Integral Polynomials Research Articles

Conditions for tractability of the weighted $$L_p$$-discrepancy and integration in non-homogeneous tensor product spaces

AbstractWe study tractability properties of the weighted $$L_p$$ L p -discrepancy, where the weights model the influence of different coordinates. A discrepancy is said to be tractable if the minimal number N of points, such that the discrepancy is less than the initial discrepancy times an error threshold $$\varepsilon $$ ε , does not grow exponentially fast with the dimension and $$\varepsilon ^{-1}$$ ε - 1 . In this case there are various notions of tractabilities used in order to classify the exact rate. In the present paper we prove matching sufficient conditions (upper bounds) and neccessary conditions (lower bounds) for polynomial and weak tractability for all $$p \in (1, \infty )$$ p ∈ ( 1 , ∞ ) and we prove a necessary condition for strong polynomial tractability. The proofs of the lower bounds are based on a powerful general result for the information complexity of integration with positive quadrature rules for tensor product spaces. As a second application of this general result we consider the integration of polynomials of degree at most 2.

Polynomial dynamical systems associated with the KdV hierarchy

In 1974, S.P. Novikov introduced the stationary n-equations of the Korteweg–de Vries hierarchy, namely the n-Novikov equations. These are associated with integrable polynomial dynamical systems, with polynomial 2n integrals, in ℂ3n. In this paper, we construct an infinite-dimensional polynomial dynamical system that is universal for all dynamical systems corresponding to the n-Novikov equations. Thus, we solve the well-known problem of the relationship between the n-Novikov equations for different n.

A subspace‐adaptive weights cubature method with application to the local hyperreduction of parameterized finite element models

AbstractThis article is concerned with quadrature/cubature rules able to deal with multiple subspaces of functions, in such a way that the integration points are common for all the subspaces, yet the (nonnegative) weights are tailored to each specific subspace. These subspace‐adaptive weights cubature rules can be used to accelerate computational mechanics applications requiring efficiently evaluating spatial integrals whose integrand function dynamically switches between multiple pre‐computed subspaces. One of such applications is local hyperreduced‐order modeling (HROM), in which the solution manifold is approximately represented as a collection of basis matrices, each basis matrix corresponding to a different region in parameter space. The proposed optimization framework is discrete in terms of the location of the integration points, in the sense that such points are selected among the Gauss points of a given finite element mesh, and the target subspaces of functions are represented by orthogonal basis matrices constructed from the values of the functions at such Gauss points, using the singular value decomposition (SVD). This discrete framework allows us to treat also problems in which the integrals are approximated as a weighted sum of the contribution of each finite element, as in the energy‐conserving sampling and weighting method of C. Farhat and co‐workers. Two distinct solution strategies are examined. The first one is a greedy strategy based on an enhanced version of the empirical cubature method (ECM) developed by the authors elsewhere (we call it the subspace‐adaptive weights ECM, SAW‐ECM for short), while the second method is based on a convexification of the cubature problem so that it can be addressed by linear programming algorithms. We show in a toy problem involving integration of polynomial functions that the SAW‐ECM clearly outperforms the other method both in terms of computational cost and optimality. On the other hand, we illustrate the performance of the SAW‐ECM in the construction of a local HROMs in a highly nonlinear equilibrium problem (large strains regime). We demonstrate that, provided that the subspace‐transition errors are negligible, the error associated to hyperreduction using adaptive weights can be controlled by the truncation tolerances of the SVDs used for determining the basis matrices. We also show that the number of integration points decreases notably as the number of subspaces increases, and that, in the limiting case of using as many subspaces as snapshots, the SAW‐ECM delivers rules with a number of integration points only dependent on the intrinsic dimensionality of the solution manifold and the degree of overlapping required to avoid subspace‐transition errors. The Python source codes of the proposed SAW‐ECM are openly accessible in the public repository https://github.com/Rbravo555/localECM.

