Optimum stratification

Summary This paper considers the problem of optimum stratification on a concomitant variablex when the form of the regression of the estimation variabley on the concomitant variablex as also the form of the variance functionV(y|x) are known. Minimal equations giving optimum strata boundaries have been obtained for Neyman and proportional allocations. Since the minimal equations cannot be solved easily, various methods of finding approximate solutions have been given. A comparison of approximate solutions with the exact solutions is made for certain density functions.     

Optimum stratification with ratio and regression methods of estimation

Summary The paper considers the problem of optimum stratification on an auxiliary variablex when the information on the auxiliary variablex is also used to estimate the population mean using ratio or regression methods of estimation. Assuming the form of the regression of the estimation variabley on the auxiliary variablex as also the form of the conditional variance function V(y/x), the problem of determining optimum strata boundaries (OSB) is shown to be a particular case of optimum stratification on the auxiliary variable for stratified simple random sampling estimate. A numerical investigation has also been made to study the amount of gain in efficiency that can be brough about by stratifying the population.     

Einige unstetige stochastische Prozesse

Summary Stochastic processes of the following type are considered. At random time points, the variablex(t) jumps fromy tox, say. The heightsx–y of the jumps have a given distributionG ^*(x–y) that may depend ony ort. Between the jumps,x(t) is a solution to a given differential equationdx/dt=x(x, t). We look for the distributionF(x, t) ofx at timet>0,F(x, 0) being given. In the stationary case, stable distributions are investigated.If there is a lower boundaryx ₀ and ifF(x ₀)>0, the problem is similar to the queueing problem. We solve it in the stationary case with integral equations of the Volterra type. Other problems can be transformed to differential equations for the moment generating functions. These equations are partial in the non stationary and ordinary in the stationary case.     

A Family of Estimators of Finite-Population Distribution Function Using Auxiliary Information

This paper considers the problem of estimating the finite-population distribution function and quantiles with the use of auxiliary information at the estimation stage of a survey. We propose the families of estimators of the distribution function of the study variate y using the knowledge of the distribution function of the auxiliary variate x. In addition to ratio, product and difference type estimators, many other estimators are identified as members of the proposed families. For these families the approximate variances are derived, and in addition, the optimum estimator is identified along with its approximate variance. Estimators based on the estimated optimum values of the unknown parameters used to minimize the variance are also given with their properties. Further, the family of estimators of a finite-population distribution function using two-phase sampling is given, and its properties are investigated.      

Approximation of the perturbation equations of a quasi-linear hyperbolic system in the neighborhood of a bicharacteristic

In 1956 Whitham gave a nonlinear theory for computing the intensity of an acoustic pulse of an arbitrary shape. The theory has been used very successfully in computing the intensity of the sonic bang produced by a supersonic plane. [4.] derived an approximate quasi-linear equation for the propagation of a short wave in a compressible medium. These two methods are essentially nonlinear approximations of the perturbation equations of the system of gas-dynamic equations in the neighborhood of a bicharacteristic curve (or rays) for weak unsteady disturbances superimposed on a given steady solution. In this paper we have derived an approximate quasi-linear equation which is an approximation of perturbation equations in the neighborhood of a bicharacteristic curve for a weak pulse governed by a general system of first order quasi-linear partial differential equations in m + 1 independent variables (t, x₁,…, x_m) and derived Gubkin's result as a particular case when the system of equations consists of the equations of an unsteady motion of a compressible gas. We have also discussed the form of the approximate equation describing the waves propagating upsteam in an arbitrary multidimensional transonic flow.     

Stratified simple random sampling and prior distributions

Summary The problem of stratification with proportional and optimum allocations in the case of simple random sampling has been examined in the light of an appropriate super-population model and a formal proof has been provided here for arranging the auxiliary character in increasing order of magnitude for stratification in the case of proportional allocation and also it is shown here that the same may not be necessary in the case of optimum allocation. However, if the coefficient of variation with respect to the auxiliary variate is same in each stratum the necessity of arraging the auxiliary character in increasing order of magnitude for stratification is established. The results are illustrated with respect to empirical examples. Also, some comparisons among different estimators have been made under the super-population model.     

