Finite Polynomial Research Articles

Probability measure-valued polynomial diffusions

We introduce a class of probability measure-valued diffusions, coined polynomial, of which the well-known Fleming–Viot process is a particular example. The defining property of finite dimensional polynomial processes considered in [8, 21] is transferred to this infinite dimensional setting. This leads to a representation of conditional marginal moments via a finite dimensional linear PDE, whose spatial dimension corresponds to the degree of the moment. As a result, the tractability of finite dimensional polynomial processes are preserved in this setting. We also obtain a representation of the corresponding extended generators, and prove well-posedness of the associated martingale problems. In particular, uniqueness is obtained from the duality relationship with the PDEs mentioned above.

Generalised Linear Cepstral Models for the Spectrum of a Time Series

The paper introduces the class of generalised linear models with Box-Cox link for the spectrum of a time series. The Box-Cox transformation of the spectral density is represented as a finite Fourier polynomial, with coefficients, that we term generalised cepstral coefficients, providing a complete characterisation of the properties of the random process. The link function depends on a power transformation parameter and encompasses the exponential model (logarithmic link), the autoregressive model (inverse link), and the moving average model (identity link). One of the merits of this model class is the possibility of nesting alternative spectral estimation methods under the same likelihood-based framework, so that the selection of a particular parametric spectrum amounts to estimating the transformation parameter. We also show that the generalised cepstral coefficients are a one to one function of the inverse partial autocorrelations of the process, which can be used to evaluate the mutual information between the past and the future of the process.

About the decision regularized equations of perturbed motion body

The article deals with determination of the second- and higher-order perturbations in Cartesian coordinates and body motion velocity constituents. A special perturbed motion differential equations system is constructed. The right-hand sides of this system are finite polynomials relative to an independent regularizing variable. This allows constructing a single algorithm to determine the second and higher order perturbations in the form of finite polynomials relative to some regularizing variables that are chosen at each approximation step. Following the calculations results with the use of the developed method, the coefficients of approximating polynomials representing rectangular coordinates and components of the regularized body speed were obtained. Comparison with the results of numerical integration of the equations of disturbed motion shows close agreement of the results. The developed methods make it possible to calculate, by the approximating polynomials, any intermediate point of the motion trajectory of the body.

Distribution of postcritically finite polynomials III: Combinatorial continuity

In the first part of the present paper, we continue our study of distribution of postcritically finite parameters in the moduli space of polynomials: we show the equidistribution of Misiurewicz parameters with prescribed combinatorics toward the bifurcation measure. Our results essentially rely on a combinatorial description of the escape locus and of the bifurcation measure developped by Kiwi and Dujardin-Favre. In the second part of the paper, we construct a bifurcation measure for the connectedness locus of the quadratic anti-holomorphic family which is supported by a strict subset of the boundary of the Tricorn. We also establish an approximation property by Misiurewicz parameters in the spirit of the previous one. Finally, we answer a question of Kiwi, exhibiting in the moduli space of degree 4 polynomials, non-trivial Impression of specific combinatorics.

About choosing the form of perturbed body motion differential equations system

The article deals with determination of the second- and higher-order perturbations in Cartesian coordinates and body motion velocity constituents. A special perturbed motion differential equations system is constructed. The right-hand sides of this system are finite polynomials relative to an independent regularizing variable. This allows constructing a single algorithm to determine the second and higher order perturbations in the form of finite polynomials relative to some regularizing variables that are chosen at each approximation step. Following the calculations results with the use of the developed method, the coefficients of approximating polynomials representing rectangular coordinates and components of the regularized body speed were obtained. Comparison with the results of numerical integration of the equations of disturbed motion shows close agreement of the results. The developed methods make it possible to calculate, by the approximating polynomials, any intermediate point of the motion trajectory of the body.

A non-uniform Haar wavelet method for numerically solving two-dimensional convection-dominated equations and two-dimensional near singular elliptic equations

A non-uniform Haar wavelet based collocation method has been developed in this paper for two-dimensional convection dominated equations and two-dimensional near singular elliptic partial differential equations, in which traditional Haar wavelet method produces oscillatory solutions or low accurate solutions. The main idea behind the proposed method is to transform the computation of numerical solution of considered partial differential equations to computation of solution of a linear system of equations. This process is done by discretizing space variables with non-uniform Haar wavelets. To confirm efficiency of the proposed method seven benchmark problems are solved and the obtained results are compared with exact solutions and with local meshless methods, finite element method, finite difference method and polynomial collocation method. Numerical experiments show that the proposed method gives convincing results even in less number of collocation nodes.

