Optimal boundary regularity for minimal surfaces with a free boundary

Let x(w), w=u+iv B, be a minimal surface in ³ which is bounded by a configuration , S consisting of an arc and of a surface S with boundary. Suppose also that x(w) is area minimizing with respect to , S. Under appropriate regularity assumptions on and S, we can prove that the first derivatives of x(u, v) are Hölder continuous with the exponent =1/2 up to the free part of B which is mapped by x(w) into S. An example shows that this regularity result is optimal.     

On equivalence of rearrangements of the Haar system in dyadic Hardy and BMO spaces

H={h ₁,I } — , . : , I ¦(I)¦=¦I¦, ¦I¦ — I. H H ={h _(I),I} . , , . L ^p .

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Dedicated to Professor B. Szökefalvi-Nagy on his 75th birthday

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This research was supported in part by MTA-NSF Grants INT-8400708 and 8620153.     

On near dimensions in Steiner systems with parallelism and matroids

In this paper we introduce two near dimensions d_m, d in a 3-(v,q,1) Steiner system with parallelism S, and we shall prove that d(S) d_m(S)-1; moreover we provide examples of matroids, starting from particular Steiner systems with parallelism.     

Charakterisierung semisimplizialer Mengen durch Unterteilung und Graphen

We define a subdivision functor G for semisimplicial sets such that GXGY implies XY for all pairs of semisimplicial sets X, Y and (GX)¹(GY)¹ implies XY, too, but only, as far as we know, for pairs of weakly degenerate semisimplicial sets X, Y. These results are analogous to theorems on simplicial complexes which have been proved by Finney [1] and Segal [6].

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In der Literatur werden semisimpliziale Mengen auch oft als complete semisimplicial complexes, abgekürzt css complexes bezeichnet. Wir halten uns hier im wesentlichen an die Terminologie von K. Lamotke [5].     

On Sets of Planes in Projective Spaces Intersecting Mutually in One Point

Let be a projective space. In this paper we consider sets of planes of such that any two planes of intersect in exactly one point. Our investigation will lead to a classification of these sets in most cases. There are the following two main results:- If is a set of planes of a projective space intersecting mutually in one point, then the set of intersection points spans a subspace of dimension 6. There are up to isomorphism only three sets where this dimension is 6. These sets are related to the Fano plane.- If is a set of planes of PG(d,q) intersecting mutually in one point, and if q3, 3(q²+q+1), then is either contained in a Klein quadric in PG(5,q), or is a dual partial spread in PG(4,q), or all elements of pass through a common point.     

Comparison of Normal Linear Experiments by Quadratic Forms

Let X and Y be observation vectors in normal linear experiments =N(A, V) and F = N(B, W). We write > Fif for any quadratic form YGY there exists a quadratic formXHX such that E(XHX) = E(Y'GY) and var(X'HX) var(Y'GY).The relation > is characterized by the matrices A, B, V and W. Moreoversome connections with known orderings of linear experiments are given.     

On the Binary Codes of Steiner Triple Systems

The binary code spanned by the rows of the point byblock incidence matrix of a Steiner triple system STS(v)is studied. A sufficient condition for such a code to containa unique equivalence class of STS(v)'s of maximalrank within the code is proved. The code of the classical Steinertriple system defined by the lines in PG(n-1,2)(n3), or AG(n,3) (n3) is shown to contain exactly v codewordsof weight r=(v-1)/2, hence the system is characterizedby its code. In addition, the code of the projective STS(2ⁿ-1)is characterized as the unique (up to equivalence) binary linearcode with the given parameters and weight distribution. In general,the number of STS(v)'s contained in the code dependson the geometry of the codewords of weight r. Itis demonstrated that the ovals and hyperovals of the definingSTS(v) play a crucial role in this geometry. Thisrelation is utilized for the construction of some infinite classesof Steiner triple systems without ovals.     

Exact error bound of approximation by interpolating splines inL-metric on the classesW p r (1≦p

Корнейчук  Н. П. 《Analysis Mathematica》1977,3(2):109-117

On summability of subsequences of partial sums of trigonometric Fourier series

n- (n1) fL ^p ([–, ] ⁿ ),=1 = (L C) . , , f([–, ] ⁿ ).     

Eine Bemerkung über a-priori-fixpunkte nicht-expansiver Abbildungen

For a certain class of convex sets K in normed spaces a generalized center isdefined such that every non-expansive mapping :KK with co (K)K has this center as a fixed point.     

