Approximate Identities in Spaces of all Absolutely Continuous Measures on Locally Compact Semigroups

Let G be a locally compact group. Then M_a (G), the space of all absolutely continuous measures on G, has a bounded approximate identity. Baker and Baker proved that (S) (the space of all measures M(S) so that maps x _x *|| and x ||*_x are weak continuous from a locally compact semigroup S into M(S)) is closed under absolutely continuity and has an approximate identity. The main purpose of this paper is to show that similar results hold true for a locally compact semigroup S and M_a(S) the space of all absolutely continuous measures on S.     

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.     

Sur l'homotopie rationnelle des espaces fonctionnels

Let X be a nilpotent space such that it exists k1 with H^p (X,) = 0 p > k and H^k (X,) 0, let Y be a (m–1)-connected space with mk+2, then the rational homotopy Lie algebra of Y^X (resp. is isomorphic as Lie algebra, to H^* (X,) (_* (Y) ) (resp.⁺ (X,) (_* (Y) )). If X is formal and Y -formal, then the spaces Y^X and are -formal. Furthermore, if dim _* (Y) is infinite and dim H^* (Y,Q) is finite, then the sequence of Betti numbers of grows exponentially.     

Admissibility of linear estimators of regression coefficients under quadratic loss

For the general fixed effects linear model:Y=X+, N(0,V),V0, we obtain the necessary and sufficient conditions forLY+a to be admissible for a linear estimable functionS in the class of all estimators under the loss function (d -S)D(d -S), whereD0 is known. For the general random effects linear model: =XV ₁₁ X+XV ₁₂+V ₂₁ X+V ₂₂0, we also get the necessary and sufficient conditions forLY+a to be admissible for a linear estimable functionS+Q in the class of all estimators under the loss function (d -S -Q)D(d -S -Q), whereD0 is known.     

Algebraic systems of quadratic forms of number fields and function fields

In this paper we examine for which Witt classes ,..., _n over a number field or a function fieldF there exist a finite extensionL/F and ₂,..., _n L^* such thatT _L/F ()=1 andTr _L/F (_i)=_i fori=2,...n.     

The Parametric Buffer Phenomenon for a Singularly Perturbed Telegraph Equation with a Pendulum Nonlinearity

We consider the boundary-value problem u _tt + u _t + (1 + cos2)sin u =² u _xx, u _x|_x=0=u_x|_x==0, where 0<1, =(1+)t, ,> 0, and the sign of is arbitrary. It is proved that for an appropriate choice of the external parameters and and for sufficiently small the number of exponentially stable solutions 2-periodic in can be made equal to an arbitrary predefined number.     

Syzygies of Veronese Embeddings

We prove that the Veronese embedding _O ⁿ (d): ^{n N} with n2, d3 does not satisfy property N _p (according to Green and Lazarsfeld) if p3d–2. We make the conjecture that also the converse holds. This is true for n=2 and for n=d=3.     

A Hubert space characterization among function spaces

— [0,1] ,E — - e=1 [0,1]. I — _E =1, E=L ₂ x _e =xL ₂ x E.

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This work was prepared when the second author was a visiting professor of the CNR at the University of Firenze. He was supported by the Soros International Fund.     

Weighted Voronoi tessellation with a poisson field of centroids

One considers a weighted Voronoi tessellation generated by a system of n Poisson fields of centroids. One investigates the composition and the boundary field of this structure. It is proved that the boundary field intensity between grains of types i and j (1ijn) is equal to where p_i is the volume part, _i>0 is the weight (for the comparison of distances) of the i-th component, c is a scaling factor. One obtains formulas for the boundary field intensities in plane and linear sections of the structure: . Estimators for the parameters p_i and _i from the observation of the linear section are proposed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 153, pp. 160–172, 1986.     

On the number of simplexes of subdivisions of finite complexes

Combinatorial invariants of a finite simplicial complex K are considered that are functions of the number _i(K) of Simplexes of dimension i of this complex. The main result is Theorem 2, which gives the necessary and sufficient condition for two complexes K and L to have subdivisions K' and L' such that _i(K')=_i(L') for 0 . The theorem yields a corollary: if the polyhedra ¦K¦ and ¦L¦ are homeomorphic, then there exist subdivisions K' and L' such that _i(K')=_i(L') for i0.Translated from Matematicheskie Zametki, Vol. 3, No. 5, pp. 511–522, May, 1968.     

