On Tyler's M-Functional of Scatter in High Dimension

Let y¹, y²,..., yⁿ R^q be independent, identically distributed random vectors with nonsingular covariance matrix , and let S = S(y¹,..., yⁿ) be an estimator for . A quantity of particular interest is the condition number of ^-1 S. If the yⁱ are Gaussian and S is the sample covariance matrix, the condition number of ^-1 S, i.e. the ratio of its extreme eigenvalues, equals 1 + O^p((q/n)^1/2) as q and q/n 0. The present paper shows that the same result can be achieved with two estimators based on Tyler's (1987, Ann. Statist., 15, 234-251) M-functional of scatter, assuming only elliptical symmetry of (yⁱ) or less. The main tool is a linear expansion for this M-functional which holds uniformly in the dimension q. As a by-product we obtain continuous Fréchet-differentiability with respect to weak convergence.     

Superconvergence for galerkin methods for the two point boundary problem via local projections

The two point boundary problemy'-a(x)y–b(x)y=-f(x), o<x<1,y(0)=y(1)=0, is first solved approximately by the standard Galerkin method, (Y, ) + (aY+bY, )=(f, ), ₁ ⁰ (r, ), for a function Y ₁ ⁰ (r, ), the space ofC ¹-piecewise--degree-polynomials vanishing atx=0 andx=1 and having knots at {x ₀ ,x ₁ , ...,x _M }=. ThenY is projected locally into a polynomial of higher degree by means of one of several projections. It is then shown that higher-order convergence results locally, provided thaty is locally smooth and is quasi-uniform.This research was supported in part by the National Science Foundation.     

Spectral analysis of biorthogonal expansions generated by Muckenhoupt weights

An arbitrary Muckenhoupt A₂-weight w² on the special contour ( 1/2) generates a function Y_,w (, t), which for =1, w²(z) 1 coincides with the exponential exp{it}. In the paper, with the aid of B. S. Pavlov's geometric approach, one obtains criteria for the unconditional basis property of families of functions of the form {y,w(_k,t):_k} in the space L₂(0, ). The analytic foundation of the constructions is a generalization of M. M. Dzhrbashyan's certain results (power weight) to the case of arbitrary Muckenhoupt A₂- weights.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 190, pp. 34–80, 1991.     

Monotonicity properties of zeros of generalized Airy functions

Summary We show, among other things, that the positive zeros of a solution ofy +x y=0,y(0)=0 decrease to 1 as increases, 0.

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Sommario Si dimostra, tra l'altro, che gli zeri positivi d'una soiuzione diy +x y=0,y(0)=0 decrescono al limite 1, quando cresce, 0.

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To the memory of Milo Háik

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This research was supported by grants from the Natural Sciences and Engineering Research Council (Canada) and Consiglio Nazionale delle Ricerche (Italy). Some of the work was done while the second-named author was visiting the Department of Mathematics, University of Torino.     

Symmetrische Permutationsmengen

A permutation set (M, I) consisting of a setM and a set of permutations ofM, is calledsymmetric, if for any two permutations, the existence of anx M with (x) (x) and ^–1 (x) = ^–1 (x) implies ^–1 = ^–1 , andsharply 3-transitive, if for any two triples (x ₁,x ₂,x ₃), (y ₁,y ₂,y ₃) M ³ with|{x ₁,x ₂,x ₃ }| = |{y ₁,y ₂,y ₃ }| = 3 there is exactly one permutation with(x ₁) =y ₁,(x ₂) =y ₂,(x ₃) =y ₃. The following theorem will be proved.THEOREM.Let (M, ) be a sharply 3-transitive symmetric permutation set with |M|3, such that contains the identity. Then is a group and there is a commutative field K such that and the projective linear group PGL(2, K) are isomorphic.     

