Rotating Disk Shape: Is the Equation Nonlinear?

1989 ◽  
Vol 111 (4) ◽  
pp. 456-458
Author(s):  
R. R. Jettappa
Keyword(s):  
Differential Equation ◽  
Linear Equation ◽  
Rotating Disk ◽  
Variable Coefficients ◽  
Thin Disk ◽  
Second Order ◽  
First Order ◽  
Governing Differential Equation ◽  
Centrifugal Loading

The determination of the shape of a rotating disk under centrifugal loading is considered. It is shown that the governing differential equation for the shape of a rotating thin disk is reducible to a linear equation of second order with variable coefficients. However, the form of this equation is such that it can be treated as an equation of first order thereby facilitating the integration by quadratures. All this is possible without any additional mathematical assumptions so that the results are exact within the limitations of the thin disk theory.

scholarly journals ON THE TRANSFORMATION OF THE LINEAR DIFFERENTIAL EQUATION INTO A SYSTEM OF THE FIRST ORDER EQUATIONS

2019 ◽  
pp. 71-75
Author(s):  
M.I. Ayzatsky
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Linear Equation ◽  
Second Order ◽  
Dimensional System ◽  
Third Order ◽  
Order Linear ◽  
First Order ◽  
The Third ◽  
Linear Differential

The generalization of the transformation of the linear differential equation into a system of the first order equations is presented. The proposed transformation gives possibility to get new forms of the N-dimensional system of first order equations that can be useful for analysis of the solutions of the N-th-order differential equations. In particular, for the third-order linear equation the nonlinear second-order equation that plays the same role as the Riccati equation for second-order linear equation is obtained.

4. On the Linear Differential Equation of the Second Order

1878 ◽  
Vol 9 ◽  
pp. 93-98 ◽  
Author(s):  
Tait
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Linear Equation ◽  
Arbitrary Constant ◽  
Second Order ◽  
General Linear ◽  
First Order ◽  
Non Linear Equation ◽  
Non Linear ◽  
Linear Differential

This paper contains the substance of investigations made for the most part many years ago, but recalled to me during last summer by a question started by Sir W. Thomson, connected with Laplace's theory of the tides.A comparison is instituted between the results of various processes employed to reduce the general linear differential equation of the second order to a non-linear equation of the first order. The relation between these equations seems to be most easily shown by the following obvious process, which I lit upon while seeking to integrate the reduced equation by finding how the arbitrary constant ought to be involved in its integral.

ANOMALY FREE SUPERGRAVITY IN D=10: I) THE BIANCHI IDENTITIES AND THE BOSONIC LAGRANGIAN

1988 ◽  
Vol 03 (04) ◽  
pp. 953-1021 ◽  
Author(s):  
RICCARDO D’AURIA ◽  
PIETRO FRÉ ◽  
MARIO RACITI ◽  
FRANCO RIVA
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Variational Equation ◽  
Second Order ◽  
Effective Theory ◽  
Spin Connection ◽  
Bianchi Identities ◽  
First Order ◽  
Interaction Terms ◽  
Starting Point

