Authors:Ping Kang Abstract: Suppose that G is a finite group and H is a subgroup of G. H is said to be an ss-quasinormal subgroup of G if there is a subgroup B of G such that \(G=HB\) and H permutes with every Sylow subgroup of B; H is said to be c-normal in G if G has a normal subgroup T such that \(G=HT\) and , where \(H_{G}\) is the normal core of H in G. We fix in every non-cyclic Sylow subgroup P of G some subgroup D satisfying \(1< D < P \) and study the structure of G under the assumption that every subgroup H of P with \( H = D \) is either ss-quasinormal or c-normal in G. Some recent results are generalized and unified. PubDate: 2017-11-08 DOI: 10.1007/s40879-017-0196-7

Authors:Alastair Craw; James Green Abstract: We prove that every toric quiver flag variety Y is isomorphic to a fine moduli space of cyclic modules over the algebra \(\mathrm{End}(T)\) for some tilting bundle T on Y. This generalises the well-known fact that can be recovered from the endomorphism algebra of . PubDate: 2017-11-07 DOI: 10.1007/s40879-017-0194-9

Authors:Sorin V. Sabau; Kazuhiro Shibuya; Ryozo Yoshikawa Abstract: We study the behavior of the geodesics of strong Kropina spaces. The global and local aspects of geodesics theory are discussed. Our theory is illustrated with several examples. PubDate: 2017-11-02 DOI: 10.1007/s40879-017-0189-6

Authors:Andrew du Plessis; Charles Terence Clegg Wall Abstract: We recall the classical construction and theory of invariants for the case of binary quintics, describe the moduli space, and identify the curves in it defined by quintics having symmetry. We describe the real case, and identify the number of real roots depending on the point in moduli space. Our main interest is in five curves of binary quintics defined as linear sections of plane curves with infinite symmetry groups: these play a role in the canonical stratification of jet space, so we describe their singularities and count their intersections. All this is done in the classical case. Thereafter we analyse the changes to be made to the whole theory when we work in characteristic 2. PubDate: 2017-11-01 DOI: 10.1007/s40879-017-0187-8

Authors:Michael Kapovich; Bernhard Leeb; Joan Porti Abstract: We study infinite covolume discrete subgroups of higher rank semisimple Lie groups, motivated by understanding basic properties of Anosov subgroups from various viewpoints (geometric, coarse geometric and dynamical). The class of Anosov subgroups constitutes a natural generalization of convex cocompact subgroups of rank one Lie groups to higher rank. Our main goal is to give several new equivalent characterizations for this important class of discrete subgroups. Our characterizations capture “rank one behavior” of Anosov subgroups and are direct generalizations of rank one equivalents to convex cocompactness. Along the way, we considerably simplify the original definition, avoiding the geodesic flow. We also show that the Anosov condition can be relaxed further by requiring only non-uniform unbounded expansion along the (quasi)geodesics in the group. PubDate: 2017-10-30 DOI: 10.1007/s40879-017-0192-y

Authors:Simeon Ball Abstract: An arc is a set of vectors of the k-dimensional vector space over the finite field with q elements \({\mathbb {F}}_q\) , in which every subset of size k is a basis of the space, i.e. every k-subset is a set of linearly independent vectors. Given an arc G in a space of odd characteristic, we prove that there is an upper bound on the largest arc containing G. The bound is not an explicit bound but is obtained by computing properties of a matrix constructed from G. In some cases we can also determine the largest arc containing G, or at least determine the hyperplanes which contain exactly \(k-2\) vectors of the large arc. The theorems contained in this article may provide new tools in the computational classification and construction of large arcs. PubDate: 2017-10-30 DOI: 10.1007/s40879-017-0193-x

Authors:Emilia Mezzetti Abstract: In an article of 1967 Edge gave a description of some beautiful geometric properties of the Kummer surface complete intersection of three quadrics in . Working on it, Dye proved in 1982 and 1992 that all its osculating spaces have dimension less than the expected 5. Here we discuss these results, also in the light of some recent result about varieties with hypo-osculating behaviour. PubDate: 2017-10-24 DOI: 10.1007/s40879-017-0191-z

Authors:Carlos Eduardo Durán; Henrique Vitório Abstract: We express invariants of Finsler manifolds in a geometrical way by means of using moving planes and their associated Jacobi curves, which are curves in a fixed homogeneous Grassmannian manifold. We also use this language to review some applications. PubDate: 2017-10-24 DOI: 10.1007/s40879-017-0190-0

Authors:Sophie Marques Abstract: We find all the generic polynomials for geometric \(\ell \) -cyclic function field extensions over the finite fields \(\mathbb {F}_q\) where \(q= p^n\) , p prime integer such that . PubDate: 2017-10-24 DOI: 10.1007/s40879-017-0188-7

Authors:Miguel Angel Javaloyes; Miguel Sánchez Abstract: We provide an application of the recently introduced wind Riemmanian structures to the understanding of Randers metrics of constant flag curvature (CFC). Any wind Riemannian structure is constructed from a Riemannian metric and a vector field with no restriction on its norm. The local and global classifications for wind Riemannian structures of CFC are obtained, generalizing both, the celebrated classification of the CFC Randers case by Bao, Robles and Shen, and its extension to the Kropina case by Yoshikawa, Okubo and Sabau. Remarkably, any incomplete CFC Randers metric can be extended globally to a complete wind Riemannian structure of CFC, providing then a neat interpretation of the Randers global classification. PubDate: 2017-10-16 DOI: 10.1007/s40879-017-0186-9

