Related papers: Chain rule for pointwise Lipschitz mappings

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…

We show that derivations of the differential structure of a subcartesian space satisfy the chain rule and have maximal integral curves.

The cutoff phenomenon was recently shown to systematically follow from non-negative curvature and the product condition, for all Markov diffusions. The proof crucially relied on a classical \emph{chain rule} satisfied by the carr\'e du…

Many modern numerical methods in computational science and engineering rely on derivatives of mathematical models for the phenomena under investigation. The computation of these derivatives often represents the bottleneck in terms of…

We give an alternative proof of the general chain rule for functions of bounded variation ([ADM90]), which allows to compute the distributional differential of $\varphi\circ F$, where $\varphi\in \mathrm{LIP}(\mathbb{R}^m)$ and…

Rademacher theorem states that every Lipschitz function on the Euclidean space is differentiable almost everywhere, where "almost everywhere" refers to the Lebesgue measure. In this paper we prove a differentiability result of similar type,…

We define the tangential derivative, a notion of directional derivative which is invariant under diffeomorphisms. In particular this derivative is invariant under changes of chart and is thus well-defined for functions defined on a…

Starting from the Riemann-Liouville derivative, many authors have built their own notion of fractional derivative in order to avoid some classical difficulties like a non zero derivative for a constant function or a rather complicated…

We prove the equivalence of two seemingly very different ways of generalising Rademacher's theorem to metric measure spaces. One such generalisation is based upon the notion of forming partial derivatives along a very rich structure of…

We establish the Fa\`a di Bruno formula, in the sense of almost everywhere equality, for derivatives of the composed function $f \circ g$, for all function $f : R \rightarrow R$ such that $f$ acts on $W^m_p(R^n)$ by composition, and all $g…

One shortcoming of the chain rule is that it does not iterate: it gives the derivative of f(g(x)), but not (directly) the second or higher-order derivatives. We present iterated differentials and a version of the multivariable chain rule…

Let $X$ be a perfect, compact subset of the complex plane, and let $D^{(1)}(X)$ denote the (complex) algebra of continuously complex-differentiable functions on $X$. Then $D^{(1)}(X)$ is a normed algebra of functions but, in some cases,…

We show that no matter what subset of a normed space is given, a typical 1-Lipschitz mapping into a Banach space is non-differentiable at a typical point of the set in a very strong sense: the derivative ratio approximates, on arbitrary…

For Lipschitz maps between a metric measure space and a metric space, combining the ideas of Kirchheim's metric differentiability and Cheeger's differentiable structures leads to a Rademacher-type theorem for a notion of metric…

The work considers a system of fractional order partial differential equations. The existence and uniqueness theorems for the classical solution of initial-boundary value problems are proved in two cases: 1) the right-hand side of the…

In this paper, we study equality-type Clarke subdifferential chain rules of matrix factorization and factorization machine. Specifically, we show for these problems that provided the latent dimension is larger than some multiple of the…

This paper addresses the study of novel constructions of variational analysis and generalized differentiation that are appropriate for characterizing robust stability properties of constrained set-valued mappings/multifunctions between…

Let ${\mathfrak M}=({\mathcal M},\rho)$ be a metric space and let $X$ be a Banach space. Let $F$ be a set-valued mapping from ${\mathcal M}$ into the family ${\mathcal K}_m(X)$ of all compact convex subsets of $X$ of dimension at most $m$.…

In the United States, regions are frequently divided into districts for the purpose of electing representatives. How the districts are drawn can affect who's elected, and drawing districts to give an advantage to a certain group is known as…

Rademacher's Theorem can be interpreted as an almost-everywhere \emph{little-$o$ improvement principle}: if a function admits a uniform pointwise first-order Lipschitz control at every point, then this control improves to a vanishing one at…