Search results for "26A24"
showing 3 items of 3 documents
On the fractional probabilistic Taylor's and mean value theorems
2016
In order to develop certain fractional probabilistic analogues of Taylor's theorem and mean value theorem, we introduce the nth-order fractional equilibrium distribution in terms of the Weyl fractional integral and investigate its main properties. Specifically, we show a characterization result by which the nth-order fractional equilibrium distribution is identical to the starting distribution if and only if it is exponential. The nth-order fractional equilibrium density is then used to prove a fractional probabilistic Taylor's theorem based on derivatives of Riemann-Liouville type. A fractional analogue of the probabilistic mean value theorem is thus developed for pairs of nonnegative rand…
The forgotten mathematical legacy of Peano
2019
International audience; The formulations that Peano gave to many mathematical notions at the end of the 19th century were so perfect and modern that they have become standard today. A formal language of logic that he created, enabled him to perceive mathematics with great precision and depth. He described mathematics axiomatically basing the reasoning exclusively on logical and set-theoretical primitive terms and properties, which was revolutionary at that time. Yet, numerous Peano’s contributions remain either unremembered or underestimated.
Tangential behavior of functions and conical densities of Hausdorff measures.
2005
We construct a $C^1$-function $f\colon [0,1]\to \mathbb{R}$ such that for almost all $x\in(0,1)$, there is $r>0$ for which $f(y)>f(x)+f'(x)(y-x)$ when $y\in(x,x+r)$ and $f(y)< f(x)+f'(x)(y-x)$ when $y\in(x-r,x)$. The existence of such functions is related to a problem concerning conical density properties of Hausdorff measures on $\mathbb{R}^n$. We also discuss the tangential behavior of typical $C^1$-functions, using an improvement of Jarnik's theorem on essential derived numbers