Solving regularized equations in the form of polynomials

The article is devoted to the determination of second-order perturbations in rectangular coordinates and components of the body motion to be under study. The main difficulty in solving this problem was the choice of a system of differential equations of perturbed motion, the coefficients of the projections of the perturbing acceleration are entire functions with respect to the independent regularizing variable. This circumstance allows constructing a unified algorithm for determining perturbations of the second and higher order in the form of finite polynomials with respect to some regularizing variables that are selected at each stage of approximation. The number of approximations is determined by the given accuracy. It is rigorously proven that the introduction of a new regularizing variable provides a representation of the right-hand sides of the system of differential equations of perturbed motion by finite polynomials. Special points are used to reduce the degree of approximating polynomials, as well as to choose regularizing variables.

The Use of a Grid Structure for Reconstructing and Forecasting the Value of Real Estate in Selected Measurement Epochs

The absence of sufficiently long time series of data relating to real estate prices in a selected location prevents accurate analyses and the development of precise forecasts that play an important role in a market economy. New methods and solutions are being sought to address this problem. This paper proposes an original method for reconstructing, forecasting and archiving data relating to real estate value. The proposed method involves a GRID (regular square nets) structure and it relies on the prices quoted in successive years (epochs) of measurement in a selected object. Irregularly distributed measurement data (real estate prices) acquired in successive years are transformed into a regular GRID structure to develop digital surface models that describe the distribution of data. The nodes of the GRID structure are described with the coefficients of an approximating polynomial to reconstruct and forecast real estate value in a specific location at any point in time. A GRID structure supports a comparison of changes in real estate value over time in a given node or group of nodes selected from successive measurement epochs. Individual coefficients of an approximating polynomial are generated, allocated to selected nodes, and automatically adapted to local changes in value. As a result, the observed changes can be described in a given period of time. Source data covering multiple epochs are replaced with a single file containing coefficients of approximating polynomials to reduce the size of the stored datasets and facilitate data management.

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.

Design procedure of first-order perturbations for collisions trajectories with a perturbing body

The article is devoted to the determination of firstand secondorder perturbations in rectangular coordinates and velocity components of body motion. Special differential equation system of perturbed motion is constructed. The right-hand sides of this system are finitesimal polynomials in powers of an independent regularizing variate. This allows constructing a single algorithm to determine firstand second-order perturbations in the form of finitesimal polynomials in powers of regularizing variates that are chosen at each approximation step. Following the calculations results with the developed method use, the coefficients of approximating polynomials representing rectangular coordinates and components of the regularized body speed were obtained. Comparison with the numerical results of the disturbed motion equations shows their close agreement. The developed method make it possible to calculate any visa point of body motion by the approximating polynomials.

On the heuristic of approximating polynomials over finite fields by random mappings

The behavior of iterations of functions is frequently approximated by the Brent–Pollard heuristic, where one treats functions as random mappings. We aim at understanding this heuristic and focus on the expected rho length of a node of the functional graph of a polynomial over a finite field. Since the distribution of preimage sizes of a class of functions appears to play a central role in its average rho length, we survey the known results for polynomials over finite fields giving new proofs and improving one of the cases for quartic polynomials. We discuss the effectiveness of the heuristic for many classes of polynomials by comparing our experimental results with the known estimates for different random mapping models. We prove that the distribution of preimage sizes of general polynomials and mappings have similar asymptotic properties, including the same asymptotic average coalescence. The combination of these results and our experiments suggests that these polynomials behave like random mappings, extending a heuristic that was known only for degree [Formula: see text]. We show numerically that the behavior of Chebyshev polynomials of degree [Formula: see text] over finite fields present a sharp contrast when compared to other polynomials in their respective classes.

Bicubic splines and biquartic polynomials

The paper proposes a new efficient approach to computation of interpolating spline surfaces. The generalization of an unexpected property, noticed while approximating polynomials of degree four by cubic ones, confirmed that a similar interrelation property exists between biquartic and bicubic polynomial surfaces as well. We prove that a 2×2-component C1 -class bicubic Hermite spline will be of class C2 if an equispaced grid is used and the coefficients of the spline components are computed from a corresponding biquartic polynomial. It implies that a 2×2 uniform clamped spline surface can be constructed without solving any equation. The applicability of this biquartic polynomials based approach to reducing dimensionalitywhile computing spline surfaces is demonstrated on an example.