Polynomial Density Research Articles

On a problem of Pinkus and Wajnryb regarding density of multivariate polynomials

On a problem of Pinkus and Wajnryb regarding density of multivariate polynomials

The density of shifted and affine Eisenstein polynomials

In this paper we provide a complete answer to a question by Heyman and Shparlinski concerning the natural density of polynomials which are irreducible by Eisenstein’s criterion after applying some shift. The main tool we use is a local to global principle for density computations over a free Z \mathbb {Z} -module of finite rank.

A compressible lattice Boltzmann finite volume model for high subsonic and transonic flows on regular lattices

A multi-dimensional double distribution function thermal lattice Boltzmann model has been developed to simulate fully compressible flows at moderate Mach number. The lattice Boltzmann equation is temporally and spatially discretizated by an asymptotic preserving finite volume scheme. The micro-velocities discretization is adopted on regular low-symmetry lattices (D1Q3, D2Q9, D3Q15, D3Q19, D3Q27). The third-order Hermite polynomial density distribution function on low-symmetry lattices is used to solve the flow field, while a second-order energy distribution is employed to compute the temperature field. The fully compressible Navier–Stokes equations are recovered by standard order Gauss–Hermite polynomial expansions of Maxwell distribution with cubic correction terms, which are added by an external force expressed in orthogonal polynomials form. The proposed model is validated considering several benchmark cases, namely the Sod shock tube, thermal Couette flow and two-dimensional Riemann problem. The numerical results are in very good agreement with both analytical solution and reference results.

Component Caching in Hybrid Domains with Piecewise Polynomial Densities

Counting the models of a propositional formula is an important problem: for example, it serves as the backbone of probabilistic inference by weighted model counting. A key algorithmic insight is component caching (CC), in which disjoint components of a formula, generated dynamically during a DPLL search, are cached so that they only have to be solved once. In the recent years, driven by SMT technology and probabilistic inference in hybrid domains, there is an increasing interest in counting the models of linear arithmetic sentences. To date, however, solvers for these are block-clause implementations, which are nonviable on large problem instances. In this paper, as a first step in extending CC to hybrid domains, we show how propositional CC systems can be leveraged when limited to piecewise polynomial densities. Our experiments demonstrate a large gap in performance when compared to existing approaches based on a variety of block-clause strategies.

Improved Convergence Rates for Lasserre-Type Hierarchies of Upper Bounds for Box-Constrained Polynomial Optimization

We consider the problem of minimizing a given $n$-variate polynomial $f$ over the hypercube $[-1,1]^n$. An idea introduced by Lasserre, is to find a probability distribution on $[-1,1]^n$ with polynomial density function $h$ (of given degree $r$) that minimizes the expectation $\int_{[-1,1]^n} f(x)h(x)d\mu(x)$, where $d\mu(x)$ is a fixed, finite Borel measure supported on $[-1,1]^n$. It is known that, for the Lebesgue measure $d\mu(x) = dx$, one may show an error bound $O(1/\sqrt{r})$ if $h$ is a sum-of-squares density, and an $O(1/r)$ error bound if $h$ is the density of a beta distribution. In this paper, we show an error bound of $O(1/r^2)$, if $d\mu(x) = \left( \prod_{i=1}^n \sqrt{1-x_i^2} \right)^{-1}$ (the well-known measure in the study of orthogonal polynomials), and $h$ has a Schmudgen-type representation with respect to $[-1,1]^n$, which is a more general condition than a sum of squares. The convergence rate analysis relies on the theory of polynomial kernels and, in particular, on Jackson kerne...

Counting zero kernel pairs over a finite field

Helmke et al. have recently given a formula for the number of reachable pairs of matrices over a finite field. We give a new and elementary proof of the same formula by solving the equivalent problem of determining the number of so called zero kernel pairs over a finite field. We show that the problem is equivalent to certain other enumeration problems and outline a connection with some recent results of Guo and Yang on the natural density of rectangular unimodular matrices over Fq[x]. We also propose a new conjecture on the density of unimodular matrix polynomials.

Fourier forward modeling of vector and tensor gravity fields due to prismatic bodies with variable density contrast

Prismatic bodies with variable density contrasts are very useful in modeling and inversion of gravity anomalies for sedimentary basins, in which some simple mathematical functions offer better approximations for the density-depth relationship of the basin fill than a constant density model. We have explored Fourier-domain modeling of vector and tensor gravity anomalies due to 2D and 3D prismatic bodies following a wide class of variable density functions. We placed the emphasis on general polynomial models because they provided a flexible way to approximate arbitrary density functions. The recently proposed Gauss-fast Fourier transform method was applied to improve the accuracy of inverse Fourier transform. The numerical performance of the presented method was examined by comparing with space-domain analytical, seminumerical, or completely numerical solutions. Singularity and numerical stability of the algorithm were also analyzed with ranges of numerical stability for polynomial density models of different orders specified. Truncation errors of Fourier forward methods were estimated quantitatively using relevant model parameters, and the behavior was observed through theoretical analysis and model tests. Synthetic models, of 2D and 3D prisms with variable density contrasts, found the accuracy, efficiency, and flexibility of the method. We also interpreted a real data example of the gravity profile across the San Jacinto graben.

