The Expected Number of Maximal Points of the Convolution of Two 2-D Distributions

The Maximal points in a set S are those that are not dominated by any other point in S. Such points arise in multiple application settings and are called by a variety of different names, e.g., maxima, Pareto optimums, skylines. Their ubiquity has inspired a large literature on the expected number of maxima in a set S of n points chosen IID from some distribution. Most such results assume that the underlying distribution is uniform over some spatial region and strongly use this uniformity in their analysis. This research was initially motivated by the question of how this expected number changes if the input distribution is perturbed by random noise. More specifically, let B_p denote the uniform distribution from the 2-dimensional unit ball in the metric L_p. Let delta B_q denote the 2-dimensional L_q-ball, of radius delta and B_p + delta B_q be the convolution of the two distributions, i.e., a point v in B_p is reported with an error chosen from delta B_q. The question is how the expected number of maxima change as a function of delta. Although the original motivation is for small delta, the problem is well defined for any delta and our analysis treats the general case. More specifically, we study, as a function of n,delta, the expected number of maximal points when the n points in S are chosen IID from distributions of the type B_p + delta B_q where p,q in {1,2,infty} for delta > 0 and also of the type B_infty + delta B_q where q in [1,infty) for delta > 0. For fixed p,q we show that this function changes "smoothly" as a function of delta but that this smooth behavior sometimes transitions unexpectedly between different growth behaviors

Max-Cut and Max-Bisection are NP-hard on unit disk graphs

We prove that the Max-Cut and Max-Bisection problems are NP-hard on unit disk graphs. We also show that $\lambda$-precision graphs are planar for $\lambda$ > 1 / \sqrt{2}$

Absorption Time of the Moran Process

The Moran process models the spread of mutations in populations on graphs. We investigate the absorption time of the process, which is the time taken for a mutation introduced at a randomly chosen vertex to either spread to the whole population, or to become extinct. It is known that the expected absorption time for an advantageous mutation is O(n^4) on an n-vertex undirected graph, which allows the behaviour of the process on undirected graphs to be analysed using the Markov chain Monte Carlo method. We show that this does not extend to directed graphs by exhibiting an infinite family of directed graphs for which the expected absorption time is exponential in the number of vertices. However, for regular directed graphs, we show that the expected absorption time is Omega(n log n) and O(n^2). We exhibit families of graphs matching these bounds and give improved bounds for other families of graphs, based on isoperimetric number. Our results are obtained via stochastic dominations which we demonstrate by establishing a coupling in a related continuous-time model. The coupling also implies several natural domination results regarding the fixation probability of the original (discrete-time) process, resolving a conjecture of Shakarian, Roos and Johnson.Comment: minor change

Geoshow3D, una aplicación de cartografía dinámica 3D

Peer Reviewe

Productivity, Digital Footprint and Sustainability in the Textile and Clothing Industry

[EN] In recent years, there has been a shift from the linear economic model on which the textile and clothing industry is based to a more sustainable model. However, to date, limited research on the relationship between sustainability commitment and firm productivity has focused on the textile and clothing industry. This study addresses this gap and aims to explore whether the digital footprint of small and medium-sized textile companies in terms of their sustainable performance is related to their productivity. To this end, the paper proposes an innovative model to monitor the companies’ commitment to sustainable issues by analyzing online data retrieved from their corporate websites. This information is merged with balance sheet data to examine the impact of sustainability practices, capital and human capital on productivity. The estimated firm’s total factor productivity is explained as a function of the sustainability digital footprint measures and additional control variables for a sample of 315 textile firms located in the region of Comunidad Valenciana, Spain.This work was partially funded by MCIN/AEI/10.13039/501100011033 under grant PID2019-107765RB-I00.Domenech, J.; Garcia-Bernabeu, A.; Diaz-Garcia, P. (2023). Productivity, Digital Footprint and Sustainability in the Textile and Clothing Industry. Editorial Universitat Politècnica de València. 319-326. https://doi.org/10.4995/CARMA2023.2023.1644631932