G\^ ateaux and Hadamard differentiability via directional differentiability

Let $X$ be a separable Banach space, $Y$ a Banach space and $f: X \to Y$ an arbitrary mapping. Then the following implication holds at each point $x \in X$ except a $\sigma$-directionally porous set: If the one-sided Hadamard directional derivative $f'_{H+}(x,u)$ exists in all directions $u$ from a set $S_x \subset X$ whose linear span is dense in $X$, then $f$ is Hadamard differentiable at $x$. This theorem improves and generalizes a recent result of A.D. Ioffe, in which the linear span of $S_x$ equals $X$ and $Y = \R$. An analogous theorem, in which $f$ is pointwise Lipschitz, and which deals with the usual one-sided derivatives and G\^ ateaux differentiability is also proved. It generalizes a result of D. Preiss and the author, in which $f$ is supposed to be Lipschitz

Properties of Hadamard directional derivatives: Denjoy-Young-Saks theorem for functions on Banach spaces

The classical Denjoy-Young-Saks theorem on Dini derivatives of arbitrary functions $f: \R \to \R$ was extended by U.S. Haslam-Jones (1932) and A.J. Ward (1935) to arbitrary functions on $\R^2$. This extension gives the strongest relation among upper and lower Hadamard directional derivatives $f^+_H (x,v)$, $f^-_H (x,v)$ ($v \in X$) which holds almost everywhere for an arbitrary function $f:\R^2\to \R$. Our main result extends the theorem of Haslam-Jones and Ward to functions on separable Banach spaces

Around the Fed: Safety first?

Related link(s): http://www.richmondfed.org/publications/research/region_focus/2009/summer/around_the_fed_weblinks.cfmEconomics

Curves in Banach spaces which allow a C2C^2 parametrization

We give a complete characterization of those $f: [0,1] \to X$ (where $X$ is a Banach space which admits an equivalent Fr\'echet smooth norm) which allow an equivalent $C^2$ parametrization. For $X=\R$, a characterization is well-known. However, even in the case $X=\R^2$, several quite new ideas are needed. Moreover, the very close case of parametrizations with a bounded second derivative is solved.Comment: 22 pages; We split the original paper into two parts. This is the first part ($C^2$ paths), the second part (paths with finite convexity) will appear late

Advancing immunity : what is the role for policy in the private decision to vaccinate children?

Related links: http://www.richmondfed.org/publications/research/region_focus/2010/q2/feature2_weblinks.cfmPublic health - Economic aspects