McShane-Whitney extensions in constructive analysis

Within Bishop-style constructive mathematics we study the classical McShane-Whitney theorem on the extendability of real-valued Lipschitz functions defined on a subset of a metric space. Using a formulation similar to the formulation of McShane-Whitney theorem, we show that the Lipschitz real-valued functions on a totally bounded space are uniformly dense in the set of uniformly continuous functions. Through the introduced notion of a McShane-Whitney pair we describe the constructive content of the original McShane-Whitney extension and examine how the properties of a Lipschitz function defined on the subspace of the pair extend to its McShane-Whitney extensions on the space of the pair. Similar McShane-Whitney pairs and extensions are established for H\"{o}lder functions and $\nu$-continuous functions, where $\nu$ is a modulus of continuity. A Lipschitz version of a fundamental corollary of the Hahn-Banach theorem, and the approximate McShane-Whitney theorem are shown

Connected Choice and the Brouwer Fixed Point Theorem

We study the computational content of the Brouwer Fixed Point Theorem in the Weihrauch lattice. Connected choice is the operation that finds a point in a non-empty connected closed set given by negative information. One of our main results is that for any fixed dimension the Brouwer Fixed Point Theorem of that dimension is computably equivalent to connected choice of the Euclidean unit cube of the same dimension. Another main result is that connected choice is complete for dimension greater than or equal to two in the sense that it is computably equivalent to Weak K\H{o}nig's Lemma. While we can present two independent proofs for dimension three and upwards that are either based on a simple geometric construction or a combinatorial argument, the proof for dimension two is based on a more involved inverse limit construction. The connected choice operation in dimension one is known to be equivalent to the Intermediate Value Theorem; we prove that this problem is not idempotent in contrast to the case of dimension two and upwards. We also prove that Lipschitz continuity with Lipschitz constants strictly larger than one does not simplify finding fixed points. Finally, we prove that finding a connectedness component of a closed subset of the Euclidean unit cube of any dimension greater or equal to one is equivalent to Weak K\H{o}nig's Lemma. In order to describe these results, we introduce a representation of closed subsets of the unit cube by trees of rational complexes.Comment: 36 page

Mutual Dimension

We define the lower and upper mutual dimensions $mdim(x:y)$ and $Mdim(x:y)$ between any two points $x$ and $y$ in Euclidean space. Intuitively these are the lower and upper densities of the algorithmic information shared by $x$ and $y$. We show that these quantities satisfy the main desiderata for a satisfactory measure of mutual algorithmic information. Our main theorem, the data processing inequality for mutual dimension, says that, if $f:\mathbb{R}^m \rightarrow \mathbb{R}^n$ is computable and Lipschitz, then the inequalities $mdim(f(x):y) \leq mdim(x:y)$ and $Mdim(f(x):y) \leq Mdim(x:y)$ hold for all $x \in \mathbb{R}^m$ and $y \in \mathbb{R}^t$. We use this inequality and related inequalities that we prove in like fashion to establish conditions under which various classes of computable functions on Euclidean space preserve or otherwise transform mutual dimensions between points.Comment: This article is 29 pages and has been submitted to ACM Transactions on Computation Theory. A preliminary version of part of this material was reported at the 2013 Symposium on Theoretical Aspects of Computer Science in Kiel, German

Unified treatment of fractional integral inequalities via linear functionals

In the paper we prove several inequalities involving two isotonic linear functionals. We consider inequalities for functions with variable bounds, for Lipschitz and H\" older type functions etc. These results give us an elegant method for obtaining a number of inequalities for various kinds of fractional integral operators such as for the Riemann-Liouville fractional integral operator, the Hadamard fractional integral operator, fractional hyperqeometric integral and corresponding q-integrals

On the structure of ∞\infty-Harmonic maps

Let $H \in C^2(\mathbb{R}^{N \times n})$, $H\geq 0$. The PDE system \label{1} A_\infty u \, :=\, \Big(H_P \otimes H_P + H [H_P]^\bot H_{PP} \Big)(Du) : D^2 u\, = \, 0 \tag{1} arises as the ``Euler-Lagrange PDE" of vectorial variational problems for the functional $E_{\infty}(u,\Omega) = \| H(Du) \|_{L^\infty(\Omega)}$ defined on maps $u : \Omega \subseteq \mathbb{R}^n \longrightarrow \mathbb{R}^N$. \eqref{1} first appeared in the author's recent work \cite{K3}. The scalar case though has a long history initiated by Aronsson in \cite{A1}. Herein we study the solutions of \eqref{1} with emphasis on the case of $n=2\leq N$ with $H$ the Euclidean norm on $\mathbb{R}^{N \times n}$, which we call the ``$\infty$-Laplacian". By establishing a rigidity theorem for rank-one maps of independent interest, we analyse a phenomenon of separation of the solutions to phases with qualitatively different behaviour. As a corollary, we extend to $N \geq 2$ the Aronsson-Evans-Yu theorem regarding non-existence of zeros of $|Du|$ and prove a Maximum Principle. We further characterise all $H$ for which \eqref{1} is elliptic and also study the initial value problem for the ODE system arising for $n=1$ but with $H(\cdot,u,u')$ depending on all the arguments.Comment: 30 pages, 10 figures, revised including referees' comments, (Communications in PDE

A derivative for complex Lipschitz maps with generalised Cauchy–Riemann equations

AbstractWe introduce the Lipschitz derivative or the L-derivative of a locally Lipschitz complex map: it is a Scott continuous, compact and convex set-valued map that extends the classical derivative to the bigger class of locally Lipschitz maps and allows an extension of the fundamental theorem of calculus and a new generalisation of Cauchy–Riemann equations to these maps, which form a continuous Scott domain. We show that a complex Lipschitz map is analytic in an open set if and only if its L-derivative is a singleton at all points in the open set. The calculus of the L-derivative for sum, product and composition of maps is derived. The notion of contour integration is extended to Scott continuous, non-empty compact, convex valued functions on the complex plane, and by using the L-derivative, the fundamental theorem of contour integration is extended to these functions

Asymptotic Control for a Class of Piecewise Deterministic Markov Processes Associated to Temperate Viruses

We aim at characterizing the asymptotic behavior of value functions in the control of piece-wise deterministic Markov processes (PDMP) of switch type under nonexpansive assumptions. For a particular class of processes inspired by temperate viruses, we show that uniform limits of discounted problems as the discount decreases to zero and time-averaged problems as the time horizon increases to infinity exist and coincide. The arguments allow the limit value to depend on initial configuration of the system and do not require dissipative properties on the dynamics. The approach strongly relies on viscosity techniques, linear programming arguments and coupling via random measures associated to PDMP. As an intermediate step in our approach, we present the approximation of discounted value functions when using piecewise constant (in time) open-loop policies.Comment: In this revised version, statements of the main results are gathered in Section 3. Proofs of the main results (Theorem 4 and Theorem 7) make the object of separate sections (Section 5, resp. Section 6). The biological example makes the object of Section 4. Notations are gathered in Subsection 2.1. This is the final version to be published in SICO