Difference sets in abelian groups ofp-rank two

Under a technical assumption that pertains to the so-called self-conjugacy, we prove: if an abelian groupG ofp-rank two,p a prime, admits a (nontrivial) (v, k, ) difference setD, then for each for some subgroupC _p ofG of orderp. Consequently,k(p=1), with equality only ifF=1/p D , whereD is the image ofD under the canonical homomorphism fromG ontoG/E (E being the unique elementary abelian subgroup ofG of orderp ²), is a (v/p ²,k/p, ) difference set inG/E. As applications, we establish the nonexistence of (i) (96, 20, 4) difference sets in ₄ x ₈ x ₃, (ii) (640, 72, 8) difference sets in ₈ x ₁₆ x ₅ and (iii) (320, 88, 24) difference sets in ₈ x ₈ x ₅. The first one fills a missing entry in Lander's table [6] and the other two in Kopilovich's table [5] (all with the answer no). We also point out the connection of the parameter sets in (i) above with the Turyn-type bounds [10] for the McFarland difference sets [9].Research partially supported by NSA Grant #904-92-H-3057 and by NSF Grant # NCR-9200265.     

The rhombidodecahedral tessellation of 3-space and a particular 15-vertex triangulation of the 3-dimensional torus

It is well known that the unique 7-vertex triangulation of the 2-dimensional torus S¹×S¹ is a consequence of the relationship between two hexagonal lattices in the euclidean plane: it is just the quotient of the triangular tessellation of the plane by a translation group. Each vertex star is a regular hexagon and the symmetry group of this triangulation is the affine group A(1,₇) in one dimension over ₇. In this paper we describe a particular 15-vertex triangulation of the 3-dimensional torus S¹×S¹×S¹ whose symmetry group is the affine group A(1,₁₅) and which is similarly related to two lattices in euclidean 3-space: it is just the quotient of a particular tessellation of 3-space by a translation group. Each vertex star happens to be a rhombidodecahedron, the dual of a (semiregular) cuboctahedron.     

Topological hochschild homology of extensions of ?/p? by polynomials, and of ?[x]/(xn) and ?[x]/(xn?1)

Topological Hochschild homology is calculated for the rings /p[x]/(f(x)) (where p is prime and f(x) /p[x] any polynomial), [x]/(x ⁿ) and [x]/(x ⁿ–1). A spectral sequence argument is used for calculating the homology of the topological Hochschild homology spectrum, from which its stable homotopy structure can be read off since the spectrum is known for a priori reasons to be a restricted product of Eilenberg-MacLane spectra.     

On the boundedness of Weyl sums

Leta be irrational and letf:[0,1] be Riemann-integrable with integral zero. Letf _n (x) denote the Weyl sumf _n (x):= _k=0 ^n–1 f({x k>}),x/[0,1[,n. We prove criteria for the boundedness of the sequence (f _n ) _n1 and discuss the relation of this question to irregularities of the distribution of sequences.     

On a Theorem of Grothendieck

A smooth projective morphism p : T S to a smooth variety S is considered. In particular, the following result is proved. The total direct image Rp _*(/n) of the constant étale sheaf /n is locally (in Zariski topology) quasiisomorphic to a bounded complex on S that consists of locally constant, constructible étale sheaves of /n-modules. Bibliography: 2 titles.     

On the number of relations in the cohomology of a fixed point set

If _p (respectively S¹) acts on a space, then under certain conditions the _p (respectively Q) cohomology of each component of of the fixed point set can be generated as an algebra by at most the same number of elements as the cohomology of the space itself. It is shown below that the ideal of relations among these generators is generated by at most the number of generators in the ideal of relations defining the cohomology of the original space.     

