Uniform approximation and Bernstein polynomials with coefficients in the unit interval

Among the applications of the Bernstein polynomials in one variable is their use in solving problems associated with stochastic computing. Taking as a starting point the notion of stochastic logic in the sense of Qian-Riedel-Rosenberg, the aim of this paper is to investigate some necessary and sufficient conditions for guaranteeing whether polynomial operations can be implemented with stochastic logic based on multivariate Bernstein polynomials with coefficients in the unit interval.

This study develops the method of constructing a multi-parametric distribution on the unit interval using Bernstein polynomials and investigates its key statistical properties. The performance of the proposed distribution’s parameter estimators was evaluated by Monte Carlo simulation under both maximum-likelihood (MLE) and minimum-distance frameworks, with the MLE approach generally yielding lower estimation error. A real data example illustrates the advantages of the construction method. The results reveal that the Bernstein distribution provides superior goodness-of-fit to the dataset compared with competing models.

Mahalanobis distance may be used as a measure of the disparity between an individual’s profile of scores and the average profile of a population of controls. The degree to which the individual’s profile is unusual can then be equated to the proportion of the population who would have a larger Mahalanobis distance than the individual. Several estimators of this proportion are examined. These include plug-in maximum likelihood estimators, medians, the posterior mean from a Bayesian probability matching prior, an estimator derived from a Taylor expansion, and two forms of polynomial approximation, one based on Bernstein polynomial and one on a quadrature method. Simulations show that some estimators, including the commonly-used plug-in maximum likelihood estimators, can have substantial bias for small or moderate sample sizes. The polynomial approximations yield estimators that have low bias, with the quadrature method marginally to be preferred over Bernstein polynomials. However, the polynomial estimators sometimes yield infeasible estimates that are outside the 0–1 range. While none of the estimators are perfectly unbiased, the median estimators match their definition; in simulations their estimates of the proportion have a median error close to zero. The standard median estimator can give unrealistically small estimates (including 0) and an adjustment is proposed that ensures estimates are always credible. This latter estimator has much to recommend it when unbiasedness is not of paramount importance, while the quadrature method is recommended when bias is the dominant issue.

In 2010, Long and Zeng introduced a new generalization of the Bernstein polynomials that depends on a parameter and called -Bernstein polynomials. After that, in 2018, Lain and Zhou studied the uniform convergence for these -polynomials and obtained a Voronovaskaja-type asymptotic formula in ordinary approximation. This paper studies the convergence theorem and gives two Voronovaskaja-type asymptotic formulas of the sequence of -Bernstein polynomials in both ordinary and simultaneous approximations. For this purpose, we discuss the possibility of finding the recurrence relations of the -th order moment for these polynomials and evaluate the values of -Bernstein for the functions , is a non-negative integer

Even though many neurochips have been developed and investigated, the best suitable way for implementation has not been known clearly. Our approach is to exploit stochastic logic for various operations required for neural functions. The advantage of stochastic logic is that complex operations can be implemented with a few ordinary logic gates. On the other hand, the operation speed is not so fast since stochastic logic requires certain accumulation time for averaging. However, a huge integration can be achieved and its reliability is high because all of operations are done on digital circuits. Furthermore, we propose a nonmonotonic neuron realized by stochastic logic, since the nonmonotonic property is efficient for the performance enhancement in association and learning. In this paper, we show the circuit design and measurement results of a neurochip comprising 50 neurons are shown. The advantages of nonmonotonic and stochastic properties are shown clearly.

Polynomial operators and local smoothness classes on the unit interval

Dual Bernstein polynomials of one or two variables have proved to be very useful in obtaining Bezier form of the L 2-solution of the problem of best polynomial approximation of Bezier curve or surface. In this connection, the Bezier coefficients of dual Bernstein polynomials are to be evaluated at a reasonable cost. In this paper, a set of recurrence relations satisfied by the Bezier coefficients of dual bivariate Bernstein polynomials is derived and an efficient algorithm for evaluation of these coefficients is proposed. Applications of this result to some approximation problems of Computer Aided Geometric Design (CAGD) are discussed.

