Some analytical properties of γ-convex functions on the real line

This paper deals with the analytical properties of yt4157u005882lq7/xxlarge947.gif" alt="gamma" align="MIDDLE" BORDER="0">-convex functions, which are defined as those functions satisfying the inequalityf(x ₁ ^{yt4157u005882lq7/xxlarge8242.gif" alt="prime" align="BASELINE" BORDER="0">} )+f(x ₂ ^{yt4157u005882lq7/xxlarge8242.gif" alt="prime" align="BASELINE" BORDER="0">} )yt4157u005882lq7/xxlarge8804.gif" alt="le" align="MIDDLE" BORDER="0">f(x ₁)+f(x ₂), forx _i ^{yt4157u005882lq7/xxlarge8242.gif" alt="prime" align="BASELINE" BORDER="0">} yt4157u005882lq7/xxlarge8712.gif" alt="isin" align="MIDDLE" BORDER="0">[x ₁,x ₂], |x _i –x _i ^{yt4157u005882lq7/xxlarge8242.gif" alt="prime" align="BASELINE" BORDER="0">} |=yt4157u005882lq7/xxlarge947.gif" alt="gamma" align="MIDDLE" BORDER="0">, i=1,2, whenever |x ₁–x ₂|>yt4157u005882lq7/xxlarge947.gif" alt="gamma" align="MIDDLE" BORDER="0">, for some given positive yt4157u005882lq7/xxlarge947.gif" alt="gamma" align="MIDDLE" BORDER="0">. This class contains all convex functions and all periodic functions with period yt4157u005882lq7/xxlarge947.gif" alt="gamma" align="MIDDLE" BORDER="0">. In general, yt4157u005882lq7/xxlarge947.gif" alt="gamma" align="MIDDLE" BORDER="0">-convex functions do not have ideal properties as convex functions. For instance, there exist yt4157u005882lq7/xxlarge947.gif" alt="gamma" align="MIDDLE" BORDER="0">-convex functions which are totally discontinuous or not locally bounded. But yt4157u005882lq7/xxlarge947.gif" alt="gamma" align="MIDDLE" BORDER="0">-convex functions possess so-called conservation properties, meaning good properties which remain true on every bounded interval or even on the entire domain, if only they hold true on an arbitrary closed interval with length yt4157u005882lq7/xxlarge947.gif" alt="gamma" align="MIDDLE" BORDER="0">. It is shown that boundedness, bounded variation, integrability, continuity, and differentiability almost everywhere are conservation properties of yt4157u005882lq7/xxlarge947.gif" alt="gamma" align="MIDDLE" BORDER="0">-convex functions on the real line. However, yt4157u005882lq7/xxlarge947.gif" alt="gamma" align="MIDDLE" BORDER="0">-convex functions have also infection properties, meaning bad properties which propagate to other points, once they appear somewhere (for example, discontinuity). Some equivalent properties of yt4157u005882lq7/xxlarge947.gif" alt="gamma" align="MIDDLE" BORDER="0">-convexity are given. Ways for generating and representing yt4157u005882lq7/xxlarge947.gif" alt="gamma" align="MIDDLE" BORDER="0">-convex functions are described.This research was supported by the Deutsche Forschungsgemeinschaft. The first author thanks Prof. Dr. E. Zeidler and Prof. Dr. H. G. Bock for their hospitality and valuable support.     

Microfoam formation in a capillary

The ultrasound-induced formation of bubble clusters may be of interest as a therapeutic means. If the clusters behave as one entity, i.e., one mega-bubble, its ultrasonic manipulation towards a boundary is straightforward and quick. If the clusters can be forced to accumulate to a microfoam, entire vessels might be blocked on purpose using an ultrasound contrast agent and a sound source.In this paper, we analyse how ultrasound contrast agent clusters are formed in a capillary and what happens to the clusters if sonication is continued, using continuous driving frequencies in the range 1-10 MHz. Furthermore, we show high-speed camera footage of microbubble clustering phenomena.We observed the following stages of microfoam formation within a dense population of microbubbles before ultrasound arrival. After the sonication started, contrast microbubbles collided, forming small clusters, owing to secondary radiation forces. These clusters coalesced within the space of a quarter of the ultrasonic wavelength, owing to primary radiation forces. The resulting microfoams translated in the direction of the ultrasound field, hitting the capillary wall, also owing to primary radiation forces.We have demonstrated that as soon as the bubble clusters are formed and as long as they are in the sound field, they behave as one entity. At our acoustic settings, it takes seconds to force the bubble clusters to positions approximately a quarter wavelength apart. It also just takes seconds to drive the clusters towards the capillary wall.Subjecting an ultrasound contrast agent of given concentration to a continuous low-amplitude signal makes it cluster to a microfoam of known position and known size, allowing for sonic manipulation.