Interpolation by holomorphic automorphisms and embeddings in Cn

Let n > 1 and let C ⁿ denote the complex n-dimensional Euclidean space. We prove several jet-interpolation results for nowhere degenerate entire mappings F:C ⁿ →C ⁿ and for holomorphic automorphisms of C ⁿ on discrete subsets of C ⁿ.We also prove an interpolation theorem for proper holomorphic embeddings of Stein manifolds into C ⁿ.For each closed complex submanifold (or subvariety) M ⊂ C ⁿ of complex dimension m < n we construct a domain Ω ⊂C ⁿ containing M and a biholomorphic map F: Ω → C ⁿ onto C ⁿ with J F ≡ 1such that F(M) intersects the image of any nondegenerate entire map G:C ^n−m →C ⁿ at infinitely many points. If m = n − 1, we construct F as above such that C ⁿ ∖F(M) is hyperbolic. In particular, for each m ≥ 1we construct proper holomorphic embeddings F:C ^m →C ^m−1 such that the complement C ^m+1 ∖F(C ^m )is hyperbolic.     

Two composition methods for solving certain systems of linear equations

Summary This note is concerned with the following problem: Given a systemA·x=b of linear equations and knowing that certains of its subsystemsA ¹·x ¹=b ¹, ...,A ^m ·x ^m =b ^m can be solved uniquely what can be said about the regularity ofA and how to find the solutionx fromx ¹, ...,x ^m ? This question is of particular interest for establishing methods computing certain linear or quasilinear sequence transformations recursively [7, 13, 15].Work performed under NATO Research Grant 027-81     

Notes on Free Monadic Boolean Algebras

If F B(2 ⁿ – 1) denotes the Boolean algebra with 2 ⁿ – 1 free generators and P(2 ⁿ ) is the Cartesian product of 2 ⁿ Boolean algebras all equal to F B(2 ⁿ – 1), we define on P(2 ⁿ ) an existential quantifier by means of a relatively complete Boolean subalgebra of P(2 ⁿ ) and we prove that (P(2ⁿ),) is the monadic Boolean algebra with n free generators. Every element of P(2 ⁿ ) is a 2 ⁿ -tuple whose coordinates are in F B(2 ⁿ – 1); in particular, so are the n generators of P(2 ⁿ ). We indicate in this work the coordinates of the n generators of P(2 ⁿ ).     

Varietà di Segre e ovoidi dello spazio polare Q+(7, q)

Summary In this paper, for q even, we construct an ovoid O ₃ and a spread S of the finite classical polar space Q⁺(7, q) determinated by a hyperbolic quadric Q⁺ of PG(7, q) such that there is a subgroup of PGO ₊ ⁸ (q) isomorphic to PGL₂(q ³), which maps O ₃ in itself and S in S and is 3-transitive on O ₃ and on S; for q>2, S is not a Desarguesian spread of Q⁺(7, q) and O ₃ is a Desarguesian ovoid.

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Varietà di Segre e ovoidi dello spazio polare Q⁺(7, q)

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Al Prof. Adriano Barlotti in occasione del suo 60^o compleanno     

The torsion of a submanifold of a Riemannian manifold

For a submanifoldM ⁿ of a Riemannian manifoldM ^q, the concept of a torsion bivector at the point x M ⁿ for given one- and two-dimensional directions fromT _x M ⁿ is introduced using only the first and second fundamental forms ofM ⁿ. Its relation to the concept of Gaussian torsion is then established. It is proved that: 1) equality to zero of the torsion bivector is necessary and, whenM ⁿ is a nondevelopable surface of a space of constant curvature with nonzero second fundamental form, is also sufficient for the "flattening" ofM ⁿ into some totally geodesicM ⁿ⁺¹ inM ^q; 2) when n = 2, the independence of the nonzero torsion bivector of direction characterizes a minimalM ² inM ^q.Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 39–42, 1991.     

On projective manifolds with two P-bundle structures

We investigate whether a projective manifold V which is a P ¹-bundle over a projective manifold X with the same homology type of P ^r can have another P-bundle structure over some projective manifold Y; moreover, in the affirmative case we find restrictions on the topology of Y. Among the corollaries we prove that if V is a P ^l-bundle over P ⁴ and over a 4-fold Y not of general type, then either V = P ⁴ x P ⁴, V = P(T P ⁴), or, possibly, Y is a rational Fano 4-fold many properties of which are known. Further generalizations naturally arising from the geometry of flag manifolds are discussed.Work partially supported by M.P.I. of the Italian Government.Both authors are members of G.N.S.A.G.A. of the Italian C.N.R.     

Existence and uniqueness of complete constant mean curvature surfaces at infinity ofH 3

A set of conditions are given, each equivalent to the constancy of mean curvature of a surface in H ³.It is shown that analogs of these equivalences exist for surfaces in S _∞ ² ,the bounding ideal sphere of H ³,leading to a notion of constant mean curvature at infinity of H ³.A parametrization of all complete constant mean curvature surfaces at infinity of H ³ is given by holomorphic quadratic differentials on Ĉ,C, and D.     

On directional convexity

Motivated by problems from calculus of variations and partial differential equations, we investigate geometric properties of D-convexity. A function f: R ^d → R is called D-convex, where D is a set of vectors in R ^d, if its restriction to each line parallel to a nonzero v ∈ D is convex. The D-convex hull of a compact set A ⊂ R ^d, denoted by co^D(A), is the intersection of the zero sets of all nonnegative D-convex functions that are zero on A. It also equals the zero set of the D-convex envelope of the distance function of A. We give an example of an n-point set A ⊂ R ² where the D-convex envelope of the distance function is exponentially close to zero at points lying relatively far from co ^D(A), showing that the definition of the D-convex hull can be very nonrobust. For separate convexity in R ³ (where D is the orthonormal basis of R ³), we construct arbitrarily large finite sets A with co ^D(A) ≠ A whose proper subsets are all equal to their D-convex hull. This implies the existence of analogous sets for rank-one convexity and for quasiconvexity on 3 × 3 (or larger) matrices. This research was supported by Charles University Grants No. 158/99 and 159/99.     

