Subsystem entropy for ? sofic shifts

We prove that any ?^d shift of finite type with positive topological entropy has a family of subsystems of finite type whose entropies are dense in the interval from zero to the entropy of the original shift. We show a similar result for ?^d sofic shifts, and also show every ?^d sofic shift can be covered by a ?^d shift of finite type arbitrarily close in entropy.     

Forbidden Families of Geometric Permutations in ℝd

A geometric permutation induced by a transversal line of a finite family ℱ of disjoint convex sets in ℝ^d is the order in which the transversal meets the members of the family. We prove that for each natural k, each family of k permutations is realizable (as a family of geometric permutations of some ℱ) in ℝ^d for d ≥ 2k – 1, but there is a family of k permutations which is non-realizable in ℝ^d for d ≤ 2k – 2.     

Positive undecidable numberings in the Ershov hierarchy

A sufficient condition is given under which an infinite computable family of Σ^-1 _a -sets has computable positive but undecidable numberings, where a is a notation for a nonzero computable ordinal. This extends a theorem proved for finite levels of the Ershov hierarchy in [1]. As a consequence, it is stated that the family of all Σ^-1 _a -sets has a computable positive undecidable numbering. In addition, for every ordinal notation a > 1, an infinite family of Σ^-1 _a -sets is constructed which possesses a computable positive numbering but has no computable Friedberg numberings. This answers the question of whether such families exist at any—finite or infinite—level of the Ershov hierarchy, which was originally raised by Badaev and Goncharov only for the finite levels bigger than 1.     

Piercing Families of Convex Sets with the <Emphasis Type="Italic">d</Emphasis>-Intersection Property in ℝ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript>

In this paper, we consider finite families of convex sets in ℝ ^d such that every d or fewer sets of the family have a common point. For some families of this type, we give upper bounds on the size of a finite set intersecting all sets of the family. This research was supported by the Russian Foundation for the Basic Research Grants No. 03-01-00801 and 06-01-00648, and by the President of the Russian Federation Grant No. MK-5724.2006.1.     

The topology of flows near a dicritical singularity

In this paper, we prove a topological finite determinacy theorem for a generic family of C^∞ vector fields at a dicritical singularity in any dimension.     

The Maximal Number of Geometric Permutations for n Disjoint Translates of a Convex Set in ℝ Is Ω(n)

A geometric permutation induced by a transversal line of a finite family of disjoint convex sets in ℝ^d is the order in which the transversal meets the members of the family. It is known that the maximal number of geometric permutations in families of n disjoint translates of a convex set in ℝ³ is 3. We prove that for d ≥ 3 the maximal number of geometric permutations for such families in ℝ^d is Ω(n).     

Irregular elliptic surfaces of degree 12 in projective fourspace

So far a few families of smooth irregular surfaces are known to exist in ?⁴ up to pullbacks by suitable finite morphisms from ?⁴ onto ?⁴ itself. In this paper we present two different constructions of irregular smooth minimal elliptic surfaces of degree 12 in ?⁴. The first is a monad construction while the other uses liaison. The family constructed via liaison includes the surfaces of the first family as a special case. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Zero-one laws in finite W1-algebras

The authors study the notion of independence in the noncommutative analogue of a probability space, namely, a finite W¹-algebra with trace τ. Two definitions of independence, both generalizing the classical definition, are considered, and examples are given to illustrate that the definitions are inequivalent. For one of these notions of independence, the structure of a finite W¹-algebra generated by an independent family of subalgebras is determined. As a consequence, the authors obtain extensions of the Kolmogorov and Hewitt-Savage zero-one laws.     

Hyperbolic Structures on the Configuration Space of Six Points in the Projective Line

The oriented configuration space X⁺₆ of six points on the real projective line is a noncompact three-dimensional manifold which admits a unique complete hyperbolic structure of finite volume with ten cusps. On the other hand, it decomposes naturally into 120 cells each of which can be interpreted as the set of equiangular hexagons with unit area. Similar hyperbolic structures can be obtained by considering nonequiangular hexagons so that the standard hyperbolic structure on X⁺₆ is at the center of a five parameter family of hyperbolic structures of finite volume. This paper contributes to investigations of the properties of this family. In particular, we exhibit two real analytic maps from the set of prescribed angles of hexagons into R¹⁰ whose components are the traces of the monodromies at the ten cusps. We show that this map has maximal rank 5 at the center.     

On the number of conjugacy classes of a finite solvable group II

We prove that a finite solvable group G has at least (49p+1)/60 conjugacy classes whenever p is a prime such that p² divides the order of G. We also construct an infinite family of finite solvable groups, where this bound is attained.     

On the Milito-Cruz adaptive control scheme for Markov chains

Milito and Cruz have introduced a novel adaptive control scheme for finite Markov chains when a finite parametrized family of possible transition matrices is available. The scheme involves the minimization of a composite functional of the observed history of the process incorporating both control and estimation aspects. We prove the a.s. optimality of a similar scheme when the state space is countable and the parameter space a compact subset ofR ^d .     

