Showing papers in "Manuscripta Mathematica in 1972"
TL;DR: In this paper, a linear representation of a given category is given by a map V associating with any morphism ϕ: a→e of K a linear vector space map V(ϕ): V(a)→V(e).
Abstract: LetK be the structure got by forgetting the composition law of morphisms in a given category. A linear representation ofK is given by a map V associating with any morphism ϕ: a→e ofK a linear vector space map V(ϕ): V(a)→V(e). We classify thoseK having only finitely many isomorphy classes of indecomposable linear representations. This classification is related to an old paper by Yoshii [3].
862 citations
TL;DR: In this paper, a bijective correspondence between normal Hopf ideals and sub-Hopf algebras of a commutative Hopf algebra over a field k was shown.
Abstract: In this paper we show a bijective correspondence between normal Hopf ideals and sub-Hopf algebras of a commutative Hopf algebra over a field k. This gives a purely algebraic proof of the fundamental theorem [2, III, §3, no7] of the theory of affine k-groups.
136 citations
TL;DR: A simple proof of D. Sullivan's theorem about the Euler-Poincare characteristic of (X, X-x), for real or complex analytic spaces, as well as some extensions and generalisations are given in this article.
Abstract: Using Milnor's fibration theorem [5] for analytic hypersurfaces and the “conic structure lemma”, one gets informations on local homological properties of analytic spaces, as also on the topological behaviour of algebraic sets at ∞. A simple proof of D. Sullivan's theorem about the Euler-Poincare characteristic of (X, X-x), for X a real or complex analytic space, as well as some extensions and generalisations are given.
96 citations
63 citations
TL;DR: In this article, the stability of critical points of a real valued C1 function f on a Finsler-manifold M under small perturbations of f was studied.
Abstract: We study the stability of critical points of a real valued C1 function f on a Finsler-manifold M under small perturbations of f. We give a topological description of certain (possibly degenerate) critical levels of f and show that for a certain class of functions g on M the function f+g has in a prescribed neighbourhood of the critical level of f a set of critical points the category of which is bounded below by an integer given by the topological description of that critical level of f.
38 citations
TL;DR: In this paper, a theory of algebraic De Rham cohomology and homology for arbitrary schemes over a field of characteristic zero has been proposed, which can be seen as a generalization of theorems of Lefschetz and Barth on projective varieties.
Abstract: We announce the development of a theory of algebraic De Rham cohomology and homology for arbitrary schemes over a field of characteristic zero. Over the complex numbers, this theory is equivalent to singular cohomology. Applications include generalizations of theorems of Lefschetz and Barth on the cohomology of projective varieties.
29 citations
TL;DR: In this article, generalized p-Bernoulli numbers defined by the integral on open and compact subsets of the rational p-adic field with values in an algebraically closed and complete extension of the Laurent-series are studied.
Abstract: In part 1 [12] there is defined an integral on open and compact subsets of the rational p-adic field
$$
ot Q_p$$
with values in an algebraically closed and complete extension of
$$
ot Q_p$$
. In this part we will compute so-called generalized p-Bernoulli numbers defined by
$$\int\limits_{|u| = 1} { u^k du for k \in \mathbb{Z}}$$
and prove their properties, which are important for the integration of Laurent-series. Furthermore we study several p-adic functions defined by the integral.
27 citations
TL;DR: In this article, it was shown that all direct image sheaves of polynomials are Y[T]-coherent, which is an analogue of Grothendieck's result in algebraic geometry.
Abstract: We prove here a theorem, which generalizes Grauert's comparison theorem ([4], Hauptsatz IIa; cf. also Knorr [7], Vergleichssatz) and which is an analogue of a Grothendieck's result in Algebraic Geometry ([6], Chap. III., 4.1.5). The proof makes essential use of a coherence theorem for sheaves of polynomials: Let X,Y be complex spaces, π: X→Y a proper holomorphic map and T=(T1,...,TN) a system of indeterminates. Then, for everyOX[T] graded sheafm, all direct image sheaves Rqπ*m areY[T]-coherent. The proof is as in [2].
