The curve selection lemma and the Morse-Sard theorem

We use an inequality due to Bochnak and Lojasiewicz, which follows from the Curve Selection Lemma of real algebraic geometry in order to prove that, given a C(r) function f : U subset of R(m) -> R, we have lim(y -> xy is an element of crit(f)) vertical bar f(y) - f(x)vertical bar/vertical bar y - x vertical bar(r) = 0, for all x is an element of crit(f)` boolean AND U, where crit( f) = {x is an element of U vertical bar df ( x) = 0}. This shows that the so-called Morse decomposition of the critical set, used in the classical proof of the Morse-Sard theorem, is not necessary: the conclusion of the Morse decomposition lemma holds for the whole critical set. We use this result to give a simple proof of the classical Morse-Sard theorem ( with sharp differentiability assumptions).

Experimental micromorphology in Tierra del Fuego (Argentina): building a reference collection for the study of shell middens in cold climates

Commonly used in archaeological contexts, micromorphology did not see a parallel advance in the field of experimental archaeology. Drawing from early work conducted in the 1990`s on ethnohistoric sites in the Beagle Channel, we analyze a set of 25 thin sections taken from control features and experimental tests. The control features include animal pathways and environmental contexts (beach samples, forest litter, soils from the proximities of archaeological sites), while the experimental samples comprise anthropic structures, such as hearths, and valves of Mytilus edulis (the most important component of shell middens in the region) heated from 200 degrees C to 800 degrees C. Their micromorphological study constitutes a modern analogue to assist archaeologists studying site formation and ethnographical settings in cold climates, with particular emphasis on shell midden contexts. (c) 2010 Elsevier Ltd. All rights reserved.

THE ENDOPARASITE PILOSTYLES ULEI (APODANTHACEAE - CUCURBITALES) INFLUENCES WOOD STRUCTURE IN THREE HOST SPECIES OF MIMOSA

Pilostyles species (Apodanthaceae) are endoparasites in stems of the plant family Fabaceae. The body comprises masses of parenchyma in the host bark and cortex, with sinkers, comprising groups of twisted tracheal elements surrounded by parenchyma that enter the secondary xylem of the host plant. Here we report for the first time the effects of Pilostyles parasitism on host secondary xylem. We obtained healthy and parasitized stems from Mimosa foliolosa, M. maguirei and M. setosa and compared vessel element length, fiber length, vessel diameter and vessel frequency, measured through digital imaging. Also, tree height and girth were compared between healthy and parasitized M. setosa. When parasitized, plant size, vessel diameter, vessel element length and fiber length are all less than in healthy plants. Also, vessel frequency is greater and vessels are narrower in parasitized stems. These responses to parasitism are similar to those observed in stressed plants. Thus, hosts respond to the parasite by changing its wood micromorphology in favour of increased hydraulic safety.

Evolutionary patterns in neotropical Helieae (Gentianaceae): evidence from morphology, chloroplast and nuclear DNA sequences

Parsimony-based phylogenetic analyses of the neotropical tribe Helieae (Gentianaceae) are presented, including 22 of the 23 genera and 60 species. This study is based on data from morphology, palynology, and seed micromorphology (127 structural characters), and DNA sequences (matK, trnL intron, ITS). Phylogenetic reconstructions based on ITS and morphology provided the greatest resolution, morphological data further helping to tentatively place several taxa for which DNA was not available (Celiantha, Lagenanthus, Rogersonanthus, Roraimaea, Senaea, Sipapoantha, Zonanthus). Celiantha, Prepusa and Senaea together appear as the sister clade to the rest of Helieae. The remainder of Helieae is largely divided into two large subclades, the Macrocarpaea subclade and the Symbolanthus subclade. The first subclade includes Macrocarpaea, sister to Chorisepalum, Tochia, and Zonanthus. Irlbachia and Neblinantha are placed as sisters to the Symbolanthus subclade, which includes Aripuana, Calolisianthus, Chelonanthus, Helia, Lagenanthus, Lehmanniella, Purdieanthus, Rogersonanthus, Roraimaea, Sipapoantha, and symbolanthus. Generic-level polyphyly is detected in Chelonanthus and Irlbachia. Evolution of morphological characters is discussed, and new pollen and seed characters are evaluated for the first time in a combined morphological-molecular phylogenetic analysis.

