Extended dyer-lashof algebras and modular coinvariants

Given an increasing sequence of integersN=(0,n ₁,n ₂,...), a functorG _N is constructed from the category of based spaces of the homotopy type ofCW compleces and based maps to a subcategory of in analogy to May's approximation modelC. A family of homology operationsRN is associated toG _N and its algebraic structure is described in terms of modular coinvariants of parabolic subgroups.     

Counting the relatively finite factors of a Bernoulli shift

LetT acting on (Ω, ℱ, μ) be afinite entropy Bernoulli shift. AT invariant factor is “relatively finite” if a.e. fiber of has a finite, hence constantk, number of points. We say two factors , “sit the same” if there is a measurable measure preserving map ϕ with and . We show here that up to sitting the same there are only finitely many relatively finite factors withk point fibers in a Bernoulli shift, and that they are classified by a certain algebraic structure in the symmetric group onk-points. This work was supported in part by the Miller Institute for Basic Research, U. C. Berkeley.     

Lower bounds at infinity for solutions of differential equations with constant coefficients

Two extensions of a classical theorem of Rellich are proved: (1) LetP=P(−iϖ/ϖx) be a partial differential operator with constant coefficients in , let the manifolds contained in have codimension ≧k>0, and denote by Γ an open cone in intersecting each normal plane of every such manifold. If ,Pu=0 and it follows thatu=0. (2) Assume in addition that each irreducibe lfactor ofP van shes on a real hypersurface and that Γ contains both normal directions at some such point. If andP(D) u has compact support, the same condition withk=1 implies thatu has compact support. In both results the hypotheses on the cone Γ and on the operatorP are minimal.     

Interpolation in Logiken monotoner systeme

A monotone structure ( ;μ) consists of a structure and a monotone systemμ over the domain of .L(Q ⁿ ) is , enlarged by a newn-ary quantifierQ ⁿ . says in ( ;μ) that there isU ∈μ such thatϕ[ā] is valid in ( ;μ) for allā ∈U ⁿ . If is a class of monotone structures, means thatϕ is valid in all expansions of monotone structures in . We show for the class of all ultrafilters that interpolation with respect to holds forL(Q ⁿ ) exactly in casen=1. Then we prove for a large class of (e.g. the class of topological groups) thatL(Q ⁿ ) satisfies interpolation with respect to for alln ≧ 1. Counterexamples indicate that the class of is sharp in some sense. Finally the results are carried over to certain topological structures and the interior quantifiersI ⁿ instead ofQ ⁿ , thereby generalizing results of Makowsky/Ziegler and Sgro, and to a multidimensional type of monotone structures including uniform spaces.     

Grassmann manifold of aJH-algebra

We study the set of rankp idempotents in a topologically simple Hilbert Jordan algebra (JH-algebra for short). To produce the differential geometric structure on, we establish Jordan algebraic results concerning the structure of some two-generator subalgebras. We identify geodesics, the Riemannian distance and the sectional curvature of by using the Jordan algebraic structure.     

On the existence of integral currents with prescribed mean curvature vector

Given an integralm-currentT ₀ in ℝ ^m+k and a tensorH of typ (m, 1) on ℝ ^m+k with values orthogonal to each of its arguments we prove the existence of an integralm-currentT with boundary ∂T=∂T ₀ having prescribed mean curvature vectorH, i. e. is a solution of      

A decomposition of 2-weak vertex-packing polytopes

The 2-weak vertex-packing polytope of a loopless graphG withd vertices is the subset of the unitd-cube satisfyingx _i +x _j ≤1 for every edge (i,j) ofG. The dilation by 2 of this polytope is a polytope with integral vertices. We triangulate with lattice simplices of minimal volume and label the maximal simplices with elements of the hyperoctahedral groupB _d . This labeling gives rise to a shelling of the triangulation of , where theh-vector of (and the Ehrharth ^*-vector of can be computed as a descent statistic on a subset ofB _d defined in terms ofG. A recursive way of computing theh-vector of is also given, and a recursive formula for the volume of . This work was partially supported by grants from the Icelandic Council of Science and the Royal Swedish Academy of Sciences, respectively.     

On Galois isomorphisms between ideals in extensions of local fields

LetL/K be a totally ramified, finite abelian extension of local fields, let and be the valuation rings, and letG be the Galois group. We consider the powers of the maximal ideal of as modules over the group ring . We show that, ifG has orderp ^m (withp the residue field characteristic), ifG is not cyclic (or ifG has orderp), and if a certain mild hypothesis on the ramification ofL/K holds, then and are isomorphic iffr≡r′ modp ^m . We also give a generalisation of this result to certain extensions not ofp-power degree, and show that, in the casep=2, the hypotheses thatG is abelian and not cyclic can be removed.     

Structures of polyhedra determined by submodular functions on crossing families

The present paper shows that for any submodular functionf on a crossing family with , if the polyhedron is nonempty, then there exist a unique distributive lattice with and a unique submodular function with such thatB(f) coincides with the base polyhedron associated with the submodular system . Here, iff is integer-valued, thenf ₁ is also integer-valued. Based on this fact, we also show the relationship between the independent-flow problem considered by the author and the minimum cost flow problem considered by J. Edmonds and R. Giles.     

Maße für zyklische symmetrien ebener konvexer körper

Let

denote a finite group of rotations of

A realvalued functional f defined on a subfamily

of the family of all convex bodies in

is called a measure of symmetry with respect to

if (1) 0?(K)?1 for all

; (2) f(K)=1 iff K is symmetric with respect to

; (3)

for all

and for all orientation preserving similarties τ of

; (4) f:

is continuous.

A special measure of symmetry with respect to

,

, is introduced. The following theorems hold: Let

denote the cyclic group of proper rotations of

generated by the rotation about 2π/k with the origin as center, k?3 an integer, and let

denote the family of all plane centrally symmetric convex bodies containing at least two points.

1. 1.

   for k?3, where c_k is a positive real number which is explicitly calculated.

2. 2.

   iff K is a segment. For K=3,4 better results can be obtained. One tool for these improvements is:

3. 3.

   About every plane convex body with interior points an equilateral triangle can be circumscribed such that its center of gravity lies in the interior of K.