Polynomial Density Research Articles

Three-dimensional Motion Skeleton Reconstruction Algorithm for Gymnastic Dancing Movements

Aiming at the problem of inaccurate matching results in the traditional three-dimensional reconstruction algorithm of gymnastic skeleton, a three-dimensional motion skeleton reconstruction algorithm of gymnastic dance action is proposed. Taking the center of gravity of the human body as the origin, the position of other nodes in the camera coordinate system relative to the center point of the human skeleton model is calculated, and the human skeleton data collection is completed through action division and posture feature calculation. Polynomial density is introduced into the integration of convolution surface, and the human body model of convolution surface is established according to convolution surface. By using the method of binary parameter matching, the accuracy of the matching results is improved, and the three-dimensional skeleton of gymnastic dance movement is reconstructed. The experimental results show that the fitting degree between the proposed method and the actual reconstruction result is 99.8%, and the reconstruction result of this algorithm has high accuracy.

Lpdensity: Local Polynomial Density Estimation and Inference

Density estimation and inference methods are widely used in empirical work. When the underlying distribution has compact support, conventional kernel-based density estimators are no longer consistent near or at the boundary because of their well-known boundary bias. Alternative smoothing methods are available to handle boundary points in density estimation, but they all require additional tuning parameter choices or other typically ad hoc modifications depending on the evaluation point and/or approach considered. This article discusses the R and Stata package lpdensity implementing a novel local polynomial density estimator proposed and studied in Cattaneo, Jansson, and Ma (2020, 2022), which is boundary adaptive and involves only one tuning parameter. The methods implemented also cover local polynomial estimation of the cumulative distribution function and density derivatives. In addition to point estimation and graphical procedures, the package offers consistent variance estimators, mean squared error optimal bandwidth selection, robust bias-corrected inference, and confidence bands construction, among other features. A comparison with other density estimation packages available in R using a Monte Carlo experiment is provided.

On the density of irreducible polynomials which generate k-free polynomials over function fields

Let M∈Fq[t] be a polynomial, and let k≥2 be an integer. In this note, we will compute the asymptotic density of irreducible monic polynomials P∈Fq[t] for which P+M is not divisible by the kth power of any irreducible polynomial.

Idempotents and moment problem for discrete measure

In this paper, we investigate the full multidimensional moment problem for discrete measure using Vasilescu's idempotent approach based on Λ-multiplicative elements with respect to the associated square positive Riesz functional Λ. We give a sufficient condition for the existence of a discrete integral representation of the Riesz functional Λ, which turns out to be necessary in the bounded shift space case (in fact, it suffices to assume the density of polynomials in the corresponding L2-space). We pay special attention to Λ-multiplicative elements, providing several criteria guaranteeing that they are characteristic functions of single point sets. We also give an example showing that Λ-multiplicative elements may not be characteristic functions of single point sets.

О плотности многочленов в специальном пространстве целых функций экспоненциального типа

The traditional approach to solving a specific problem of spectral synthesis in a complex domain involves reducing it to the problem of local description of closed submodules in a certain space of entire functions. The last problem is split into checking the stability and saturation of the submodule under study. This approach turned out to be very effective, for example, in the study of submodules of local rank 1 and in the study of submodules in topological modules associated with unbounded convex domains. Recent studies on spectral synthesis in the complex domain are based on a different scheme. This scheme involves reducing the problem of local description to checking the density of polynomials in a special module of entire functions of exponential type. Moreover, the space under study is a separable locally convex space of type (LN)*. Polynomial approximation in such a space is understood by us as sequential approximation, that is, we are talking about the approximation of space elements by ordinary (not generalized) sequences of polynomials. In this article, we study a special locally convex module of entire vector functions over the ring of polynomials in the degree of the independent variable. The theorem proved in the article can serve as a source of new results on spectral synthesis in the complex domain.

Comparison of estimation methods for the density of autoregressive parameter in aggregated AR(1) processes

The article investigates the properties of two alternative disaggregation methods. First one, proposed in Chong (2006), is based on the assumption of polynomial autoregressive parameter density. Second one, proposed in Leipus et al. (2006), uses the approximation of the density by the means of Gegenbauer polynomials. Examining results of Monte-Carlo simulations it is shown that none of the methods was found to outperform another. Chong’s method is narrowed by the class of polynomial densities, and the secondmethod is not effective in the presence of common innovations.Bothmethodswork correctly under assumptions proposed in the corresponding articles.

