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作者：Bjorner, Anders; Ekedahl, Torsten
摘要：Let (W, S) be a crystallographic Coxeter group (this includes all finite and affine Weyl groups), and let J subset of S. Let W-J denote the set of minimal coset representatives modulo the parabolic subgroup W-J. For w is an element of W-J, let f(i)(w,J) denote the number of elements of length i below w in Bruhat order on W-J (with notation simplified to f(i)(w) in the case when W-J = W). We show that 0 <= i < j <= l(w)-i implies f(i)(w,J) <= f(j)(w,J). Also, the case of equalities f(i)(w) = f(...
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作者：Behrend, Kai
摘要：We prove that Donaldson-Thomas type invariants are equal to weighted Euler characteristics of their moduli spaces. In particular, such invariants depend only on the scheme structure of the moduli space, not the symmetric obstruction theory used to define them. We also introduce new invariants generalizing Donaldson-Thomas type invariants to moduli problems with open moduli space. These are useful for computing Donaldson-Thomas type invariants over stratifications.
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作者：Avila, Artur; Kahn, Jeremy; Lyubich, Mikhail; Shen, Weixiao
摘要：We prove that any unicritical polynomial f(c) : z bar right arrow z(d) + c which is at most finitely renormalizable and has only repelling periodic points is combinatorially rigid. This implies that the connectedness locus (the Multibrot set) is locally connected at the corresponding parameter values and generalizes Yoccoz's Theorem for quadratics to the higher degree case.
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作者：Avila, Artur; Jitomirskaya, Svetlana
摘要：We prove the conjecture (known as the Ten Martini Problem after Kac and Simon) that the spectrum of the almost Mathieu operator is a Cantor set for all nonzero values of the coupling and all irrational frequencies.
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作者：Ru, Min
摘要：This paper establishes a defect relation for algebraically nondegenerate holomorphic mappings into an arbitrary nonsingular complex projective variety V (rather than just the projective space) intersecting possible nonlinear hypersurfaces, extending the result of H. Cartan.
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作者：Biss, Daniel K.
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作者：Fefferman, Charles; Klartag, Bo'az
摘要：Suppose we are given a finite subset E subset of R-n and a function f : E -> R. How to extend f to a C-m function F : R-n -> R with C-m norm of the smallest possible order of magnitude? In this paper and in [20] we tackle this question from the perspective of theoretical computer science. We exhibit algorithms for constructing such an extension function F, and for computing the order of magnitude of its C-m norm. The running time of our algorithms is never more than CN log N, where N is the ca...
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作者：Ramadas, T. R.
摘要：Let a reductive group G act oil a projective variety chi(+), and suppose given a lift of the action to an ample line bundle (theta) over cap. By definition, all G-invariant sections of (theta) over cap vanish on the nonsemistable locus chi(nss)(+). Taking an appropriate normal derivative defines a map H-0(chi(+),(theta) over cap)(G) -> H-0(S-mu,V-mu)(G), where V-mu is a G-vector bundle on a G-variety S-mu. We call this the Harder-Narasimhan trace. Applying this to the Geometric Invariant Theor...
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作者：Savin, Ovidiu
摘要：We consider local minimizers of the Ginzburg-Landau energy functional integral 1/2 vertical bar del u vertical bar(2) + 1/4(1-u(2))(2)dx and prove that, if the 0 level set is included in a flat cylinder then, in the interior, it is included in a flatter cylinder. As a consequence we prove a conjecture of De Giorgi which states that level sets of global solutions of Delta u = u(3) - u such that vertical bar u vertical bar <= 1, partial derivative(n)u > 0, lim(xn ->+/-infinity) u(x', x(n)) = +/-...
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作者：Goldston, Daniel A.; Pintz, Janos; Yildirim, Cem Y.
摘要：We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the Elliott-Halberstam conjecture, we prove that there are infinitely often primes differing by 16 or less. Even a much weaker conjecture implies that there are infinitely often primes a bounded distance apart. Unconditionally, we prove that there exist consecutive primes which are closer than any arbitrar...
