Polynomial Equation Of Degree Research Articles

Solving a Polynomial Equation: Some History and Recent Progress

The classical problem of solving an nth degree polynomial equation has substantially influenced the development of mathematics throughout the centuries and still has several important applications to the theory and practice of present-day computing. We briefly recall the history of the algorithmic approach to this problem and then review some successful solution algorithms. We end by outlining some algorithms of 1995 that solve this problem at a surprisingly low computational cost.

Effects of chronic hypoxia, normoxia and hyperoxia on larval development in the beetle Tenebrio molitor

The effect of chronic hypoxia and hyperoxia on survival, developmental time, body mass, growth and moulting in Tenebrio molitor larvae were studied. Larvae were reared from eggs to pupae at 25 °C, 70% relative humidity, in 3 different oxygen environments: normoxia (21%), hypoxia (10%) and hyperoxia (40%) balanced by nitrogen (at sea level). 20% of larvae in hypoxia, and 96% in normoxia and hyperoxia survived to pupation. It was estimated that hypoxic, normoxic and hyperoxic larvae moulted 12.3, 5.8 and 3.9 times, respectively, between day 38 and the beginning of pupation. Body mass growth patterns were fitted to 3rd degree polynomial equations. The lowest growth rate was found in the hypoxic group. No differences in growth rate during the steep mass increase phase were found between the hyperoxic and the normoxic larvae. Despite a low total number of moults in hyperoxia, larvae finally reached a body mass similar to those reared in normoxia. This suggests that oxygen availability affects moulting. Our results indicate that the trigger for moulting is related to the atmospheric oxygen availability and to the size of the tracheal system, which must supply sufficient oxygen to meet tissue demands.

Biplot display for diagnostic in a two-level regression model for growth curve analysis

This work presents a formulation of the hierarchical general linear model with two levels for growth curve modelling. This formulation considers a Kth degree polynomial equation for each individual growth curve like a one-level regression equation. In the second level the regression coefficients are modelled considering q explanatory variables at the individual level, this procedure permits an explication of variability between growth curves. The second level of the model is expressed as a multivariate linear regression model, obtaining a multivariate residuals matrix for the second level. The biplot is proposed as a graphical tool that permits to do diagnostics using this residual matrix. An example of yield growth curves of an orthogonal centered design matrix on a randomized blocks experimental design is used for presenting an illustration of these proposals.

AN EFFICIENT COMPUTATIONAL SCHEME FOR THE VIBRATION ANALYSIS OF HIGH TENSION CABLE NETS

An approach base on group theory is described for calculating the eigenvalues and hence natural circular frequencies of vibration of cable nets consisting of two families of high tensioned cables, each cable lying in a vertical plane, and with the projections of cables on the horizontal plane comprising two perpendicular sets of lines. The cable net systems are assumed to have ndegrees of freedom in the form of vertical motions of masses concentrated at the cable intersections. After briefly outlining the linear cable net theory that forms the basis for the illustration of the proposed group-theoretic approach, those concepts of symmetry groups and representation theory that are fundamental to the present development are summarized. The actual computational scheme is then outlined, followed by a step-by-step illustration of the proposed procedure, the proposed method makes use of the full symmetry of the cable network in a systematic and highly efficient manner: the problem is decomposed into mutually independent subspaces spanned by symmetry adapted variables, for which the required eigenvalues are simply obtained through the solution of a small number of polynomial equations each of degree a fraction of n(instead of through the solution of a single polynomial equations of degree n, as yielded by a conventional analysis), resulting is substantial simplifications in the computation of the natural frequencies of vibration of the cable network. Once eigenvalues have been obtained, calculation of the system eigenvector components is carried out on the basis of parent subspaces of the eigenvalues, with mode shapes of the cable net following through a relatively trivial final step.

Computation of packet response time via Lagrange series expansion

Packets are processed in bulk at the nodes of a computer communication network. A complete analytical solution of the packet response time in a node processor is derived. The node processor is modelled as an M/M[K]/1 queue. In this queue, the packets arrive according to a Poisson process, and the service times are exponentially distributed. In each service period the number of packets served is the minimum of K and the number of packets present in the queue. In existing literature it is assumed implicitly that a root of a polynomial equation, lying in the interval (0, 1), will be evaluated numerically. A Lagrange series expansion of a root of this polynomial equation of degree (K+1) is derived. The proposed solution technique can be used for evaluating analytically other Markovian bulk service queues, and also the Er/M/1 queue.