Bayesian nonparametrics for directional statistics

We introduce a density basis of the trigonometric polynomials that is suitable to mixture modelling. Statistical and geometric properties are derived, suggesting it as a circular analogue to the Bernstein polynomial densities. Nonparametric priors are constructed using this basis and a simulation study shows that the use of the resulting Bayes estimator may provide gains over comparable circular density estimators previously suggested in the literature.From a theoretical point of view, we propose a general prior specification framework for density estimation on compact metric space using sieve priors. This is tailored to density bases such as the one considered herein and may also be used to exploit their particular shape-preserving properties. Furthermore, strong posterior consistency is shown to hold under notably weak regularity assumptions and adaptive convergence rates are obtained in terms of the approximation properties of positive linear operators generating our models.

Convergence of Bieberbach polynomials: Keldysh’s theorems and Mergelyan’s conjecture

Results due to Keldysh on the convergence of Bieberbach polynomials and the density of polynomials in spaces of analytic functions are considered. Their further development and relevance in the contemporary context of constructive complex analysis are discussed. Particular focus is placed on Mergelyan’s conjecture on the rate of convergence in a domain with smooth boundary, which is still open. Bibliography: 20 titles.

Elliptical harmonic method for gravity forward modelling of 2D bodies

The elliptical harmonic method for gravity forward computation of two-dimensional (2D) bodies with constant, polynomial and exponential density distributions is presented. This paper gives the elliptical harmonic expansions for the gravity field of 2D bodies mainly including the gravitational attractions and gradients, and the conversions from the surface integral expressions of the elliptical harmonic coefficients and the surface integral of the density function into line integrals using Gauss divergence theorem for evaluation of the gravity field. Due to the requirements of the Gauss divergence and Stokes theorems that the boundary surface or curve of the body is piecewise smooth and the vector field as well as its first-order partial derivatives are continuous, the effects of the discontinuities of the elliptical coordinates on the integral conversion using Gauss divergence and Stokes theorems are considered. The integrands of the converted line integrals are the analytical expressions with respect to the coordinates of the internal point of the 2D body for the constant and polynomial density models, but are not elementary functions for the exponential density model that the integrands can be solved by approximating the density function with polynomial function. The numerical experiments with two tested bodies including a rectangular cylinder body with quadratic density contrast varying with depth and a 26-sided polygon body with quadratic density contrast varying in both horizontal and vertical directions show the convergence, accuracy and stability of the elliptical harmonic algorithm. Compared with the circular harmonic approach, the elliptical harmonic approach has smaller non-convergence region between the boundary of the 2D body and the reference ellipse and is convergent faster for most of the external observation points.

The moment-SOS hierarchy and the Christoffel–Darboux kernel

We consider the global minimization of a polynomial on a compact set \(\mathbf {B}\). We show that each step of the Moment-SOS hierarchy has a nice and simple interpretation that complements the usual one. Namely, it computes coefficients of a polynomial in an orthonormal basis of \(L^2(\mathbf {B},\mu )\) where \(\mu \) is an arbitrary reference measure whose support is exactly \(\mathbf {B}\). The resulting polynomial is a certain density (with respect to \(\mu \)) of some signed measure on \(\mathbf {B}\). When some relaxation is exact (which generically takes place) the coefficients of the optimal polynomial density are values of orthonormal polynomials at the global minimizer and the optimal (signed) density is simply related to the Christoffel–Darboux (CD) kernel and the Christoffel function associated with \(\mu \). In contrast to the hierarchy of upper bounds which computes positive densities, the global optimum can be achieved exactly as integration against a polynomial (signed) density because the CD-kernel is a reproducing kernel, and so can mimic a Dirac measure (as long as finitely many moments are concerned).

Density of polynomials in certain weighted Dirichlet type spaces

We study weighted Dirichlet type spaces in the unit disk. We prove that analytic polynomials are dense in weighted Dirichlet type spaces if the (non-radial) weight function is super-biharmonic and satisfies a growth condition up to the boundary.

