Looking for vertex number one

Given an instance of the preferential attachment graph $G_n=([n],E_n)$, we would like to find vertex 1, using only 'local' information about the graph; that is, by exploring the neighborhoods of small sets of vertices. Borgs et. al gave an an algorithm which runs in time $O(\log^4 n)$, which is local in the sense that at each step, it needs only to search the neighborhood of a set of vertices of size $O(\log^4 n)$. We give an algorithm to find vertex 1, which w.h.p. runs in time $O(\omega\log n)$ and which is local in the strongest sense of operating only on neighborhoods of single vertices. Here $\omega=\omega(n)$ is any function that goes to infinity with $n$.Comment: As accepted for AA

The Isomorphism Problem for Computable Abelian p-Groups of Bounded Length

Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider only countable members. This paper explores such a notion for classes of computable structures by working out a sequence of examples. We follow recent work by Goncharov and Knight in using the degree of the isomorphism problem for a class to distinguish classifiable classes from non-classifiable. In this paper, we calculate the degree of the isomorphism problem for Abelian $p$-groups of bounded Ulm length. The result is a sequence of classes whose isomorphism problems are cofinal in the hyperarithmetical hierarchy. In the process, new back-and-forth relations on such groups are calculated.Comment: 15 page

The Lefthanded Local Lemma characterizes chordal dependency graphs

Shearer gave a general theorem characterizing the family \LLL of dependency graphs labeled with probabilities $p_v$ which have the property that for any family of events with a dependency graph from \LLL (whose vertex-labels are upper bounds on the probabilities of the events), there is a positive probability that none of the events from the family occur. We show that, unlike the standard Lov\'asz Local Lemma---which is less powerful than Shearer's condition on every nonempty graph---a recently proved `Lefthanded' version of the Local Lemma is equivalent to Shearer's condition for all chordal graphs. This also leads to a simple and efficient algorithm to check whether a given labeled chordal graph is in \LLL.Comment: 12 pages, 1 figur

PAC Learning, VC Dimension, and the Arithmetic Hierarchy

We compute that the index set of PAC-learnable concept classes is $m$-complete $\Sigma^0_3$ within the set of indices for all concept classes of a reasonable form. All concept classes considered are computable enumerations of computable $\Pi^0_1$ classes, in a sense made precise here. This family of concept classes is sufficient to cover all standard examples, and also has the property that PAC learnability is equivalent to finite VC dimension

The biosynthesis of galactolipids in red clover (Trifolium pratense) : a thesis presented in partial fulfilment of the requirements for the degree of Master of Science in Biochemistry

Galactolipids occur in all cells capable of the photosynthetic evolution of oxygen. The main galactolipids are monogalactosyl diglyceride (MGDG), digalactosyldiglyceride (DGDG) and sulfolipid (SL), MGDG being the most abundant. These three together form the major nonpigment lipids in both leaves and green algae, accounting for about 40% of the total lipids. Nearly all of the cellular MGDG of plants is present in the chloroplasts and is concentrated in the lamellae. Galactolipids contain unusually high percentages of polyunsaturated acids (MGDG contains greater than 90% in some plants), the major one being α-linolenic acid (9,12,15-octadecatrienoic acid; 18:3). Thus in higher plants most of the MGDG is monogalactosyl dilinolenoyl glycerol. MGDG may be the actual substrate of the desaturation reactions of α-linolenic biosynthesis

Shem, Shem, I Know Your Name: Tracing the Significance of Naming in Genesis

Graduate Textual or Investigativ