Core entropy of polynomials with a critical point of maximal order

This paper discusses some properties of the topological entropy of the systems generated by polynomials of degree $ d$ with two critical points. A partial order in the parameter space is defined. The monotonicity of the topological entropy of postcritically finite polynomials of degree $ d$ acting on Hubbard tree is generalized.

Finite sections: A functional analytic perspective on approximation methods

We review approximation methods and their stability and applicability. We then focus on the finite section method and Galerkin methods and show that on separable Hilbert spaces either one can be interpreted as the other. In the end we demonstrate that well‐known methods such as the finite element method and polynomial chaos expansion are particular examples of the finite section method; their applicability can therefore be studied via the latter.

On hp-streamline diffusion and Nitsche schemes for the relativistic Vlasov-Maxwell system

We study stability and convergence of hp-streamline diffusion (SD) finite element, and Nitsche's schemes for the three dimensional, relativistic (3 spatial dimension and 3 velocities), time dependent Vlasov-Maxwell system and Maxwell's equations, respectively. For the hp scheme for the Vlasov-Maxwell system, assuming that the exact solution is in the Sobolev space HS+1(Omega), we derive global a priori error bound of order O(h/p)(s+1/2), where h(= max(K) h(K)) is the mesh parameter and p(= max(K) p(K)) is the spectral order. This estimate is based on the local version with h(K) = diam K being the diameter of the phase-space-time element K and pR-is the spectral order (the degree of approximating finite element polynomial) for K. As for the Nitsche's scheme, by a simple calculus of the field equations, first we convert the Maxwell's system to an elliptic type equation. Then, combining the Nitsche's method for the spatial discretization with a second order time scheme, we obtain optimal convergence of O(h(2) +k(2)), where h is the spatial mesh size and k is the time step. Here, as in the classical literature, the second order time scheme requires higher order regularity assumptions. Numerical justification of the results, in lower dimensions, is presented and is also the subject of a forthcoming computational work [22].

Physics-informed machine learning approach for augmenting turbulence models: A comprehensive framework

Reynolds-averaged Navier-Stokes (RANS) equations are widely used in engineering turbulent flow simulations. However, RANS predictions may have large discrepancies due to the uncertainties in modeled Reynolds stresses. Recently, Wang et al. demonstrated that machine learning can be used to improve the RANS modeled Reynolds stresses by leveraging data from high fidelity simulations (Physics informed machine learning approach for reconstructing Reynolds stress modeling discrepancies based on DNS data. Physical Review Fluids. 2, 034603, 2017). However, solving for mean flows from the improved Reynolds stresses still poses significant challenges due to potential ill-conditioning of RANS equations with Reynolds stress closures. Enabling improved predictions of mean velocities are of profound practical importance, because often the velocity and its derived quantities (QoIs, e.g., drag, lift, surface friction), and not the Reynolds stress itself, are of ultimate interest in RANS simulations. To this end, we present a comprehensive framework for augmenting turbulence models with physics-informed machine learning, illustrating a complete workflow from identification of input features to final prediction of mean velocities. This work has two innovations. First, we demonstrate a systematic procedure to generate mean flow features based on the integrity basis for mean flow tensors. Second, we propose using machine learning to predict linear and nonlinear parts of the Reynolds stress tensor separately. Inspired by the finite polynomial representation of tensors in classical turbulence modeling, such a decomposition is instrumental in overcoming the ill-conditioning of RANS equations. Numerical tests demonstrated merits of the proposed framework.

Gravitational waves from spinning binary black holes at the leading post-Newtonian orders at all orders in spin

We determine the binding energy, the total gravitational wave energy flux, and the gravitational wave modes for a binary of rapidly spinning black holes, working in linearized gravity and at leading orders in the orbital velocity, but to all orders in the black holes' spins. Though the spins are treated nonperturbatively, surprisingly, the binding energy and the flux are given by simple analytical expressions which are finite (respectively third- and fifth-order) polynomials in the spins. Our final results are restricted to the important case of quasi-circular orbits with the black holes' spins aligned with the orbital angular momentum.

