**ESTRUCTURAS ORDENADAS Y TOPOLOGIA: TOPOLOGIA SIN PUNTOS, TOPOLOGIA RETICULO-VALUADA Y NO COMMUTATIVA, Y REPRESENTACIONES NUMERICAS**

MTM2015-63608-P

•

**Nombre agencia financiadora**Ministerio de Economía y Competitividad

**Acrónimo agencia financiadora**MINECO

**Programa**Programa Estatal de Fomento de la Investigación Científica y Técnica de Excelencia

**Subprograma**Subprograma Estatal de Generación del Conocimiento

**Convocatoria**Proyectos de I+D dentro del Subprograma Estatal de Generación del Conocimiento (2015)

**Año convocatoria**2015

**Unidad de gestión**Dirección General de Investigación Científica y Técnica

**Centro beneficiario**UNIVERSIDAD DEL PAIS VASCO EUSKAL HERRIKO UNIBERTSITATEA

**Centro realización**FACULTAD DE CIENCIA Y TECNOLOGÍA - DEPARTAMENTO DE MATEMÁTICA APLICADA, ESTADÍSTICA E INVESTIGACIÓN OPERATIVA

**Identificador persistente**http://dx.doi.org/10.13039/501100003329

## Publicaciones

Found(s) 21 result(s)

Found(s) 1 page(s)

Found(s) 1 page(s)

#### On fuzzy uniformities induced by a fuzzy metric space

##### RiuNet. Repositorio Institucional de la Universitat Politécnica de Valéncia

- Gutierrez Garcia, Javier
- Rodríguez López, Jesús|||0000-0001-5141-9977
- Romaguera Bonilla, Salvador

[EN] Different types of fuzzy uniformities have been introduced in the literature standing out the notions due to Hutton, Hohle and Lowen. The main purpose of this paper is to study several methods to endow a fuzzy metric space (X, M, *), in the sense of George and Veeramani, with a probabilistic uniformity and with a Hutton [0, 1](-quasi)-uniformity. We will show the functorial behavior of these constructions as well as its relation with respect to Lowen's functors and Katsaras's functors, which establish a relationship between the categories of probabilistic uniformities and Hutton [0, 1](-quasi)-uniformities with the category of classical uniformities respectively. Furthermore, we also study the fuzzy topologies associated with these fuzzy uniformities. (C) 2017 Elsevier B.V. All rights reserved., The first named author acknowledges the support of the grants MTM2015-63608-P (MINECO/FEDER, UE) and IT974-16 (Basque Government).The second named author is supported by the grant MTM2015-64373-P (MINECO/FEDER, UE).

#### Pointwise aggregation of maps: its structural functional equation and some applications to social choice theory

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Miguel Turullols, Laura de
- Campión Arrastia, María Jesús
- Induráin Eraso, Esteban
- Paternain Dallo, Daniel
- Candeal, Juan Carlos

We study a structural functional equation that is directly related to the pointwise aggregation of a finite number of maps from a given nonempty set into another. First we establish links between pointwise aggregation and invariance properties. Then, paying attention to the particular case of aggregation operators of a finite number of real-valued functions, we characterize several special kinds of aggregation operators as strictly monotone modifications of projections. As a case study, we introduce a first approach of type-2fuzzy sets via fusion operators. We develop some applications and possible uses related to the analysis of properties of social evaluation functionals in social choice, showing that those functionals can actually be described by using methods that derive from this setting., This work has been supported by the research projects ECO2012-34828,

MTM2012-37894-C02-02, MTM2015-63608-P, TIN2013-47605-P, TIN2011-29520 (Spain) and the Research Services of the Public University of Navarre.

MTM2012-37894-C02-02, MTM2015-63608-P, TIN2013-47605-P, TIN2011-29520 (Spain) and the Research Services of the Public University of Navarre.