Eisenstein polynomials over function fields

In this paper we compute the density of monic and non-monic Eisenstein polynomials of fixed degree having entries in an integrally closed subring of a function field over a finite field.

Positive density of integer polynomials with some prescribed properties

In this paper, we show that various kinds of integer polynomials with prescribed properties of their roots have positive density. For example, we prove that almost all integer polynomials have exactly one or two roots with maximal modulus. We also show that for any positive integer n and any set of n distinct points symmetric with respect to the real line, there is a positive density of integer polynomials of degree n, height at most H and Galois group Sn whose roots are close to the given n points.

Density of polynomials in classes of functions on products of planar domains

For planar domains Ωi, i∈I, where I is an arbitrary set, we give sufficient conditions for the density of polynomials in A(∏i∈IΩi) as well as in A∞(∏i∈IΩi), endowed with their natural topologies. We also characterize the uniform limits, with respect to the chordal metric, of polynomials on ∏i∈IΩ‾i. Moreover, we discuss sets of uniqueness.

The Generalized Pascal Triangle and the Matrix Variate Jensen-Logistic Distribution

This article defines the so called Generalized Matrix Variate Jensen-Logistic distribution. The relevant applications of this class of distributions in Configuration Shape Theory consist of a more efficient computation, supported by the corresponding inference. This demands the solution of two important problems: (1) the development of analytical and efficient formulae for their k-th derivatives and (2) the use of the derivatives to transform the configuration density into a polynomial density under some special matrix Kummer relation, indexed in this case by the Jensen-Logistic kernel. In this article, we solve these problems by deriving a simple formula for the k-th derivative of the density function, avoiding the usual partition theory framework and using a generalization of Pascal triangles. Then we apply the results by obtaining the associated Jensen-Logistic Kummer relations and the configuration polynomial density in the setting of Statistical Shape Theory.

Explicit constructions of extremal graphs and new multivariate cryptosystems

New multivariate cryptosystems are introduced. Sequences f(n) of bijective polynomial transformations of bijective multivariate transformations of affine spaces Kn, n = 2, 3, ... , where K is a finite commutative ring with special properties, are used for the constructions of cryptosystems. On axiomatic level, the concept of a family of multivariate maps with invertible decomposition is proposed. Such decomposition is used as private key in a public key infrastructure. Requirements of polynomiality of degree and density allow to estimate the complexity of encryption procedure for a public user. The concepts of stable family and family of increasing order are motivated by studies of discrete logarithm problem in Cremona group. Statement on the existence of families of multivariate maps of polynomial degree and polynomial density with the invertible decomposition is formulated. We observe known explicit constructions of special families of multivariate maps. They correspond to explicit constructions of families of nonlinear algebraic graphs of increasing girth which appeared in Extremal Graph Theory. The families are generated by pseudorandom walks on graphs. This fact ensures the existence of invertible decomposition; a certain girth property guarantees the increase of order for the family of multivariate maps, good expansion properties of families of graphs lead to good mixing properties of graph based private key algorithms. We describe the general schemes of cryptographic applications of such families (public key infrastructure, symbolic Diffie—Hellman protocol, functional versions of El Gamal algorithm).

An exact general remeshing scheme applied to physically conservative voxelization

We present an exact general remeshing scheme to compute analytic integrals of polynomial functions over the intersections between convex polyhedral cells of old and new meshes. In physics applications this allows one to ensure global mass, momentum, and energy conservation while applying higher-order polynomial interpolation. We elaborate on applications of our algorithm arising in the analysis of cosmological N-body data, computer graphics, and continuum mechanics problems.We focus on the particular case of remeshing tetrahedral cells onto a Cartesian grid such that the volume integral of the polynomial density function given on the input mesh is guaranteed to equal the corresponding integral over the output mesh. We refer to this as “physically conservative voxelization.”At the core of our method is an algorithm for intersecting two convex polyhedra by successively clipping one against the faces of the other. This algorithm is an implementation of the ideas presented abstractly by Sugihara [48], who suggests using the planar graph representations of convex polyhedra to ensure topological consistency of the output. This makes our implementation robust to geometric degeneracy in the input. We employ a simplicial decomposition to calculate moment integrals up to quadratic order over the resulting intersection domain.We also address practical issues arising in a software implementation, including numerical stability in geometric calculations, management of cancellation errors, and extension to two dimensions. In a comparison to recent work, we show substantial performance gains. We provide a C implementation intended to be a fast, accurate, and robust tool for geometric calculations on polyhedral mesh elements.

