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Modeling, prediction, and recognition tasks depend on the proper representation of the objective curves and surfaces. Polynomial functions have been proved to be a powerful tool for representing curves and surfaces. Until now, various methods have been used for polynomial fitting. With a recent boom in neural networks, researchers have attempted to solve polynomial fitting by using this end-to-end model, which has a powerful fitting ability. However, the current neural network-based methods are poor in stability and slow in convergence speed. In this article, we develop a novel neural network-based method, called Encoder-X, for polynomial fitting, which can solve not only the explicit polynomial fitting but also the implicit polynomial fitting. The method regards polynomial coefficients as the feature value of raw data in a polynomial space expression and therefore polynomial fitting can be achieved by a special autoencoder. The entire model consists of an encoder defined by a neural network and a decoder defined by a polynomial mathematical expression. We input sampling points into an encoder to obtain polynomial coefficients and then input them into a decoder to output the predicted function value. The error between the predicted function value and the true function value can update parameters in the encoder. The results prove that this method is better than the compared methods in terms of stability, convergence, and accuracy. In addition, Encoder-X can be used for solving other mathematical modeling tasks.

In this paper, we introduce the $ q $-analogus of generalized differential operator involving $ q $-Mittag-Leffler function in open unit disk \begin{document}$ \begin{equation*} E = \left \{ z:z\in \mathbb{C\ \ }\text{ and} \ \ \left \vert z\right \vert and define new subclass of analytic and bi-univalent functions. By applying the Faber polynomial expansion method, we then determined general coefficient bounds $ |a_{n}| $, for $ n\geq 3 $. We also highlight some known consequences of our main results.

Cutting force is essential for studying cutting mechanism and improving cutting process. In order to predict cutting force, the cutting force was measured by experiments; cutter runout was measured by dial indicator; the actual undeformed cutting thickness was obtained in presence of cutter runout. Then a mechanical cutting force model was established. Three kinds of cutting force coefficients, linear, exponential, and polynomial coefficients, are solved and compared. The results show that the exponential model has the highest generality and accuracy. The validation experiment illustrates deviation rates between the predicted values and measured values are mostly

A P-recursive sequence is a sequence which satisfies a linear recurrence relation with polynomial coefficients. For a P-recursive sequence {an}n≥0 and its power {anm}n≥0, we give a method to constr...

The aim of the paper is to find polynomial parametric equations of rectifying Salkowski curves in Minkowski 3-space, via a serial approach. These curves are characterized by according to their curvature; in particular those curves with constant curvature functions and linear harmonic curvature functions are fully characterized. Then, the equations of the rectifying Salkowski curves are obtained as serial solutions of differential equations with third-order polynomial coefficients.

The precision in predicting the Earth Orientation Parameters by the least squares method is statistically analyzed. The prediction involves assessment of the polynomial coefficients approximating the evolution of the polar motion and the irregularity of Earth’s rotation. Optimal approximating polynomials and approximation intervals corresponding to the greatest statistical accuracy of prediction over the specified time interval are determined. The proposed algorithms are analyzed on the basis of historical data regarding the evolution of the Earth Orientation Parameters over many years, and the accuracy levels attained in prediction are identified.

This paper is devoted to the matrix representation of ordinary symmetric differential operators with polynomial coefficients on the whole axis. We prove that in this case, generalized Jacobi matrices appear. We examine the problem of defect indexes for ordinary differential operators and generalized Jacobi matrices corresponding to them in the spaces L2(−∞,+∞) and l2, respectively, and analyze the spectra of self-adjoint extensions of these operators (if they exist). This method allows one to detect new classes of entire differential operators of minimal type (in the sense of M. G. Krein) with certain defect numbers. In this case, the defect numbers of these operators can be not only less than or equal, but also greater than the order of the corresponding differential expressions. In particular, we construct examples of entire differential operators of minimal type that are generated by irregular differential expressions.

The mass surface of odd nuclei is studied in the form of second-order polynomials in deviations from N and Z of an odd nucleus. It is shown that the polynomial coefficients depend on the group of atomic nuclei in the vicinity of N and Z according to which these coefficients are determined.

In this paper, we introduce and investigate two new subclasses of analytic and bi-univalent functions defined in the open unit disc. We use the Faber polynomial expansions to find upper bounds for the $n$th$~ n\geq 3 $ Taylor-Maclaurin coefficients $\left\vert a_{n}\right\vert $ of functions belong to these new subclasses with $a_{k}=0$ for $2\leq k\leq n-1$, also we find non-sharp estimates on the first two coefficients $\left\vert a_{2}\right\vert $ and $\left\vert a_{3}\right\vert $. The results, which are presented in this paper, would generalize those in related earlier works of several authors.

