Nonzero degree maps between closed orientable three-manifolds

This paper adresses the following problem: Given a closed orientable three-manifold , are there at most finitely many closed orientable three-manifolds 1-dominated by ? We solve this question for the class of closed orientable graph manifolds. More precisely the main result of this paper asserts that any closed orientable graph manifold 1-dominates at most finitely many orientable closed three-manifolds satisfying the Poincaré-Thurston Geometrization Conjecture. To prove this result we state a more general theorem for Haken manifolds which says that any closed orientable three-manifold 1-dominates at most finitely many Haken manifolds whose Gromov simplicial volume is sufficiently close to that of .

     

The uniform companion for large differential fields of characteristic 0

We show that there is a theory UC of differential fields (in several commuting derivatives) of characteristic , which serves as a model companion for every theory of large and differential fields extending a model complete theory of pure fields. As an application, we introduce differentially closed ordered fields, differentially closed p-adic fields and differentially closed pseudo-finite fields.

     

On the spectral kernel of symmetric extensions

In a Hilbert space H we consider closed and symmetric operators A and à with closed ranges such that AÃ. We prove a necessary and sufficient condition for the existence of a closed and symmetric operator B with ABà the range of which is not closed. We show that this condition can be fulfilled and, by the way, we get a counter example to the assertion that the continuous part of the spectral kernel of a symmetric operator is contained in the corresponding part of a symmetric extension, as is claimed in the books of Achieser-Glasmann [1], Neumark [2] and Smirnow [3].     

Lipschitz Kernel of a Closed Set-Valued Map

This paper deals with Lipschitz selections of set-valued maps with closed graphs. First, we characterize Lipschitzianity of a closed set-valued map in the differential games framework in terms of a discriminating property of its graph. This allows us to consider the -Lipschitz kernel of a given set-valued map as the largest -Lipschitz closed set-valued map contained in the initial one, to derive an algorithm to compute the collection of Lipschitz selections, and to extend the Pasch–Hausdorff envelope to set-valued maps.     

On the Lusternik-Schnirelman theory of a real cohomology class

Farber developed a Lusternik-Schnirelman theory for finite CW-complexes X and cohomology classes H ¹ (X;). This theory has similar properties as the classical Lusternik-Schnirelman theory. In particular in [7] Farber defines a homotopy invariant cat(X,) as a generalization of the Lusternik-Schnirelman category. If X is a closed smooth manifold this invariant relates to the number of zeros of a closed 1-form representing . Namely, a closed 1-form representing which admits a gradient-like vector field with no homoclinic cycles has at least cat(X,) zeros. In this paper we define an invariant F(X,) for closed smooth manifolds X which gives the least number of zeros a closed 1-form representing can have such that it admits a gradient-like vector field without homoclinic cycles and give estimations for this number. Mathematics Subject Classification (2000): Primary 37C29; Secondary 58E05     

Hyperbolic characteristics on star-shaped hypersurfaces

In this paper, we study the stability of closed characteristics on a starshaped compact smooth hypersurface Σ in ²ⁿ. We show that the Maslov-type mean index of such a closed characteristic is independent of the choice of the Hamiltonian functions, and prove that on Σ either there are infinitely many closed characteristics, or there exists at least one nonhyperbolic closed characteristic, provided every closed characteristic possesses its Maslov-type mean index greater than 2 when n is odd, and greater than 1 when n is even.     

Algebras of operators and closed range

We consider operators such that every operator in the norm closed algebra generated by has closed range. Examples in a triangular AF algebra are constructed.

     

Complete lattices of subgroups in compact groups

Summary LetG be a compact group and a sublattice of the lattice of all closed subgroups ofG. In Proposition 1 it is shown that is a complete lattice if it is a closed subset of the spaceG ^c of all closed non empty subsets ofG. In general the converse of this fact is not true (Example 3), but the following result can be obtained (Theorem 5): If is complete and if each element of is normalized by the connected component of the identity ofG, then is a closed, totally disconnected subset ofG ^c. We mention the following corollary: IfG is totally disconnected or abelian, then is complete if and only if it is a closed subset ofG ^c.While writing this paper the author was a fellow of the National Research Council (A 7171).     

Algebraically closed and existentially closed Abelian lattice-ordered groups

If \({\mathcal{G}}\) is an Abelian lattice-ordered (l-) group, then \({\mathcal{G}}\) is algebraically (existentially) closed just in case every finite system of l-group equations (equations and inequations), involving elements of \({\mathcal{G}}\), that is solvable in some Abelian l-group extending \({\mathcal{G}}\) is solvable already in \({\mathcal{G}}\). This paper establishes two systems of axioms for algebraically (existentially) closed Abelian l-groups, one more convenient for modeltheoretic applications and the other, discovered by Weispfenning, more convenient for algebraic applications. Among the model-theoretic applications are quantifierelimination results for various kinds of existential formulas, a new proof of the amalgamation property for Abelian l-groups, Nullstellensätze in Abelian l-groups, and the display of continuum-many elementary-equivalence classes of existentially closed Archimedean l-groups. The algebraic applications include demonstrations that the class of algebraically closed Abelian l-groups is a torsion class closed under arbitrary products, that the class of l-ideals of existentially closed Abelian l-groups is a radical class closed under binary products, and that various classes of existentially closed Abelian l-groups are closed under bounded Boolean products.     

