Highly saturated packings and reduced coverings

We introduce and study certain notions which might serve as substitutes for maximum density packings and minimum density coverings. A body is a compact connected set which is the closure of its interior. A packingP with congruent replicas of a bodyK isn-saturated if non–1 members of it can be replaced withn replicas ofK, and it is completely saturated if it isn-saturated for eachn1. Similarly, a coveringC with congruent replicas of a bodyK isn-reduced if non members of it can be replaced byn–1 replicas ofK without uncovering a portion of the space, and its is completely reduced if it isn-reduced for eachn1. We prove that every bodyK ind-dimensional Euclidean or hyperbolic space admits both ann-saturated packing and ann-reduced covering with replicas ofK. Under some assumptions onKE ^d (somewhat weaker than convexity), we prove the existence of completely saturated packings and completely reduced coverings, but in general, the problem of existence of completely saturated packings, and completely reduced coverings remains unsolved. Also, we investigate some problems related to the the densities ofn-saturated packings andn-reduced coverings. Among other things, we prove that there exists an upper bound for the density of ad+2-reduced covering ofE ^d with congruent balls, and we produce some density bounds for then-saturated packings andn-reduced coverings of the plane with congruent circles.     

The average number of normals through a point in a convex body and a related Euler-type identity

Anormal to a convex bodyK at a pointx in the boundary ofK is a ray with endpoint atx, perpendicular to support planeH ofK atx, and contained in the halfspace bounded byH that containsK. This work establishes bounds on the average number ofnormals through a point inK forK a polytope and forK with sufficiently smooth boundary. The integrals used to establish these bounds lead to an Euler-type relation.     

On Tate-Shafarevich groups over galois extensions

LetA be an abelian variety defined over a number fieldK. LetL be a finite Galois extension ofK with Galois groupG and let III(A/K) and III(A/L) denote, respectively, the Tate-Shafarevich groups ofA overK and ofA overL. Assuming these groups are finite, we compute [III(A/L) ^G ]/[III(A/K)] and [III(A/K)]/[N(III(A/L))], where [X] is the order of a finite abelian groupX. Especially, whenL is a quadratic extension ofK, we derive a simple formula relating [III(A/L)], [III(A/K)], and [III(A ^x/K)] whereA x is the twist ofA by the non-trivial characterχ ofG.     

Konstruktion regulärer Hüllen konstanter Breite

LetM be a compact, convex set of diameter 2 inE ^d. There exists a bodyK of constant width 2 containingM such that every symmetry ofM is one ofK and every singular boundary point ofK is a boundary point ofM, for which the set of antipodes inK is the convex hull of the antipodes, which are already inM.

---

---

Mit 1 Abbildung     

Counting models in universal Horn classes

Definen _K (λ) to be either ω, or the number of non-isomorphic models inK having cardinality α, whichever cardinal is larger. This paper contains a proof that for a congruence modular variety ⋎ of algebras of countable similarity type, there are only six possible functionsn _⋎. It is also proved that ifn _K (λ)≠2^λ for some λ, andK is a universal Horn class of models for a countable language, thenK must satisfy two conditions, one of which is quite restrictive and requires that the members ofK are all in a certain sense Abelian. Presented by B. Jonsson.     

The frattini subgroup of the absolute Galois group of a local field

LetK be a local field,T the maximal tamely ramified extension ofK, F the fixed field inK _sof the Frattini subgroup ofG(K), andJ the compositum of all minimal Galois extensions ofK containingT. The main result of the paper is thatF=J. IfK is a global field andK _solv is the maximal prosolvable extension ofK, then the Frattini group of % MathType!End!2!1!(K _solv/K) is trivial. Partially supported by a grant from the G.I.F., the German-Israeli Foundation for Scientific Research and Development.     

On component factors

LetK be a connected graph. A spanning subgraphF ofG is called aK-factor if every component ofF is isomorphic toK. On the existence ofK-factors we show the following theorem: LetG andK be connected graphs andp be an integer. Suppose|G| = n|K| and 1 <p < n. Also suppose every induced connected subgraph of orderp|K| has aK-factor. ThenG has aK-factor.     

Iwasawa invariants of imaginary quadratic fields

LetK be an imaginary quadratic field andp an odd prime which splits inK. We study the Iwasawa invariants for ℤ _p -extensions ofK. This is motivated in part by a recent result of Sands. The main result is the following. Assumep does not divide the class number ofK. LetK _∞ be a ℤ _p -extension ofK. SupposeK _∞ is not totally ramified at the primes abovep. Then the μ-invariant forK _∞/K vanishes. We also show that if μ=0 for all ℤ _p -extensions ofK, then the λ-invariant is bounded asK _∞ runs through all such extensions.     

On the minimal convex annulus of a planar convex body

In this paper we introduce the notion of a minimal convex annulusK (C) of a convex bodyC, generalizing the concept of a minimal circular annulus. Then we prove the existence — as for the minimal circular annulus — of a Radon partition of the set of contact points of the boundaries ofK (C) andC. Subsequently, the uniqueness ofK (C) is shown. Finally, it is proven that, for typicalC, the boundary ofC has precisely two points in common with each component of the boundary ofK (C).     

