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作者：Hugelmeyer, Cole
摘要：We prove that for every smooth Jordan curve gamma, if X is the set of all r is an element of [0, 1] so that there is an inscribed rectangle in gamma of aspect ratio tan(r . pi/4), then the Lebesgue measure of X is at least 1/3. To do this, we study sets of disjoint homologically nontrivial projective planes smoothly embedded in R x RP3. We prove that any such set of projective planes can be equipped with a natural total ordering. We then combine this total ordering with Kemperman's theorem in ...
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作者：Kaluba, Marek; Kielak, Dawid; Nowak, Piotr W.
摘要：We prove that Aut(F-n) has Kazhdan's property (T) for every n >= 6. Together with a previous result of Kaluba, Nowak, and Ozawa, this gives the same statement for n >= 5. We also provide explicit lower bounds for the Kazhdan constants of SAut(F-n) (with n >= 6) and of SLn (Z) (with n >= 3) with respect to natural generating sets. In the latter case, these bounds improve upon previously known lower bounds whenever n > 6.
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作者：Greene, Joshua Evan; Lobb, Andrew
摘要：For every smooth Jordan curve gamma and rectangle R in the Euclidean plane, we show that there exists a rectangle similar to R whose vertices lie on gamma. The proof relies on the theorem of Shevchishin and Nemirovski that the Klein bottle does not admit a smooth Lagrangian embedding in C-2.
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作者：Burness, Timothy C.; Guralnick, Robert M.; Harper, Scott
摘要：A group G is said to 3/2-generated if every nontrivial element belongs to a generating pair. It is easy to see that if G has this property, then every proper quotient of G is cyclic. In this paper we prove that the converse is true for finite groups, which settles a conjecture of Breuer, Guralnick and Kantor from 2008. In fact, we prove a much stronger result, which solves a problem posed by Brenner and Wiegold in 1975. Namely, if G is a finite group and every proper quotient of G is cyclic, t...
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作者：Birkar, Caucher
摘要：We study log canonical thresholds (also called global log canonical threshold or alpha-invariant) of R-linear systems. We prove existence of positive lower bounds in different settings, in particular, proving a conjecture of Ambro. We then show that the Borisov-Alexeev-Borisov conjecture holds; that is, given a natural number d and a positive real number epsilon, the set of Fano varieties of dimension d with epsilon-log canonical singularities forms a bounded family. This implies that biration...
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作者：Harvey, David; van der Hoeven, Joris
摘要：We present an algorithm that computes the product of two n-bit integers in O (n log n) bit operations, thus confirming a conjecture of Schonhage and Strassen from 1971. Our complexity analysis takes place in the multitape Turing machine model, with integers encoded in the usual binary representation. Central to the new algorithm is a novel Gaussian resampling technique that enables us to reduce the integer multiplication problem to a collection of multidimensional discrete Fourier transforms o...
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作者：Beuzart-Plessis, Raphael; Liu, Yifeng; Zhang, Wei; Zhu, Xinwen
摘要：We introduce a new technique for isolating components on the spectral side of the trace formula. By applying it to the Jacquet-Rallis relative trace formula, we complete the proof of the global Gan-Gross-Prasad conjecture and its refinement Ichino-Ikeda conjecture for U(n) x U(n + 1) in the stable case.
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作者：Oh, Sung-Jin; Tataru, Daniel
摘要：This article represents the fourth and final part of a four-paper sequence whose aim is to prove the Threshold Conjecture as well as the more general Dichotomy Theorem for the energy critical 4 + 1-dimensional hyperbolic Yang Mills equation. The Threshold Theorem asserts that topologically trivial solutions with energy below twice the ground state energy are global and scatter. The Dichotomy Theorem applies to solutions in arbitrary topological class with large energy, and provides two exclusi...
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作者：Cheeger, Jeff; Jiang, Wenshuai; Naber, Aaron
摘要：This paper is concerned with the structure of Gromov-Hausdorff limit d spaces (M-i(n), gi, pi) ->(dGH )(X-n ,d,p) of Riemannian manifolds satisfying a uniform lower Ricci curvature bound RicM(i)(n) >= -(n - 1) as well as the noncollapsing assumption Vol(B-1(p(i))) > v > 0. In such cases, there is a filtration of the singular set, S-0 subset of S-1 center dot center dot center dot Sn-1 := S, where S-k := {x is an element of X : no tangent cone at x is (k +1)-symmetric}. Equivalently, S-k is the...