The flattenings of the layers of rotating planets and satellites deformed by a tidal potential

We consider the Clairaut theory of the equilibrium ellipsoidal figures for differentiated non-homogeneous bodies in non-synchronous rotation adding to it a tidal deformation due to the presence of an external gravitational force. We assume that the body is a fluid formed by $n$ homogeneous layers of ellipsoidal shape and we calculate the external polar flattenings and the mean radius of each layer, or, equivalently, their semiaxes. To first order in the flattenings, the general solution can be written as $\epsilon_k={\cal H}_k*\epsilon_h$ and $\mu_k={\cal H}_k*\mu_h$, where $\cal{H}_k$ is a characteristic coefficient for each layer which only depends on the internal structure of the body and $\epsilon_h, \mu_h$ are the flattenings of the equivalent homogeneous problem. For the continuous case, we study the Clairaut differential equation for the flattening profile, using the Radau transformation to find the boundary conditions when the tidal potential is added. Finally, the theory is applied to several examples: i) a body composed of two homogeneous layers; ii) bodies with simple polynomial density distribution laws and iii) bodies following a polytropic pressure-density law.

Number fields and function fields: coalescences, contrasts and emerging applications.

The similarity between the density of the primes and the density of irreducible polynomials defined over a finite field of q elements was first observed by Gauss. Since then, many other analogies have been uncovered between arithmetic in number fields and in function fields defined over a finite field. Although an active area of interaction for the past half century at least, the language and techniques used in analytic number theory and in the function field setting are quite different, and this has frustrated interchanges between the two areas. This situation is currently changing, and there has been substantial progress on a number of problems stimulated by bringing together ideas from each field. We here introduce the papers published in this Theo Murphy meeting issue, where some of the recent developments are explained.

Noised image segmentation based on rough set and orthogonal polynomial density model

In order to segment a noised image, a method is proposed based on the rough set and orthogonal polynomial density model, in which the nonparametric mixture model can accurately fit the image gray distribution and the rough set can deal with the inaccuracy and uncertainty problems. First, the nonparametric mixture density model is constructed based on the upper and lower approximations of the rough set which can address the problem of over-relying on the prior presumption. Second, the nonparametric expectation-maximization is used to estimate the mixture model parameters. Finally, image pixels are classified according to Bayesian criterion. Experiments on different datasets show that our method is effective in solving the problem of model mismatch, restraining the noise, and preserving the boundary for the noised image segmentation.

A Bayesian hierarchical model for estimating and partitioning Bernstein polynomial density functions

A Bayesian hierarchical model for simultaneously estimating and partitioning probability density functions is presented. Individual density functions are flexibly modeled using Bernstein densities, which are mixtures of beta densities whose parameters depend only on the number of mixture components. A prior distribution is placed on the number of mixture components, and the mixture weights are expressed as increments of a distribution function G. A Dirichlet process prior is placed on G and the parameters of the Dirichlet process, the baseline distribution and the precision parameter, are treated as random. A mixture of a product of beta densities is used to partition subjects into groups, with subjects in the same group sharing information via a common baseline distribution. Inference is carried out using Markov chain Monte Carlo. A computing algorithm based on the constructive definition of the Dirichlet process is offered, for both a fixed number of groups and an unknown number of groups. When the number of groups is unknown, a birth–death algorithm is used to make inference regarding the number of groups. The model is demonstrated using radiologist-specific distributions of percent mammographic density.

The Gravity Anomaly of a 2D Polygonal Body Having Density Contrast Given by Polynomial Functions

An analytical solution is presented for the gravity anomaly produced by a 2D body whose geometrical shape is arbitrary and where the density contrast is a polynomial function in both the horizontal and vertical directions. Approximating the real shape of the body by a polygon, the solution is expressed as sum of algebraic quantities that depend only upon the coordinates of the vertices of the polygon and upon the polynomial density function. The solution presented in the paper, which refers to a third-order polynomial function as a maximum, exhibits an intrinsic symmetry that naturally suggests its extension to the case of higher-order polynomials describing the density contrast. Furthermore, the gravity anomaly is evaluated at an arbitrary point that does not necessarily coincide with the origin of the reference frame in which the density function is assigned. Invoking recent results of potential theory, the solution derived in the paper is shown to be singularity-free and numerically robust. The accuracy and effectiveness of the proposed approach is witnessed by the numerical comparisons with examples derived from the existing literature.

On shifted Eisenstein polynomials

We study polynomials with integer coefficients which become Eisenstein polynomials after the additive shift of a variable. We call such polynomials shifted Eisenstein polynomials. We determine an upper bound on the maximum shift that is needed given a shifted Eisenstein polynomial and also provide a lower bound on the density of shifted Eisenstein polynomials, which is strictly greater than the density of classical Eisenstein polynomials. We also show that the number of irreducible degree $$n$$ polynomials that are not shifted Eisenstein polynomials is infinite. We conclude with some numerical results on the densities of shifted Eisenstein polynomials.

Localization of Zeros for Cauchy Transforms

We study the localization of zeros of Cauchy transforms of discrete measures on the real line. This question is motivated by the theory of canonical systems of differential equations. In particular, we prove that the spaces of Cauchy transforms having the localization property are in one-to-one correspondence with the canonical systems of special type, namely, those whose Hamiltonians consist only of indivisible intervals accumulating on the left. Various aspects of the localization phenomena are studied in details. Connections with the density of polynomials and other topics in analysis are discussed.