Approximation in the mean by bounded analytic functions

Let 1<p< and . LetC _q denote the Bessel capacity in the plane. Let be the set of homomorphisms ofH (G) such that (z)= and letNP denote the set of points in G for which is not a peak set forH (G). In this note, we show that ifC _q (NP)=0, thenH (G) is dense inL _a ^p (G), the Bergman space overG.Partially supported by NSF DMS-9401234     

Partial t-Spreads in PG(2t+1,q)

This article first of all discusses the problem of the cardinality of maximal partial spreads in PG(3,q), q square, q>4. Let r be an integer such that 2rq+1 and such that every blocking set of PG(2,q) with at most q+r points contains a Baer subplane. If S is a maximal partial spread of PG(3,q) with q ²-1-r lines, then r=s( +1) for an integer s2 and the set of points of PG(3,q) not covered byS is the disjoint union of s Baer subgeometriesPG(3, ). We also discuss maximal partial spreads in PG(3,p ³), p=p ₀ ^h , p ₀ prime, p ₀ 5, h 1, p 5. We show that if p is non-square, then the minimal possible deficiency of such a spread is equal to p ²+p+1, and that if such a maximal partial spread exists, then the set of points of PG(3,p ³) not covered by the lines of the spread is a projected subgeometryPG(5,p) in PG(3,p ³). In PG(3,p ³),p square, for maximal partial spreads of deficiency p ²+p+1, the combined results from the preceding two cases occur. In the final section, we discuss t-spreads in PG(2t+1,q), q square or q a non-square cube power. In the former case, we show that for small deficiencies , the set of holes is a disjoint union of subgeometries PG(2t+1, ), which implies that 0 (mod +1) and, when (2t+1)( -1) <q-1, that 2( +1). In the latter case, the set of holes is the disjoint union of projected subgeometries PG(3t+2, ) and this implies 0 (mod q ^2/3+q ^1/3+1). A more general result is also presented.     

Замечания о порядке погрещности интерполяции

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On the distribution of square-full and cube-full integers

LetN _r (x) be the number ofr-full integers x and let _r (x) be the error term in the asymptotic formula forN _r (x). Under Riemann's hypothesis, we prove the estimates ₂(x)x^1/7+, ₃(x)x^97/804+(>0), which improve those of Cao and Nowak. We also investigate the distribution ofr-full andl-free numbers in short intervals (r=2,3). Our results sharpen Krätzel's estimates.     

Über geschlossene affine Zwangläufe in der Ebene

We consider integral coverings y:{1,2,..,} of an affine plane which occur when is moved under a continuous periodic affine motion(t):. One can distinguish normal points × , i.e. is constant in a certain neighborhood of x, and singular points. If (x) is the number of times x passes through its orbit (t)x all normal points x have (x)=1, and the set of all singular points consists of a number of isolated points and lines. If (x) is the tangent rotation number of the orbit of x all singular points lie on the moving pole curve.     

A remark on normal derivations of Hilbert-Schmidt type

LetB (H) denote the algebra of operators on the separable Hilbert spaceH. LetC ₂ denote the (Hilbert) space of Hilbert-Schmidt operators onH, with norm .₂ defined by S ₂ ² =(S,S)=tr(SS ^*). GivenA, B B (H), define the derivationC (A, B):B(H)B(H) byC(A, B)X=AX-XB. We show that C(A,B)X+S ₂ ² =C(A,B)X ₂ ² +S ₂ ² holds for allXB(H) and for everySC ₂ such thatC(A, B)S=0 if and only if reducesA, ker S reducesB, andA | S and B| ker S are unitarily equivalent normal operators. We also show that ifA, BB(H) are contractions andR(A, B)B(H)B(H) is defined byR(A, B)X=AXB-X, thenSC ₂ andR(A, B)S=0 imply R(A,B)X+S ₂ ² =R(A,B)X ₂ ² +S ₂ ² for allXB(H).     

