Forcing a sparse minor

This paper addresses the following question for a given graph $H$: what is the minimum number $f(H)$ such that every graph with average degree at least $f(H)$ contains $H$ as a minor? Due to connections with Hadwiger's Conjecture, this question has been studied in depth when $H$ is a complete graph. Kostochka and Thomason independently proved that $f(K_t)=ct\sqrt{\ln t}$. More generally, Myers and Thomason determined $f(H)$ when $H$ has a super-linear number of edges. We focus on the case when $H$ has a linear number of edges. Our main result, which complements the result of Myers and Thomason, states that if $H$ has $t$ vertices and average degree $d$ at least some absolute constant, then $f(H)\leq 3.895\sqrt{\ln d}\,t$. Furthermore, motivated by the case when $H$ has small average degree, we prove that if $H$ has $t$ vertices and $q$ edges, then $f(H) \leq t+6.291q$ (where the coefficient of 1 in the $t$ term is best possible)

Polynomial treewidth forces a large grid-like-minor

Robertson and Seymour proved that every graph with sufficiently large treewidth contains a large grid minor. However, the best known bound on the treewidth that forces an $\ell\times\ell$ grid minor is exponential in $\ell$. It is unknown whether polynomial treewidth suffices. We prove a result in this direction. A \emph{grid-like-minor of order} $\ell$ in a graph $G$ is a set of paths in $G$ whose intersection graph is bipartite and contains a $K_{\ell}$-minor. For example, the rows and columns of the $\ell\times\ell$ grid are a grid-like-minor of order $\ell+1$. We prove that polynomial treewidth forces a large grid-like-minor. In particular, every graph with treewidth at least $c\ell^4\sqrt{\log\ell}$ has a grid-like-minor of order $\ell$. As an application of this result, we prove that the cartesian product $G\square K_2$ contains a $K_{\ell}$-minor whenever $G$ has treewidth at least $c\ell^4\sqrt{\log\ell}$.Comment: v2: The bound in the main result has been improved by using the Lovasz Local Lemma. v3: minor improvements, v4: final section rewritte

Emission from Very Small Grains and PAH Molecules in Monte Carlo Radiation Transfer Codes: Application to the Edge-On Disk of Gomez's Hamburger

We have modeled optical to far infrared images, photometry, and spectroscopy of the object known as Gomez's Hamburger. We reproduce the images and spectrum with an edge-on disk of mass 0.3M_sun and radius 1600AU, surrounding an A0 III star at a distance of 280pc. Our mass estimate is in excellent agreement with recent CO observations. However, our distance determination is more than an order of magnitude smaller than previous analyses which inaccurately interpreted the optical spectrum. To accurately model the infrared spectrum we have extended our Monte Carlo radiation transfer codes to include emission from polycyclic aromatic hydrocarbon (PAH) molecules and very small grains (VSG). We do this using pre-computed PAH/VSG emissivity files for a wide range of values of the mean intensity of the exciting radiation field. When Monte Carlo energy packets are absorbed by PAHs/VSGs we reprocess them to other wavelengths by sampling from the emissivity files, thus simulating the absorption and re-emission process without reproducing lengthy computations of statistical equilibrium, excitation and de-excitation in the complex many level molecules. Using emissivity lookup tables in our Monte Carlo codes gives the flexibility to use the latest grain physics calculations of PAH/VSG emissivity and opacity that are being continually updated in the light of higher resolution infrared spectra. We find our approach gives a good representation of the observed PAH spectrum from the disk of Gomez's Hamburger. Our models also indicate the PAHs/VSGs in the disk have a larger scaleheight than larger radiative equilibrium grains, providing evidence for dust coagulation and settling to the midplane.Comment: ApJ accepte