Using a theorem by Bonora-Pasti and Tonin on the existence of a solution for D=10N=1 Bianchi identities in the presence of a Lorentz Chern Simons term, we find an explicit parametrization of the superspace curvatures. Our solution depends only on one free parameter which can be reabsorbed in a field redefinition of the dilaton and of the gravitello. We emphasize that the essential point which enables us to obtain a closed form for the curvature parametrizations and hence for the supersymmetry transformation rules is the use of first order formalism. The spin connection is known once the torsion is known. This latter, rather than being identified with Hµνρ as it is usually done in the literature, is related to it by a differential equation which reduces to the algebraic relation Hµνρ = - 3Tµνρ e4/3σ only at γ1=0 (γ1 being proportional to κ/g2). The solution of the Bianchi identities exhibited in this paper corresponds to a D=10 anomaly free supergravity (AFS). This theory is unique in first order formalism but corresponds to various theories in second order formalism. Indeed the torsion equation is a differential equation which, in order to be solved must be supplemented with boundary conditions. One wonders whether supplemented with a judicious choice of boundary conditions for the torsion equation, AFS yields all the interaction terms found in the effective theory of the heterotic string (ETHS). In this respect two remarks are in order. Firstly it appears that solving the torsion equation iteratively with Tµνρ = -1/3Hµνρ e-4/3σ as starting point all the terms of ETHS except those with a ζ(3) coefficient show up. (Whether the coefficient agree is still to be checked.) Secondly, as shown in this paper the rheonomic solution of the super Poincaré Bianchi identities is unique. Hence additional interaction terms can be added to the Lagrangian only by modifying the rheonomic parametrization of the [Formula: see text]-curvature. The only assumption made in our paper is that [Formula: see text] has at most ψ∧ψ∧V components (sector (1,2)). Correspondingly the only room left for a modification of the present theory is the addition of a (0, 3) part in the rheonomic parametrization of [Formula: see text]. When this work was already finished a conjecture was published by Lechner Pasti and Tonin that such a generalization of AFS might exist and be responsible for the ζ(3) missing term. Indeed if we were able to solve the [Formula: see text]-Bianchi with this new (0, 3)-part then the torsion equation would be modified via new terms which, in second order formalism, lead to additional gravitational interactions. The equation of motion of Anomaly Free Supergravity can be worked out from the Bianchi identities: we indicate through which steps. The corresponding Lagrangian could be constructed with the standard procedures of the rheonomy approach. In this paper we limit ourselves to the bosonic sector of such a Lagrangian and we show that it can indeed be constructed in such a way as to produce the relation between Hµνρ and Tµνρ as a variational equation.

Determination of cost accounting objects, calculation and improving the calculation of product cost in agriculture

2021 ◽  
pp. 30-39
Author(s):  
Rolan Arkhipovich Alborov ◽  
Ekaterina Leonidovna Mosunova ◽  
Elena Vyacheslavovna Zakharova ◽  
Gregory Rolanovich Alborov
Keyword(s):  
Cost Accounting ◽  
Second Order ◽  
Agricultural Products ◽  
Accounting Systems ◽  
First Order ◽  
Management Accounting Systems ◽  
Order Calculation ◽  
Livestock Products ◽  
The Cost

The article deals with the problems of calculating the cost of agricultural products in crop and livestock production, associated with the methods of production accounting and management accounting systems for production facilities used in practice by agricultural organizations. Variants of definition (selection) of cost accounting objects, objects of calculation of the first order and objects of calculation of the second order are proposed. Conceptual models for the distribution of costs between the objects of the first-order calculation, the objects of the second-order calculation and the calculation of the cost of the received types of agricultural products have been developed. Using the example of the production of the main herd of dairy cattle, it is shown that the use of old methods of calculating the cost of agricultural products is not consistent, and it is recommended to use more justified methods of calculating the cost of crop and livestock products, recommended in the new editions of the relevant guidelines of the Ministry of Agriculture of the Russian Federation.

scholarly journals XV. On a general method in dynamics; by which the study of the motions of all free systems of attracting or repelling points is reduced to the search and differentiation of one central relation, or characteristic function

1834 ◽  
Vol 124 ◽  
pp. 247-308 ◽  
Keyword(s):  
Partial Differential Equations ◽  
Characteristic Function ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Second Order ◽  
Mathematical Science ◽  
Partial Differential ◽  
First Order ◽  
Present Essay

