Matrices with the edmonds—Johnson property

A matrixA=(a _ij ) has theEdmonds—Johnson property if, for each choice of integral vectorsd ₁,d ₂,b ₁,b ₂, the convex hull of the integral solutions ofd ₁≦x≦d ₂,b ₁≦Ax≦b ₂ is obtained by adding the inequalitiescx≦|δ|, wherec is an integral vector andcx≦δ holds for each solution ofd ₁≦x≦d ₂,b ₁≦Ax≦b ₂. We characterize the Edmonds—Johnson property for integral matricesA which satisfy for each (row index)i. A corollary is that ifG is an undirected graph which does not contain any homeomorph ofK ₄ in which all triangles ofK ₄ have become odd circuits, thenG ist-perfect. This extends results of Boulala, Fonlupt, Sbihi and Uhry. First author’s research supported by the Netherlands Organization for the Advancement of Pure Research (Z.W.O.).     

Sequentially equidistant points in metric spaces

A sequence (z ₀,z ₁,z ₂,, ...,z _n, z _n+1) of points fromp=z ₀ toq=z _n+1 in a metric spaceX is said to besequentially equidistant ifd(z _i−1,z _i)=d(z _i,z _i+1) for 1≦i≦n. If there is path inX fromp toq (or if a certain weaker condition holds), then such a sequence exists, with all points distinct, for every choice ofn, while ifX is compact and connected, then such a sequence exists at least forn=2. An example is given of a dense connected subspaceS ofR ^m ,m≧2, and an uncountable dense subsetE disjoint fromS for which there is no sequentially equidistant sequence of distinct points (n ≧ 2) inS ∪E between any two points ofE. Techniques of dimension theory are utilized in the construction of these examples, as well as in the proofs of some of the positive results. Supported in part by NSF Grant DMS-8701666.     

A Higman inequality for regular near polygons

The inequality of Higman for generalized quadrangles of order (s,t) with s>1 states that t≤s ². We generalize this by proving that the intersection number c _i of a regular near 2d-gon of order (s,t) with s>1 satisfies the tight bound c _i ≤(s ²ⁱ −1)/(s ²−1), and we give properties in case of equality. It is known that hemisystems in generalized quadrangles meeting the Higman bound induce strongly regular subgraphs. We also generalize this by proving that a similar subset in regular near 2d-gons meeting the bounds would induce a distance-regular graph with classical parameters (d,b,α,β)=(d,−q,−(q+1)/2,−((−q) ^d +1)/2) with q an odd prime power.     

The Banach spaceT and the fast growing hierarchy from logic

Necessary and sufficient conditions for (t _k) to be equivalent to (t _f(k)) are given where (t _k) is the natural basis of the Banach spaceT, the so-called “Tsirelson space”. The condition is in terms of the growth rate off(k) relative to the fast growing hierarchy. Roughly speaking, the ω-th level is needed to obtain nonequivalence. In particular, primitive recursivef(k) yield equivalent basic sequences. The proof is obtained via some asymptotically sharp estimates on the divergence of finite-dimensional subspaces ofT froml ₁ ^d of the same dimension.     

Intersection theorems with geometric consequences

In this paper we prove that ifℱ is a family ofk-subsets of ann-set, μ₀, μ₁, ..., μ_s are distinct residues modp (p is a prime) such thatk ≡ μ₀ (modp) and forF ≠ F′ ≠ℱ we have |F ∩F′| ≡ μ_i (modp) for somei, 1 ≦i≦s, then |ℱ|≦( _s ⁿ ). As a consequence we show that ifR ⁿ is covered bym sets withm<(1+o(1)) (1.2) ⁿ then there is one set within which all the distances are realised. It is left open whether the same conclusion holds for compositep.     

