The condition f(x + 2h) 2f(x + h) + f(x) = o(l) (as h -* 0) at each x is equivalent to continuity for measurable functions. But there is a discontinuous function satisfying 2f(x + 2h) f(x + h) f(x) = o(l) at each x. The question of which generalized Riemann derivatives of order 0 characterize continuity is studied. In particular, a measurable function satisfying EZI1if(x + fih) _ 0 must be a po...

Lemma 8. Given an m × m symmetric matrix A and a target rank k, we let C1 contain the c1 columns of A selected by a column sampling algorithm such that the following inequality holds: ∥∥A− PC1A∥∥2F ≤ f∥∥A−Ak∥∥2F . Then we select c2 = kf −1 columns to construct C2 and c3 = (c1+ c2) −1 columns to construct C3, both using the adaptive sampling according to the residual B1 = A − PC1A and B2 = A − P...

A new family of chiral ferrocenyl P,S-ligands has been developed and successfully applied in a highly endo-selective catalytic asymmetric cycloaddition of azomethine ylides with various enones, including cyclic and acyclic α-enones. For cyclic α-enones, a [Cu(CH(3)CN)(4)]ClO(4)/(R(c),S(Fc))-2f complex catalyzed the cycloaddition to give the sole endo-cycloadducts in perfect enantioselectivities...

In this article, Figure 2F was incorrect. The correct panel is shown below. The authors sincerely apologise for this error.

The tiles of the canonical tilings T ∗(2F ) are six tetrahedra. We determine their inflation rules by the projection method.

Let H be an infinite--dimensional Hilbert space and K(H) be the set of all compact operators on H. We will adopt spectral theorem for compact self-adjoint operators, to investigate of higher derivation and higher Jordan derivation on K(H) associated with the following cauchy-Jencen type functional equation 2f(frac{T+S}{2}+R)=f(T)+f(S)+2f(R) for all T,S,Rin K(H).

Let H be an innite dimensional Hilbert space and K(H) be the set of all compactoperators on H. We will adopt spectral theorem for compact self-adjoint operators, to investigate ofhigher derivation and higher Jordan derivation on K(H) associated with the following Cauchy-Jensentype functional equation 2f((T + S)/2+ R) = f(T ) + f(S) + 2f(R) for all T, S, R are in K(...