Divisible designs and semi-regular relative difference sets from additive Hadamard cocycles

Additive Hadamard cocycles are a natural generalization of presemifields. In this paper, we study divisible designs and semi-regular relative difference sets obtained from additive Hadamard cocycles. We show that the designs obtained from additive Hadamard cocycles are flag transitive. We introduce a new product construction of Hadamard cocycles. We also study additive Hadamard cocycles whose divisible designs admit a polarity in which all points are absolute. Our main results include generalizations of a theorem of Albert and a theorem of Hiramine from presemifields to additive Hadamard cocycles. At the end, we generalize Maiorana-McFarland?s construction of bent functions to additive Hadamard cocycles.     

Topological Features of Massive Bosons on Two-Dimensional Einstein Space–Time

In this paper, we tackle the problem of constructing explicit examples of topological cocycles of Roberts’ net cohomology, as defined abstractly by Brunetti and Ruzzi. We consider the simple case of massive bosonic quantum field theory on the two-dimensional Einstein cylinder. After deriving some crucial results of the algebraic framework of quantization, we address the problem of the construction of the topological cocycles. All constructed cocycles lead to unitarily equivalent representations of the fundamental group of the circle (seen as a diffeomorphic image of all possible Cauchy surfaces). The construction is carried out using only Cauchy data and related net of local algebras on the circle.     

The giant spiral

Cocycles ofZ ^m actions on compact metric spaces provide a means for constructingR ^m actions or flows, called suspension flows. It is known that allR ^m flows with a free dense orbit have an almost one-to-one extension which is a suspension flow. Whenm=1, examples of cocycles are easy to construct; there is a one-to-one correspondence between cocycles and real valued continuous functions. However, whenm>1 the construction of examples of cocycles becomes more problematic. The only existing class of examples, the close to linear cocycles, have strong linearity properties and are well understood. In fact, when theZ ^m action is uniquely ergodic, all cocycles are close to linear. We will show that in general this need not be the case. We present a method, suggested to us by Hillel Furstenberg, for constructing examples of cocycles whenm>1 and use this method to construct a non close to linear cocycle on a minimalZ ² action.     

Evolution Equations for Markov Cocycles Obtained by Second Quantization in the Symplectic Fock Space

We derive a quantum stochastic differential equation satisfied by the unitary Markov cocycles obtained for a model situation during second quantization in the symmetric Fock space. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 186–192, January, 2006.     

Pullback attractors in nonautonomous difference equations

Nonautonomous difference equations are formulated as cocycles which generalize semigroups corresponding to autonomous difference equations. Pullback attractors are the appropriate generalization of autonomous attractors to cocycles. The existence of a pullback attractor follows when the difference equation cocycle has a pullback absorbing set. Results from the literature are outlined, including the construction of a Lyapunov function characterizing pullback attraction, and illustrated with several examples.     

On quantum stochastic differential equations

Existence and uniqueness theorems for quantum stochastic differential equations with nontrivial initial conditions are proved for coefficients with completely bounded columns. Applications are given for the case of finite-dimensional initial space or, more generally, for coefficients satisfying a finite localisability condition. Necessary and sufficient conditions are obtained for a conjugate pair of quantum stochastic cocycles on a finite-dimensional operator space to strongly satisfy such a quantum stochastic differential equation. This gives an alternative approach to quantum stochastic convolution cocycles on a coalgebra.     

Sesquilinear quantum stochastic analysis in Banach space

A theory of quantum stochastic processes in Banach space is initiated. The processes considered here consist of Banach space valued sesquilinear maps. We establish an existence and uniqueness theorem for quantum stochastic differential equations in Banach modules, show that solutions in unital Banach algebras yield stochastic cocycles, give sufficient conditions for a stochastic cocycle to satisfy such an equation, and prove a stochastic Lie–Trotter product formula. The theory is used to extend, unify and refine standard quantum stochastic analysis through different choices of Banach space, of which there are three paradigm classes: spaces of bounded Hilbert space operators, operator mapping spaces and duals of operator space coalgebras. Our results provide the basis for a general theory of quantum stochastic processes in operator spaces, of which Lévy processes on compact quantum groups is a special case.     

Cross Product Bialgebras Part II

This is the central article of a series of three papers on cross product bialgebras. We present a universal theory of bialgebra factorizations (or cross product bialgebras) with cocycles and dual cocycles. We also provide an equivalent (co-)modular (co-)cyclic formulation. All known examples as for instance bi- or smash, doublecross and bicross product bialgebras as well as double biproduct bialgebras and bicrossed or cocycle bicross product bialgebras are now united within a single theory. Furthermore our construction yields various novel types of cross product bialgebras.     

