On the Approximability and Hardness of the Minimum Connected Dominating Set with Routing Cost Constraint

In the problem of minimum connected dominating set with routing cost constraint, we are given a graph $G=(V,E)$, and the goal is to find the smallest connected dominating set $D$ of $G$ such that, for any two non-adjacent vertices $u$ and $v$ in $G$, the number of internal nodes on the shortest path between $u$ and $v$ in the subgraph of $G$ induced by $D \cup \{u,v\}$ is at most $\alpha$ times that in $G$. For general graphs, the only known previous approximability result is an $O(\log n)$-approximation algorithm ($n=|V|$) for $\alpha = 1$ by Ding et al. For any constant $\alpha > 1$, we give an $O(n^{1-\frac{1}{\alpha}}(\log n)^{\frac{1}{\alpha}})$-approximation algorithm. When $\alpha \geq 5$, we give an $O(\sqrt{n}\log n)$-approximation algorithm. Finally, we prove that, when $\alpha =2$, unless $NP \subseteq DTIME(n^{poly\log n})$, for any constant $\epsilon > 0$, the problem admits no polynomial-time $2^{\log^{1-\epsilon}n}$-approximation algorithm, improving upon the $\Omega(\log n)$ bound by Du et al. (albeit under a stronger hardness assumption)

A theorem on the absence of phase transitions in one-dimensional growth models with onsite periodic potentials

We rigorously prove that a wide class of one-dimensional growth models with onsite periodic potential, such as the discrete sine-Gordon model, have no phase transition at any temperature $T>0$. The proof relies on the spectral analysis of the transfer operator associated to the models. We show that this operator is Hilbert-Schmidt and that its maximum eigenvalue is an analytic function of temperature.Comment: 6 pages, no figures, submitted to J Phys A: Math Ge

Learning to trust: trust and attachment in early psychosis

Background Distrust and social dysfunction are characteristic in psychosis and may arise from attachment insecurity, which is elevated in the disorder. The relationship between trust and attachment in the early stages of psychosis is unknown, yet could help to understand interpersonal difficulties and disease progression. This study aimed to investigate whether trust is reduced in patients with early psychosis and whether this is accounted for by attachment avoidance and attachment anxiety. Method We used two trust games with a cooperative and unfair partner in a sample of 39 adolescents with early psychosis and 100 healthy controls. Results Patients had higher levels of attachment anxiety, but the groups did not differ in attachment avoidance. Basic trust was lower in patients than controls, as indicated by lower initial investments. During cooperation patients increased their trust towards levels of controls, i.e. they were able to learn and to override initial suspiciousness. Patients decreased their trust less than controls during unfair interactions. Anxious attachment was associated with higher basic trust and higher trust during unfair interactions and predicted trust independent of group status. Discussion Patients showed decreased basic trust but were able to learn from the trustworthy behaviour of their counterpart. Worries about the acceptance by others and low self-esteem are associated with psychosis and attachment anxiety and may explain behaviour that is focused on conciliation, rather than self-protection