On the 2-class field tower of some imaginary biquadratic number fields

Let be an imaginary biquadratic number field with Cl_k,2, the 2-class group of k, isomorphic to Z/2Z × Z/2^mZ, m > 1, with q a prime congruent to 3 mod 4 and d a square-free positive integer relatively prime to q. For a number of fields k of the above type we determine if the 2-class field tower of k has length greater than or equal to 2. To establish these results we utilize capitulation of ideal classes in the three unramified quadratic extensions of k, ambiguous class number formulas, results concerning the fundamental units of real biquadratic number fields, and criteria for imaginary quadratic number fields to have 2-class field tower length 1. 2000 Mathematics Subject Classification Primary—11R29     

Ons-sets and mutual absolute continuity of measures on homogeneous spaces

We extend a recent result of A. Jonsson about mutual absolute continuity of twoD _s -measures on ans-setF ⊂R ⁿ to the homogeneous spaces (X, d, μ) of Coifman, Weiss. Here we define Hausdorff measure, Hausdorff dimension,D _s -set andd-set relative to the measureμ. Our main result holds for so called (s, d)-sets,d ≥s, and is stronger than Jonssons result even inR ⁿ . As applications we interpret this Hausdorff dimension as a relative dimension for very regular sets and show that it in general depends strongly onμ. For this purpose we construct a strictly increasing functionf :R →R, whose measure is doubling and concentrated on a set of arbitrary small Hausdorff dimension. The extension off to a quasiconformal map of the half plane onto itself sharpens a classical example of Ahlfors-Beurling.     

The minimal number of circuits in a finite set inR 2k−1

The minimal number of circuits (i.e., minimal affinely dependent subsets) in a set ofs points inR ^2k−1 isr( ₃ ⁴ )+(k−r)( ₃ ^4⋅1 ) as conjectured by J.-P. Doignon, wheres=qk+r, 0≦r<q.     

Central localization and Gelfand-Kirillov dimension

LetR be a factor ring of the enveloping algebra of a finite dimensional Lie algebra over a fieldk. If the centre ofR, Z, consists of non-zero divisors inR, the ringR _z obtained by localizing at the non-zero elements ofZ becomes a finitely generated algebra over the fieldK which arises as the field of fractions ofZ. The Gelfand-Kirillov dimension of anR-moduleM is denotedd(M). In this paper it is shown that ifR _Z ⊗ _R M ≠ 0 thend(M) ≧d(R _Z ⊗ _R M) + tr. deg _k Z, whered (R _z ⊗M) is the Gelfand-Kirillov dimension ofR _z ⊗M) viewed as anR _z -module andR _z is viewed as a finitely generatedK-algebra (not as ak-algebra). The result is primarily of a technical nature.     

On Hille-Tamarkin operators and Schatten classes

We determine the smallest Schatten class containing all integral operators with kernels inL _p(L_{p', q})^symm, where 2 <p∞ and 1≦q≦∞. In particular, we give a negative answer to a problem posed by Arazy, Fisher, Janson and Peetre in [1]. Supported in part by DGICYT (SAB-90-0033).     

How many s-subspaces must miss a point set in PG(d, q)

The following result is well-known for finite projective spaces. The smallest cardinality of a set of points of PG(n, q) with the property that every s-subspace has a point in the set is (q ^n+1-s - 1)/(q - 1). We solve in finite projective spaces PG(n, q) the following problem. Given integers s and b with 0 ≤ s ≤ n - 1 and 1 ≤ b ≤ (q ^n+1-s - 1)/(q - 1), what is the smallest number of s-subspaces that must miss a set of b points. If d is the smallest integer such that b ≤ (q ^d+1 - 1)/(q - 1), then we shall see that the smallest number is obtained only when the b points generate a subspace of dimension d. We then also determine the smallest number of s-subspaces that must miss a set of b points of PG(n, q) which do not lie together in a subspace of dimension d. The results are obtained by geometrical and combinatorial arguments that rely on a strong algebraic result for projective planes by T. Szőnyi and Z. Weiner.     

Weights andL Φ-boundedness of the Poisson integral operator

In this paper, we get a necessary and sufficient condition on the weights (μ,v) for the Poisson integral operator to be bounded fromL _Φ(R ⁿ, v(x)dx) to weak-L _Φ(R ₊ ⁿ⁺¹ ,dμ), where Φ is anN-function satisfying the Δ₂-condition. We also find a necessary and sufficient condition on the weights (μ,v) for the Poisson integral operator to be bounded fromL _Φ(R ⁿ,v(x)dx) toL _Φ(R ₊ ⁿ⁺¹ ,dμ) under some additional condition. Partially supported by NNSF of P.R. China     

A theorem on cyclic polytopes

LetC(ν, d) represent a cyclic polytope withν vertices ind dimensions. A criterion is given for deciding whether a given subset of the vertices ofC(ν, d) is the set of vertices of some face ofC(ν, d). This enables us to determine, in a simple manner, the number ofj-faces ofC(ν, d) for each value ofj (1≦j≦d−1).     

