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Showing papers in "Monatshefte für Mathematik in 1999"


Journal ArticleDOI
TL;DR: In this paper, the complex two-plane Grassmannian with both a Kahler and a quaternionic Kahler structure was applied to the normal bundle of a real hypersurface M in G
Abstract: The complex two-plane Grassmannian G 2(C m+2 in equipped with both a Kahler and a quaternionic Kahler structure. By applying these two structures to the normal bundle of a real hypersurface M in G 2(C m+2 one gets a one- and a three-dimensional distribution on M. We classify all real hypersurfaces M in G 2 C m+2 , m≥3, for which these two distributions are invariant under the shape operator of M.

127 citations


Journal ArticleDOI
TL;DR: In this paper, it is proved that for every fixed point in the lattice, the boundary of the convex body is smooth and has nonzero curvature throughout the whole lattice.
Abstract: Denote by \(\) the number of points of the lattice \(\) in the “blown up” domain \(\), where \(\) is a convex body in \(\) (\(\)) whose boundary is smooth and has nonzero curvature throughout. It is proved that for every fixed \(\) $$$$

47 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that there exists a positive integer k 0 such that every large even integer is a sum of four squares of primes and k 0 powers of 2.
Abstract: As an extension of the Linnik-Gallagher results on the “almost Goldbach” problem, we prove, among other things, that there exists a positive integer k0 such that every large even integer is a sum of four squares of primes and k0 powers of 2.

45 citations


Journal ArticleDOI
TL;DR: The volume of the symmetric difference of a smooth convex body in and its best approximating polytope with n vertices is asymptotically a constant multiple of.
Abstract: The volume of the symmetric difference of a smooth convex body in and its best approximating polytope with n vertices is asymptotically a constant multiple of . We determine this constant and the similarly defined constant for approximation with a given number of facets by solving two isoperimetric problems for planar tilings.

26 citations


Journal ArticleDOI
TL;DR: For biprojective Banach algebras, the authors showed that the cohomological properties of a bip-jective algebra are equivalent to its bip-projectivity.
Abstract: Let A be a biprojective Banach algebra, and let A-mod-A be the category of Banach A-bimodules. In this paper, for every given \(\)-mod-A, we compute all the cohomology groups \(\). Furthermore, we give some cohomological characterizations of biprojective Banach algebras. In particular, we show that the following properties of a Banach algebra A are equivalent to its biprojectivity: (i) \(\) for all \(\)-mod -A; (ii) \(\) for all \(\)-mod-A; (iii) \(\) for all \(\)-mod-A. (Here \(\) and \(\) are, respectively, the Banach A-bimodules of left, right and double multipliers of X.)

25 citations


Journal ArticleDOI
TL;DR: In this paper, the mean square formula of the error term for a class of Arithmetical functions whose Dirichlet series satisfies a functional equation with multiple gamma factors was studied.
Abstract: Based on the method in Meurman [5], we study the mean square formula of the error term for a class of Arithmetical functions whose Dirichlet series satisfies a functional equation with multiple gamma factors. We obtain improvements on some results of Chandrasekharan and Narasimhan [1].

22 citations


Journal ArticleDOI
TL;DR: In this paper, the authors construct algebraic curves C defined over a finite prime field such that the number of rational points of C is large relative to the genus of C. The methods of construction are based on the relationship between algebraic curve and their function fields, as well as on narrow ray class extensions obtained from Drinfeld modules of rank 1.
Abstract: We construct algebraic curves C defined over a finite prime field \(\) such that the number of \(\)-rational points of C is large relative to the genus of C. The methods of construction are based on the relationship between algebraic curves and their function fields, as well as on narrow ray class extensions obtained from Drinfeld modules of rank 1.

22 citations


Journal ArticleDOI
TL;DR: In this article, discrepancy estimates in terms of the quality of corresponding quadrature formulas and bounds for potential differences were derived for measures on the unit sphere in Ω( √ d ≥ 3, d ≥ 2, d≥ 3 ).
Abstract: For measures on the unit sphere in ℝ d , d≥3, we derive discrepancy estimates in terms of the quality of corresponding quadrature formulas and in terms of bounds for potential differences.