Twisted product of cocycles and factorization of semi‐regular relative difference sets

In this article, we introduce what we call twisted Kronecker products of cocycles of finite groups and show that the twisted Kronecker product of two cocycles is a Hadamard cocycle if and only if the two cocycles themselves are Hadamard cocycles. This enables us to generalize some known results concerning products and factorizations of central semi‐regular relative difference sets. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 431–441, 2008     

Cohomology of permutative cellular automata

We examineU(d) valued cocycles for a ?²⁺ action generated by a mixing, permutative cellular automaton and show that the set of Hölder continuous cocycles (for a given Hölder order) which are cohomologous to constant cocycles is both open and closed in the appropriate topology. A continuous dimension function with values in {0, 1,…,d} is defined on cocycles; a cocycle is cohomologous to a constant precisely when the value isd. Whend=1 (the abelian case) the first (essential) cohomology group is countable. IfU(1)? circle is replaced by a finite subgroup, this cohomology group is finite.     

Group Self-Similar Stable Processes in R d

Group self-similar processes are introduced. The spectral representation of such processes in the class of multidimensional strictly stable processes is determined. The uniqueness of such representations in terms of the corresponding group actions and cocycles is established.     

Classification of cocycles of generic equivalence relations

We study cocycles of an ergodic generic countable equivalence relation ℜ modulo meager sets. Two cocycles of ℜ are called weakly equivalent if they are cohomologous up to an element of Aut ℜ. It is proved that two nontransient cocycles with values in an arbitrary countable group are weakly equivalent if and only if their generic Mackey actions are isomorphic.     

Direct sums of balanced functions,perfect nonlinear functions,and orthogonal cocycles

Determining if a direct sum of functions inherits nonlinearity properties from its direct summands is a subtle problem. Here, we correct a statement by Nyberg on inheritance of balance and we use a connection between balanced derivatives and orthogonal cocycles to generalize Nyberg's result to orthogonal cocycles. We obtain a new search criterion for PN functions and orthogonal cocycles mapping to non‐cyclic abelian groups and use it to find all the orthogonal cocycles over Z ₂^t, 2 ≤ t ≤ 4. We conjecture that any orthogonal cocycle over Z ₂^t, t ≥ 2, must be multiplicative. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 173–181, 2008     

Smooth cocycles for an irrational rotation

Explicit examples of smooth cocycles not cohomologous to constants are constructed. Necessary and sufficient conditions on the irrational numberθ are given for the existence of such cocycles. It is shown that, depending onθ, the set ofC ^r cocycles whose skew-product is ergodic is either residual or empty.     

Cocycles over interval exchange transformations and multivalued Hamiltonian flows

We consider interval exchange transformations of periodic type and construct different classes of ergodic cocycles of dimension?1 over this special class of IETs. Then using Poincaré sections we apply this construction to obtain the recurrence and ergodicity for some smooth flows on non-compact manifolds which are extensions of multivalued Hamiltonian flows on compact surfaces.     

Finding cocycles in the bicrossed product construction for Lie groups

We find an explicit formula for finding pairs of cocycles for the construction of examples of locally compact quantum groups by using the bicrossed product of Lie groups. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 11, pp. 1510–1522, November, 2007.     

Cocycles Over Abelian TNS Actions

We study extensions of higher-rank Abelian TNS actions (i.e. hyperbolic and with a special structure of the stable distributions) by compact connected Lie groups. We show that up to a constant, there are only finitely many cohomology classes. We also show the existence of cocycles over higher-rank Abelian TNS actions that are not cohomologous to constant cocycles. This is in contrast to earlier results, showing that real valued cocycles, or small Lie group valued cocycles, over higher-rank Abelian actions are cohomologous to constants.     

Stable one-dimensional quasi-periodic cocycles on unitary group

In this paper, we study the stable one-dimensional quasi-periodic C ^∞ cocycles on U(N). We prove that any such cocycle on a generic irrational rotation is a limit point of reducible cocycles. The proof is based on Krikorian’s renormalization scheme and a local result of him.     

Trivial cocycles and invariants of homology 3-spheres

We study the relationship between trivial cocycles on the Torelli group and invariants of oriented integral homology 3-spheres. We apply this study to give a new purely algebraic construction of the Casson invariant. As a by-product we get a new 2-torsion cohomology class in the second integral cohomology of the Torelli group.