Additive Volume of Sets Contained in Few Arithmetic Progressions

A conjecture of Freiman gives an exact formula for the largest volume of a finite set $A$ of integers with given cardinality $k = |A|$ and doubling $T = |2A|$. The formula is known to hold when $T \le 3k-4$, for some small range over $3k-4$ and for families of structured sets called chains. In this paper we extend the formula to sets of every dimension and prove it for sets composed of three segments, giving structural results for the extremal case. A weaker extension to sets composed of a bounded number of segments is also discussed.Comment: 16 page

Offline and online power aware resource allocation algorithms with migration and delay constraints

© . This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/In order to handle advanced mobile broadband services and Internet of Things (IoT), future Internet and 5G networks are expected to leverage the use of network virtualization, be much faster, have greater capacities, provide lower latencies, and significantly be power efficient than current mobile technologies. Therefore, this paper proposes three power aware algorithms for offline, online, and migration applications, solving the resource allocation problem within the frameworks of network function virtualization (NFV) environments in fractions of a second. The proposed algorithms target minimizing the total costs and power consumptions in the physical network through sufficiently allocating the least physical resources to host the demands of the virtual network services, and put into saving mode all other not utilized physical components. Simulations and evaluations of the offline algorithm compared to the state-of-art resulted on lower total costs by 32%. In addition to that, the online algorithm was tested through four different experiments, and the results argued that the overall power consumption of the physical network was highly dependent on the demands’ lifetimes, and the strictness of the required end-to-end delay. Regarding migrations during online, the results concluded that the proposed algorithms would be most effective when applied for maintenance and emergency conditions.Peer ReviewedPreprin

The magnetic precursor of L1448-mm: Excitation differences between ion and neutral fluids

Shock modelling predicts an electron density enhancement within the magnetic precursor of C-shocks. Previous observations of SiO, H13CO+, HN13C and H13CN toward the young L1448-mm outflow showed an over-excitation of the ion fluid that was attributed to an electron density enhancement in the precursor. We re-visit this interpretation and test if it still holds when we consider different source morphologies and kinetic temperatures for the observed molecules, and also give some insight on the spatial extent of the electron density enhancement around L1448-mm. We estimate the opacities of H13CO+ and HN13C by observing the J=3\to2 lines of rarer isotopologues to confirm that the emission is optically thin. To model the excitation of the molecules, we use the large velocity gradient (LVG) approximation with updated collisional coefficients to i) re- analyse the observations toward the positions where the over-excitation of H13CO+ has previously been observed [i.e. toward L1448- mm at offsets (0,0) and (0,-10)], and ii) to investigate if the electron density enhancement is still required for the cases of extended and compact emission, and for kinetic temperatures of up to 400 K. We also report several lines of SiO, HN13C and H13CO+ toward new positions around this outflow, to investigate the spatial extent of the over-excitation of the ions in L1448-mm. From the isotopologue observations, we find that the emission of H13CO+ and HN13C from the precursor is optically thin if this emission is extended. Using the new collisional coefficients, an electron density enhancement is still needed to explain the excitation of H13CO+ for extended emission and for gas temperatures of\le 400 K toward L1448-mm (0,-10), and possibly also toward L1448-mm (0,0). For compact emission the data cannot be fitted. We do not find any evidence for the over-excitation of the ion fluid toward the newly observed positions around L1448-mm. The observed line emission of SiO, H13CO+ and HN13C toward L1448-mm (0,0) and (0,-10) is consistent with an electron density enhancement in the precursor component, if this emission is spatially extended. This is also true for the case of high gas temperatures (\le400 K) toward the (0,-10) offset. The electron density enhancement seems to be restricted to the southern, redshifted lobe of the L1448-mm outflow. Interferometric images of the line emission of these molecules are needed to confirm the spatial extent of the over-excitation of the ions and thus, of the electron density enhancement in the magnetic precursor of L1448-mm.Comment: Accepted for publication in A&A; 9 pages, 3 figure

Variegate galaxy cluster gas content: Mean fraction, scatter, selection effects and covariance with X-ray luminosity

We use a cluster sample selected independently of the intracluster medium content with reliable masses to measure the mean gas mass fraction and its scatter, the biases of the X-ray selection on gas mass fraction, and the covariance between the X-ray luminosity and gas mass. The sample is formed by 34 galaxy clusters in the nearby ($0.050<z<0.135$) Universe, mostly with $14<\log M_{500}/M_\odot \lesssim 14.5$, and with masses calculated with the caustic technique. First, we found that integrated gas density profiles have similar shapes, extending earlier results based on subpopulations of clusters such as those that are relaxed or X-ray bright for their mass. Second, the X-ray unbiased selection of our sample allows us to unveil a variegate population of clusters; the gas mass fraction shows a scatter of $0.17\pm0.04$ dex, possibly indicating a quite variable amount of feedback from cluster to cluster, which is larger than is found in previous samples targeting subpopulations of galaxy clusters, such as relaxed or X-ray bright clusters. The similarity of the gas density profiles induces an almost scatterless relation between X-ray luminosity, gas mass, and halo mass, and modulates selection effects in the halo gas mass fraction: gas-rich clusters are preferentially included in X-ray selected samples. The almost scatterless relation also fixes the relative scatters and slopes of the $L_X-M$ and $M_{gas}-M$ relations and makes core-excised X-ray luminosities and gas masses fully covariant. Therefore, cosmological or astrophysical studies involving X-ray or SZ selected samples need to account for both selection effects and covariance of the studied quantities with X-ray luminosity/SZ strength.Comment: A&A, in press, minor language changes from previous versio

Open and closed industry clusters: The social structure of innovation

In this paper we discuss knowledge and innovation in clusters and the benefits of clustering from a knowledge-based perspective. Knowledge-based resources and innovations are important sources of competitive advantage for firms. Aware of the importance of continuously seeking new knowledge firms increasingly seek knowledge-rich locations such as specific industry clusters across the world. These are locations characterized by the concentration of firms operating in related and supporting activities, a specialized work force and a specialized institutional environment that nurtures the industry. However, it is not likely that these clusters are always locations from which the firms will be able to draw the intended knowledge benefits. The social structure of the relationships between individuals and firms determines the extent to which knowledge will be created, will flow between co-located firms and bounds the knowledge benefits the firms may capture. We finish with a discussion of the need of further examination of the network dynamics involved in an industry cluster to obtain a clearer identification of the actual positive externalities that may accrue to co-locating firms.Strategy; Industry clusters; Innovation