Energy Element Method for Three-Dimensional Vibration Analysis of Stiffened Plates with Complex Geometries

A novel numerical method, the energy element method, is proposed for the three-dimensional vibration analysis of stiffened plates with complex geometries. The problem is modeled in a cuboidal domain, and a cuboidal energy element is developed for the simulation of the structural components. To simulate the energy distribution of the structure, stiffened plates are treated as discrete energy systems, where variable stiffness is used to characterize their energy distribution in a cuboidal domain and a global admissible function is used to approximate their vibrational behavior. With extended interval integral, Gauss quadrature, variable stiffness, and Legendre polynomials used for numerical integration in the cuboidal domain, the cuboidal energy element can offer sufficient precision with sufficient Gauss points for simulating the strain energy of stiffened plates with complex geometries. The Gauss–Legendre quadrature will provide accurate integration results if the integral domain is cuboidal. Otherwise, small cuboidal energy elements are generated with denser Gauss points to capture the geometric boundaries. As the numerical model is constructed on a standard geometric domain, all energy functionals and computational procedures are standard. Three-dimensional vibration problems are investigated for variously shaped stiffened plates with straight or curvilinear stiffeners. The present results are compared with analytical, numerical, and experimental results published in the literature.

A note on the squarefree density of polynomials

AbstractThe conjectured squarefree density of an integral polynomial in variables is an Euler product which can be considered as a product of local densities. We show that a necessary and sufficient condition for to be 0 when is a polynomial in variables over the integers, is that either there is a prime such that the values of at all integer points are divisible by or the polynomial is not squarefree as a polynomial. We also show that generally the upper squarefree density satisfies .

Investigation of the accuracy of integral models of the oculo-motor system

For mathematical modeling of the human oculomotor system (OMS), integral nonlinear models are used, which simultaneously take into account the nonlinear and inertial properties of the research object. Based on the data of experimental studies of the OMS "input-output", transient and diagonal intersections of transient functions of the second and third orders are determined. To obtain experimental data, an innovative eye tracking technology is used, which allows recording eye responses to test visual stimuli. Thus test signals are displayed on the computer monitor at different distances from the starting position in the horizontal direction. The aim of the research is to study the accuracy of OMS identification using eye-tracking data by evaluating the calculation errors of multidimensional transient functions when using methods of nonlinear dynamic identification based on models in the form of Volterra series and polynomials. The object of the study is the process of nonparametric identification of the OMS based on Volterra models in the time domain. The subject of the research is algorithmic and software tools for calculating the dynamic characteristics of OMS based on eye-tracking data, analyzing the accuracy of the obtained models using two identification methods: the approximation method and the least squares method (LSM). The means of nonlinear dynamic identification of the human OMS based on Volterra series and polynomials were developed in the Python programming environment. The accuracy estimates of various OMS (linear, quadratic, and cubic) models were obtained based on data from three responses to test signals of different amplitudes. For the same test signals, the same models in the form of the Volterra series and polynomials were obtained, as these models coincide in the region of convergence of the Volterra series. The analysis of errors in the assessment of the dynamic characteristics of the OMS demonstrated that the model in the form of an integral polynomial of the second degree, which was built using the LSM based on three responses, has an accuracy twice as high as the accuracy of similar models built using the data of two responses. Thus, in further studies of the psychophysiological state of a person based on nonlinear dynamic models of the OMS according to the data of three responses, it is advisable to use a model in the form of a quadratic Volterra polynomial. Figs.: 17. Tabl. 3. Refs.: 10 titles.

Уравнения Лакса на супералгебрах Ли

It is demonstrated that the standard construction of Lax equations on Lie algebras can be extended to Lie superalgebras, with the even subspace carrying the usual Lax equations. The extended equations inherit the existence of the canonical trace polynomial integrals of motion. An extra set of integrals exists in the odd subspace, with a nontrivial homological structure of the orbit space. This establishes a curious algebraic link between integrable evolution equations, supersymmetry and the deformation theory.

