Optimal Polynomial Research Articles

Optimal order polynomial motion profile for high-precision flexible scanning platform

Optimal order polynomial motion profile for high-precision flexible scanning platform

Towards Optimal Motion Planning of Surface Vehicles in Confined Waters

Abstract This paper presents a method for motion planning of fully actuated autonomous surface vehicles in confined waters. The goal is to generate a dynamically feasible trajectory from a given initial state to a given final state without collisions of the vehicle with obstacles. Both the own vehicle and static obstacles are represented as convex polygons. A sampling-based approach is used, where multiple samples in a state space are connected by third-order polynomials. A method is proposed to compute optimal polynomials that both minimize a cost function and account for input variable constraints. Within the cost function, time is weighted together with the control effort. The obtained objective function is a scalar nonlinear function that is solved using numerical minimization techniques. Embedded in a sampling-based motion planning algorithm, the major motion planning problem is solved by many small optimizations. The result is an optimal trajectory from an initial state to a final state, taking into account dynamics and input variable constraints, and avoiding collisions of the vehicle with static obstacles. A simulation is used to demonstrate that the proposed method is basically applicable.

On the weighted tests of independence based on Bernstein empirical copula

In this article, we propose a flexible weighted test of independence based on Cramér-von Mises statistic of the Bernstein empirical copula process. We investigate the asymptotic properties of the new weighted test. A Monte Carlo study is conducted to measure the power performance of the new test under various dependence structures and also to compare the performance with some other alternative tests. The effects of the weights and the selection of optimal Bernstein polynomial degree are significantly influencing the power of the test according to the particular type of dependence. Finally, the procedure is also applied to a dataset on diabetes for illustrative purpose.

Optimal approximants and orthogonal polynomials in several variables

AbstractWe discuss the notion of optimal polynomial approximants in multivariable reproducing kernel Hilbert spaces. In particular, we analyze difficulties that arise in the multivariable case which are not present in one variable, for example, a more complicated relationship between optimal approximants and orthogonal polynomials in weighted spaces. Weakly inner functions, whose optimal approximants are all constant, provide extreme cases where nontrivial orthogonal polynomials cannot be recovered from the optimal approximants. Concrete examples are presented to illustrate the general theory and are used to disprove certain natural conjectures regarding zeros of optimal approximants in several variables.

Application of colorimetric indicators to predict the fermentation stage of kimchi.

Cabbage kimchi, a popular side dish in Korean cuisine, produces several fermentation by-products (FBPs). Kimchi is praised for its flavor, taste, and texture when suitably fermented at 0.7% to 0.9% total acidity, or a pH of approximately 4.1. Beyond this acidity level, the quality of the product decreases, negatively impacting consumers' purchase preferences. Therefore, the current study seeks to develop an optimally fermented (OptF) kimchi indicator that can be inserted into product packaging to evaluate its utility at 4 and 10 °C. A gradual change in the total color difference (ΔE) was observed during the kimchi fermentation stage, and the highest ΔE values were observed at 4 (34.87) and 10 °C (37.99), after 9 weeks. Moreover, the color-change response function value F(Xc) was more linear at 4 and 10 °C (0.981 and 0.984, respectively) compared to the ΔE over time, during kimchi fermentation. Coefficients of determination for F(Xc)-carbon dioxide (0.983), F(Xc)-pH (0.979), and F(Xc)-titratable acidity (0.974) were sufficient to meet the optimal polynomial regression model, while that for F(Xc)-lactic acid bacteria (0.881) was not. Standardized residuals of predicted data indicated that 95% of the residuals were in the range of -2.0 to 2.0. The regression analysis further suggested that the OptF kimchi indicator could be used as a kimchi fermentation indicator. PRACTICAL APPLICATION: Cabbage kimchi, a traditional Korean fermented food, produces several fermentation by-products. After the optimal fermenting stage, the sensory evaluation of cabbage kimchi and consumers' purchase preference decreases. This study describes an optimally fermented kimchi indicator and its utility at 4 and 10 °C. Our results demonstrate the ability of this indicator to predict the freshness and fermentation stage of kimchi without the need for sensory evaluation. This method could help increase the purchase preference for commercial kimchi.

Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems

We present a quantum eigenstate filtering algorithm based on quantum signal processing (QSP) and minimax polynomials. The algorithm allows us to efficiently prepare a target eigenstate of a given Hamiltonian, if we have access to an initial state with non-trivial overlap with the target eigenstate and have a reasonable lower bound for the spectral gap. We apply this algorithm to the quantum linear system problem (QLSP), and present two algorithms based on quantum adiabatic computing (AQC) and quantum Zeno effect respectively. Both algorithms prepare the final solution as a pure state, and achieves the near optimalO~(dκlog⁡(1/ϵ))query complexity for ad-sparse matrix, whereκis the condition number, andϵis the desired precision. Neither algorithm uses phase estimation or amplitude amplification.

Genetic algorithms with permutation-based representation for computing the distance of linear codes

Abstract Finding the minimum distance of linear codes is an NP-hard problem. Traditionally, this computation has been addressed by means of the design of algorithms that find, by a clever exhaustive search, a linear combination of some generating matrix rows that provides a codeword with minimum weight. Therefore, as the dimension of the code or the size of the underlying finite field increase, so it does exponentially the run time. In this work, we prove that, given a generating matrix, there exists a column permutation which leads to a reduced row echelon form containing a row whose weight is the code distance. This result enables the use of permutations as representation scheme, in contrast to the usual discrete representation, which makes the search of the optimum polynomial time dependent from the base field. In particular, we have implemented genetic and CHC algorithms using this representation as a proof of concept. Experimental results have been carried out employing codes over fields with two and eight elements, which suggests that evolutionary algorithms with our proposed permutation encoding are competitive with regard to existing methods in the literature. As a by-product, we have found and amended some inaccuracies in the Magma Computational Algebra System concerning the stored distances of some linear codes.

The influence of the coefficients of a system of wave equations coupled by velocities on its stabilization

In this work, we consider a system of two wave equations coupled by velocities in a one-dimensional space, with one boundary fractional damping. First, we show that the system is strongly asymptotically stable if and only if the coupling parameter b of the two equations is outside a discrete set of exceptional real values. Next, we show that our system is not uniformly stable. Hence, we look for a polynomial decay rate for smooth initial data. Using a frequency domain approach combined with the multiplier method, we prove that the energy decay rate is greatly influenced by the nature of the coupling parameter b, the arithmetic property of the wave propagation speed a and the order of the fractional damping $$\alpha $$ . Indeed, under the equal speed propagation condition, i.e., $$a=1$$ , we establish an optimal polynomial energy decay rate of type $$t^{-\frac{2}{{1-\alpha }}}$$ if the coupling parameter $$b\notin \pi {\mathbb {Z}}$$ and of type $$t^{-\frac{2}{{5-\alpha }}}$$ if the coupling parameter $$b\in \pi {\mathbb {Z}}$$ . Furthermore, when the wave propagates with different speeds, i.e., $$a\not =1$$ , we prove that, for any rational number $$\sqrt{a}$$ and almost all irrational numbers $$\sqrt{a}$$ , the energy of our system decays polynomially to zero like as $$t^{-\frac{2}{{5-\alpha }}}$$ . This result still holds if $$a\in {\mathbb {Q}}$$ , $$\sqrt{a}\notin {\mathbb {Q}}$$ and b small enough.

Optimal Polynomial Prediction Measures and Extremal Polynomial Growth

We show that the problem of finding the measure supported on a compact set Ksubset mathbb {C} such that the variance of the least squares predictor by polynomials of degree at most n at a point z_0in mathbb {C}^dbackslash K is a minimum is equivalent to the problem of finding the polynomial of degree at most n, bounded by 1 on K, with extremal growth at z_0. We use this to find the polynomials of extremal growth for [-1,1]subset mathbb {C} at a purely imaginary point. The related problem on the extremal growth of real polynomials was studied by Erdős (Bull Am Math Soc 53:1169–1176, 1947).

Principal Component Projection with Low-Degree Polynomials

In this paper, we consider approximations of principal component projection (PCP) without explicitly computing principal components. This problem has been studied in several recent works. The main feature of existing approaches is viewing the PCP matrix as a matrix function. This underlying function is the composition of a step function with a rational function. To find an approximate PCP, the step function is approximated by a polynomial while the rational function is evaluated by a fast ridge regression solver. In this work, we further improve this process by replacing the rational function with carefully constructed polynomials of low degree. We characterize the properties of polynomials that are suitable for approximating PCP, and then establish an optimization problem to select the optimal one from those polynomials. We show theoretically and confirm numerically that the resulting approximate PCP approach with optimal polynomials is indeed effective for approximations of principal component projection.

