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...

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 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 ...

We prove that for every convex body K with the center of mass at the origin and every ε ∈ ( 0, 12 ) , there exists a convex polytope P with at most eO(d)ε− d−1 2 vertices such that (1− ε)K ⊂ P ⊂ K.

Mohammad Zaki, MN State Univ, Mankato We define an algebraic number α as a root of an algebraic equation, a0∗x+a1∗x+· · ·+an−1∗x+an = 0 where ao, a1, · · · , an are rational integers, not all zero. We say α is an algebraic integer if a0 = 0. If an algebraic number α satisfies an algebraic equation of degree n with rational coefficients, and none of lower degree, then we say α is of degree n. If...