Hybrid journal (It can contain Open Access articles) ISSN (Print) 0033-5606 - ISSN (Online) 1464-3847 Published by Oxford University Press[419 journals]

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Authors:Humphries P. Pages: 01 - 16 Abstract: AbstractWe study various statistics regarding the distribution of the points $$ \left\{\left(\frac{d}{q},\frac{\overline{d}}{q}\right) \in \mathbb{T}^2 : d \in (\mathbb{Z}/q\mathbb{Z})^{\times}\right\} $$ as q tends to infinity. Due to non-trivial bounds for Kloosterman sums, it is known that these points equidistribute on the torus. We prove refinements of this result, including bounds for the discrepancy, small-scale equidistribution, bounds for the covering exponent associated with these points, sparse equidistribution, and mixing. PubDate: Thu, 08 Jul 2021 00:00:00 GMT DOI: 10.1093/qmath/haab015 Issue No:Vol. 73, No. 1 (2021)

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Authors:Faraco D; Guerra A. Pages: 17 - 21 Abstract: AbstractWe give a very concise proof of Ornstein’s L1 non-inequality for first- and second-order operators in two dimensions. The proof just needs a two-dimensional laminate supported on three points. PubDate: Thu, 18 Mar 2021 00:00:00 GMT DOI: 10.1093/qmath/haab016 Issue No:Vol. 73, No. 1 (2021)

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Authors:Sosnilo V. Pages: 23 - 44 Abstract: AbstractWe study regularity in the context of connective ring spectra and spectral stacks. Parallel to that, we construct a weight structure on the category of compact quasi-coherent sheaves on spectral quotient stacks of the form $X=[\operatorname{Spec} R/G]$ defined over a field, where R is a connective ${{\mathcal{E}}_\infty}$-k-algebra and G is a linearly reductive group acting on R. Under reasonable assumptions, we show that regularity of X is equivalent to regularity of R. We also show that if R is bounded, such a stack is discrete. This result can be interpreted in terms of weight structures and suggests a general phenomenon: for a symmetric monoidal stable $\infty$-category with a compatible bounded weight structure, the existence of an adjacent t-structure satisfying a strong boundedness condition should imply discreteness of the weight-heart. We also prove a gluing result for weight structures and adjacent t-structures, in the setting of a semi-orthogonal decomposition of stable $\infty$-categories. PubDate: Tue, 30 Mar 2021 00:00:00 GMT DOI: 10.1093/qmath/haab017 Issue No:Vol. 73, No. 1 (2021)

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Authors:GIMÉNEZ CONEJERO R; Giménez Conejero J. Pages: 45 - 63 Abstract: AbstractWe show three basic properties of the image Milnor number µI(f) of a germ $f\colon(\mathbb{C}^{n},S)\rightarrow(\mathbb{C}^{n+1},0)$ with isolated instability. First, we show the conservation of the image Milnor number, from which one can deduce the upper semi-continuity and the topological invariance for families. Second, we prove the weak Mond’s conjecture, which states that µI(f) = 0 if and only if f is stable. Finally, we show a conjecture by Houston that any family $f_t\colon(\mathbb{C}^{n},S)\rightarrow(\mathbb{C}^{n+1},0)$ with $\mu_I(\,f_t)$ constant is excellent in Gaffney’s sense. For technical reasons, in the last two properties, we consider only the corank 1 case. PubDate: Sat, 27 Mar 2021 00:00:00 GMT DOI: 10.1093/qmath/haab019 Issue No:Vol. 73, No. 1 (2021)

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Authors:Funk D; Mayhew D, Newman M. Pages: 65 - 92 Abstract: AbstractWe conjecture that the class of frame matroids can be characterized by a sentence in the monadic second-order logic of matroids, and we prove that there is such a characterization for the class of bicircular matroids. The proof does not depend on an excluded-minor characterization. PubDate: Mon, 29 Mar 2021 00:00:00 GMT DOI: 10.1093/qmath/haab020 Issue No:Vol. 73, No. 1 (2021)

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Authors:Baro E; De Vicente J, Otero M. Pages: 93 - 125 Abstract: AbstractWe give a classification of connected abelian locally (real) Nash groups of dimension two. We first consider Painlevé’s description of meromorphic maps admitting an algebraic addition theorem and analyse the algebraic dependence of such maps. We then give a classification of connected abelian locally complex Nash groups of dimension two, from which we deduce the corresponding real classification. As a consequence, we obtain a classification of two-dimensional abelian irreducible algebraic groups defined over $\mathbb{R}$. PubDate: Fri, 23 Apr 2021 00:00:00 GMT DOI: 10.1093/qmath/haab021 Issue No:Vol. 73, No. 1 (2021)