Bounds for the nonhomogeneous GASPT equation

Summary Weighted a priori bounds for the equation u+(/y)u_y=f(>0), in the halfplane y>0, are proved. If p>1, 0<+p^–1<1+, u has bounded support and y^µu_y0 (as y0₊), then the L^p norms of uu and yD²u are bounded by the L^p norm of yf. A boundary value problem in a rectangle is also studied in the appropriate weighted Sobolev class.Lavoro eseguito nell'ambito dell'Istituto di Analisi Globale ed Applicazioni del C.N.R.Lavoro eseguito nell'ambito del Gruppo Nazionale di Analisi Funzionale ed Applicazioni del C.N.R.     

Interpolation of spectra

By the M.Riesz Convexity Theorem, an operator T on the space of simple integrable functions into the measurable functions (on some measure space) which has continuous extensions to L^p() and L^q() , where 1 p q , also has continuous exten — sions to all L^r () , p r q . It is shown that, whenever (T_p) and (T_q) are o-dimensional (in particular, countable) then the spectra (T_r) (p r q) are pairwise identical. For q = , only w^*-continuous extensions are considered. An example due to Dayanithy shows that the conclusion fails in general.     

Superstable implicit Runge-Kutta methods for second order initial value problems

Using the well known properties of thes-stage implicit Runge-Kutta methods for first order differential equations, single step methods of arbitrary order can be obtained for the direct integration of the general second order initial value problemsy=f(x, y, y),y(x _o)=y _o,y(x _o)=y _o. These methods when applied to the test equationy+2y+ ² y=0, ,0, +>0, are superstable with the exception of a finite number of isolated values ofh. These methods can be successfully used for solving singular perturbation problems for which f/y and/or f/y are negative and large. Numerical results demonstrate the efficiency of these methods.     

Non polynomial splines and weakly singular two-point boundary value problems

In the present paper we shall consider an application of simple non-polynomial splines to a numerical solution of a weakly singular two-point boundary value problem:x ^– (x y)=f(x,y), (0<x1) subject toy(0)=0,y(1)=c ₁(1) ory(0)=c ₂,y(1)=c ₃(0<<1). Our collocation method gives a continuously differentiable approximation and isO(h ²)-convergent.     

Solution of two sequencing problems

The solution of the following problems is offered. Suppose a multiset J (¦J¦=p) is given. For each pair of elements and J, a number 1 P is given. Moreover, if 1 < x<p then x is undefined. If x=1, then x=p. Problem 1. Find the permutation ₁..._F of elements of the multiset J satisfying the following conditions. Let _i, _i=. If i,j < x, thenj <i. If i,j > x, then i<j. Such a permutation is called a PC-schedule. Problem 2. Find a PC-schedule in which the following property holds: if i < x < j, _i=, _j=, then. Such a PC-schedule is called an SC-schedule. The conditions under which these problems have solutions are studied. For their solution an algorithm of shifts is used with the complexity O(¦B(J)¦²¦J¦).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 124, pp. 44–72, 1983.     

On the convergence in a restricted sense of multiple series

Z ^d — k=(k ₁, ...,k _d) k _j,d1.d- (8), . . a _k s _m= a _k s, >0 N, min (m ₁,...,m _d)N, ¦s _m–s¦. , , >0 N, min (m ₁,...,m _d)N min (n ₁,...,n _d)N, ¦s _m–s _n. . , (8) , >0 N, max (b ₁,...,b _d) N, m — Z ^d , m1, ¦s(b, m)¦ where      

Spectral properties of Neumann Laplacian of horns

We study the Neumann Laplacian of unbounded regions in ⁿ with cusps at infinity so that the corresponding Dirichlet Laplacian has compact resolvent. Typical of our results is that of the region {(x, y)²x, y|<1} the Neumann Laplacian has absolutely continuous spectrum [0, ) of uniform multiplicity four and an infinity of eigenvaluesE _o<E ₁... and that for the region {(x, y)²y|1}, it has absolutely continuous spectrum [1/4, ) of uniform multiplicity 2 and an infinity of eigenvaluesE ₀=0<E ₁.... We use the Enss theory with a suitable asymptotic dynamics.The second author's research is partially funded under NSF grand number DMS-8801918     

