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作者：Martel, Yvan; Merle, Frank
摘要：In this paper, we give the first description of the collision of two solitons for a nonintegrable equation in a special regime. We consider solutions of the quartic gKdV equation partial derivative(t)u + partial derivative(x)(partial derivative(2)(x)u + u(4)) = 0, which behave as t -> -infinity like u(t, x) = Q(c1) (x - c(1)t) + Q(c2) (x - c(2)t) + eta(t, x), where Q(c)(x - ct) is a soliton and parallel to eta(t)parallel to(H1) << parallel to Q(c2)parallel to(H1) << parallel to Q(c1)parallel t...
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作者：Friedl, Stefan; Vidussi, Stefano
摘要：A classical result in knot theory says that for a fibered knot the Alexander polynomial is monic and that the degree equals twice the genus of the knot. This result has been generalized by various authors to twisted Alexander polynomials and fibered 3-manifolds. In this paper we show that the conditions on twisted Alexander polynomials are not only necessary but also sufficient for a 3-manifold to be fibered. By previous work of the authors this result implies that if a manifold of the form S-...
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作者：Benson, David J.; Iyengar, Srikanth B.; Krause, Henning
摘要：We classify localising subcategories of the stable module category of a finite group that are closed under tensor product with simple (or, equivalently all) modules. One application is a proof of the telescope conjecture in this context. Others include new proofs of the tensor product theorem and of the classification of thick subcategories of the finitely generated modules which avoid the use of cyclic shifted subgroups. Along the way we establish similar classifications for differential grad...
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作者：del Pino, Manuel; Kowalczyk, Michal; Wei, Juncheng
摘要：A celebrated conjecture due to De Giorgi states that any bounded solution of the equation Delta u + (1 - u(2))u = 0 in R-N with partial derivative(yN)u > 0 must be such that its level sets {u = lambda} are all hyperplanes, at least for dimension N <= 8. A counterexample for N >= 9 has long been believed to exist. Starting from a minimal graph F which is not a hyperplane, found by Bombieri, De Giorgi and Giusti in R-N, N >= 9, we prove that for any small alpha > 0 there is a bounded solution u(...
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作者：Larsen, Michael; Shalev, Aner; Pham Huu Tiep
摘要：The classical Waring problem deals with expressing every natural number as a sum of g(k) k-th powers. Recently there has been considerable interest in similar questions for non-abelian groups, and simple groups in particular. Here the k-th power word can be replaced by an arbitrary group word w not equal 1, and the goal is to express group elements as short products of values of w. We give a best possible and somewhat surprising solution for this Waring type problem for (non-abelian) finite si...
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作者：Meersseman, Laurent; Verjovsky, Alberto
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作者：Cantat, Serge
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作者：Breuillard, Emmanuel
摘要：We introduce a conjugation invariant normalized height (h) over cap (F) on finite subsets of matrices F in GL(d)((Q) over bar) and describe its properties. In particular, we prove an analogue of the Lehmer problem for this height by showing that (h) over cap (F) > epsilon whenever F generates a nonvirtually solvable subgroup of GL(d)((Q) over bar), where epsilon = epsilon(d) > 0 is an absolute constant. This can be seen as a global adelic analog of the classical Margulis Lemma from hyperbolic ...
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作者：Pila, Jonathan
摘要：We give an unconditional proof of the Andre-Oort conjecture for arbitrary products of modular curves. We establish two generalizations. The first includes the Manin-Mumford conjecture for arbitrary products of elliptic curves defined over Q as well as Lang's conjecture for torsion points in powers of the multiplicative group. The second includes the Manin Mumford conjecture for abelian varieties defined over Q. Our approach uses the theory of o-minimal structures, a part of Model Theory, and f...
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作者：Brezis, Haim; Nguyen, Hoai-Minh
摘要：H. Brezis and L. Nirenberg proved that if (g(k)) subset of C-0(S-N, S-N) and g is an element of C-0(S-N, S-N) (N >= 1) are such that g(k) -> g in BMO(S-N), then deg g(k) -> deg g. On the other hand, if g is an element of C-1(S-N, S-N), then Kronecker's formula asserts that deg = 1/vertical bar S-N vertical bar integral(SN) det(del g)d sigma. Consequently, integral(SN) det(del g(k)) d sigma converges to integral(SN) det(del g)d sigma provided g(k) -> g in BMO(S-N). In the same spirit, we consid...