On the smoothness of L p of a positive vector measure

The final publication is available at Springer via http://dx.doi.org/10.1007/s00605-014-0666-7We investigate natural sufficient conditions for a space L p(m) of pintegrable functions with respect to a positive vector measure to be smooth. Under some assumptions on the representation of the dual space of such a space, we prove that this is the case for instance if the Banach space where the vector measure takes its values is smooth. We give also some examples and show some applications of our results for determining norm attaining elements for operators between two spaces L p(m1) and Lq (m2) of positive vector measures m1 and m2.Professor Agud and professor Sanchez-Perez authors gratefully acknowledge the support of the Ministerio de Economia y Competitividad (Spain), under project #MTM2012-36740-c02-02. Professor Calabuig gratefully acknowledges the support of the Ministerio de Economia y Competitividad (Spain), under project #MTM2011-23164.Agud Albesa, L.; Calabuig Rodriguez, JM.; Sánchez Pérez, EA. (2015). On the smoothness of L p of a positive vector measure. Monatshefte für Mathematik. 178(3):329-343. https://doi.org/10.1007/s00605-014-0666-7S3293431783Beauzamy, B.: Introduction to Banach Spaces and Their Geometry. North-Holland, Amsterdam (1982)Diestel, J., Uhl, J.J.: Vector measures. In: Mathematical Surveys, vol. 15. AMS, Providence (1977)Fernández, A., Mayoral, F., Naranjo, F., Sáez, C., Sánchez-Pérez, E.A.: Spaces of p-integrable functions with respect to a vector measure. Positivity 10, 1–16 (2006)Ferrando, I., Rodríguez, J.: The weak topology on $$L^p$$ L p of a vector measure. Topol. Appl. 155(13), 1439–1444 (2008)Godefroy, G.: Boundaries of a convex set and interpolation sets. Math. Ann. 277(2), 173–184 (1987)Howard, R., Schep, A.R.: Norms of positive operators on $$L^p$$ L p -spaces. Proc. Am. Math. Soc. 109(1), 135–146 (1990)Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II. Springer, Berlin (1977)Meyer-Nieberg, P.: Banach Latticces. Universitext, Springer-Verlag, Berlin (1991)Okada, S., Ricker, W.J., Sánchez-Pérez, E.A.: Optimal Domain and Integral Extension of Operators Acting in Function Spaces. Operator Theory: Advances and Applications, vol. 180. Birkhäuser Verlag, Basel (2008)Schep, A.: Products and factors of Banach function spaces. Positivity 14(2), 301–319 (2010

Measuring the balance space sensitivity in vector optimization

Recent literature has shown that the balance space approach may be a significant a1ternative to address several topics concerning vector optimization. Although this new look also leads lo the eflicient set and, consequently, is equivalent to the classical viewpoint, it yields new results and a1gorithms, as well as new economic interpretations, that may be very useful in theoretical framevorks and practical applications. The present paper focuses on the sensitivity of The balance set. We prove a general envelope theorem that yields the sensitivity with respect to any parameter considered in the problem. Fulthermore, we provide a dual problem that characlerizes the primal balance space and its sensitivity. Finally, we a1so give the implications of our results with respect to the sensitivity of the efficient set

Challenges in measuring measles case fatality ratios in settings without vital registration

Measles, a highly infectious vaccine-preventable viral disease, is potentially fatal. Historically, measles case-fatality ratios (CFRs) have been reported to vary from 0.1% in the developed world to as high as 30% in emergency settings. Estimates of the global burden of mortality from measles, critical to prioritizing measles vaccination among other health interventions, are highly sensitive to the CFR estimates used in modeling; however, due to the lack of reliable, up-to-date data, considerable debate exists as to what CFR estimates are appropriate to use. To determine current measles CFRs in high-burden settings without vital registration we have conducted six retrospective measles mortality studies in such settings. This paper examines the methodological challenges of this work and our solutions to these challenges, including the integration of lessons from retrospective all-cause mortality studies into CFR studies, approaches to laboratory confirmation of outbreaks, and means of obtaining a representative sample of case-patients. Our experiences are relevant to those conducting retrospective CFR studies for measles or other diseases, and to those interested in all-cause mortality studies

Attachment and coping in psychosis in relation to spiritual figures

Background: Studies have found higher levels of insecure attachment in individuals with schizophrenia. Attachment theory provides a framework necessary for conceptualizing the development of interpersonal functioning. Some aspects of the attachment of the believer to his/her spiritual figure are similar to those between the child and his/her parents. The correspondence hypothesis suggests that early child-parent interactions correspond to a person's relation to a spiritual figure. The compensation hypothesis suggests that an insecure attachment history would lead to a strong religiousness/spirituality as a compensation for the lack of felt security. The aim of this study is to explore attachment models in psychosis vs. healthy controls, the relationships between attachment and psychopathology and the attachment processes related to spiritual figures. Methods: Attachment models were measured in 30 patients with psychosis and 18 controls with the AAI (Adult Attachment interview) in relationship with psychopathology. Beliefs and practices related to a spiritual figure were investigated by qualitative and quantitative analyses. Results: Patients with psychosis showed a high prevalence of insecure avoidant attachment. Spiritual entities functioned like attachment figures in two thirds of cases. Interviews revealed the transformation of internal working models within relation to a spiritual figure: a compensation process was found in 7 of the 32 subjects who showed a significant attachment to a spiritual figure. Conclusions: Attachment theory allows us to highlight one of the underlying dimensions of spiritual coping in patients with psychosis