Let A be a Banach algebra. A is called ideally amenable if for every closed ideal I of A, the first cohomology group of A with coefficients in I* is trivial. We investigate the closed ideals I for which H1 (A,I* )={0}, whenever A is weakly amenable or a biflat Banach algebra. Also we give some hereditary properties of ideal amenability.

So far, we have been considering point estimation. In this lecture, we will study interval estimation. Let X denote our data. Let θ ∈ R be our parameter of interest. Our task is to construct a data-dependent interval [l(X), r(X)] so that it contains θ with large probability. One possibility is to set l(X) = −∞ and r(X) = +∞. Such an interval will contain θ with probabity 1. Of course, the probl...

Let S be a set of n points in Rd and let t > 1 be a real number. A graph G = (S,E) is called a t-spanner for S, if for any two points p and q in S, the shortest-path distance in G between p and q is at most t|pq|, where |pq| denotes the Euclidean distance between p and q. The graph G is called θ-angle-constrained, if any two distinct edges sharing an endpoint make an angle of at least θ. It is ...

Let X be an observable random variable with unknown distribution function [Formula: see text], [Formula: see text], and let [Formula: see text] We call θ the power of moments of the random variable X. Let [Formula: see text] be a random sample of size n drawn from [Formula: see text]. In this paper we propose the following simple point estimator of θ and investigate its asymptotic properties: [...

Now, let (u, v) = Rα(θ, φ′). Here, Rα = Ry(α). We omit the z rotation since that does not affect Yl0 which has no azimuthal dependence. The vector corresponding to coordinates (u, v) is then given by   sinu cos v sinu sin v cos u   =   cosα 0 sinα 0 1 0 − sinα 0 cosα     sin θ′ cosφ′ sin θ′ sinφ′ cos θ′   =   cosα sin θ′ cosφ′ + sinα cos θ′ sin θ′ sinφ′ cosα cos θ′ + sinα sin θ′ (...

Let σ , θ be commuting involutions of the connected semisimple algebraic group G where σ , θ and G are defined over an algebraically closed field k, char k = 0. Let H := Gσ and K := Gθ be the fixed point groups. We have an action (H × K )×G → G, where ((h, k), g) 7→ hgk−1, h ∈ H , k ∈ K , g ∈ G. Let G/ (H × K ) denote the categorical quotient SpecO(G)H×K . We determine when this quotient is smo...

We compare multivariate aging notions of conditionally independent lifetimes with corresponding univariate aging notions of their one-dimensional marginal distributions. Let G(t|θ) (t ≥ 0) be a one-dimensional survival function indexed by θ; for simplicity’s sake, we assume here that θ is a non-negative scalar quantity; let, furthermore, Π be a probability distribution over [0,∞) and consider t...

let g(v,e) be a graph with p vertices and q edges. a graph g is said to have an odd mean labeling if there exists a function f : v (g) → {0, 1, 2,...,2q - 1} satisfying f is 1 - 1 and the induced map f* : e(g) → {1, 3, 5,...,2q - 1} defined by f*(uv) = (f(u) + f(v))/2 if f(u) + f(v) is evenf*(uv) = (f(u) + f(v) + 1)/2 if f(u) + f(v) is odd is a bijection. a graph that admits an odd mean lab...

and Applied Analysis 3 Let T be a positive number. In the following we take ΩT 0, T × 0, 1 . Let vl and vr be in C0 0, T , R and m0 ∈ L∞ 0, 1 , ρ0 ∈ W1,∞ 0, 1 . We set Γl {t ∈ 0, T | vl t > 0}, Γr {t ∈ 0, T | vr t < 0}. 1.3 In the following, we will always suppose that the sets Pl {t ∈ 0, T | vl t 0}, Pr {t ∈ 0, T | vr t 0} 1.4 have a finite number of connected components. Finally, let ml, ρl ∈...

This supplement contains the proofs of Theorem 2, Propositions 2 and 3, Lemma 1 and Eq.(14). 7 Proof of Theorem 2 Rather than prove Theorem 2 directly it is convenient to first prove an analogue of the Theorem in the context of multivariate Normal random vectors. This is done in section 7.1. The proof of Theorem 2 is then given in section 7.2 7.1 Confidence Bound For Multivariate Normal Vectors...