Finite Cyclicity of Graphics with a Nilpotent Singularity of Saddle or Elliptic Type

In this paper we prove finite cyclicity of several of the most generic graphics through a nilpotent point of saddle or elliptic type of codimension 3 inside C^∞ families of planar vector fields. In some cases our results are independent of the exact codimension of the point and depend only on the fact that the nilpotent point has multiplicity 3. By blowing up the family of vector fields, we obtain all the limit periodic sets. We calculate two different types of Dulac maps in the blown-up family and develop a general method to prove that some regular transition maps have a nonzero higher derivative at a point. The finite cyclicity theorems are derived by a generalized derivation-division method introduced by Roussarie.     

The structure and construction of bi-Frobenius Hom-algebras

Bi-Frobenius Hom-algebras are introduced in this paper. They provide a common generalization of finite dimensional monoidal Hom-Hopf algebras and of bi-Frobenius algebras introduced by Doi and Takeuchi. The different conditions for finite dimensional monoidal Hom-algebras to be bi-Frobenius Hom-algebras are discussed. The substructures, quotient structures as well as morphisms of bi-Frobenius Hom-algebras are also studied. In addition, the Radford’s formula for S⁴ of a bi-Frobenius Hom-algebra is shown. The semisimplicity and separability for a special class of finite dimensional bi-Frobenius Hom-algebras are researched finally, which presents a version of Maschke’s theorem for this family.     

Groups Admitting a Kantor Family and a Factorized NormalSubgroup

We study the structure of a finite groupG admitting a Kantor family (F, F ^*) of type (s, t) and a nontrivial normal subgroupX which isfactorized byF F ^*. The most interesting cases, giving necessary conditions on the structure ofG and the parameterss andt, are those where a further Kantor family is induced inX, or where a partial congruence partition is induced in the factor groupG/X. Most of the known finite generalized quadrangles can be constructed as coset geometries with respect to a Kantor family. We show that the parameters of a skew translation generalized quadrangle necessarily are powers of the same prime. Furthermore, the structure of nonabelian groups admitting a Kantor family consisting only of abelian members is considered.Dedicated to Hanfried Lenz on the occasion of his 80th birthday     

On the connected components of the space of line transversals to a family of convex sets

Let ℒ be the space of line transversals to a finite family of pairwise disjoint compact convex sets in ℝ³. We prove that each connected component of ℒ can itself be represented as the space of transversals to some finite family of pairwise disjoint compact convex sets. The research of J. E. Goodman was supported in part by NSF Grant DMS91-22065 and by NSA Grant MDA904-92-H-3069. R. Pollack's research was supported in part by NSF Grant CCR91-22103 and by NSA Grant MDA904-92-H-3075. The research of R. Wenger was supported in part by NSA Grant MDA904-93-H-3026 and by the NSF Regional Geometry Institute (Smith College, July 1993) Grant DMS90-13220.     

Equivalence and traces on C1-algebras

We introduce an equivalence relation among the positive elements in a C¹ and show that the algebra is (semi-) finite if and only if there is a separating family of (semi-) finite traces. Concentrating on simple, semi-finite C¹-algebras we relate geometrical properties in the cone of equivalence classes to functional analytic properties of the algebra, such as the number of normalized traces and their possible values on a given element. The paper may be considered as an attempt to extend Murray and von Neumann's type and equivalence theory to C¹-algebras.     

Surgery obstructions from Khovanov homology

For a 3-manifold with torus boundary admitting an appropriate involution, we show that Khovanov homology provides obstructions to certain exceptional Dehn fillings. For example, given a strongly invertible knot in S ³, we give obstructions to lens space surgeries, as well as obstructions to surgeries with finite fundamental group. These obstructions are based on homological width in Khovanov homology, and in the case of finite fundamental group depend on a calculation of the homological width for a family of Montesinos links.     

Groups acting on finite dimensional spaces with finite stabilizers

It is shown that every H -group G of type admits a finite dimensional G-CW-complex X with finite stabilizers and with the additional property that for each finite subgroup H, the fixed point subspace X ^H is contractible. This establishes conjecture (5.1.2) of [9]. The construction of X involves joining a family of spaces parametrized by the poset of non-trivial finite subgroups of G and ultimately relies on the theorem of Cornick and Kropholler that if M is a -module which is projective as a -module for all finite then M has finite projective dimension. Received: April 30, 1997     

A Note on Adjacency-Transitivity of a Graph and Its Complement

We give a necessary and sufficient condition for a graphical regular representation to be adjacency-transitive, and provide an infinite family of finite simple undirected vertex-transitive graphs Γ, such that neither Γ nor Γ ^c is adjacency-transitive. Revised: March 24, 1998     

Common transversals

Given t families, each family consisting of s finite sets, we show that if the families “separate points” in a natural way, and if the union of all the sets in all the families contains more than (s + 1)^t ? s^t?1 ? 1 elements, then a common transversal of the t families exists. In case each family is a covering family, the bound is s^t ? s^t?1. Both of these bounds are best possible. This work extends recent work of Longyear [2].