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Suppose P(x, D) is a linear partial differential operator on an open set ~ contained in R ~ and that A is a closed subset of ~. Given a class ~(~) of distributions on ~, the set A is said to be removable for ~(~) if e ach /E ~(~), which satisfies P(x, D ) / = 0 in ~ A , also satisfies P(x, D)/= 0 in ~. The problem considered in this paper is the following. Given a class ~(~) of distributions on ~, what restriction on the size of A will ensure that A is removable for ~(~). We obtain results for Lroc (~) (p ~< ~ ) , C(~), and Lipa (~). The first result of this kind was the Riemann removable singularity theorem: if a function / is holomorphic in the punctured unit disk a n d / ( z ) = o ( H -1) as z approaches zero, then / is holomorphic in the whole disk. Bochner [1] generalized Riemann's result by considering the class ~(~) of functions f on ~ such that ](x)=o(d(x, A) -q) uniformly for x in compact subsets of ~, and giving a condition on the size of A which insures tha t A is removable for ~(~) (Theorem 2.5 below). Bochner's theorem is remarkable in that the condition on the size of A only depends on the order of the operator P(x, D). The theorem applies, therefore, to systems of differential operators, such as exterior differentiation in R n and ~ (the Cauchy-Riemann operator) in C n. The same can be said for the other results in this paper. The proof of Bochner's theorem provided the motivation for our results. I t is interesting to note tha t a very general result (Corollary 2.4) f o r / ~ (~) (due to Li t tman [7]) is an easy corollary of Bochner's work. Here the condition on the singular set A is expressed in terms of Minkowski content. In section 4 the case of Ll~oc(~) is studied again, and results in section 2 are improved by replacing Minkowski content with Hausdorff measure. In addition, the cases C(~)

The purpose of the present paper is the solution of the boundary value problems for minimal surfaces when the boundaries are not, or not entirely-fixed Jordan curves but are free to move on prescribed manifolds. At the same time I shall present modifications and simplifications of my previous solution of the Plateau' and Douglas' problem for fixed boundary curves and prescribed topological structure and incidentally discuss certain features of the problem in order to clarify its relation to the theory of conformal mapping. Though based on previous publications, the paper may, except for some references, be read independently.

Jordan, von Neumann, and Wigner [5] have classified all finite dimensional Jordan algebras over the reals. The present paper is an attempt to do the same in the infinite dimensional case. The following restriction will be imposed: we assume the Jordan alge- bras are weakly closed Jordan algebras of self-adjoint operators with minimal projections acting on a Hilbert space, i.e. are irreducible JW-algebras of type i.(1) The result is then quite analogous to that in [5], except we do not get hold of the Jordan algebra ~a s of that paper, as should be expected from the work of Albert [1]. We first classify all irreducible JW-algebras of type In, n>~3 (Theorem 3.9). These algebras are roughly all seif-adjoint operators on a Hilbert space over either the reals, the complexes, or the quaternions. Then all JW-factors of type

then f(m) is said to be strongly additive. I n this paper we shall confine our a t tent ion to strongly addit ive functions. The paper falls into three sections. I n the first section we consider those s trongly addit ive functions f(m) which, after a suitable translation, possess a hmit ing distribution. Theorems 1 and 2 provide a characterization of such functions, essentially in terms of their values on the primes. A classic result of ErdSs and Wintner states t ha t an addit ive funct ion f(m) has a limiting distr ibution if and only if the two series

on the basis of the general analytic properties of the kernel K (x, ~) such as im tegrability, continuity, differentiability, analyticity and the like? The l i tera ture where this and analogous questions are t rea ted is v e r y considerable [HELLI~G~n-To~eLITZ, I]. ~ A relatively small par t of this l i terature, however, has points of contact with the present paper, the discussion of the majori ty of papers published on the subject being based on various special properties of the kernels. I t is assumed frequent ly tha t the kernel belongs to some special class of functions, or tha t it coincides with the Green 's funct ion of a different ial or integro-differential boundary value problem. Problems of this sort will be excluded f rom the scope of our paper a l though they are in teres t ing f rom a theoret ical point of view and impor tan t for the applications.