#### Weightable quasi-metrics related to fuzzy sets

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Campión Arrastia, María Jesús
- García Catalán, Olga Raquel
- Induráin Eraso, Esteban
- Valero, Óscar

We show that the definition of a fuzzy set is directly related to the existence of a weightable quasi-metric on a universe. This relationship is also explored in terms of functional equations coming either from the membership function of a fuzzy set or from the disymmetry function of a quasi-metric., This work has been partially supported by the research projects TIN2013-47605-P, TIN2014-56381-REDT, ECO2015-65031-R, MTM2015-63608-P (MINECO/ AEI-FEDER, UE), TIN2016-81731-REDT, TIN2016-77356-P (MINECO/ AEI-FEDER, UE) and the Research Services of the Public University of Navarre (Spain).

#### An axiomatic approach to finite means

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Campión Arrastia, María Jesús
- García Catalán, Olga Raquel
- Induráin Eraso, Esteban
- Candeal, Juan Carlos
- Giarlotta, Alfio

In this paper we analyze the notion of a finite mean from an axiomatic point of view. We discuss several axiomatic alternatives, with the aim of establishing a universal definition reconciling all of them and exploring theoretical links to some branches of Mathematics as well as to multidisciplinary applications., This work has been partially supported by the research projects S2013/ICE-2845, ECO2015-65031-R, MTM2015-63608-P (MINECO/ AEI-FEDER, UE), TIN2015-66471-P (MINECO/ AEI-FEDER, UE), TIN2016-77356-P (MINECO/ AEI-FEDER, UE).

**Proyecto:**MINECO, MINECO, MINECO, ES/1PE/ECO2015-65031-R, MTM2015-63608-P, TIN2015-66471-P, TIN2016-77356-P

#### Geometrical aggregation of finite fuzzy sets

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Campión Arrastia, María Jesús
- García Catalán, Olga Raquel
- Induráin Eraso, Esteban
- Lizasoain Iriso, María Inmaculada
- Raventós Pujol, Armajac
- Valero, Óscar

A fuzzy set on a finite universe can be interpreted as a vector in a unit cube. This gives rise to a huge variety of approaches in order to aggregate finite fuzzy sets or to modify a given one. We analyze several geometrical methods and discuss possible applications in a multidisciplinary setting., This work has been partially supported by the research projects TIN2013-47605-P, ECO2015-65031-R, MTM2015-63608-P (MINECO/AEI-FEDER, UE), TIN2016-77356-P (MINECO/AEI-FEDER, UE) and TIN2016-81731-REDT (LO-DISCO II and MINECO/AEI-FEDER, UE).

#### On fixed point theory in partially ordered sets and an application to asymptotic complexity of algorithms

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Estevan Muguerza, Asier
- Miñana, Juan José
- Valero, Óscar

The celebrated Kleene fixed point theorem is crucial in the mathematical modelling of recursive specifications in Denotational Semantics. In this paper we discuss whether the hypothesis of the aforementioned result can be weakened. An affirmative answer to the aforesaid inquiry is provided so that a characterization of those properties that a self-mapping must satisfy in order to guarantee that its set of fixed points is non-empty when no notion of completeness are assumed to be satisfied by the partially ordered set. Moreover, the case in which the partially ordered set is coming from a quasi-metric space is treated in depth. Finally, an application of the exposed theory is obtained. Concretely, a mathematical method to discuss the asymptotic complexity of those algorithms whose running time of computing fulfills a recurrence equation is presented. Moreover, the aforesaid method retrieves the fixed point based methods that appear in the literature for asymptotic complexity analysis of algorithms. However, our new method improves the aforesaid methods because it imposes fewer requirements than those that have been assumed in the literature and, in addition, it allows to state simultaneously upper and lower asymptotic bounds for the running time computing., A. Estevan acknowledges financial support from Spanish Ministry of Economy and Competitiveness under Grants MTM2015-63608-P (MINECO/FEDER) and ECO2015-65031. J.J. Miñana and O. Valero acknowledge financial support from Spanish Ministry of Science, Innovation and Universities under Grant PGC2018-095709-B-C21 and AEI/FEDER, UE funds. This work is also partially supported by Programa Operatiu FEDER 2014–2020 de les Illes Balears, by project PROCOE/4/2017 (Direcció General d’Innovació i Recerca, Govern de les Illes Balears) and by project ROBINS. The latter has received research funding from the EU H2020 framework under GA 779776.

