Big and little Lipschitz one sets

Given a continuous function $f: {{\mathbb R}}\to {{\mathbb R}}$ we denote the so-called "big Lip" and "little lip" functions by ${{\mathrm {Lip}}} f$ and ${{\mathrm {lip}}} f$ respectively}. In this paper we are interested in the following question. Given a set $E {\subset} {{\mathbb R}}$ is it possible to find a continuous function $f$ such that ${{\mathrm {lip}}} f=\mathbf{1}_E$ or ${{\mathrm {Lip}}} f=\mathbf{1}_E$? For monotone continuous functions we provide the rather straightforward answer. For arbitrary continuous functions the answer is much more difficult to find. We introduce the concept of uniform density type (UDT) and show that if $E$ is $G_\delta$ and UDT then there exists a continuous function $f$ satisfying ${{\mathrm {Lip}}} f =\mathbf{1}_E$, that is, $E$ is a ${{\mathrm {Lip}}} 1$ set. In the other direction we show that every ${{\mathrm {Lip}}} 1$ set is $G_\delta$ and weakly dense. We also show that the converse of this statement is not true, namely that there exist weakly dense $G_{{\delta}}$ sets which are not ${{\mathrm {Lip}}} 1$. We say that a set $E\subset \mathbb{R}$ is ${{\mathrm {lip}}} 1$ if there is a continuous function $f$ such that ${{\mathrm {lip}}} f=\mathbf{1}_E$. We introduce the concept of strongly one-sided density and show that every ${{\mathrm {lip}}} 1$ set is a strongly one-sided dense $F_\sigma$ set.Comment: This is the final preprint version accepted to appear in European Journal of Mathematic

Bilateral microform cleft lip

Microform cleft lip (MCL), also called congenital healed cleft lip or cleft lip "frustré", is a rare congenital anomaly. MCL has been described as having the characteristic appearance of a typical cleft lip which has been corrected in utero. We present a girl with bilateral microform cleft lip associated with a preauricular sinus and bilateral camptodactyly.peer-reviewe

Weakly compact composition operators on spaces of Lipschitz functions

Let $X$ be a pointed compact metric space. Assuming that $\mathrm{lip}_0(X)$ has the uniform separation property, we prove that every weakly compact composition operator on spaces of Lipschitz functions $\mathrm{Lip}_0(X)$ and $\mathrm{lip}_0(X)$ is compact.Comment: 6 page

On the enumeration of closures and environments with an application to random generation

Environments and closures are two of the main ingredients of evaluation in lambda-calculus. A closure is a pair consisting of a lambda-term and an environment, whereas an environment is a list of lambda-terms assigned to free variables. In this paper we investigate some dynamic aspects of evaluation in lambda-calculus considering the quantitative, combinatorial properties of environments and closures. Focusing on two classes of environments and closures, namely the so-called plain and closed ones, we consider the problem of their asymptotic counting and effective random generation. We provide an asymptotic approximation of the number of both plain environments and closures of size $n$. Using the associated generating functions, we construct effective samplers for both classes of combinatorial structures. Finally, we discuss the related problem of asymptotic counting and random generation of closed environemnts and closures

Strong Nash Equilibria in Games with the Lexicographical Improvement Property

We introduce a class of finite strategic games with the property that every deviation of a coalition of players that is profitable to each of its members strictly decreases the lexicographical order of a certain function defined on the set of strategy profiles. We call this property the Lexicographical Improvement Property (LIP) and show that it implies the existence of a generalized strong ordinal potential function. We use this characterization to derive existence, efficiency and fairness properties of strong Nash equilibria. We then study a class of games that generalizes congestion games with bottleneck objectives that we call bottleneck congestion games. We show that these games possess the LIP and thus the above mentioned properties. For bottleneck congestion games in networks, we identify cases in which the potential function associated with the LIP leads to polynomial time algorithms computing a strong Nash equilibrium. Finally, we investigate the LIP for infinite games. We show that the LIP does not imply the existence of a generalized strong ordinal potential, thus, the existence of SNE does not follow. Assuming that the function associated with the LIP is continuous, however, we prove existence of SNE. As a consequence, we prove that bottleneck congestion games with infinite strategy spaces and continuous cost functions possess a strong Nash equilibrium

The convergence rate of approximate solutions for nonlinear scalar conservation laws

The convergence rate is discussed of approximate solutions for the nonlinear scalar conservation law. The linear convergence theory is extended into a weak regime. The extension is based on the usual two ingredients of stability and consistency. On the one hand, the counterexamples show that one must strengthen the linearized L(sup 2)-stability requirement. It is assumed that the approximate solutions are Lip(sup +)-stable in the sense that they satisfy a one-sided Lipschitz condition, in agreement with Oleinik's E-condition for the entropy solution. On the other hand, the lack of smoothness requires to weaken the consistency requirement, which is measured in the Lip'-(semi)norm. It is proved for Lip(sup +)-stable approximate solutions, that their Lip'convergence rate to the entropy solution is of the same order as their Lip'-consistency. The Lip'-convergence rate is then converted into stronger L(sup p) convergence rate estimates

Normal functionals on Lipschitz spaces are weak∗^\ast continuous

Let $\operatorname{Lip}_0(M)$ be the space of Lipschitz functions on a complete metric space $M$ that vanish at a base point. We show that every normal functional in $\operatorname{Lip}_0(M)^\ast$ is weak$^*$ continuous, answering a question by N. Weaver.Comment: v2: Revised versio