26 citations
TL;DR: In this article, the existence of a surface with boundary curve r having constant mean curvature H (except in possible branching points) was shown for any real number H with H = 0.52.
Abstract: Let r be a recifiable closed Jordan curve in the euclidean 3-space IR3, and denote by Ar the infimum of the areas of all surfaces bounded by r. Then for every real number H with\(\left| H \right| \leqslant 0.52 \cdot \sqrt \pi /\sqrt {A_\Gamma }\) we show the existence of a surface with boundary curve r having constant mean curvature H (except in possible branching points). This improves a theorem of WENTE. Given an isolated minimal surface bounded by r for sufficiently small |H| we further prove the existence of a surface of constant mean curvature with boundary curve r which is close to the minimal surface.
17 citations
TL;DR: In this article, it was shown that an almost homogeneous Kaehlerian manifold with vanishing first Betti number is projectiv and simply connected, if and only if there is an element xo∈ X and a neighbourhood U(xo) in X with the property U⊂{g (xo); g∈Aut X}.
Abstract: Let X be a connected compact complex manifold. It is known, that the group Aut X of biholomorphic automorphisms of X is a complex Lie group. We call X an almost homogeneous manifold, if there is an element xo∈X and a neighbourhood U(xo) in X with the property U⊂{g(xo); g∈Aut X}. We state the following theorem: An almost homogeneous Kaehlerian manifold with vanishing first Betti number is projectiv and simply connected.
15 citations
TL;DR: In this paper, the authors derived a necessary requirement for the self-adjointness of a boundary value problem of ordinary differential equations, which involves only the leading coefficients of the normalized boundary conditions.
Abstract: In this note we derive a necessary requirement for the self-adjointness of a boundary value problem of ordinary differential equations, which involves only the leading coefficients of the normalized boundary conditions. As application it is shown that every self-adjoint boundary value problem of even order (and thus especially every real self-adjoint boundary value problem) is regular. The proof of this fact depends on the evaluation of a sort of generalized Vandermonde's determinant.
TL;DR: In this article, the authors give a complete characterization of closed modulesheaves of N-differentiable functions of one variable in terms of distinguished systems of generating elements, i.e., closed pointwise finitely generated idealsheaves.
Abstract: In this note we give a complete characterization of closed modulesheaves of N-differentiable functions of one variable in terms of distinguished systems of generating elements. For example closed pointwise finitely generated idealsheaves are those generated by one global analytic function in the case N=∞, by a constant in the case N<∞. Closed N-differentiable functions in the case N=∞ are exactly those satisfying a Lojasiewicz-inequality. The several variable case however is remarkably different, as is pointed out.
TL;DR: In this article, the authors give a characterization of the g-ideal, annihilator of the finite dimensional g-modules of height ≤ 2, and show that a Lie algebra g is solution of the problem if and only if its radical has dimension ≤ 1, and then g is uniserial.
Abstract: Our problem is to determine which are the finite dimensional Lie algebras such that certain undercategories of the category of finite dimensional g-modules have only a finite number of indecomposable objects, up to isomorphism. As the study of the graph of g permits us to eliminate many Lie algebras, we construct it explicitely in the solvable case and indicate how to obtain it in the general case. For this, we give a characterization of the g-ideal, annihilator of the finite dimensional g-modules of height ≤2. Then it remains two types of Lie algebras which a supplementary study eliminates also. The result is: a Lie algebra g is solution of the problem if and only if its radical has dimension ≤1, and then g is uniserial.
TL;DR: In this paper, saturation theorems for approximation processes generated by families (μt)t > 0 of complex bounded Radon measures on G and operating on a submodule of the Banach module over the convolution algebra are established.
Abstract: Let G be an arbitrary locally compact abelian group. It is the purpose of the present paper to establish saturation theorems for approximation processes generated by families (μt)t > 0 of complex bounded Radon measures on G and operating on a submodule of the Banach module Lp(G), Lp(G), over the convolution algebra
. A basic tool is the Fourier transform method and, in the case p>1 for noncompact G, its interpretation in the context of the theory of quasimeasures on G.
