Some propositions related to a dilation theorem of W. Benz

Summary This paper begins with another proof of a theorem of W. Benz [2] concerning dilations in normed linear spaces. Our proof motivates several questions which are addressed thereafter. For instance it is shown that, ifI is an open interval in ,: I ⁿ , is continuously differentiable and there exista ₁,...,a _n I such that {(a ₁,...,(a _n )} is linearly independent, then {(t): t I} contains a Hamel basis for ⁿ over .     

More on directed colimits of models

M. Richter has proved that whenever a classK of -structures has a finitary first-order axiomatization then the inclusionK Str preserves all existing directed colimits (see [7]). We will generalize this result to classes of -structures having an infinitary first-order axiomatization in a larger signature . We will also show that, as categories, these classes have a natural characterization.In honour of Dieter Pumplün on the occassion of his 60th birthday     

The regularity of Lagrangiansf(x, ξ)=254-01254-01254-01with Hölder exponents α(x)

In this paper the regularity of the Lagrangiansf(x, )=||^(x)(1< ₁(x)₂< +) is studied. Our main result: If(x) is Holder continuous, then the Lagrangianf(x, )=f(x, )=||^(x) is regular. This result gives a negative answer to a conjecture of V. Zhikov.Supported by the National Natural Science Foundation of China.     

Critical star multigraphs

A star-multigraphG is a multigraph in which there is a vertexv ⁺ which is incident with each non-simple edge. It is critical if it is connected, Class 2 and(G\e) < (G) for eache E(G). We show that, ifG is any star multigraph, then(G) (G) + 1. We investigate the edge-chromatic class of star multigraphs with at most two vertices of maximum degree. We also obtain a number of results on critical star multigraphs. We shall make use of these results in later papers.     

Approximate Inclusion-Exclusion

The Inclusion-Exclusion formula expresses the size of a union of a family of sets in terms of the sizes of intersections of all subfamilies. This paper considers approximating the size of the union when intersection sizes are known for only some of the subfamilies, or when these quantities are given to within some error, or both.In particular, we consider the case when allk-wise intersections are given for everykK. It turns out that the answer changes in a significant way aroundK=n: ifKO(n) then any approximation may err by a factor of (n/K ²), while ifK (n) it is shown how to approximate the size of the union to within a multiplicative factor of .When the sizes of all intersections are only given approximately, good bounds are derived on how well the size of the union may be approximated. Several applications for Boolean function are mentioned in conclusion.Partially supported by NSF 865727-CCR and ARO DALL03-86-K-017. Part of this work was done in U.C. Berkeley, supported by NSF CCR-8411954.     

Additional results of Zalcman's Pompeiu problem

Summary In this article we continue our study of the following problem posed by Lawrence Zalcman in 1972. LetS be the closed unit square. For eachz in the interior,S ⁰, ofS letS(z) be the largest closed square inS with centroidz, and for each in the interval (0, 1] letS _(z) be the square homothetic toS(z) with linear ratio . Iff is a continuous function such that its integral overS _(z) vanishes for allz in S⁰, is f =0? We show that the answer is yes if 3/4 < 1.     

Contraction of Exterior Powers in Characteristic 2 and the Spin Module

Let K be a field of characteristic 2 and letV be a vector space of dimension 2m over K. Let f be a non-degenerate alternating bilinear form defined on V × V. The symplectic group Sp(2m, K) acts on the exterior powers ^k V for 0 k. 2m There is a contraction map defined on the exterior algebra , which commutes with the Sp(2m, K) action and satisfies ² = 0 and ( ^k V) ^k–1 V We prove that ( ^k V)= ker ^k–1 V except when k=m+2. In the exceptional case, ( ^m+2 V) has codimension 2^m in ker ^m V and we show that the quotient module ker ^m V/ ^m+2 V is a spin module for Sp(2m,K). When K is algebraically closed, we show that this spin module occurs with multiplicity 1 in ^m V and multiplicity 0 in all other components of V.     

Rates of convergence for the approximation of dual shift-invariant systems in ℓ2(ℤ)

A shift-invariant system is a collection of functions {g_m,n} of the form g_m,n(k)=g_m(k–an). Such systems play an important role in time-frequency analysis and digital signal processing. A principal problem is to find a dual system _m,n(k)=_m(k–an) such that each functionf can be written asf= f, _m,ngm,n. The mathematical theory usually addresses this problem in infinite dimensions (typically in L² () or ²()), whereas numerical methods have to operate with a finite-dimensional model. Exploiting the link between the frame operator and Laurent operators with matrix-valued symbol, we apply the finite section method to show that the dual functions obtained by solving a finite-dimensional problem converge to the dual functions of the original infinite-dimensional problem in ²(). For compactly supported g_{m, n} (FIR filter banks) we prove an exponential rate of convergence and derive explicit expressions for the involved constants. Further we investigate under which conditions one can replace the discrete model of the finite section method by the periodic discrete model, which is used in many numerical procedures. Again we provide explicit estimates for the speed of convergence. Some remarks on tight frames complete the paper.Part of this work was done while the author was a visitor at the Department of Statistics at the Stanford University.The author has been partially supported by Erwin-Schrödinger scholarship J01388-MAT of the Austrian Science foundation FWF.     

Eigenvalues of integral operators onL 2(I) given by analytic kernels

Let {_n} _n=0 be the eigenvalue sequence of a symmetric Hilbert-Schmidt operator onL ₂(I). WhenI is an open interval, a necessary condition for {_n} _n=0 to be in the sequence space is obtained. WhenI is a closed bounded interval, sufficient conditions for {_n} _n=0 to be in the sequence space ^– are obtained.     

On some properties of multiple conjugate trigonometric series

We have obtained an estimate, in terms of partial and mixed moduli, of the continuity of deviation of the Cesáro (C, ) means ( = (₁,...,_n),_i , ₁ > –1, ) of the sequence of rectangular partial sums ofn-multiple (n>1) conjugate trigonometric series from then-multiple truncated conjugate function. This estimate implies the result on them -convergence (1) of (C, ) means (₁ > 0, ) provided that the essential conditions are imposed on the partial moduli of continuity. Finally, it is shown that them -convergence cannot be replaced by ordinary convergence.