A variational principle for eigenvalues of pencils of Hermitian matrices

LetH=(A, B) be a pair of HermitianN×N matrices. A complex number is an eigenvalue ofH ifdet(A–B)=0 (we include = ifdetB=0). For nonsingularH (i.e., for which some is not an eigenvalue), we show precisely which eigenvalues can be characterized as _k ⁺ =sup{inf{*A:*B=1,S},SS _k},S _k being the set of subspaces of C ^N of codimensionk–1.Dedicated to the memory of our friend and colleague Branko NajmanResearch supported by NSERC of Canada and the I.W.Killam FoundationProfessor Najman died suddenly while this work was at its final stage. His research was supported by the Ministry of Science of CroatiaResearch supported by NSERC of Canada     

Die Verteilung der schlichten Funktionen in einem Funktionenraum

We consider the set of regular functions . We construct a Borel measure and a class of outer measures _h onH. With these and _h we show that: (HS)=0 and _h (HS)=0, (S is the set of normed univalent functions). From _h (HS)=0 follows—forh=t —that the Hausdorff—Billingsley-dimension ofHS is zero.     

Regularity of solutions to a non homogeneous boundary value problem for general Stokes systems in R<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript><Subscript>+</Subscript>

We give a simple and very complete proof of the existence of a strong (H²) solution to the non-homogeneous problem (1.1) under the non homogeneous boundary conditions (1.6). Here we consider the half-space case = Rⁿ₊, n 3, see theorem 1.2. This regularity result was previously obtained by Solonnikov and adilov in reference [33] for the classical Stokes system (==0, g(x)=0) in the 3–D homogenous case (a=0, b=0) and a suitable open subset of R^3.Partly supported by CMAF/UL and FCT.     

Zur Vollständigkeit der Induktiven Gruppoide der Partiellen Automorphismen von Algebren

For any algebraA let(A) be the set of partial automorphisms (isomorphisms between subalgebras). With the natural multiplication it is an inductive groupoid in the sense of Ehresmann.(A) is complete iff every subset of(A) which is compatible with the semi-ordering has an upper bound. The fact, whether(A) is complete or not, depends on the defining operations ofA. For every direct familyF = (, (A ) ,()_, , of algebras such that all ,, with are one-to-one functions, the direct limit is complete iff all(A) are complete. We give some theorems on the decomposition of inductive groupoids, and employ them in proving the completeness of(A) to variousA.In particular, we obtain that, in case whenG is a finite group,(G) is complete iffG is either cyclic or direct product of a noncyclic group of order 4 and a cyclic group of odd order. For finite acyclic ringsR and finite fieldsK the inductive groupoids(R) and(K) are complete.Further we deal with the question, to what extent algebras are determined by their inductive groupoids. (An algebraA of a classS is defined by(A) iff, for any algebraB of the classS, isomorphism between(A) and(B) implies isomorphism betweenA andB.) Particular attention is paid to finite groups. In general, algebras of classesS are not defined within the classS by their inductive groupoids.

     

A Remark on Bernstein,Prokhorov, Bennett,Hoeffding, and Talagrand Inequalities

Let X ₁,..., X_n be independent random variables such that {X_j 1}=1 and E X _j=0 for all j. We prove an upper bound for the tail probabilities of the sum M _n=X₁+...+ X_n. Namely, we prove the inequality {M _nx} 3.7 {S_n x}, where S _n=₁+...+ _n is a sum of centered independent identically distributed Bernoulli random variables such that E S _n ² =ME M _n ² and {_k=1}=E S _n ² /(n+E S _n ² ) for all k (we call a random variable Bernoulli if it assumes at most two values). The inequality holds for x at which the survival function x{S _nx} has a jump down. For remaining x, the inequality still holds provided that we interpolate the function between the adjacent jump points linearly or log-linearly. If necessary, in order to estimate {S _nx} one can use special bounds for binomial probabilities. Up to the factor at most 2.375, the inequality is final. The inequality improves the classical Bernstein, Prokhorov, Bennett, Hoeffding, Talagrand, and other bounds.     

Large-Deviation Local Theorem for Additive Functionals of a Markov Gaussian Field. I

We prove a local limit theorem (LLT) on Cramer-type large deviations for sums S _V = _t V ( _t ), where _t , t Z , 1, is a Markov Gaussian random field, V Z , and is a bounded Borel function. We get an estimate from below for the variance of S _V and construct two classes of functions , for which the LLT of large deviations holds.     