#### Partial representations of orderings

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Bosi, Gianni
- Zuanon, Magali
- Estevan Muguerza, Asier

In the present paper a new concept of representability is introduced, which can be applied to not total and also to intransitive relations (semiorders in particular). This idea tries to represent the orderings in the simplest manner, avoiding any unnecessary information. For this purpose, the new concept of representability is developed by means of partial functions, so that other common definitions of representability (i.e. (Richter-Peleg) multi-utility, Scott-Suppes representability, … ) are now particular cases in which the partial functions are actually functions. The paper also presents a collection of examples and propositions showing the advantages of this kind of representations, particularly in the case of partial orders and semiorders, as well as some results showing the connections between distinct kinds of representations., Asier Estevan acknowledges financial support from the Ministry of Economy and Competitiveness of Spain under grants MTM2012-37894-C02-02, MTM2015-63608-P and ECO2015-65031 as well as from the Basque Government under grant IT974-16. Gianni Bosi and Magali E. Zuanon acknowledge financial support from the Istituto Nazionale di Alta Matematica 'F. Severi' (Italy).

#### Orness for idempotent aggregation functions

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Legarreta, Leire
- Mardones Pérez, Iraide
- Lizasoain Iriso, María Inmaculada

Aggregation functions are mathematical operators that merge given data in order to obtain a global value that preserves the information given by the data as much as possible. In most practical applications, this value is expected to be between the infimum and the supremum of the given data, which is guaranteed only when the aggregation functions are idempotent. Ordered weighted averaging (OWA) operators are particular cases of this kind of function, with the particularity that the obtained global value depends on neither the source nor the expert that provides each datum, but only on the set of values. They have been classified by means of the orness—a measurement of the proximity of an OWA operator to the OR-operator. In this paper, the concept of orness is extended to the framework of idempotent aggregation functions defined both on the real unit interval and on a complete lattice with a local finiteness condition., This work has been partially supported by the research projects MTM2015-63608-P of the Spanish Government and IT974-16 of the Basque Government.

**Proyecto:**MINECO//MTM2015-63608-P

#### Topologies for semicontinuous Richter–Peleg multi-utilities

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Bosi, Gianni
- Estevan Muguerza, Asier
- Raventós Pujol, Armajac

The present paper gives a topological solution to representability problems related to multi-utility, in the field of Decision Theory. Necessary and sufficient topologies for the existence of a semicontinuous and finite Richter–Peleg multi-utility for a preorder are studied. It is well known that, given a preorder on a topological space, if there is a lower (upper) semicontinuous Richter–Peleg multi-utility, then the topology of the space must be finer than the Upper (resp. Lower) topology. However, this condition fails to be sufficient. Instead of search for properties that must be satisfied by the preorder, we study finer topologies which are necessary or/and sufficient for the existence of semicontinuous representations. We prove that Scott topology must be contained in the topology of the space in case there exists a finite lower semicontinuous Richter–Peleg multi-utility. However, the existence of this representation cannot be guaranteed. A sufficient condition is given by means of Alexandroff’s topology, for that, we prove that more order implies less Alexandroff’s topology, as well as the converse. Finally, the paper is implemented with a topological study of the maximal elements., Asier Estevan acknowledges financial support from the Ministry of Economy and Competitiveness of Spain under Grants MTM2015-63608-P and ECO2015-65031. Gianni Bosi acknowledges financial support from the Istituto Nazionale di Alta Matematica 'F. Severi' (Italy). Armajac Raventos acknowledges financial support from the Ministry of Economy and Competitiveness of Spain under Grant ECO2015-65031.

