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.     

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.     

On semi-fredholm properties of a boundary value problem inR + n

The paper considers a boundary value problem with the help of the smallest closed extensionL ^∼ :H _k →H _{k 0}×B _{h 1}×...×B _{h N} of a linear operatorL :C ₍₀₎ ^∞ (R ₊ ⁿ ) →L(R ₊ ⁿ )×L(R ⁿ⁻¹)×...×L(R ⁿ⁻¹). Here the spacesH _k (the spaces ℬ _h ) are appropriate subspaces ofD′(R ₊ ⁿ ) (ofD′(R ⁿ⁻¹), resp.),L(R ₊ ⁿ ) andC ₍₀₎ ^∞ (R ₊ ⁿ )) denotes the linear space of smooth functionsR ⁿ →C, which are restrictions onR ₊ ⁿ of a function from the Schwartz classL (fromC ₀ ^∞ , resp.),L(R ⁿ⁻¹) is the Schwartz class of functionsR ⁿ⁻¹ →C andL is constructed by pseudo-differential operators. Criteria for the closedness of the rangeR(L ^∼) and for the uniqueness of solutionsL ^∼ U=F are expressed. In addition, ana priori estimate for the corresponding boundary value problem is established.     

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.     

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     

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 ).     

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.     

Mapping Zr into Zs with Maximal Contraction

Given any bijection f: Z ^r →f: Z ^s with s≥ r , easy volume comparisons show that there must be a universal constant K>0 (depending only on r and s ) and infinitely many pairs of points x,y∈ Z ^r such that || f(x)-f(y)|| > K|| x-y|| ^r/s . This puts a bound on how much contraction can be achieved for any such bijection. We show that, conversely, for any s≥ r there is a bijection f: Z ^r →Z ^s and a constant C>0 such that for all x,y∈ Z ^r we have || f(x)-f(y)|| <C|| x-y|| ^r/s . Phrased differently there is a bijection f: Z ^r →Z ^s which shrinks the distance between the images of any two points as much as possible, up to a constant factor. This generalizes a construction in fractal image processing and answers in the affirmative a question of Michael Freedman. Received May 15, 1996.     

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     

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.     

Frobenius problem for semigroups \mathsf{S}(d_{1},d_{2},d_{3})

We find the matrix representation of the set Δ(d ³), where d ³=(d ₁,d ₂,d ₃), of integers that are unrepresentable by d ₁,d ₂,d ₃ and develop a diagrammatic procedure for calculating the generating function Φ(d ³;z) for the set Δ(d ³). We find the Frobenius number F(d ³), the genus G(d ³), and the Hilbert series H(d ³;z) of a graded subring for nonsymmetric and symmetric semigroups and enhance the lower bounds of F(d ³) for symmetric and nonsymmetric semigroups.      

Quantified propositional calculus and a second-order theory for NC1

Let H be a proof system for quantified propositional calculus (QPC). We define the Σ^q_j-witnessing problem for H to be: given a prenex Σ^q_j-formula A, an H-proof of A, and a truth assignment to the free variables in A, find a witness for the outermost existential quantifiers in A. We point out that the Σ^q₁-witnessing problems for the systems G^*₁and G₁ are complete for polynomial time and PLS (polynomial local search), respectively. We introduce and study the systems G^*₀ and G₀, in which cuts are restricted to quantifier-free formulas, and prove that the Σ^q_j-witnessing problem for each is complete for NC¹. Our proof involves proving a polynomial time version of Gentzen’s midsequent theorem for G^*₀ and proving that G₀-proofs are TC⁰-recognizable. We also introduce QPC systems for TC⁰ and prove witnessing theorems for them. We introduce a finitely axiomatizable second-order system VNC¹ of bounded arithmetic which we prove isomorphic to Arai’s first order theory AID + Σ^b ₀-CA for uniform NC¹. We describe simple translations of VNC¹ proofs of all bounded theorems to polynomial size families of G^*₀ proofs. From this and the above theorem we get alternative proofs of the NC¹ witnessing theorems for VNC¹ and AID.This research was carried while this author was a student at the University of Toronto.     

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     

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.     

Diffeomorphic types of complements of nice point arrangements in CP<Superscript><Emphasis Type="Italic">l</Emphasis></Superscript>

We use a new method to study arrangement in CP ^l , define a class of nice point arrangements and show that if two nice point arrangements have the same combinatorics, then their complements are diffeomorphic to each other. In particular, the moduli space of nice point arrangements with same combinatorics in CP ^l is connected. It generalizes the result on point arrangements in CP ³ to point arrangements in CP ^l for any l.     

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.     

Curvature Estimates for Stable Minimal Hypersurfaces in R4 and R5

We obtain curvature estimates for certain stable minimalhypersurfaces in R ⁴ and R ⁵without using volume bounds. It follows that if M is acomplete stable minimal hypersurface in R ⁴ orR ⁵, then M is a hyperplane whenM intersects each extrinsic ball in, at most,N-components.     

Commuting relations for Hausdorff operators and Hilbert transforms on real Hardy spaces

We consider Hausdorff operators generated by a function ϕ integrable in Lebesgue"s sense on either R or R ², and acting on the real Hardy space H ¹(R), or the product Hardy space H ¹¹(R×R), or one of the hybrid Hardy spaces H ¹⁰(R ²) and H ⁰¹(R ²), respectively. We give a necessary and sufficient condition in terms of ϕ that the Hausdorff operator generated by it commutes with the corresponding Hilbert transform. This revised version was published online in June 2006 with corrections to the Cover Date.