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Authors:Jolliffe L; Martin S. Pages: 127 - 148 Abstract: AbstractOne of the most useful tools for calculating the decomposition numbers of the symmetric group is Schaper’s sum formula. The utility of this formula for a given Specht module can be improved by knowing the Schaper number of the corresponding partition. Fayers gives a characterization of those partitions whose Schaper number is at least two. In this paper, we shall demonstrate how this knowledge can be used to calculate some decomposition numbers before extending this result with the hope of allowing more decomposition numbers to be calculated in the future. For p = 2 we shall give a complete characterization of partitions whose Schaper number is at least three, and those whose Schaper number at least four. We also present a list of necessary conditions for a partition to have Schaper number at least three for odd primes and a conjecture on the sufficiency of these conditions. PubDate: Sat, 24 Apr 2021 00:00:00 GMT DOI: 10.1093/qmath/haab023 Issue No:Vol. 73, No. 1 (2021)

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Authors:Dartyge C; Feutrie D, Tenenbaum G. Pages: 149 - 174 Abstract: AbstractA natural integer is called y-ultrafriable if none of the prime powers occurring in its canonical decomposition exceed y. We investigate the distribution of y-ultrafriable integers not exceeding x among arithmetic progressions to the modulus q. Given a sufficiently small, positive constant ɛ, we obtain uniform estimates valid for $q\leqslant y^{c/\log_2y}$ whenever $\log y\leqslant (\log x)^\varepsilon$, and for $q\leqslant \sqrt{y}$ if $(\log x)^{2+\varepsilon}\leqslant y\leqslant x$. PubDate: Wed, 09 Jun 2021 00:00:00 GMT DOI: 10.1093/qmath/haab025 Issue No:Vol. 73, No. 1 (2021)

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Authors:Morishige N. Pages: 175 - 212 Abstract: AbstractThe Banana manifold $X_{Ban}$ is a compact Calabi–Yau threefold constructed as the conifold resolution of the fiber product of a generic rational elliptic surface with itself, which was first studied by Bryan. We compute Katz’s genus 0 Gopakumar–Vafa invariants of fiber curve classes on the Banana manifold $X_{Ban}\to \mathbf{P} ^1$. The weak Jacobi form of weight −2 and index 1 is the associated generating function for these genus 0 Gopakumar–Vafa invariants. The invariants are shown to be an actual count of structure sheaves of certain possibly non-reduced genus 0 curves on the universal cover of the singular fibers of $X_{Ban}\to\mathbf{P}^1$. PubDate: Wed, 19 May 2021 00:00:00 GMT DOI: 10.1093/qmath/haab026 Issue No:Vol. 73, No. 1 (2021)

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Authors:Duarte L; Oliveira L. Pages: 213 - 237 Abstract: AbstractWe extend to Banach space nest algebras the theory of essential supports and support function pairs of their bimodules, thereby obtaining Banach space counterparts of long-established results for Hilbert space nest algebras. Namely, given a Banach space nest algebra $\mathcal{A}$, we characterize the maximal and the minimal $\mathcal{A}$-bimodules having a given essential support function or support function pair. These characterizations are complete except for the minimal $\mathcal{A}$-bimodule corresponding to a support function pair, in which case we make some headway. We also show that the weakly closed bimodules of a Banach space nest algebra are exactly those that are reflexive operator spaces. To this end, we crucially prove that reflexive bimodules determine uniquely a certain class of admissible support functions. PubDate: Mon, 31 May 2021 00:00:00 GMT DOI: 10.1093/qmath/haab028 Issue No:Vol. 73, No. 1 (2021)

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Authors:Nicolini M. Pages: 239 - 259 Abstract: AbstractWe give a one-parameter family of examples of shrinking Laplacian solitons, which are the second known solutions to the closed G2-Laplacian flow with a finite-time singularity. The torsion forms and the Laplacian and Ricci operators of a large family of G2-structures on different Lie groups are also studied. We apply these formulas to prove that, under a suitable extra condition, there is no closed eigenform for the Laplacian on such family. PubDate: Fri, 18 Jun 2021 00:00:00 GMT DOI: 10.1093/qmath/haab029 Issue No:Vol. 73, No. 1 (2021)

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Authors:de Giovanni F; Trombetti M, Wehrfritz B. Pages: 261 - 275 Abstract: AbstractA classical theorem of Reinhold Baer shows that any group which is finite over its k-th centre has a finite (k + 1)-th term of its lower central series. Although the converse statement is false in general, Philip Hall proved that for any group G the finiteness of γk + 1(G) implies that the index $ G:\zeta_{2k}(G) $ is finite. Similar situations have been investigated when finiteness is replaced by suitable weaker conditions. Moreover, it was proved by Yuri${\text{\u{\i}}}$ Merzljakov that Baer’s theorem and its potential converse do hold for linear groups. The aim of this paper is to obtain results of the latter type for several other finiteness conditions.Finally, although a result of Baer type does not hold for the class of soluble-by-finite reduced minimax groups, we prove that for this class a theorem of Hall type is true in arbitrary groups. PubDate: Sat, 26 Jun 2021 00:00:00 GMT DOI: 10.1093/qmath/haab030 Issue No:Vol. 73, No. 1 (2021)