Envelope viscosity solutions of first- and second-order PDEs with u-dependence

General envelope methods are introduced which may be used to embed equations with u-dependence into equations without solution dependence. Furthermore, these methods present a rigorous way to consider so-called nodal solutions. That is, if w(t,x,z) is the viscosity solution of some pde, the nodal solution of an associated pde is a function u(t,x) so that w(t,x,u(t,x)) = 0. Examples are given to first- and second-order pdes arising in optimal control, differential games, minimal time problems, scalar conservation laws, geometric-type equations, and forward backward stochastic control.     

Generalized three-point difference schemes of high-order accuracy for systems of second-order nonlinear ordinary differential equations

For systems of second-order nonlinear ordinary differential equations with the Dirichlet boundary conditions, we develop generalized three-point difference schemes of high-order accuracy on a nonuniform grid. The construction of the suggested schemes requires solving four auxiliary Cauchy problems (two problems for systems of nonlinear ordinary differential equations and two problems for matrix linear ordinary differential equations) on the intervals [x _j−1, x _j ] (forward) and [x _j , x _j+1] (backward) at each grid point; this is done at each step by any single-step method of accuracy order $ \bar m $ \bar m = 2[(m+1)/2]. (Here m is a given positive integer, and [·] is the integer part of a number.) We prove that such three-point difference schemes have the accuracy order $ \bar m $ \bar m for the approximation to both the solution u of the boundary value problem and the flux K(x)d u/dx at the grid points.     

Nonlinear operator equations with a functional perturbation of the argument of neutral type

We construct small solutions x(t) → 0 as t → 0 of nonlinear operator equations F(x(t), x(α(t)),t) = 0 with a functional perturbation α(t) of the argument. By the Newton diagram method, we reduce the problem to quasilinear operator equations with a functional perturbation of the argument. We show that the solutions of such equations can have not only algebraic but also logarithmic branching points and contain free parameters. The number of free parameters and the form of the solution depend on the properties of the Jordan structure of the operator coefficients of the equation.     

On improvement of summability properties of solutions of some degenerate nonlinear fourth-order equations

We consider a class of degenerate nonlinear elliptic fourth-order equations in divergence form. Coefficients of the equations satisfy a strengthened ellipticity condition involving two weighted functions. The right-hand side F?(x,?u) of the equations depends on the unknown function u. Under some conditions, including L^m -summability of F(·, 0) with m close to 1 and restrictions on the exponent σ of the growth of F?(x,?u) with respect to u, we establish existence of W-solutions of the Dirichlet problem for the given equations and describe the dependence of summability properties of these solutions on m and σ.     

Error estimates for Galerkin methods for quasilinear parabolic and elliptic differential equations in divergence form

Summary Conditions are given for the validity of long range quasi optimal error estimates of the Galerkin method for quasilinear parabolic equations. If the corresponding nonlinear elliptic operator satisfies a coercivity condition, the stationary problem may be approximated by the numerical solution of the parabolic problem when the time variablet is large enough, starting with an arbitrary initial function. Some practical applications are discussed.     

Acceleration methods based on convergence tests

Summary In this paper we propose an acceleration method based on a general convergence test and depending on an auxiliary sequence (x _n). For different choices of (x _n) we obtain some known and some new transformations and for each one sufficient conditions for acceleration are given.     

Evaluation de l'incertitude sur la solution d'un système lineaire

The evaluation of the accuracy of solutions of linear algebraic systems by numerical method is not easily accomplished on account of the finite precision of computer arithmetic.In this paper, we present a numerical method which determines for each variablex _i , the numberC _i of exact significant figures.     

On self-duality of ring extensions over quasi-frobenius rings

ABSTRACT:

Let R be a zero-dimensional SFT-ring. It is proved that the minimal prime ideals of the formal power series ring A=R[[x ₁, …, x_n ]] are the ideals of the form [[x ₁, …, x_n ]], where is a (minimal) prime of R. It follows that A has Krull dimension n and is catenarian. If R?T where T is also a zero-dimensional SFT-ring, the lying-over, going-up, incomparable, and going-down properties are studied for the extension A?T[[x ₁, …, x_n ]].     