The sequence of codimensions of PI-algebras

Bounds and asymptotic formulas are given for the size of the irreducible representations of the symmetric groups. These are applied to obtain information on the identities and codimension sequencec _n(R) of a PI-algebraR, of a PI-algebraR of characteristic zero, e.g., the “ultimate” width of the hook in which the diagrams of the cocharacters ofR lies is <=(lim c _n (R)^1/n ) ² , and lim c_n(R)^1/n≦ 2(lim c_n(R)^1/n)² for rings with no right (or left) total annihilators.     

The structure of injective covers of special modules

In this paper, we prove that the injective cover of theR-moduleE(R/B)/R/B for a prime ideal B ofR is the direct sum of copies ofE(R/B) for prime ideals D ⊃ B, and if B is maximal, the injective cover is a finite sum of copies ofE(R/B). For a finitely generatedR-moduleM withn generators andG an injectiveR-module, we argue that the natural mapG ⁿ →G ⁿ/Hom _R (M, G) is an injective precover if Ext _R ¹ (M, R) = 0, and that the converse holds ifG is an injective cogenerator ofR. Consequently, for a maximal ideal R ofR, depth_R R ≧ 2 if and only if the natural mapE(R/R) →E(R/R)/R/R is an injective cover and depth_R R > 0.     

Contact Elements on Fibered Manifolds

For every product preserving bundle functor T ^μ on fibered manifolds, we describe the underlying functor of any order (r, s, q), s ≥ r ≤ q. We define the bundle K_k,l^r,s,q YK_{k,l}^{r,s,q} Y of (k, l)-dimensional contact elements of the order (r, s, q) on a fibered manifold Y and we characterize its elements geometrically. Then we study the bundle of general contact elements of type μ. We also determine all natural transformations of K_k,l^r,s,q YK_{k,l}^{r,s,q} Y into itself and of T( K_k,l^r,s,q Y )T\left( {K_{k,l}^{r,s,q} Y} \right) into itself and we find all natural operators lifting projectable vector fields and horizontal one-forms from Y to K_k,l^r,s,q YK_{k,l}^{r,s,q} Y .     

An extremal problem for Dirichlet-finite holomorphic functions on Riemann surfaces

Iff is a nonconstant holomorphic function with finite Dirichlet integralD(f) on a Riemann surfaceR, then |f|² has the least harmonic majorantf ₂ onR. We show Σf ₂(a ≦π ⁻¹ D(f)), wherea runs over all the roots off = 0 onR. The equality holds if and only iff is of type ℬℓ₁ fromR onto a disk of center 0. A consideration is proposed for the non-Euclidean case.     

Optimal linear perfect hash families with small parameters

A linear (q^d, q, t)‐perfect hash family of size s consists of a vector space V of order q^d over a field F of order q and a sequence ?₁,…,?_s of linear functions from V to F with the following property: for all t subsets X ? V, there exists i ∈ {1,·,s} such that ?_i is injective when restricted to F. A linear (q^d, q, t)‐perfect hash family of minimal size d( – 1) is said to be optimal. In this paper, we prove that optimal linear (q², q, 4)‐perfect hash families exist only for q = 11 and for all prime powers q > 13 and we give constructions for these values of q. © 2004 Wiley Periodicals, Inc. J Comb Designs 12: 311–324, 2004     

Infinite T?plitz–Lipschitz matrices and operators

We introduce a class of infinite matrices (A_ss￠, s, s￠ ? \mathbbZ^d){(A_{ss\prime}, s, s\prime \in \mathbb{Z}^d)} , which are asymptotically (as |s| + |s′| → ∞) close to Hankel–T?plitz matrices. We prove that this class forms an algebra, and that flow-maps of nonautonomous linear equations with coefficients from the class also belong to it.