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Authors:Shubin A. Pages: 277 - 310 Abstract: AbstractWe continue to study the distribution of prime numbers p, satisfying the condition $\{p^{\alpha} \} \in I \subset [0; 1)$, in arithmetic progressions. In this paper, we prove an analogue of Bombieri–Vinogradov theorem for 0 < α < 1/9 with the level of distribution $\theta = 2/5 - (3/5) \alpha$, which improves the previous result corresponding to $\theta \leqslant 1/3$. PubDate: Mon, 28 Jun 2021 00:00:00 GMT DOI: 10.1093/qmath/haab031 Issue No:Vol. 73, No. 1 (2021)

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Authors:Maldague D. Pages: 311 - 331 Abstract: AbstractWe present a certain regularized version of Brascamp–Lieb inequalities studied by Bennett, Carbery, Christ and Tao. Using the induction-on-scales method of Guth, these regularized inequalities lead to a generalization of the multilinear Kakeya inequality. PubDate: Thu, 15 Jul 2021 00:00:00 GMT DOI: 10.1093/qmath/haab032 Issue No:Vol. 73, No. 1 (2021)

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Authors:Wang S. Pages: 333 - 347 Abstract: AbstractWe compare the smooth concordance invariants Upsilon, phi and epsilon. Previous work gave examples of knots with one of the Upsilon and phi invariants zero but the epsilon invariant nonzero. We build an infinite family of linearly independent knots with both the Upsilon and phi invariants zero but the epsilon invariant nonzero. This provides examples of knots with arbitrarily large concordance genus but vanishing bounds from the Upsilon and phi invariants. PubDate: Sat, 17 Jul 2021 00:00:00 GMT DOI: 10.1093/qmath/haab033 Issue No:Vol. 73, No. 1 (2021)

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Authors:Atarihuana Y; García J, Hidalgo R, et al. Pages: 349 - 369 Abstract: AbstractThe theory of dessins d’enfants on compact Riemann surfaces, which are bipartite maps on compact orientable surfaces, are combinatorial objects used to study branched covers between compact Riemann surfaces and the absolute Galois group of the field of rational numbers. In this paper, we show how this theory is naturally extended to non-compact orientable surfaces and, in particular, we observe that the Loch Ness monster (LNM; the surface of infinite genus with exactly one end) admits infinitely many regular dessins d’enfants (either chiral or reflexive). In addition, we study different holomorphic structures on the LNM, which come from homology covers of compact Riemann surfaces, and infinite hyperelliptic and infinite superelliptic curves. PubDate: Fri, 23 Jul 2021 00:00:00 GMT DOI: 10.1093/qmath/haab034 Issue No:Vol. 73, No. 1 (2021)

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Authors:De Schepper A; Schillewaert J, Van Maldeghem H, et al. Pages: 369 - 394 Abstract: AbstractHjelmslev–Moufang (HM) planes are point-line geometries related to the exceptional algebraic groups of type $\mathsf{E}_6$. More generally, point-line geometries related to spherical Tits buildings—Lie incidence geometries—are the prominent examples of parapolar spaces: axiomatically defined geometries consisting of points, lines and symplecta (structures isomorphic to polar spaces). In this paper we classify the parapolar spaces with a similar behaviour as the HM planes, in the sense that their symplecta never have a non-empty intersection. Under standard assumptions, we obtain that the only such parapolar spaces are exactly given by the HM planes and their close relatives (arising from taking certain restrictions). On the one hand, this work complements the algebraic approach to HM planes using Jordan algebras and due to Faulkner in his book ‘The Role of Nonassociative Algebra in Projective Geometry’, published by the American Mathematical Society in 2014; on the other hand, it provides a new tool for classification and characterization problems in the general theory of parapolar spaces. PubDate: Mon, 25 Oct 2021 00:00:00 GMT DOI: 10.1093/qmath/haab043 Issue No:Vol. 73, No. 1 (2021)

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Authors:De Schepper A; Schillewaert J, Van Maldeghem H, et al. Pages: 395 - 395 Abstract: Scientific Research Flanders PubDate: Tue, 14 Dec 2021 00:00:00 GMT DOI: 10.1093/qmath/haab063 Issue No:Vol. 73, No. 1 (2021)