-
作者：Wang, Qian
摘要：We prove the local-in-time well-posedness for the solution of the compressible Euler equations in 3-D for the Cauchy data of the velocity, density and vorticity (upsilon, rho, w) is an element of H-s x H-s x H-s('), 2 < s' < s. The result extends the sharp results of Smith-Tataru and of Wang, established in the irrotational case, i.e., w = 0, which is known to be optimal for s > 2. At the opposite extreme, in the incompressible case, i.e., with a constant density, the result is known to hold f...
-
作者：Liu, Yuchen; Xu, Chenyang; Zhuang, Ziquan
摘要：We prove that on any log Fano pair of dimension n whose stability threshold is less than n+1/n, any valuation computing the stability threshold has a finitely generated associated graded ring. Together with earlier works, this implies that (a) a log Fano pair is uniformly K-stable (resp. reduced uniformly K-stable) if and only if it is K-stable (resp. K-polystable); (b) the K-moduli spaces are proper and projective; and combining with the previously known equivalence between the existence of K...
-
作者：Sawin, Will; Shusterman, Mark
摘要：Using geometric methods, we improve on the function field version of the Burgess bound and show that, when restricted to certain special subspaces, the Mobius function over F-q[T] can be mimicked by Dirichlet characters. Combining these, we obtain a level of distribution close to 1 for the Mobius function in arithmetic progressions and resolve Chowla's k-point correlation conjecture with large uniformity in the shifts. Using a function field variant of a result by Fouvry-Michel on exponential ...
-
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 ...
-
作者：Witaszek, Jakub
摘要：We develop techniques of mimicking the Frobenius action in the study of universal homeomorphisms in mixed characteristic. As a consequence, we show a mixed characteristic Keel's base point free theorem obtaining applications towards the mixed characteristic Minimal Model Program, we generalise Kollar's theorem on the existence of quotients by finite equivalence relations to mixed characteristic, and we provide a new proof of the existence of quotients by affine group schemes.
-
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.
-
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.
-
作者：Brown, Aaron; Fisher, David; Hurtado, Sebastian
摘要：We prove several cases of Zimmer's conjecture for actions of higher-rank, cocompact lattices on low-dimensional manifolds. For example, if 1-' is a cocompact lattice in SL(n, R), M is a compact manifold, and omega a volume form on M, we show that any homomorphism alpha: 1-'-+ Diff(M) has finite image if the dimension of M is less than n - 1 and that any homomor-phism alpha: 1-'-+ Diff(M, omega) has finite image if the dimension of M is less than n. The key step in the proof is to show that any...
-
作者：Merle, Frank; Raphael, Pierre; Rodnianski, Igor; Szeftel, Jeremie
摘要：In this paper, which continues our investigation of strong singularity formation in compressible fluids, we consider the compressible three-dimensional Navier-Stokes and Euler equations. In a suitable regime of barotropic laws, we construct a set of finite energy smooth initial data for which the corresponding solutions to both equations implode (with infinite density) at a later time at a point, and completely describe the associated formation of singularity. An essential step in the proof is...
-
作者：Krause, Ben; Mirek, Mariusz; Tao, Terence
摘要：We establish convergence in norm and pointwise almost everywhere for the non-conventional (in the sense of Furstenberg) bilinear polynomial ergodic averages AN(f,g)(x) := 1/N Sigma(N)(n=1) f(T(n)x)g(T(P(n))x) as N -> infinity, where T: X -> X is a measure-preserving transformation of a sigma-finite measure space (X, mu), P(n) is an element of Z[n] is a polynomial of degree d >= 2, and f is an element of L-P1 (X), g is an element of L-P2 (X) for some p1 ,p 2 > 1 with 1/p1 + 1/p2 <= 1. We also e...
