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.

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).

Closing the Clymene ocean and bending a Brasiliano belt: Evidence for the Cambrian formation of Gondwana, southeast Amazon craton

We report new paleomagnetic and geochronological data from Ediacaran rift-drift carbonates in the Paraguai belt at the southern end of the suture zone between the Amazon craton and the Sao Francisco and Rio de Plata cratons, South America. Early thrusting resulted in remagnetization ca. 528 +/- 36 Ma or later; the mean age is established by (40)Ar/(39)Ar encapsulation dating of mixed authigenic and detrital illite from remagnetized carbonates from the unmetamorphosed fold-thrust belt. This remagnetization overlaps with a 525 Ma Gondwana reference pole. Metamorphic illite from the slate belt yields (40)Ar/(39)Ar ages of 496-484 Ma, the timing of peak regional metamorphism. Oroclinal bending of the Paraguai belt was caused by a 90 degrees clockwise rotation of the east-west limb after ca. 528 Ma, probably reflecting the irregular margin of the southeast Amazon craton. The age of the Paraguai belt overlaps with that of the Pampean orogeny farther south along the western margin of the Rio de Plata craton, suggesting a coeval closure for the Clymene ocean separating the Amazon craton from the Sao Francisco and Rio de Plata cratons.

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.

Synthesis of substituted alpha,beta-unsaturated delta-lactones from vinyl tellurides

A new approach for the synthesis of alpha,beta-unsaturated delta-lactones, a unit present in many natural products with interesting biological activities is described. The approach was based on the use of a vinyl telluride, and it is complementary to the methods using ring-closing metathesis. The sequence was performed in good overall yield with retention of the Z-double bond geometry. (C) 2011 Elsevier Ltd. All rights reserved.

Mode of cembranoid action on embryonic muscle acetylcholine receptor

The mechanism of eupalmerin acetate (EUAC) actions on the embryonic muscle nicotinic acetylcholine receptor (nAChR) in BC3H-1 cells was studied by using whole-cell and single-channel patch-clamp current measurements. With whole-cell currents, EUAC did not act as an agonist on this receptor. Coapplication of 30 mu M EUAC with 50 mu M, 100 N, or 500 mu M carbamoylcholine (CCh) reversibly inhibited the current amplitude, whereas, with 20 mu M CCh, current was increased above control values in the presence of EUAC. EUAC concentration curves (0.01-40 N) obtained with 100 mu M and 500 mu M CCh displayed slope coefficients, n(H), significantly smaller than one, suggesting that EUAC bound to several sites with widely differing affinities on the receptor molecule. The apparent rate of receptor desensitization in the presence of EUAC and CCh was either slower than or equal to that obtained with CCh alone. The major finding from single-channel studies was that EUAC did not affect single-channel conductance or the ability of CCh to interact with the receptor. Instead, EUAC acted by increasing the channel closing rate constant. The results are not consistent with the competitive model for EUAC inhibition, with the sequential open-channel block model, or with inhibition by increased desensitization. The data are best accounted for by a model in which EUAC acts by closed-channel block at low concentrations, by positive modulation at intermediate concentrations, and by negative allosteric modulation of the open channel at high concentrations. (c) 2007 Wiley-Liss, Inc.