The theoretical development of the laws of motion of bodies is a problem of such interest and importance, that it has engaged the attention of all the most eminent mathematicians, since the invention of dynamics as a mathematical science by Galileo, and especially since the wonderful extension which was given to that science by Newton. Among the successors of those illustrious men, Lagrange has perhaps done more than any other analyst, to give extent and harmony to such deductive researches, by showing that the most varied consequences respecting the motions of systems of bodies may be derived from one radical formula; the beauty of the method so suiting the dignity of the results, as to make of his great work a kind of scientific poem. But the science of force, or of power acting by law in space and time, has undergone already another revolution, and has become already more dynamic, by having almost dismissed the conceptions of solidity and cohesion, and those other material ties, or geometrically imaginable conditions, which Lagrange so happily reasoned on, and by tending more and more to resolve all connexions and actions of bodies into attractions and repulsions of points: and while the science is advancing thus in one direction by the improvement of physical views, it may advance in another direction also by the invention of mathematical methods. And the method proposed in the present essay, for the deductive study of the motions of attracting or repelling systems, will perhaps be received with indulgence, as an attempt to assist in carrying forward so high an inquiry. In the methods commonly employed, the determination of the motion of a free point in space, under the influence of accelerating forces, depends on the integration of three equations in ordinary differentials of the second order; and the determination of the motions of a system of free points, attracting or repelling one another, depends on the integration of a system of such equations, in number threefold the number of the attracting or repelling points, unless we previously diminish by unity this latter number, by considering only relative motions. Thus, in the solar system, when we consider only the mutual attractions of the sun and of the ten known planets, the determination of the motions of the latter about the former is reduced, by the usual methods, to the integration of a system of thirty ordinary differential equations of the second order, between the coordinates and the time; or, by a transformation of Lagrange, to the integration of a system of sixty ordinary differential equations of the first order, between the time and the elliptic elements: by which integrations, the thirty varying coordinates, or the sixty varying elements, are to be found as functions of the time. In the method of the present essay, this problem is reduced to the search and differentiation of a single function, which satisfies two partial differential equations of the first order and of the second degree: and every other dynamical problem, respecting the motions of any system, however numerous, of attracting or repelling points, (even if we suppose those points restricted by any conditions of connexion consistent with the law of living force,) is reduced, in like manner, to the study of one central function, of which the form marks out and characterizes the properties of the moving system, and is to be determined by a pair of partial differential equations of the first order, combined with some simple considerations. The difficulty is therefore at least transferred from the integration of many equations of one class to the integration of two of another: and even if it should be thought that no practical facility is gained, yet an intellectual pleasure may result from the reduction of the most complex and, probably., of all researches respecting the forces and motions of body, to the study of one characteristic function, the unfolding of one central relation.

scholarly journals Weierstrass elliptic difference equations

1987 ◽  
Vol 35 (1) ◽  
pp. 43-48 ◽  
Author(s):  
Renfrey B. Potts
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Difference Equations ◽  
Elliptic Function ◽  
Order Differential Equation ◽  
Second Order ◽  
Elliptic Difference ◽  
First Order ◽  
Second Order Differential Equation ◽  
Weierstrass Elliptic Function

The Weierstrass elliptic function satisfies a nonlinear first order and a nonlinear second order differential equation. It is shown that these differential equations can be discretized in such a way that the solutions of the resulting difference equations exactly coincide with the corresponding values of the elliptic function.

Rapid X-Ray Fluorescence Determination of Phosphorus in Geologic Samples

1971 ◽  
Vol 25 (1) ◽  
pp. 41-43 ◽  
Author(s):  
Brent P. Fabbi
Keyword(s):  
Pulse Height ◽  
Second Order ◽  
Analytical Line ◽  
X Ray ◽  
First Order ◽  
Peak Overlap ◽  
Fluorescence Determination ◽  
Geologic Samples

Calcium and Kβ1 and Kβ5 both have second order (II) lines that interfere spectrally with the phosphorus Kα1 first order (I) analytical line in the x-ray fluorescence determination. By combining maximum pulse height discrimination with a mathematical correction for peak overlap, phosphorus can be accurately determined in a wide variety of geologic samples.