20 citations


Journal ArticleDOI
TL;DR: In this article, a contraction of the sphere, considered as the homogeneous space, to the Heisenberg group is defined, and infinite dimensional irreducible unitary representations of the group are defined.
Abstract: A contraction of the sphere , considered as the homogeneous space , to the Heisenberg group is defined The infinite dimensional irreducible unitary representations of Heisenberg group are then shown to be the limits of the irreducible representations of which are class-1 with respect to Our results generalise the earlier results of Fulvio Ricci

19 citations


Journal ArticleDOI
TL;DR: In this paper, an analogue of the Krein formula in the case that ℋ is a degenerated inner product space is given. And the set of parameters is determined by a kernel condition.
Abstract: Let S be a symmetric operator with defect index (1,1) in a Pontryagin space ℋ. The Krein formula establishes a bijective correspondence between the generalized resolvents of S and the set of Nevanlinna functions as parameters. We give an analogue of the Krein formula in the case that ℋ is a degenerated inner product space. The set of parameters is determined by a kernel condition. These results are applied to some classical interpolation problems with singular data.

18 citations


Journal ArticleDOI
TL;DR: In this paper, a non-trivial estimate of the sum of the von Mangoldt function and Mobius function is given, for the first time, in terms of the total sum.
Abstract: Let \(\) and \(\) be the von Mangoldt function and Mobius function, respectively, x real and y“small” compared with x. This paper gives, for the first time, a non-trivial estimate of the sum $$$$

Journal ArticleDOI
TL;DR: In this article, the authors considered the problem of coloring an n-point set P in a convex polytope A such that for any d-dimensional interval B, the number of red points in differs from the numbers of blue points in by at most Δ, where should be as small as possible.
Abstract: We consider so-called Tusnady’s problem in dimension d: Given an n-point set P in R d , color the points of P red or blue in such a way that for any d-dimensional interval B, the number of red points in differs from the number of blue points in by at most Δ, where should be as small as possible. We slightly improve previous results of Beck, Bohus, and Srinivasan by showing that , with a simple proof. The same asymptotic bound is shown for an analogous problem where B is allowed to be any translated and scaled copy of a fixed convex polytope A in R d . Here the constant of proportionality depends on A and we give an explicit estimate. The same asymptotic bounds also follow for the Lebesgue-measure discrepancy, which improves and simplifies results of Beck and of Karolyi.

Journal ArticleDOI
TL;DR: In this article, the authors evaluate some interesting families of infinite series by analyzing known identities involving generalized hypergeometric series and several special cases of the main results are shown to be related to earlier works on the subject.
Abstract: The authors evaluate some interesting families of infinite series by analyzing known identities involving generalized hypergeometric series. Several special cases of the main results are shown to be related to earlier works on the subject.

Journal ArticleDOI
TL;DR: In this article, it was shown that the basic cohomology of a singular Riemannian foliation of a compact manifold is a topological invariant and is finite dimensional.
Abstract: In this short note we prove that the basic cohomology of a singular Riemannian foliation of a compact manifold is a topological invariant and is finite dimensional.

Journal ArticleDOI
TL;DR: In this paper, it was shown that if points in nontrivial Gleason parts of a uniform Banach algebra have unique representing measures, then the weak and the norm topology coincide on the spectrum.
Abstract: If points in nontrivial Gleason parts of a uniform Banach algebra have unique representing measures, then the weak and the norm topology coincide on the spectrum. We derive from this several consequences about weakly compact homomorphisms and discuss the case of other uniform Banach algebras arising in complex infinite dimensional analysis.

Journal ArticleDOI
TL;DR: In this paper, the Boettcher function of a power series is introduced and investigated and the principal aim of this paper is to prove some results going back to J. F. Ritt in this general setting.
Abstract: We study composition of power series and polynomials over algebraically closed fields of arbitrary characteristic. The so-called Boettcher function of a power series is introduced and investigated. It is the principal aim of this paper to prove some results going back to J. F. Ritt in this general setting. In particular, we determine the pairs of permutable polynomials and characterize polynomials which satisfy a certain rational functional equation and polynomials which have a common iterate.

Journal ArticleDOI
TL;DR: For the real Hardy spaces, the authors show the Hardy type integral inequalities, and applying the inequalities they shall establish the Hardy's inequalities with respect to Hankel transforms, and apply the inequalities to prove the hardness of the Hardy spaces.
Abstract: For the real Hardy spaces \(\), we shall show the Hardy type integral inequalities, and applying the inequalities we shall establish the Hardy’s inequalities with respect to Hankel transforms.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the Osgood-thue theorem about Diophantine approximation in function fields holds under a more general condition when the ground field is finite.
Abstract: We prove that the Osgood-Thue Theorem, about Diophantine Approximation in function fields, holds under a more general condition when the ground field is finite.