The Weierstrass polynomial approximation theorem plays a central role in showing approximation capability of polynomial-based higher-order feedforward networks. The Bernstein polynomials are a family of polynomials which satisfy the Weierstrass polynomial theorem. Baldi (1991) interpreted the Bernstein polynomials in connectionist framework and argued that they can be viewed as a model for biological bell-shaped receptive fields. However, due to their strict constraint of equally-spaced input points, their connectionist interpretation has some problems in the real-world setting. In this paper, we present the modified Bernstein polynomials which have lesser constraints on the position of input points. The modified Bernstein polynomials can directly utilize given input/output data with a minor constraint on the position of the input points. We present theorems that show the approximation capability of the modified Bernstein polynomials. The relationship between the modified Bernstein polynomials and other higher-order feedforward network approaches is also discussed. >

Bernstein polynomial approximation of continuous function has a slower rate of convergence compared to other approximation methods. “The fact seems to have precluded any numerical application of Bernstein polynomials from having been made. Perhaps they will find application when the properties of the approximant in the large are of more importance than the closeness of the approximation.”—remarked P.J. Davis in his 1963 book, Interpolation and Approximation. This paper presents a direct approximation method for nonlinear optimal control problems with mixed input and state constraints based on Bernstein polynomial approximation. We provide a rigorous analysis showing that the proposed method yields consistent approximations of time-continuous optimal control problems and can be used for costate estimation of the optimal control problems. This result leads to the formulation of the Covector Mapping Theorem for Bernstein polynomial approximation. Finally, we explore the numerical and geometric properties of Bernstein polynomials, and illustrate the advantages of the proposed approximation method through several numerical examples.

Georg Cantor (1848-1914) was a German mathematician who worked on the foundations of set theory. He published the Cantor set in 1883. The Cantor set, along with the other fractals discussed in this chapter, are referred to as mathematical monsters by Mandelbrot because of their ambiguous features. The Cantor set is produced by starting with a unit interval [0, 1] and taking out the center open interval (1/3, 2/3). An open interval does not include the end points and is shown with rounded parenthesis. A closed interval includes the end points and is shown with squared parenthesis. This process leaves two closed intervals [0,1/3] and [2/3,1]. The same procedure is repeated on each of these two intervals. This procedure is repeated an infinite number of times (Peitgen, Jurgens, and Saupe, 1992). The result is an infinite number of clustered points. Figure 2.1 shows the construction process for the Cantor set through a few of the infinite steps.

One of the models of discrete analog of the Renyi problem known as the “parking problem” has been considered. Let n and i be integers, n ≥ 0, and 0 ≤ i ≤ n–1. Open interval (i, i + 1), where i is a random variable taking values 0, 1, 2, …, and n–1 for all n ≥ 2 with equal probability, is placed on interval [0, n]. If n < 2, we say that the interval cannot be placed. After placing the first interval, two free intervals [0, i] and [i + 1, n] are formed, which are filled with intervals of unit length according to the same rule, independently of each other, etc. When the filling of [0, n] with unit intervals is completed, the distance between any two neighboring intervals does not exceed 1. Let Xn be the number of placed intervals. This paper analyzes the asymptotic behavior of moments of random variable Xn. Unlike the classical case, exact expressions for the first moments can be found.

On Selected Properties of Uninorm Petri Nets and Their Application in Modeling Knowledge-Based Systems

Bernstein polynomials play an important role in computer graphics. For instance they are used as a basis for Bézier curves. The envelope of the Bernstein polynomial of degree is given by . This Demonstration shows Bernstein polynomials and their envelopes.;

To transmit the teaching of digital electronics today, considerations are outlined which must be kept in mind. Digital electronics is evolving quickly and its techniques and tools have revolutionized the manner of analyzing simulating, synthesizing and verifying digital systems. Microelectronic development is more and more dependant on the technology and on design methodology. Standard hardware description languages (Verilog and VHDL), together with simulation and synthesis tools are some of the drivers behind microelectronic development. The constructivist model in the teaching-learning process proposes: significant learning as opposed to memorizing; structuring and sequencing of content; learning through guided discovery and a spiral or recurring procedure. These circumstances, together with pedagogical concern and the knowledge and experience that the authors have acquired as digital electronics teaching Professors have resulted in the formulation of a curricular proposal to transmit this updated teaching method. Using this teaching model as a base, the aim of the proposal is to have the student studying digital electronics acquire the theory and practices (know-how) from the beginning by using current design methodologies and CAD and EDA tools that the students will use in their professional future. Another objective of this method is to provide students with multimedia applications as a learning resource.

The paper concerns the uniform polynomial approximation of a function f, continuous on the unit Euclidean sphere of ℝ3 and known only at a finite number of points that are somehow uniformly distributed on the sphere. First, we focus on least squares polynomial approximation and prove that the related Lebesgue constants w.r.t. the uniform norm grow at the optimal rate. Then, we consider delayed arithmetic means of least squares polynomials whose degrees vary from n − m up to n + m, being m = θn for any fixed parameter 0 < θ < 1. As n tends to infinity, we prove that these polynomials uniformly converge to f at the near-best polynomial approximation rate. Moreover, for fixed n, by using the same data points, we can further improve the approximation by suitably modulating the action ray m determined by the parameter θ. Some numerical experiments are given to illustrate the theoretical results.