Gruson-Jensen Duality for Idempotent Rings

It is known that if R is a ring with identity, and S and A ^op are the functor rings associated to the categories Mod(R) and Mod(R ^op ), respectively, then there is a duality between the categories of finitely presented objects of Mod(S ^op ) and Mod(A). We prove here this result in a more general case, namely when R is an idempotent ring, not necessarily having an identity, and when the categories Mod(R) of torsionfree and unitary right R-modules and Mod(R ^op ) of torsionfree and unitary left R-modules are locally finitely presented.     

Operators on the Lattice of Pseudovarieties of Finite Semigroups

L (F) of pseudovarieties of finite semigroups that attempts to take full advantage of the underlying lattice structure, Auinger, Hall and the present authors recently introduced fourteen complete congruences on L (F). Such congruences provide a framework from which to study L (F) both locally and globally. For each such congruence ρ and each U∈L (F) the ρ-class of U is an interval [U ^ρ, U _ρ]. This provides a family of operators of the form U⇒Uρ on L (F) that reveal important relationships between elements of L (F). Various aspects of these operators are considered including characterizations of U ^ρ, bases of pseudoidentities for U ^ρ, instances of commutativity (U ^ρ)^σ = U ^σ)^ρ, as well as the semigroups generated by certain pairs of such operators.     

Homomorphisms into N *

We study the properties of continuous homomorphisms from β S into N ^* and from N ^* into N ^* . We show that the image C of N ^* under a continuous homomorphism which does not arise from the continuous extension of a homomorphism mapping N to itself, has the property that C+C is a singleton.     

Coding of a translation of the two-dimensional torus

Let α be in the two-dimensional torus T ² = R ²/Z ². Assume that the translation map T : x → x + α acts ergodically. We present a symbolic coding of the map T which shares several properties with the Sturmian coding of a one-dimensional translation. The symbolic dynamical system is metrically isomorphic to the geometric dynamical system (T ², T). The coding is of quadratic growth complexity and 2-balanced. Moreover, there is a geometric underpinning, the coding is related to a fundamental domain for the action of Z ² on R ² and also to bounded remainder sets.      

The Dimension of the Hilbert Scheme of Special Threefolds #

The Hilbert scheme of 3-folds in ? ⁿ , n ≥  6 , that are scrolls over ? ² or over a smooth quadric surface Q  ? ? ³ or that are quadric or cubic fibrations over ? ¹ is studied. All known such threefolds of degree 7  ≤ d ≤  11 are shown to correspond to smooth points of an irreducible component of their Hilbert scheme, whose dimension is computed.     

Almost periodic measures on the torus

Given a skew product flow (T,T ²) on the two torus, we construct a family of flows onT ³ parametrized by elements of the circleT. We show that under a certain condition on (T,T ²) almost every flow in this family is strictly ergodic. This is used to characterize minimal subsets of the flow (T,P(T ²)) induced byT on the space of probability measures onT ². Using a result of M. Herman, we give an example to show that this characterization does not hold for everyT. To the memory of Shlomo Horowitz     

Vector Lattice Powers: f-Algebras and Functional Calculus

Let s ∈ {2.3,…} and E be an Archimedean vector lattice. We prove that there exists a unique pair (E ^? ,?), where E ^? is an Archimedean vector lattice and ?:E× ··· ×E (s times) → E ^? is a symmetric lattice s-morphism, such that for every Archimedean vector lattice F and every symmetric lattice s-morphism T:E × ··· × E (s times) → F, there exists a unique lattice homomorphism T ^? :E ^?  → F such that T = T ^? ○?. We give two approaches to construct (E ^? ,?) based on f-algebras and functional calculus, respectively, provided that E is also uniformly complete.     

Coding of a translation of the two-dimensional torus

Let α be in the two-dimensional torus T ² = R ²/Z ². Assume that the translation map T : x → x + α acts ergodically. We present a symbolic coding of the map T which shares several properties with the Sturmian coding of a one-dimensional translation. The symbolic dynamical system is metrically isomorphic to the geometric dynamical system (T ², T). The coding is of quadratic growth complexity and 2-balanced. Moreover, there is a geometric underpinning, the coding is related to a fundamental domain for the action of Z ² on R ² and also to bounded remainder sets.     

Interpolating sequences in the ball ofC n

LetB be the unit ball ofC ⁿ , I give necessary conditions on sequenceS of points inB to beH ^∞(B) interpolating in term of aC ⁿ valued holomorphic function zero onS (a substitute for the interpolating Blaschke product). These conditions are sufficient to prove that the sequenceS is interpolating for ∩ _p>1 (B) and is also interpolating forH ^p (B) for 1≤p<∞.     

Stochastic Dynamics of Compact Spins: Ergodicity and Irreducibility

Stochastic dynamics associated with Gibbs measures on M ^{Z d} , where M is a compact Riemannian manifold and Z ^d is an integer lattice, is considered. Equivalence of its L ²-ergodicity and the extremality of the corresponding Gibbs measure is proved.     

Smooth 4-folds which contain a P1-bundle¶as an ample divisor

We show that if a smooth projective 4-fold M contains an ample divisor A which is P ¹-bundle π :A→S over a smooth projective surface S, π is extended to a P ²-bundle π :S→S, unless $A$ is isomorphic to P ²×P ¹. Received: 28 September 1998 / Revised version: 16 August 1999