Completions of epsilon-dense partial Latin squares

A classical question in combinatorics is the following: given a partial Latin square $P$, when can we complete $P$ to a Latin square $L$? In this paper, we investigate the class of textbf{$epsilon$-dense partial Latin squares}: partial Latin squares in which each symbol, row, and column contains no more than $epsilon n$-many nonblank cells. Based on a conjecture of Nash-Williams, Daykin and H"aggkvist conjectured that all $frac{1}{4}$-dense partial Latin squares are completable. In this paper, we will discuss the proof methods and results used in previous attempts to resolve this conjecture, introduce a novel technique derived from a paper by Jacobson and Matthews on generating random Latin squares, and use this novel technique to study $ epsilon$-dense partial Latin squares that contain no more than $delta n^2$ filled cells in total.

In Chapter 2, we construct completions for all $ epsilon$-dense partial Latin squares containing no more than $delta n^2$ filled cells in total, given that $epsilon < frac{1}{12}, delta < frac{ left(1-12epsilonright)^{2}}{10409}$. In particular, we show that all $9.8 cdot 10^{-5}$-dense partial Latin squares are completable. In Chapter 4, we augment these results by roughly a factor of two using some probabilistic techniques. These results improve prior work by Gustavsson, which required $epsilon = delta leq 10^{-7}$, as well as Chetwynd and H"aggkvist, which required $epsilon = delta = 10^{-5}$, $n$ even and greater than $10^7$.

If we omit the probabilistic techniques noted above, we further show that such completions can always be found in polynomial time. This contrasts a result of Colbourn, which states that completing arbitrary partial Latin squares is an NP-complete task. In Chapter 3, we strengthen Colbourn's result to the claim that completing an arbitrary $left(frac{1}{2} + epsilonright)$-dense partial Latin square is NP-complete, for any $epsilon > 0$.

Colbourn's result hinges heavily on a connection between triangulations of tripartite graphs and Latin squares. Motivated by this, we use our results on Latin squares to prove that any tripartite graph $G = (V_1, V_2, V_3)$ such that begin{itemize} item $|V_1| = |V_2| = |V_3| = n$, item For every vertex $v in V_i$, $deg_+(v) = deg_-(v) geq (1- epsilon)n,$ and item $|E(G)| > (1 - delta)cdot 3n^2$ end{itemize} admits a triangulation, if $epsilon < frac{1}{132}$, $delta < frac{(1 -132epsilon)^2 }{83272}$. In particular, this holds when $epsilon = delta=1.197 cdot 10^{-5}$.

This strengthens results of Gustavsson, which requires $epsilon = delta = 10^{-7}$.

In an unrelated vein, Chapter 6 explores the class of textbf{quasirandom graphs}, a notion first introduced by Chung, Graham and Wilson cite{chung1989quasi} in 1989. Roughly speaking, a sequence of graphs is called "quasirandom"' if it has a number of properties possessed by the random graph, all of which turn out to be equivalent. In this chapter, we study possible extensions of these results to random $k$-edge colorings, and create an analogue of Chung, Graham and Wilson's result for such colorings.

Payments for Water Ecosystem Services in Latin America: Evidence from Reported Experience

27 p.

Basque legacy in the New World: on the surnames of Latin American presidents

[EN] In this article we explain the etymology of the surnames of Basque origin that some presidents of Latin American countries have or have had in the past. These family names were created in the language called Euskara, in the Basque Country (Europe), and then, when some of the people who bore them emigrated to America, they brought their surnames with them. Most of the family names studied here are either oiconymic or toponymic, but it must be kept in mind that the oiconymic ones are, very often, based on house-nicknames, that is, they are anthroponymic in the first place. As far as possible, we have related the surname, when its origin is oiconymic or toponymic, to its source, i.e. to the house or place where it was created.

International transport costs and the margins of Intra-Latin American maritime trade.

This paper focuses on the analysis of the relationship between maritime trade and transport cost in Latin America. The analysis is based on disaggregated (SITC 5 digit level) trade data for intra Latin maritime trade routes over the period 1999-2004. The research contributes to the literature by disentangling the effects of transport costs on the range of traded goods (extensive margin) and the traded volumes of goods (intensive margin) of international trade in order to test some of the predictions of the trade theories that introduce firm heterogeneity in productivity, as well as fixed costs of exporting. Recent investigations show that spatial frictions (distance) reduce trade mainly by trimming the number of shipments and that most firms ship only to geographically proximate customers, instead of shipping to many destinations in quantities that decrease in distance. Our analyses confirm these findings and show that the opposite pattern is observed for ad-valorem freight rates that reduce aggregate trade values mainly by reducing the volume of imported goods (intensive margin).

Client-ship and Citizenship in Latin America

Taylor, L. (2004). Client-ship and Citizenship in Latin America. Bulletin of Latin American Research. 23(2), pp.213-227. RAE2008

The geography of Celtic personal names in the Latin inscriptions of the Roman Empire

Raybould, Marilynne, and Sims-Williams, Patrick, The geography of Celtic personal names in the Latin inscriptions of the Roman Empire (Aberystwyth: CMCS publications, 2007) RAE2008

A Corpus of Latin Inscriptions of the Roman Empire containing Celtic personal names

Raybould, M. and Sims-Williams, P. (2007). A Corpus of Latin Inscriptions of the Roman Empire containing Celtic personal names. Aberystwyth: CMCS publications. RAE2008

Les manuscrits latins de la Chirurgia d'Albucasis et la lexicographie du latin m?di?val

Trotter, David, 'Les manuscrits latins de la Chirurgia d'Albucasis et la lexicographie du latin m?di?val', Archivum Latinitatis Medii Aevi (Bulletin Du Cange) (2001) 59(1) pp.181-202 RAE2008

Christian cooperation in Latin America

http://www.archive.org/details/christiancoopera00inmauoft/