Multidimensional Polynomial Research Articles

A stable computational approach to analyze semi‐relativistic behavior of fractional evolutionary problems

AbstractThe current work is related to the development of a hybrid semi‐spectral computational scheme based on multidimensional Chelyshkov polynomials (CPs). One dimensional traditional CPs have been efficiently converted to multidimensional Chelyshkov polynomials (MDCPs) and used to construct some novel operational matrices of fractional‐order derivatives. A method is developed via obtained polynomials as the Chelyshkov polynomial method (CPM) and further coupled with a finite difference scheme (FDM) and named as a semi‐spectral method (SSM). The method is used to evaluate the results of nonlinear evolutionary problems of fractional order where the space domain is collocated via CPM and the time variable is discretized through FDM. The stability analysis of the proposed scheme is proved to show its mathematical authenticity. Additionally, the reported study is evidence that the proposed methodology is reasonable in terms of accuracy, efficiency and low‐cost to tackle the nonlinear problem in higher dimensions while could be further modified for other class of dynamical problems.

UNIVARIATE AND MULTIVARIATE POLYNOMIAL REGRESSION CONSTRUCTION FROM A REDUNDANT REPRESENTATION USING AN ACTIVE EXPERIMENT

We consider the problem of a multidimensional polynomial regression construction from a given redundant representation based on the results of an active experiment. Redundant representation means inclusion in it the members which are possibly absent in the structure of the studied regression. Thus, we have a problem not only to estimate the values of the unknown coefficients of multidimensional polynomial regression from the results of an active experiment, but also to eliminate the redundant members from its redundant representation. The solution to this problem is based on: (a) obtaining new properties of the coefficients of normalized orthogonal polynomials of Forsythe; (b) possibility of reducing the problem of estimating the unknown coefficients for nonlinear members of multivariate polynomial regression to the problem of estimating the coefficients for the set of univariate polynomial regressions and solving the corresponding systems of linear equalities; (c) using the method to eliminate the redundant members of multidimensional nonlinear polynomial regression which organically includes both the methodology of cluster analysis and the main idea of the group method of data handling – dividing the experimental data into two sets, one of which is not used to estimate unknown coefficients of multidimensional polynomial regression given by a redundant representation.

Orthonormal Bases of Multidimensional Trigonometric Polynomials, Consisting of Translations of One of Them

Orthonormal bases of consecutive shifts of one polynomial are constructed in some spaces of multidimensional trigonometric polynomials. The methods for constructing Parseval frames of consecutive translations of a polynomial in wider classes of multidimensional trigonometric polynomials are proposed.

Dynamic response and sensitivity analysis for mechanical systems with clearance joints and parameter uncertainties using Chebyshev polynomials method

A mechanical system with clearance joints exhibits non-linear characteristics. Even a small variation of one parameter may lead to a drastic change of the overall system response. The coupling of clearance and uncertainty would therefore significantly influence the system performance. In this paper, an analysis method for dynamic response and parameter sensitivity of mechanical systems is proposed with the consideration of both clearance joints and uncertainties. In this method, elements of a revolute clearance joint are modeled as colliding bodies, where continuous contact force model and modified friction force model are employed to describe the impact-contact behavior. The clearance joint model is then coupled with the system motion equations to obtain the system dynamic response. Furthermore, by introducing the multi-dimensional Chebyshev polynomials approach, dependence of the system dynamic response on its parameters can be established. The bounds of the dynamic response could be obtained by using the interval operations, and the parameter sensitivity is interpreted as the rate of the change of the system dynamic response with respect to the parameter variation, revealing the contribution of each parameter on the dynamic performance. Finally, dynamics of a crank-slider mechanism with clearance and uncertainties is investigated. Results show that a larger clearance would cause severe oscillation in the response bounds and larger expansion of the interval length, and clearance effect restraint zones can be obtained through the sensitivity analysis, where the expansion of the response interval length could be reduced.

