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We introduce a graph Ramsey game called Ramsey, Paper, Scissors. This game has two players, Proposer and Decider. Starting from an empty graph on $n$ vertices, on each turn Proposer proposes a potential edge and Decider simultaneously decides (without knowing Proposer's choice) whether to add it to the graph. Proposer cannot propose an edge which would create a triangle in the graph. The game ends when Proposer has no legal moves remaining, and Proposer wins if the final graph has independence number at least $s$. We prove a threshold phenomenon exists for this game by exhibiting randomized strategies for both players that are optimal up to constants. Namely, there exist constants $0B\sqrt{n}\log{n}$. This is a factor of $\Theta(\sqrt{\log{n}})$ larger than the lower bound coming from the off-diagonal Ramsey number $r(3,s)$.

The purpose of the present note is to point out mathematical mistakes in some published papers on boundary value problems to prevent usage of mistaken results and methods by mathematicians. Copyright © 2013 John Wiley & Sons, Ltd.

We show abstract versions for Banach couples of several limiting compact interpolation theorems established by Edmunds and Opic for couples of Lp spaces.

We investigate the Chow groups of projective determinantal varieties and those of their strata of matrices of fixed rank, using Chern class computations (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We give an example of an affinoid curve without analytic continuation. We use this to produce an example of an affinoid morphism that cannot be flattened by a finite sequence of local blow-ups. Thus the global rigid analogue of Hironaka's complex analytic flattening theorem given by T. Gardener and H. Schoutens, in Theorem 2.3 of ?Flattening and subanalytic sets in rigid analytic geometry?, Proc. London Math. Soc (3) 83 (2001) 681?707, is not true. Since this is a key step in the proof of the affinoid elimination theorem (loc. cit. Theorem 3.12), that proof contains a serious gap. We also give an example of an affinoid subset of the plane that is not the image under a proper rigid analytic map of a set that is globally semianalytic in the domain of that map. This clarifies the relationship among several natural categories of rigid subanalytic sets.

A new property of a complete intersection in a graded Noetherian ring is proved; this result, in conjunction with the Jacobian criterion, is then applied to the description of the loci of singularities in a family of affine varieties introduced in a well-known paper on forms of many variables.

Let c e IN. For d e ℤ with gcd (d, c) = 1 let δ(d, c) be defined by d · δ(d, c) 1 mod c, 1 ⩽ δ(d, c) ⩽ c. Let s, t e IN with 1 ⩽ s ⩽ c, 1 ⩽ t ⩽ c. The main result is that for arbitrary fixed e > 0, but uniformly over c, s and t Equivalently, the points (d, δ(c,d)) are approximately uniformly distributed in [0, c] × [0, c]: The two-dimensional discrepancy of {(d,δ (c, d)): 1 ⩽ d ⩽ c and (c, d)= 1} in [0,c] × [0,c] is Oe(ce-1/2).

In a preceding paper of Jin and Wang the authors describe computations concerning differential systems of the form x=y, y=-x+a 1 x 2 +a 2 xy+a 3 y 2 +a 4 x x3 +a 5 x 2 y+a 6 xy 2 +a 7 y 3 . We give proofs of the results suggested by these computations and provide a full description of the bifurcation of small-amplitude limit cycles in a subclass of these systemes