Members of Random Closed Sets

We establish that effective continued fraction dimension originally defined using s -gales [ 21 ] is robust, but surprisingly, that the effective continued fraction dimension and effective (base- b ) Hausdorff dimension of the same real can be unequal in general. We initially provide an equivalent characterization of continued fraction dimension using Kolmogorov complexity. We also prove new bounds on the Lebesgue measure of continued fraction cylinders, which may be of independent interest. We apply these bounds to reveal an unexpected behavior of continued fraction dimension. It is known that effective dimension is invariant with respect to base conversion [ 8 ]. We also know that Martin-Löf randomness and computable randomness are invariant not only with respect to base conversion, but also with respect to the continued fraction representation [ 21 ]. In contrast, for any \(0 \lt \varepsilon \lt 0.5\) , we prove the existence of a real whose effective Hausdorff dimension is less than \(\varepsilon\) but whose effective continued fraction dimension is greater than or equal to 0.5. This phenomenon is related to the “non-faithfulness” of certain families of covers [ 1 , 23 ]. We also establish that, for any real number, the effective continued fraction dimension of the real number is always greater than or equal to its effective Hausdorff dimension.

Martin-Löf randomness and Galton–Watson processes

We generalize the classical theorem by Jarnik and Besicovitch on the irrationality exponents of real numbers and Hausdorff dimension and show that the two notions are independent. For any real number a greater than or equal to 2 and any non-negative real b be less than or equal to 2 / a, we show that there is a Cantor-like set with Hausdorff dimension equal to b such that, with respect to its uniform measure, almost all real numbers have irrationality exponent equal to a. We give an analogous result relating the irrationality exponent and the effective Hausdorff dimension of individual real numbers. We prove that there is a Cantor-like set such that, with respect to its uniform measure, almost all elements in the set have effective Hausdorff dimension equal to b and irrationality exponent equal to a. In each case, we obtain the desired set as a distinguished path in a tree of Cantor sets.

Monotonous betting strategies in warped casinos

We study the thresholds for the emergence of various properties in random subgraphs of $(\mathbb N, <)$. In particular, we give sharp sufficient conditions for the existence of (finite or infinite) cliques and paths in a random subgraph. No specific ass

We study protected nodes in various classes of random rooted trees by putting them in the general context of fringe subtrees introduced by Aldous (1991). Several types of random trees are considered: simply generated trees (or conditioned Galton-Watson trees), which includes several cases treated separately by other authors, binary search trees and random recursive trees. This gives unified and simple proofs of several earlier results, as well as new results.

Itô’s excursion theory and random trees

Semikernels modulo F and kernels in digraphs

The properties of the polar sets are discussed for a real-valued (N, d)-fractional Brownian sheet with Hurst index. Sufficient conditions and necessary conditions for a compact set to be polar for the fractional Brownian sheet are proved. The infimum of Hausdorff dimensions of its polar sets are also obtained by means of constructing a Cantor-type set to connect its Hausdorff dimension and capacity.

We work on a Galton-Watson tree with random weights, in the so-called "subdiffusive" regime.We study the rate of decay of the conductance between the root and the n-th level of the tree, as n goes to infinity, by a mostly analytic method.It turns out the order of magnitude of the expectation of this conductance can be less than 1/n (in contrast with the results of Addario-Berry-Broutin-Lugosi and Chen-Hu-Lin), depending on the value of the second zero of the characteristic function associated to the model.We also prove the almost sure (and in L p for some p > 1) convergence of this conductance divided by its expectation towards the limit of the additive martingale.

We consider infinite Galton-Watson trees without leaves together with i.i.d.~random variables called marks on each of their vertices. We define a class of flow rules on marked Galton-Watson trees for which we are able, under some algebraic assumptions, to build explicit invariant measures. We apply this result, together with the ergodic theory on Galton-Watson trees developed in \cite{LPP95}, to the computation of Hausdorff dimensions of harmonic measures in two cases. The first one is the harmonic measure of the (transient) $λ$-biased random walk on Galton-Watson trees, for which the invariant measure and the dimension were not explicitly known. The second case is a model of random walk on a Galton-Watson trees with random lengths for which we compute the dimensions of the harmonic measure and show dimension drop phenomenon for the natural metric on the boundary and another metric that depends on the random lengths.

Innovation is highly demanded for achieving competitiveness and sustainability in value chains. However, few empirical studies have been conducted to identify causal conditions and measurements of innovation capacity in value chains, particularly in developing countries. Combing concepts and methods from value chain upgrading, technological capability and sustainability-oriented innovation system literature, this study aims at identifying a condition or combination of conditions for promoting innovation capacity for inclusive sustainable value chains. It then empirically analyzes the necessary conditions and identifies multiple paths for developing innovation capacity by taking up the cases of two local and two globally linked value chains from Ethiopia. The study also develops a comprehensive innovation capacity evaluation model by combining different capability building strategies and learning mechanisms, and applies a comprehensive fuzzy evaluation method for measuring the level of innovation capacity of the four value chains. The study identifies eight main dimensions of innovation capacity for sustainable value chains, which are categorized into four groups in order to identify necessary and sufficient conditions. The study finds that the simultaneous presence of technological upgrading, value chain restructuring and green governance reforms is sufficient conditions for the development of innovation capacity. We also find that public-private partnership (innovation platform) is necessary condition for achieving outstanding value chain innovation capacity. By developing and applying a fuzzy comprehensive evaluation model for measuring value chain innovation capacity and fuzzy set qualitative comparative analysis for identifying necessary and sufficient conditions for innovation capacity development in a sustainable value chain, the study makes an important methodological contribution to existing literature. It also provides relevant insight for policy makers and practitioners in designing strategies and policy instruments for achieving a high degree of innovation capacity.

ABSTRACTWe prove asymptotic normality for the number of fringe subtrees isomorphic to any given tree in uniformly random trees with given vertex degrees. As applications, we also prove corresponding results for random labeled trees with given vertex degrees, for random simply generated trees (or conditioned Galton–Watson trees), and for additive functionals. The key tool for our work is an extension to the multivariate setting of a theorem by Gao and Wormald (2004), which provides a way to show asymptotic normality by analyzing the behavior of sufficiently high factorial moments.

On the estimation of the mean density of random closed sets

In this article a collection of random self-similar fractal dendrites is constructed, and their Hausdorff dimension is calculated. Previous results determining this quantity for random self-similar structures have relied on geometrical properties of an underlying metric space or the scaling factors being bounded uniformly away from 0. However, using a percolative argument, and taking advantage of the tree-like structure of the sets considered here, it is shown that conditions such as these are not necessary. The scaling factors of the recursively defined structures in consideration form what is known as a multiplicative cascade, and results about the height of this random object are also obtained.