Journal ArticleDOI
TL;DR: In this paper, the authors considered the case of a semisimple compact projective plane and showed that it is either a Moufang-Hughes plane or isomorphic to Spin9 (ℝ, r), r∈{0, 1}.
Abstract: This paper deals with the so-called Salzmann program aiming to classify special geometries according to their automorphism groups. Here, topological connected compact projective planes are considered. If finite-dimensional, such planes are of dimension 2, 4, 8, or 16. The classical example of a 16-dimensional, compact projective plane is the projective plane over the octonions with 78-dimensional automorphism group E6(−26). A 16-dimensional, compact projective plane ? admitting an automorphism group of dimension 41 or more is clasical, [23] 87.5 and 87.7. For the special case of a semisimple group Δ acting on ? the same result can be obtained if dim \(\), see [22]. Our aim is to lower this bound. We show: if Δ is semisimple and dim \(\), then ? is either classical or a Moufang-Hughes plane or Δ is isomorphic to Spin9 (ℝ, r), r∈{0, 1}. The proof consists of two parts. In [16] it has been shown that Δ is in fact almost simple or isomorphic to SL3?ċSpin3ℝ. In the underlying paper we can therefore restrict our considerations to the case that Δ is almost simple, and the corresponding planes are classified.

Journal ArticleDOI
TL;DR: In this article, it was shown that ν is strongly τ-decomposable if and only if ν possesses a finite logarithmic moment with respect to a homogeneous norm on the tangent space.
Abstract: Let ? be a simply connected nilpotent Lie group with Lie Algebra ? and let τ be a contraction on ?. A probability measure μ on ? is strongly τ-decomposable iff it is representable as the limit of \(\) for some probability ν on ?. We show that such a limit exists if and only if ν possesses a finite logarithmic moment with respect to a homogeneous norm on ?. This result is then generalized to the class of selfdecomposable laws on ?. We also show that selfdecomposable laws on ? correspond in a 1–1 way to operator selfdecomposable laws on the tangent space ?.

Journal ArticleDOI
TL;DR: In this paper, the authors consider patterns of particular events in sequences of trials, some independent and others Markovian, and find matrix recursions for the number of sequences of length n avoiding a specific pattern, and the associated probability of this event is evaluated.
Abstract: This paper considers patterns of particular events in sequences of trials, some independent and others Markovian. Matrix recursions are found for the number of sequences of length n avoiding a specific pattern, and the associated probability of this event is evaluated. A Markov chain method for the study of such problems is outlined, and is illustrated in various cases. Finally, configurations of length 3 in Bernoulli trials are examined as an example.

Journal ArticleDOI
TL;DR: In this article, the authors classify one-function skew-symmetric and super-switching differential operators by four combinatorial identities related to hyperbolic functions, and classify all one-and super-witching functions by the same identities.
Abstract: In this paper, we classify all one-function skew-symmetric and super skew-symmetric differential operators by four combinatorial identities related to hyperbolic functions.

Journal ArticleDOI
TL;DR: In this article, it was shown that if G is a union of σ-admissible neighbourhoods of the identity, relative to X, then every bounded σharmonic function on X is constant, for spread out σ, for each connected component of a [SIN]-group and if G acts strictly transitively on a splittable metric space X are constant.
Abstract: Given a locally compact group G acting on a locally compact space X and a probability measure σ on G, a real Borel function f on X is called σ-harmonic if it satisfies the convolution equation . We give conditions for the absence of nonconstant bounded harmonic functions. We show that, if G is a union of σ-admissible neighbourhoods of the identity, relative to X, then every bounded σ-harmonic function on X is constant. Consequently, for spread out σ, the bounded σ-harmonic functions are constant on each connected component of a [SIN]-group and, if G acts strictly transitively on a splittable metric space X, then the bounded σ-harmonic functions on X are constant which extends Furstenberg’s result for connected semisimple Lie groups.

