Slowly oscillating continuity.
We characterise the set on which an infinitely differentiable function can be locally polynomial.
Mathematics Subject Classification: 26A33, 34A25, 45D05, 45E10We consider ordinary fractional differential equations with Caputo-type differential operators with smooth right-hand sides. In various places in the literature one can find the statement that such equations cannot have smooth solutions. We prove that this is wrong, and we give a full characterization of the situations where smooth solutions exist. The results can be extended to a class of weakly singular Volterra integral equations.
We give a characterization of those probability measures on the real line which satisfy certain Sobolev inequalities. Our starting point is a simpler approach to the Bobkov-Götze characterization of measures satisfying a logarithmic Sobolev inequality. As an application of the criterion we present a soft proof of the Latała-Oleszkiewicz inequality for exponential measures, and describe the measures on the line which have the same property. New concentration inequalities for product measures follow....
We define a Sobolev space by means of a generalized Poincaré inequality and relate it to a corresponding space based on upper gradients.
In this note we prove that the unique concave t-norm is Minimum and, among the class of triangular functions that have the family of unit step-functions as idempotent elements, the unique concave triangular function is piM.