#### Aggregation of individual rankings through fusion functions: criticism and optimality analysis

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Bustince Sola, Humberto
- Bedregal, Benjamin
- Campión Arrastia, María Jesús
- Fernández Fernández, Francisco Javier
- Induráin Eraso, Esteban
- Raventós Pujol, Armajac
- Silva, Ivanoska da
- Santiago, Regivan

Throughout this paper, our main idea is to analyze from a theoretical and normative point of view different methods to aggregate individual rankings. To do so, first we introduce the concept of a general mean on an abstract set. This new concept conciliates the social choice where well-known impossibility results as the Arrovian ones are encountered and the decision-making approaches where the necessity of fusing rankings is unavoidable. Moreover it gives rise to a reasonable definition of the concept of a ranking fusion function that does indeed satisfy the axioms of a general mean. Then we will introduce some methods to build ranking fusion functions, paying a special attention to the use of score functions, and pointing out the equivalence between ranking and scoring. To conclude, we prove that any ranking fusion function introduces a partial order on rankings implemented on a finite set of alternatives. Therefore, this allows us to compare rankings and different methods of aggregation, so that in practice one should look for the maximal elements with respect to such orders defined on rankings IEEE., This work is partially supported by the research projects ECO2015-65031-R, MTM2015-63608-P (MINECO/ AEI-FEDER, UE), TIN2016-77356-P (MINECO/ AEI-FEDER, UE), PID2019-108392GB-I00 (AEI/10.13039/501100011033), and Brazilian National Council for Scientific and Technological Development CNPq (Proc. 307781/2016-0).

#### Decomposition and arrow-like aggregation of fuzzy preferences

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Raventós Pujol, Armajac
- Campión Arrastia, María Jesús
- Induráin Eraso, Esteban

We analyze the concept of a fuzzy preference on a set of alternatives, and how it can be decomposed in a triplet of new fuzzy binary relations that represent strict preference, weak preference and indifference. In this setting, we analyze the problem of aggregation of individual fuzzy preferences in a society into a global one that represents the whole society and accomplishes a shortlist of common-sense properties in the spirit of the Arrovian model for crisp preferences. We introduce a new technique that allows us to control a fuzzy preference by means of five crisp binary relations. This leads to an Arrovian impossibility theorem in this particular fuzzy setting., This work is partially supported by the research projects MTM2015-63608-P and TIN2016-77356-P (MINECO/ AEI-FEDER, UE).

#### Open questions in utility theory

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Campión Arrastia, María Jesús
- Induráin Eraso, Esteban

Throughout this paper, our main idea is to explore different classical questions arising in Utility Theory, with a particular attention to those that lean on numerical representations of preference orderings. We intend to present a survey of open questions in that discipline, also showing the state-of-art of the corresponding literature., This work is partially supported by the research projects ECO2015-65031-R, MTM2015-63608-P (MINECO/ AEI-FEDER, UE), and TIN2016-77356-P (MINECO/ AEI-FEDER, UE).

#### Automatic detection of high-voltage power lines in LiDAR surveys using data mining techniques

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Chasco Hernández, Daniel
- García Morales, Víctor
- Sanz Delgado, José Antonio
- Álvarez-Mozos, Jesús

