Clique Factors: Extremal and Probabilistic Perspectives

A K_r-factor in a graph G is a collection of vertex-disjoint copies of K_r covering the vertex set of G. In this thesis, we investigate these fundamental objects in three settings that lie at the intersection of extremal and probabilistic combinatorics. Firstly, we explore pseudorandom graphs. An n-vertex graph is said to be (p,β)-bijumbled if for any vertex sets A, B ⊆ V (G), we have e( A, B) = p| A||B| ± β√|A||B|. We prove that for any 3 ≤ r ∈ N and c > 0 there exists an ε > 0 such that any n-vertex (p, β)-bijumbled graph with n ∈ rN, δ(G) ≥ c p n and β ≤ ε p^{r −1} n, contains a K_r -factor. This implies a corresponding result for the stronger pseudorandom notion of (n, d, λ)-graphs. For the case of K_3-factors, this result resolves a conjecture of Krivelevich, Sudakov and Szabó from 2004 and it is tight due to a pseudorandom triangle-free construction of Alon. In fact, in this case even more is true: as a corollary to this result, we can conclude that the same condition of β = o( p^2n) actually guarantees that a (p, β)-bijumbled graph G contains every graph on n vertices with maximum degree at most 2. Secondly, we explore the notion of robustness for K_3-factors. For a graph G and p ∈ [0, 1], we denote by G_p the random sparsification of G obtained by keeping each edge of G independently, with probability p. We show that there exists a C > 0 such that if p ≥ C (log n)^{1/3}n^{−2/3} and G is an n-vertex graph with n ∈ 3N and δ(G) ≥ 2n/3 , then with high probability G_p contains a K_3-factor. Both the minimum degree condition and the probability condition, up to the choice of C, are tight. Our result can be viewed as a common strengthening of the classical extremal theorem of Corrádi and Hajnal, corresponding to p = 1 in our result, and the famous probabilistic theorem of Johansson, Kahn and Vu establishing the threshold for the appearance of K_3-factors (and indeed all K_r -factors) in G (n, p), corresponding to G = K_n in our result. It also implies a first lower bound on the number of K_3-factors in graphs with minimum degree at least 2n/3, which gets close to the truth. Lastly, we consider the setting of randomly perturbed graphs; a model introduced by Bohman, Frieze and Martin, where one starts with a dense graph and then adds random edges to it. Specifically, given any fixed 0 < α < 1 − 1/r we determine how many random edges one must add to an n-vertex graph G with δ(G) ≥ α n to ensure that, with high probability, the resulting graph contains a K_r -factor. As one increases α we demonstrate that the number of random edges required ‘jumps’ at regular intervals, and within these intervals our result is best-possible. This work therefore bridges the gap between the seminal work of Johansson, Kahn and Vu mentioned above, which resolves the purely random case, i.e., α = 0, and that of Hajnal and Szemerédi (and Corrádi and Hajnal for r = 3) showing that when α ≥ 1 − 1/r the initial graph already hosts the desired K_r -factor.Ein K_r -Faktor in einem Graphen G ist eine