Learning deep kernels in the space of monotone conjunctive polynomials

• The monotone Conjunctive Polynomial feature space is here presented. • Homogeneous polynomial kernels can be decomposed as a combination of monotone conjunctive kernels. • The optimal monotone Conjunctive Polynomial kernel can be learned via Multiple Kernel Learning . • The proposed approach proved to be robust to overfitting. • The monotone Conjunctive Polynomial kernel has a deep structure. Dot-product kernels is a large family of kernel functions based on dot-product between examples. A recent result states that any dot-product kernel can be decomposed as a non-negative linear combination of homogeneous polynomial kernels of different degrees, and it is possible to learn the coefficients of the combination by exploiting the Multiple Kernel Learning (MKL) paradigm. In this paper it is proved that, under mild conditions, any homogeneous polynomial kernel defined on binary valued data can be decomposed in a parametrized finite linear non-negative combination of monotone conjunctive kernels. MKL has been employed to learn the parameters of the combination. Furthermore, we show that our solution produces a deep kernel whose feature space consists of hierarchically organized features of increasing complexity. We also emphasize the connection between our solution and existing deep kernel learning frameworks. A wide empirical assessment is presented to evaluate the proposed framework, and to compare it against the baselines on several categorical and binary datasets.

L&T bags EPC contract for NTPC's CO2-to-methanol demo plant

Accurate information of path followed by vehicles at several real-life scenarios is essential for modeling and testing the optimal path for autonomous vehicles, analyzing the lateral dynamics such as lane-changing and overtaking, vehicle dynamics at the horizontal curves, intersections, and roundabouts. Recently, the availability of such data is plenty, thanks to the advancements in drone and computer-vision technologies. Nevertheless, extracting useful information from the observed data requires certain pre-processing that can handle the noises added at several stages of the data collection. The adequacy of the existing off-the-shelf smoothing techniques (e.g., Moving average, Savitzky–Golay, Kalman filter) for denoising the trajectory data of the modern era is certainly a question. The motivation for the present study is twofold. Firstly, the challenges of contemporary trajectory data collection and associated error patterns are relatively unknown, and thus an appropriate denoising technique needs to be developed. Secondly, many of the existing smoothing techniques have chosen invariant parameters across all the vehicle paths, even though the error patterns are dissimilar across the vehicles and, in the longitudinal and lateral directions. Thus, an adaptive smoothing needs to be applied for each observed vehicle path and its components. The present study proposes an adaptive, and data-driven path smoothing technique that can be universally applied to the video-based path data obtained from any camera platform (either drone or fixed camera setup). The proposed reconstruction framework works in three-stages; in the first stage, the data is prepared by resampling any missing data. Also, any outlier in the observed path data is identified and removed at this stage. In the second stage, a Recursively Ensembled Low-Pass (RELP) filter is proposed to handle the ’heavily tailed’ noise found in the video-processing based trajectory data, particularly for drone videos. A robust adaptive Gaussian kernel smoothing is applied in the third stage to have a localized reconstruction. For Kernel smoothing, the smoothing parameters such as the optimal bandwidth and polynomial order are estimated using the proposed grid-search algorithm. The parameter estimation process ensures that the bias and variance are perfectly traded-off to achieve a smooth vehicle path. The performance evaluation of the proposed method shows an internally and physically consistent trajectory reconstruction.

Vehicle path reconstruction using Recursively Ensembled Low-pass filter (RELP) and adaptive tri-cubic kernel smoother

