We shall introduce a p-local geometry Δp(G) for the pair (G, p) of a finite group G and a prime divisor p of the order of G, which is constructed by ‘maximal parabolic like’ subgroups of G. Under the natural hypothesis, Δp(G) behaves very much like the building associated with a group of Lie type in characteristic p. We shall also show that, under some hypothesis, Δp(G) is homotopy equivalent to the subgroup complex [Bscr ]cenp(G) which is a more essential part of the p-radical complex [Bscr ]p(G). Some of the p-local sporadic geometries can be understood well in our system.

Let G be a compact, simply-connected, simple Lie group and T ⊂ G a maximal torus. The purpose of this paper is to study the connection between various fibrations over BG (where G is a compact, simply-connected, simple Lie group) associated to the adjoint representation and homotopy colimits over poset categories [Cscr ], hocolim[Cscr ]BGI where GI are certain connected maximal rank subgroups of G.

Let G be a finite group; by an algebraic 2-complex over G we mean an exact sequence of Z[G]-modules of the form:

E = (0 → J → E2 → E1 → E0 → Z → 0)

where Er is finitely generated free over Z[G] for 0 [les ] r [les ] 2. The module J is determined up to stability by the fact of appearing in such an exact sequence; we denote its stable class by Ω3(Z); E is said to be minimal when rkZ(J) attains the minimum possible value within Ω3(Z).

Suppose that (X,ω) is a symplectic manifold and that there exists a Liouville vector field V defined in a neighbourhood of and transverse to M = ∂X. Then V induces a contact form α = ιVω[mid ]M on M which determines the germ of ω along M. One should think of the contact manifold (M,ξ = ker α) as controlling the behaviour of ω ‘at infinity’. If V points out of X along M then we call (X,ω) a convex filling of (M,ξ), and if V points into X along M then we call (X,ω) a concave filling of (M,ξ).

Using twistor techniques we shall show that there is a hypercomplex structure in the neighbourhood of the zero section of the tangent bundle TX of any complex manifold X with a real-analytic torsion-free connection compatible with the complex structure whose curvature is of type (1, 1). The zero section is totally geodesic and the Obata connection restricts to the given connection on the zero section.

We also prove an analogous result for vector bundles: any vector bundle with real-analytic connection whose curvature is of type (1, 1) over X can be extended to a hyperholomorphic bundle over a neighbourhood of the zero section of TX.

Let Si: ℝd → ℝd for i = 1, …, n be contracting similarities, and let (p1, …, pn) be a probability vector. Let K and μ be the self-similar set and the self-similar measure associated with (Si,pi)i. For q ∈ ℝ and r > 0, define the qth covering moment and the qth packing moment of μ by

[formula here]

where the infimum is taken over all r-spanning subsets E of K, and the supremum is taken over all r-separated subsets F of K. If the Open Set Condition (OSC) is satisfied then it is well known that

[formula here]

where β(q) is defined by [sum ]ipqirβi(q) = 1 (here ri denotes the Lipschitz constant of Si). Assuming the OSC, we determine the exact rate of convergence in (*): there exist multiplicatively periodic functions πq, Πq: (0,∞) → ℝ such that

[formula here]

where ε(r) → 0 as r[searr ]0. As an application of (**) we show that the empirical multi-fractal moment measures converges weakly:

[formula here]

where, for each positive r, Er is a (suitable) minimal r-spanning subset of K and Fr is a (suitable) maximal r-separated subset of K, and [Hscr ]q,β(q)μ and [Pscr ]q,β(q)μ are the multifractal Hausdorff measure and the multifractal packing measure, respectively.

Let {W(t):t [ges ] 0} denote a Wiener process, and set [Sscr ] for the unit ball of the reproducing kernel Hilbert space pertaining to the restriction of W on [0,1], with Hilbert norm [mid ] · [mid ]H. Gorn and Lifshits [8] have shown that, whenever f ∈ [Sscr ] fulfills [mid ] f [mid ]H = 1 and has Lebesgue derivative of bounded variation, the rate of clustering of (2h log(1/h))−½(W(t + h·) − W(t)) to f is of the order O((log(1/h))−2/3. In this paper, we show that the set of exceptional points in [0,1] where this rate is reached constitutes a random fractal whose Hausdorff–Besicovitch measure is evaluated.

In this work we generalize Markov's inequality for any derivative of a polynomial on a real Hilbert space and provide estimates for the second and third derivatives of a polynomial on a real Banach space. Our result on a real Hilbert space answers a question raised by L. A. Harris in his commentary on problem 74 in the Scottish Book [20]. We also provide generalizations of previously obtained inequalities of the Bernstein and Markov-type for polynomials with curved majorants on a real Hilbert space.

We present, starting from a set of canonical axioms, a complete classification of the notions of non-commutative stochastic independence. Our result originates from a first contribution and a conjecture by M. Schürmann and is based on a fundamental paper by R. Speicher.