Factorization intok-dimensional linear bijections

LetV be a vector space,k withkdimV andS _k{GL(V)|dimV(–1)=k}. ThenS _k generates GL _f (V){GL(V)|V(-1) is finite-dimensional} (with the exception that dimV=2=k and the field is GF2). We study the length problem in GL _f (V) withS _k as set of generators.     

Some Classes of L 1-Convergence of Fourier Series

In this paper equivalent classes of the classes M and S _pr , p > 1, > 0. r { 0,1,2, ... ,[]} defined by Shuyun [3] are obtained. Then, it is shown that the class S _pr , 1 > p 2, 0, r {0,1,2,...,[]} is a subclass of BVC _r , where S _pr is the equivalent class of the Shuyun's class S _pr , BV is the class of null sequences of bounded variation and C _r is the extension of the Garrett--Stanojevic class. As a corollary of this result, we have obtained the theorem, proved in [7].     

ℒ2 sets which are almost starshaped

Let S be a subset of R ^d . The set S is said to be an set if and only if for every two points x and y of S, there exists some z S such that [x, z] [z, y] S. Clearly every starshaped set is an set, yet the converse is false and introduces an interesting question: Under what conditions will an set S be almost starshaped; that is, when will there exist a convex subset C of S such that every point of S sees some point of C via SThis paper provides one answer to the question above, and we have the following result: Let S be a closed planar set, S simply connected, and assume that the set Q of points of local nonconvexity of S is finite. If some point p of S see each member of Q via S, then there is a convex subset C of S such that every point of S sees some point of C via S.     

On a Property of Subexponential Distributions

Let F be a distribution function (d.f.) on [0, ) with finite first moment m >0. We define the integrated tail distribution function F ₁ of F by F ₁(t)=m^-1 ₀ ^t (1- F(u))du, t0. In this paper, we obtain sufficient conditions under which implications FSF ₁S and F ₁S FS hold, where S is the class of subexponential distributions.     

Nested Optimal λ-Packings and λ-Coverings of Pairs with Triples

We say that an optimal -packing ( -covering) of pairs with triples can be nested if it is possible to add one point to each block such that an optimal 2 -packing (2 -covering) of pairs with quadruples is obtained. A packing (covering) of order v can exist only if v 4 and v / 3 (v - 1) / 2 = v / 4 2(v - 1) / 3 ( v / 3 (v - 1) / 2 = v / 4 2 (v - 1) / 3 ). It is shown in this paper that this condition is also sufficient, except possibly for v = 20.     

Some mildly wild circles in Sn arising from algebraic K-Theory

We study wild embeddings of S ¹ in S ⁿ which are tame in a sense introduced by Quinn. We show that if is a finitely presented group with H ₁()=H ₂()=0, then any finiteness obstruction K ₀() can be realized on the complement of such an embedded S ¹. We also realize trivially symmetric K _–1() obstructions on the complements of such embeddings. For trivially symmetric , the embeddings constructed are shown to be isotopy homogeneous.     

Equivalence Bimodule Between Non-Commutative Tori

The non-commutative torus C ^*(ⁿ,) is realized as the C*-algebra of sections of a locally trivial C*-algebra bundle over S with fibres isomorphic to C ^*n/S, ₁) for a totally skew multiplier ₁ on ⁿ/S. D. Poguntke [9] proved that A is stably isomorphic to C(S) C(^*( Zⁿ/S, ₁) C(S) A M_kl( C) for a simple non-commutative torus A and an integer kl. It is well-known that a stable isomorphism of two separable C*-algebras is equivalent to the existence of equivalence bimodule between them. We construct an A-C(S) A-equivalence bimodule.     

On a problem about covering lines by squares

LetS be the square [0,n]² of side lengthn and letS = {S ₁, ...,S _t} be a set of unit squares lying insideS, whose sides are parallel to those ofS.S is called a line cover, if every line intersectingS also intersects someS _i S. Let(n) denote the minimum cardinality of a line cover, and let(n) be defined in the same way, except that we restrict our attention to lines which are parallel to either one of the axes or one of the diagonals ofS. It has been conjectured by Fejes Tóth that(n)=2n+O(1) and by Bárány and Füredi that(n)=3/2n+O(1). We will prove that, instead,(n)=4/3n+O(1) and, as to Fejes Tóth's conjecture, we will exhibit a noninteger solution to a related LP-relaxation, which has size equal to 3/2n+O(1).