Accurate information of path followed by vehicles at several real-life scenarios is essential for modeling and testing the optimal path for autonomous vehicles, analyzing the lateral dynamics such as lane-changing and overtaking, vehicle dynamics at the horizontal curves, intersections, and roundabouts. Recently, the availability of such data is plenty, thanks to the advancements in drone and computer-vision technologies. Nevertheless, extracting useful information from the observed data requires certain pre-processing that can handle the noises added at several stages of the data collection. The adequacy of the existing off-the-shelf smoothing techniques (e.g., Moving average, Savitzky–Golay, Kalman filter) for denoising the trajectory data of the modern era is certainly a question. The motivation for the present study is twofold. Firstly, the challenges of contemporary trajectory data collection and associated error patterns are relatively unknown, and thus an appropriate denoising technique needs to be developed. Secondly, many of the existing smoothing techniques have chosen invariant parameters across all the vehicle paths, even though the error patterns are dissimilar across the vehicles and, in the longitudinal and lateral directions. Thus, an adaptive smoothing needs to be applied for each observed vehicle path and its components. The present study proposes an adaptive, and data-driven path smoothing technique that can be universally applied to the video-based path data obtained from any camera platform (either drone or fixed camera setup). The proposed reconstruction framework works in three-stages; in the first stage, the data is prepared by resampling any missing data. Also, any outlier in the observed path data is identified and removed at this stage. In the second stage, a Recursively Ensembled Low-Pass (RELP) filter is proposed to handle the ’heavily tailed’ noise found in the video-processing based trajectory data, particularly for drone videos. A robust adaptive Gaussian kernel smoothing is applied in the third stage to have a localized reconstruction. For Kernel smoothing, the smoothing parameters such as the optimal bandwidth and polynomial order are estimated using the proposed grid-search algorithm. The parameter estimation process ensures that the bias and variance are perfectly traded-off to achieve a smooth vehicle path. The performance evaluation of the proposed method shows an internally and physically consistent trajectory reconstruction.

Optimal polynomial stability of a string with locally distributed Kelvin–Voigt damping and nonsmooth coefficient at the interface

The purpose of this paper is to investigate the stabilization of one‐dimensional wave equation with localized internal viscoelastic damping of Kelvin–Voigt type in a bounded domain. In this work, we consider only one damping Kelvin–Voigt mechanism acting in the internal of the body. Using frequency domain arguments combined with the multiplier method, we prove that the energy of the system has the optimal decay of type t−4.

Polynomial approach to cyclicity for weighted $$\ell ^p_A$$

In previous works, an approach to the study of cyclic functions in reproducing kernel Hilbert spaces has been presented, based on the study of so called optimal polynomial approximants. In the present article, we extend such approach to the (non-Hilbert) case of spaces of analytic functions whose Taylor coefficients are in $$\ell ^p(\omega )$$ , for some weight $$\omega $$ . When $$\omega =\{(k+1)^\alpha \}_{k\in \mathbb {N}}$$ , for a fixed $$\alpha \in \mathbb {R}$$ , we derive a characterization of the cyclicity of polynomial functions and, when $$1<p<\infty $$ , we obtain sharp rates of convergence of the optimal norms.

IGLU 2.0: A New Wearable for Accurate Non-Invasive Continuous Serum Glucose Measurement in IoMT Framework

To best of the authors knowledge, this article presents the first-ever non-invasive glucometer that takes into account serum glucose for high accuracy. In case of blood glucose measurement, serum glucose value has always been considered precise blood glucose value during prandial modes. Serum glucose can be measured in laboratory and more stable glucose level compare to capillary glucose. However, this invasive approach is not convenient for frequent measurement. Sometimes, Conventional invasive blood glucose measurement may be responsible for cause of trauma and chance of blood related infections. To overcome this issue, in the current paper, we propose a novel Internet-of-Medical (IoMT) enabled glucometer for non-invasive precise serum glucose measurement. In this work, a near-infrared (NIR) spectroscopic technique has been used for glucose measurement. The novel device called iGLU 2.0 is based on optical detection and precise machine learning (ML) regression models. The optimal multiple polynomial regression and deep neural network models have been presented to analyze the precise measurement. The glucose values of serum are saved on cloud through open IoT platform for endocrinologist at remote location. To validate iGLU 2.0, Mean Absolute Relative Difference (mARD) and Average Error (AvgE) are obtained 6.07% and 6.09%, respectively from predicted blood glucose values for capillary glucose. For serum glucose, mARD and AvgE are found 4.86% and 4.88%, respectively. These results represent that the proposed non-invasive glucose measurement device is more precise for serum glucose compared to capillary glucose.