On odd points and the volume of lattice polyhedra

Denote by _n ³ ,n 2, the lattice consisting of all pointsx in ³ such thatnx belongs to the fundamental lattice ³ of points with integer coordinates. Letl _n be the subset of _n ³ consisting of all points whose coordinates are odd multiples of 1/n. The purpose of this paper is to give several new Pick-type formulae for the volume of three-dimensional lattice polyhedra, that is, polyhedra with vertices in ³. Our formulae are in terms of numbers of only thel _n-points belonging to a lattice polyhedronP in contrast to already known formulae which employ numbers of all the _n ³ -points inP. On our way to establishing the formulae we show that the number of points froml _n belonging to a three-dimensional lattice polyhedronP has some polynomiality properties similar to those of the well-known Ehrhart polynomial expressing the number of points of _n ³ inP. The paper contains also some comments on a problem of finding a volume formula which would employ only the setsl _n and which would be applicable to lattice polyhedra in arbitrary dimensions.Research partially supported by KBN Grant 2 P03A 008 10.     

A Construction of Partial Difference Sets in {\mathbb{Z}}_{p^{2 \times } } {\mathbb{Z}}_{p^{2 \times \cdot \cdot \cdot \times } } {\mathbb{Z}}_{p^2 }

In this paper, we give a construction of partial difference sets in _p ² x _p ² x ... x _p ²using some finite local rings.Dedicated to Hanfried Lenz on the occasion of his 80th birthdayThe work of this paper was done when the authors visited the University of Hong Kong.     

Inverting the Motivic Bott Element

We prove a version for motivic cohomology of Thomason's theorem on Bott-periodic K-theory, namely, that for a field k containing the nth roots of unity, the mod n motivic cohomology of a smooth k-scheme agrees with mod n étale cohomology, after inverting the element in H⁰(k,(1)) corresponding to a primitive nth root of unity.     

The elements of K2(ℤs)

Let S={p₁,...,p_s} be a set of rational primes, . One has K₂(_s)K₂(_su{2} and we want to assume 2 S. It is snown that every element of K₂(_S) is a Dehnis-Stein-symbol <a,b>, 1+ab being a unit of _S.Here b can be determined concretely, depending only on S, and we obtain a normal form of the elements of K₂(Q) as Steinberg-symbols, which is unique in some way and expresses the quadratic reciprocity law.     

Central Limit Theorem for a Class of Nonhomogeneous Random Walks

A spatially nonhomogeneous random walk _t on the grid =^m X ⁿ is considered. Let _t ⁰ be a random walk homogeneous in time and space, and let _t be obtained from it by changing transition probabilities on the set A= X ⁿ, || < , so that the walk remains homogeneous only with respect to the subgroup ⁿ of the group . It is shown that if >m 2 or the drift is distinct from zero, then the central limit theorem holds for _t.     

Representations of finite p-groups over the ring of formal power series with integral p-adic coefficients

A classification is given of the representations of the cyclic group of order p over the ring _p [[x]] of formal power series with integral p-adic coefficients and it is proved that in the other cases the problem of representations of a finite, p-group over _P [[x₁,...x_n]] is wild.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 6, pp. 753–765, June, 1992.     

Counting Sets With Small Sumset, And The Clique Number Of Random Cayley Graphs

Given a set A /N we may form a Cayley sum graph G _A on vertex set /N by joining i to j if and only if i+j A. We investigate the extent to which performing this construction with a random set A simulates the generation of a random graph, proving that the clique number of G _A is almost surely O(logN). This shows that Cayley sum graphs can furnish good examples of Ramsey graphs. To prove this result we must study the specific structure of set addition on /N. Indeed, we also show that the clique number of a random Cayley sum graph on =(/2) ⁿ is almost surely not O(log ||).* Supported by a grant from the Engineering and Physical Sciences Research Council of the UK and a Fellowship of Trinity College Cambridge.     

Moyennes uniformes et moyennes suivant une marche aléatoire

Summary Let be a bounded function on such that converges towards l as n goes to infinity, uniformly with respect to m. Let {X _n} be a random walk on , not concentrated on a proper subgroup of Then, with probability 1, converges towards l as n goes to infinity. The result also holds for any countable abelian group instead of . Other modes of convergence are considered (Cesaro convergence of order >1/2). The Cesaro convergence of expressions such that (X _n) (X _n+1) is also investigated.     

