We study contact 3-manifolds using the techniques of sub-Riemannian geometry and geometric measure theory, in particular establishing properties of their Lipschitz homotopy groups. We prove a biLipschitz version of the Theorem of Darboux: a contact $(2n+1)$-manifold endowed with a sub-Riemannian structure is locally biLipschitz equivalent to the Heisenberg group $\mathbb H^n$ with its Carnot-Carathéodory metric. Then each contact $(2n+1)$-manifold endowed with a sub-Riemannian structure is purely $k$-unrectifiable for $k>n$. We then extend results of Dejarnette et al. [5] and Wenger and Young [20] on the Lipschitz homotopy groups of $\mathbb H^1$ to an arbitrary contact 3-manifold endowed with a Carnot-Carathéodory metric, namely that for any contact 3-manifold the first Lipschitz homotopy group is uncountably generated and all higher Lipschitz homotopy groups are trivial. Therefore, in the sense of Lipschitz homotopy groups, a contact 3-manifold is a $K(\pi,1)$-space with an uncountably generated first homotopy group. Along the way, we prove that each open distributional embedding between purely 2-unrectifiable sub-Riemannian manifolds induces an injective map on the associated first Lipschitz homotopy groups. Therefore, each open subset of a contact 3-manifold determines an uncountable subgroup of the first Lipschitz homotopy group of the contact 3-manifold.

A function F:[a,b]→X is said to be an SL function if it satisfies the Strong Lusin (SL) condition given as follows: for every θ-nbd U and a set E⊂[a,b] of measure zero, there exists a gauge δ such that for every δ-fine partial partition D={([xi-1,xi],ti):1≤i≤n} of [a,b] with ti∈E, there exist θ-nbds U1,U2,…,Un such that ∑i=1nUi⊆V and F(xi)-F(xi-1)∈Ui for each i=1,2,…,n. In this paper, we introduce the SL integral of a function taking values on a locally convex topological vector space (LCTVS). Further, we show that this integral is equivalent to a stronger version of the Henstock integral.

In this survey we discuss the connection between commutator operators and functions of bounded mean oscillation, and at the same time outline the parallel story of the real Hardy space and weak factorization of these functions. We provide motivation as to why these questions are interesting and highlight the many different methods of proof that exist for proving results of the type discussed.

In the first part of this paper we investigate four types of convergence of sequences of functions in metric spaces which preserve continuity. Besides a consideration of locally, quasi and continuously uniform convergence, we introduce the notion of semi uniform convergence. We discuss how all types of convergence are related both to each other and to pointwise convergence, and illustrate their behavior by examples. Moreover, we show how some of the types of convergence can be used to characterize compactness of the domains the functions under consideration live in. In the second part we investigate sequences of composition operators in the space \\(BV\\) of functions of bounded variation in the sense of Jordan. We give criteria under which such sequences converge locally uniformly and semi uniformly and present a new and short proof for the fact that composition operators which map the space \\(BV\\) into itself are automatically continuous.

Let \(E\) be a subset of a doubling metric space \((X,d)\). We prove that for any \(s\in [0, \dim_{A}E]\), where \(\dim_{A}\) denotes the Assouad dimension, there exists a subset \(F\) of \(E\) such that \(\dim_{A}F=s\). We also show that the same statement holds for the lower dimension \(\dim_L\).

We give upper and lower bounds for the Hausdorff dimensions for a class of graph-directed measures when its underlying directed graph is the infinite \\(N\\)-ary tree. These measures are different from graph-directed self-similar measures driven by finite directed graphs and are not necessarily Gibbs measures. However our class contains several measures appearing in fractal geometry and functional equations, specifically, measures defined by restrictions of non-constant harmonic functions on the two-dimensional Sierpínski gasket, the Kusuoka energy measures on it, and, measures defined by solutions of de Rham’s functional equations driven by linear fractional transformations.

Like his colleagues de Prony, Petit, and Poisson at the Ecole Polytechnique,\nCauchy used infinitesimals in the Leibniz-Euler tradition both in his research\nand teaching. Cauchy applied infinitesimals in an 1826 work in differential\ngeometry where infinitesimals are used neither as variable quantities nor as\nsequences but rather as numbers. He also applied infinitesimals in an 1832\narticle on integral geometry, similarly as numbers. We explore these and other\napplications of Cauchy's infinitesimals as used in his textbooks and research\narticles.\n An attentive reading of Cauchy's work challenges received views on Cauchy's\nrole in the history of analysis and geometry. We demonstrate the viability of\nCauchy's infinitesimal techniques in fields as diverse as geometric\nprobability, differential geometry, elasticity, Dirac delta functions,\ncontinuity and convergence.\n Keywords: Cauchy--Crofton formula; center of curvature; continuity;\ninfinitesimals; integral geometry; limite; standard part; de Prony; Poisson\n

Sondow and co-workers have employed a key change of variables in order to evaluate double integrals over the unit square \([0,1] \times [0,1]\) in exact closed-form. Motivated by their results, I introduce here a change of variables which creates a ‘bridge’ between integrals of the form \(\,\int_0^1\!\!\int_0^1{f(x \cdot y)~dx \, dy}\,\) and single integrals of the form \(\int_0^1{f(p)\,\ln{p}~d p}\). This allows for prompt closed-form evaluations of several interesting integrals, including some of those investigated recently by Sampedro. I also show that the bridge holds when the intervals of integration are changed from \([0,1]\) to \([1,\infty)\). Finally, a generalization for higher dimensions is proved, which reveals an interesting link of those integrals to Mellin’s transform.

In this paper, we introduce and study a certain type of sub-semigroup of \(\mathbb{R}/\mathbb{Z}\) which turns out to be closely related to Szemeredi’s theorem on arithmetic progressions.