Data-driven Arbitrary Polynomial Chaos Expansion on Uncertainty Quantification for Real-time Hybrid Simulation Under Stochastic Ground Motions

Uncertainties in real-time hybrid simulation include structural parameters and ground motion. Uncertain parameters often do not follow common distribution types. Data-driven arbitrary polynomial chaos constructs optimal orthogonal polynomial basis based on the sample data without distribution assumption. In this study, the data-driven polynomial chaos is compared with other generalized polynomial chaos from the aspects of the rate of error convergence when applied for uncertainty quantification of real-time hybrid simulation. Moreover, uncertainties of ground motion are considered in the RTHS problem to represent the scenarios with more complex input variables. Different statistical indicators are utilized to evaluate the accuracy of the alternative model in comparison with the Monte Carlo simulation results. Compared with generalized polynomial chaos, the data-driven arbitrary polynomial chaos presents potential for uncertainty quantification of real-time hybrid simulation with approximate or better accuracy. Actuator delay in RTHS could change the sensitivity of model output to the random variables.

Interval uncertain optimization for damping fluctuation of a segmented electromagnetic buffer under intensive impact load

Aiming at the problems of demagnetization effect of electromagnetic buffer (EMB) caused by high velocity under intensive impact load and the difficulty and error of machining composite thin-walled long tube, a segmented EMB is proposed. The inner tube and air-gap are divided into initial segments and the traversing segments. Through theoretical analysis, impact test and simulation, it can be found that the RRF curve has two peaks. Firstly, in order to reduce the resultant resistance force (RRF) peaks, the sensitivity analysis based on optimal Latin hypercube design (OLHD) and polynomial regression was performed. The results show that the smallest contribution ratio to the dynamic response is the seventh and ninth segments of the inner tube, which are less than 1%. Then, fully considering the uncertain factors, important parameters are selected for uncertain optimization after sensitivity analysis. The interval order and interval probability degree methods are used to establish interval uncertain optimization model of the RRF considering robustness. The model was solved using an interval nested optimization method based on radial basis function (RBF) neural network. Finally, the Pareto front is obtained and numerical simulation is performed to verify the optimal value. It indicates that the two kinds of RRF peak is obviously reduced, and the optimization object and strategy are effective.

Boundary Behavior of Optimal Polynomial Approximants

In this paper, we provide an efficient method for computing the Taylor coefficients of $$1-p_n f$$ , where $$p_n$$ denotes the optimal polynomial approximant of degree n to 1/f in a Hilbert space $$H^2_\omega $$ of analytic functions over the unit disc $$\mathbb {D}$$ , and f is a polynomial of degree d with d simple zeros. As a consequence, we show that in many of the spaces $$H^2_\omega $$ , the sequence $$\{1-p_nf\}_{n\in \mathbb {N}}$$ is uniformly bounded on the closed unit disc and, if f has no zeros inside $$\mathbb {D}$$ , the sequence $$\{1-p_nf \}$$ converges uniformly to 0 on compact subsets of the complement of the zeros of f in $$\overline{\mathbb {D}}, $$ and we obtain precise estimates on the rate of convergence on compacta. We also obtain the precise constant in the rate of decay of the norm of $$1 - p_n f$$ in the previously unknown case of a function with a single zero of multiplicity greater than 1, when the weights are given by $$\omega _k = (k+1)^{\alpha }$$ for $$\alpha \le 1$$ .

Discrete-binary configuration of metaheuristic algorithms for RFM optimization in the case of limited GCPs

Metaheuristic algorithms have been widely used in determining optimum rational polynomial coefficients (RPCs). By eliminating a number of unnecessary RPCs, these algorithms increase the accuracy of geometric correction of high-resolution satellite images. To this end, these algorithms use ordinary least squares and a number of ground control points (GCPs) to estimate RPCs. Due to the cost of GCP collection, using limited GCPs has become an attractive topic in various research studies. A configuration for RPC estimation using metaheuristic algorithms, namely, discrete-binary configuration for rational function model (DBRFM), is presented to find the optimal number and combination of RPCs in the case of limited GCPs. Based on the fact that the maximum number of RPCs is twice the number of GCPs, the particle/chromosome in the proposed configuration is composed of two binary and discrete parts. This configuration not only is compatible with the nature of the metaheuristic algorithms but also significantly reduces the search space. The proposed method has been tested on various types of remotely sensed data sets. The results of the experiments indicate the superiority of the DBRFM in comparison with the traditional binary configuration of metaheuristic algorithms.