Properties of a jointly ergodic action of the direct product of two groups

An ergodic action a of the direct product of and , not isomorphic to a product of actions of and G, is constructed, such that the actions of and G separately are not ergodic. The actions of on its ergodic components are metrically isomorphic if and only if these components are taken into one another by the action of G. Finally, the centralizerC (×G) is such thatC (×G)/(×G)₂.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 5, pp. 684–688, May, 1991.     

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.     

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.     

Almost block independence and bernoullicity of ℤ d -actions by automorphisms of compact abelian groups

Summary We prove that a ^d -action by automorphisms of a compact, abelian group is Bernoulli if and only if it has completely positive entropy. The key ingredients of the proof are the extension of certain notions of asymptotic block independence from -actions to ^d -action and their equivalence with Bernoullicity, and a surprisingly close link between one of these asymptotic block independence properties for ^d -actions by automorphisms of compact, abelian groups and the product formula for valuations on global fields.Oblatum 20-X-1994     

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.     

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.     

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.     

Some mildly wild circles in Sn arising from algebraic K-Theory

We study wild embeddings of S ¹ in S ⁿ which are tame in a sense introduced by Quinn. We show that if is a finitely presented group with H ₁()=H ₂()=0, then any finiteness obstruction K ₀() can be realized on the complement of such an embedded S ¹. We also realize trivially symmetric K _–1() obstructions on the complements of such embeddings. For trivially symmetric , the embeddings constructed are shown to be isotopy homogeneous.     

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.     

The Cohomology of SL(3, ?[1/2])

We compute the cohomology of SL(3, [1/2]) with coefficients in the prime fields and in the integers. On the way we obtain the cohomology of certain mod-2 congruence subgroups of SL(3, ) with coefficients in p for p>2. Finally we compute the cohomology of GL(3, [1/2]).     

Uniqueness of the infinite component in a random graph with applications to percolation and spin glasses

Summary We extend the theorem of Burton and Keane on uniqueness of the infinite component in dependent percolation to cover random graphs on ^d or ^d × with long-range edges. We also study a short-range percolation model related to nearest-neighbor spin glasses on ^d or on a slab ^d × {0,...K} and prove both that percolation occurs and that the infinite component is unique forV=²×{0,1} or larger.A.G. was partially supported from AFOSR through grant no. 90-0090     

Some Criteria for Multivalently Starlikeness

Let S_n(p)(p, n N) be the class of functions f() = ^p + a_p+n^p+n + which are p-valently starlike in the unit disk. Some sufficient conditions for a function f() to be in the class S_n(p) are given.AMS Subject Classification (2000): primary 30C45     

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.     

ApproximatingK*(?) through degree five

We approximate theK-theory spectrum of the integers using a spectrum level rank filtration. By means of a certain poset spectral sequence, we explicitly compute the first three subquotients of this filtration. Assuming a conjecture about the filtration's rate of convergence, we conclude thatK ₄()=0 andK ₅() is a copy of (the Borel summand) plus two-torsion of order at most eight.     

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.     

The Eta Invariant and the Connective K-Theory of the Classifying Space for Cyclic 2 Groups

We use the eta invariant to study the connective K-theory groups ko _m (B ) of the classifying space for the cyclic group where - 2 2.     

Summation formulas for general Kloosterman sums

N. V. Kuznetsov's summation formula is generalized to the case of a discrete subgroup GSL₂() and a system of multiplicators , satisfying certain not too restrictive conditions. In the arithmetic cases, when G is a congruence-subgroup in SL₂(), these conditions are satisfied. N. V. Kuznetsov's formula has been proved for the case G=SL₂()., =1.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 82, pp. 103–135, 1979.     

On random walks arising in queueing systems: ergodicity and transience via quadratic forms as lyapounov functions — Part I

A simple and quite general approach is proposed to derive criteria for transience and ergodicity of a certain class of irreducibleN-dimensional Markov chains in ₊ ^N assuming a boundedness condition on the second moment of the jumps. The method consists in constructing convenient smooth supermartingales outside some compact set. The Lyapounov functions introduced belong to the set of quadratic forms in ₊ ^N and do not always have a definite sign. Existence and construction of these forms is shown to be basically equivalent to finding vectors satisfying a system of linear inequalities.Part I is restricted toN=2, in which case a complete characterization is obtained for the type of random walks analyzed by Malyshev and Mensikov, thus relaxing their condition of boundedness of the jumps. The motivation for this work is partly from a large class of queueing systems that give rise to random walks in ₊ ^N      

Generating Elements of the Annihilating Ideal for Modular Symbols

In the paper, the concrete generators are found for the annihilating ideal of the modular symbols associated with SL(s;). Using this result, the (co)homology of subgroups of finite index is evaluated for SL(s;).     

On the Hantzsche-Wendt manifold

We explicitly compute the outer automorphism group Out ₁ M of the fundamental group of the Hantzsche — Wendt manifoldM. It is an extension 1(₂)³Out₁ MS ₃₂1, but not the semidirect product (₂)³(S ₃₂) as claimed in [3] (see also [4]). As a consequence, we get a quick algebraic computation of the symmetry groups of the Borromean rings and the figure-8-knot.