Uncertain Frequency Responses of Clamp-Pipeline Systems Using an Interval-Based Method

Due to manufacturing errors and material deteriorations in the metal rubber of clamps, the clamp stiffness of the pipeline is uncertain. This paper presents a non-intrusive multi-dimensional Chebyshev polynomial approximation method (M-CPAM) to evaluate uncertain characteristics of the frequency response function (FRF) of the clamp-pipeline system (CPS), where the clamp stiffness parameters are taken as unknown but bounded interval variables. Firstly, a finite element model of the clamp-pipeline system is established. Secondly, the variance-based global sensitivity analysis is implemented to determine significant stiffness parameters. Then, the uncertain intervals of the clamp stiffness are measured by experiments and the dispersion of the clamp stiffness is described. Finally, based on the measured stiffness interval, the uncertain frequency responses of the CPS under different tightening torques are analyzed by the proposed M-CPAM, and the effectiveness of simulation results is verified by experiments. Compared with the results obtained from the Monte Carlo simulation, the experimental measurements, and the polynomial chaos expansion, the proposed M-CPAM provides a more accurate, time-saving and practical method for solving the uncertain frequency responses of the CPS with interval stiffness variables. The results show that the clamp stiffness has great dispersion under the same tightening torque. A frequency shift phenomenon will be observed when the clamp stiffness is uncertain. Moreover, the dispersion of the frequency response of the CPS tends to be concentrated with the increase of the tightening torques.

On the multidimensional zeta functions associated with theta functions, and the multidimensional Appell polynomials

We define and study the multidimensional Appell polynomials associated with theta functions. For the trivial theta functions, we obtain the various well‐known Appell polynomials. Many other interesting examples are given. To push our study, by Mellin transform, we introduce and investigate the multidimensional zeta functions associated with thetas functions and prove that the multidimensional Appell polynomials are special values at the nonpositive integers of these zeta functions. Using zeta functions techniques, among others, we prove an induction formula for multidimensional Appell polynomials. The last part of this paper is devoted to spectral zeta functions and its generalization associated with Laplacians on compact Riemannian manifolds. From this generalization, we construct new Appell polynomials associated with Riemannan manifolds of finite dimensions.

Quantification of uncertainty in metallic elements subjected to fatigue

This investigation analyzes the propagation of the uncertainty of the number of life cycles to fatigue failure, taking into account the strain life method, considering as random variables some parameters of the material and loading in a steel plate under fatigue, with constant and variable loading through multidimensional Hermite polynomials. The application of series of multidimensional Hermite polynomials allowed the prediction of the randomness of the output vector. This research has shown that a multidimensional Hermite polynomial adequately estimates the propagation of the uncertainty of the input random variables. The results showed that variations in material and loading parameters can generate important failure probabilities and allowed quantifying the variability in the number of life cycles for fatigue failure steel parts

Stochastic Delay Characterization for Multicoupled RLC Interconnects Under Process Variations

The characterization of interconnect delay metrics in terms of process variations is an important but complicated task for statistical timing analysis in today’s integrated circuits (ICs). This paper presents a stochastic delay characterization framework for multicoupled interconnects in the presence of process variation. The proposed method starts with deriving the stochastic nodal equations for the RLC network that models multicoupled interconnects. By employing polynomial chaos (PC) expansion, a nonsampling-based stochastic prediction method, we further represent the voltage responses of network nodes as a series of orthogonal polynomials of random variables. During the expansion procedure, we use an adaptive approximation algorithm to reduce the number of required sampling points. We then use a stochastic collocation method to estimate the coefficients in the PC expansion model. With the voltage response determined as an expression of a multi-dimensional polynomial of random variables, the stochastic properties of the delay of multicoupled interconnects can be predicted. The proposed method not only takes into account the strong correlations among process variations, but also extracts an explicit delay representation for multicoupled interconnects in terms of process variations. Experimental results demonstrate that the delay characteristics predicted by the proposed method match well with the results by the brute-force Monte-Carlo method. Moreover, a significant speedup over the Monte-Carlo method has been achieved by the proposed delay characterization framework.

Multiband Linearization Technique for Broadband Signal With Multiple Closely Spaced Bands

Under the scenario of noncontiguous carrier aggregation with closely spaced multiple bands, the interband modulation products after power amplifying will overlap with the bands of interest. This leads to linearization performance degradation for the conventional three-band digital predistortion (DPD) method which only accounts for in-band intermodulation (IMD) and cross modulation. To resolve the spectral overlaps, a multiband linearization technique combining the direct power amplifier (PA) model extraction and the indirect DPD learning strategies is proposed. A multidimensional memory polynomial (MD-MP) model is first presented to jointly extract: 1) the in-band IMD; 2) the cross modulation; and 3) the interband modulation products from the observations for the multiple bands of interest to yield an improved frequency/time selective modeling. The extracted MD-MP models are then used in the indirect learning structure instead of the physically filtered observations to estimate the multiband DPD coefficients, given the fact that the observations spectrally overlap with each other which would result in interference for DPD coefficients estimation with direct filtering. Moreover, a novel MD cubic spline (MD-CS) basis is applied for the first time to the considered three-band scenario while the principal component analysis is used to alleviate any ill-conditioning problem during the model extraction. Improved PA modeling and DPD linearization performances are verified through experiments performed on three groups of wide-bandwidth noncontiguous aggregated signals covering a variety of application scenarios.