TL;DR: In this paper, an upper bound of n+4 for the chromatic number of the pseudo-two-manifold S2(n) is established for n>0.
Abstract: An upper bound of n+4 for the chromatic number of the pseudo-two-manifold S2(n) is established for n>0. Since S2(n) is just the result of identifying 2n distinct points of the two-sphere S2 in n pairs, S2(0) is the two-sphere and this is the only case not establiblished in this paper; this is precisely the four-colour conjecture. Embeddings are displayed in two low order cases, n=1 and n=2, indicating that the upper bound is in fact an equality there.
TL;DR: The linear hull of a Tchebyshev system is called a Haarspace, and every Haar-space of periodic functions has odd dimension as discussed by the authors, and it is shown that under certain conditions an n-dimensional Haar space of periodic function contains Ui, i=1,3,...,n, with U1⊆U3⊂U3,U4,U5,U6,U7,U8,U9,U10,U11,U12,U13,U14,U15,U
Abstract: The linear hull of a Tchebyshev system is called a Haarspace. Every Haar-space of periodic functions has odd dimension. It is shown that under certain conditions an n-dimensional Haar-space of periodic functions contains i-dimensional Haar-spaces Ui, i=1,3,...,n, with U1⊂U3⊂...⊂Un=U.
TL;DR: In this article, the results of R. Courant and N. Davids concerning the existence of minimal surfaces with free boundaries in Euclidean space are generalized to a large class of Riemannian manifolds.
Abstract: Using the direct methods of calculus of variations, C.B. Morrey solved Plateau's problem in Riemannian manifolds. In the present paper, the results of R. Courant and N. Davids concerning the existence of minimal surfaces with free boundaries in Euclidean space are generalized to a large class of Riemannian manifolds. Furthermore, I show that the solutions are regular at the boundary.
TL;DR: In this article, a sufficient Legendre-and Weierstras-condition for geodesic field theory is derived and a regularity condition analogous to the HADAMARD sufficient-legendre condition is derived.
Abstract: The theory of geodesic fields of VAGNER [12] and LIESEN [10] evidently yields the best results in comparison with the field theories of the LEPAGEan bundle. We try to elucidate the theory form a FINSLER-geometric point of view and to make it better applicable. A regularity condition analogous to the HADAMARD sufficient Legendre-condition is deduced, which is less restrictive than the conditions of the original CARATHEODORY and de DONDER-WEYL field theories. Local geodesic fields are constructed. We give a sufficient Legendre-and Weierstras-condition and finally discuss the scope of field theory.
TL;DR: In this paper, it is shown that a family of probability measures on a manifold is dominated if it is compact with respect to each sub-region of the manifold, and that theoretical assumptions are needed to obtain domination.
Abstract: It is well known that each dominated family of probability measures is compact in the sense of Pitcher. In this paper the converse direction is investigated. It is shown that a family
of probability measures on a
-field
is dominated if it is compact with respect to each sub-
-field
. If
is only compact with respect to the basic
-field
settheoretical assumptions are needed to obtain domination.
TL;DR: In this paper, it was shown that for certain solutions of the linear differential equation with no polynomial coefficients, there exists a constant r0>0 such that we can replace N by n+α for |a|> r0.
Abstract: Every solution w of the linear differential equation
(*)
$$L_n (w) = w^{(n)} + a_{n - 1^{w^{(n - 1)} } } + \ldots + a_0 w = 0$$
with polynomial coefficients aj is a polynomial or an entire function of finite order λ>0. In this paper we prove the following theorem: Let w be a solution of (*) and no polynomial. Let further λ be the order of w and na (R, 1/(w−c)) the number of the zeros in the disc |z−a| 41/λ
$$n_a \left( {L|a|^{1 - \lambda } ,1/(w - c)} \right) \leqq N.$$
It is also shown, that for certain solutions of (*) there exists a constant r0>0 such that we can replace N by n+α for |a|> r0. α0 is the degree of the polynomial a0. An important tool for the proofs is the index of an entire function.
