We study the concentration of measure in metric-measurable (mm)-spaces. define notion locus a flag sequence Some examples infinite group action on an dimensional compact and non-compact manifold show role played by trajectory locus. also provide some applications physics, which emphasize gravitational effects.

In most markets, buyers differ in their ability or willingness to switch supplier. This note proposes a novel industry concentration measure that takes this heterogeneity into account. The index increases the share of captive sales, coincides with Hirschman–Herfindahl Index when none are captive, and ”pure monopoly” value 1 all captive.

Abstract: We give a simple development of the concentration properties of compound Poisson measures on the nonnegative integers. A new modification of the Herbst argument is applied to an appropriate modified logarithmic-Sobolev inequality to derive new concentration bounds. When the measure of interest does not have finite exponential moments, these bounds exhibit optimal polynomial decay. Sim...

Our understanding of geometry relies heavily on the ability we have to draw graphs and reason about them. This is why we have a good intuitive understanding of 2 and 3−dimensional spaces. This is not the case for highdimensional geometry. The observations we make from low-dimensional cases can not be extrapolated to higher dimensions. Concentration of measure, the study of the distribution of m...