Multi-Dimensional Chebyshev Polynomials: A Non-Conventional Approach

Abstract Chebyshev polynomials are traditionally applied to the approximation theory where are used in polynomial interpolation and also in the study of di erential equations, in particular in some special cases of Sturm-Liouville di erential equation. Many of the operational techniques presented, by using suitable integral transforms, via a symbolic approach to the Laplace transform, allow us to introduce polynomials recognized belonging to the families of Chebyshev of multi-dimensional type. The non-standard approach come out from the theory of multi-index Hermite polynomials, in particular by using the concepts and the related formalism of translation operators.

Recurrent neural networks as approximators of non-linear filters operators

In cases that the mathematical model of a device is complicated or it can not be constructed because of the absence of sufficient information about a researched object, the approach based on the description of unique relationship between the sets of input and output signals is used. The point is the approximation of a non-linear operator, establishing the unique input-output mapping, by mathematical constructions (multidimensional polynomials, regression models, neural networks). Demand to reach the high accuracy of approximation often arises in practice. Recurrent neural networks possessing the properties of dynamics and nonlinearity are considered as mathematical models within the framework of the input-output approach. The non-linear filters synthesis is the constructing of mathematical models using the available sets of input and output signals. Non-linear filters serve for cancelling non-Gaussian noise from distorted signals. As an example, images distorted by the impulse noise is recovered with the help of non-linear filtering performed by different kinds of neural networks.

Variational estimates for discrete operators modeled on multi-dimensional polynomial subsets of primes

We prove the extensions of Birkhoff’s and Cotlar’s ergodic theorems to multi-dimensional polynomial subsets of prime numbers mathbb {P}^k. We deduce them from ell ^pbig (mathbb {Z}^dbig )-boundedness of r-variational seminorms for the corresponding discrete operators of Radon type, where p > 1 and r > 2.

Spurious weather effects

AbstractRainfall is exogenous to human actions and hence popular as an exogenous source of variation. But it is also spatially correlated. This can generate spurious relationships between rainfall and other spatially correlated outcomes. As an illustration, rainfall on almost any day of the year has seemingly high predictive power of electoral turnout in Norwegian municipalities. In Monte Carlo analyses, I find that standard tests reject true null hypotheses in as much as 99% of cases. Standard approaches to estimating consistent standard errors do not solve the problem. Instead, I suggest controlling for spatial and spatiotemporal trends using multidimensional polynomials.

Degenerate Bernstein polynomials

Here we consider the degenerate Bernstein polynomials as a degenerate version of Bernstein polynomials, which are motivated by Simsek’s recent work ‘Generating functions for unification of the multidimensional Bernstein polynomials and their applications’ (Simsek in Filomat 30(7):1683–1689, 2016, Math Methods Appl Sci 1–12, 2018) and Carlitz’s degenerate Bernoulli polynomials. We derived their generating function, symmetric identities, recurrence relations, and some connections with generalized falling factorial polynomials, higher-order degenerate Bernoulli polynomials and degenerate Stirling numbers of the second kind.

Multidimensional Bernstein polynomials and Bezier curves: Analysis of machine learning algorithm for facial expression recognition based on curvature

In this paper, by using partial derivative formulas of generating functions for the multidimensional unification of the Bernstein basis functions and their functional equations, we derive derivative formulas and identities for these basis functions and their generating functions. We also give a conjecture and some open questions related to not only subdivision property of these basis functions, but also solutions of a higher-order special differential equations. Moreover, we provide an implementation for a real world problem of human facial expression recognition with the help of curvature of Bezier curves whose machine learning supported by statistical evaluations on feature vectors using in the aforementioned machine learning algorithm.

Low Latency Area-Efficient Distributed Arithmetic Based Multi-Rate Filter Architecture for SDR Receivers

In software defined radio (SDR) receivers, sample rate conversion (SRC) and channelization are two computational intensive tasks. Coefficient-less cascaded-integrator-comb (CIC) filters achieve SRC with low computational complexity, but the design of its gain droop compensation filter involves coefficients. These coefficients vary with the change in radio standards. In this paper, an architecture for variable digital filter (VDF) for gain droop compensation employing a set of fixed coefficient sub-filters and multi-dimensional polynomials in terms of spectral parameters is realized based on distributed arithmetic (DA). As the coefficients in the sub-filters are fixed, the proposed method uses ROM-based LUTs giving rise to low computational complexity. The proposed DA–VDF filter is synthesized on an application specific integrated circuit (ASIC) employing CMOS 90[Formula: see text]nm technology using Synopsis Design Complier. The proposed architecture achieves low latency at a reduced area delay product (ADP) of 78% and an efficiency of 72% in energy per sample (EPS) when compared with the conventional MAC-based architecture.