On C(r)-closing for flows on orientable and non-orientable 2-manifolds

We provide an affirmative answer to the C(r)-Closing Lemma, r >= 2, for a large class of flows defined on every closed surface.

Weak hypergraph regularity and linear hypergraphs

We consider conditions which allow the embedding of linear hypergraphs of fixed size. In particular, we prove that any k-uniform hypergraph H of positive uniform density contains all linear k-uniform hypergraphs of a given size. More precisely, we show that for all integers l >= k >= 2 and every d > 0 there exists Q > 0 for which the following holds: if His a sufficiently large k-uniform hypergraph with the property that the density of H induced on every vertex subset of size on is at least d, then H contains every linear k-uniform hypergraph F with l vertices. The main ingredient in the proof of this result is a counting lemma for linear hypergraphs, which establishes that the straightforward extension of graph epsilon-regularity to hypergraphs suffices for counting linear hypergraphs. We also consider some related problems. (C) 2009 Elsevier Inc. All rights reserved.

Sparse partition universal graphs for graphs of bounded degree

In 1983, Chvatal, Trotter and the two senior authors proved that for any Delta there exists a constant B such that, for any n, any 2-colouring of the edges of the complete graph K(N) with N >= Bn vertices yields a monochromatic copy of any graph H that has n vertices and maximum degree Delta. We prove that the complete graph may be replaced by a sparser graph G that has N vertices and O(N(2-1/Delta)log(1/Delta)N) edges, with N = [B`n] for some constant B` that depends only on Delta. Consequently, the so-called size-Ramsey number of any H with n vertices and maximum degree Delta is O(n(2-1/Delta)log(1/Delta)n) Our approach is based on random graphs; in fact, we show that the classical Erdos-Renyi random graph with the numerical parameters above satisfies a stronger partition property with high probability, namely, that any 2-colouring of its edges contains a monochromatic universal graph for the class of graphs on n vertices and maximum degree Delta. The main tool in our proof is the regularity method, adapted to a suitable sparse setting. The novel ingredient developed here is an embedding strategy that allows one to embed bounded degree graphs of linear order in certain pseudorandom graphs. Crucial to our proof is the fact that regularity is typically inherited at a scale that is much finer than the scale at which it is assumed. (C) 2011 Elsevier Inc. All rights reserved.

An eigenvalue problem for the biharmonic operator on Z(2)-symmetric regions

In this work we show that the eigenvalues of the Dirichlet problem for the biharmonic operator are generically simple in the set Of Z(2)-symmetric regions of R-n, n >= 2, with a suitable topology. To accomplish this, we combine Baire`s lemma, a generalised version of the transversality theorem, due to Henry [Perturbation of the boundary in boundary value problems of PDEs, London Mathematical Society Lecture Note Series 318 (Cambridge University Press, 2005)], and the method of rapidly oscillating functions developed in [A. L. Pereira and M. C. Pereira, Mat. Contemp. 27 (2004) 225-241].

The 3-colored Ramsey number of even cycles

Denote by R(L, L, L) the minimum integer N such that any 3-coloring of the edges of the complete graph on N vertices contains a monochromatic copy of a graph L. Bondy and Erdos conjectured that when L is the cycle C(n) on n vertices, R(C(n), C(n), C(n)) = 4n - 3 for every odd n > 3. Luczak proved that if n is odd, then R(C(n), C(n), C(n)) = 4n + o(n), as n -> infinity, and Kohayakawa, Simonovits and Skokan confirmed the Bondy-Erdos conjecture for all sufficiently large values of n. Figaj and Luczak determined an asymptotic result for the `complementary` case where the cycles are even: they showed that for even n, we have R(C(n), C(n), C(n)) = 2n + o(n), as n -> infinity. In this paper, we prove that there exists n I such that for every even n >= n(1), R(C(n), C(n), C(n)) = 2n. (C) 2009 Elsevier Inc. All rights reserved.