Parametric Polynomial Research Articles

ANALYTICAL SOLUTION OF THE FREQUENCY OF THE LOAD OSCILLATION AT AN ARBITRARY GIRDER NODE IN THE SYSTEM MAPLE

Introduction. We study the oscillations of a massive load on a planar statically definable symmetric truss of a regular type with parallel belts. Truss weight is not included. Free vertical oscillations are considered. The stiffness of the truss rods is assumed to be the same, the deformations are elastic. Lattice of the truss is double with descending braces and racks. New in the formulation and solution of the problem is the analytical form of the solution, which makes it possible in practice to easily evaluate the frequency characteristics of the structure depending on an arbitrary number of truss panels and the location of the load. Materials and methods. The operators and methods of the system of computer mathematics Maple are used. To determine the forces in the rods, the knotting method is used. The common terms of the sequence of coefficients of solutions for different numbers of panels are obtained from solving linear homogeneous recurrent equations of various order, obtained by special operators of the Maple system. Dependence on two arbitrary natural parameters is revealed in two stages. First, solutions for fixed load positions are found, then these solutions are summarized into one final formula for frequency. Results. By a series of individual solutions to the problem of load oscillation using the double induction method, it was possible to find common members of all sequences. The solution is polynomial in both natural parameters. Graphs constructed for particular cases, showed the adequacy of the approach. The discontinuous non-monotonic nature of the intermittent change depending on the number of truss panels and some other features of the solution are noted. Conclusions. It is shown that the induction method, previously applicable mainly to statics problems with one parameter (number of truss panels), is fully operational to the problems of the oscillations of system with two natural parameters. It should be noted that significant labor costs and a significant increase in the time symbolic transformations in such tasks

Influence of Krawtchouk and Charlier moment’s parameters on image reconstruction and classification

The discrete orthogonal moments such as Krawtchouk and Charlier moments have been used successfully in the field of image analysis. These moments are based on discrete orthogonal polynomials that depend on the locality parameters (p for Krawtchouk and a1 for Charlier). However, the choice of the value of these parameters during the image analysis remains a problem without solutions. In the absence of an analytical study to make these choices, we will study experimentally, in this article, the influence of the variation of the parameters of discrete orthogonal polynomials on the analysis and in particular on the reconstruction and classification of the image. The results of the simulation show the influence of the variation of discrete orthogonal moments parameters of Krawtchouk and Charlier on the quality of reconstruction of the binary, grayscale and colors images in order to make the choice of the numerical value of each parameter. This allows us to obtain a range of parameter values that will be exploited in the classification task of image forms using the invariant moments of Krawtchouk and Charlier.

A NOVEL MODEL FOR ORIENTATION OF LINEAR OPTICAL SATELLITE IMAGES BASED IN THE ADAPTATION OF THE ORBIT-ATTITUDE MODEL

In this paper a novel model to orient a pushbroom linear optical satellite image is proposed. This one is based in the adaptation of the Orbit-Attitude model for use of the Modified UCL Kepler platform model. It has only the components of satellite position and velocity as unknowns. This implies not only the reduction of unknowns but also the elimination of initial adjustments of the orbit to estimate the polynomial parameters. In order to validate the model, four experiments were conducted using one HRC-CBERS 2B image. Two distributions of ground controls points (GCPs) were tested, 70 and 35 GCPs. A quantity of 43 check points (CPs) was used to analyze the planimetric accuracies of the orientations. For comparisons purpose the image was also oriented with platform model using 2nd order polynomial. The results showed that the planimetric accuracy difference in the use of the two models is 4 and 6 centimeters in the two mentioned configurations of GCPs, respectively. However, the main advantage of applying the proposed model is the direct use of ephemeris without the necessity of interpolations and preadjustments, which makes the orientation process simpler.