Orthogonalpolynome in x und q ?x als L?sungen von reellen q -Operatorgleichungen zweiter Ordnung

 The explicit form and the three term recurrance relation for all the polynomial solutions in x and q ^−x from q-operator equations of second order are given. The known 17 q-orthogonal polynomial systems are special cases of 7 comprehensive q-systems.     

A geometric approach to Euler’s addition formula A direct formulation of the addition law for elliptic curves given in?terms of Weierstra? quartics

Plane quartic curves given by equations of the form y ²=P(x) with polynomials P of degree 4 represent singular models of elliptic curves which are directly related to elliptic integrals in the form studied by Euler and for which he developed his famous addition formulas. For cubic curves, the well-known secant and tangent construction establishes an immediate connection of addition formulas for the corresponding elliptic integrals with the structure of an algebraic group. The situation for quartic curves is considerably more complicated due to the presence of the singularity. We present a geometric construction, similar in spirit to the secant method for cubic curves, which defines an addition law on a quartic elliptic curve given by rational functions. Furthermore, we show how this addition on the curve itself corresponds to the addition in the (generalized) Jacobian variety of the curve, and we show how any addition formula for elliptic integrals of the form ò(1/?{P(x)}) dx\int (1/\sqrt{P(x)})\,\mathrm{d}x with a quartic polynomial P can be derived directly from this addition law.     

Weight of the comprehension axiom in a theory based on logic without contractions

A set-theoretic LST system based on a logic without rules of contraction of identical formulas in the antecedent or succedent of sequents is introduced. The set-theoretic axioms used are the comprehension principle,

Enhanced Group Analysis and Exact Solutions of Variable Coefficient Semilinear Diffusion Equations with a Power Source

A new approach to group classification problems and more general investigations on transformational properties of classes of differential equations is proposed. It is based on mappings between classes of differential equations, generated by families of point transformations. A class of variable coefficient (1+1)-dimensional semilinear reaction–diffusion equations of the general form f(x)u _t =(g(x)u _x ) _x +h(x)u ^m (m≠0,1) is studied from the symmetry point of view in the framework of the approach proposed. The singular subclass of the equations with m=2 is singled out. The group classifications of the entire class, the singular subclass and their images are performed with respect to both the corresponding (generalized extended) equivalence groups and all point transformations. The set of admissible transformations of the imaged class is exhaustively described in the general case m≠2. The procedure of classification of nonclassical symmetries, which involves mappings between classes of differential equations, is discussed. Wide families of new exact solutions are also constructed for equations from the classes under consideration by the classical method of Lie reductions and by generation of new solutions from known ones for other equations with point transformations of different kinds (such as additional equivalence transformations and mappings between classes of equations).     

A note on the polynomial solution of a class of dual integral equations arising in mixed boundary value problems of elasticity

Summary The present paper is concerned with finding an effective polynomial solution to a class of dual integral equations which arise in many mixed boundary value problems in the theory of elasticity. The dual integral equations are first transformed into a Fredholm integration equation of the second kind via an auxiliary function, which is next reduced to an infinite system of linear algebraic equations by representing the unknown auxiliary function in the form of an infinite series of Jacobi polynomials. The approximate solution of this infinite system of equations can be obtained by a suitable truncation. It is shown that the unknown function involving the dual integral equations can also be expressed in the form of an infinite series of Jacobi polynomials with the same expansion coefficients with no numerical integration involved. The main advantage of the present approach is that the solution of the dual integral equations thus obtained is numerically more stable than that obtained by reducing themdirectly into an infinite system of equations, insofar as the expansion coefficients are determined essentially by solving asecond kind integral equation.     

Orthogonalpolynome in x und q −x als Lösungen von reellen q-Operatorgleichungen zweiter Ordnung

 The explicit form and the three term recurrance relation for all the polynomial solutions in x and q ^−x from q-operator equations of second order are given. The known 17 q-orthogonal polynomial systems are special cases of 7 comprehensive q-systems. (Eingegangen 27. M?rz 2000; in revidierter Fassung 2. November 2000)