Sticky flows on the circle and their noises

This paper gives a construction of sticky flows on the circle. Sticky flows give examples of stochastic flows of kernels that interpolates between Arratias coalescing flow and the deterministic diffusion flow. They are associated with systems of sticky independent Brownian particles on the circle, for some fixed parameter of stickyness. It is proved that the noise generated by Brownian sticky flows is black. A new proof of the fact that the noise of Arratias coalescing flow is black is given.     

Representation theory of 2-groups on Kapranov and Voevodsky's 2-vector spaces

In this paper the representation theory of 2-groups in 2-categories is considered, focusing the attention on the 2-category Rep2MatK(G) of representations of a 2-group G in (a version of) Kapranov and Voevodsky's 2-category of 2-vector spaces over a field K. The set of equivalence classes of such representations is computed in terms of the invariants π₀(G), π₁(G) and [α]∈H³(π₀(G),π₁(G)) classifying G, and the categories of intertwiners are described in terms of categories of vector bundles endowed with a projective action. In particular, it is shown that the monoidal category of finite dimensional linear representations (more generally, the category of [z]-projective representations, for any given cohomology class [z]∈H²(π₀(G),K^∗)) of the first homotopy group π₀(G) as well as its category of representations on finite sets both live in Rep2MatK(G), the first as the monoidal category of endomorphisms of the trivial representation (more generally, as the category of intertwiners between suitable 1-dimensional representations) and the second as a non-full subcategory of the homotopy category of Rep2MatK(G).     

K2 and K3 of the circle

Let k be $\text{Z}\text{[}\text{1}\text{2}\text{]}$, $\text{Q}$ or $\text{R}$, and set $\text{A =}\text{k[x,y]}\text{(x}{}^2\text{+ y}{}^2\text{? 1)}$. We compute K₂(A) and K₃(A). Our method is to construct a map $\text{? : K}{}_1\text{(k[i])→K}{}_{\mathrm{1\; +\; 1}}\text{(A)}$ and compare this to a localization sequence.We give three applications. We show that ? accounts for the primitive elements in K₂(A), and compare our results to computations of Bloch [1] for group schemes. Secondly, we consider the problem of basepoint independence, and indicate the interplay of geometry upon the K-theory of affine schemes obtained by glueing points of Spec(A). Third, we can iterate the construction to compute the K-theory of the torus ring A ?_kA.     

On a problem of linear extrapolation for homogeneous and isotropic random fields from the observations on a circle

We shall find a linear extrapolation formula for a homogeneous and isotropic plane random field which is observed on n concentric circles. Necessary and sufficient conditions are given for error-free extrapolation of a field with respect to a circle to its center.Translated from Matematicheskie Zametki, Vol. 4, No. 5, pp. 589–598, November, 1968.In concluding the author thanks M. I. Yadrenko for posing the problem and for help during the work.     

Lifting the 3-dimensional invariant of 2-plane fields on 3-manifolds

Homotopy classes of plane fields on 3-manifolds have been classified using a 2-dimensional invariant Γ and a 3-dimensional invariant θ by R. Gompf. Under regular covering maps, Γ lifts in the natural way. The lifting property of θ remained unresolved. In this paper, we present the lifting property of θ together with applications to Lens spaces. The applications help in specifying the liftings of the contact structures of the Lens space L(p,1) when lifted to S³.     

On the action of a circle on manifolds

The bordism groups of actions of a circle are calculated in the complex case and (in particular) in the oriented case; it is proved that certain systems of elements in them are linearly independent.Translated from Matematicheskie Zametki, Vol. 10, No. 5, pp. 511–518, November, 1971.In conclusion the author expresses his gratitude to S. P. Novikov for his assistance.     

On circle groups of nilpotent rings of charateristic 2

We prove that every nilpotent group of class 2 and exponent 4 is the circle group of a nilpotent ring of index 3 and characteristic 2.Dedicated to Professor László Fuchs on his 70th birthdayResearch supported by the Hungarian National Foundation for Scientific Research grant No. T 014279     

On a Toeplitz Representation of the H1/2 Norm

Some representations of the H1/2 norm are used as Schur complement preconditioner in PCG based domain decomposition algorithms for elliptic problems. These norm representations are efficient preconditioners but the corresponding matrices are dense, so they need FFT algorithm for matrix-vector multiplications. Here we give a new matrix representation of this norm by a special Toeplitz matrix. It contains only O(log(n)) different entries at each row, where n is the number of rows and so a matrix-vector computation can be done by O(nlog(n)) arithmetic operation without using FFT algorithm. The special properties of this matrix assure that it can be used as preconditioner. This is proved by estimating spectral equivalence constants and this fact has also been verified by numerical tests.     

On foliated circle bundles over closed orientable 3-manifolds

We show that there exists a family of smooth orientable circle bundles over closed orientable 3-manifolds each of which has a codimension-one foliation transverse to the fibres of class C ⁰ but has none of class C ³ . There arises a necessary condition induced from the Milnor-Wood inequality for the existence of a foliation transverse to the fibres of an orientable circle bundle over a closed orientable 3-manifold. We show that with some exceptions this necessary condition is also sufficient for the existence of a smooth transverse foliation if the base space is a closed Seifert fibred manifold. Received: May 13, 1996     

Mixing for some classes of special flows over rotations of the circle

Landau Institute of Theoretical Physics, RAN. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 26, No. 3, pp. 1–21, July–September, 1992.     

On Peixoto's conjecture for flows on non-orientable 2-manifolds

Contrary to the case of vector fields on orientable compact -manifolds, there is a smooth vector field on a non-orientable compact -manifold with a dense orbit (and therefore without closed orbits) whose phase portrait -up to topological equivalence- remains intact under a one-parameter family of twist perturbations localized in a flow box of