The structure of a stationary Gaussian process near a local maximum with a prescribed height u has been explored in several papers by the present author, see [5]–[7], which include results for moderate u as well as for u→±∞. In this paper we generalize these results to a homogeneous Gaussian field {ξ(t) t ∈ Rn}, with mean zero and the covariance function r(t). The local structure of a Gaussian field near a high maximum has also been studied by Nosko, [8], [9], who obtains results of a slightly different type. ¶ In Section 1 it is shown that if ξ has a local maximum with height u at 0 then ξ(t) can be expressed as $\xi _u (t) = uA(t) - \xi _u^\prime b(t) + \Delta (t),$ ¶ WhereA(t) and b(t) are certain functions, θu is a random vector, and Δ(t) is a non-homogeneous Gaussian field. Actually ξu(t) is the old process ξ(t) conditioned in the horizontal window sense to have a local maximum with height u for t=0; see [4] for terminology. ¶ In Section 2 we examine the process ξu(t) as u→−∞, and show that, after suitable normalizations, it tends to a fourth degree polynomial in t1…, tn with random coefficients. This result is quite analogous with the one-dimensional case. ¶ In Section 3 we study the locations of the local minima of ξu(t) as u → ∞. In the non-isotropic case r(t) may have a local minimum at some point t0. Then it is shown in 3.2 that ξu(t) will have a local minimum at some point τu near t0, and that τu-t0 after a normalization is asymptotically n-variate normal as u→∞. This is in accordance with the one-dimensional case.

The paper deals with the possibility to solve the heat equation backwards in time. More specifically, we treat the following problem. Given the temperature at a finite number of points of a homogeneous bar, how old can the heat distribution be? In the case that the temperature is given at equidistant points x1, the problem is completely solved. In the case of nonequidistant xi we find an upper bound for the age. Such a bound is also obtained when the information about the heat distribution is given by the value of a finite number of linear functionals.

In this paper we introduce, and undertake the study of a class of Banach algebras associated with a locally compact group G. These algebras are related to the two-sided Laplace transform in the same way that the group algebra LI(G) and the measure algebra M(G) are related to the Fourier transform. In the following paragraph, we indicate the nature of some of our final results by exposing them in the simplest nontrivial case. If A is a compact convex subset of R" let s denote the space of measurable functions on R a for which II/II~=SRI~'(~)I~A(~),Z~

of /~ and a respectively (Lloyd, 1952). These est imates may be called best unbiased estimates. A serious drawback of the solution is tha t in most cases it involves very time-consuming numerical calculations. The object of this paper is to show that , under general conditions, it is possible to find a convenient approximation to the best solution which m a y be te rmed a nearly best unbiased estimate. The variance of this est imate is, as some examples will show, often very little in excess of the minimum variance. The method presupposes tha t t h e means (but not the covariances) of the variables x~ are known. By a slight modification of the method it may be used also when neither the means nor the covariances are known. The resulting estimates will be called nearly best, nearly unbiased estimates. Both types of estimates mentioned above m a y be derived from a theorem given in the next section.

In the first part of this paper are considered real polynomialsP(ζ), ζ∈R n, complete and nondegenerate in the sense that there is a set of (even) multi-indices α j ,j=1,...,N, such that, for |ζ|>K, ζ real, $$cP(\xi ) \leqslant \sum {\xi ^{\alpha j} } \leqslant CP(\xi ).$$ (See V. P. Mihailov,Soviet Math. Dokl. 164 (1965), MR 32: 6047). It is then proved by an explicit computation, for every given even multi-index γ, that there are a real number θ>0 and an integerr, 0≤r

Generators and defining relations for the group A n of automorphisms of a free group of rank n were derived by J . Nielsen [11]. For n = 2, this is a fairly easy task, but for n ~> 3 it requires very difficult combinatorial arguments which have not been simplified since the appearance of Nielsen's paper. In order to obtain an easier approach to the investigation of An and a bet ter understanding of its structure, it seems natural to s tudy its subgroups. For all n, the elements of An which induce the identical automorphism in the commutator quotient group Fn/F'~ form a normal subgroup K of An. Bachmuth [1] calls this the group of IA automorphisms of F n. Magnus [8] showed tha t this subgroup is generated by the automorphisms Ktj : ai ~ a ja i a i 1