Weighted Polynomial Approximation for Weights with Slowly Varying Extremal Density

Polynomial approximation by weighted polynomials of the form wⁿ(x) P_n(x) is investigated on closed subsets of the real line. It is known that the possibility of approximation is closely related to the density of an extremal measure associated with w via a weighted energy problem. It is also known that if in a neighborhood of a point x₀ this density is continuous and positive, then, in that neighborhood, any continuous function can be approximated. The aim of the present paper is twofold. On the one hand it is shown that the same approximation theorem is true if in a neighborhood of x₀ the density is slowly varying and is bounded away from 0. This allows singularities of logarithmic types. On the other hand, we also show that under some mild conditions, if the density at x₀ is slowly varying, then approximation is still possible even if the density vanishes at x₀ . This is the first positive result for approximation with a vanishing density.     

Theoremes de Viabilite Pour les Inclusions Differentielles du Second Ordre

This paper gives conditions ensuring the existence for an initial value (x ₀,v ₀) of a solution to the second order differential inclusionx″(t) ∈F[x(t),x′(t)],x(0)=x ₀,x′(0)=v ₀ such thatx(t) ∈K for allt whereK is a nonempty given subset ofR ⁿ .      

On the identification of degenerate indices in the nonlinear complementarity problem with the proximal point algorithm

In this paper we focus on the problem of identifying the index sets P(x):=i|x_i>0, N(x):={i|F_i(x)>0 and C(x):=i|x_i=F_i(x)=0} for a solution x of the monotone nonlinear complementarity problem NCP(F). The correct identification of these sets is important from both theoretical and practical points of view. Such an identification enables us to remove complementarity conditions from the NCP and locally reduce the NCP to a system which can be dealt with more easily. We present a new technique that utilizes a sequence generated by the proximal point algorithm (PPA). Using the superlinear convergence property of PPA, we show that the proposed technique can identify the correct index sets without assuming the nondegeneracy and the local uniqueness of the solution.This work was supported in part by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture of Japan.Mathematics Subject Classification (2000): 90C33, 65K10     

A Minimax Problem Admitting the Equioscillation Characterization of Bernstein and Erd?s

This paper shows that under certain conditions a solution of the minimax problem min_{a<x1<…<xn<b} max_1in+1 f_i(x₁, …, x_n) admits the equioscillation characterizations of Bernstein and Erd s and has strong uniqueness. This problem includes as a particular example the optimal Lagrange interpolation problem.     

Minimal non-permutative pseudovarieties of semigroups III

Various types of permutativity conditions are important in the study of (pseudo)varieties of semigroups. In previous papers of this series, minimal pseudovarieties failing certain permutativity conditions were determined. This theme is pursued here in relation to a family of conditions which appear in the investigation of power pseudovarieties of semigroups, namely, for eachk0, we consider the conditionC _k on a semigroup that it satisfy the identityx ₁ ...x _k y ₁ y ₂ z ₁ ...z _n=x ₁ ...x _k y ₂ y ₁ z ₁ ...z _n for somen. For all minimal pseudovarieties failing C_k, a generator and finite basis of identities are specified.This work was supported, in part, by NSERC Grant A4044.Presented by Boris Schein.     

Isolated singularities of Monge-Ampère equations

Letz=z(x, y) be a real-valued twice continuously differentiable solution of the elliptic Monge-Ampère equationAr+2Bs+Ct+rt – s ²=E in the punctured disk 0<(x–x ₀)²+(y–y ₀)²<². Assume thatq is continuous at (x₀, y₀). Our aim is to give sufficient conditions on the coefficientsA,..., E which ensure that the singularity (x ₀,y ₀) is removable. This generalizes an earlier result of Jörgens (Math. Ann. 129 (1955), 330–344).     

Approaching an extinction point in one-dimensional semilinear heat equations with strong absorption

This paper deals with the Cauchy problem u_t − u_xx + u^p = 0; − ∞ < x < + ∞, t>0, u(x, 0) = u₀(x); − ∞ < x < + ∞, where 0 < p < 1 and u₀(x) is continuous, nonnegative, and bounded. In this case, solutions are known to vanish in a finite time T, and interfaces separating the regions where u(x, t) > 0 and u(x, t) = 0 appear when t is close to T. We describe here all possible asymptotic behaviours of solutions and interfaces near an extinction point as the extinction time is approached. We also give conditions under which some of these behaviours actually occur.     