Authors:Hervé Gaussier; Harish Seshadri Abstract: We present classical results and recent developments concerning the Kobayashi metric on domains in \(\mathbb {C}^n\) . In particular, we discuss the questions of holomorphicity of isometries and Gromov hyperbolicity of the metric. PubDate: 2017-10-11 DOI: 10.1007/s40879-017-0177-x

Authors:Valery Gritsenko; Haowu Wang Abstract: We show that the eighth power of the Jacobi triple product is a Jacobi–Eisenstein series of weight 4 and index 4 and we calculate its Fourier coefficients. As applications we obtain explicit formulas for the eighth powers of theta-constants of arbitrary order and the Fourier coefficients of the Ramanujan \({\Delta }\) -function \({\Delta }(\tau )=\eta ^{24}(\tau ),\eta ^{12}(\tau )\) and \(\eta ^{8}(\tau )\) in terms of Cohen’s numbers H(3, N) and H(5, N). We give new formulas for the number of representations of integers as sums of eight higher figurate numbers. We also calculate the sixteenth and the twenty-fourth powers of the Jacobi theta-series using the basic Jacobi forms. PubDate: 2017-10-10 DOI: 10.1007/s40879-017-0185-x

Authors:Ivan Babenko; Daniel Massart Abstract: We define Dirichlet type series associated with homology length spectra of Riemannian, or Finsler, manifolds, or polyhedra, and investigate some of their analytical properties. As a consequence we obtain an inequality analogous to Gromov’s classical intersystolic inequality, but taking the whole homology length spectrum into account rather than just the systole. PubDate: 2017-10-06 DOI: 10.1007/s40879-017-0181-1

Authors:Dusa McDuff Abstract: This note constructs sharp obstructions for stabilized symplectic embeddings of an ellipsoid into a ball, in the case when the initial four-dimensional ellipsoid has ‘eccentricity’ of the form \(3\ell -1\) for some integer \(\ell \) . PubDate: 2017-10-06 DOI: 10.1007/s40879-017-0184-y

Authors:Ivan Arzhantsev; Lukas Braun; Jürgen Hausen; Milena Wrobel Abstract: Looking at the well understood case of log terminal surface singularities, one observes that each of them is the quotient of a factorial one by a finite solvable group. The derived series of this group reflects an iteration of Cox rings of surface singularities. We extend this picture to log terminal singularities in any dimension coming with a torus action of complexity one. In this setting, the previously finite groups become solvable torus extensions. As explicit examples, we investigate compound du Val threefold singularities. We give a complete classification and exhibit all the possible chains of iterated Cox rings. PubDate: 2017-10-04 DOI: 10.1007/s40879-017-0179-8

Authors:Ichiro Shimada Abstract: We describe explicitly the correspondence of Edge between the set of planes contained in the Fermat cubic fourfold in characteristic 2, and the set of lattice points T of the Leech lattice such that OABT is a regular tetrahedron, where O is the origin of , and A and B are fixed points of such that OAB is a regular triangle of edge length 2. Using this description, we present Conway’s isomorphism from to in terms of matrices. PubDate: 2017-09-27 DOI: 10.1007/s40879-017-0183-z

Authors:Anton Betten; James W. P. Hirschfeld; Fatma Karaoglu Abstract: In Hirschfeld (J Austral Math Soc 4(1):83–89, 1964), the existence of the cubic surface which arises from a double-six over the finite field of order four was considered. In Hirschfeld (Rend Mat Appl 26:115–152, 1967), the existence and the properties of the cubic surfaces over the finite fields of odd and even order was discussed and classified over the fields of order seven, eight, nine. In this paper, cubic surfaces with twenty-seven lines over the finite field of thirteen elements are classified. PubDate: 2017-09-27 DOI: 10.1007/s40879-017-0182-0

Authors:Vladimir V. Tkachuk Abstract: We establish that a monolithic compact space X is not scattered if and only if has a dense subset without non-trivial convergent sequences. Besides, for any cardinal \(\kappa \geqslant \mathfrak {c}\) , the space \(\mathbb {R}^\kappa \) has a dense subspace without non-trivial convergent sequences. If X is an uncountable \(\sigma \) -compact space of countable weight, then any dense set has a dense subspace without non-trivial convergent sequences. We also prove that for any countably compact sequential space X, if has a dense k-subspace, then X is scattered. PubDate: 2017-09-22 DOI: 10.1007/s40879-017-0180-2

Authors:Travis Willse Abstract: In his 1910 “Five Variables” paper, Cartan solved the equivalence problem for the geometry of (2, 3, 5) distributions and in doing so demonstrated an intimate link between this geometry and the exceptional simple Lie groups of type \(\mathrm{G}_2\) . He claimed to produce a local classification of all such (complex) distributions which have infinitesimal symmetry algebra of dimension at least 6 (and which satisfy a natural uniformity condition), but in 2013 Doubrov and Govorov showed that this classification misses a particular distribution \(\mathbf {E}\) . We produce a closed form for the Fefferman–Graham ambient metric \({\smash {{\smash {\widetilde{g}}}}}_{\mathbf {E}}\) of the conformal class induced by (a real form of) \(\mathbf {E}\) , expanding the small catalogue of known explicit, closed-form ambient metrics. We show that the holonomy group of \(\smash {{\smash {\widetilde{g}}}_{\mathbf {E}}}\) is the exceptional group \({\smash {\mathrm{G}}_2^*}\) and use that metric to give explicitly a projective structure with normal projective holonomy equal to that group. We also present some simple but apparently novel observations about ambient metrics of general left-invariant conformal structures that were used in the determination of the explicit formula for \(\smash {{\smash {\widetilde{g}}}_{\mathbf {E}}}\) . PubDate: 2017-09-12 DOI: 10.1007/s40879-017-0178-9