Second-degree Equations Research Articles

CRITERION FOR SL(2,Z)-MATRIX TO BE CONJUGATE TO ITS INVERSE

This paper gives a necessary and sufficient condition for a matrix in SL(2, Z) to be conjugate to its inverse. This condition reduces the determination of the conjugation to solving some indeterminate equation of second degree. It yields an algorithm to determine this conjugation in finite steps based on the elementary number theory.

On optimal pairwise linear classifiers for normal distributions: the two-dimensional case

Optimal Bayesian linear classifiers have been studied in the literature for many decades. We demonstrate that all the known results consider only the scenario when the quadratic polynomial has coincident roots. Indeed, we present a complete analysis of the case when the optimal classifier between two normally distributed classes is pairwise and linear. We focus on some special cases of the normal distribution with nonequal covariance matrices. We determine the conditions that the mean vectors and covariance matrices have to satisfy in order to obtain the optimal pairwise linear classifier. As opposed to the state of the art, in all the cases discussed here, the linear classifier is given by a pair of straight lines, which is a particular case of the general equation of second degree. We also provide some empirical results, using synthetic data for the Minsky's paradox case, and demonstrated that the linear classifier achieves very good performance. Finally, we have tested our approach on real life data obtained from the UCI machine learning repository. The empirical results that we obtained show the superiority of our scheme over the traditional Fisher's discriminant classifier.

New integrable equations of fourth order and higher degree related to Cosgrove’s equation

We give a general formulation of the algorithm of Fokas and Ablowitz, which then allows us to obtain transformations for nth order ordinary differential equations, to equations of the same order but perhaps of higher degree. Previously this algorithm has been used to obtain transformations for the six second order equations defining new transcendental functions discovered by Painlevé and co-workers, either to other equations in the Painlevé classification or to equations of second order and second degree. As an example of our approach we consider a new fourth order ordinary differential equation due to Cosgrove which is believed to define a new transcendent. We obtain transformations relating this equation to other fourth order ordinary differential equations, of degrees ⩾2. All of these transformations, as well as the corresponding higher degree differential equations, all of which have the Painlevé property, are new.

Some statistical properties of bright galaxies with a uv excess

The results of a statistical investigation of bright galaxies with a UV excess are given. The data obtained for them are compared with those for normal galaxies. A list of bright galaxies (mpg ≤14m.5 ) with a UV excess has been compiled, numbering 461. Their absolute stellar magnitudes and average geometrical linear diameters were determined. The relationships between Mpg and log D are given for galaxies with a UV excess and normal galaxies. The graphs of the relationships are expressed by second-degree equations. It is shown that the UV evolutionary stage of the galaxies has no influence on the relationship between Mpg and log D.

Systems of Two First-Order Second-Degree Differential Equations of the Painlevé Type

Systems of Two First-Order Second-Degree Differential Equations of the Painlevé Type

Second-order second-degree Painlevé equations related with Painlevé I–VI equations and Fuchsian-type transformations

One-to-one correspondence between the Painlevé I–VI equations and certain second-order second-degree equations of Painlevé type is investigated. The transformation between the Painlevé equations and second-order second-degree equations is the one involving the Fuchsian-type equation.

Second-order second-degree Painlevé equations related to Painlevé IV,V,VI equations

The algorithmic method introduced by Fokas and Ablowitz to investigate the transformation properties of Painlevé equations is used to obtain one-to-one correspondence between the Painlevé IV, V and VI equations and the second-order second-degree equations of Painlevé type.

Nets of lines depending on a parameter and sufficient conditions for their convergence

We study criteria for the global regularity of a net of lines defined on the plane or a part of it by an ordinary differential equation of first order and second degree, nets depending on a parameter, and questions of convergence on the parameter. We use an analytic technique connected with hyperbolic systems of equations of a special form. We give applications to surfaces of negative Gaussian curvature and to hyperbolic Monge-Ampère equations.

Evaluation of Filter Coefficients λ and σ for Iron Removal by Rapid Sand Filtration

Abstract Studies wore conducted on a sand bed filter using stock suspension of iron sulfate. By measuring precisely the amount of iron oxide deposited in each layer of sand the coefficients of filter, namely the impediment modulus λ and the specific deposit coefficient σ, are evaluated. Also the phenomenon of “breakthrough” has been explained. From the results obtained it is found that an exponential equation of second degree can adequately describe the process of iron removal by rapid sand filtration.

