Various types of stability are introduced in DEFINITION 2. A one-parameter family of discretizations, {B(k)}, is said to be (1) stable if and only if for any T>0 there exist positive constants C and ko such that ||S(fe)|| g C for all 0<k<k0 and all positive integers n such that nk S T, (2) strictly stable if and only if there exists positive constants k0 and C such that ||S(fe)|| g C for all 0 ...

If X is a separable Banach space, we consider the existence of non-trivial twisted sums 0 → C(K) → Y → X → 0, where K = [0, 1] or ωω . For the case K = [0, 1] we show that there exists a twisted sum whose quotient map is strictly singular if and only if X contains no copy of `1. If K = ωω we prove an analogue of a theorem of Johnson and Zippin (for K = [0, 1]) by showing that all such twisted s...

The k-error linear complexity of a periodic sequence of period N is deened as the smallest linear complexity that can be obtained by changing k or fewer bits of the sequence per period. This paper shows a relationship between the linear complexity and the minimum value k for which the k-error linear complexity is strictly less than the linear complexity.

Let f be a smooth real function with strictly monotone first k derivatives. We show that for finite set A, ?A + A? ?K?A?, $$\left| {{2^k}f(A) - ({2^k} 1)f(A)} \right|{ \gg _k}\,{\left| A \right|^{k 1 o(1)}}/{K^{{O_k}(1)}}.$$ deduce several new sum-product type implications, e.g. A+A being small implies unbounded growth many enough times iterated product ? A.

We prove that some quantized Arnold cat maps are entropic K-systems. This result was formulated by H. Narnhofer [1], but the fact that the optimal decomposition for the multi-channel entropy constructed there is not strictly local was not appropriately taken care of. We propose a strictly local decomposition based on a construction of Voiculescu.

We characterize β-strongly normalizing λ-terms by means of a non-idempotent intersection type system. More precisely, we first define a memory calculus K together with a non-idempotent intersection type system K, and we show that a K-term t is typable in K if and only if t is K-strongly normalizing. We then show that β-strong normalization is equivalent to K-strong normalization. We conclude si...

approximation error A linear operator L H H on a Hilbert space H is said to be self adjoint if for all f g H hLf gi hf Lgi It is said to be positive resp strictly positive if it is self adjoint and for all non trivial f H hLf fi resp hLf fi The next result the Spectral Theorem for compact operators see Section of for a proof will be useful in this and the next chapter Theorem Let L be a compact...

In the Edgeworth non-tatonnement process, trade occurs if there exists some coalition of agents able to make a Paretoimproving trade among themselves at current prices. It is known that the size of such coalitions is bounded by the number of commodities and that, provided all agents always have strictly positive endowments, bilateral trade suffices. These results are generalized: Let there be h...