Ma Dolores López

In this paper, we deal with a challenging problem in Discrete Geometry. The problem is to maximize the number of halving lines for finite sets in the plane. With respect to this, we improve the lower bounds for the maximum number of halving lines for sets in the plane of n points (h_n), for some values of n with n ≤ 100, expanding the range of the values of n for which the authors achieved an improvement in preliminary works. To do this, we strengthen previous results of the authors about recursive bounds concerning h_n. These improvements are important due to the relation of the number of halving lines for finite sets with another classical problem in Discrete Geometry, the rectilinear crossing number problem.

Causal graphs are powerful instruments for structuring information and analyzing the extent to which an observed effect can be attributed to a given cause. In this work, we demonstrate how edge-weighted causal graphs can be used to quantify whether a sentence acts as a direct or indirect cause. We introduce methods to identify the causal path with the highest total strength between two concepts and, given a specific context, to determine the pair of concepts with the strongest causal connection. In addition, we propose centrality measures that incorporate causality scores and graph weights, enabling the ranking of sentences by causal importance and identifying the most relevant ones.

In this paper, we give examples that improve the lower bound on the maximum number of halving lines for sets in the plane with 35, 59, 95, and 97 points and, as a consequence, we improve the current best upper bound of the rectilinear crossing number for sets in the plane with 35, 59, 95, and 97 points, provided that a conjecture included in the literature is true. As another consequence, we also improve the lower bound on the maximum number of halving pseudolines for sets in the plane with 35 points. These examples, and the recursive bounds for the maximum number of halving lines for sets with an odd number of points achieved, give a new insight in the study of the rectilinear crossing number problem, one of the most challenging tasks in Discrete Geometry. With respect to this problem, it is conjectured that, for all n multiples of 3, there are 3-symmetric sets of n points for which the rectilinear crossing number is attained.

We give two new ways of calculating the probability of a chord of circumference randomly selected being larger than the side of an equilateral triangle inscribed in the circumference (this problem is known as the Bertrand paradox). The first one employs an immersion in R⁴, and the second one uses a direct probability measure over the set of chords.

Information about the paper titled "LINKS BETWEEN MATHEMATICS EDUCATION AND DANCE" at IATED Digital Library

Debris cones represent a typical landform in temperate high mountain regions. These features show representative periglacial and nival processes in the frame of a dynamic equilibrium. In this work, two debris cones (ahead A and B) located in the “La Vueltona” study area in the Picos de Europa mountain range (N Spain) are analysed. The dynamics of these landforms was studied for a period of 10 years (2009-2018) by means of Digital Elevation Models (DEM) surveyed using a Terrestrial Laser Scanner (TLS). We used the steepest slope line (SSL calculated in the 2009 DEM) to understand the lineal dynamics of the studied debris cones for the whole period. Statistical methods based on multivariate data were used to model the altitude of every location along the SSL for the year 2018. These statistical models used as input the topographic data from 2009 to 2017. The modelled and surveyed altitude for the year 2018 were compared, showing a coherent performance of the modelling approach, with errors within the typical uncertainty of the surveying technique. The Vueltona debris cones showed a wavy dynamics along the SSL, which may be useful to predict their evolution in the short-term. The dynamics model could be extrapolated to other debris cones in temperate high mountain areas.

The study presented makes an original, new and exhaustive analysis of the adaptation of a classical board game which has been named Binary Who is Who? This proposal shows a very useful tool for the consolidation of mathematical concepts related to the study of real-valued functions that are treated in the different levels of the teaching of mathematics (first and second year of superior secondary studies and the first years of some university degrees). The use of games as a means for learning is the authors’ proposal. The aim is to offer teachers the chance of using the games as a method of teaching mathematical concepts, as well as a motivating instrument for them. This game has been created to be played face-to-face in the classroom and it has also been programmed to create a video game which allows the students to play virtually.

This article discusses a theoretical construction based on the graph theory to rework the space of potential partitions in envy-free distribution. This work has the objective of applying Sperner’s lemma to the distribution of three rotating shifts for three workers who are to cover a 24 h job position in a company. As a novel feature, worker’s preferences have been modeled as functions of probability for the three shifts, according to salary offers for said shifts. Envy-free allocation was achieved, since each worker received their preferred shift without the need for negotiation between agents in conflict. Adaptation to the type of dynamic situations that arise with rotating shifts, as well as the consideration of probabilistic preferences by workers are some of the main novelties of this work.