Retention behaviour and molecular structure of diols in reversed-phase liquid chromatography

Correlations between the retention behaviour of diols in reversed-phase liquid chromatography and their molecular structures have been studied using hydrophobicity (log P) and topological indices (χ R, χ V R , W, W*, J, J*, C T). It has been shown that the retention behaviour of diols in an eluent of acetonitrile-water (50:50) is best described by a polynomial equation of second degree involving hydrophobicity. Topological indices fail to predict the elution sequence of diols but the use of a parameterization scheme for topological indices calculated from the distance matrix improves the results. The combination of log P with the Wiener index and the first-order valence molecular connectivity indices gives reliable predicted capacity factors.

Solving the Kinematics of the Most General Six- and Five-Degree-of-Freedom Manipulators by Continuation Methods

This paper presents a unique approach to the kinematic analysis of the most general six-degree-of-freedom, six-revolute-joint manipulators. Previously, the problem of computing all possible configurations of a manipulator corresponding to a given hand position was approached by reducing the problem to that of solving a high degree polynomial equation in one variable. In this paper it is shown that the problem can be reduced to that of solving a system of eight second-degree equations in eight unknowns. It is further demonstrated that this second-degree system can be routinely solved using a continuation algorithm. To complete the general analysis, a second numerical method—a continuation heuristic—is shown to generate partial solution sets quickly. Finally, in some special cases, closed form solutions were obtained for some commonly used industrial manipulators. The results can be applied to the analysis of both six and five-degree-of-freedom manipulators composed of mixed revolute and prismatic joints. The numerical stability of continuation on small second-degree systems opens the way for routine use in offline robot programming applications.

One-sided type-D gravitational instantons

It is shown that for every one-sided type-D gravitational instanton, Einstein's vacuum equations can be reduced locally to a single second-order, nonlinear differential equation of second degree of one real function.

An analysis of world records in three types of locomotion.

The record history of running, swimming and ice-skating, over various distances, was analyzed. A mean period of about 66 years for the 18 male events and of about 50 years for the 14 female events was studied. Over a given distance the velocity (v) was related to the dates of the records minus 1900 (T) according to polynomial functions like: v = a0 + a1T + a2T2 + ..... + anTn. In 21 out of the 32 events equations of first or second degree fitted the experimental data. The mean correlation coefficient was 0.979 +/- 0.019 (+/- S.D.). The ratio between predicted (vlp) and actual value (vl) of the last records was 0.999 +/- 0.010. For T corresponding to v1 (Tl), the rate of record growth was slowing down in 5 events. Hence up to June 1981 a tendency towards an asymptotic v was not yet a general phenomenon. At Tl the range of the relative rate of increase of v (dvp/dT . vlp) was 0.9 . 10(-3) per year (800 m - female running) and 12.4 . 10(-3) per year (800 m - female swimming). dv/dT . vlp in swimming and skating was similar in both sexes but 4 times faster than in male running. Less marked differences were found for female running. A lowering of the cost of transport was probably the main reason of the fast growth of swimming and skating records. The numerical constants calculated from linear regression of v versus the time of the races over different distances did not seem to have a clear physiological meaning, as reported in the previous literature.

Morphometric evaluation of the number of exocrine pancreatic cells during early postnatal growth in the rat

The number of the various cell categories of the exocrine pancreas of the rat was evaluated by morphometric methods in paraffin sections of glands from rats aged 2, 5, 15, 20 and 33 days. The evolution of these cell types could be properly expressed by equations of the type y = aoek.x, where y = cell number, and x = age in days. The time necessary for each cell type to duplicate was thus obtained. The percentages (y') of acinar, intercalated duct and connective tissue cells also proved to be age-dependent and could be expressed by second-degree equations.