On Fourier multipliers between Besov spaces with 0 <p o ≤ min{1,p 1}

, 0<0 min{1,p1}.     

On the Cesàro summability of orthogonal sequences

(C, ). , . 0<<1. 1) - ( _k ), _k =a _k , (C, ), . 2) , , (C, ) ; _k = =¦a _k ¦.     

Translation triple systems

We prove the following result. Let S be a Steiner triple system embedded in the projective plane of order n, such that r=n+1, and such that there exists a line l of exterior to S. Let G be a collineation group of fixing S, fixing l and transitive on the blocks of S. Then n=3 and S=l=AG(2, 3), and G contains the group of translations of S with respect to l.This research was supported by NSERC Grant A3485.     

The integrability of conjugate functions

For any functionf of L(0, 2), we prove that there is a function L(0, 2) such that ¦(x)¦ = ¦f(x)¦ almost everywhere and L(0, 2), where is the conjugate of.Translated from Matematicheskie Zametki, Vol. 4, No. 4, pp. 461–465, October, 1968.     

Parabolic subgroups of twisted Chevalley groups over a semilocal ring

It is shown that, under minor additional assumptions, the standard parabolic subgroups of a Chevalley group G (, R) of twisted type =A_l,l odd, D_l, E₆ over a commutative semilocal ring R with involution are in one-to-one correspondence with the -invariant parabolic nets of ideals of R of type , i.e., with the sets, of ideals of R such that: (l) whenever; (2) = for all ; (3) =R for > 0. For Chevalley groups of normal types, analogous results were obtained in Ref. Zh. Mat. 1976, 10A151; 1977, 10A 301; 1978, 6A476.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 94, pp. 21–36, 1979.     

Holomorphic Solutions of Some Functional Equations II

We study a certain type of functional equation, which is of significance from the view point of systems of difference equations. Let the characteristic values of the system be and The case that either || > 1 or 0 < || < 1 has been treated in a former paper. The case that = 1, || = 1 with 1 will be given in another paper. The present note deals with the case = = 1, the most difficult case.AMS Subject Classification (1991): 39A10 39B05     

Plane Partitions and Characters of the Symmetric Group

In this paper we show that the existence of plane partitions, which are minimal in a sense to be defined, yields minimal irreducible summands in the Kronecker product of two irreducible characters of the symmetric group S(n). The minimality of the summands refers to the dominance order of partitions of n. The multiplicity of a minimal summand equals the number of pairs of Littlewood-Richardson multitableaux of shape (, ), conjugate content and type . We also give lower and upper bounds for these numbers.     

A representation theorem for weak automorphisms of a universal algebra

Given a group G and a descending chainG ₀,G ₁,...,G _n, of normal subgroups ofG, we prove that there exists a universal algebra , such that the chain ...W_n( )...W₁( }) W₀( )W( ) is isomorphic to the chain ...G _n ...G ₁G ₀G, where W( ) is the group of weak automorphisms of , and W_n( ) is the group of weak automorphisms of that leaves alln-ary operations fixed.We also prove that there are an infinite number of non-isomorphic algebras that satisfy the above.These results are a generalization of those proved by J. Sichler, in the special case when G=G₀, and G₁=G₂=...=G_n=....Presented by J. Mycielski.This paper comprises part of the author's doctoral dissertation at the University of Notre Dame in 1983. The author wishes to express her deep gratitude to Professor Abraham Goetz for suggesting this problem, for being extremely generous with his time and experience, and for giving her his constant encouragement. The author also thanks the reviewer for his helpful comments.     

Complete problems withL-samplable distributions

The average case complexity classes P, L-samplable and NL, L-samplable are defined. We show that Deterministic Bounded Halting is complete for P, L-samplable and that Graph Reachability is complete for NL-samplable, both problems with a universal logspace samplable distribution.     

Sur les Solutions D'un Opérateur Différentiel Singulier Semi-Linéaire

This paper deals with the following Dirichlet problem Lu = 1A ( Au – qu = – f ( , u ) on ] 0, [ , u, ( 0 ) = 0, u ( ) = 0, where ] 0, + ], q 0 is continuous on [ 0, [ × ] 0, + [ ] 0, + [ is continuous and A satisfies some appropriate conditions. The main result is the existence and the uniqueness of a strictly positive regular solution of the problem ( ). Moreover, we study the behaviour of this solution in a neighbourhood of . Our approach is based on the use of the Green's function of the homogeneous equation and Schauder's fixed point theorem.