Bulletin of the Brazilian Mathematical Society, New Series

Chen, Xiaojing; Chu, Wenchang

doi: 10.1007/s00574-020-00210-zpmid: N/A

We establish q-analogues for three summation formulae about the λ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\lambda $$\end{document}-extended Catalan numbers.

Anđelić, Milica; Koledin, Tamara; Stanić, Zoran

doi: 10.1007/s00574-020-00211-ypmid: N/A

We give an upper bound on the largest eigenvalue of the signless Laplacian matrix of a Hamiltonian graph. This bound is applied to obtain sufficient spectral conditions for the non-existence of Hamiltonian cycles. Under certain additional assumptions we provide a polynomial time decisive spectral criterion for the Hamiltonicity of a given graph with sufficiently large minimum vertex degree.

Campillo, Antonio; Olivares, Jorge

doi: 10.1007/s00574-020-00214-9pmid: N/A

This paper deals with foliations by curves [s] of degree r≥2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ r \ge 2 $$\end{document} on P2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ \mathbb {P}^2 $$\end{document} with isolated singularities S, called non-degenerate if S is reduced and otherwise degenerate. Say that [s] is uniquely determined by a zero-dimensional Y⊂S\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ Y \subset S $$\end{document} if [s] is the unique foliation that vanishes on Y and say that Y is minimal for [s] if, moreover, the degree of Y is the minimal possible to do so. Previous work of the authors show that every non-degenerate foliation in degrees 2≤r≤5\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ 2 \le r \le 5 $$\end{document} does have a minimal subscheme and that the set of non-degenerate foliations of degree r≥6\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ r \ge 6 $$\end{document} that have a minimal subscheme contains a Zariski-open subset of the space of foliations of degree r. For non-degenerate foliations [s] we present both characterizations and sufficient conditions for [s] to have a minimal subscheme. We also give examples of degenerate foliations of degree r≥7\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ r \ge 7 $$\end{document} that do not have a minimal subscheme at all.

Mase, Makiko

doi: 10.1007/s00574-020-00215-8pmid: N/A

We study a relation between coupling introduced by Ebeling (Kodai Math J 29:319-336, 2006) and the polytope duality among families of K3 surfaces.

Leandro, Benedito; Pina, Romildo; dos Santos, João Paulo

doi: 10.1007/s00574-020-00216-7pmid: N/A

In this paper, we classify the Einstein hypersurfaces of Sn×R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {S}^n \times \mathbb {R}$$\end{document} and Hn×R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {H}^n \times \mathbb {R}$$\end{document}. We use the characterization of the hypersurfaces of Sn×R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {S}^n \times \mathbb {R}$$\end{document} and Hn×R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {H}^n \times \mathbb {R}$$\end{document} whose tangent component of the unit vector field spanning the factor R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {R}$$\end{document} is a principal direction and the theory of isoparametric hypersurfaces of space forms to show that Einstein hypersurfaces of Sn×R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {S}^n \times \mathbb {R}$$\end{document} and Hn×R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {H}^n \times \mathbb {R}$$\end{document} must have constant sectional curvature.

Casonatto, C.; Fuster, M. C. Romero; Atique, R. Wik

doi: 10.1007/s00574-020-00217-6pmid: N/A

We describe the generic geometry of the 3D-crosscap (image of a stable map of a 3-manifold into R4\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {R}^4$$\end{document}) by means of the simultaneous analysis of the generic singularities of height and squared distanced functions on the flag composed by the 3-manifold, the surface of double points and the crosscaps curve at any point of this curve.

Loeza, Luis; Cano, Angel

doi: 10.1007/s00574-020-00218-5pmid: N/A

In this note we show that the exceptional algebraic set for an infinite discrete group in PSL(3,C)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$PSL(3,{{\mathbb {C}}})$$\end{document} should be a finite union of: complex lines, copies of the Veronese curve or copies of the cubic xy2-z3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$xy^2-z^3$$\end{document}.

Dajczer, Marcos; Jimenez, Miguel Ibieta

doi: 10.1007/s00574-020-00220-xpmid: N/A

This paper deals with the subject of infinitesimal variations of Euclidean submanifolds with arbitrary dimension and codimension. The main goal is to establish a Fundamental theorem for these geometric objects. Similar to the theory of isometric immersions in Euclidean space, we prove that a system of three equations for a certain pair of tensors are the integrability conditions for the differential equation that determines the infinitesimal variations. In addition, we give some rigidity results when the submanifold is intrinsically a Riemannian product of manifolds.

Deolindo-Silva, Jorge Luiz

doi: 10.1007/s00574-020-00221-wpmid: N/A

We establish cross-ratio invariants for surfaces in 4-space in an analogous way to Uribe-Vargas’s work for surfaces in 3-space. We study the geometric locii of local and multi-local singularities of orthogonal projections of the surface to 3-space. Cross-ratio invariants at P3(c)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$P_3(c)$$\end{document}-points are used to recover two moduli in the 4-jet of a projective parametrization of the surface and identify the stable configurations of the asymptotic curves of the surface.

Dória, Cayo; Paula, Gisele Teixeira

doi: 10.1007/s00574-020-00222-9pmid: N/A

We study the action of Bianchi groups on the hyperbolic 3-space H3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {H}^3 $$\end{document}. Given the standard fundamental domain for this action and any point in H3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ \mathbb {H}^3 $$\end{document}, we show that there exists an element in the group which sends the given point into the fundamental domain such that its height is bounded by a quadratic function on the coordinates of the point. This generalizes and establishes a sharp version of a similar result of Habegger and Pila for the action of the Modular group on the hyperbolic plane. Our main theorem can be applied in the reduction theory of binary Hermitian forms with entries in the ring of integers of quadratic imaginary fields. We also show that the asymptotic behavior of the number of elements in a fixed Bianchi group with height at most T is biquadratic in T.