Yield limit dependence of titanic alloy VТ14 from the stress state parameters

The stress state mode has influence on the yield limit of construction materials that causes the necessity of account of this phenomenon at determination of this material resource. A quantitative characterization of the stress state can be given by means of two parameters which describe, accordingly, by the volume constituent of tensor and stress deviator mode. Influence of the volume constituent of tensor is characterized by the stress triaxiality parameter, and stress deviator mode - by the Lode parameter, Lode corners or normalized third deviatoric stress invariant. In paper is proposed the yield limit dependences as second degree polynomials of the stress state parameters. These dependences are concreted for the titanic alloy VТ14 on the basis of experiments with the loading of tubular specimen by axial force and intrinsic pressure. Supposed, that the stress state in the wall of specimen corresponded to what exists in on his middle surface. At the calculation of the stress intensity not only axial and circuitous but also radial stresses token into account, that resulted in reduction of divergence of yield limits at the different stress state from 12,4 to 10,2 %. The results of calculations showed that the best from the polynomials is equation with contain members: the first degree from both stress state parameters, multiplicity of these parameters and second degree from the stress triaxiality parameter. From the point of view of statistical meaningfulness appeared the best equation that as stress deviator parameter contained the module of the normalized third deviatoric stress invariant. The level of meaningfulness of this equalization on the criterion of Fisher and constants on the criterion of Student exceeded a size 94 %. The application of the proposed regression equation for describing the dependence of the yield strength of a titanium alloy VT14 on parameters of the stress state type makes it possible to make fuller use of the resource capabilities of this material.

Excitation control for three-dimensional Peregrine solution and combined breather of a partially nonlocal variable-coefficient nonlinear Schrödinger equation

A (2 + 1)-dimensional variable-coefficient partially nonlocal nonlinear Schrodinger equation is considered, and analytical Peregrine solution (PS) and combined Akhmediev breather (AB) are presented from a reduced transformation and the Darboux transformation method. Based on these analytical solutions, the excitation control for three-dimensional PS and combined AB is investigated by comparing values between the maximum value of the effective propagation distance $$\zeta _{\max }$$ and the crest position $$\zeta _{0}$$ (for the first-order PS and second-order PS) or the crest position $$\zeta _n(n=1,2,3)$$ (for the combined AB). If $$\zeta _{\max } \zeta _{0}$$, the anterior excitation, crest excitation, and tail excitation of the first-order PS and second-order PS can be studied, respectively. Similarly, if $$\zeta _{\max } \zeta _{n}$$, the anterior excitation, crest excitation, and tail excitation of the first first-order PS, the second-order PS and the second first-order PS in the combined AB can be discussed, respectively. The $$k+1$$-layer excited structures for the first-order PS, second-order PS, and the combined AB can be constructed for the fixed value of the Hermite polynomial parameter k.

Time-Optimal Trajectory Planning along Parametric Polynomial Lane-Change Curves with Bounded Velocity and Acceleration: Simulations for a Unicycle Based on Numerical Integration

G2 lane-change path imposes symmetric conditions on the path geometric properties. This paper presents the comparative study of time-optimal velocities to minimize the time needed for traversal of three planar symmetric parametric polynomial lane-change paths followed by an autonomous vehicle, assuming that the neighboring lane is free. A simulated model based on unicycle that accounts for the acceleration and velocity bounds and is particularly simple for generating the time-optimal path parameterization of each lane-change path is adopted. We base the time-optimal trajectory simulations on numerical integration on a path basis under two different end conditions representing sufficient and restricted steering spaces with remarkable difference in allowable maximum curvature. The rest-to-rest lane-change maneuvering simulations highlight the effect of the most relevant path geometric properties on minimal travel time: a faster lane-change curve such as a quintic Bezier curve followed by a unicycle tends to be shorter in route length and lower in maximum curvature to have achievable highest speed at the maximum curvature points. The results have implications to path selection for parallel parking and allow the design of continuous acceleration profile via time scaling for smooth, faster motion along a given path. This could provide a reference for on-road lane-change trajectory planning along a given path other than parametric polynomials for significantly more complex, complete higher-dimensional highly nonlinear dynamic model of autonomous ground vehicle considering aerodynamic forces, tire and friction forces of tire-ground interaction, and terrain topology in real-world.

Solving geoinformatics parametric polynomial systems using the improved Dixon resultant

Improvements in computational and observational technologies in geoinformatics, e.g., the use of laser scanners that produce huge point cloud data sets, or the proliferation of global navigation satellite systems (GNSS) and unmanned aircraft vehicles (UAVs), have brought with them the challenges of handling and processing this “big data”. These call for improvement or development of better processing algorithms. One way to do that is integration of symbolically presolved sub-algorithms to speed up computations. Using examples of interest from real geoinformatic problems, we will discuss the Dixon-EDF resultant as an improved resultant method for the symbolic solution of parametric polynomial systems. We will briefly describe the method itself, then discuss geoinformatics problems arising in minimum distance mapping (MDM), parameter transformations, and pose estimation essential for resection. Dixon-EDF is then compared to older notions of “Dixon resultant”, and to several respected implementations of Grobner bases algorithms on several systems. The improved algorithm, Dixon-EDF, is found to be greatly superior, usually by orders of magnitude, in both CPU usage and RAM usage. It can solve geoinformatics problems on which the other methods fail, making symbolic solution of parametric systems feasible for many problems.