Interpolation and best polynomial approximation in the domain ofp-adic integers

, , . , . Lip

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The authors are indebted to Professor R. Bojanic for his valuable remarks and suggestions, especially for the simplification of the proof of Theorem 4.     

On the summability of Fourier series with the method of lacunary arithmetic means

. f- ,S _n (f)— . {n _k }, n _k+1/n _k >1+ck ^– ,— , 0<1/2, f ₀, .     

The Spectral Function and Principal Eigenvalues for Schrödinger Operators

Let m , 0 m₊ in Kato's class. We investigate the spectral function s( + m) where s( + m) denotes the upper bound of the spectrum of the Schrödinger operator + m. In particular, we determine its derivative at 0. If m_- is sufficiently large, we show that there exists a unique ₁ > 0 such that s( + ₁m) = 0. Under suitable conditions on m⁺ it follows that 0 is an eigenvalue of + ₁m with positive eigenfunction.     

OnC 2-diffeomorphisms of the circle which are of type III1

For every irrational number [0, 1) which is not of constant type we construct aC ²-diffeomorphism of the circle with rotation number which is of type III₁. This diffeomorphism can be chosen arbitrarily close to the rotationR . Our methods also allow us to construct, for every Liouville number [0, 1), aC -diffeomorphism of the circle with rotation number which is of type III₁.     

A Note on Thin P-Polynomial and Dual-Thin Q-Polynomial Symmetric Association Schemes

Let Y denote a d-class symmetric association scheme, with d 3. We show the following: If Y admits a P-polynomial structure with intersection numbers p _ij ^h and Y is 1-thin with respect to at least one vertex, then p _ll ^l =0 p _li ⁱ =0 1 i - 1. If Y admits a Q-polynomial structure with Krein parameters q _ij ^h , and Y is dual 1-thin with respect to at least one vertex, then q _ll ^l = 0 q _li ⁱ = 01 i d-1.     

Finite Representability of ℓp in Quotients of Banach Spaces

A typical result of the paper states that if X is a Banach space with a basis and for some 1pq, the spaces _p and _q are finitely block representable in every block subspace of X, then every block subspace of X admits a block quotient Z such that for every r[p,q], the space _r is finitely block representable in Z. Results of a similar nature are also established for ^N _p-block-sequences and asymptotic spaces.     

On quadratic spline interpolation

Using the quadratic spline interpolates(x) fitting the data (x _i,y _i), 0in and satisfying the end conditions_o=y_o, we give formulae approximatingy andy at selected knots by orders up toO(h ⁴).     

Limit distribution of the last exit time for stationary random sequences

Summary For a strictly stationary random sequence (X _i) _i0 we find sufficient conditions such that the distribution of the last exit time t = max{i X _i>i} (>0) tends weakly to a nondegenerate limit distribution as 0.     

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.     

Necessary conditions for the existence of divisible designs

BOSE and CONNOR [2] proved that a symmetric regular divisible design with w classes of sizes g and joining numbers ₁ and ₂ must satisfy for every prime p the arithmetic condition (d₁, (–1)^sw)_p(d₂,(–l)^tg^w)_p=1, where d₁=k²–v₂, d₂= k–₁ s=(w-1)(w-2)/2, t=(v-w)(v-w-1)/2 and (*,*) is the Hilbert symbol. We show that if in addition _{1 2} and the design is fully symmetric divisible then (d₁, (–1)^s w)_p=(d₂, (–1)^tg^w)=1. Our assumption is by a result of CONNOR [5] fulfilled, if d₁ and ₁–₂ are relatively prime. Thus, we can exclude parameters not accessible to the Bose-Connor-Theorem. Our result can be derived from a theorem of RAGHAVARAO [9], and we give the precise assumptions of this theorem. We also discuss arithmetic restrictions for divisible designs which satisfy diverse other rules for the intersection numbers and generalize a result of DEMBOWSKI [6; 2.1.11].Dedicated to Professor Benz on occasion of his sixtieth birthday