Uniqueness of Solutions of Some Nonlocal Boundary-Value Problems for Operator-Differential Equations on a Finite Segment

For the equation L ₀ x(t) + L ₁ x ⁽¹⁾(t) + ... + L _n x ⁽ⁿ⁾(t) = 0, where L _k, k = 0, 1, ... , n, are operators acting in a Banach space, we formulate conditions under which a solution x(t) that satisfies some nonlocal homogeneous boundary conditions is equal to zero.     

Tikhonov regularization of nonlinear III-posed problems in hilbert scales

In this paper we consider nonlinear ill-posed problems F(x) = y ₀, where x and y ₀ are elements of Hilbert spaces X and Y, respectively. We solve these problems by Tikhonov regularization in a Hilbert scale. This means that the regularizing norm is stronger than the norm in X. Smoothness conditions are given that guarantee convergence rates with respect to the data noise in the original norm in X. We also propose a variant of Tikhonov regularization that yields these rates without needing the knowledge of the smoothness conditions. In this variant F is allowed to be known only approximately and X can be approximated by a finite-dimensional subspace. Finally, we illustrate the required conditions for a simple parameter estimation problem for regularization in Sobolev spaces.     

Spreading Properties and Complex Dynamics for Monostable Reaction–Diffusion Equations

This paper is concerned with the study of the large-time behavior of the solutions u of a class of one-dimensional reaction–diffusion equations with monostable reaction terms f, including in particular the classical Fisher-KPP nonlinearities. The nonnegative initial data u ₀(x) are chiefly assumed to be exponentially bounded as x tends to + ∞ and separated away from the unstable steady state 0 as x tends to ? ∞. On the one hand, we give some conditions on u ₀ which guarantee that, for some λ > 0, the quantity c _λ = λ +f′(0)/λ is the asymptotic spreading speed, in the sense that lim  _t→+∞ u(t, ct) = 1 (the stable steady state) if c < c _λ and lim  _t→+∞ u(t, ct) = 0 if c > c _λ. These conditions are fulfilled in particular when u ₀(x) e ^λx is asymptotically periodic as x → + ∞. On the other hand, we also construct examples where the spreading speed is not uniquely determined. Namely, we show the existence of classes of initial conditions u ₀ for which the ω-limit set of u(t, ct + x) as t tends to + ∞ is equal to the whole interval [0, 1] for all x ∈ ? and for all speeds c belonging to a given interval (γ₁, γ₂) with large enough γ₁ < γ₂ ≤ + ∞.     

Convergence analysis of iterative sequences for a pair of mappings in Banach spaces

Let C be a nonempty closed convex subset of a real Banach space E. Let S ： C→ C be a quasi-nonexpansive mapping, let T ： C→C be an asymptotically demicontractive and uniformly Lipschitzian mapping, and let F ：= {x ∈C ： Sx = x and Tx = x}≠Ф Let {xn}n≥0 be the sequence generated irom an arbitrary x0∈Cby xn＋i=（1-cn）Sxn＋cnT^nxn, n≥0.We prove the necessary and sufficient conditions for the strong convergence of the iterative sequence {xn} to an element of F. These extend and improve the recent results of Moore and Nnoli.     

Quadruples in the four-number game with large termination times

The discrete dynamical system of absolute differences defined by the map Ψ(x ₁, x ₂, x ₃, x ₄) = (| x ₂ - x ₁ |, | x ₁ - x ₁|, |x ₁ - x ₁ |, |x ₁ - x ₄ |) has been studied by many authors and one of the interesting questions is how to locate quadruples which converge to the fixed point (0, 0, 0, 0) in large numbers of steps. An elementary method is offered for obtaining such quadruples. The method is also able to find quadruples that will not converge to (0, 0, 0, 0).     

Extended Aitken acceleration

The ordinary Aitken's ²-process uses three consecutive valuesx ₀,x ₁ andx ₂ of a linearly converging sequencex _i+1=(x _i ) to give an accelerated value with an error of order 2 in the error inx ₀. It is shown how the process may be extended in a simple way, very similar to the repeated Richardson'sh ²-extrapolation, to give an accelerated value with error of any orderk, using only thek+1 first order valuesx ₀,x ₁, ...,x _k .     

