On unconditional bases in certain Banach function spaces

Let X be a Banach space, L ([0,1])XL ¹([0,1]), with an unconditional basis. By the well-known stability property in X, there exists a unconditional basis {f _n} _m=1 , where f _n in C([0,1]), nN. In this paper, we introduce the notion that X ^*has the singularity property of X ^*at a point t ₀[0,1]. It is proved that if X ^*has the singularity property at a point t ₀ [0,1], then there exists no orthonormal, fundamental system in C([0,1]) which forms an unconditional basis in X.     

Ouverts d"harmonicité pour les fonctions séparément harmoniques

Let D ^N , G ^M be two open sets, E D and F G two compact sets which satisfy the condition (H) (that is a harmonic condition similar to Leja"s condition). We find an open set ^N+M such that each separately harmonic function f : X : = (D× F) (E × G) (i.e.: for all x in E, f(x,.) is harmonic on G; for all y in F, f(., y) is harmonic on D) extends to a harmonic function on .     

Pyramids in the complex projective plane

We study the critical points of the diameter functional on the n-fold Cartesian product of the complex projective plane C P ² with the Fubini-Study metric. Such critical points arise in the calculation of a metric invariant called the filling radius, and are akin to the critical points of the distance function. We study a special family of such critical points, P _kC P ¹C P ², k=1,2... We show that P _k is a local minimum of by verifying the positivity of the Hessian of (a smooth approximation to) at P _k. For this purpose, we use Shirokov's law of cosines and the holonomy of the normal bundle of C P ¹C P ². We also exhibit a critical point of , given by a subset which is not contained in any totally geodesic submanifold of C P ².     

Existence of fixed points of nonexpansive mappings satisfying the Rothe condition

In this paper we consider the following situation: H is a Hilbert space, A is a nonempty bounded closed (not necessarily convex) subset of H, f:D HH is a nonexpansive mapping and A D. In the basic result (Theorem 1) it is shown that in this situation the nonexpansive map f has a fixed point in A, if f satisfies the Rothe condition on A:f(A)A.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 122, pp. 5–12, 1982.     

Composantes de dimension maximale d'un analogue du lieu de Noether-Lefschetz (Components of Maximal Dimension of an Analogue of the Noether-Lefschetz Locus)

Let X ⁴ be a smooth hypersurface of degree d 5, and let S X be a smooth hyperplane section. Assume that there exists a non trivial cycle Z Pic(X) of degree 0, whose image in CH₁(X) is in the kernel of the Abel–Jacobi map. The family of couples (X, S) containing such Z is a countable union of analytic varieties. We show that it has a unique component of maximal dimension, which is exaclty the locus of couples (X, S) satisfying the following condition: There exists a line S and a plane P ⁴ such that P X = , and Z = – dh, where h is the class of the hyperplane section in CH₁(S). The image of Z in CH₁(X) is thus 0. This construction provides evidence for a conjecture by Nori which predicts that the Abel–Jacobi map for 1–cycles on X is injective.     

Strong solution for an elastic-plastic plate

Given a regular bounded open set R ²,, >0 andg L ^q () withq>2, we prove, under compatibility and safe load conditions ong, the existence of a minimizing pair for the functional, over closed setsK ² and functionsu C⁰( ) C²(/K); here ¦[Du]¦ denotes the jump ofDu acrossK and ¹ is the 1-dimensional Hausdorff measure.Dedicated to Enrico Magenes for his 70th birthday     

The Conditions of Analytic Extendability of Functions Defined on a Part of the Boundary of Circular Domains

The paper is devoted to the conditions of existence of the holomorphic expansion, in a domain D, of a function f C (M) defined on a set M with a positive measure of the Shilov boundary S of the complete circular domain D C ⁿ . These conditions are as follows: integrals of the product of f by certain polynomials that are orthogonal holomorphic functions tend to 0.     

The existence of probability measures with specified projections

Let X and Y be locally compact-compact topological spaces, F X×Y is closed, and P(F) is the set of all Borel probability measures on F. For us to find, for the pair of probability measures (_x, _y P (X)×P(Y), a probability measure P(F) such that _X = _X ^–1 , _Y = _Y ^–1 it is necessary and sufficient that, for any pair of Borel sets A X, B Y for which (A× B) F=Ø, the condition _XA+ _YB 1 holds.Translated from Matematicheskie Zametki, Vol. 14, No. 4, pp. 573–576, October, 1973.     

On Remez-type inequalities for polynomials in R m and C m

Remez-type inequalities provide estimates for the size of polynomials on given sets KR ^m (or C ^m ) when the magnitude of polynomials on largeldquo subsets of K is known. We shall study this question on smooth sets K in R ^m and C ^m and show how the smoothness of K effects the estimates.     

The caratheodory number for thek-core

Thek-core of the setS ⁿ is the intersection of the convex hull of all setsA S with ¦SA¦<-k. The Caratheodory number of thek-core is the smallest integerf (d,k) with the property thatx core _{kS, S} ⁿ implies the existence of a subsetT S such thatx core_kT and ¦T¦f (d, k). In this paper various properties off(d, k) are established.Research of this author was partially supported by Hungarian National Science Foundation grant no. 1812.     

A Comparison of Two Spaces of Generalized White Noise Functionals

In the literature (see [5, 6, 8]) there are two families of spaces called Kondratiev spaces: (_c)^± and (S _c)^± for 0 1. We investigate the relation between the spaces and show that they are topologically isomorphic when (^d) L2 (^d) (^d) is the underlying Gel'fand triple for (_c)^±. In this case we also give the explicit relation between the S-transform and -transform on (_c)^-1 and (S _c)^-1, respectively.     