scholarly journals Noncanonical Neutral DDEs of Second-Order: New Sufficient Conditions for Oscillation

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2026
Author(s):  
Awatif A. Hindi ◽  
Osama Moaaz ◽  
Clemente Cesarano ◽  
Wedad R. Alharbi ◽  
Mohamed A. Abdou
Keyword(s):  
Differential Equation ◽  
Sufficient Conditions ◽  
Second Order ◽  
Neutral Differential Equation ◽  
First Order ◽  
Riccati Substitution

In this paper, new oscillation conditions for the 2nd-order noncanonical neutral differential equation (a0t((ut+a1tug0t)′)β)′+a2tuβg1t=0, where t≥t0, are established. Using Riccati substitution and comparison with an equation of the first-order, we obtain criteria that ensure the oscillation of the studied equation. Furthermore, we complement and improve the previous results in the literature.

scholarly journals Kinetika Reaksi Hidrolisis Pati Biji Alpukat (Persea americana Mill) dengan Katalis HCl

2020 ◽  
Vol 4 (1) ◽  
pp. 120-131
Author(s):  
Sitti Rahmawati ◽  
Asnila Asnila ◽  
Suherman Suherman ◽  
Paulus Hengky Abram
Keyword(s):  
Reaction Order ◽  
Second Order ◽  
Hydrolysis Reaction ◽  
Starch Hydrolysis ◽  
First Order ◽  
Graph Method ◽  
Kinetics Of ◽  
Hydrolysis Of ◽  
Avocado Seed

One of the plants that can be used as raw material for making sugar is plants that contain starch content such as avocado seeds. This study aims to determine the reaction order, the reaction rate constant from the hydrolysis of avocado seed starch using HCl. The method of this research is to determine the optimum concentration of HCl hydrolysis reaction from avocado seed starch using various concentrations of HCl (0.5 M; 1 M; 1.5 M; 2 M; 2.5 M) at the optimum temperature and stirring time (90oC for 70 minute). The hydrolysis process was followed by neutralization using 5 M NaOH solution and evaporated to obtain concentrated glucose, glucose was analyzed qualitatively and quantitatively by the Benedict method and the phenol sulfuric acid method. Based on the results of the maximum glucose levels obtained from the hydrolysis of variations in the concentration of HCl avocado seed starch, HCl 1.5 M. Furthermore, determine the kinetics of the starch hydrolysis reaction using time variations (30, 40, 50, 60 and 70) minutes at 90oC and concentrations The HCl 1.5 M. reaction order is determined by the intral method and the graph method. Determination of the first order graph method is done by plotting the value of ln [A] versus time, while the second order by plotting the value of 1 / [A] versus time. The first order with a 93% confidence level was obtained from the value of R2 = 0.9312, while the second order was 85% obtained from the value of R2 = 0.8581. Determination of the order of the integral method k value tends to remain in the first order formula with an average of k = 0.01962 minutes-1. Based on the two methods, it can be determined that the kinetics of the avocado seed starch hydrolysis reaction follows a first-order reaction.

scholarly journals Chasing the Critical Wetting Transition. An Effective Interface Potential Method

Materials ◽  
2021 ◽  
Vol 14 (23) ◽  
pp. 7138
Author(s):  
Paweł Bryk ◽  
Artur P. Terzyk
Keyword(s):  
Potential Method ◽  
Second Order ◽  
Wetting Transition ◽  
First Order ◽  
Methods Of Determination ◽  
Critical Wetting ◽  
Interface Potential ◽  
Order Continuous

Wettablity is one of the important characteristics defining a given surface. Here we show that the effective interface potential method of determining the wetting temperature, originally proposed by MacDowell and Müller for the surfaces exhibiting the first order wetting transition, can also be used to estimate the wetting temperature of the second order (continuous) wetting transition. Some selected other methods of determination of the wetting temperature are also discussed.

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