A Generalized Maximum Pseudo-Likelihood Estimator for Noisy Markov Fields

In this paper we present an asymptotic estimator, obtained by observing a noisy image, for the parameters of both a stationary Markov random field and an independent Bernoulli noise. We first estimate the parameter of the noise by solving a polynomial equation of moderate degree (about 6-7 in the one-dimensional Ising model and about 10-15 in the two-dimensional Ising model, for instance) and then apply the maximum pseudo-likelihood method after removing the noise. Our method requires no extra simulation and is likely to be applicable to any Markov random field, in any dimension. Here we present the general theory and some examples in one dimension; more interesting examples in two dimensions will be discussed at length in a companion paper.

Forward displacement analysis of the decahedral variable geometry truss manipulator

A forward displacement analysis in closed form is performed for the decahedral variable geometry truss manipulator. Based on the simplified geometry model, a system of the constraint equations are formulated. With elimination, this system of equations can be reduced to a 48th degree polynomial equation in one output variable.

Symphysis-fundus height: construction of a new Swedish reference curve, based on ultrasonically dated pregnancies.

Repeated measurements of the symphysis-fundus (SF) distance is generally regarded as an acceptable screening instrument for antenatal detection of intrauterine growth retardation (IUGR). In the present investigation an SF curve is constructed, based on measurements from 1650 Swedish women with ultrasound dated pregnancies. The SF measurements were included in a cross-sectional stratified manner; only one measurement from each woman was used and the same number of measurements (n = 75) was used for each gestational week from 20 through 41. A regression analysis by a polynomial equation of third degree was applied on the means and standard deviations of the SF measurements in each week respectively to construct a graph of fundal height. This SF curve was compared with the SF curve in common use in Sweden (constructed by Westin); in the new curve the mean measurement was about one cm higher up to 37 gestational weeks, after which time the curves almost coincided. Neither did the mean SF height differ between smokers and non-smokers nor between nulliparous and parous women. The mean SF difference between the heaviest women (> or = 79 kg) and the lightest (> or = 53 kg) was about two cm throughout pregnancy. The validity of the constructed SF curve as a screening instrument for IUGR needs to be investigated in a different study.

Prediction of reformate research octane number by FT-i.r. spectroscopy

Fourier transform infrared (FT-i.r.) spectroscopy was used to calculate the research octane number (RON) of naphtha feed and reformate during the course of performance evaluation of reforming catalysts. Five absorption regions that correspond to aliphatic and aromatic bands within the mid-infrared region were utilized for developing the correlation for RON estimation. Statistical methods were used to derive third degree polynomial equations which could give a better estimate of engine RON. The technique was found to give excellent correlation ( R 2 = 0.95) with the engine RON data.

Seasonal changes of understorey herbage yield in relation to light intensity and soil moisture content in a Pinus pinaster plantation

The relationship of understorey herbage yield with light intensity and soil moisture content was investigated in a 20-year-old Pinus pinaster plantation in Kilkis of Macedonia, northern Greece, in three seasons (fall, winter and spring). The plantation was thinned at three density levels, high, medium and low (2500, 1750 and 1000 trees/ha respectively) and it was seeded with Dactylis glomerata. In addition, two fertilization levels (0 and 110 N+150 P2O5 kg/ha) and grazing intensity levels (0 and 0.8 sheep/ha/yr) were applied in a split-split plot experimental design. It was found that herbage yield was highest in the ungrazed and fertilized low tree density stands in all seasons. In the medium tree density stand, although light was lacking, fertilized plots produced higher herbage yield as compared to unfertilized low tree density stands while fertilization is of no use in close stands if herbage production is the goal. Animals seemed to prefer fertilized and medium density stands. Third degree polynomial equations were the best to explain herbage variation through the seasons both in fertilized and unfertilized plots.

