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作者：Ding, Jian; Sly, Allan; Sun, Nike
摘要：We establish the satisfiability threshold for random k-SAT for all k >= k(0), with k(0) an absolute constant. That is, there exists a limiting density alpha(sat) (k) such that a random k-SAT formula of clause density alpha is with high probability satisfiable for alpha < alpha(sat), and unsatisfiable for alpha > alpha(sat). We show that the threshold alpha(sat)(k) is given explicitly by the one-step replica symmetry breaking prediction from statistical physics. The proof develops a new analyti...
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作者：Albritton, Dallas; Brue, Elia; Colombo, Maria
摘要：In a seminal work, Leray (1934) demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. We exhibit two distinct Leray solutions with zero initial velocity and identical body force. Our approach is to construct a ???background??? solution which is unstable for the Navier-Stokes dynamics in similarity variables; its similarity profile is a smooth, compactly supported vortex ring whose cross-section is a modification of the unstable two-dimensional ...
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作者：Judge, Chris; Mondal, Sugata
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作者：Raskin, Sam
摘要：We prove the rank 1 case of a conjecture of Frenkel-Gaitsgory: critical level Kac-Moody representations with regular central characters localize onto the affine Grassmannian. The method uses an analogue in local geometric Langlands of the existence of Whittaker models for most representations of GL(2) over a non-Archimedean field.
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作者：Bialy, Misha; Mironov, Andrey E.
摘要：In this paper we prove the Birkhoff-Poritsky conjecture for centrally-symmetric C-2-smooth convex planar billiards. We assume that the domain A between the invariant curve of 4-periodic orbits and the boundary of the phase cylinder is foliated by C-0-invariant curves. Under this assumption we prove that the billiard curve is an ellipse. For the original Birkhoff-Poritsky formulation we show that if a neighborhood of the boundary of billiard domain has a C-1-smooth foliation by convex caustics ...
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作者：Avila, Artur; Lyubich, Mikhail
摘要：We construct Feigenbaum quadratic-like maps with a Julia set of positive Lebesgue measure. Indeed, in the quadratic family Pc : z -> z2 + c the corresponding set of parameters c is shown to have positive Hausdorff dimension. Our examples include renormalization fixed points, and the corresponding quadratic polynomials in their stable manifold are the first known rational maps for which the hyperbolic dimension is different from the Hausdorff dimension of the Julia set.
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作者：Moreira, Carlos Gustavo T. de A.; Zamudio, Alex Mauricio
摘要：This is an erratum for the paper Stable intersections of regular Cantor sets with large Hausdorff dimensions by Moreira and Yoccoz. We show how to fix a flaw - a bad choice of parameters - in the proof of the scale recurrence lemma. This lemma is an important step towards establishing the main theorem.
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作者：Milman, Emanuel; Neeman, Joe
摘要：We establish the Gaussian Multi-Bubble Conjecture: the least Gaussianweighted perimeter way to decompose Rn into q cells of prescribed (positive) Gaussian measure when 2 < q < n+ 1, is to use a simplicial cluster, obtained from the Voronoi cells of q equidistant points. Moreover, we prove that simplicial clusters are the unique isoperimetric minimizers (up to nullsets). In particular, the case q = 3 confirms the Gaussian Double-Bubble Conjecture: the unique least Gaussian-weighted perimeter wa...
