Let X be a normed linear space, x ∈ X an element of norm one, and ε > 0 and δ(x,ε) the local modulus of convexity of X . We denote by ρ(x,ε) the greatest ρ≥ 0 such that for each closed linear subspace M of X the quotient mapping Q : X → X/M maps the open ε-neighbourhood of x in U onto a set containing the open ρ-neighbourhood of Q(x) in Q(U). It is known that ρ(x,ε) ≥ (2/3)δ(x,ε). We prove that...

The Casimir energy of a dilute dielectric cylinder, with the same light-velocity as in its surrounding medium, is evaluated exactly to first order in ξ 2 = ε 1 − ε 2 ε 1 + ε 2 2 (where ε 1 , ε 2 are the dielectric constants of the cylinder and of its environment), and numerically to higher orders in ξ 2. The first part is carried out using addition formulas for Bessel functions, and no Debye ex...

The direct CP violation parameter Re(ε′/ε) has been measured from the decay rates of neutral kaons into two pions using the NA48 detector at the CERN SPS. With 3.3 million KL → ππ events collected during the 1998 and 1999 running periods, a result of Re(ε′/ε) = (15.0 ± 2.7)×10−4 has been obtained. The result combined with the published 1997 sample is Re(ε′/ε) = (15.3± 2.6)×10−4.

We analyze the CP violating ratio ε/ε in the Standard Model in view of the new KTeV results. We review the present status of the most important non-perturbative parameters B (1/2) 6 , B (3/2) 8 , B̂K and of the strange quark mass ms. We also briefly discuss the issues of final state interactions and renormalization scheme dependence. Updating the values of the CKM parameters, of mt and Λ (4) MS ...

In the present paper we introduce a generalization of complete invariance property (CIP) for metric spaces, which will call ε-approximated (ε-ACIP). For our goals, use so called degree nondensifiability (DND) which, roughly speaking, measures (in specified sense) distance from bounded space to its class Peano continua. Our main result relates ε-ACIP with DND and, in particular, proves that dens...

Denote by T (X) the semigroup of full transformations on a set X. For ε ∈ T (X), the centralizer of ε is a subsemigroup of T (X) defined by C(ε) = {α ∈ T (X) : αε = εα}. It is well known that C(idX) = T (X) is a regular semigroup. By a theorem proved by J.M. Howie in 1966, we know that if X is finite, then the subsemigroup generated by the idempotents of C(idX) contains all non-invertible trans...

and Applied Analysis 3 Proposition 3. For any constant R ≥ 0 there exists a positive constant K = K(R) such that, for any initial data u 1 (0), u 2 (0) with ‖u i (0)‖ Hη,ε ≤ R, i = 1, 2 one has 󵄩󵄩󵄩󵄩 S ε,η (t)u 1 (0) − S ε,η (t)u 2 (0) 󵄩󵄩󵄩󵄩Hη,ε ≤ e (K 2 /ε 2 )t󵄩󵄩󵄩󵄩u1 (0) − u2 (0) 󵄩󵄩󵄩Hη,ε , (20) where S ε,η (t) is the solution semigroup of the problem (1). Proof. Let u 1 , u 2 , two solutions of ...

We study the set of solutions of the complex Ginzburg-Landau equation in R, d < 3. We consider the global attracting set (i.e., the forward map of the set of bounded initial data), and restrict it to a cube Q L of side L. We cover this set by a (minimal) number N QL (ε) of balls of radius ε in L∞(Q L ). We show that the Kolmogorov ε-entropy per unit length, H ε = lim L→∞ L −d log N QL (ε) exist...

ε-support vector regression (ε-SVR) can be converted into an unconstrained convex and non-smooth quadratic programming problem. It is not solved by the traditional algorithm. In order to solve this non-smooth problem, a class of piecewise smooth functions is introduced to approximate the ε-insensitive loss function of ε-SVR, which generates a ε-piecewise smooth support vector regression (ε-dPWS...