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作者：Beresnevich, Victor
摘要：This work is motivated by problems on simultaneous Diophantine approximation on manifolds, namely, establishing Khintchine and Jarnik type theorems for submanifolds of R-n. These problems have attracted a lot of interest since Kleinbock and Margulis proved a related conjecture of Alan Baker and V. G. Sprinauk. They have been settled for planar curves but remain open in higher dimensions. In this paper, Khintchine and Jarnik type divergence theorems are established for arbitrary analytic nondeg...
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作者：Kaveh, Kiumars; Khovanskii, A. G.
摘要：Generalizing the notion of Newton polytope, we define the Newton-Okounkov body, respectively, for semigroups of integral points, graded algebras and linear series on varieties. We prove that any semigroup in the lattice Z(n) is asymptotically approximated by the semigroup of all the points is a sublattice and lying in a convex cone. Applying this we obtain several results. We show that for a large class of graded algebras, the Hilbert functions have polynomial growth and their growth coefficie...
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作者：Polymath, D. H. J.
摘要：The Hales-Jewett theorem asserts that for every r and every k there exists n such that every r-colouring of the n-dimensional grid {1, ... , k}(n) contains a monochromatic combinatorial line. This result is a generalization of van der Waerden's theorem, and it is one of the fundamental results of Ramsey theory. The theorem of van der Waerden has a famous density version, conjectured by Erdos and Turan in 1936, proved by Szemeredi in 1975, and given a different proof by Furstenberg in 1977. The...
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作者：Wooley, Trevor D.
摘要：We obtain estimates for Vinogradov's integral that for the first time approach those conjectured to be the best possible. Several applications of these new bounds are provided. In particular, the conjectured asymptotic formula in Waring's problem holds for sums of 8 kth powers of natural numbers whenever s >= 2k(2) + 2k - 3.
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作者：Haglund, Frederic; Wise, Daniel T.
摘要：We prove that certain compact cube complexes have special finite covers. This means they have finite covers whose fundamental groups are quasiconvex sub-groups of right-angled Artin groups. As a result we obtain, linearity and the separability of quasiconvex subgroups, for the groups we consider. Our result applies in particular to compact negatively curved cube complexes whose hyperplanes don't self-intersect. For cube complexes with word-hyperbolic fundamental group, we are able to show that...
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作者：Johnson, W. B.; Szankowski, A.
摘要：If X is a Banach space such that the isomorphism constant to l(2)(n) from n-dimensional subspaces grows sufficiently slowly as n -> infinity, then X has the approximation property. A consequence of this is that there is a Banach space X with a symmetric basis but not isomorphic to l(2) so that all subspaces of X have the approximation property. This answers a problem raised in 1980 [7]. An application of the main result is that there is a separable Banach space X that is not isomorphic to a Hi...
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作者：Fernandez, Luis
摘要：We prove the conjecture, posed in 1993 by Bolton and Woodward, that the dimension of the space of harmonic maps from the 2-sphere to the 2n-sphere is 2d + n(2). We also give an explicit algebraic method to construct all harmonic maps from the 2-sphere to the m-sphere.
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作者：Marshall, T. H.; Martin, G. J.
摘要：This paper represents the final step in solving the problem, posed by Siegel in 1945, of determining the minimal co-volume lattices of hyperbolic 3-space HI (also Problem 3.60 (F) in the Kirby problem list from 1993). Here we identify the two smallest co-volume lattices. Both these groups are two-generator arithmetic lattices, generated by two elements of finite orders 2 and 3. Their co-volumes are 0.0390 ... and 0.0408 ...; the precise values are given in terms of the Dedekind zeta function o...
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作者：Santos, Francisco
摘要：The Hirsch Conjecture (1957) stated that the graph of a d-dimensional polytope with n facets cannot have (combinatorial) diameter greater than n-d. That is, any two vertices of the polytope can be connected by a path of at most n - d edges. This paper presents the first counterexample to the conjecture. Our polytope has dimension 43 and 86 facets. It is obtained from a 5-dimensional polytope with 48 facets that violates a certain generalization of the d-step conjecture of Klee and Walkup.
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作者：Croot, Ernie; Granville, Andrew; Pemantle, Robin; Tetali, Prasad
摘要：In the fastest-performing integer factoring algorithms, one creates a sequence of integers (in a pseudo-random way) and wishes to rapidly determine a subsequence whose product is a square. In 1994 Pomerance stated the following problem which encapsulates all of the key issues: Select integers a(1), a(2), ... , at random from the interval [1,x], until some (nonempty) subsequence has product equal to a square. Find a good estimate for the expected stopping time of this process. A good solution s...
