On the local and global comparison of generalized Bajraktarevi\'c means

Given two continuous functions $f,g:I\to\mathbb{R}$ such that $g$ is positive and $f/g$ is strictly monotone, a measurable space $(T,A)$, a measurable family of $d$-variable means $m: I^d\times T\to I$, and a probability measure $\mu$ on the measurable sets $A$, the $d$-variable mean $M_{f,g,m;\mu}:I^d\to I$ is defined by $$M_{f,g,m;\mu}(\pmb{x}) :=\left(\frac{f}{g}\right)^{-1}\left( \frac{\int_T f\big(m(x_1,\dots,x_d,t)\big) d\mu(t)} {\int_T g\big(m(x_1,\dots,x_d,t)\big) d\mu(t)}\right) \qquad(\pmb{x}=(x_1,\dots,x_d)\in I^d).$$ The aim of this paper is to study the local and global comparison problem of these means, i.e., to find conditions for the generating functions $(f,g)$ and $(h,k)$, for the families of means $m$ and $n$, and for the measures $\mu,\nu$ such that the comparison inequality $$M_{f,g,m;\mu}(\pmb{x})\leq M_{h,k,n;\nu}(\pmb{x}) \qquad(\pmb{x}\in I^d)$$ be satisfied

The maximal energy of classes of integral circulant graphs

The energy of a graph is the sum of the moduli of the eigenvalues of its adjacency matrix. We study the energy of integral circulant graphs, also called gcd graphs, which can be characterized by their vertex count $n$ and a set $\cal D$ of divisors of $n$ in such a way that they have vertex set $\mathbb{Z}_n$ and edge set ${{a,b}: a,b\in\mathbb{Z}_n, \gcd(a-b,n)\in {\cal D}}$. For a fixed prime power $n=p^s$ and a fixed divisor set size $|{\cal D}| =r$, we analyze the maximal energy among all matching integral circulant graphs. Let $p^{a_1} < p^{a_2} < ... < p^{a_r}$ be the elements of ${\cal D}$. It turns out that the differences $d_i=a_{i+1}-a_{i}$ between the exponents of an energy maximal divisor set must satisfy certain balance conditions: (i) either all $d_i$ equal $q:=\frac{s-1}{r-1}$, or at most the two differences $[q]$ and $[q+1]$ may occur; %(for a certain $d$ depending on $r$ and $s$) (ii) there are rules governing the sequence $d_1,...,d_{r-1}$ of consecutive differences. For particular choices of $s$ and $r$ these conditions already guarantee maximal energy and its value can be computed explicitly.Comment: Discrete Applied Mathematics (2012

Las respuestas a los conflictos de pareja desde una perspectiva de género: la inteligencia emocional como patrón diferencial

Conflicts that arise in romantic relationships can be one of the greatest causes of suffering when they are not faced appropriately. Through two studies, this research analyses the differential effect of gender and the influence of emotional intelligence on conflict-facing responses, as well as their consequences at an individual and relationship level. In Study 1, we examined the existence of gender differences during conflicts through four independent samples of subjects (N = 727). After meta-analysing the results, we observed that women responded more expressively and loyally to conflict, while men had a more negligent response. In our observations in Study 2 (N = 185), emotional intelligence became a key factor in promoting a constructive response to face conflicts that is adopted by both women and men. Moreover, emotional intelligence finally favours their psychological well-being and satisfaction with the relationship. This research shows the importance of emotional skills in confronting conflicts that originate in intimate contexts such as romantic relationships and their consequences to both men and women.Los conflictos que surgen en las relaciones de pareja pueden ser una de las mayores causas de sufrimiento cuando no se afrontan idóneamente. A través de dos estudios, esta investigación analiza el efecto diferencial del género así como la influencia de la inteligencia emocional en el afrontamiento de los conflictos de pareja y sus consecuencias a nivel individual y relacional. En el estudio 1 se examina la existencia de diferencias de género en el afrontamiento de conflictos a través de 4 muestras independientes de sujetos (N = 727). Tras metaanalizar los resultados de las mismas se observa que las mujeres responden de manera más expresiva y leal ante los conflictos, mientras que los hombres emplean una respuesta más negligente. En el estudio 2 (N = 185) se comprueba como la inteligencia emocional se convierte en un factor clave al promover el afrontamiento constructivo de conflictos adoptado tanto por mujeres como por hombres, lo que favorece tanto su bienestar psicológico como la satisfacción con la relación. Esta investigación muestra la importancia de las habilidades emocionales ante los conflictos que se originan en los contextos más íntimos, como son las relaciones de pareja, y sus consecuencias tanto en hombres como en mujeres.This paper was made possible thanks to the financing provided by the Spanish Ministry of Education, Culture, and Sports through a university teaching training grant (FPU16/03023), and by the projects “Macrosocial realities (economic crisis and social class) and psychosocial processes: Trust, welfare, altruism, and politics” (Ref. PSI-2017- 83966-R) (MINECO/AEI/FEDER/UE) and “New ways of gender violence: Risk and protector factors for psychosocial well-being” (Ref. PSI2017-84703-R) (MINECO/AEI/FEDER/UE)

Quasi-Monte Carlo rules for numerical integration over the unit sphere S2\mathbb{S}^2

We study numerical integration on the unit sphere $\mathbb{S}^2 \subset \mathbb{R}^3$ using equal weight quadrature rules, where the weights are such that constant functions are integrated exactly. The quadrature points are constructed by lifting a $(0,m,2)$-net given in the unit square $[0,1]^2$ to the sphere $\mathbb{S}^2$ by means of an area preserving map. A similar approach has previously been suggested by Cui and Freeden [SIAM J. Sci. Comput. 18 (1997), no. 2]. We prove three results. The first one is that the construction is (almost) optimal with respect to discrepancies based on spherical rectangles. Further we prove that the point set is asymptotically uniformly distributed on $\mathbb{S}^2$. And finally, we prove an upper bound on the spherical cap $L_2$-discrepancy of order $N^{-1/2} (\log N)^{1/2}$ (where $N$ denotes the number of points). This slightly improves upon the bound on the spherical cap $L_2$-discrepancy of the construction by Lubotzky, Phillips and Sarnak [Comm. Pure Appl. Math. 39 (1986), 149--186]. Numerical results suggest that the $(0,m,2)$-nets lifted to the sphere $\mathbb{S}^2$ have spherical cap $L_2$-discrepancy converging with the optimal order of $N^{-3/4}$

Point sets on the sphere S2\mathbb{S}^2 with small spherical cap discrepancy

In this paper we study the geometric discrepancy of explicit constructions of uniformly distributed points on the two-dimensional unit sphere. We show that the spherical cap discrepancy of random point sets, of spherical digital nets and of spherical Fibonacci lattices converges with order $N^{-1/2}$. Such point sets are therefore useful for numerical integration and other computational simulations. The proof uses an area-preserving Lambert map. A detailed analysis of the level curves and sets of the pre-images of spherical caps under this map is given