In this note we show that for analytic semigroups the so-called Weiss condition of uniform boundedness of the operators Re(λ) 1/2C(λ+A)−1, Re(λ) > 0 on the complex right half plane and weak Lebesgue L2,∞–admissibility are equivalent. Moreover, we show that the weak Lebesgue norm is best possible in the sense that it is the endpoint for the ’Weiss conjecture’ within the scale of Lorentz spaces L...

• Banach-Alaoglu: compactness of polars • Variant Banach-Steinhaus/uniform boundedness • Second polars • Weak boundedness implies boundedness • Weak-to-strong differentiability The comparison of weak and strong differentiability is due to Grothendieck, although the original sources are not widely available.

and Applied Analysis 3 2. Results Solutions to 1.1 subject to conditions 1.2 are meant in the following weak sense. Definition 2.1. A function u ∈ Lmloc Q ∩ L p loc Q is called a weak solution to 1.1 if ∫ Q |u|φ dP ∫ S u 0, t2;x φ 0, t2;x dP2 ∫ S u t1, 0;x φ t1, 0;x dP1 − ∫

Reliability and validity are crucial issues in research. Reliability is concerned with lowered quality of data due to random errors in measurement. Validity is concerned with whether variables measure what they purport to measure or are biased by other systemic factors, excluding random errors. Strength of correlation is limited by reliability. Given the problems in measuring reliability and va...

In the preceding chapters, the effects of lensing were so strong as to leave an unmistakable imprint on a specific source, allowing a detailed treatment. However, only the densest regions of the universe are able to provide such a spectacular lensing effect. To study more representative regions of the universe, we must examine large numbers of sources statistically. This is the domain of weak l...

This paper is devoted to clarify the algorithmic definition of the weak stability boundary in the framework of the planar Restricted Three Body Problem. The role of the invariant hyperbolic manifolds associated to the central manifolds of the libration points L1 and L2, as boundary of the weak stability region, is shown.

We explain how any cofibrantly generated weak factorisation system on a category may be equipped with a universally and canonically determined choice of cofibrant replacement. We then apply this to the theory of weak ω-categories, showing that the universal and canonical cofibrant replacement of the operad for strict ω-categories is precisely Leinster’s operad for weak ω-categories.

Double no-touch options, contracts which pay out a fixed amount provided an underlying asset remains within a given interval, are commonly traded, particularly in FX markets. In this work, we establish model-free bounds on the price of these options based on the prices of more liquidly traded options (call and digital call options). Key steps are the construction of superand sub-hedging strateg...