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作者：An, Xinliang
摘要：Employing the Einstein-scalar field system, we demonstrate an approach for proving high co-dimensional nonlinear instability of naked-singularity solutions as constructed by Christodoulou in 1994. We further investigate the censorship of Christodoulou's naked singularity and show that a tiny anisotropic perturbation arising from the outgoing characteristic initial data would lead to the emergence of an anisotropic apparent horizon, which covers and censors the naked singularity. Our approach a...
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作者：Bhargava, Manjul
摘要：Of the (2H+1)(n) monic integer polynomials f(x)=x(n)+a1x(n-1)+...+a(n) with max{|a(1)|,...,|a(n)|} <= H, how many have associated Galois group that is not the full symmetric group S-n?There are clearly >> Hn-1 such polynomials, as may be obtained by setting a(n)=0. In 1936, van der Waerden conjectured that O(Hn-1) should in fact also be the correct upper bound for the count of such polynomials. The conjecture has been known previously for degrees n <= 4, due to work of van der Waerden and Chow...
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作者：Poels, Anthony
摘要：In his seminal 1961 paper, Wirsing studied how well a given transcendental real number xi can be approximated by algebraic numbers alpha of degree at most n for a given positive integer n , in terms of the so-called naive height H(alpha) of alpha . He showed that the supremum omega & lowast;n(xi) of all omega for which infinitely many such alpha have |xi- alpha| <= H(alpha)-omega-1 is at least (n + 1)/2. He also asked if we could even have omega & lowast;n(xi) >= nas it is generally expected. ...
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作者：Shankar, Ravi; Yuan, Yu
摘要：We derive a priori interior Hessian estimates and interior regularity for the sigma 2 equation in dimension four. Our method provides both a new proof for the corresponding three dimensional results and a Hessian estimate for smooth solutions satisfying a dynamic semi-convexity condition in higher (n >= 5) dimensions.
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作者：Prasad, Gopal
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作者：Jaikin-Zapirain, Andrei; Linton, Marco
摘要：We prove that one-relator groups are coherent, solving a well-known problem of Gilbert Baumslag. Our proof strategy is readily applicable to many classes of groups of cohomological dimension two. We show that fundamental groups of two complexes with non-positive immersions are homologically coherent, we show that groups with staggered presentations and many Coxeter groups are coherent and we show that group algebras over fields of characteristic zero of groups with reducible presentations with...
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作者：Manolescu, Ciprian; Ozsvath, Peter s.; Thurston, Dylan p.
摘要：We give combinatorial descriptions of the Heegaard Floer homology groups for arbitrary three-manifolds (with coefficients in Z/2Z). The descriptions are based on presenting the three-manifold as an integer surgery on a link in the three-sphere, and then using a grid diagram for the link. We also give combinatorial descriptions of the mod 2 Ozsv & aacute;th-Szab & oacute; mixed invariants of closed four-manifolds, also in terms of grid diagrams.
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作者：Yang, Lei
摘要：In this paper, we will prove an effective version of Ratner's equidistribution theorem for unipotent orbits in SL(3, R)/SL(3, Z) with a natural Diophantine condition.
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作者：Ruhstorfer, Lucas
摘要：In this paper we consider the inductive Alperin-McKay condition for quasi-isolated 2-blocks of exceptional groups of Lie type. Thereby, we complete the proof of the Alperin-McKay conjecture and Brauer's height zero conjecture for the prime 2.
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作者：Haag, Summer; Kertzer, Clyde; Rickards, James; Stange, Katherine E.
摘要：In a primitive integral Apollonian circle packing, the curvatures that appear must fall into one of six or eight residue classes modulo 24. The local-global conjecture states that every sufficiently large integer in one of these residue classes appears as a curvature in the packing. We prove that this conjecture is false for many packings, by proving that certain quadratic and quartic families are missed. The new obstructions are a property of the thin Apollonian group (and not its Zariski clo...