Identification of parametric models in the frequency-domain through the subspace framework under LMI constraints

In this paper, an algorithm to identify parametric systems with an affine (or polynomial) parameter dependence through the subspace framework is proposed. It stands as an extension of the standard subspace-based algorithm which is well established in the linear time invariant (LTI) case. The formulation is close to the LTI identification scheme and simply involves frequency-domain data obtained at different operating points (the parameters are frozen during each experiment). The proposed algorithm allows to identify directly a parameter-dependent model instead of interpolating multiple local models as in traditional local approaches. Another contribution is that it is possible to impose the poles location through linear matrix inequalities (LMI) constraints, extending what has been done in the LTI case. This technique is applied to a numerical example and to real industrial frequency-domain data originating from an open-channel flow simulation for hydroelectricity production.

Maximum Likelihood Estimation of Symmetric Group-Based Models via Numerical Algebraic Geometry

Phylogenetic models admit polynomial parametrization maps in terms of the root distribution and transition probabilities along the edges of the phylogenetic tree. For symmetric continuous-time group-based models, Matsen studied the polynomial inequalities that characterize the joint probabilities in the image of these parametrizations (Matsen in IEEE/ACM Trans Comput Biol Bioinform 6:89–95, 2009). We employ this description for maximum likelihood estimation via numerical algebraic geometry. In particular, we explore an example where the maximum likelihood estimate does not exist, which would be difficult to discover without using algebraic methods.

On Multivariate Hermitian Quadratic Forms

Multivariate Hermitian quadratic forms play an important role in the real quantifier elimination algorithm based on the computation of comprehensive Grobner systems introduced by V. Weispfenning and further improved by us. Our algorithm needs the computation of a certain type of saturation ideal in a parametric polynomial ring. In this paper, we study multivariate Hermitian quadratic forms in more detail and show several facts which have special importance in a parametric polynomial ring. Our results enable us to have an efficient method to compute the saturation ideal, which brings us a drastic improvement of our real quantifier elimination software.

A new age-period-cohort model for cancer surveillance research

We develop a new age-period-cohort model for cancer surveillance research; the theory and methods are broadly applicable. In the new model, cohort deviations are weighted to account for the variable number of periods that each cohort is observed. Weighting ensures that the fitted rates can be naturally expressed as a function of age × a function of period × a function of cohort. Furthermore, the age, period, and cohort deviations are split into orthogonal quadratic components plus higher-order terms. These decompositions enable powerful combination significance tests of first- and second-order age, period, and cohort effects. The regression parameters of the orthogonal quadratic polynomials (global curvatures) quantify how fast on average the trends in the rates are changing. Importantly, the global curvature for cohort determines the least squares slope of the expected annual percentage changes by age group versus age (local drifts), thereby providing a powerful one-degree-of-freedom test of age-period interactions. We introduce new estimable functions, including age gradients that quantify the rate of change of the longitudinal and cross-sectional age curves at each attained age, and gradient shifts that quantify how the cross-sectional age trend varies by period. We illustrate the new model using nationally representative multiple myeloma incidence. Comprehensive proofs are given in technical appendices. We provide an R package.

PRM130 - OPEN-SOURCE SOFTWARE FOR DEVELOPING COMPUTATIONALLY EFFICIENT COST-EFFECTIVENESS MODELS TO EVALUATE SEQUENTIAL TREATMENT STRATEGIES IN ONCOLOGY