Algorithmic aspects of branched coverings IV/V. Expanding maps

Thurston maps are branched self-coverings of the sphere whose critical points have finite forward orbits. We give combinatorial and algebraic characterizations of Thurston maps that are isotopic to expanding maps as maps and as maps with contracting biset. We prove that every Thurston map decomposes along a unique minimal multicurve into Levy-free and finite-order pieces, and this decomposition is algorithmically computable. Each of these pieces admits a geometric structure. We apply these results to matings of post-critically finite polynomials, extending a criterion by Mary Rees and Tan Lei: they are expanding if and only if they do not admit a cycle of periodic rays.

Energy efficiency control of pneumatic actuator systems through nonlinear dynamic optimization

Abstract As one of the most widely used power sources in the engineering field, compressed air is especially common in industrial applications. In a traditional pneumatic circuit, a three-position five-way valve is used to control the air inlet and exhaust with a single pressure, which has low efficiency. This paper proposed a novel bridge-type circuit with variable pressure control using four on-off valves to improve the energy efficiency of compressed air for pneumatic actuators. The key idea of the method was to use air expansion energy to do work. Reducing the volume of consumed air was the goal, and a numerical method was presented to calculate the optimal on-off time sequence of the four valves using nonlinear dynamic optimization. The sequence was obtained based on the combination of a finite element polynomial configuration method and a reduced space sequential quadratic programming algorithm. Then, experimental validation was conducted to illustrate the energy efficiency, including pneumatic cylinder motion test under different conditions. Compared with the traditional circuit, the experimental results show that the bridge-type circuit put forward by this paper can significantly improve the energy efficiency of pneumatic driving systems by 50–70%. Additionally, the bridge-type circuit allows the piston to move smoothly to the end of the stroke, which can take the place of the function of cushion component and speed control valve of the traditional circuit.

On the adequacy of the polynomial approximation to the exponential growth curve model

Exponential growth curves are usually approximated with a finite order polynomial curve in the study of trending curves. This is done because the most popular way of removing the trend component is by differencing. This paper shows that the trend curve cannot be removed by differencing when the trend curve is the exponential growth curve. Exponential curves are transcendental functions which can be reduced to a finite order polynomial by Maclaurin series expansion. The objective of this paper is to examine the adequacy of the polynomial approximation of the exponential growth curve with respect to its growth rate and sample size. The coefficients of the associated polynomial curve were obtained theoretically by the use of Maclaurin series expansion method. Next, exponential growth curves with varying growth rates and sample sizes were simulated. Adequate polynomials were fitted to the simulated exponential growth curves and the coefficients obtained were compared with the theoretical coefficients using absolute error and paired tests. Results obtained show that adequacy depend on both growth rate and sample size. For the purpose of statistical analysis, the highest sample size of 28 is not useful, especially in times series analysis where the demand of samples of 60 or more is made.

A large arboreal Galois representation for a cubic postcritically finite polynomial

We give a complete description of the arboreal Galois representation of a certain postcritically finite cubic polynomial over a large class of number fields and for a large class of basepoints. This is the first such example that is not conjugate to a power map, Chebyshev polynomial, or Lattès map. The associated Galois action on an infinite ternary rooted tree has Hausdorff dimension bounded strictly between that of the infinite wreath product of cyclic groups and that of the infinite wreath product of symmetric groups. We deduce a zero-density result for prime divisors in an orbit under this polynomial. We also obtain a zero-density result for the set of places of convergence of Newton’s method for a certain cubic polynomial, thus resolving the first nontrivial case of a conjecture of Faber and Voloch.

Quantum effects and quantum chaos in multidimensional tunneling.

