In a recent paper Cohen, Liu and Yu introduce the Brauer algebra of type C. We show that this algebra is an iterated inflation of hyperoctahedral groups, and that it is cellularly stratified. This allows us to give an indexing set of the standard modules, results on decomposition numbers, and the conditions under which the algebra is quasi-hereditary.

Let (, d, μ) be a metric measure space and satisfy the so-called upper doubling condition and the geometrical doubling condition. We introduce the atomic Hardy space H1(μ) and prove that its dual space is the known space RBMO(μ) in this context. Using this duality, we establish a criterion for the boundedness of linear operators from H1(μ) to any Banach space. As an application of this criterion, we obtain the boundedness of Calderón–Zygmund operators from H1(μ) to L1(μ).

We consider the Schrödinger operator α on the half-line with a periodic background potential and the Wigner–von Neumann potential of Coulomb type: csin(2ωx + δ)/(x + 1). It is known that the continuous spectrum of the operator α has the same band-gap structure as the free periodic operator, whereas in each band of the absolutely continuous spectrum there exist two points (so-called critical or resonance) where the operator α has a subordinate solution, which can be either an eigenvalue or a “half-bound” state. The phenomenon of an embedded eigenvalue is unstable under the change of the boundary condition as well as under the local change of the potential, in other words, it is not generic. We prove that in the general case the spectral density of the operator α has power-like zeroes at critical points (i.e., the absolutely continuous spectrum has pseudogaps). This phenomenon is stable in the above-mentioned sense.

Let h: 2 → 2 be a dissipative and orientation preserving homeomorphism having an asymptotically stable fixed point. Let U be the region of attraction and assume that it is proper and unbounded. Using Carathéodory's prime ends theory one can associate a rotation number, ρ, to h|U. We prove that any map in the above conditions and with ρ ∉ induces a Denjoy homeomorphism in the circle of prime ends. We also present some explicit examples of maps in this class and we show that, if the infinity point is accessible by an arc in U, ρ ∉ if and only if Per(h) = Fix(h) = {p}.

Klyachko, in 1974, considered the tensor and Lie powers of the natural module for the general linear group over a field of characteristic 0 and showed that nearly all of the irreducible submodules of the rth tensor power also occur up to isomorphism as submodules of the rth Lie power. Here we prove an analogue for infinite fields of prime characteristic by showing, with some restrictions on r, that nearly all of the indecomposable direct summands of the rth tensor power also occur up to isomorphism as summands of the rth Lie power.

We give examples of isospectral non-isometric surfaces of genus two and three with variable curvature, as well as hyperbolic orbifolds of genus two. We apply the first result to construct isospectral potentials on a Riemann surface of genus two.

We construct triplets of killed Brownian motions to obtain the Brunn–Minkowski inequalities concerning the solutions of the equation (1/2)Δψ − h ψ = g on three interrelated compact sets in Euclidean space. These, in particular, include inequalities relating to the solutions of the Schrödinger equation and the Poisson equation on the three compact convex sets and an inequality relating to harmonic functions.

Recently, Dasbach, Futer, Kalfagianni, Lin and Stoltzfus extended the notion of a Tait graph by associating a set of ribbon graphs (or, equivalently, cellularly embedded graphs) to a link diagram. Here we focus on Seifert graphs, which are the ribbon graphs of a knot or link diagram that arise from Seifert states. We provide a characterization of Seifert graphs in terms of Eulerian subgraphs. This characterization can be viewed as a refinement of the fact that Seifert graphs are bipartite. We go on to examine the family of ribbon graphs that arises by forming the parallels of a link diagram and determine how the genus of the ribbon graph of a r-fold parallel of a link diagram is related to that of the original link diagram.

Let q ≥ 3, 2 ≤ r ≤ φ(q) and a1, . . ., ar be distinct residue classes modulo q that are relatively prime to q. Assuming the Generalized Riemann Hypothesis (GRH) and the Linear Independence Hypothesis (LI), M. Rubinstein and P. Sarnak [11] showed that the vector-valued function Eq;a1, . . ., ar(x) = (E(x;q,a1), . . ., E(x;q,ar)), where , has a limiting distribution μq;a1, . . ., ar which is absolutely continuous on . Furthermore, they proved that for r fixed, μq;a1, . . ., ar tends to a multidimensional Gaussian as q → ∞. In the present paper, we determine the exact rate of this convergence, and investigate the asymptotic behavior of the large deviations of μq;a1, . . ., ar.

Let E be a second-countable, locally compact, Hausdorff groupoid equipped with an action of such that G: = E/ is a principal groupoid with Haar system λ. The twisted groupoid C*-algebra C*(E; G, λ) is a quotient of the C*-algebra of E obtained by completing the space of -equivariant functions on E. We show that C*(E; G, λ) is postliminal if and only if the orbit space of G is T0 and that C*(E; G, λ) is liminal if and only if the orbit space is T1. We also show that C*(E; G, λ) has bounded trace if and only if G is integrable and that C*(E; G, λ) is a Fell algebra if and only if G is Cartan.

Let be a second-countable, locally compact, Hausdorff groupoid with Haar system λ and continuously varying, abelian isotropy groups. Let be the isotropy groupoid and : = /. Using the results about twisted groupoid C*-algebras, we show that the C*-algebra C*(, λ) has bounded trace if and only if is integrable and that C*(, λ) is a Fell algebra if and only if is Cartan. We illustrate our theorems with examples of groupoids associated to directed graphs.