In this paper, we obtain the upper bounds for the n-th (n ≥ 1) coefficients for meromorphic bi-subordinate functions of complex order by using Faber polynomial expansions. The results, which are presented in this paper, would generalize those in related works of several earlier authors.

To share the same visual content between color vision deficiencies (CVD) and normal-vision people, attempts have been made to allocate the two visual experiences of a binocular display (wearing and not wearing glasses) to CVD and normal-vision audiences. However, existing approaches only work for still images. Although state-of-the-art temporal filtering techniques can be applied to smooth the per-frame generated content, they may fail to maintain the multiple binocular constraints needed in our applications, and even worse, sometimes introduce color inconsistency (same color regions map to different colors). In this paper, we propose to train a neural network to predict the temporal coherent polynomial coefficients in the domain of global color decomposition. This indirect formulation solves the color inconsistency problem. Our key challenge is to design a neural network to predict the temporal coherent coefficients, while maintaining all required binocular constraints. Our method is evaluated on various videos and all metrics confirm that it outperforms all existing solutions.

In this paper, a simple and computationally efficient approach is proposed to predict the cement strength. It is based on the mathematical concept of covariance matrix and polynomial coefficients. ...

On établit une décomposition de l’homologie stable des groupes d’automorphismes des groupes libres à coefficients polynomiaux contravariants en termes d’homologie des foncteurs. Elle permet plusieurs calculs explicites, qui recoupent des résultats établis de manière indépendante par O. Randal-Williams et généralisent certains d’entre eux. Nos méthodes reposent sur l’examen d’extensions de Kan dérivées associées à plusieurs catégories de groupes libres, la généralisation d’un critère d’annulation homologique à coefficients polynomiaux dû à Scorichenko, le théorème de Galatius identifiant l’homologie stable des groupes d’automorphismes des groupes libres à celle des groupes symétriques, la machinerie des $\unicode[STIX]{x1D6E4}$-espaces et le scindement de Snaith.

The aim of this paper is twofold. First, we propose a q-analogue of polynomial coeffcients , associated with the vector a = (a0, … , as), which are defined as the k-th coefficients of (a0 +...

We prove that the sequence $$\left( e_n^{\Gamma }\right) _{n\in \mathbb {N}}$$ of numbers of excursions in a three-quadrant cone corresponding to a non-singular step set $$\Gamma \subset \{0, \pm 1\}^2$$ with an infinite group does not satisfy any nontrivial linear recurrence with polynomial coefficients. This gives a positive answer to a question asked by M. Bousquet-Melou about D-finiteness of the trivariate generating function of the numbers of walks with given length and prescribed ending point. In the process, we determine the asymptotics of $$e_n^{\Gamma }$$ , $$n\rightarrow \infty $$ , for the 74 non-singular two-dimensional models, giving the first complete computation of excursions asymptotics in a non-convex cone. Moreover, using potential theoretic comparison arguments, we give the asymptotics of the number of walks avoiding the negative quadrant and of length n for all non-singular step sets having zero drift.

Integral formulae for the coefficients of cyclotomic and polygonal polynomials were recently obtained in [2] and [3]. In this paper, we define and study a family of polynomials depending on an integer sequence m1,?, mn,?, and on a sequence of complex numbers z1,?, zn, ? of modulus one. We investigate some particular instances such as: extended cyclotomic, extended polygonal-type, and multinomial polynomials, for which we obtain formulae for the coefficients. Some novel related integer sequences are also derived.

The problem of characterizing the sensitivity indices of structural performance considering random and interval uncertainty variables simultaneously is analyzed. Data-driven polynomial chaos expansion (PCE) is established to treat random variables with arbitrary input uncertainty information, and subsequently extended to consider interval uncertainty variables simultaneously. An adaptive-sparse polynomial chaos expansion calculation algorithm is proposed to determine the appropriate polynomial coefficients based on the available uncertainty information of random and interval uncertainty variables. Subsequently, the local and total sensitivity indices of random and interval uncertainty variables are calculated directly using the constructed PCE model. Two numerical and engineering examples are presented to demonstrate the effectiveness of the proposed approach. Compared with the results of the Monte Carlo simulation method, the proposed approach yields excellent results with significantly reduced computational effort in the three examples.