Controllability and decentralized stabilization of the Kolmogorov forward equation for Markov chains

In this paper, we provide several results on controllability and stabilizability properties of the Kolmogorov forward equation of a continuous-time Markov chain (CTMC) evolving on a finite state space, with the transition rates defined as the control parameters. First, we show that any target probability distribution can be reached asymptotically using time-varying control parameters. Second, we characterize all stationary distributions that are stabilizable using time-independent control parameters. For bidirected graphs, we construct rational and polynomial density feedback laws that stabilize stationary distributions while satisfying the additional constraint that the feedback law takes zero value at equilibrium. This last result enables the construction of decentralized density feedback controllers, using tools from linear systems theory and sum-of-squares based polynomial optimization, that stabilize a swarm of robots modeled as a CTMC to a target state distribution with no state-switching at equilibrium. In addition to these results, we prove a sufficient condition under which the classical rank conditions for controllability can be generalized to forward equations with non-negativity constraints on the control inputs. We apply this result to prove local controllability of a forward equation in which only a small subset of the transition rates are the control inputs. Lastly, we extend our feedback stabilization results to stationary distributions that have a strongly connected support.

Weighted L^2 Version of Mergelyan and Carleman Approximation

We study the density of polynomials in H^2(E,varphi ), the space of square integrable functions with respect to mathrm{e}^{-varphi }mathrm{d}m and holomorphic on the interior of E in {mathbb {C}}, where varphi is a subharmonic function and dm is a measure on E. We give a result where E is the union of a Lipschitz graph and a Carathéodory domain, which we state as a weighted L^2-version of the Mergelyan theorem. We also prove a weighted L^2-version of the Carleman theorem.

Recursive Analytical Formulae of Gravitational Fields and Gradient Tensors for Polyhedral Bodies with Polynomial Density Contrasts of Arbitrary Non-negative Integer Orders

Exact computation of the gravitational field and gravitational gradient tensor for a general mass body is a core routine to model the density structure of the Earth. In this study, we report on the existence of closed-form solutions of the gravitational potential, gravitational field and gravitational gradient tensor for a general polyhedral mass body with a polynomial density function of arbitrary non-negative integer orders that can simultaneously vary in both horizontal and vertical directions. Our closed-form solutions of the gravitational potential and the gravitational field are singularity-free, which implies that the observation sites can have arbitrary geometric relationships with polyhedral mass source bodies. However, weak logarithmic singularities exist on the edges of polyhedra for the gravitational gradient tensor. A simple prismatic mass body with polynomial density contrast varying in the vertical direction and a complicated dodecahedral mass body with quartic-order density contrasts were tested to verify the accuracy of the newly derived closed-form solutions. For the gravitational potential, gravitational fields and gradient tensors, our closed-form solutions are in excellent agreement with previously published analytical solutions and Gaussian numerical quadrature solutions.

On square-free values of large polynomials over the rational function field

AbstractWe investigate the density of square-free values of polynomials with large coefficients over the rational function field 𝔽q[t]. Some interesting questions answered as special cases of our results include the density of square-free polynomials in short intervals, and an asymptotic for the number of representations of a large polynomial N as a sum of a k-th power of a small polynomial and a square-free polynomial.

A Characterization of Polynomial Density on Curves via Matrix Algebra

In this work, our aim is to obtain conditions to assure polynomial approximation in Hilbert spaces L 2 ( μ ) , with μ a compactly supported measure in the complex plane, in terms of properties of the associated moment matrix with the measure μ . To do it, in the more general context of Hermitian positive semidefinite matrices, we introduce two indexes, γ ( M ) and λ ( M ) , associated with different optimization problems concerning theses matrices. Our main result is a characterization of density of polynomials in the case of measures supported on Jordan curves with non-empty interior using the index γ and other specific index related to it. Moreover, we provide a new point of view of bounded point evaluations associated with a measure in terms of the index γ that will allow us to give an alternative proof of Thomson’s theorem, by using these matrix indexes. We point out that our techniques are based in matrix algebra tools in the framework of Hermitian positive definite matrices and in the computation of certain indexes related to some optimization problems for infinite matrices.

