Variations on Cops and Robbers

We consider several variants of the classical Cops and Robbers game. We treat the version where the robber can move R > 1 edges at a time, establishing a general upper bound of N / \alpha ^{(1-o(1))\sqrt{log_\alpha N}}, where \alpha = 1 + 1/R, thus generalizing the best known upper bound for the classical case R = 1 due to Lu and Peng. We also show that in this case, the cop number of an N-vertex graph can be as large as N^{1 - 1/(R-2)} for finite R, but linear in N if R is infinite. For R = 1, we study the directed graph version of the problem, and show that the cop number of any strongly connected digraph on N vertices is at most O(N(log log N)^2/log N). Our approach is based on expansion.Comment: 18 page

Packing tight Hamilton cycles in 3-uniform hypergraphs

Let H be a 3-uniform hypergraph with N vertices. A tight Hamilton cycle C \subset H is a collection of N edges for which there is an ordering of the vertices v_1, ..., v_N such that every triple of consecutive vertices {v_i, v_{i+1}, v_{i+2}} is an edge of C (indices are considered modulo N). We develop new techniques which enable us to prove that under certain natural pseudo-random conditions, almost all edges of H can be covered by edge-disjoint tight Hamilton cycles, for N divisible by 4. Consequently, we derive the corollary that random 3-uniform hypergraphs can be almost completely packed with tight Hamilton cycles w.h.p., for N divisible by 4 and P not too small. Along the way, we develop a similar result for packing Hamilton cycles in pseudo-random digraphs with even numbers of vertices.Comment: 31 pages, 1 figur

Ramsey games with giants

The classical result in the theory of random graphs, proved by Erdos and Renyi in 1960, concerns the threshold for the appearance of the giant component in the random graph process. We consider a variant of this problem, with a Ramsey flavor. Now, each random edge that arrives in the sequence of rounds must be colored with one of R colors. The goal can be either to create a giant component in every color class, or alternatively, to avoid it in every color. One can analyze the offline or online setting for this problem. In this paper, we consider all these variants and provide nontrivial upper and lower bounds; in certain cases (like online avoidance) the obtained bounds are asymptotically tight.Comment: 29 pages; minor revision

Packing Hamilton Cycles Online

It is known that w.h.p. the hitting time $\tau_{2\sigma}$ for the random graph process to have minimum degree $2\sigma$ coincides with the hitting time for $\sigma$ edge disjoint Hamilton cycles. In this paper we prove an online version of this property. We show that, for a fixed integer $\sigma\geq 2$, if random edges of $K_n$ are presented one by one then w.h.p. it is possible to color the edges online with $\sigma$ colors so that at time $\tau_{2\sigma}$, each color class is Hamiltonian.Comment: Minor change

First detection in gamma-rays of a young radio galaxy: Fermi-LAT observations of the Compact Symmetric Object PKS 1718-649

We report the $\gamma$-ray detection of a young radio galaxy, PKS 1718$-$649, belonging to the class of Compact Symmetric Objects (CSOs), with the Large Area Telescope (LAT) on board the {\it Fermi} satellite. The third {\it Fermi} Gamma-ray LAT catalog (3FGL) includes an unassociated $\gamma$-ray source, 3FGL J1728.0$-$6446, located close to PKS 1718$-$649. Using the latest Pass 8 calibration, we confirm that the best fit $1 \sigma$ position of the $\gamma$-ray source is compatible with the radio location of PKS 1718$-$649. Cross-matching of the $\gamma$-ray source position with the positions of blazar sources from several catalogs yields negative results. Thus, we conclude that PKS 1718$-$649 is the most likely counterpart to the unassociated LAT source. We obtain a detection test statistics TS$\sim 36$ ($>$5$\sigma$) with a best fit photon spectral index $\Gamma=$2.9$\pm$0.3 and a 0.1-100 GeV photon flux density $F_{\rm 0.1-100GeV}=$(11.5$\pm$0.3)$\times{\rm 10^{-9}}$ ph cm$^{-2}$ s$^{-1}$. We argue that the linear size ($\sim$2 pc), the kinematic age ($\sim$100 years), and the source distance ($z=0.014$) make PKS 1718$-$649 an ideal candidate for $\gamma$-ray detection in the framework of the model proposing that the most compact and the youngest CSOs can efficiently produce GeV radiation via inverse-Compton scattering of the ambient photon fields by the radio lobe non-thermal electrons. Thus, our detection of the source in $\gamma$-rays establishes young radio galaxies as a distinct class of extragalactic high-energy emitters, and yields an unique insight on the physical conditions in compact radio lobes interacting with the interstellar medium of the host galaxy.Comment: 7 pages, 2 figures, accepted for publication in ApJ Letter