Propriété de stabilité de la fonction extrémale relative

Nous donnons une caractérisation des domaines DX pour lesquels la fonction extrémale relative ^*(,E,D) a la propriété de stabilité pour tout ED, i.e. lim _k^*(,E,D _k )=^*(,E,D), ED. Ensuite, nous étudions la relation entre cette propriété et les enveloppes pluripolaires. Nous concluons par quelques remarques sur la propriété de stabilité lim _k^*(,E _k ,D)=^*(,E,D).     

Fine Analysis of the Quasi-Orderings on the Power Set

We pursue the fine analysis of the quasi-orderings and on the power set of a quasi-ordering (Q,). We set X Y if every xX is majorized in by some yY, and X Y if every yY is minorized in by some xX. We show that both these quasi-orderings are -wqo if and only if the original quasi-ordering is ( )-wqo. For this holds also restricted to finite subsets, thus providing an example of a finitary operation on quasi-orderings which does not preserve wqo but preserves bqo.     

Perturbation of the Laplacian by the Coulomb potential and a point interaction in L p (ℝ)3

Perturbations of -+/|x| (with >0) by a point interaction centered at zero are defined in L ^p(³). This is done for 3/20 (³{0}), such that the extension is the negative generator of an analytic semigroup on L ^p(³).     

The distortion of Hilbert space

The unit sphere of Hilbert space, ₂, is shown to contain a remarkable sequence of nearly orthogonal sets. Precisely, there exist a sequence of sets of norm one elements of ₂, (C _i ) _i=1 , and reals _i 0 so that a) each setC _i has nonempty intersection with every infinite dimensional closed subspace of ₂ and b) forij,xC, andyC _j , |x, y|<_{min(i, j)} E. Odell was partially supported by NSF and TARP. Th. Schlumprecht was partially supported by NSF and LEQSF.     

On certain functional derivatives

The functional derivative J _y =F _y – (d/dx)F _y of the functionalJ[y] = _a ^b F(x,y,y)dx may be computed by the limit J _y (x) = lim ₀(J/), where is the area under a positive local variation atx, provided the height of the variation vanishes faster than the square of its width. This justifies the use of this limit by Gelfand and Fomin (Ref. 1).This work was supported in part by The National Research Council of Canada, Grant No. A-8744. The author would like to thank Dr. F. H. Hsu for discussions.     

On the type of spectral operators and the non-spectrality of several differential operators on Lp

A spectral operator, not necessarily bounded, which is the infinitesimal generator of a strongly continuous group of operators {U(t)|t} where U(t) = (|t|^K), is of type k N_o, i.e. T = S + N, where S is spectral of scalar type, N is bounded, N^k+1 = O and S and N are commuting. This result yields a simple proof of the non-spectrality of certain differential operators on L^p, p 2, which are known to be selfadjoint for p = 2.     

On the semimartingale representation of reflecting Brownian motion in a cusp

Summary LetC be the symmetric cusp {(x, y)²:–x yx ,x0} where >1. In this paper we decide whether or not reflecting Brownian motion inC has a semimartingale representation. Here the reflecting Brownian motion has directions of reflection that make constant angles with the unit inward normals to the boundary. Our results carry through for a wide class of asymmetric cusps too.     

Fast Summation of Power Series with Coefficients Analytic at Infinity

We propose a fast summation algorithm for slowly convergent power series of the form _{j=j 0} z ^j _j j _i=1 ^s (j+ _i )^– _i , where R, _i 0 and _i C, 1is, are known parameters, and _j =(j), being a given real or complex function, analytic at infinity. Such series embody many cases treated by specific methods in the recent literature on acceleration. Our approach rests on explicit asymptotic summation, started from the efficient numerical computation of the Laurent coefficients of . The effectiveness of the resulting method, termed ASM (Asymptotic Summation Method), is shown by several numerical tests.