Sammlung von Knoten-disjunkten Kopien von K_r , die die Knotenmenge von G überdecken. Wir untersuchen diese Objekte in drei Kontexten, die an der Schnittstelle zwischen extremaler und probabilistischer Kombinatorik liegen. Zuerst untersuchen wir Pseudozufallsgraphen. Ein Graph heißt (p,β)-bijumbled, wenn für beliebige Knotenmengen A, B ⊆ V (G) gilt e( A, B) = p| A||B| ± β√|A||B|. Wir beweisen, dass es für jedes 3 ≤ r ∈ N und c > 0 ein ε > 0 gibt, so dass jeder n-Knoten (p, β)-bijumbled Graph mit n ∈ rN, δ(G) ≥ c p n und β ≤ ε p^{r −1} n, einen K_r -Faktor enthält. Dies impliziert ein entsprechendes Ergebnis für den stärkeren Pseudozufallsbegriff von (n, d, λ)-Graphen. Im Fall von K_3-Faktoren, löst dieses Ergebnis eine Vermutung von Krivelevich, Sudakov und Szabó aus dem Jahr 2004 und ist durch eine pseudozufällige K_3-freie Konstruktion von Alon bestmöglich. Tatsächlich ist in diesem Fall noch mehr wahr: als Korollar dieses Ergebnisses können wir schließen, dass die gleiche Bedingung von β = o( p^2n) garantiert, dass ein (p, β)-bijumbled Graph G jeden Graphen mit maximalem Grad 2 enthält. Zweitens untersuchen wir den Begriff der Robustheit für K_3-Faktoren. Für einen Graphen G und p ∈ [0, 1] bezeichnen wir mit G_p die zufällige Sparsifizierung von G, die man erhält, indem man jede Kante von G unabhängig von den anderen Kanten mit einer Wahrscheinlichkeit p behält. Wir zeigen, dass, wenn p ≥ C (log n)^{1/3}n^{−2/3} und G ein n-Knoten-Graph mit n ∈ 3N und δ(G) ≥ 2n/3 ist, G_pmit hoher Wahrscheinlichkeit (mhW) einen K_3-Faktor enthält. Sowohl die Bedingung des minimalen Grades als auch die Wahrscheinlichkeitsbedingung sind bestmöglich. Unser Ergebnis ist eine Verstärkung des klassischen extremalen Satzes von Corrádi und Hajnal, entsprechend p = 1 in unserem Ergebnis, und des berühmten probabilistischen Satzes von Johansson, Kahn und Vu, der den Schwellenwert für das Auftreten eines K_3-Faktors (und aller K_r -Faktoren) in G (n, p) festlegt, entsprechend G = K_n in unserem Ergebnis. Es impliziert auch eine erste untere Schranke für die Anzahl der K_3-Faktoren in Graphen mit einem minimalen Grad von mindestens 2n/3, die der Wahrheit nahe kommt. Schließlich betrachten wir die Situation von zufällig gestörten Graphen; ein Modell, bei dem man mit einem dichten Graphen beginnt und dann zufällige Kanten hinzufügt. Wir bestimmen, bei gegebenem 0 < α < 1 − 1/r, wie viele zufällige Kanten man zu einem n-Knoten-Graphen G mit δ(G) ≥ α n hinzufügen muss, um sicherzustellen, dass der resultierende Graph mhW einen K_r -Faktor enthält. Wir zeigen, dass, wenn man α erhöht, die Anzahl der benötigten Zufallskanten in regelmäßigen Abständen “springt", und innerhalb dieser Abstände unser Ergebnis bestmöglich ist. Diese Arbeit schließt somit die Lücke zwischen der oben erwähnten bahnbrechenden Arbeit von Johansson, Kahn und Vu, die den rein zufälligen Fall, d.h. α = 0, löst, und der Arbeit von Hajnal und Szemerédi (und Corrádi und Hajnal für r = 3), die zeigt, dass der ursprüngliche Graph bereits den gewünschten K_r -Faktor enthält, wenn α ≥ 1 − 1/r ist