English

We introduce variable exponent Fock spaces and study some of their basic properties such as boundedness of evaluation functionals, density of polynomials, boundedness of a Bergman-type projection and duality. We also prove that under the global log-Holder condition, the variable exponent Fock spaces coincide with the classical ones.

A spectral-domain approach for gravity forward modelling of 2D bodies

We provide the spectral-domain solutions for the gravity forward modelling of a 2D body with constant, polynomial and exponential density distributions, including the gravitational attraction and its arbitrary-order derivatives. The gravity effects may be directly obtained from the derivatives of a constructed scalar quantity and the surface integral of the density function over the 2D body. Then, the surface integrals of the spectral expansion coefficients of the scalar and the density function are converted to the line integrals along the boundary of the mass body using the Gauss divergence theorem and thus can be evaluated by numerical integration. The surface integrals for the exponential density model are expressed as the infinite expansions of line integrals and converge fast. Approximating the mass body to a 2D polygon, the surface integrals can be evaluated by simple analytic formulas for the constant density model and by linear recursive relations for the polynomial density model. The numerical implementation shows that the spectral-domain algorithm of this paper can produce high accurate forward results, e.g. 13–16 digits achievable precision for the gravity vector. Although the spectral-domain method is only suitable for forward computation of the external gravity effects of the 2D body, it is numerically stable for arbitrary observation point outside the smallest enclosed circle. The closed-form solutions for the 2D body with constant or polynomial density distribution are high precision to evaluate the gravity anomaly at the point near or inside the body, but may be lower precision when the computation point moves away from the body. The spectral-domain algorithm can handle the polynomial and exponential density model in both horizontal and vertical directions.

Distributionally robust optimization with polynomial densities: theory, models and algorithms

In distributionally robust optimization the probability distribution of the uncertain problem parameters is itself uncertain, and a fictitious adversary, e.g., nature, chooses the worst distribution from within a known ambiguity set. A common shortcoming of most existing distributionally robust optimization models is that their ambiguity sets contain pathological discrete distributions that give nature too much freedom to inflict damage. We thus introduce a new class of ambiguity sets that contain only distributions with sum-of-squares (SOS) polynomial density functions of known degrees. We show that these ambiguity sets are highly expressive as they conveniently accommodate distributional information about higher-order moments, conditional probabilities, conditional moments or marginal distributions. Exploiting the theoretical properties of a measure-based hierarchy for polynomial optimization due to Lasserre (SIAM J Optim 21(3):864–885, 2011), we prove that certain worst-case expectation constraints are polynomial-time solvable under these new ambiguity sets. We also show how SOS densities can be used to approximately solve the general problem of moments. We showcase the applicability of the proposed approach in the context of a stylized portfolio optimization problem and a risk aggregation problem of an insurance company.

Analytical Solutions of Gravity Vector and Gravity Gradient Tensor Caused by a 2D Polygonal Body with a 2D Polynomial Density Contrast

In this paper, analytical solutions are presented for the gravity vector and gravity gradient tensor at any point produced by a 2D body whose cross-section is an arbitrary polygon and the density contrast is a 2D arbitrary-order polynomial function varying in both horizontal and vertical directions. In addition, we analyze the singularity of our expressions. For the gravity vector, the singularity points only exist at the vertices of the polygon. But for the gravity gradient tensor, there are two situations: (1) if every side of the polygon is not parallel to z-axis, the singularity points will only exist at the vertices of the polygon; (2) if there is any side parallel to z-axis in the polygon, all the points on the line passing through the side parallel to z-axis will become singularity points. To avoid this singularity, observation points can be moved from the singularity points by a minimal distance. Besides, the analytic expressions are validated compared with conventional method that sums up the gravity effects of a series of units with uniform densities, with the numerical stability also being evaluated through numerical tests. What is more, applications with some numerical examples and effective models show that our analytical solution within the range of numerical stability is superior in computational accuracy and efficiency to the conventional method that sums up the gravity effects of a series of units with uniform densities. In a word, our expressions provide an effective method for computing the gravity vector and gravity gradient tensor of an irregular 2D body with complicated density variation.