A discrete spectral method for time fractional fourth-order 2D diffusion-wave equation involving [formula omitted]-Caputo fractional derivative

In this work, the ψ-Caputo fractional derivative, as a generalization of the classical Caputo fractional derivative in which the fractional derivative of a sufficiently differentiable function is defined with respect to another strictly increasing function, ψ, is utilized to define the time fractional fourth-order 2D diffusion-wave equation. A Chebyshev–Gauss–Lobatto scheme is developed to solve this equation. In this way, a new operational matrix for the ψ-Riemann–Liouville fractional integration of the Chebyshev polynomials is derived. The solution of the equation is obtained by determining the solution of the algebraic system extracted from approximation the ψ-Caputo fractional derivative term via a finite discrete Chebyshev series and employing the expressed operational matrix. The validity of the established approach is examined by solving two examples.

Fourier–Gegenbauer pseudospectral method for solving periodic fractional optimal control problems

This paper introduces a new, highly accurate model for periodic fractional optimal control problems (PFOCPs) based on Riemann–Liouville and Caputo fractional derivatives (FDs) with sliding fixed memory lengths. This new model enables periodic solutions in fractional-order control models by preserving the periodicity of the fractional derivatives. Additionally, we develop a novel numerical solution method based on Fourier and Gegenbauer pseudospectral approaches. Our method simplifies PFOCPs into constrained nonlinear programming problems (NLPs), readily solvable by standard NLP solvers. Some key innovations of this study include: (i) The derivation of a simplified FD calculation formula to transform periodic FD calculations into evaluations of trigonometric Lagrange interpolating polynomial integrals, efficiently computed using Gegenbauer quadratures. (ii) The introduction of the αth-order fractional integration matrix based on Fourier and Gegenbauer pseudospectral approximations, which proves to be highly effective for computing periodic FDs. A priori error analysis is provided to predict the accuracy of the Fourier–Gegenbauer-based FD approximations. Benchmark PFOCP results validate the performance of the proposed pseudospectral method.

Square-free values of random polynomials

The question of whether or not a given integral polynomial takes infinitely many square-free values has only been addressed unconditionally for polynomials of degree at most 3. We address this question, on average, for polynomials of arbitrary degree.

Non-polynomial q-Askey Scheme: Integral Representations, Eigenfunction Properties, and Polynomial Limits

AbstractWe construct a non-polynomial generalization of the q-Askey scheme. Whereas the elements of the q-Askey scheme are given by q-hypergeometric series, the elements of the non-polynomial scheme are given by contour integrals, whose integrands are built from Ruijsenaars’ hyperbolic gamma function. Alternatively, the integrands can be expressed in terms of Faddeev’s quantum dilogarithm, Woronowicz’s quantum exponential, or Kurokawa’s double sine function. We present the basic properties of all the elements of the scheme, including their integral representations, joint eigenfunction properties, and polynomial limits.

Implementation of Polynomial Functions to Improve the Accuracy of Machine Learning Models in Predicting the Corrosion Inhibition Efficiency of Pyridine-Quinoline Compounds as Corrosion Inhibitors

Historically, the exploration of corrosion inhibitor technology has relied extensively on experimental methodologies, which are inherently associated with substantial costs, prolonged durations, and significant resource utilization. However, the emergence of ML approaches has recently garnered attention as a promising avenue for investigating potential materials with corrosion inhibition properties. This study endeavors to enhance the predictive capacity of ML models by leveraging polynomial functions. Specifically, the investigation focuses on assessing the effectiveness of pyridinequinoline compounds in mitigating corrosion. Diverse ML models were systematically evaluated, integrating polynomial functions to augment their predictive capabilities. The integration of polynomial functions notably amplifies the predictive accuracy across all tested models. Notably, the SVR model emerges as the most adept, exhibiting R² of 0.936 and RMSE of 0.093. The outcomes of this inquiry underscore a significant enhancement in predictive accuracy facilitated by the incorporation of polynomial functions within ML models. The proposed SVR model stands out as a robust tool for prognosticating the corrosion inhibition potential of pyridine-quinoline compounds. This pioneering approach contributes invaluable insights into advancing machine learning methodologies geared toward designing and engineering materials with promising corrosion inhibition properties.
 Keywords: machine learning, polynomial, corrosion inhibition, pyridine-quinoline

