Optimum stratification in sampling with varying probabilities

Summary The paper considers the problem of optimum stratification on an auxiliary variablex when the units from the different strate are selected with probability proportional to the value of the auxiliary variable. Under a super-population set-up assuming the form, of the regression of the estimation variabley on the auxiliary variablex as also the form of the variance functionV(y/x), minimal equations giving optimum strata boundaries have been obtained for the Neyman allocation method. As the minimal equations cannot be solved easily, methods to find approximate solutions have been given. A numerical illustration has also been given to study the effect of optimum stratification.     

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.     

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,

Central limit theorem for integrated square error of kernel estimators of spherical density

LetX ₁,…,X _n be iid observations of a random variableX with probability density functionf(x) on the q-dimensional unit sphere Ω_q in R^q+1,q ⩾ 1. Let be a kernel estimator off(x). In this paper we establish a central limit theorem for integrated square error off _n under some mild conditions.     

Quadratically constrained least squares and quadratic problems

Summary We consider the following problem: Compute a vectorx such that Ax–b₂=min, subject to the constraint x₂=. A new approach to this problem based on Gauss quadrature is given. The method is especially well suited when the dimensions ofA are large and the matrix is sparse.It is also possible to extend this technique to a constrained quadratic form: For a symmetric matrixA we consider the minimization ofx ^T A x–2b ^T x subject to the constraint x₂=.Some numerical examples are given.This work was in part supported by the National Science Foundation under Grant DCR-8412314 and by the National Institute of Standards and Technology under Grant 60NANB9D0908.     

Long-time behaviour for weakly damped driven nonlinear Schrödinger equations in ℝ N,N≤3

We study the long-time behaviour of solutions to nonlinear Schrödinger equations with a zero order dissipation and an additional external force, when the space variablex varies over ^N ,N3. We prove that the long-time behaviour is described by a maximal compact attractor for the strong topology ofH ¹ ( ^N ).     

A Note on Primes and Goldbach Numbers in Short Intervals

Let J(N, H) be the Selberg integral and E(x, T) the error term in Kaczorowski-Perelli's weighted form of the classical explicit formula. We prove that the estimate J(N, H) = o(H2 N) is connected with an appropriate estimate of _N ^2N| E(x,T)² dx, uniformly for H and T in some ranges. Moreover, assuming a suitable bound for _N ^2N| E(x,T)|² dx, we also obtain, for all sufficiently large N and H (log N)11/12, that every interval [N,N + H] contains H Goldbach numbers.     

A variable dimension algorithm with the Dantzig-Wolfe decomposition for structured stationary point problems

Given a set ofR ⁿ and a functionf from intoR ⁿ we consider a problem of finding a pointx ^* in such that(x–x ^*) ^t f(x ^*)0 holds for every pointx in. This problem is called the stationary point problem and the pointx ^* is called a stationary point. We present a variable dimension algorithm for solving the stationary point problem with an affine functionf on a polytope defined by constraints of linear equations and inequalities. We propose a system of equations whose solution set contains a piecewise linear path connecting a trivial starting point in with a stationary point. The path can be followed by solving a series of linear programs which inherit the structure of constraints of. The linear programs are solved efficiently with the Dantzig-Wolfe decomposition method by exploiting fully the structure.Part of this research was carried out when the first author was supported by the Center for Economic Research, Tilburg University, The Netherlands and the third author was supported by the Alexander von Humboldt-Foundation, Federal Republic of Germany.     

On the entropic regularization method for solving min-max problems with applications

Consider a min-max problem in the form of min _xX max_1im {f _i (x)}. It is well-known that the non-differentiability of the max functionF(x) max_1im {f _i (x)} presents difficulty in finding an optimal solution. An entropic regularization procedure provides a smooth approximationF _p(x) that uniformly converges toF(x) overX with a difference bounded by ln(m)/p, forp > 0. In this way, withp being sufficiently large, minimizing the smooth functionF _p(x) overX provides a very accurate solution to the min-max problem. The same procedure can be applied to solve systems of inequalities, linear programming problems, and constrained min-max problems.This research work was supported in part by the 1995 NCSC-Cray Research Grant and the National Textile Center Research Grant S95-2.     

Regularity of ideals and their radicals

In this paper we compare the regularityreg I of a homogeneous idealI⊂K[x ₁,...,x _n] with that of its radical. We prove that regI ≥ reg ifR/I is a BuchsbaumR-module or ifI is a monomial ideal. We also prove the same result when defines a non-singular curve inP ³ under some additional hypotheses     

A note on smoothed estimating functions

The kernel estimate of regression function in likelihood based models has been studied in Staniswalis (1989,J. Amer. Statist. Assoc.,84, 276–283). The notion of optimal estimation for the nonparametric kernel estimation of semimartingale intensity (t) is proposed. The goal is to arrive at a nonparametric estimate of ₀=(t ₀) for a fixed pointt ₀ [0, 1]. We consider the estimator that is a solution of the smoothed optimal estimating equation is the optimal estimating function as in Thavaneswaran and Thompson (1986,J. Appl. Probab.,23, 409–417).     