hesim is an R package for cost-effectiveness modeling that provides a general framework for integrating statistical analysis for parameter estimation with simulation techniques for economic evaluation. Cost-effectiveness models are built by specifying a model structure, which consists of a set of statistical models for disease progression, utility, and costs as a function of estimated parameters and input data describing the patient population and treatment strategies. The package currently supports N-state partitioned survival models (PSMs), but PSMs are limited because they do not account for structural relationships between survival endpoints. Our aim was to extend hesim for the economic evaluation of sequential treatment strategies in oncology. We developed software for simulating semi-Markov and inhomogeneous continuous time state transition models. Hazard functions for semi-Markov and inhomogeneous models are estimated using “clock-reset” and “clock-forward” multi-state models, respectively. State membership in semi-Markov models is simulated using a patient-level simulation and transition probabilities in inhomogeneous models are estimated using the Aalen-Johansen estimator. In patient-level simulations, time on each treatment can be simulated from time-to-event distributions. Survival functions for state transitions and treatment duration can be fit using parametric, spline, or fractional polynomial models. Separate survival models can be joined together (e.g., pre/post follow-up) to assess assumptions during the extrapolation period. The core code is written in C++ to facilitate probabilistic sensitivity analysis (PSA) and patient-level simulation. The software is illustrated with a cost-effectiveness analysis of sequential treatment strategies in oncology along with a new method for multi-state network meta-analysis with summary-level data. A new toolkit for developing state-transition models for cost-effectiveness modeling in oncology is available. State transition models with hesim are computationally efficient and allow users to model sequential treatment strategies, conduct scenario analyses for the extrapolation period, and conduct PSAs.

Propagation of electromagnetic waves through non-uniform dielectric layers

The propagation of TE- and TM-polarized electromagnetic (EM) waves through a dielectric layer with spatial variation of the refractive index is reported in this paper. The relative permittivity of the layer is assumed to be graded along the thickness direction, and its spatial variation is described by a quartic polynomial. The corresponding mode equations are Helmholtz equations with variable coefficients that can be transformed to a triconfluent Heun equation, a local exact solution of which is expressed in terms of triconfluent Heun functions. The solutions cover many particular cases owing to a variability of four optional parameters (coefficients) of the quartic polynomial. The general local solution for a TE-polarized EM wave is employed for the calculation of transmission properties of a periodic one-dimensional photonic crystal using the Floquet theory.

Identification of Cascade Dynamic Nonlinear Systems: A Bargaining-Game-Theory-Based Approach

Cascade dynamic nonlinear systems can describe a wide class of engineering problems, but little efforts have been devoted to the identification of such systems so far. One of the difficulties comes from its non-convex characteristic. In this paper, the identification of a general cascade dynamic nonlinear system is rearranged and transformed into a convex problem involving a double-input single-output nonlinear system. In order to limit the estimate error at the frequencies of interest and to overcome the singularity problem incurred in the least-square-based methods, the identification problem is, thereafter, decomposed into a multi-objective optimization problem, in which the objective functions are defined in terms of the spectra of the unbiased error function at the frequencies of interest and are expressed as a first-order polynomial of the model parameters to be identified. The coefficients of the first-order polynomial are derived in an explicit expression involving the system input and the measured noised output. To tackle the convergence performance of the multi-objective optimization problem, the bargaining game theory is used to model the interactions and the competitions among multiple objectives defined at the frequencies of interest. Using the game-theory-based approach, both the global information and the local information are taken into account in the optimization, which leads to an obvious improvement of the convergence performance. Numerical studies demonstrate that the proposed bargaining-game-theory-based algorithm is effective and efficient for the multi-objective optimization problem, and so is the identification of the cascade dynamic nonlinear systems.

Regular Perturbation of Inviscid Burger Equation in a Traffic Flow Problem

The Inviscid Burger Equation is one of the models which can be applied to one-way traffic flow problem. This model was constructed using transport equation in which the flux density following The Greenshield’s approximation. On its development, distinct flux densities were performed by varying the approximation’s model. This cause changing the velocity value. In this paper, we would like to analyze the perturbation problem in the velocity value for inviscid burger equations using multiple scale method. First, we expanded the solution into polynomial of the perturbation parameter and substitute it into the equation. Next, we obtained new equations based on the identical power of it which are quasilinear and linear equations. The new equations were simply solved using Lagrange-Charpit’s method. Since the exact solution of inviscid burger equation can be found using the characteristic method, therefore we also compared both methods. The results show that both solutions are similar.

Structural nonlinearity and mass identification with a nonparametric model using limited acceleration measurements

