and Applied Analysis 3 For any Banach space B, the space Lp(0, T;B) consists of all strongly measurable functions u : [0, T] → B equipped with the norm ‖u‖ L p (0,T;B) := (∫ T 0 ‖u (t)‖PBdt) 1/p < ∞ (12) for 1 ≤ p < ∞, and ‖u‖ L ∞ (0,T;B) = ess sup t∈[0,T] ‖u(t)‖B < ∞. (13) And the space C([0, T];B) denotes the set of continuous functions u : [0, T] → B with ‖u‖ C([0,T];B) = max t∈[0,T] ‖u(t)‖B...

We characterize quantum limits and semi-classical measures corresponding to sequences of eigenfunctions for systems coupled harmonic oscillators with arbitrary frequencies. The structure the set turns out depend strongly on arithmetic relations between frequencies each decoupled oscillator. In particular, we show that as soon these are not rational multiples a fixed fundamental frequency, is co...

We study the formation and dynamical evolution of clusters with multiple stellar generations. Observational studies have found that some globular clusters host a population of second generation (SG) stars which show chemical anomalies and must have formed from gas containing matter processed in the envelopes of first generation (FG) cluster stars. We study the SG formation process by means of 1...

let r be a ring and v be a matrix valuation on r. it is shown that, there exists a correspondence between matrix valuations on r and some special subsets ?(mvpr) of the set of all square matrices over r, analogous to the correspondence between invariant valuation rings and abelian valuation functions on a division ring. furthermore, based on malcolmson’s localization, an alternative proof for t...