Axial Hysteretic Behavior of Steel Members

A procedure for predicting the force-deformation behavior of axially-loaded members with intermediate slenderness ratios is developed for use in computer-aided inelastic dynamic analyses of frame structures. While the analytical solutions predict the general cyclic force-deformation behavior of axially-loaded members, the quantitative behavior of actual members may not be predicted with a high level of accuracy. A rigorous formulation that permits zones of plastic behavior, or experimental results, could be used to develop empirical results. The elastic bending of a member exerts an appreciable influence on its post-buckling force-deformation behavior and increases in importance as the slenderness ratio increases. Complete second-degree equations afford an excellent representation of the cyclic force-deformation behavior emanating from the analytical solutions. With second-degree equations and computer analysis, computational time may become excessive.

Dose dependence of growth response to human growth hormone in growth hormone deficiency.

A trial of the relative effect on growth of 20 IU/week and 10 IU/week of human growth hormone has been made in 38 patients with "isolated" growth hormone deficiency over 1 year of treatment, 18 patients over 2 years and 10 over 3 years, and in 17 patients with surgically treated craniopharyngiomata over 1 year. The velocity of height growth in the first year of treatment, compared with a full year of pre-treatment control, was 1.3 times as great in both groups of patients on the larger dose as it was in those on the smaller one. Second-degree equations fitted to the treatment catch-up curve gave estimates of 1.7 cm more height gained on the larger dose by the end of the first year, 2.7 cm by the end of the second, and 3.4 cm by the end of the third. Adjusting treatment increment by covariance for bone age at the beginning of treatment, pre-treatment velocity, and body surface area did not alter these mean differences. Bone age velocity during treatment was the same in both treatment groups (mean 1.09 "years"/year in the first year); thus we anticipate a gain in final adult height of the order of 10 cm from employing the larger dose. The decrease in skin folds occurring on treatment, however, was no different with the larger than with the smaller dose. This reinforces previous observations that the short-term metabolic and longer-term auxologic effects of hGH are not necessarily related.

Viscoplastic properties of asphaltic bitumen

In the present paper, various relationships between stress and strain rate have been derived. They are interpreted by rheological models, which may express the mechanical behaviour of the tested material. The experimental curves are compared with theoretical relationships corresponding to various presented rheological models. From these models, the theoretical rheological equations have been derived, which may express the behaviour of bitumens exhibiting viscous and plastic properties. Some experimental curves are evaluated by the procedure of least squares and the empirical equations of the first and second degree between stress and strain rate are found. These equations contain the equilibrated yield stresses, the coefficient of viscosity and the second-order coefficient of viscosity. The influence of temperature on various samples has been observed. In the most general case, the bitumen exhibits viscous and plastic properties, which may be represented by rheological models consisting of viscous and plastic elements. The paper deals with the theoretical and experimental investigation of the rheological behaviour of the asphaltic bitumen. The results of rheological tests are evaluated by the procedure of least squares and various empirical equations between shear stress and shear-strain rate are found. The experimental curves are compared with theoretical relationships derived on base of rheological models consisting of many groups of plastic and viscous elements.

連続溶融亜鉛メッキ操業における鋼板と溶融亜鉛合金間の反応量の推測方法について

1) Arithmetrical formulae have been derived on the amounts of dssolved iron into a galvanizing bath and iron in alloy layers in the continuous sheet galvanizing process influenced by working conduction, such as the aluminium content in the bath, the line-speed and the temperature of the sheet.First the quantities of dissolved iron ΔW1 (g/m2) and iron in the alloy layer ΔW2 (g/m2) are studied in relation to three separate variables; the aluminium content in the bath, y (%), the immersion time, t (sec), and the temperature of the sheet, θ(°C). Then equations of first degree or second degree are derived from these studies, and these equations are combined to arithmetrical formulae using the common conditions in these studies. For example, the formulae for the bath containing 0.12-0.14% Al is as follows:2) A study has been carried out to estimate the aluminium concentration in the coating on the basis of the quantity of iron in the alloy layer. The aluminium concentration in the alloy layer is lower than that estimated from the equilibrium diagram.

Second Degree Equations as Sections of a Cone

Second Degree Equations as Sections of a Cone

On a variant of the matrix transformation method for approximate solution of partial differential equations of second degree

On a variant of the matrix transformation method for approximate solution of partial differential equations of second degree