TL;DR: For a certain class of convex sets K in normed spaces, a generalized center is defined such that every non-expansive mapping ϕ:K→K with co ϕ(K)⊃K has this center as a fixed point as discussed by the authors.
Abstract: For a certain class of convex sets K in normed spaces a generalized center isdefined such that every non-expansive mapping ϕ:K→K with co ϕ(K)⊃K has this center as a fixed point.
TL;DR: In this article, it was shown that there is no Stiefel-Whitnev class Wn in BSJ satisfving Wu formulas, even for n≤13.
Abstract: Let BSJ denote the “quotient” of\(BSO\xrightarrow{{\psi ^3 - 1}}BSO\) localized at 2. It had been thought that the Adams conjecture might deloop to give a diagram
realizing the J-homomorphism. We show this to be impossible by showing there do not exist Stiefel-Whitnev class Wn (even for n≤13) in BSJ satisfving the Wu formulas.
TL;DR: In this article, the connection between the fiber cohomology groups Hi(X(y),ℑ(y)) and the direct image sheaves Riπ*ℓ was studied.
Abstract: Let π: X→Y be a relative analytic space and ℑ an бX module. The object of this paper is to collect some results on the connection between the fibre cohomology groups Hi(X(y),ℑ(y)) and the direct image sheaves Riπ*ℑ. This generalizes results of Kodaira-Spencer and Grauert-Riemen-schneider giving a unified approach based on the methods of [9].
TL;DR: In this article, it was shown that the assumption that the regularity of the conditional probability of P cannot be replaced by a fixed compact system cannot be satisfied by the assumptions that each condition is approximable by fixed compact systems.
Abstract: If a
-field
is sufficient for a family
of probability measures defined on a
-field
then there exist regular determinations of the conditional probability of P, given
, which are independent of the special measure
, provided that
is
-regular. A counterexample shows that the
-regularity of
cannot be replaced by the assumption that each
is approximable by a fixed compact system. In particular if a
-field
is sufficient for a family
of probability measures defined on a separable
-field
and if each
admits a regular conditional probability, given
, a common regular conditional probability, given
, need not exist.
TL;DR: The strong completeness theorem for intuitionistic type theory with cut rule relative to full Kripke-valuations was proved in this article, which is a semantical equivalence of cutelimination.
Abstract: Sections 1 to 4 contain the strong completeness theorem of cut-free intuitionistic type theory relative to semi-Kripke-valuations, which gives the following corollaries: (1) The strong completeness theorem for intuitionistic type theory with cut rule relative to full Kripke-valuations. (2) A semantical eqivalent of cutelimination in intuitionistic type theory. Section 5 contains a proof of the cutelimination theorem and some corollaries to it.
TL;DR: In this article, a new convergence proof for free steering minimization methods was given, where the function f to be minimized was assumed to be strictly convex and continously differentiable.
Abstract: A first convergence proof for free steering minimization methods was given by SCHECHTER [3] The function f to be minimized was assumed by him to be twice continously differentiable and strictly convex These conditions were weakened by ELKIN [1], who considered continously differentiable and uniformly convex functions and who employed a slight modification of SCHECHTER's convergence proof The following article gives a new proof for convergence of free steering methods We assume f to be strictly convex and continously differentiable
TL;DR: In this article, it is shown that the conditional expectations of conditional expectations will converge in Po-measure to the conditional expectation of the conditional fields of a given conditional expectation in the Po-Measure to fo.
Abstract: Let Pn, n∈IN∪{0}, be probability measures on a
-fieldA; fn, n∈IN∪{0}, be a family of uniformly boundedA-measurable functions andAn, n∈IN, be a sequence of sub-
-fields ofA, increasing or decreasing to the
-fieldAo. It is shown in this paper that the conditional expectations
converge in Po-measure to
with k, n, m → ∞, if Pn|A, n∈IN, converges uniformly to Pn|A and fn, n∈IN, converges in Po-measure to fo.