Abstract: Let C = C(X) be the unital C*-algebra of all continuous functions on a finite CW complex X and let A be a unital simple C*-algebra with tracial rank at most one. We show that two unital monomorphisms φ,ψ: C → A are asymptotically unitarily equivalent, i.e., there exists a continuous path of unitaries {u
t
: t ∈ [0, 1)} ⊂ A such that lim
t→1
u*
t
φ(f)u
t
= ψ(f) for all f ∈ C(X) if and only if [φ] = [ψ] in KK(C,A), τ ◦φ = τ ◦ψ for all τ ∈ T(A), and φ
† = ψ
†, where T(A) is the simplex of tracial states of A and φ
†, ψ
†: U
∞(C)/DU
∞(C) → U
∞(A)/DU
∞(A) are the induced homomorphisms and where U
∞(A) = ∪
k=1
∞
U(M
k
(A)) and U
∞(C) = ∪
k=1
∞
U(M
k
(C)) are usual infinite unitary groups, respectively, and DU
∞(A) and DU
∞(C) are the commutator subgroups of U
∞(A) and U
∞(C), respectively. We actually prove a more general result for the case in which C is any general unital AH-algebra. PubDate: 2015-07-01

Abstract: Let G × Σ(1) × Σ(1) be a free, properly discontinuous and cellular action of a group G on a finite-dimensional CW-complex Σ(1) that has the homotopy type of the circle. We determine all virtually cyclic groups G that act on ∑(1) together with the induced action G → Aut(H
1(Σ(1), Z)), and we classify the orbit spaces Σ(1)/G.
We, then, study the same questions for certain families of groups. First, we consider the family of groups with vcd ⩽ 1 which includes semi-direct products ℤ
n
⋊ F and F ⋊ ℤ
n
and amalgamated products of finite groups with bounded orders since these groups have vcd = 1. Next, we study locally cyclic groups consisting of subgroups of the rationals ℚ with vcd ⩽ 2 and subgroups of the quotient ℚ/ℤ with vcd = ∞. The results obtained depend upon the subfamily in question. In particular, for an action of any subgroup of ℚ/ℤ there is only one orbit space up to homotopy and the induced action on H
1(Σ(1), ℤ) is trivial. PubDate: 2015-07-01

Abstract: This paper deals with sheaves of differential operators on noncommutative algebras, in a manner related to the classical theory of D-modules. The sheaves are defined by quotienting the tensor algebra of vector fields (suitably deformed by a covariant derivative). As an example we can obtain enveloping algebra like relations for Hopf algebras with differential structures which are not bicovariant. Symbols of differential operators are defined, but not studied. These sheaves are shown to be in the center of a category of bimodules with flat bimodule covariant derivatives. Also holomorphic differential operators are considered. PubDate: 2015-07-01

Abstract: The notion of locally ideal liquid, similar to locally ideal fluid, is introduced. The distribution of locally ideal liquid is presented and compared with the one given by the van der Waals equation. The notion of phase transition from the metastable liquid state to the metastable solid state (glass) is introduced. PubDate: 2015-07-01

Abstract: In the spirit of noncommutative geometry, we prove a full-fledged “relative index” type theorem that compares certain elements of the Kasparov KK-group KK(A,B). PubDate: 2015-07-01

Abstract: We consider the index problem for a wide class of nonlocal elliptic operators on a smooth closed manifold, namely, differential operators with shifts induced by the action of a (not necessarily periodic) isometric diffeomorphism. The key to the solution is the method of uniformization. To the nonlocal problem we assign a pseudodifferential operator, with the same index, acting on the sections of an infinite-dimensional vector bundle on a compact manifold. We then determine the index in terms of topological invariants of the symbol, using the Atiyah—Singer index theorem. PubDate: 2015-07-01

Abstract: The goal of this paper is to study some cases of not Col-divisible balanced three-dimensional polytopes and to calculate the corresponding E-groups and K-groups. PubDate: 2015-07-01

Abstract: We compute the K-theory of noncommutative Bieberbach manifolds, which are fixed point C* subalgebras of a three-dimensional noncommutative torus by a free action of a cyclic group ℤ
N
, N = 2, 3, 4, 6. PubDate: 2015-07-01

Abstract: As is known from previous research, the study of the structure and counting of Reidemeister classes (twisted conjugacy classes) of an automorphism ϕ: G → G, i.e., the classes x ∼ gxϕ(g
−1), is closely related to the study of the corresponding twisted inner representation of a discrete group G, i.e., a representation on ℓ2(G) corresponding to the action g ↦ xgϕ(x
−1) (x, g ∈ G) of G on itself. In the present paper, we study twisted inner representations from a more general point of view, but the questions under consideration are still close to important relations with Reidemeister classes. PubDate: 2015-07-01