High-Load, Hybrid Si-ROMP Reagents

The combination of norbornenyl-tagged (Nb-tagged) silica particles and functionalized Nb-tagged monomers for the generation of hybrid Si-ROMP reagents and scavengers is reported. Specifically Si-ROMP-derived bis-acid chloride, dichlorotriazine and triphenylphosphine scavenger/reagents have been grafted from the surface of silica particles utilizating surface-initiated, ring-opening metathesis polymerization (ROMP). These hybridpolymeric materials combine the physical properties of current immobilized silica reagents and represent a key advancement in load by merging the inherent tunable properties of the ROMP-derived oligomers with silica supports for application in parallel synthesis

Vascular biomarkers derived from dynamic contrast-enhanced MRI predict response of vestibular schwannoma to antiangiogenic therapy in type 2 neurofibromatosis.

BACKGROUND: Antiangiogenic therapy of vestibular schwannoma (VS) in type 2 neurofibromatosis can produce tumor shrinkage with response rates of 40%–60%. This study examines the predictive value of parameter-derived MRI in this setting. METHODS: Twelve patients with 20 VSs were recruited. Each had at least one rapidly growing tumor. Patients were treated with bevacizumab, 5 mg/kg every 2 weeks. Patients with stable or reduced VS volume were maintained at 2.5–5 mg every 4 weeks after 6 months. Those who failed treatment had their bevacizumab discontinued. Dynamic contrast-enhanced (DCE) MRI performed prior to treatment using a high temporal resolution technique, and data were analyzed to allow measurement of contrast transfer coefficient (K(trans)), vascular fraction (v(p)), extravascular-extracellular fraction (v(e)). Relaxation rate (R1(N)) was measured using a variable flip angle technique. Apparent diffusional coefficient (ADC) was calculated from diffusion-weighted imaging. The predictive power of microvascular parameters and ADC were examined using logistic regression modeling. RESULTS: Responding tumors were larger (P < .001), had lower R1(N) (P < .001), and higher K(trans) (P < .05) and ADC (P < .01). They showed increases in R1(N) (P < .01) and reduction of K(trans) (P < .01) and ADC (P < .01). Modeling to predict response demonstrated significant independent predictive power for R1(N) (Β = − 0.327, P < .001), and K(trans) (Β = 0.156, P < .05). Modeling to predict percentage change in tumor volume at 90 days identified baseline tumor volume (Β = 5.503, P < .05), R1(N) (Β = − 5.844, P < .05), and K(trans) (Β = 5.622, P < .05) as independent significant predictors. CONCLUSIONS: In patients with type 2 neurofibromatosis, biomarkers from DCE-MRI are predictive of VS volume response to inhibition of vascular endothelial growth factor inhibition

Reaction Pairing: A Diversity-Oriented Synthesis Strategy for the Synthesis of Diverse Benzofused Sultams

A reaction pairing strategy centered on utilization of a reaction triad (sulfonylation, SNAr addition and Mitsunobu alkylation) generating skeletally diverse benzofused tricyclic and bicyclic sultams is reported. Pairing sulfonylation and SNAr reactions yields bridged, tricyclic and bicyclic benzofused sultams. Application of the Mitsunobu reaction in a sulfonylation–Mitsunobu–SNAr pairing allows access to benzo-oxathiazocine-1,1-dioxides, while a simple change in the order of pairing to sulfonylation–SNAr–Mitsunobu affords structurally different, benzofused bridged tricyclic sultams