A connectedness criterion for ℓ-adic galois representations-adic galois representations

To every compatible system of Galois representations of a global fieldK, there is associated a natural invariantK ^conn, the smallest extension ofK over which the associated algebraic monodromy groups become connected. We present a purely field-theoretic construction ofK ^conn for all Galois representations arising from cohomology. Partially supported by the Sloan Foundation and by NSF Grant DMS94-00833.     

Linearly independent zeros of quadratic forms over number-fields

LetK be an algebraic number-field of degree [K:Q] =n 1 and letO denote some fixed order ofK. Let, be a quadratic form which represents zero for some. For the special caseK =Q,O =Z, theorems ofCassels and ofDavenport provide estimates for the magnitude (in terms of the coefficients off(x)) of a zero and of a pair of linearly independent zeros off, respectively. Recently,Raghavan extendedCassels' result to arbitraryK. In this article, a new proof ofDavenport's theorem for a pair of linearly independent zeros is given which not only provides explicit constants in the estimates but also extends to generalK. A refinement of this proof leads to effectively computable bounds for rational representations of a numbern0 byf.     

Convex surfaces which intersect each congruent copy of themselves in a connected set

LetK ⊂R ³ be a three-dimensional convex body such that, for every isometry ρ ofR ³, the boundaries ofK and ρK meet in a connected set. ThenK is a parallel set of some possibly degenerate linesegment.     

A generalization of nets and bases

In this paper the notions ofK-nets andK-bases are introduced and the corresponding cardinal functions,K-netweight andK-weight, are studied. Spaces with smallK-nets orK-bases are in some sense close to compact spaces. It turns out that the ordinary weight and net-weight can be easily expressed in terms of theK-versions and some auxiliary functions, moreover under some restrictions, weight and net-weight actually coincide with the hereditary modifications ofK-weight andK-netweight, respectively.     

Realization of cartesian closed topological hulls

The cartesian closed topological (CCT) hull of a concrete categoryK is the smallest full CCT extension ofK. A general method for describing the CCT hulls is presented and applied to render unified proofs for the basic examples ofK: topological spaces, uniform spaces, pretopological spaces, compact T₂ spaces, metrizable spaces, and completely regular spaces.     

L’Extension Abelienne D’une Categorie semi-abelienne

We show that if (K,L) is a semi-abelian category, there exists an abelian categoryK ^x with the followings properties:

Characterization off-vectors of families of convex sets inR d Part I: Necessity of Eckhoff’s conditions

LetK=K ₁,...,K_n be a family ofn convex sets inR ^d . For 0≦i<n denote byf _i the number of subfamilies ofK of sizei+1 with non-empty intersection. The vectorf(K) is called thef-vectors ofK. In 1973 Eckhoff proposed a characterization of the set off-vectors of finite families of convex sets inR ^d by a system of inequalities. Here we prove the necessity of Eckhoff's inequalities. The proof uses exterior algebra techniques. We introduce a notion of generalized homology groups for simplicial complexes. These groups play a crucial role in the proof, and may be of some independent interest.     

On existence of sets of distinct representatives for families of subsets of a multiset

IfX is a family ofl-subsets of a multiset, then there is a set of distinct representatives of the members ofX among the (l-k)-subsets of the members ofX provided the cardinality ofX does not exceed a certain bound. An algorithm for calculating this bound is given.     

The parameterized local deduction theorem for quasivarieties of algebras and its application

Let be an algebraic type. To each classK of -algebras a consequence relation _K defined on the set of -equations is assigned. Some weak forms of the deduction theorem for _K and their algebraic counterparts are investigated. The (relative) congruence extension property (CEP) and its variants are discussed.CEP is shown to be equivalent to a parameter-free form of the deduction theorem for the consequence _K .CEP has a strong impact on the structure ofK: for many quasivarietiesK,CEP implies thatK is actually a variety. This phenomenon is thoroughly discussed in Section 5. We also discuss first-order definability of relative principal congruences. This property is equivalent to the fact that the deduction theorem for _K is determined by a finite family of finite sets of equations. The following quasivarietal generalization of McKenzie's [26] finite basis theorem is proved:LetK be quasivariety of algebras of finite type in which the principalK-congruences are definable. ThenK is finitely axiomatizable iff either the classK _FSI (of all relatively finitely subdirectly irreducible members ofK) or the class K_SI (of all relatively subdirectly irreducible members ofK) is strictly elementary.Applications of the theory to Heyting, interior, Sugihara, and ukasiewicz algebras are provided.Presented by R. McKenzie.The paper was presented in a talk given to the Conference on Algebraic Logic, Budapest, August 1988.     

On the composition length of finite primitive linear groups

LetG be a finite primitive linear group over a fieldK, whereK is a finite field or a number field. We bound the composition length ofG in terms of the dimension of the underlying vector space and of the degree ofK over its prime subfield. As a byproduct, we prove a result of number theory which bounds the number of prime factors (counting multiplicities) ofq ⁿ−1, whereq, n>1 are integers, improving a result of Turull and Zame [6].     

The extension property for compact convex sets

A closed convex subsetQ of a compact convex setK is said to have the extension property if every continuous affine function onQ can be extended to a continuous affine function onK. It is proved that the extension property is equivalent to the existence of a numberN such that is any direction in whichQ has positive width, the ratio of the width ofK to the width ofQ is less thanN.