Abstract: This paper is devoted to analyzing two approaches to characteristic classes of transitive Lie algebroids. The first approach is due to Kubarski [5] and is a version of the Chern-Weil homomorphism. The second approach is related to the so-called categorical characteristic classes (see, e.g., [6]). The construction of transitive Lie algebroids due to Mackenzie [1] can be considered as a homotopy functor T LA
g from the category of smooth manifolds to the transitive Lie algebroids. The functor T LA
g assigns to every smooth manifold M the set T LA
g(M) of all transitive algebroids with a chosen structural finite-dimensional Lie algebra g. Hence, one can construct [2, 3] a classifying space B
g such that the family of all transitive Lie algebroids with the chosen Lie algebra g over the manifold M is in one-to-one correspondence with the family of homotopy classes of continuous maps [M, B
g]: T LA
g(M) ≈ [M, B
g]. This enables us to describe characteristic classes of transitive Lie algebroids from the point of view of a natural transformation of functors similar to the classical abstract characteristic classes for vector bundles and to compare them with those derived from the Chern-Weil type homomorphism by Kubarski [5]. As a matter of fact, we show that the Chern-Weil type homomorphism by Kubarski does not cover all characteristic classes from the categorical point of view. PubDate: 2015-07-01

Abstract: We extend the Chern character construction of Neshveyev and Tuset to a map whose values lie in Hopf cyclic homology with coefficients, generalizing their definition of K-theory as well. We also introduce the sheaf of equivariant K-theory (with and without coefficients) similar to the equivariant cohomology of Block and Getzler. This construction is much more geometric (it is defined only for the case in which the Hopf algebra and the Hopfmodule algebra are both algebras of functions on some spaces). Thus, we give a geometric definition of the corresponding Chern character, which takes values in a version of Block—Getzler’s sheaf of equivariant cohomology. PubDate: 2015-07-01

Abstract: A fluid flow along a plate with small irregularities on the surface is considered for large Reynolds numbers. The boundary layer has a double-deck structure, i.e., both a thin boundary layer and the classical Prandtl boundary layer are present. It is proved that the solution of the boundary-value problem thus obtained exists and is unique in the Prandtl boundary layer, and the stability of the solution is investigated at large times. The results of numerical modeling are given. Supported by the Basic Research Program of the National Research University “Higher School of Economics.” PubDate: 2015-04-01

Abstract: A periodic system of domains coupled by small windows is considered. In a domain of this kind, we study the band spectrum of a Schrödinger operator subjected to the Neumann condition. We show that, near every isolated eigenvalue of a similar operator in the periodicity cell, there are several nonintersecting bands of the spectrum for the perturbed operator. We also discuss the position of points at which the band functions attain the edges of each band. PubDate: 2015-04-01

Abstract: All nonequivalent integrable evolution equations of the fifth order of the form
\(u_t = D_x \tfrac{{\delta H}}
{{\delta u}}\)
are found. PubDate: 2015-04-01

Abstract: This paper deals with the coarsening operation in dynamical systems where the phase space with a finite invariant measure is partitioned into measurable pieces and the summable function transferred by the phase flow is averaged over these pieces at each instant of time. Letting the time tend to infinity and then refining the partition, we arrive at a modernization of the von Neumann ergodic theorem, which is useful for the purposes of nonequilibrium statistical mechanics. In particular, for fine-grained partitions, we obtain the law of increment of coarse entropy for systems approaching the state of statistical equilibrium. PubDate: 2015-04-01

Abstract: Using the Smirnov-Sobolev approach, we reduce the nonstationary problem of diffraction of a plane wave by an impedance wedge to the Hilbert problem on a half-plane. The index of the Hilbert problem is equal to one. The Hilbert problem is “solved in quadratures.” In another way and in another form, a similar diffraction problem was solved earlier by Popandopulos [J. Aust. Math. Soc. 1 (1), 97–106 (1961)]. PubDate: 2015-04-01

Abstract: Error estimates for homogenization in L
2- and H
1-norms for an equation with rapidly oscillating quasiperiodic coefficients are studied. PubDate: 2015-04-01

Abstract: A singularly perturbed periodic problem for a parabolic reaction-advection-diffusion equation at low advection is studied. The case when there is an internal transition layer under unbalanced nonlinearity is considered. An asymptotic expansion of a solution is constructed. To substantiate the asymptotics thus constructed, the asymptotic method of differential inequalities is used. The Lyapunov asymptotic stability of a periodic solution is studied; the proof uses the Krein-Rutman theorem. PubDate: 2015-04-01

Abstract: Exact solutions of the linear water-wave problem describing oblique water waves trapped by a submerged horizontal cylinder of small (but otherwise arbitrary) cross-section are constructed in the form of a convergent series in powers of the small parameter characterizing the “thinness” of the cylinder. The terms of this series are expressed through the solutions of the exterior Neumann problem for the Laplace equation describing the flow of unbounded fluid past the cylinder. PubDate: 2015-04-01