The correct classification of power lines in LiDAR point clouds has attracted the interest of the mapping community in the last years. The objective of this research is the detection and automatic extraction of high-voltage transmission lines from LiDAR data using data mining techniques. With this aim, a Single Photon LiDAR (SPL) survey acquired over the region of Navarre (Spain) in 2017 was used, with a mean point density of 14 pt/m2. Different data mining techniques were evaluated, including decision trees (C4.5 and CART) and ensemble learning algorithms (Random Forests, Bagging and AdaBoost). Fifteen test sites were studied corresponding to areas with high-voltage power lines over different conditions regarding the underlying vegetation and topography. For these sites 92,104 LiDAR points were identified as power lines and more than 4M points as not power lines using existing cartography. This dataset was randomly split in train and test sets and then balanced two obtain a similar amount of data for the two classes. The results obtained show the importance of balancing the training data with improvements in accuracy of ~10% with respect to the imbalanced case. Accuracies higher than 87% were obtained in all balanced cases, with particularly successful results for ensemble learning techniques, being AdaBoost the technique with the highest accuracy 91%. These results suggest that the combination of SPL surveys and data mining tools can be successfully used for the operational mapping of high voltage power lines., The Government of Navarre and Tracasa are acknowledged for the provision of the SPL LiDAR data, the TerraScan software license and their expertise.

#### Searching for a Debreu’s open gap lemma for semiorders

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Estevan Muguerza, Asier

In 1956 R. D. Luce introduced the notion of a semiorder to deal with indifference relations in the representation of a preference. During several years the problem of finding a utility function was studied until a representability characterization was found. However, there was almost no results on the continuity of the representation. A similar result to Debreu’s Lemma, but for semiorders was never achieved. In the present paper we propose a characterization for the existence of a continuous representation (in the sense of Scott-Suppes) for bounded semiorders. As a matter of fact, the weaker but more manageable concept of ε-continuity is properly introduced for semiorders. As a consequence of this study, a version of the Debreu’s Open Gap Lemma is presented (but now for the case of semiorders) just as a conjecture, which would allow to remove the open-closed and closed-open gaps of a subset S ⊆ R, but now keeping the constant threshold, so that x + 1 < y if and only if g(x) + 1 < g(y) (x, y ∈ S)., The author acknowledges financial support from the Ministry of Economy and Competitiveness of Spain under grants MTM2015-63608-P and ECO2015-65031.

#### A survey on the mathematical foundations of axiomatic entropy: representability and orderings

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Campión Arrastia, María Jesús
- Gómez Polo, Cristina
- Induráin Eraso, Esteban
- Raventós Pujol, Armajac

Different abstract versions of entropy, encountered in science, are interpreted in the light of numerical representations of several ordered structures, as total-preorders, interval-orders and semiorders. Intransitivities, other aspects of entropy as competitive systems, additivity, etc., are also viewed in terms of representability of algebraic structures endowed with some compatible ordering. A particular attention is paid to the problem of the construction of an entropy function or their mathematical equivalents. Multidisciplinary comparisons to other similar frameworks are also discussed, pointing out the mathematical foundations., This work was partially supported by the research projects ECO2015-65031-R, MTM2015-63608-P (MINECO/AEI-FEDER, UE), 0011-1365-2016-000060, PC003-004-005 DRUG-MAG, TIN2016-77356-P (MINECO/AEI-FEDER, UE) and 0011-1365-2017-000118.

#### Binary relations coming from solutions of functional equations: orderings and fuzzy subsets

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Campión Arrastia, María Jesús
- Miguel Turullols, Laura de
- García Catalán, Olga Raquel
- Induráin Eraso, Esteban
- Abrísqueta Usaola, Francisco Javier

Electronic version of an article published as International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems Vol. 25, Suppl. 1 (December 2017) 19-42 DOI: 10.1142/S0218488517400025 © World Scientific Publishing Company http://www.worldscientific.com/worldscinet/ijufks, We analyze the main properties of binary relations, defined on a nonempty set, that arise in a natural way when dealing with real-valued functions that satisfy certain classical functional equations on two variables. We also consider the converse setting, namely, given binary relations that accomplish some typical properties, we study whether or not they come from solutions of some functional equation. Applications to the numerical

representability theory of ordered structures are also furnished as a by-product. Further interpretations of this approach as well as possible generalizations to the fuzzy setting are also commented. In particular, we discuss how the values taken for bivariate functions that are bounded solutions of some classical functional equations define, in a natural way, fuzzy binary relations on a set., This work has been partially supported by the research projects MTM2012-37894-C02-02, TIN2013-47605-P,

