Throughout we let Γ be a discrete group. For f : Γ → C and each s ∈ Γ we define the left translation action by (s.f)(t) = f(s−1t). Definition 1.1. A group Γ is amenable is there exists a state μ on l∞(Γ) which is invariant under the left translation action: for all s ∈ Γ and f ∈ l∞(Γ), μ(s.f) = μ(f). Example 1.2. Finite groups are amenable: take the state which sends χ{s} to 1 |Γ| for each s ∈ ...

A novel framework for describing suffusion in cohesionless soil, incorporating ideally gap-graded is presented this paper. The key assumption of the proposed simulation that an erodible particle flow induced primarily by drag force. multiphase seepage-soil phenomena conducted based on framework. validity method checked through a past laboratory experiments, which variation grain size distributi...

Let $ (A,| cdot |) $ be a real Banach algebra, a complex algebra $ A_mathbb{C} $ be a complexification of $ A $ and $ | | cdot | | $ be an algebra norm on  $ A_mathbb{C}  $  satisfying a simple condition together with the norm $ | cdot | $ on $ A$.  In this paper we first show that $ A^* $ is a real Banach $ A^{**}$-module if and only if $ (A_mathbb{C})^* $ is a complex Banach $ (A_mathbb{C})^{...

In multi-label classification, where the evaluation of predictions is less straightforward than in single-label various meaningful, though different, loss functions have been proposed. Ideally, learning algorithm should be customizable towards a specific choice performance measure. Modern implementations boosting, most prominently gradient boosted decision trees, appear to appealing from this p...

An overexpanded jet in a truncated ideally contoured nozzle is found to feature tonal behavior. The flow field investigated understand its origin and show how it modifies side-load properties. temporal spatial organization of wall pressure velocity are first experimentally characterized based on synchronized acquisition both wall-pressure along rings probes located within the high-rate time-res...

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There are two notions of amenability for discrete equivalence relations. The \global" amenability (which is usually referred to just as \amenability") is the property of existence of leafwise invariant means, which, by a theorem of Connes{Feldman{Weiss, is equivalent to hyperrniteness, or, to being the orbit equivalence relation of a Z-action. The notion of \local" amenability applies to equiva...