Necessary and sufficient conditions for a variety of Leibniz algebras to have polynomial growth

We study the behavior of the codimension sequence of polynomial identities of Leibniz algebras over a field of characteristic 0. We prove that a variety V has polynomial growth if and only if the condition

Défaut d’estimation a priori pour un problème de dirichlet

Résumé  Dans un célèbre papier ([3]), B. GIDAS et J.SPRUCK ont utilisé-sous des hypothèses adéquates- la technique du “blow up” pour montrer que les solutionsu ∈C ⁰ ∩C ¹ (Ω) du problème admettent une estimation a priori dansC ⁰ . Dans ce travail, on montre que, si les solutionsu sont juste supposéesC ⁰ , une telle estimation a priori n’existe plus. In a famous paper ([3]), B. GIDAS and J. SPRUCK used a “blow-up” argument to show that, under appropriate assumptions, all the solutionsu ∈C ⁰ ∩C ¹ (Ω) of the problem admit an a priori estimate inC ⁰ . In this work, we show that, if one supposes the solutions are only inC ⁰ , such an a priori estimate does not hold.     

Multiplicity of standing wave solutions of nonlinear Schrödinger equations with electromagnetic fields

In this paper, we are concerned with the multiplicity of standing wave solutions of nonlinear Schr?dinger equations with electromagnetic fields

Some Problems for Cubic Nonlinear Equations on a Half-Line

The paper deals with a problem of developing an inverse-scattering based formalism for solving problems for the cubic nonlinear (or the modified Korteweg–de Vries (KdV)) equations: q _t +q _xxx +6q ² q _x =0, 0x<, –<t<,q _t +q _xxx –6q ² q _x =0, with the given initial and boundary conditions: q(x,0)=q(x),q(0,t)=p(t), p(t)L ₁(–,). The relation between the solution of the initial-boundary value problem (1), (3), (4) and that of the KdV equation on the half-line is shown. The Cauchy problem for the cubic nonlinear equation: q _t +q _xxx –6|q|² q _x =0, 0x<, –<t<, with the given initial condition (3) is considered also. Here we solve the above problems on the half-line 0x< but with –<t<.     

On Stabilization of the Solution of the Cauchy Problem for a Parabolic Equation with a Lower Order Coefficient and a Growing Initial Function

The paper deals with the study of sufficient conditions on the coefficients of a second-order partial differential equation, ensuring the stabilization to zero of the solution of the Cauchy problem uniformly in x on any compact set K ^N for any continuous initial function u ₀(x), growing at infinity not faster than some power of |x| ^m , m > 0.     

Continuity properties of the normal cone to the level sets of a quasiconvex function

Two main properties of the subgradient mapping of convex functions are transposed for quasiconvex ones. The continuity of the functionxf(x)^–1f(x) on the domain where it is defined is deduced from some continuity properties of the normal coneN to the level sets of the quasiconvex functionf. We also prove that, under a pseudoconvexity-type condition, the normal coneN(x) to the set {x:f(x)f(x)} can be expressed as the convex hull of the limits of type {N(x _n)}, where {x _n} is a sequence converging tox and contained in a dense subsetD. In particular, whenf is pseudoconvex,D can be taken equal to the set of points wheref is differentiable.This research was completed while the second author was on a sabbatical leave at the University of Montreal and was supported by a NSERC grant. It has its origin in the doctoral thesis of the first author (Ref. 1), prepared under the direction of the second author.The authors are grateful to an anonymous referee and C. Zalinescu for their helpful remarks on a previous version of this paper.     

On polynomial automorphisms of spheres

Let K be an infinite field with characteristic different from two and (K) the n-sphere over K. We show that ambient polynomial automorphisms of (K) preserve the quadratic form x ₀² + ⋯ + x _n ² and the group Aut ((K ⁿ⁺¹, (K)) of such automorphisms of (K) is isomorphic to the (n + 1)-orthogonal group O(n + 1, K) provided K is real. Next, the restriction map Aut (K ³, (K)) → Aut ( (K)) yields a surjection provided K is an algebraically closed field as well. Furthermore, for any such a field K, there is an imbedding

Computational complexity of norm-maximization

This paper discusses the problem of maximizing a quasiconvex function over a convex polytopeP inn-space that is presented as the intersection of a finite number of halfspaces. The problem is known to beNP-hard (for variablen) when is thep ^th power of the classicalp-norm. The present reexamination of the problem establishesNP-hardness for a wider class of functions, and for thep-norm it proves theNP-hardness of maximization overn-dimensionalparallelotopes that are centered at the origin or have a vertex there. This in turn implies theNP-hardness of {–1, 1}-maximization and {0, 1}-maximization of a positive definite quadratic form. On the good side, there is an efficient algorithm for maximizing the Euclidean norm over an arbitraryrectangular parallelotope.The authors are indebted to J. O'Rourke, P, Pardalos and R. Freund for useful references. The second and third authors are indebted to the Institute for Mathematics and its Applications in Minneapolis, where much of this paper was written: they acknowledge additional support from the Alexander von Humboldt Stiftung and the National Science Foundation, respectively.     

Hyperinvariant Subspace Problem for Quasinilpotent Operators

In this paper we consider the hyperinvariant subspace problem for quasinilpotent operators. Let denote the class of quasinilpotent quasiaffinities Q in such that Q ^* Q has an infinite dimensional reducing subspace M with Q ^* Q| _M compact. It was known that if every quasinilpotent operator in has a nontrivial hyperinvariant subspace, then every quasinilpotent operator has a nontrivial hyperinvariant subspace. Thus it suffices to solve the hyperinvariant subspace problem for elements in . The purpose of this paper is to provide sufficient conditions for elements in to have nontrivial hyperinvariant subspaces. We also introduce the notion of “stability” of extremal vectors to give partial solutions to the hyperinvariant subspace problem.