Let F : X → X be a C 2 loc map in a Banach space X, and A be its Frèchet derivative at the element w := w ε , which solves the problem (*) ˙ w = −A −1 ε (F (w) + εw), w(0) = w 0 , where A ε := A + εI. Assume that A −1 ε ≤ cε −k , 0 < k ≤ 1, 0 < ε > ε 0. Then (*) has a unique global solution, w(t), there exists w(∞), and (* *) F (w(∞)) + εw(∞) = 0. Thus the DSM (Dynamical Systems Method) is just...

We quantify the important effect of strong final state interactions in the weak K → 2π amplitudes, using the measured π-π phase shifts with J = 0 and I = 0, 2. The main results of this analysis, with their implications for ε/ε and the ∆I = 1/2 rule, have been already presented in a previous paper [1]. Here we provide a detailed formal derivation of those results and further discuss the Standard...

Proof of Observation 1. Before the proof, we remind that what we want to prove is the following statement: (∀ε ∈ R) ε > 0⇒ ((∃N ∈ N)(∀n ∈ N) n ≥ N ⇒ |an − x | < ε). To this end, choose arbitrary positive real ε > 0 and pick a positive rational ε′ with 0 < ε′ < ε by utilizing the Archimedean property. Then there exists N (depending on ε′ and hence on ε) such that j, k ≥ N ⇒ |a j − ak | < ε′. Now...

We give an efficient randomized algorithm for approximating an arbitrary element of SU(2) by a product of Clifford+T operators, up to any given error threshold ε > 0. Under a mild hypothesis on the distribution of primes, the algorithm’s expected runtime is polynomial in log(1/ε). If the operator to be approximated is a z-rotation, the resulting gate sequence has T -count K + 4 log 2 (1/ε), whe...

and Applied Analysis 3 where xi k , i 1, 2 stand for the density of the prey and the predator at time k, respectively. ui k , i 1, 2 are the control variables at time k. h2 is a positive constant, denoting the constant of capturing half-saturation. In this paper, we are concerned with the effects of the almost periodicity of ecological and environmental parameters and time delays on the global ...

The objective of this paper is to evaluate the predictive capability of three turbulence models for the simulation of unsteady cavitating flows around a 2D Clark-y hydrofoil. Three turbulence models were standard k-ε model, hybrid model of density correction model (DCM) and filter-based model (FBM) and an improved partially-averaged Navier-Stokes model (PANS) based on k-ε model. Using the above...

We analyze the CP violating ratio ε/ε and rare K and B decays in the MSSM with minimal flavour and CP violation, including NLO QCD corrections and imposing constraints on the supersymmetric parameters coming from ε, B0 d,s − B̄0 d,s mixings, B → Xsγ, ∆̺ in the electroweak precision studies and from the lower bound on the neutral Higgs mass. We provide a compendium of phenomenologically relevant f...

The following is a proof which is independent of this characterisation. First assume that ‖ ‖ is non-archimedean. Let x, y ∈ K. Using that ‖ ‖ extends | | we then obtain |x + y| = ‖x + y‖ ≤ max{‖x‖, ‖y‖} = max{|x|, |y|} which shows that | | is non-archimedean. Now assume that | | is non-archimedean. Let x, y ∈ K̂. Let ε > 0. Since K is dense in K̂ there exist u, v ∈ K such that ‖x − u‖ < ε and ‖y...

Let P be a set of n points in R. In the projective clustering problem, given k, q and norm ρ ∈ [1,∞], we have to compute a set F of k q-dimensional flats such that ( ∑ p∈P d(p,F)) is minimized; here d(p,F) represents the (Euclidean) distance of p to the closest flat in F . We let f k (P, ρ) denote the minimal value and interpret f k (P,∞) to be maxr∈P d(r,F). When ρ = 1, 2 and ∞ and q = 0, the ...