Journal ArticleDOI
Werner Kratz1
TL;DR: In this article, the main results of this paper state optimal constants for estimates of successive minima in two dimensions under a constraint on the denominator, while these inequalities are known for every dimension, best possible constants within these estimates are unknown for any dimension larger than one and remain unknown for all dimensions larger than two.
Abstract: The main results of this paper state optimal constants for estimates of so-called successive minima in two dimensions under a constraint on the denominator. While these inequalities are known for every dimension, best possible constants within these estimates are, of course, notknown for any dimension larger than one and remain unknown for all dimensions larger than two.

Journal ArticleDOI
TL;DR: In this article, the authors study constant mean curvature compact surfaces immersed in hyperbolic space with non-empty boundary and prove that the only H-surfaces with boundary circular and 0≤∣H∣≤1, are the umbilical examples.
Abstract: We study constant mean curvature compact surfaces immersed in hyperbolic space with non-empty boundary (=H-surfaces). We prove that the only H-surfaces with boundary circular and 0≤∣H∣≤1, are the umbilical examples. When the surface is embedded, conditions to be umbilical are given. Finally, we characterize umbilical surfaces bounded by a circle among all H-discs with small area.

Journal ArticleDOI
Marco Reni1
TL;DR: In this article, it was shown that any Sylow subgroup of G of odd order is either cyclic or the direct sum of two cyclic groups and if G has odd order, then it splits as a semidirect product of a subgroup A and a normal subgroup B such that B acts freely and there exist some simple closed curves in the homology 3-sphere which are fixed pointwise by some non-trivial element of A.
Abstract: If a finite group acts freely on a homology 3-sphere, then it has periodic cohomology. To say that a finite group F has periodic cohomology is equivalent to say that any Sylow subgroup of F of odd order is cyclic and a Sylow 2-subgroup of F is either cyclic or a quaternion group. In this paper we consider more generally smooth actions of finite groups G on homology 3-spheres which may have fixed points. We prove that any Sylow subgroup of G of odd order is either cyclic or the direct sum of two cyclic groups. Moreover, we show that if G has odd order, then it splits as a semidirect product of a subgroup A and a normal subgroup B such that B acts freely and there exist some simple closed curves in the homology 3-sphere which are fixed pointwise by some non-trivial element of A. We discuss the relation between these algebraic results and some classical constructions of the theory of 3-manifolds.

Journal ArticleDOI
TL;DR: In this article, the class of spaces which are countable unions of zero-dimensional sets and with the larger class of Haver's C-spaces are assumed to be separable and metrizable.
Abstract: This paper deals with the class of spaces which are countable unions of zero-dimensional sets and with the larger class of Haver’s C-spaces. All spaces are assumed to be separable and metrizable. We are concerned with various aspects of universality of these classes when they are combined with the covering analogue for σ-compactness defined by Menger and the rational dimension introduced by Menger and Nobeling. A solution of a problem of S. D. Iliadis [16] concerning universal spaces for rational dimension will result.

Journal ArticleDOI
TL;DR: In this paper, the first n nonzero Neumann eigenvalues of the Laplacian on Ω satisfy, where is a computable constant depending only on and, Ω being the volume of Ω.
Abstract: Let M be an n-dimensional simply connected Hadamard manifold with Ricci curvature satisfying and be a bounded domain having smooth boundary. In this paper, we prove that the first n nonzero Neumann eigenvalues of the Laplacian on Ω satisfy , where is a computable constant depending only on and , Ω being the volume of Ω. This result generalizes the corresponding estimate for bounded domains in a Euclidean space obtained recently by M. S. Ashbaugh and R. D. Benguria.

Journal ArticleDOI
Jie Wu1
TL;DR: In this article, the authors give three estimates on bilinear exponential sums of type I application, where the A j and A j are effective constants and t(?) denotes the number of unitary factors of a finite abelian group.
Abstract: We give three estimates on bilinear exponential sums of type I As application we prove that , where the A j are effective constants and t(?) denotes the number of unitary factors of a finite abelian group ? This improves on the previous result

Journal ArticleDOI
TL;DR: In this article, a necessary and sufficient condition on the partial quotients of two real quadratic irrational numbers is provided to insure that they are elements of the same number field over ℚ.
Abstract: Here we provide a necessary and sufficient condition on the partial quotients of two real quadratic irrational numbers to insure that they are elements of the same quadratic number field over ℚ. Such a condition has implications to simultaneous diophantine approximation. In particular, we deduce an improvement to Dirichlet’s Theorem in this context which, as an immediate consequence, shows the Littlewood Conjecture to hold for all numbers α and β both from \(\). Specifically, for all such pairs we have \(\).