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     

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.     

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.     

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.     

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.     

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.     

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.     

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]).     

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.     

Fields with vanishing K2. Torsion in H1(X,K2) and Ch2(x)

This paper describes fields F of nonzero characteristic with the property that for all finite extensions E/F K₂E=0. We consider a somewhat wider class of fields which includes finite and separably closed fields. For smooth projective varieties X over such a field we show that the groups H¹(X, K₂){} and H²(X_et, (2)), NH³(X_et, (2)) and Ch²(X){} are isomorphic. These results are applied to describe the groups SK₁ of a smooth affine curve over such a field.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 116, pp. 108–118, 1982.     

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 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      

Structure of semisimple differentials and p-class groups in ℤp-extensions

For a wildly ramified p-extension E/F of algebraic function fields of one variable in an algebraically closed field k of characteristic p with Galois Group G, Nakajima obtained two exact sequences which determined implicitly the structure of the holomorphic semisimple differentials as k[G]-module. In this paper, in many cases, e.g., if there is a fully ramified prime, the structure is determined explicitly. Analogous results are obtained for p-extensions of _p-fields of CM-type. In the latter situation, if E/F is unramified, the structure of the minus part of the p-class group of E is determined as _p[G]-module.     

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 Semisimple Hopf Algebras of Dimension 2 m

In this paper we classify all nontrivial semisimple Hopf algebras of dimension 2 ^{n +1} with the group of grouplikes isomorphic to _{2 n–1}×₂. Moreover, we extend some results on irreducible representations from groups to semisimple Hopf algebras and prove that certain semisimple Hopf algebras, including the ones classified in this paper, satisfy the generalized power map property.     

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 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.     

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     

Projective manifolds with the same homology as ℙ k

Complex projective threefolds having the same integral homology groups as ³ are classified. This classification is used to improve a result of Fujita on manifolds whose integral cohomology ring is the same as that of ^k and applies to the problem of classifying polarized manifolds (X, A), A being a nonsingular hypersurface such thatH _i (A, )H _i (X, ) fori2 dimA.Partially supported by M.P.I. of the Italian Government. Both authors are members of G.N.S.A.G.A. of the Italian C.N.R.     

Generators of the group of principal units of a cyclic p-extension of a regular local field

Suppose k is a local field that is an extension of the field of p -adic numbers of degree n and does not contain a primitive p -th root of 1, and suppose K/k is a cyclic p-extension with Galois group G. The group E of principal units of K is a multiplicatively written module over the group ring =_p[G], where _p is the ring of p-adic integers. It was shown by Borevich (Ref. Zh. Mat., 1965, 3A256) that the -module E has a system of n+1 generators, of which n–1 are free and two are connected by certain relations. In the present paper these -generators are constructed explicitly and their arithmetical characteristics indicated.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 71, pp. 16–23, 1977.