A Resource-Based View Of International Human Resources: Toward A Framework of Integrative and Creative Capabilities

Drawing on organizational learning and MNC perspectives, we extend the resource-based view to address how international human resource management provides sustainable competitive advantage. We develop a framework that emphasizes and extends traditional assumptions of the resource-based view by identifying the learning capabilities necessary for a complex and changing global environment. These capabilities address how MNCs might both create new HR practices in response to local environments and integrate existing HR practices from other parts of the firm (affiliates, regional headquarters, and global headquarters). In an effort to understand the nature of such capabilities, we discuss aspects of human capital, social capital, and organizational capital that might be linked to their development. Page

The Mid-Infrared Spectrum of the Zodiacal and Exozodiacal Light

The zodiacal light is the dominant source of the mid-infrared sky brightness seen from Earth, and exozodiacal light is the dominant emission from planetary and debris systems around other stars. We observed the zodiacal light spectrum with ISOCAM over 5-16 over a wide range of orientations relative to the Sun and the ecliptic. We present theoretical models for a wide range of particle size distributions and compositions. The observed temperature is as expected for large (>10 um radius), low-albedo (< 0.08), rapidly-rotating, grey particles 1 AU from the Sun. In addition to the continuum, we detect a weak excess in the 9-11 um range, with an amplitude of 6% of the continuum. The shape of the feature can be matched by a mixture of silicates: amorphous forsterite/olivine, dirty crystalline olivine, and a hydrous silicate (montmorillonite). The presence of hydrous silicate suggests the parent bodies of those particles were formed in the inner solar nebula. Large particles dominate the size distribution, but at least some small particles (radii ~1 um) are required to produce the silicate emission feature. To compare the properties of zodiacal dust to dust around other main sequence stars, we reanalyzed the exozodiacal light spectrum for Beta Pic. The exozodiacal spectra are dominated by cold dust, with emission peaking in the far-infrared, while the zodiacal spectrum peaks around 20 um. The shape of the silicate feature from Beta Pic is nearly identical to that derived from the ISO spectrum of 51 Oph; both exozodiacal features are very different from that of the zodiacal light. The exozodiacal features are roughly triangular, peaking at 10.3 um while the zodiacal feature is more boxy.Comment: accepted to Icaru

Red Eyes on Wolf-Rayet Stars: 60 New Discoveries via Infrared Color Selection

We have spectroscopically identified 60 Galactic Wolf-Rayet (WR) stars, including 38 nitrogen types (WN) and 22 carbon types (WC). Using photometry from the Spitzer/GLIMPSE and 2MASS databases, the WRs were selected via a method we have established that exploits their unique infrared colors, which is mainly the result of excess radiation from free-free scattering within their dense ionized winds. The selection criteria has been refined since our last report, and now yields WRs at a rate of ~20% in spectroscopic follow-up of candidates that comprise a broad color space defined by the color distribution of all known WRs having B>14 mag. However, there are subregions within the broad color space which yield WRs at a rate of >50%. Cross-correlation of WR candidates with archival X-ray point-source catalogs increases the WR detection rate of the broad color space to ~40%; ten new WR X-ray sources have been found, in addition to a previously unrecognized X-ray counterpart to a known WR. The extinction values, distances, and galactocentric radii of all new WRs are calculated using the method of spectroscopic parallax. Although the majority of the new WRs have no obvious association with stellar clusters, two WC8 stars reside in a previously unknown massive-star cluster that lies near the intersection of the Scutum-Centaurus Arm and the Galaxy's bar, in which five OB supergiants were also identified. In addition, two WC and four WN stars were identified in association with the stellar clusters Danks 1 and 2. A WN9 star has also been associated with the cluster [DBS2003] 179. This work brings the total number of known Galactic WRs to 476, or ~7-8% of the total empirically estimated population. An examination of their Galactic distribution reveals a tracing of spiral arms and an enhanced WR surface density toward several massive-star formation sites (abridged).Comment: Accepted to the Astronomical Journal on May 20, 2011. Document is 39 pages, including 20 figures and 8 table

Measuring η Carinae's High Mass Ejecta in the Infrared and Sub-millimeter

I address uncertainties on the spatial distribution and mass of the dust formed in η Carinae's Homunculus nebula with data being combined from several space- and ground-based facilities spanning near-infrared to sub-mm wavelengths, in terms of observational constraints and modeling. Until these aspects are better understood, the mass loss history and mechanisms responsible for η Car's enormous eruption(s) remain poorly constrained

Newfoundland fisheries [report of the General Assembly Committee; covers 1549-1845]

An historical overview of the Newfoundland fisheries carried out by the British, French and Americans.Periodical covers have been included with many of the digitized articles whether they relate to them or not. Extra pages not part of the article in question may also be present