Structural nonlinearity identification is critical for post-event damage detection or condition evaluation of engineering structures after strong dynamic excitation such as earthquake where structural nonlinear behaviour should be considered. Structural nonlinear restoring force provides direct indicator describing structural damage initiation and development procedure. Considering the availability of structural dynamic response measurement and the difficulty in defining a parametric model for structural nonlinearity and in estimating structure mass accurately in practice, in this article, a time-domain structural nonlinear restoring force and mass identification approach for multi-degree-of-freedom structures under incomplete excitation using limited acceleration measurements but without using any parametric models of structural nonlinear restoring force is proposed. At first, a memory fading extended Kalman filter with a weighted globl iteration (MF-EKF-WGI) is used to identify the location of nonlinearities and then a Chebyshev polynomial nonparametric model is introduced to model the nonlinear restoring force. The unscented Kalman filter is used to identify the structural responses and the parameters of the Chebyshev polynomial to describe structural nonlinearity. Numerical and experimental studies with a four-storey frame model structure equipped with a magnetorheological damper, which is employed to mimic structural nonlinear behaviour, under impact excitations are carried out to validate the performance of the proposed approach using acceleration measurements at certain degrees of freedom. Numerical and experimental results show that the proposed approach is capable of identifying both structural nonlinear restoring force and mass with acceptable accuracy even with a very rough initial mass estimation. The proposed time-domain identification approach can be used to detect structural damage initiation and development process and to evaluate energy consumption quantitatively of engineering structures under dynamic loadings.

Predictive models of the galaxies’ movement speeds and accelerations of movement on applying the Doppler Effect

This article discusses the theoretical assumptions for the development of a predictive model of galaxy movement speed and acceleration on applying the measurement data of the parameter z in the formula for the Doppler Effect. Galaxy movement speed and acceleration are understood as multidimensional continuous values when the values of vectors of speed and acceleration are calculated according to the redshift parameter z measurements in the relevant Universe time moments. Galaxy movement acceleration is obtained as a partial derivative of galaxy speed with respect to the time t of oscillations of the received redshift signal. The theoretical expressions of the functions of galaxy speed and acceleration are obtained by approximating in the form of a polynomial when the values of the polynomial parameters are established by the least squares method on applying operators of the Matlab set of programs.

Measuring Vapor and Liquid Concentrations for Binary and Ternary Systems in a Microbubble Distillation Unit via Gas Sensors

A cost effective, fast, and accurate technique was needed to measure the vapor composition of a binary system (ethanol-water) and also that of a liquid composition in a ternary system (acetic acid-acetol–water) in a microbubble distillation unit. Cheap TGS-series gas sensors were used for this purpose with both calibrations and measurements carried out in a specially designed chamber. A single parameter polynomial regression was fitted to the binary system, and a two parameter polynomial with an interaction term was fitted to the ternary system. The correlation coefficient, R-squared, was found to be greater than 0.99 for both systems, thus validating the implementation of this novel sensor.

Galois realizations with inertia groups of order two

There are several variants of the inverse Galois problem which involve restrictions on ramification. In this paper, we give sufficient conditions that a given finite group [Formula: see text] occurs infinitely often as a Galois group over the rationals [Formula: see text] with all nontrivial inertia groups of order [Formula: see text]. Notably any such realization of [Formula: see text] can be translated up to a quadratic field over which the corresponding realization of [Formula: see text] is unramified. The sufficient conditions are imposed on a parametric polynomial with Galois group [Formula: see text] — if such a polynomial is available — and the infinitely many realizations come from infinitely many specializations of the parameter in the polynomial. This will be applied to the three finite simple groups [Formula: see text], [Formula: see text] and [Formula: see text]. Finally, the applications to [Formula: see text] and [Formula: see text] are used to prove the existence of infinitely many optimally intersective realizations of these groups over the rational numbers (proved for [Formula: see text] by the first author in [J. König, On intersective polynomials with nonsolvable Galois group, Comm. Alg. 46(6) (2018) 2405–2416.

Secondary concentrator optimization of a linear Fresnel reflector using Bezier polynomial parametrization

Linear Fresnel reflector (LFR) consists a promising solar concentrating technology for medium and high temperatures (150–400 °C). Usually, these collectors are designed with a secondary reflector in order to maximize the exploitation of the incident solar irradiation. The objective of this work is to optimize the secondary reflector of an LFR using Bezier polynomial parametrization which is something new in this domain. This parametrization is based on the movement of some control points (three in this case) in order to determine the optimum secondary reflector geometry. According to the final results, the optimum optical efficiency of the LFR is found to be 72.84%, while the optical efficiency in the initial design is 61.01%. Moreover, the optical efficiency is calculated for different solar angles and it is found that the optimized concentrator has superior performance compared to the initial design in all the cases. Furthermore, it is found that the optimum design leads to a relatively uniform heat flux distribution over the absorber periphery compared to the initial design. The last step is the determination of the thermal efficiency performance curve and it is found 20.50% mean thermal efficiency enhancement with the optimum design.