A Variant of the Erd\H{o}s-S\'os Conjecture

A well-known conjecture of Erd\H{o}s and S\'os states that every graph with average degree exceeding $m-1$ contains every tree with $m$ edges as a subgraph. We propose a variant of this conjecture, which states that every graph of maximum degree exceeding $m$ and minimum degree at least $\lfloor \frac{2m}{3}\rfloor$ contains every tree with $m$ edges. As evidence for our conjecture we show (i) for every $m$ there is a $g(m)$ such that the weakening of the conjecture obtained by replacing $m$ by $g(m)$ holds, and (ii) there is a $\gamma>0$ such that the weakening of the conjecture obtained by replacing $\lfloor \frac{2m}{3}\rfloor$ by $(1-\gamma)m$ holds

Informing the ‘early years’ agenda in Scotland: understanding infant feeding patterns using linked datasets

Background: Providing infants with the ‘best possible start in life’ is a priority for the Scottish Government. This is reflected in policy and health promotion strategies to increase breast feeding, which gives the best source of nutrients for healthy infant growth and development. However, the rate of breast feeding in Scotland remains one of the lowest in Europe. Information is needed to provide a better understanding of infant feeding and its impact on child health. This paper describes the development of a unique population-wide resource created to explore infant feeding and child health in Scotland. Methods: Descriptive and multivariate analyses of linked routine/administrative maternal and infant health records for 731 595 infants born in Scotland between 1997 and 2009. Results: A linked dataset was created containing a wide range of background, parental, maternal, birth and health service characteristics for a representative sample of infants born in Scotland over the study period. There was high coverage and completeness of infant feeding and other demographic, maternal and infant records. The results confirmed the importance of an enabling environment—cultural, family, health service and other maternal and infant health-related factors—in increasing the likelihood to breast feed. Conclusions: Using the linked dataset, it was possible to investigate the determinants of breast feeding for a representative sample of Scottish infants born between 1997 and 2009. The linked dataset is an important resource that has potential uses in research, policy design and targeting intervention programmes

A linear-time algorithm for finding a complete graph minor in a dense graph

Let g(t) be the minimum number such that every graph G with average degree d(G) \geq g(t) contains a K_{t}-minor. Such a function is known to exist, as originally shown by Mader. Kostochka and Thomason independently proved that g(t) \in \Theta(t*sqrt{log t}). This article shows that for all fixed \epsilon > 0 and fixed sufficiently large t \geq t(\epsilon), if d(G) \geq (2+\epsilon)g(t) then we can find this K_{t}-minor in linear time. This improves a previous result by Reed and Wood who gave a linear-time algorithm when d(G) \geq 2^{t-2}.Comment: 6 pages, 0 figures; Clarification added in several places, no change to arguments or result

Use of Open Networks and Delay-Tolerant Protocol to Decrease WAN Latency of EOS near Real-Time Data

Since 1999, NASA's Earth Observing System Data Operations System (EDOS) project at Goddard Space Flight Center (GSFC) has provided high-rate data capture, level zero processing, and product distribution services for a majority of NASA's EOS (Earth Observing System) high-rate missions, including Terra, Aqua, Aura, ICESat, EO-1, SMAP, and OCO-2. EDOS high-rate science and engineering (150-300 Mbps) data-driven capture systems are deployed at 7 worldwide ground stations which are connected via both private (closed) and public (open) wide area networks (WANs) to the centralized EDOS Level Zero Processing Facility (LZPF) located at GSFC, where the data is processed and Level 0 products are distributed to users worldwide. All data transferred over the open networks to GSFC traverse an IPSec tunnel, providing the same level of security as a VPN connection. EDOS produces both time-based and near real-time products (session-based). Near real-time data products are produced from a single ground station contact; time-based products are produced from multiple ground station contacts. EDOS is the primary supplier of EOS Level 0 data to the NASA near real-time user community known as the Land, Atmosphere Near real-time Capability for EOS (LANCE). For the past few years, EDOS has streamlined its systems to reduce WAN latency for near real-time data delivery, including implementing Quality of Service (QoS), expanding closed network bandwidth, adding open network connections with more bandwidth, and implementing a delay-tolerant protocol to mitigate long round-trip times to remote ground stations

EDOS: 20 Years or Level 0 Processing for Terra

No abstract availabl