Solving polynomial systems over non-fields and applications to modular polynomial factoring

We study the problem of solving a system of m polynomials in n variables over the ring of integers modulo a prime-power pk. The problem over finite fields is well studied in varied parameter settings. For small characteristic p=2, Lokshtanov et al. (SODA'17) initiated the study, for degree d=2 systems, to improve the exhaustive search complexity of O(2n)⋅poly(m,n) to O(20.8765n)⋅poly(m,n); which currently is improved to O(20.6943n)⋅poly(m,n) in Dinur (SODA'21). For large p but constant n, Huang and Wong (FOCS'96) gave a randomized poly(d,m,log⁡p) time algorithm. Note that for growing n, system-solving is known to be intractable even with p=2 and degree d=2.We devise a randomized poly(d,m,log⁡p)-time algorithm to find a root of a given system of m integral polynomials of degrees bounded by d, in n variables, modulo a prime power pk; when n+k is constant. In a way, we extend the efficient algorithm of Huang and Wong (FOCS'96) for system-solving over Galois fields (i.e., characteristic p) to system-solving over Galois rings (i.e., characteristic pk); when k>1 is constant. The challenge here is to find a lift of singularFp-roots (exponentially many); as there is no efficient general way known in algebraic-geometry for resolving singularities.Our algorithm has applications to factoring univariate polynomials over Galois rings. Given f∈Z[x] and a prime-power pk (k≥2), finding factors of fmodpk has a curious state-of-the-art. It is solved for large k by p-adic factoring algorithms (von zur Gathen, Hartlieb, ISSAC'96); but unsolved for small k. In particular, no nontrivial factoring method is known for k≥5 (Dwivedi, Mittal, Saxena, ISSAC'19). One issue is that degree-δ factors of f(x)modpk could be exponentially many, as soon as k≥2. We give the first randomized poly(deg⁡(f),log⁡p)-time algorithm to find a degree-δ factor of f(x)modpk, when k+δ is constant. Our method has potential application in algebraic coding theory. In particular, extending algebraic geometric and Reed-Solomon codes to Galois rings could enable new and improved bounds on their underlying efficiency parameters.

Study of PARP inhibitors for breast cancer based on enhanced multiple kernel function SVR with PSO.

PARP1 is one of six enzymes required for the highly error-prone DNA repair pathway microhomology-mediated end joining (MMEJ) and needs to be inhibited when over-expressed. In order to study the PARP1 inhibitory effect of fused tetracyclic or pentacyclic dihydrodiazepinoindolone derivatives (FTPDDs) by quantitative structure-activity relationship technique, six models were established by four kinds of methods, heuristic method, gene expression programming, random forester, and support vector regression with single, double, and triple kernel function respectively. The single, double, and triple kernel functions were RBF kernel function, the integration of RBF and polynomial kernel functions, and the integration of RBF, polynomial, and linear kernel functions respectively. The problem of multi-parameter optimization introduced in the support vector regression model was solved by the particle swarm optimization algorithm. Among the models, the model established by support vector regression with triple kernel function, in which the optimal R 2 and RMSE of training set and test set were 0.9353, 0.9348 and 0.0157, 0.0288, and R2 cv of training set and test set were 0.9090 and 0.8971, shows the strongest prediction ability and robustness. The method of support vector regression with triple kernel function is a great promotion in the field of quantitative structure-activity relationship, which will contribute a lot to designing and screening new drug molecules. The information contained in the model can provide important factors that guide drug design. Based on these factors, six new FTPDDs have been designed. Using molecular docking experiments to determine the properties of new derivatives, the new drug was ultimately successfully designed.