ACC-derived Ethylene Production, a Sensitive Test for Seed Vigor

ACC-derived ethylene production was used as an index of seed vigor of lettuce (Lactuca sativa L.), cabbage [Brassica oleracea (Capitata Group)], tomato (Lycopersicon esculentum Mill.), snap bean (Phaseolus vulgaris L.), and sweet corn (Zea mays L.) seeds. Seeds were aged at 40C and 93% relative humidity over saturated solution of KH2PO4 for various times to obtain seeds of differing vigor. Naturally aged lettuce seeds, differing in vigor, were also used. Depending on the seed type, 0.25 to 2 mm ACC (saturating dose) was needed to produce maximal amounts of ethylene. Seeds in the presence of ACC produced a much larger amount of ethylene than those in the absence of ACC, the ACC-derived ethylene could be detected before germination, and ACC had no adverse effect on germination. ACC-derived ethylene production paralleled vigor loss as determined by a decrease in percentage germination over a soak period required for complete germination of nonaged seeds (16 hours for lettuce, 24 hours for cabbage, and 48 hours for tomato and sweet corn), an increase in mean germination time (determined for lettuce only), and a decrease in seedling growth (determined for snap bean only). Second degree polynomial and logarithmic equations generated for the relationship of ACC-derived ethylene production to germination or growth parameters following seed aging, provided good to excellent fit. As a vigor test, the ACC-ethylene procedure has several advantages over the non-ACC ethylene procedure: It improves the sensitivity of the test by enhancing ethylene production, permits detection of small differences in vigor, and allows detection of ethylene before germination within a few hours of soaking. Chemical name used: 1-aminocyclopropane-1-carboxylic acid (ACC).

339 ACC-DERIVED ETHYLENE PRODUCTION, A RAPID TEST FOR SEED VIGOR

1-aminocyclopropane-1-carboxylic acid (ACC) was used to measure seed vigor of lettuce, cabbage, tomato, snap bean and sweet corn seeds. Accelerated aging at 40C and 93% RH over saturated solution of KH2PO4 and natural aging under ambient storage conditions (5-7C, 28 to 60% RH) were used to obtain seeds of differing vigor levels. Depending on the type of seeds, the amount of ACC needed for maximal ethylene production (saturating dose) ranged from 0.25 to 2mM. Seeds produced much larger amounts of ethylene in the presence than in absence of ACC, the ACC-derived ethylene was detected prior to germination, and ACC had no adverse effect on germination. ACC-derived ethylene production paralleled vigor loss as determined by percentage germination, mean germination time (for lettuce only) and seedling growth (for snap bean only). Second degree polynomial and logarithmic equations generated for the relationship of ACC-derived ethylene production to germination or growth parameters following seed aging, provided good to excellent fit. As a vigor test, the ACC-ethylene procedure has several advantages over the non-ACC ethylene procedure: it greatly improves the sensitivity of the test by enhancing ethylene production, it permits detection of small differences in vigor levels and it allows detection of ethylene prior to germination within a few hours of soaking.

Maximum likeihood estimation of an intraclass correlation in a bivariate normal distribution with missing observations

The maximum likeihood estimate is considered for an intraclass correlation coefficent in a bivariate normal distribution when some observations on either of the varibles are missuing. The estimate is given as the soulution of a polynomial equation of degree seven. An approximate confidence interval and a test procedure for the intraclass correlation are constricted based on an asymptotic variance stabilizing transformation of the resulting estimator. The distributional results are also considered under violation of the normality assumption. A Monte Carlo study was performed to examine the finite sample properties of the maximum likelihood estimator and to evaluate the proposed procedures for hypotheses testing and interval estimation.

Closed-Form Direct Position Analysis of a 5–5 Parallel Mechanism

The paper presents the closed form direct displacement analysis for a class of Stewart platform-type parallel mechanisms whose general feature consists of six legs which meet five distinct points both in the base and in the movable output link. Out of the two possible arrangements, only one is here analyzed in detail. Given a set of actuator displacements the analysis provides all the possible locations of the platform relative to the base. The analysis results in a 40th degree polynomial equation in one unknown. The roots of the equation provide in the complex field forty closures of the mechanism. This new result has been numerically verified by the inverse displacement analysis.

Non-conservative dynamic stiffness analysis of thin-walled structures

A dynamic stiffness method is introduced to analyse thin-walled structures under the influence of a follower axial force and an in-plane moment. When harmonic oscillation is concerned, both the spatial and temporal time discretization errors are eliminated to give an exact solution in the classical sense. When warping effects are included, the governing differential equations correspond to a matrix polynomial eigenproblem of third-order matrices and degree four. The determinant equation is expanded analytically to give a scalar polynomial equation of degree 12 providing 12 integration constants for the 12 nodal displacements of the thin-walled beam member (excluding the uncoupled axial displacements). The generalized nodal forces are related to the nodal displacements analytically resulting in the exact dynamic stiffness matrix. The natural boundary conditions are modified to cater for the non-conservative loads. Numerical examples show that the interaction diagram of natural frequency against the constant in-plane moment does not have a monotonic change of slope. This is due to the fact that the constant in-plane moment softens the flexural modes while hardening the torsional modes. In some cases, isola loops are possible in the interaction diagram. Examples on plane frames are also given.