ECO2015-65031-R, MTM2015-63608-P (MINECO/FEDER), TIN2016-77356-P and the Research Services of the Public University of Navarre (Spain).

representability theory of ordered structures are also furnished as a by-product. Further interpretations of this approach as well as possible generalizations to the fuzzy setting are also commented. In particular, we discuss how the values taken for bivariate functions that are bounded solutions of some classical functional equations define, in a natural way, fuzzy binary relations on a set., This work has been partially supported by the research projects MTM2012-37894-C02-02, TIN2013-47605-P,

ECO2015-65031-R, MTM2015-63608-P (MINECO/FEDER), TIN2016-77356-P and the Research Services of the Public University of Navarre (Spain).

#### Multidistances and inequality measures on abstract sets: an axiomatic approach

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Campión Arrastia, María Jesús
- Induráin Eraso, Esteban
- Raventós Pujol, Armajac
- Díaz, Irene
- Martín, Javier
- Mayor, Gaspar
- Montes Rodríguez, Susana

Starting from the notion of a multidistance, we formalize, through a suitable system of axioms, the concept of an inequality measure defined on a nonempty set with no additional structure implemented a priori. Among inequality measures, apart from multidistances we pay special attention to dispersions, and study their main features. Classical concepts will be generalized to this abstract setting. Multidistances are then revisited, and some new methods to generate them are implemented. A wide spectrum of interdisciplinary applications is outlined in the final section., This work is partially supported by the research projects ECO2015-65031-R , MTM2015-63608-P , TIN2016-77356-P , TIN2017-87600-P (MINECO/AEI-FEDER, UE) and PID2019-108392GB-I00 (AEI/10.13039/501100011033)

#### Continuous representations of interval orders by means of two continuous functions

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Bosi, Gianni
- Estevan Muguerza, Asier

In this paper, we provide a characterization of the existence of a representation of an interval order on a topological space in the general case by means of a pair of continuous functions, when neither the functions nor the topological space are required to satisfy any particular assumptions. Such a characterization is based on a suitable continuity assumption of the binary relation, called weak continuity. In this way, we generalize all the previous results on the continuous representability of interval orders, and also of total preorders, as particular cases., Asier Estevan acknowledges financial support from the Ministry of Economy and Competitiveness of Spain under grants MTM2015-63608-P and ECO2015-65031. Gianni Bosi acknowledges financial support from the Istituto Nazionale di Alta Matematica 'F. Sever' (Italy).

#### Assigning numerical scores to linguistic expressions

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Campión Arrastia, María Jesús
- Falcó Díaz de Cerio, Edurne
- Induráin Eraso, Esteban
- García Lapresta, José

In this paper, we study different methods of scoring linguistic expressions defined on a finite set, in the search for a linear order that ranks all those possible expressions. Among them, particular attention is paid to the canonical extension, and its representability through distances in a graph plus some suitable penalization of imprecision. The relationship between this setting and the classical problems of numerical representability of orderings, as well as extension of orderings from a set to a superset is also explored. Finally, aggregation procedures of qualitative rankings and scorings are also analyzed., This work has been partially supported by the Spanish Ministerio de Economía y Competitividad (projects ECO2015-65031-R, MTM2015-63608-P, ECO2016-77900-P and TIN2016-77356-P), ERDF and the Research Services of the Universidad Pública de Navarra (Spain). Thanks are also given to Remedios Marín and Íñigo Arozarena (Departamento de Tecnología de Alimentos, Universidad Pública de Navarra, Pamplona, Spain) for their valuable suggestions and comments.