On zero-divisors of near-rings of polynomials

In this paper, we are interested to study zero-divisor properties of a 0-symmetric nearring of polynomials R₀[x], when R is a commutative ring. We show that for a reduced ring R, the set of all zero-divisors of R₀[x], namely Z(R₀[x]), is an ideal of R₀[x] if and only if Z(R) is an ideal of R and R has Property (A). For a non-reduced ring R, it is shown that Z(R₀[x]) is an ideal of Z(R₀[x]) if and only if ann_R({a, b}) ∩ N i?(R) ≠ 0, for each a, b ∈ Z(R). We also investigate the interplay between the algebraic properties of a 0-symmetric nearring of polynomials R₀[x] and the graph-theoretic properties of its zero-divisor graph. The undirected zero-divisor graph of R₀[x] is the graph Γ(R₀[x]) such that the vertices of Γ(R₀[x]) are all the non-zero zero-divisors of R₀[x] and two distinct vertices f and g are connected by an edge if and only if f ? g = 0 or g ? f = 0. Among other results, we give a complete characterization of the possible diameters of Γ(R₀[x]) in terms of the ideals of R. These results are somewhat surprising since, in contrast to the polynomial ring case, the near-ring of polynomials has substitution for its “multiplication” operation.     

Identities in rings with involutions

A ringR with an involutiona →a* which satisfies a polynomial identityp[x ₁,…,x _d ;x*₁, …,x* _d ]=0 satisfies- an identity which does not include thex*. This generalizes the result of [1] where the symmetric elements ofR were assumed to satisfy an identity.     

On a weak sum theorem in dimension theory

A metric spaceX = U _i-0 ^x X is constructed such thatX _o={x _o} consists of a single pointx _o , X _i , i=0, 1, 2, … are disjoint and closed,X _i , i=1, 2, … are open, indX _i =0 fori=0, 1, … and indX=1. The above space (proved to be, in some sense, most simple) shows also that the dimension ind of a metric space can be raised by adjoining of a single point, a fact proved recently by E.K. Van Douwen and by T. Przymusiński. Some maximality property of the family {X; IndX=0} is proved and conditions implyingP-ind=P-Ind are given. This is part of a research thesis at the Technion, Israel Institute of Technology, towards an M.Sc. degree, directed by Professor M. Reichaw.     

On Kernel polynomials and self-perturbation of orthogonal polynomials

Given an orthogonal polynomial system {Q _n (x)} _n=0 ^∞, define another polynomial system by where α _n are complex numbers and t is a positive integer. We find conditions for {P _n (x)} _n=0 ^∞ to be an orthogonal polynomial system. When t=1 and α₁≠0, it turns out that {Q _n (x)} _n=0 ^∞ must be kernel polynomials for {P _n (x)} _n=0 ^∞ for which we study, in detail, the location of zeros and semi-classical character. Received: November 25, 1999; in final form: April 6, 2000?Published online: June 22, 2001     

Optimal iterative processes for root-finding

Letf ₀(x) be a function of one variable with a simple zero atr ₀. An iteration scheme is said to be locally convergent if, for some initial approximationsx ₁, ...,x _s nearr ₀ and all functionsf which are sufficiently close (in a certain sense) tof ₀, the scheme generates a sequence {x _k} which lies nearr ₀ and converges to a zeror off. The order of convergence of the scheme is the infimum of the order of convergence of {x _k} for all such functionsf. We study iteration schemes which are locally convergent and use only evaluations off,f, ...,f ^[d] atx ₁, ...,x _k–1 to determinex _k, and we show that no such scheme has order greater thand+2. This bound is the best possible, for it is attained by certain schemes based on polynomial interpolation.This work was supported (in part) by the Office of Naval Research under contract numbers N0014-69-C-0023 and N0014-71-C-0112.     

Estimation of the Density of Hypoelliptic Diffusion Processes with Application to an Extended Itô's Formula

We prove a uniform bound for the density, p _t (x), of the solution at time t(0, 1] of a 1-dimensional stochastic differential equation, under hypoellipticity conditions. A similar bound is obtained for an expression involving the distributional derivative (with respect to x) of p _t (x). These results are applied to extend the Itô formula to the composition of a function (satisfying slight regularity conditions) with a hypoelliptic diffusion process in the spirit of the work of Föllmer et al. ⁽⁵⁾