This paper reports the algebraic derivations of the quintic polynomial equation whose solution gives the coordinates of generalized Burmester points (GBPs). Denoted with λ1 and λ2 the ratios of the first and second rate of change of curvature to curvature, respectively, the paths traced by GBPs have prescribed values of such ratios. When λ 1 = λ 2 = 0 , GBPs reduce to the (four) classical Burmester points. Our derivations, based on the properties of Cesaro’s intrinsic geometry, led to a concise algebraic form of such polynomial coefficients. This availability allows expanding the field of application of Bottema’s instantaneous invariants in higher-order mechanical approximation of any algebraic or parametric curve.

Finding graph measures with high discrimination power has been triggered by searching for so-called complete graph invariants. In a series of papers, we have already investigated highly discriminating measures to distinguish graphs (networks) based on their topology. In this paper, we propose an approach where the graph measures are based on the roots of random graph polynomials. The polynomial coefficients have been defined by utilizing information functionals which capture structural information of the underlying networks. Our numerical results obtained by employing exhaustively generated graphs reveal that the new approach outperforms earlier results in the literature.

This paper is a survey of the recent results of the author on the complex zeros of solutions to linear homogeneous second order ordinary differential equations with polynomial coefficients. In particular, estimates for the sums and products of the zeros are derived. These estimates give us bounds for the function counting the zeros of solutions and information about the zero-free domain. Some other applications of the obtained estimates for the sums and products of the zeros are also discussed. In addition, we investigate the variation of the zeros of solutions under perturbations of the coefficients. Illustrative examples are also presented. A part of the results presented in the paper is new.

In this chapter we detail more general aspects of ACSV. To begin, we combine techniques from Chapter 5 (ACSV in the smooth case) and Chapter 8 (ACSV when the singular set is a hyperplane arrangement) to study functions whose singular sets are locally hyperplane arrangements; asymptotics are determined through the use of multivariate residues, leading to Theorems 9.1 and 9.2. Following this we survey current aspects of ACSV, including new work making use of tools from topology, leading up to Theorem 9.3 on the asymptotic contributions of (potentially non-minimal) critical points. Combining these topological methods with the computational results for D-finite functions discussed in Chapter 2 gives a powerful approach to the connection problem for asymptotics of sequences satisfying linear recurrence relations with polynomial coefficients.

In this paper we study regularity of partial differential equations with polynomial coefficients in non isotropic Beurling spaces of ultradifferentiable functions of global type. We study the action of transformations of Gabor and Wigner type in such spaces and we prove that a suitable representation of Wigner type allows to prove regularity for classes of operators that do not have classical hypoellipticity properties.

Described sensor TN-485 measurement error minimization during the primary and periodical calibration method is. Provided examples for calculating calibrations coefficients, calculating sensor measurement error, approximating errors function using the second order polynomial, calculating the polynomial coefficients, calculating sensor measurement error after the calibration.

Direction-of-arrival estimation of non-Gaussian signals is studied in this paper by incorporating fourth-order cumulant with an array of arbitrary configurations. The fourth-order cumulant based closed form polynomial coefficients have been provided. The corresponding higher-order (more than fourth-order) statistics based method can also be derived similarly. The computational complexity analysis is also given. Simulation results are included to illustrate the performance of the proposed method.

We describe the resurgence properties of some partition functions corresponding to field theories in dimension 0. We show that these functions satisfy linear differential equations with polynomial coefficients and then use elementary stability results or holonomic functions to prove resurgence properties, enhancing previously known results on growth estimates for the formal series involved, which had been obtained through a delicate combinatorics., Comment: 38 pages, 7 figures

We give a short survey of result on continuous (resp. continuous semialgebraic orregulous) solutions of linear equations with polynomial coefficients

As a rule, search algorithms for those solutions of differential equations and systems that belong to a fixed class of functions are designed so that nonexistence of solutions of the desired type is detected only in the last stages of the algorithm. In some cases, performing additional tests on the intermediate results makes it possible to stop the algorithm as soon as these tests imply that no solutions of the desired type exist. We will consider these questions in connection with the search for rational solutions of linear homogeneous differential systems with polynomial coefficients. (Some approaches are already known for the case of scalar equations.)

This paper is devoted to the study of second order linear difference equations with polynomial coefficients subject to multipoint boundary conditions. We provide necessary and sufficient conditions for the existence and uniqueness of solutions and find the unique solution in closed form by using factorization techniques.