**Proyecto:**MINECO, MINECO, ES, ES/1PE, 1PE/ECO2015-65031-R, MTM2015-63608-P, ECO2016-77900, TIN2016-77356

#### Changes in muscle power after usual care or early structured exercise intervention in acutely hospitalized older adults

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- López Sáez de Asteasu, Mikel
- Martínez Velilla, Nicolás
- Zambom Ferraresi, Fabrício
- Ramírez Vélez, Robinson
- Casas Herrero, Álvaro
- Galbete Jiménez, Arkaitz
- Izquierdo Redín, Mikel
- García Hermoso, Antonio
- Cadore, Eduardo L.

Background: A classic consequence of short-term bed rest in older adults is the significant loss in skeletal muscle mass and muscle strength that underlies the accelerated physical performance deficits. Structured exercise programmes applied during acute hospitalization can prevent muscle function deterioration. Methods: A single-blind randomized clinical trial conducted in an acute care for elders unit in a tertiary public hospital in Navarre (Spain). Three hundred seventy hospitalized patients [56.5% female patients; mean age (standard deviation) 87.3 (4.9) years] were randomly allocated to an exercise intervention (n = 185) or a control (n = 185) group (usual care). The intervention consisted of a multicomponent exercise training programme performed during 5–7 consecutive days (2 sessions/day). The usual-care group received habitual hospital care, which included physical rehabilitation when needed. The main endpoints were change in maximal dynamic strength (i.e. leg-press, chest-press, and knee extension exercises) and maximal isometric knee extensors and hip flexors strength from baseline to discharge. Changes in muscle power output at submaximal and maximal loads were also measured after the intervention. Results: The physical exercise programme provided significant benefits over usual care. At discharge, the exercise group showed a mean increase of 19.6 kg [95% confidence interval (CI), 16.0, 23.2; P < 0.001] on the one-repetition maximum (1RM) in the leg-press exercise, 5.7 kg (95% CI, 4.7, 6.8; P < 0.001) on the 1RM in the chest-press exercise, and 9.4 kg (95% CI, 7.3, 11.5; P < 0.001) on the 1RM in the knee extension exercise over usual-care group. There were improvements in the intervention group also in the isometric maximal knee extension strength [14.8 Newtons (N); 95% CI, 11.2, 18.5 vs. −7.8 N; 95% CI, −11.0, −3.5 in the control group; P < 0.001] and the hip flexion strength (13.6 N; 95% CI, 10.7, 16.5 vs. −7.2 N; 95% CI, −10.1, −4.3; P < 0.001). Significant benefits were also observed in the exercise group for the muscle power output at submaximal loads (i.e. 30% 1RM, 45% 1RM, 60% 1RM, and 75% 1RM; all P < 0.001) over usual-care group. Conclusions: An individualized, multicomponent exercise training programme, with special emphasis on muscle power training, proved to be an effective therapy for improving muscle power output of lower limbs at submaximal loads and maximal muscle strength in older patients during acute hospitalization., This study was funded by a Gobierno de Navarra project grant (Resolución 2186/2014) and acknowledged with the 'Beca Ortiz de Landazuri' for the best research clinical project in 2014, as well as by a research grant PI17/01814 of the Ministerio de Economía, Industria y Competitivad (ISCIII, FEDER). The authors thank Fundación Miguel Servet (Navarrabiomed) for its support during the implementation of the trial, as well as Fundación Caja Navarra and Fundación la Caixa. R.R.-V. is funded in part by a postdoctotal fellowship grant ID 420/2019 of the Public University of Navarre, Spain.

#### An axiomatic approach to finite means

##### Zaguán. Repositorio Digital de la Universidad de Zaragoza

- Campión, M.J.
- Candeal, J.C.
- Catalán, R.G.
- Giarlotta, A.
- Greco, S.
- Induráin, E.
- Montero, J.

In this paper we analyze the notion of a finite mean from an axiomatic point of view. We discuss several axiomatic alternatives, with the aim of establishing a universal definition reconciling all of them and exploring theoretical links to some branches of Mathematics as well as to multidisciplinary applications.