Computation of Almost Split Sequences with Applications to Relatively Projective and Prinjective Modules

Let be an Artin algebra, let mod be the category of finitely generated -modules, and let Amod be a contravariantly finite and extension closed subcategory. For an indecomposable and not Ext-projective module CA, we compute the almost split sequence 0ABC0 in A from the almost split sequence 0DTrCEC0 in mod. Since the computation is particularly simple if the minimal right A-approximation of DTrC is indecomposable for all indecomposable and not Ext-projective CA, we manufacture subcategories A with the desired property using orthogonal subcategories. The method of orthogonal subcategories is applied to compute almost split sequences for relatively projective and prinjective modules.     

Biharmonic maps from <Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">4</Emphasis></Superscript> into a Riemannian manifold

For a domain R⁴ and a compact Riemannian manifold NR^k without boundary, if uW^2,2(,N) is an extrinsic (or intrinsic, respectively) biharmonic map, then uC(,N).in final form: 1 August 2003     

On the Infinitesimal Rigidity of Homogeneous Varieties

Let XP be a variety (respectively an open subset of an analytic submanifold) and let xX be a point where all integer valued differential invariants are locally constant. We show that if the projective second fundamental form of X at x is isomorphic to the second fundamental form of a point of a Segre P× P, n,m2, a Grassmaniann G(2,n+2), n4, or the Cayley plane OP², then X is the corresponding homogeneous variety (resp. an open subset of the corresponding homogeneous variety). The case of the Segre P²×P² had been conjectured by Griffiths and Harris in [GH]. If the projective second fundamental form of X at x is isomorphic to the second fundamental form of a point of a Veronese v₂(P) and the Fubini cubic form of X at x is zero, then X=v₂ (P) (resp. an open subset of v₂(P)). All these results are valid in the real or complex analytic categories and locally in the C category if one assumes the hypotheses hold in a neighborhood of any point x. As a byproduct, we show that the systems of quadrics I₂(P P) S²C, I₂(P¹× P) S²C and I₂(S₅) S²C¹⁶ are stable in the sense that if A S^* is an analytic family such that for t0,AA, then A₀A. We also make some observations related to the Fulton–:Hansen connectedness theorem.     

Rectangular convexity of convex domains of constant width

An E R ² is r-convex if for every x, y E there exists a closed rectangle R such that x, y R and R E. Several results about r-convexity appeared in [1]. Its authors formulated a conjecture about conditions for a compact, convex set in R ² to be r-convex. We prove this conjecture in the case of convex domains of constant width.     

Drachennetze mit geradlinigen Hauptdiagonalen

A C^s-net of curves N (s1) [3] in a regular C^s-2-surface Eⁿ (n2) is called a C^s-kite- net [4] if N and the net N₁ of its angular bisecting curves form a pair of diagonal nets [1] in such a way that each mesh of N-curves possessing two N₁-diagonals shows, with respect to one of these (calledmain diagonal), the same symmetry of angles and lengths as a rectilinear kite in E². Referring to the fact that the main diagonals of any C^s-kite-net N (s2) are geodesics in [5], we ask in this paper for all C^s-kite-nets and, more generally, C^s-D-nets [5] (s1) withstraight main diagonals. This leads, among other results, to a characterization of the skew ruled surfaces in Eⁿ (n3) with constant parameter of distribution and the constant striction /2.

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Herrn Professor Dr. WERNER BURAU zum 70. Geburtstag gewidmet     

On simplexes inscribed in a hypersurface

The sufficient conditions are obtained for the existence, on a hyper surface M Rⁿ, of k points whose convex hull forms a (k–1)-dimensional simplex, homothetic to a given simplex Rⁿ. In particular, it is shown that if M is a smooth hypersurface, homeomorphic to a sphere, such points will exist for any simplex Rⁿ. The proofs are based on simple topological considerations. There are six references.Translated from Matematicheskie Zametki, Vol. 5, No. 1, pp. 81–89, January, 1969.     

A Theorem on Fine Connectedness

We prove that E is a p-fine domain whenever R ⁿ is a p-fine domain, E R ⁿ is p-polar, and 1 < p n. By a p-fine domain we understand an open connected set in the p-fine topology, i.e. in the coarsest topology making all p-superharmonic functions continuous. As an application of our main result, we establish a general version of minimum principle.     

Über den algorithmischen Nachweis von Ungleichungen I

We shall develop a method to prove inequalities in a unified manner. The idea is as follows: It is quite often possible to find a continuous functional : ⁿ , such that the left- and the right-hand side of a given inequality can be written in the form (u)(v) for suitable points,v=v(u). If one now constructs a map ^{n n} , which is functional increasing (i.e. for each x ⁿ (which is not a fixed point of ) the inequality (x)<((x)) should hold) one specially gets the chain (u)( u))( ²(u))... ⁿ (u)). Under quite general conditions one finds that the sequence { ⁿ (u)} _n converges tov=v(u). As a consequence one obtains the inequality (u)(v).     

Diagonale Netze aus Schmieg- und Krümmungslinien I

In this paper we develop the theory of nets of curves in a regular C^r-2-surface Eⁿ (r1, n2) using the concept C^s-net (of curves); the term diagonal nets of curves defined by W. BPLASCHKE [2] in E² is generalized accordingly. A regular C^r-surface E³ (r2) of negative GAUSSian curvature is called a (C^r-)D_SK-surface if its asymptotic lines (S-lines) and lines of curvature (K-lines) locally form a pair of diagonal nets. For the C³-D_SK-surfaces a criterion is given and distinct categories are determined, in particular all those C³-D_SK-surfaces in which the S- and K-lines can be arranged as (curvilinear) kites, respectively parallelograms and their diagonals.

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Auszugsweise vorgetragen auf der Geometrietagung in Oberwolfach (1.10.l974).