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作者：Jiang, Zilin; Tidor, Jonathan; Yao, Yuan; Zhang, Shengtong; Zhao, Yufei
摘要：Solving a longstanding problem on equiangular lines, we determine, for each given fixed angle and in all sufficiently large dimensions, the maximum number of lines pairwise separated by the given angle. Fix 0 < alpha < 1. Let N-alpha(d) denote the maximum number of lines through the origin in R-d with pairwise common angle arccos alpha. Let k denote the minimum number (if it exists) of vertices in a graph whose adjacency matrix has spectral radius exactly (1 - alpha)/(2 alpha). If k < infinity...
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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：Jaye, Benjamin; Tolsa, Xavier; Villa, Michele
摘要：In this paper we provide a proof of the Carleson epsilon(2)-conjecture. This result yields a characterization (up to exceptional sets of zero length) of the tangent points of a Jordan curve in terms of the finiteness of the associated Carleson epsilon(2)-square function.
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作者：Gardam, Giles
摘要：The unit conjecture, commonly attributed to Kaplansky, predicts that if K is a field and G is a torsion-free group, then the only units of the group ring K[G] are the trivial units, that is, the non-zero scalar multiples of group elements. We give a concrete counterexample to this conjecture; the group is virtually abelian and the field is order two.
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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：Fintzen, Jessica
摘要：Let k be a non-archimedean local field with residual characteristic p. Let G be a connected reductive group over k that splits over a tamely ramified field extension of k. Suppose p does not divide the order of the Weyl group of G. Then we show that every smooth irreducible complex representation of G(k) contains an .6-type of the form constructed by Kim-Yu and that every irreducible supercuspidal representation arises from Yu's construction. This improves an earlier result of Kim, which held ...
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作者：Heuts, Gijs
摘要：We consider a homotopy theory obtained from that of pointed spaces by inverting the maps inducing isomorphisms in v(n)-periodic homotopy groups. The case n = 0 corresponds to rational homotopy theory. In analogy with Quillen's results in the rational case, we prove that this v(n)-periodic homotopy theory is equivalent to the homotopy theory of Lie algebras in T(n)-local spectra. We also compare it to the homotopy theory of commutative coalgebras in T(n)-local spectra, where it turns out there ...
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作者：Brieussel, Jeremie; Zheng, Tianyi
摘要：We give a solution to the inverse problem (given a prescribed function, find a corresponding group) for large classes of speed, entropy, isoperimetric profile, return probability and L-p-compression functions of finitely generated groups. For smaller classes, we give solutions among solvable groups of exponential volume growth. As corollaries, we prove a recent conjecture of Amir on joint evaluation of speed and entropy exponents and we obtain a new proof of the existence of uncountably many p...
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作者：Alweiss, Ryan; Lovett, Shachar; Wu, Kewen; Zhang, Jiapeng
摘要：A sunflower with r petals is a collection of r sets so that the intersection of each pair is equal to the intersection of all of them. Erdos and Rado proved the sunflower lemma: for any fixed r, any family of sets of size w, with at least about w(w) sets, must contain a sunflower with r petals. The famous sunflower conjecture states that the bound on the number of sets can be improved to c(w) for some constant c. In this paper, we improve the bound to about (log w)(w). In fact, we prove the re...
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作者：Li, Chao; Liu, Yifeng
摘要：In this article, we study the Chow group of the motive associated to a tempered global L-packet pi of unitary groups of even rank with respect to a CM extension, whose global root number is -1. We show that, under some restrictions on the ramification of pi, if the central derivative L'(1/2, pi) is nonvanishing, then the pi-nearly isotypic localization of the Chow group of a certain unitary Shimura variety over its reflex field does not vanish. This proves part of the Beilinson-Bloch conjectur...
