE SOLUTION TO STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY A FRACTIONAL BROWNIAN MOTION WITH HURST PARAMETER H ∈ (⅓, ½)

In this paper we establish precise estimates for the supremum norm for the solution of a dynamical system driven by a Hölder continuous function of order between ⅓ and ½ using the techniques of fractional calculus. As an application we deduce the existence of moments for the solutions to stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H ∈(⅓, ½) and we obtain an estimate for the supremum norm of the Malliavin derivative.

Weighted error estimates for approximation by Cesàro means of Fourier-Jacobi series in spaces of locally continuous functions

We establish sufficient conditions on the parameter θ > 0 of the Cesaro means of Fourier-Jacobi series in spaces of locally continuous functions in order to have bounded weighted norm. For θ ≥ 1, a Stechkin type error estimate for the order of convergence will also be given.

Properties of ( θ, s)-continuous functions

Joseph and Kwack [Proc. Amer. Math. Soc. 80 (1980) 341–348] introduced the notion of ( θ, s)-continuous functions in order to investigate S-closed spaces due to Thompson [Proc. Amer. Math. Soc. 60 (1976) 335–338]. In this paper, further properties of ( θ, s)-continuous functions are obtained and relationships between ( θ, s)-continuity, contra-continuity and regular set-connectedness defined by Dontchev et al. [Internat. J. Math. Math. Sci. 19 (1996) 303–310 and elsewhere] are investigated.

On α-quasi-irresolute functions

Joseph and Kwack [9] introduced the notion of (θ,s)-continuous functions in order to investigateS-closed spaces due to Thompson [32]. In [26], the present authors investigated further properties of (θ,s)-continuous functions. In this paper, we introduce a new class of functions called α-quasi-irresolute functions which is weaker than (θ,s)-continuous and improve some results established in [26].

Taylor's formula and preservation of generalized convexity for positive linear operators

In this paper, we consider positive linear operators L representable in terms of stochastic processes Z having right-continuous non-decreasing paths. We introduce the equivalent notions of derived operator and derived process of order n of L and Z, respectively. When acting on absolutely continuous functions of order n, we obtain a Taylor's formula of the same order for such operators, thus extending to a positive linear operator setting the classical Taylor's formula for differentiable functions. It is also shown that the operators satisfying Taylor's formula are those which preserve generalized convexity of order n. We illustrate the preceding results by considering discrete time processes, counting and renewal processes, centred subordinators and the Yule birth process.

Taylor's formula and preservation of generalized convexity for positive linear operators

In this paper, we consider positive linear operators L representable in terms of stochastic processes Z having right-continuous non-decreasing paths. We introduce the equivalent notions of derived operator and derived process of order n of L and Z, respectively. When acting on absolutely continuous functions of order n, we obtain a Taylor's formula of the same order for such operators, thus extending to a positive linear operator setting the classical Taylor's formula for differentiable functions. It is also shown that the operators satisfying Taylor's formula are those which preserve generalized convexity of order n. We illustrate the preceding results by considering discrete time processes, counting and renewal processes, centred subordinators and the Yule birth process.

WHEN IS CUTTING CHAOTIC?

An extended analysis of the cutting model analyzed by Grabec, a four dimensional non-linear dynamical system, shows that there is but one fixed point, which is unstable for reasonable values of damping. A calculation of the Lyapunov exponents and dimension shows that the model system is chaotic for a range of cutting forces, and non-chaotic otherwise. The Grabec model is discontinuous, and it was necessary to replace the discontinuities by sharply varying continuous functions in order to calculate the Lyapunov exponents.

Sensor-Actuator Placement for Flexible Structures with Actuator Dynamics

A novel approach for placement of sensors and actuators in control of flexible space structures is developed. Using an approximation of the control forces and output measurements by spatially continuous functions, the approach follows a nonlinear programming technique to determine optimal locations for sensors and actuators. Two different criteria are considered for the placement of sensors and actuators. The first criterion optimizes the location of the sensors and actuators in order to move the transmission zeros of the system farther to the left of the imaginary axis. The second criterion, however, places the sensors and actuators to optimize a function of the singular values of the Hankel matrix, which includes both measures of controllabili ty and observability. Moreover, the effect of actuator dynamics in the placement of sensors and actuators is investigated. ENSOR and actuator placement is an integral part of a control design, particularly in the control of flexible struc- tures. Although engineering judgment and trial-and-error analysis are quite often used to determine the optimal locations of sensors and actuators, there have been various attempts to develop systematic means of achieving it. Among a few of these, Chen and Seinfeld1 used an optimization- based ap- proach wherein optimal locations of measurement outputs are computed from a finite set of possible locations with the aid of integer programming. Skelton and Chiu2 have considered an order reduction approach such that the inputs and outputs are reduced from a large set so as to make the smallest possible perturbation in a performance metric. Horta and Juang3 pro- posed a sequential linear programming approach for truss structures in which optimal numbers as well as locations of actuators are obtained from a large set of actuators. A com- mon point of reference in most of these approaches is the use of a performance metric. To date, various measures of con- trollability and observability,45'7 controllability and spill- over,6 and energy-based metrics2 have been considered as per- formance metrics in the literature. Of recent efforts, Maghami and Joshi8 provide a method in which the discrete nature of the sensor and actuator placement problem is transformed into a continuous nonlinear programming problem with the aid of functional approximation of the control forces and output measurements. Lim9 proposes an approach for placing actua- tors and sensors that is based on projecting eigenvectors into the intersection subspace of the controllability and observabil- ity subspaces for each sensor-actuator pair. In this paper, a new approach for sensor and actuator place- ment in the control of flexible space structures is developed. The approach follows the development of Ref. 8, which re- duces the problem of sensor-actuator positioning to a solution of a nonlinear programming optimization. The discrete con- trol forces and output measurements are approximated by spatially continuous functions in order to avoid the disconti- nuity problems caused when actuators and sensors are recon-

On the evaluation of one-dimensional Cauchy principal value integrals by rules based on cubic spline interpolation

In this note we consider the numerical evaluation of one dimensional Cauchy principal value integrals of the form $$\rlap{--} \smallint _a^b \frac{{k(x)f(x)}}{{x - \lambda }}dx, a< \lambda< b,$$ by rules obtained by “subtracting out” the singularity and then applying product quadratures based on cubic spline interpolation at equally spaced nodes. Convergence results are established for Holder continuous functions of order, μ, 0<μ≤1, and asymptotic rates are obtained for functionsf≠C k [a, b],k=1, 2, 3 or 4. Some comparisons with other methods and numerical examples are also given.

Cpo's of measures for nondeterminism

In this approach to the semantics of nondeterminism, we introduce and study the complete partial order (cpo) of probability distributions on a domain. The approach avoids considering equivalent subsets, used in theory of powerdomains, which may lead to some unwelcome identifications. These results show that the class of probability distributions on a cpo is itself a cpo and that every probability distribution is the lub of an increasing sequence of ‘finite’ probability distributions. We introduce the probabilistic extensions of continuous functions in order to extend the ‘usual’ continuous functions on this new domain. On the other hand the structure of this cpo suggests introducing an operation called ‘random selection’, which is the counterpart of the ‘OR’ (or union) operation, commonly used in nondeterministic programs. The paper then studies the ‘naturalness’ of these extended notions and treats the question of continuity, which is of prime importance in the Scott theory of fixed point semantics.