On the multiplicity of α as an eigenvalue of Aα(G) of graphs with pendant vertices

Let $G$ be a simple undirected graph. Let $0\leq \alpha \leq 1$. Let $$A_{\alpha}(G)= \alpha D(G) + (1-\alpha) A(G)$$ where $D(G)$ and $A(G)$ are the diagonal matrix of the vertex degrees of $G$ and the adjacency matrix of $G$, respectively. Let $p(G)>0$ and $q(G)$ be the number of pendant vertices and quasi-pendant vertices of $G$, respectively. Let $m_{G}(\alpha)$ be the multiplicity of $\alpha$ as eigenvalue of $A_{\alpha}(G)$. It is proved that \begin{equation*} m_{G}(\alpha) \geq p(G) - q(G) \end{equation*} with equality if each internal vertex is a quasi-pendant vertex. If there is at least one internal vertex which is not a quasi-pendant vertex, the equality \begin{equation*} m_{G}(\alpha)= p(G)-q(G)+m_{N}(\alpha) \end{equation*} is determined in which $m_{N}(\alpha)$ is the multiplicity of $\alpha$ as eigenvalue of the matrix $N$. This matrix is obtained from $A_{\alpha}(G)$ taking the entries corresponding to the internal vertices which are non quasi-pendant vertices. These results are applied to search for the multiplicity of $\alpha$ as eigenvalue of $A_{\alpha}(G)$ when $G$ is a path, a caterpillar, a circular caterpillar, a generalized Bethe tree or a Bethe tree. For the Bethe tree case, a simple formula for the nullity is given

Migración y educación en Chile: ¿Es suficiente la respuesta educativa en el actual contexto de diversidad cultural?

Current migratory movements involving Latin American and Caribbean countries have impacted Chile in recent years and progressively increased. Foreigners must to adapt to a new sociocultural and educational context, situation that generates imbalances both for foreigners and for native citizens. The main objective of this article is to carry out a critical analysis of the current migratory situation in Chile, and how education is addressing requirements derived from this condition. In addition, general guidelines are proposed for a better social and educational integration of the immigrant population considering the principles of inclusion

Evaluation of the performance of Euro-CORDEX Regional Climate Models for assessing hydrological climate change impacts in Great Britain: A comparison of different spatial resolutions and quantile mapping bias correction methods

Regional Climate Models (RCMs) are an essential tool for analysing regional climate change impacts, such as hydrological change, as they provide simulations with more small-scale details and expected smaller errors than global climate models. There has been much effort to increase the spatial resolution and simulation skill of RCMs (i.e. through bias correction), yet the extent to which this improves the projection of hydrological change is unclear. Here, we evaluate the skill of five reanalysis-driven Euro-CORDEX RCMs in simulating precipitation and temperature, and as drivers of a hydrological model to simulate river flow on four UK catchments covering different physical, climatic and hydrological characteristics. We use a comprehensive range of evaluation indices for aspects of the distribution such as means and extremes, as well as for the structure of time series. We test whether high-resolution RCMs provide added value, through analysis of two RCM resolutions, 0.44° (50 km) and 0.11° (12.5 km), which are also bias-corrected employing the parametric quantile-mapping (QM) method, using the normal distribution for temperature, and the Gamma (GQM) and Double Gamma (DGQM) distributions for precipitation. The performance of these is considered for a range of meteorological variables and for the skill in simulating hydrological impacts at the catchment scale. In a small catchment with complex topography, the 0.11° RCMs clearly outperform their 0.44° version for precipitation and temperature, but when used in combination with the hydrological model, fail to capture the observed river flow distribution. In the other (larger) catchments, only one high-resolution RCM consistently outperforms its low-resolution version, implying that in general there is no added value from using the high-resolution RCMs in those catchments. Both resolutions produce river flow simulations that cover the observed flow duration curve, but the ensemble spread is large and therefore the simulations are difficult to use in practice. GQM decreases most of the simulation biases, except for extreme precipitation and high flows, which are further decreased by DGQM, which also reduces the multi-model simulation spread. Bias correction does not improve the representation of daily temporal variability measured by the Nash-Sutcliffe Efficiency Index, but it does for monthly variability, in particular when applying DGQM, which reduces most of the simulation biases. Overall, an increase in RCM resolution does not imply a better simulation of hydrology and bias-correction represents an alternative to ease decision-making

ANÁLISIS CLÍNICO Y ECONÓMICO DE LA HISTERECTOMÍA ABDOMINAL VERSUS LA HISTERECTOMÍA VAGINAL EN EL HOSPITAL CLÍNICO DE LA UNIVERSIDAD DE CHILE: REVISIÓN DE 2.338 CASOS

Antecedentes: La histerectomía es una intervención quirúrgica frecuente y de alto costo. Después de la operación cesárea, es el segundo procedimiento quirúrgico más frecuente en ginecología. Objetivos: Evaluar la relación costo/efectividad de la histerectomía vaginal versus la histerectomía abdominal. Método: Estudio retrospectivo observacional, en 2.338 pacientes sometidas a histerectomía por patología benigna en el Servicio de Ginecología del Hospital Clínico de la Universidad de Chile, en el período comprendido entre enero de 1997 a diciembre de 2005. Se analizaron variables clínicas y de costo entre las dos vías de abordaje. Resultados: La edad de las pacientes fluctuó entre 27 y 86 años. De las 2.338 histerectomías, la vía vaginal correspondió al 36,9% y la abdominal en el 63,1%; con una tasa de complicaciones de 29,8/100 mujeres operadas vaginalmente y de 42,8/100 mujeres operadas abdominalmente. Conclusión: Este estudio sugiere que el abordaje vaginal se asoció a tiempos operatorios más cortos, menor número de complicaciones y a costos más bajos que la vía abdomina

The white dwarf binary pathways survey IV: three close white dwarf binaries with G-type secondary stars

Constraints from surveys of post common envelope binaries (PCEBs) consisting of a white dwarf plus an M-dwarf companion have led to significant progress in our understanding of the formation of close white dwarf binary stars with low-mass companions. The white dwarf binary pathways project aims at extending these previous surveys to larger secondary masses, i.e. secondary stars of spectral type AFGK. Here we present the discovery and observational characterization of three PCEBs with G-type secondary stars and orbital periods between 1.2 and 2.5 days. Using our own tools as well as MESA we estimate the evolutionary history of the binary stars and predict their future. We find a large range of possible evolutionary histories for all three systems and identify no indications for differences in common envelope evolution compared to PCEBs with lower mass secondary stars. Despite their similarities in orbital period and secondary spectral type, we estimate that the future of the three systems are very different: TYC 4962-1205-1 is a progenitor of a cataclysmic variable system with an evolved donor star, TYC 4700-815-1 will run into dynamically unstable mass transfer that will cause the two stars to merge, and TYC 1380-957-1 may appear as super soft source before becoming a rather typical cataclysmic variable star