Subdivision algorithms with nonnegative masks generally converge

In this paper, a subdivision scheme consists of an operator froml () tol () determined by a doubly infinite sequence, called the mask. This operator convolutes, in a certain sense, sequences l () with the mask, thus producing a new sequence inl (). Moreover, this new sequence is placed on a finer grid. If we iterate this process with a positive mask infinitely many times, it is known that this process will produce a continuous function, which we callf . In this paper, we consider the extent to which non-negative masks yield similar results. An important application of subdivision schemes in computer graphics is the generation of curves and surfaces from an initial sequence.     

Equivalence between the dilation and lifting properties of an ordered group through multiplicative families of isometries. A version of the commutant lifting theorem on some lexicographic groups

Let be a locally compact abelian ordered group. has the dilation property if a special extension of the Naimark dilation theorem holds for and it has the commutant lifting property if a natural extension of the Sz.-Nagy — Foias commutant lifting theorem holds for .We prove that these two conditions are equivalent and we give another necessary and sufficient condition in terms of unitary extensions of multiplicative families of partial isometries.A version of the commutant lifting theorem is given for the groups ⁿ and × ⁿ with the lexicographic order and the natural topologies.Both authors were partially supported by the CDCH of the Universidad Central de Venezuela, and by CONICIT grant G-97000668.     

The Lattice of Interpretability Types of Cantor Varieties

For integers 1 m < n, a Cantor variety with m basic n-ary operations _i and n basic m-ary operations _k is a variety of algebras defined by identities _k(₁( ), ... , _m( )) = _k and i(₁( ), ... ,_n( )) = y _i, where = (x ₁., ... , x _n) and = (y ₁, ... , y _m). We prove that interpretability types of Cantor varieties form a distributive lattice, , which is dual to the direct product ₁ × ₂ of a lattice, ₁, of positive integers respecting the natural linear ordering and a lattice, ₂, of positive integers with divisibility. The lattice is an upper subsemilattice of the lattice of all interpretability types of varieties of algebras.     

1-formes fermées singulières et groupe fondamental

Summary We study the influence of the ₁ of a closed manifoldM ⁿ (n3) on the foliations ofM defined by closed differential 1-forms with Morse singularities (of index 0,n). Every nonexact form is cohomologous to a weakly complete one, that is one whose leaf space is of the same type as that of a nonsingular form. Generically, a form has compact leaves or is weakly complete. If ₁ M has no quotient isomorphic to *, then every nonexact form onM is weakly complete. We also say a form is complete if every path inM is homotopic to either a path transverse to or a path contained in a leaf of . Completeness of depends only on its de R ham cohomology class. The set of complete cohomology classes depends only on ₁ M and is related to finitely generated normal subgroups of ₁ M with quotient . If ₁ M is nilpotent (or even polycyclic), every nonexact form onM is complete. On irreducible 3-manifolds, a form is complete iff it is cohomologous to a nonsingular one.     

Compact groups having almost discrete orbit hypergroups

LetG be a compact group of automorphism acting continuously on a compact groupH. Then the orbit spaceH ^G is a compact hypergroup. We characterize, all solvable groupsH and compact automorphism groupsG for whichH ^G is almost discrete, i.e.,H ^G is homeomorphic to the one-point-compactification of . It turns out that thenH is isomorphic either to the infinite direct product (p) of the cyclic groups (p) or to _p ⁿ ( _p the group of allp-adic numbers) for some primep and some . The almost discrete orbit hypergroupsH ^G are determined explicitly for some examples.     

Compressing Coefficients While Preserving Ideals in K-Theory for C*-Algebras

An invariant based on orderedK-theory with coefficients in _n>1 /n and an infinite number of natural transformations has proved to be necessary and sufficient to classify a large class of nonsimple C* -algebras. In this paper, we expose and explain the relations between the order structure and the ideals of the C* -algebras in question.As an application, we give a new complete invariant for a large class of approximately subhomogeneous C*-algebras. The invariant is based on ordered K-theory with coefficients in /. This invariant is more compact (hence, easier to compute) than the invariant mentioned above, and its use requires computation of only four natural transformations.