Quartic integral polynomial Pell equations

In this paper we classify all monic, quartic, polynomials d(x)∈Z[x] for which the Pell equationf(x)2−d(x)g(x)2=1 has a non-trivial solution with f(x),g(x)∈Z[x].

Explicit calculations for Sono’s multidimensional sieve of 𝐸₂-numbers

We derive explicit formulas for integrals of certain symmetric polynomials used in Keiju Sono’s multidimensional sieve of E 2 E_2 -numbers, i.e., integers which are products of two distinct primes. We use these computations to produce the currently best-known bounds for gaps between multiple E 2 E_2 -numbers. For example, we show there are infinitely many occurrences of four E 2 E_2 -numbers within a gap size of 94 94 unconditionally and within a gap size of 32 32 assuming the Elliott-Halberstam conjecture for primes and E 2 E_2 -numbers.

Almost multiplicity free subgroups of compact Lie groups and polynomial integrability of sub-Riemannian geodesic flows

Almost multiplicity free subgroups of compact Lie groups and polynomial integrability of sub-Riemannian geodesic flows

Об интегральном подходе при использовании метода коллокации

The article describes a matrix method of polynomial Chebyshev approximation using an integral approach to construct a solution to a nonhomogeneous fourth-order differential equation with mixed boundary conditions of the first kind. The proposed method is based on the expansion of the fourth-order derivative of the desired function into a series in terms of Chebyshev polynomials of the first kind and the representation of the partial sum of this series as a product of matrices whose elements are, respectively, the Chebyshev polynomials and the coefficients in this expansion. In this paper, using analytical formulas for calculating integrals of Chebyshev polynomials, we obtain a representation of the desired function in terms of the product of the matrices defined above. The use of points of extrema and zeros of Chebyshev polynomials of the first kind as nodes, as well as the properties of the sums of products of Chebyshev polynomials at these points, made it possible to reduce the boundary value problem by the collocation method to a system of inhomogeneous linear algebraic equations with a sparse matrix of this system. It is shown that the solution constructed in this way satisfies the differential equation at all nodes, including the boundary ones, in contrast to the approximate solution obtained by approximating the exact solution in the form of a finite sum of the Chebyshev series. The effectiveness of the proposed method is demonstrated by considering a boundary value problem with a known analytical solution. The convergence analysis of the constructed solution is carried out.

A New Approximation Method for Multi-Pantograph Type Delay Differential Equations Using Boubaker Polynomials

In this paper, a new approaching technique is offered to unravel multi-pantograph-type delay differential equations. The suggested new method is a collocation method based on integration and Boubaker polynomials. As the main idea of the method, the process starts by approaching the first derivative function in the equation in the form of truncated Boubaker series. Then this approximating form is composed in the matrix form. The unknown function is then obtained by integrating the approximate derivative function and expressing it as a matrix. Using the approximation, the matrix forms for the proportionally delayed terms in the equation are derived. In addition, operational matrix forms are constructed for convenience in the method. By using these matrix forms and matrix operations, the problem is reduced to a system of algebraic linear equations. The method is illustrated through numerical implementations and compared with existing techniques in the literature. The results demonstrate the effectiveness and reliability of the proposed approach, highlighting its superiority over other methods.

AN ALGEBRAIC INTERPRETATION OF THE SUPER CATALAN NUMBERS

AbstractWe extend the notion of polynomial integration over an arbitrary circle C in the Euclidean geometry over general fields $\mathbb {F}$ of characteristic zero as a normalised $\mathbb {F}$ -linear functional on $\mathbb {F}[\alpha _1, \alpha _2]$ that maps polynomials that evaluate to zero on C to zero and is $\mathrm {SO}(2,\mathbb {F})$ -invariant. This allows us to not only build a purely algebraic integration theory in an elementary way, but also give the super Catalan numbers $$ \begin{align*} S(m,n) = \frac{(2m)!(2n)!}{m!n!(m+n)!} \end{align*} $$ an algebraic interpretation in terms of values of this algebraic integral over some circle applied to the monomials $\alpha _1^{2m}\alpha _2^{2n}$ .