The ground-state energy splitting due to tunneling in two-dimensional double wells of the form V(x,y)=(x^{2}-R^{2})^{2}/8R^{2}+x^{2}-R^{2}/R^{2}γy+ω^{2}/2y^{2} is calculated. Several results are reported. First, we give a systematic WKB expansion of the splitting in series in powers of R^{-2}, each term of the series being a finite polynomial in γ^{2}. We find an ascending sequence of the values of the parameter γ characterizing the curvature of the classical path, for which the successive corrections to the leading order vanish. This effect arises because curvature of the path and quantum nature of motion cancel each other; it does not appear for one-dimensional double wells. Second, we find that for large curvatures, such as for those describing the proton transfer in a malonaldehyde and hydroxalate anion, this expansion is of no practical use. Thus, the WKB expansion is reordered to a strong coupling form, each term of the series in powers of R^{-2} being an infinite series in powers of γ[over ¯]^{2}, γ[over ¯]=γ/R. Third, we find that the radius of convergence of the series is determined by the singularity at γ[over ¯]_{s}=ω/2. At the singularity the system changes its character from being a double well to become a single well. Close to this singularity the classical action and its first quantum correction are found to be nonanalytic functions of γ[over ¯], most likely of the form [1-(γ[over ¯]/γ[over ¯]_{s})^{2}]^{α}, where α=1/2 and α=-1/2 for the classical action and its first quantum correction, respectively. Since in the semiclassical regime of large R the splitting is exponentially dependent on the value of the classical action and its first quantum correction, close to the singularity we establish strong sensitivity of the splitting on slight variations of the parameter γ[over ¯] entering the Hamiltonian linearly.

Quantitative properties of the non-properness set of a polynomial map

Let f be a generically finite polynomial map f: mathbb {C}^nrightarrow mathbb {C}^m of algebraic degree d. Motivated by the study of the Jacobian Conjecture, we prove that the set S_f of non-properness of f is covered by parametric curves of degree at most d-1. This bound is best possible. Moreover, we prove that if Xsubset mathbb {R}^n is a closed algebraic set covered by parametric curves, and f: Xrightarrow mathbb {R}^m is a generically finite polynomial map, then the set S_f of non-properness of f is also covered by parametric curves. Moreover, if X is covered by parametric curves of degree at most d_1, and the map f has degree d_2, then the set S_f is covered by parametric curves of degree at most 2d_1d_2. As an application of this result we show a real version of the Białynicki-Birula theorem: Let G be a real, non-trivial, connected, unipotent group which acts effectively and polynomially on a connected smooth algebraic variety Xsubset mathbb {R}^n. Then the set Fix(G) of fixed points has no isolated points.

On Schur Superfunctors

We introduce super-analogues of the Schur functors defined by Akin, Buchsbaum and Weyman. These Schur superfunctors may be viewed as characteristic-free analogues of finite dimensional irreducible polynomial representations of the Lie superalgebra 𝔤𝔩(m|n) studied by Berele and Regev. Our construction realizes Schur superfunctors as objects of a certain category of strict polynomial superfunctors. We show that Schur superfunctors are indecomposable objects of this category. In characteristic zero, these correspond to the set of all simple supermodules for the Schur superalgebra, S(m|n, d), for any m, n, d ⩾ 0. We also provide decompositions of Schur bisuperfunctors in terms of tensor products of skew Schur superfunctors.

Critical dynamics of variable-separated affine correspondences

We examine affine correspondences of the form g(y)=f(x), for f and g polynomials satisfying deg(g) < deg(f), with the property that every critical point of the correspondence admits at least one finite forward orbit. In the case g(y)=y, this reduces to the study of post-critically finite polynomials, and our main result extends earlier finiteness results of the author. Specifically, we show that the collection of such correspondences of a given bidegree coincides with a subset of the parameter space of bounded Weil height. We also show that there are no non-trivial holomorphic families of correspondences with the above-described property.

Nonrelativistic anti-Snyder model and some applications

In this paper, we examine the (2[Formula: see text]+[Formula: see text]1)-dimensional Dirac equation in a homogeneous magnetic field under the nonrelativistic anti-Snyder model which is relevant to doubly/deformed special relativity (DSR) since it exhibits an intrinsic upper bound of the momentum of free particles. After setting up the formalism, exact eigensolutions are derived in momentum space representation and they are expressed in terms of finite orthogonal Romanovski polynomials. There is a finite maximum number of allowable bound states [Formula: see text] due to the orthogonality of the polynomials and the maximum energy is truncated at [Formula: see text]. Similar to the minimal length case, the degeneracy of the Dirac–Landau levels in anti-Snyder model are modified and there are states that do not exist in the ordinary quantum mechanics limit [Formula: see text]. By taking [Formula: see text], we explore the motion of effective massless charged fermions in graphene-like material and obtained a maximum bound of deformed parameter [Formula: see text]. Furthermore, we consider the modified energy dispersion relations and its application in describing the behavior of neutrinos oscillation under modified commutation relations.