Bootstrap percolation on homogeneous trees has 2 phase transitions

We study the threshold theta bootstrap percolation model on the homogeneous tree with degree b + 1, 2 <= theta <= b, and initial density p. It is known that there exists a nontrivial critical value for p, which we call p(f), such that a) for p > p(f), the final bootstrapped configuration is fully occupied for almost every initial configuration, and b) if p < p(f) , then for almost every initial configuration, the final bootstrapped configuration has density of occupied vertices less than 1. In this paper, we establish the existence of a distinct critical value for p, p(c), such that 0 < p(c) < p(f), with the following properties: 1) if p <= p(c), then for almost every initial configuration there is no infinite cluster of occupied vertices in the final bootstrapped configuration; 2) if p > p(c), then for almost every initial configuration there are infinite clusters of occupied vertices in the final bootstrapped configuration. Moreover, we show that 3) for p < p(c), the distribution of the occupied cluster size in the final bootstrapped configuration has an exponential tail; 4) at p = p(c), the expected occupied cluster size in the final bootstrapped configuration is infinite; 5) the probability of percolation of occupied vertices in the final bootstrapped configuration is continuous on [0, p(f)] and analytic on (p(c), p(f) ), admitting an analytic continuation from the right at p (c) and, only in the case theta = b, also from the left at p(f).

Extensions of homogeneous polynomials ON c(0)(l(2)(i))

We show that a 2-homogeneous polynomial on the complex Banach space c(0)(l(2)(i)) is norm attaining if and only if it is finite (i.e, depends only on finite coordinates). As the consequence, we show that there exists a unique norm-preserving extension for norm-attaining 2-homogeneous polynomials on c(0)(l(2)(i)).

UNIQUENESS OF THE EXTENSION OF 2-HOMOGENEOUS POLYNOMIALS

Homogeneous polynomials of degree 2 on the complex Banach space c(0)(l(n)(2)) are shown to have unique norm-preserving extension to the bidual space. This is done by using M-projections and extends the analogous result for c(0) proved by P.-K. Lin.

Tailored Media for Homogeneous Cellulose Chemistry: Ionic Liquid/Co-Solvent Mixtures

Co-solvents can minimize two of the major problems associated with the use of ionic liquids (ILs) as solvents for homogeneous derivatization of cellulose: high viscosity and limited miscibility with non-polar reagents or reaction products. Thus, the effects of 18 solvents and 3 binary solvent mixtures on cellulose solutions in three ILs were systematically studied with respect to the solution phase behavior. The applicable limits of these mixtures were evaluated and general guidelines for the use of co-solvents in cellulose chemistry could be advanced: Appropriate co-solvents should have EN T values (normalized empirical polarity) > 0.3, very low ``acidity`` (alpha < 0.5), and relatively high ""basicity`` (beta >= 0.4). Moreover, novel promising co-solvents and binary co-solvent mixtures were identified.

Trade, openness and domestic conflict: an empirical investigation for latin America

This paper explores whether there is an empirical relationship between trade, openness and domestic conflict for Latin America based on the analytical framework of Garfinkel, Skaperdas and Syropoulos (2004). Using ordinal regressions and Markov switching models for seventeen countries, we identify the factors responsible for the initiation and sustenance of domestic conflict. Our overall results suggest that: (i) increased trade openness reduces domestic conflict intensities but (ii) over dependence on agricultural exports, along with poor socio-political performance, lead to sustenance of low intensity conflicts. We also analyze conflict duration using proportional hazard models and find that over-reliance on agricultural exports plays the main role in conflict sustenance after controlling for socio-political factors.

The eight variable homogeneous degree three bent functions

We determine the affine equivalence classes of the eight variable degree three homogeneous bent functions using a new algorithm. Our algorithm applies to general bent functions and can systematically determine the automorphism groups. We provide a partial verification of the computer enumeration of bent functions by Meng et al.

Jorge Olivares, Allen Family Professor of Latin American Literature

Jorge Olivares, Allen Family Professor of Latin American Literature reading Sexual injustice : Supreme Court decisions from Griswold to Roe by Marc Stein

Place of latin in the church

On the evening of Wednesday 13th September 2006, Solemn Vespers began in the historic 13th century chapel of Merton College Oxford, in much the same way as it would have done at the very first sung Vespers in this beautiful space, designed for the classical Roman liturgy already very well established in its language and forms when this chapel was first built 400 years before the English Reformation. For the next four days, the Roman liturgy, through Lauds, Solemn High Mass, Vespers and Compline, were celebrated in a place that – apart from one Novus Ordo Mass in English – had not seen the glories of this Liturgy celebrated in Latin since the mid 1500s. For those of us present at the 11th International Colloquium of the International Centre for Liturgical Studies (CIEL) dedicated to “The Genius of the Roman Liturgy: Historical Diversity and Spiritual Reach”, this was not just an incredibly moving experience, though it was certainly this, but an affirmation of the timelessness and spiritual heritage of the Latin liturgy.

Crusader churches in the Latin East : expressions of medieval theology

The Latin invaders of the Eastern Mediterranean in the 12th-14th Centuries were on a mission to retrieve and protect the Christian Holy Land from Muslim occupation. They encountered a consistent, Eastern approach to the architectural expression of the Christian faith which the physical remains of their churches show they adopted. Previously, I have shown how it can be deduced from the archaeological remains of churches from the 4th-6th C that early church architecture was influenced by the theological ideas of the period. This paper argues that the Eastern approach to church architecture as adopted by the Crusaders was compatible with the medieval European theological context and can be seen as a legitimate expression of medieval theology.