Closed categories and the theory of proofs

The main aim of this article is to suggest a translation of the simplest concepts of category theory into the language of (structural) theory of proofs, to use this translation to simplify the proofs of some known results [1], and to obtain two new ones: the coherence theorem for canonical morphisms in (nonmonoidal, nonsymmetric) closed categories [2], and the solution of the problem of equality of canonical morphisms. Extensions of these results to monoidal closed, symmetric closed, and monoidal symmetric closed categories are sketched. The decision procedure is obtained by means of a correct and faithful translation of canonical morphisms into an expansion of the -language, which has the tools for a special account of superfluous premises of implications (the thinning rule). The expansions of the -language which have so far appeared in the literature have not possessed this facility.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Institute, im. V. A. Steklova AN SSSR, Vol. 68, pp. 83–114, 1977.     

Operator algebras generated by Boolean algebras of projections in Montel spaces

It is shown that the classical result of W. Bade, stating that the uniformly closed operator algebra generated by a complete Boolean algebra of projections in a Banach space coincides with the weak operator closed algebra that it generates, is also valid (if suitably interpreted) in Montel spaces, Schwartz spaces and other related spaces.     

On sequentially closed subsets of the real line in

We show:

• (i) iff every countable product of sequential metric spaces (sequentially closed subsets are closed) is a sequential metric space iff every complete metric space is Cantor complete.
• (ii) Every infinite subset X of has a countably infinite subset iff every infinite sequentially closed subset of includes an infinite closed subset.
• (iii) The statement “ is sequential” is equivalent to each one of the following propositions:

• (a) Every sequentially closed subset A of includes a countable cofinal subset C,
• (b) for every sequentially closed subset A of , is a meager subset of ,
• (c) for every sequentially closed subset A of , ,
• (d) every sequentially closed subset of is separable,
• (e) every sequentially closed subset of is Cantor complete,
• (f) every complete subspace of is Cantor complete.

     

The de Schipper formula and squares of Riesz spaces

In this paper we introduce and study the square mean and the geometric mean in Riesz spaces. We prove that every geometric mean closed Riesz space is square mean closed and give a counterexample to the converse. We define for positive a, b in a square mean closed Riesz space E an addition via the formulaab=sup {(cos x)a + (sin x)b: 0 x 2π},which goes back to a formula by de Schipper. In case that E is geometric mean closed this turns the mldeflying set of the positive cone of E into a lattice ordered semigroup, which in turn is the positive cone ofa Riesz space E^□. We prove, under the additional condition that E is geometric mean closed, that E^□ is Riesz isomorphic to the square of E as introduced earlier by Buskes and van Rooij.     

Minimal number of singular fibers in a Lefschetz fibration

There exists a (relatively minimal) genus Lefschetz fibration with only one singular fiber over a closed (Riemann) surface of genus iff and . The singular fiber can be chosen to be reducible or irreducible. Other results are that every Dehn twist on a closed surface of genus at least three is a product of two commutators and no Dehn twist on any closed surface is equal to a single commutator.

     

A construction of real closed fields

We introduce a new construction of real closed fields by using an elementary extension of an ordered field with an integer part satisfying . This method can be extend to a finite extension of an ordered field with an integer part satisfying . In general, a field obtained from our construction is either real closed or algebraically closed, so an analogy of Ostrowski's dichotomy holds. Moreover we investigate recursive saturation of an o‐minimal extension of a real closed field by finitely many function symbols.     

Convexity and the Average Curvature of Plane Curves

The average curvature of a rectifiable closed curve in R2 is its total absolute curvature divided by its length. If a rectifiable closed curve in R2 is contained in the interior of a convex set D then its average curvature is at least as large as the average curvature of the simple closed curve D which bounds the convex set.     

Remarks on star countable discrete closed spaces

In this paper, we prove the following statements:

1. There exists a Tychonoff star countable discrete closed, pseudocompact space having a regular-closed subspace which is not star countable.
2. Every separable space can be embedded into an absolutely star countable discrete closed space as a closed subspace.
3. Assuming $2^{\aleph _0 } = 2^{\aleph _1 } $ , there exists a normal absolutely star countable discrete closed space having a regular-closed subspace which is not star countable.

     

Boundaries of rotation sets for homeomorphisms of the -torus

We construct a diffeomorphism of the 3-torus whose rotation set is not closed. We prove that the rotation set of a homeomorphism of the -torus contains the extreme points of its closed convex hull. Finally, we show that each pseudo-rotation set is closed for torus homeomorphisms.