Jose Manuel Calabuig

This study introduces a novel physics-inspired framework to quantify defensive pressure in football from tracking data. We model defender–attacker interactions as a variable-mass dynamical system, translating Newtonian mechanics into operational metrics that combine spatial configuration and motion. From this formulation we derive interpretable quantities at dyad, player, and team level, including a Center of Pressure (CP), Defensive Momentum, Defensive Force, and Defensive Work. We illustrate the framework in a single-match proof-of-concept using professional optical tracking data, analysing both full-match behaviour and football-specific phases such as counter-pressing, set-pieces, and throw-ins. Results show how the proposed metrics separate persistent spatial constraint (pressure) from energetically demanding defensive actions (work), enable identification of high-cost match-ups and workload concentration, and support time-resolved descriptions of coordinated pressing sequences. The framework provides a transferable, mechanically grounded toolkit for applied defensive performance analysis and motivates future validation on larger datasets.

Este trabajo presenta una sesión tipo taller dirigida a estudiantes universitarios, cuyo objetivo es introducir los fundamentos de la simulación de datos sintéticos a través de una aplicación práctica en ciencia animal. Usando como ejemplo el comportamiento de cabras registradas mediante sensores de acelerometría triaxial, se explican dos métodos matemáticos clave de simulación: el método de Montecarlo, aplicado tanto a distribuciones discretas no paramétricas —para simular la secuencia de actividades— como a distribuciones continuas paramétricas —para modelar su duración—, y los modelos VAR —para generar señales de aceleración realistas—. La propuesta permite comprender cómo la simulación puede suplir la escasez de datos reales y apoyar el desarrollo de modelos predictivos.

Given a Banach lattice L, the space of lattice Lipschitz operators on L has been introduced as a natural Lipschitz generalization of the linear notions of diagonal operator and multiplication operator on Banach function lattices. It is a particular space of superposition operators on Banach lattices. Motivated by certain procedures in Reinforcement Learning based on McShane–Whitney extensions of Lipschitz maps, this class has proven to be useful also in the classical context of Mathematical Analysis. In this paper, we discuss the properties of such operators when defined on spaces of continuous functions, focusing attention on the functional bounds for the pointwise Lipschitz inequalities defining the lattice Lipschitz operators, the representation theorems for these operators as vector-valued functions, and the corresponding dual spaces. Finally, and with possible applications in Artificial Intelligence in mind, we provide a McShane–Whitney extension theorem for these operators.

After reviewing various notions of symmetry in graph theory, which are typically defined by the connections between vertices, we demonstrate that traditional concepts of symmetry, such as vertex transitivity, can be too restrictive for certain applications. For instance, in some areas of graph analysis, symmetry based on metric properties (such as average distances between vertices) may be more appropriate, particularly in social network analysis or economic fraud detection. This paper focuses on developing metric-based symmetry concepts by introducing mathematical analysis tools, all related to the central idea of the distance distribution function, to group vertices according to their distance-related properties within the graph. In particular, we prove several results that show, under certain compactness properties for the set of distribution functions of all the vertices in an infinite graph, that it is always possible to group these vertices into a finite number of classes with the desired accuracy based on distances.

Performance analysis, utilizing video technology and recent technological advancements in soccer stadiums, provides a wealth of data, including player trajectories and real-time game statistics, which are crucial for tactical evaluation and decision-making by coaches and players. These data allow for the definition of metrics that not only enrich the experience for soccer fans through enhanced visual displays but also empower coaching staff and managers to make informed, real-time decisions that directly impact match outcomes. Ultimately, these data serve as a pivotal tool for improving team strategy based on comprehensive post-match data analysis. In this article, we present a mathematical model to study the concept of pressure between players and, subsequently, between teams. We first explore the concept in a fixed frame of a match, determining what we call influence areas between players. We introduce the unit pressure function and analyze the total number of pressure interactions. Then, we apply these concepts to football matches, considering various factors such as players and the radius of the area of influence and examining pressure efficiency through mean unitary pressure. Lastly, a real case study is presented, showcasing visualizations like a heatmap matrix displaying individual and collective pressure, as well as the team pressure balance.

Performance analysis, utilizing video technology and recent technological advancements in soccer stadiums, provides a wealth of data, including player trajectories and real-time game statistics, which are crucial for tactical evaluation and decision-making by coaches and players. This data allows for the definition of metrics that not only enriches the experience for soccer fans through enhanced visual displays, but also empowers coaching staff and managers to make informed, real-time decisions that directly impact match outcomes. Ultimately, it serves as a pivotal tool for improving team strategy based on comprehensive post-match data analysis. In this article, we present a mathematical model to study the concept of pressure between players and, subsequently, between teams. We first explore the concept in a fixed frame of a match, determining what we call {\sl influence areas} between players. We introduce the unit pressure function and analyze the total number of pressure interactions. Then, we apply these concepts to football matches, considering various factors such as players and the radius of the area of influence, examining pressure efficiency through mean unitary pressure. Lastly, a real case study is presented, showcasing visualizations like a heatmap matrix displaying individual and collective pressure, as well as team pressure balance.

Consider a finite directed graph without cycles in which the arrows are weighted by positive weights. We present an algorithm for the computation of a new distance, called path-length-weighted distance, which has proven useful for graph analysis in the context of fraud detection. The idea is that the new distance explicitly takes into account the size of the paths in the calculations. It has the distinct characteristic that, when calculated along the same path, it may result in a shorter distance between far-apart vertices than between adjacent ones. This property can be particularly useful for modeling scenarios where the connections between vertices are obscured by numerous intermediate vertices, such as in cases of financial fraud. For example, to hide dirty money from financial authorities, fraudsters often use multiple institutions, banks, and intermediaries between the source of the money and its final recipient. Our distance would serve to make such situations explicit. Thus, although our algorithm is based on arguments similar to those at work for the Bellman–Ford and Dijkstra methods, it is in fact essentially different, since the calculation formula contains a weight that explicitly depends on the number of intermediate vertices. This fact totally conditions the algorithm, because longer paths could provide shorter distances—contrary to the classical algorithms mentioned above. We lay out the appropriate framework for its computation, showing the constraints and requirements for its use, along with some illustrative examples.