Explicit General Solution Research Articles

General solution and classification of conformal motions in static spherical spacetimes

We find the complete and explicit general solution of the conformal Killing equation in static spherically symmetric spacetimes, thus unifying and generalizing previous special cases. For non-conformally-flat spacetimes, there are at most two proper conformal motions. There are three classes of such spacetimes, and one or both of the conformal Killing vectors is non-inheriting. One of the classes includes self-similar spacetimes (i.e. with a homothetic motion). The conformally flat spacetimes (including the Schwarzschild interior metric) fall into three classes, and their eleven proper conformal Killing vectors are given in full. The only spacetimes with conformal motion that are regular at the centre are conformally flat. An addendum for this article has been published in 1996 Class. Quantum Grav. 13 317

A general structure for the unbiased estimate of the parameter's estimable function in linear regression

A general structure for the unbiased estimate (UE) T(y) of the parameter B's estimable function (EF) g(β) in linear regression model y = Xβ + e is investigated, where e is an error with normal distribution. For this general problem, by constituting a set of eigenfunctions to solve a certain integral equations, the explicit general solution of the DE T(y) is obtained under some certain conditions. Necessary and sufficient conditions for uniqueness, linearity and optimality of the DE are considered

Explicit general solution of the Quasi-Newton equation with sparsity and symmetry

We consider the problem of solving the Quasi-Newton equation under the additional condition that some elements of the solution take prescribed values. The general solution is obtained in a straightforward way by using the Huang algorithm of the ABS class. It is given in explicit form without the need of solving intermediate linear systems. Particular cases include the general symmetric or unsymmetric full or sparse update. In the sparse symmetric case we show that while positive definiteness cannot generally be forced a weaker condition called quasi positive definiteness can be satisfied under a very mild condition. This condition guarantees that all eigenvalues, except at most two, are positive.

A new solution of Schmutzer's projective relativistic cosmology

Schmutzer's cosmological model on the basis of his projective unified field theory leads to a system of differential equations equivalent to a set of highly complicated Abel differential equations [1, 2]. This paper is concerned with the properties of this coupled system of differential equations. The main results of this paper are as follows, (i) We have found the explicit general solution of the cosmological equations for the caseγ=±5/2,ɛ= ±1. (ii) The procedure of solving the general caseγ arbitraryɛ=0 is easily within this formalism, (iii) After having found the solution withγ arbitrary,ɛ=±1, one gets the corresponding solutionγ= -γ, ɛ=±1. (iv) In general the coupled system of differential equations describing Schmutzer's cosmological model is equivalent to a single generalized Emden differential equation and it is identical to Emden's differential equation for special values of the parameterγ.

Approximate steady-state modeling of solar trough collectors

This paper deals with a static model of the absorber tube of a solar trough collector. The essential difficulty concerning the steady-state analysis resides in the integration of a nonlinear differential equation which has no explicit gener alsolution. The analysis of some particular solutions leads to the formulation of two general properties of the physical behavior of the tube. Using these solutions as reference points on the space of all solutions, the differential equation is reformulated. Subsequently, an approximated solution is proposed which allows the statement of an algebraic equation relating, with high precision, all variables of interest. The approximated solution thus obtained suggests two more levels of approximation, one of which gives rise to a simple formula, relating input and output variables, which can be used as a feed-forward controller.

Explicit non-'t Hooft rational solutions of the Atiyah-Drinfeld-Hitchin-Manin instanton matrix equations for SU(2 r)

The Atiyah-Drinfeld-Hitchin-Manin instanton matrix equations may be written as a pair, one quadratic and one linear. The explicit general solution of the quadratic equation is given in terms of rational functions of free parameters, thus reducing the ADHM equations to a single equation. For SU(2 r), large families of explicit solutions, rational in free parameters, are constructed by this method. These new families are generally about twice as large as the previously known 't Hooft families, and contain non-'t Hooft solutions for every r ⩾ 1.

Alternative integration procedure for scale‐invariant ordinary differential equations

For an ordinary differential equation invariant under a one‐parameter group of scale transformations x → λx, y → λαy, y′ → λα−1y′, y″ → λα−2y″, etc., it is shown by example that an explicit analytical general solution may be obtainable in parametric form in terms of the scale‐invariant variable . This alternative integration may go through, as it does for the example equation y″ = kxy−2y′, in cases for which the customary dependent and independent variables (x−αy) and (ℓnx) do not yield an analytically integrable transformed equation.

An explicit general solution in linear fractional programming

AbstractA complete analysis and explicit solution is presented for the problem of linear fractional programming with interval programming constraints whose matrix is of full row rank. The analysis proceeds by simple transformation to canonical form, exploitation of the Farkas‐Minkowki lemma and the duality relationships which emerge from the Charnes‐Cooper linear programming equivalent for general linear fractional programming. The formulations as well as the proofs and the transformations provided by our general linear fractional programming theory are here employed to provide a substantial simplification for this class of cases. The augmentation developing the explicit solution is presented, for clarity, in an algorithmic format.

On the Fixed-Receiving-Point Problem in Channels with a Constant Sound-Speed Gradient

In a channel in which the sound speed varies linearly with depth, the general determination of the initial ray angle for which a ray will intercept a fixed receiving point is inherently implicit, and an explicit expression is feasible only for certain source-receiver configurations. However, an explicit general solution has been obtained in other studies.

First-Order Meson Wave Equations

A method is developed for the simple representation of the symbols introduced in the generalization of Dirac's first-order wave equation for the electron. This enables one to obtain an explicit general solution of the generalized equations. The sign of the energy is discussed in order to determine which of the various possible representations are physically admissible.