Semicontinuous alcoholic fermentation of sugar cane blackstrap molasses by pressed yeast

A third degree polynomial equation correlates the fermentation time and the volume fraction of inoculum in semicontinuous ethanol fermentation of sugar cane blackstrap molasses by pressed yeast. The volume of the inoculum varied from 40% to 92% of the reactor working volume. The highest ethanol productivity was obtained when the volume of the inoculum was 58% of the working volume.

DISCRETE AND CONTINUOUS AMBIGUITIES IN THE ISOSPIN ANALYSIS OF THE B→πK DECAYS

A polynomial equation of the fifth degree, which solves completely the theoretical problem of the isospin analysis of the B→πK decays, is derived. It is shown that the problem has in general an even number (two or four) of non-trivially-related solutions. There are six particular cases of null measure, but which might be relevant due to the experimental errors, in which the problem has a continuous set of solutions. It is concluded that the isospin analysis, even if experimentally feasible, has a large probability of being inconclusive.

Dynamic stiffness analysis of thin-walled structures

A dynamic stiffness method is introduced to analyse thin-walled structures to reduce spatial discretisation errors. Where harmonic oscillation is concerned, time discretisation errors are also eliminated to give an exact solution in a classic sense. Constant axial forces and in-plane moments are included for dynamic buckling analysis. When warping effects are included, the governing differential equations correspond to a matrix polynomial eigenproblem of order 3 matrices and degree 4. The determinant equation is expanded analytically to give a scalar polynomial equation of degree 12 providing 12 integration constants for the 12 nodal displacements of the thin-walled beam member (excluding the uncoupled axial displacements). The generalised nodal forces are related to the nodal displacements analytically resulting in the exact dynamic stiffness matrix. Numerical examples show that the interaction diagram of natural frequency against the constant in-plane moment do not have monotonic change of slope. This is due to the fact that the constant in-plane moment softens the flexural modes while hardening the torsional modes. Examples on frames are also given.

Mixtures of tight-binding enzyme inhibitors. Kinetic analysis by a recursive rate equation

When two or more tight-binding inhibitors are present in an enzyme assay, the equation that relates the initial velocity v to the concentration of reactants cannot be written in an algebraically explicit form. Rather, for n inhibitors it is an implicit polynomial equation of degree n + 1 with respect to v. The complexity of the polynomial coefficients dramatically increases with each added inhibitor. Solving the transcendental rate equation by traditional methods of numerical mathematics has proven tedious because of the sensitivity of these methods to initial estimates and because of the existence of multiple roots. However, the equation can be rearranged into a convenient recursive form, one in which the velocity appears on both sides and the solution is found iteratively. The algebraic form of the recursive rate equation is remarkably simple and differs from the rate equation for classical rather than tight-binding inhibition only by an added term. The numerical stability and the speed of convergence were tested on the case of two competitive inhibitors. Initial estimates of velocity that spanned 12 orders of magnitude converged within five iterations. The velocities computed with the recursive method for a single tight-binding inhibitor were identical with the values predicted by the Morrison equation. The method is used to analyze experimental data for the inhibition of rat liver dihydrofolate reductase by mixtures of the anticancer drug methotrexate and its metabolic precursor form, methotrexate-α-aspartate (a prodrug)

A Complete Solution for the Inverse Kinematic Problem of the General 6R Robot Manipulator

The inverse kinematic problem of the general 6R robot manipulator is completely solved by means of a 16th degree polynomial equation in the tangent of the half-angle of a revolute joint. An algorithm is developed to compute the desired joint angles of all possible configurations of the kinematic chain for a given position of the end-effector. Examples for robots with maximal 16 different configurations show that the polynomial degree 16 is the lowest possible for the general 6R robot manipulator. Further, a numerical method for the determination of the boundaries of the workspace and its subspaces with different numbers of configurations is developed. These boundaries indicate the singular positions of the end-effector.