Generating functions for unification of the multidimensional Bernstein polynomials and their applications

The aim of this paper is to construct generating functions for m‐dimensional unification of the Bernstein basis functions. We give some properties of these functions. We also give derivative formulas and a recurrence relation of the m‐dimensional unification of the Bernstein basis functions with help of their generating functions. By combining the m‐dimensional unification of the Bernstein basis functions with m variable functions on simplex and cube, we give m‐dimensional unification of the Bernstein operator. Furthermore, by applying integrals method including the Riemann integral, the q‐integral, and the p‐adic integral to some identities for the (q‐) Bernstein basis functions, we derive some combinatorial sums including the Bernoulli numbers and Euler numbers and also the Stirling numbers and the Cauchy numbers (the Bernoulli numbers of the second kind).

Numerical Procedures for Random Differential Equations

Some methodological approaches based on generalized polynomial chaos for linear differential equations with random parameters following various types of distribution laws are proposed. Mainly, an internal random coefficients method ‘IRCM’ is elaborated for a large number of random parameters. A procedure to build a new polynomial chaos basis and a connection between the one-dimensional and multidimensional polynomials are developed. This allows handling easily random parameters with various laws. A compact matrix formulation is given and the required matrices and scalar products are explicitly presented. For random excitations with an arbitrary number of uncertain variables, the IRCM is couplet to the superposition method leading to successive random differential equations with the same main random operator and right-hand sides depending only on one random parameter. This methodological approach leads to equations with a reduced number of random variables and thus to a large reduction of CPU time and memory required for the numerical solution. The conditional expectation method is also elaborated for reference solutions as well as the Monte-Carlo procedure. The applicability and effectiveness of the developed methods are demonstrated by some numerical examples.

Control of Systems Exhibiting Input Multiplicities using Dual Nonlinear MPC

Control of a system exhibiting input multiplicity at an optimum (singular) operating point poses a challenging control problem due to loss of invertibility and change in the sign of the steady state gain in the neighbourhood of the optimum. In this work, for controlling systems exhibiting input multiplicities, we develop a novel adaptive dual NMPC (ADNMPC) formulation based on a Wiener model which is parameterized using orthonormal basis filters (OBF). The static nonlinear output map in the Wiener model is constructed using multidimensional quadratic polynomials. The OBF poles are chosen through an off-line identification exercise and the parameters of the static nonlinear map are updated on-line using recursive least squares algorithm. Similar to Kumar et al. (2017), by introducing concept of excitation horizon, the objective function in the NMPC formulation is modified to include terms that are sensitive to the parameter covariance. The proposed formulation provides su¢cient degrees of freedom to shape the probing signals. E¢cacy of the proposed approach is demonstrated by simulating problem of controlling a continuously operated fermenter system at its optimum operating point. Analysis of the simulation results shows that the proposed ADNMPC scheme judiciously injects perturbations into the process as and when required. The probing perturbations subside when the parameter estimates stabilize. In particular, it is observed that the intensity of the perturbation increase as the excitation horizon increases. The proposed on-line model adaptation also ensures better control performance with reference to a non-adaptive NMPC controller.

Multidimensional realisation theory and polynomial system solving

ABSTRACTMultidimensional systems are becoming increasingly important as they provide a promising tool for estimation, simulation and control, while going beyond the traditional setting of one-dimensional systems. The analysis of multidimensional systems is linked to multivariate polynomials, and is therefore more difficult than the well-known analysis of one-dimensional systems, which is linked to univariate polynomials. In the current paper, we relate the realisation theory for overdetermined autonomous multidimensional systems to the problem of solving a system of polynomial equations. We show that basic notions of linear algebra suffice to analyse and solve the problem. The difference equations are associated with a Macaulay matrix formulation, and it is shown that the null space of the Macaulay matrix is a multidimensional observability matrix. Application of the classical shift trick from realisation theory allows for the computation of the corresponding system matrices in a multidimensional state-space setting. This reduces the task of solving a system of polynomial equations to computing an eigenvalue decomposition. We study the occurrence of multiple solutions, as well as the existence and analysis of solutions at infinity, which allow for an interpretation in terms of multidimensional descriptor systems.