2-Dimensional complete self-shrinkers in $$\mathbf {R}^3$$

It is our purpose to study complete self-shrinkers in Euclidean space. By making use of the generalized maximum principle for \(\mathcal {L}\)-operator, we give a complete classification for 2-dimensional complete self-shrinkers with constant squared norm of the second fundamental form in \(\mathbb R^3\). Ding and Xin (Trans Am Math Soc 366:5067–5085, 2014) have proved this result under the assumption of polynomial volume growth, which is removed in our theorem.     

完全3- 一致超图的一类填充问题和覆盖问题

设Γ 是一些单t- 一致超图的集合. 填充设计P_λ(t, Γ, v) (或覆盖设计C_λ(t, Γ, v)) 是一个二元有序组(X, B), 其中X 是完全t- 一致超图λK_v^(t) 的顶点集, B 是λK_v^(t) 的一些子超图的集合, 要求每个子超图都同构于Γ 中的某一个超图, 每个子超图称为是一个区组, 并且满足λK_v^(t) 中的每一条边至多(或至少) 含在B 的λ 个区组中. 给定参数t, v, λ, Γ, 填充设计P_λ(t, Γ, v) 的最大可能的区组数称为填充数, 记为d_λ(t, Γ, v); 覆盖设计C_λ(t, Γ, v) 的最小可能的区组数称为覆盖数, 记为C_λ(t, Γ, v). 本文将确定Γ 中仅含超图K₄⁽³⁾ + e 时的d_λ(t, Γ, v) 和C_λ(t, Γ, v) 的精确值.     

Congruences modulo $$10^3$$ for Euler numbers

For a non-negative integer \(n\), let \(E_n\) be the \(n\) th Euler number. In this note, for any positive integer \(n\), we prove the following congruences:

$$\begin{aligned} {\left\{ \begin{array}{ll} E_{4n} \equiv 380n-375 \pmod {10^3}, \\ E_{4n+2} \equiv -460n+399 \pmod {10^3}. \end{array}\right. } \end{aligned}$$

Our proof is based on induction on \(n\) and elementary direct calculations.

     

Cremona transformations of $${\mathbb P}^4$$ coming from complete intersection of quadrics

We consider Cremona transformations \(\phi {:}\,{\mathbb P}^4\dashrightarrow {\mathbb P}^4\) which factorize through projections of a smooth complete intersection of quadrics in \({\mathbb P}^7\). We prove there are three types of such transformations according to the relative position of the centers of projection. Moreover, in order to fix one of these three types, we give a geometric characterization of Cremona transformations of \({\mathbb P}^n\) which act birationally on the set of hyperplanes passing through a point.     

Special canal surfaces of $$\mathbb{S}^3$$

Canal surfaces defined as envelopes of 1-parameter families of spheres, can be characterized by the vanishing of one of the conformal principal curvatures. We distinguish special canals which are characterized by the fact that the non-vanishing conformal principal curvature is constant along the characteristic circles and show that they are conformally equivalent to either surfaces of revolution, or to cones over plane curves, or to cylinders over plane curves, so they are isothermic.     

On homogeneous hypersurfaces in $${\mathbb {C}}^3$$

We consider a family \(M_t^n\), with \(n\geqslant 2\), \(t>1\), of real hypersurfaces in a complex affine n-dimensional quadric arising in connection with the classification of homogeneous compact simply connected real-analytic hypersurfaces in  \({\mathbb {C}}^n\) due to Morimoto and Nagano. To finalize their classification, one needs to resolve the problem of the embeddability of \(M_t^n\) in  \({\mathbb {C}}^n\) for \(n=3,7\). In our earlier article we showed that \(M_t^7\) is not embeddable in  \({\mathbb {C}}^7\) for every t and that \(M_t^3\) is embeddable in  \({\mathbb {C}}^3\) for all \(1<t<1+10^{-6}\). In the present paper, we improve on the latter result by showing that the embeddability of \(M_t^3\) in fact takes place for \(1<t<\sqrt{(2+\sqrt{2})/3}\). This is achieved by analyzing the explicit totally real embedding of the sphere \(S^3\) in \({\mathbb {C}}^3\) constructed by Ahern and Rudin. For \(t\geqslant {\sqrt{(2+\sqrt{2})/3}}\), the problem of the embeddability of \(M_t^3\) remains open.     

Symmetric factorizations of the complete uniform hypergraph

Crystal graphs, in the sense of Kashiwara, carry a natural monoid structure given by identifying words labelling vertices that appear in the same position of isomorphic components of the crystal. In the particular case of the crystal graph for the q-analogue of the special linear Lie algebra \(\mathfrak {sl}_{n}\), this monoid is the celebrated plactic monoid, whose elements can be identified with Young tableaux. The crystal graph and the so-called Kashiwara operators interact beautifully with the combinatorics of Young tableaux and with the Robinson–Schensted–Knuth correspondence and so provide powerful combinatorial tools to work with them. This paper constructs an analogous ‘quasi-crystal’ structure for the hypoplactic monoid, whose elements can be identified with quasi-ribbon tableaux and whose connection with the theory of quasi-symmetric functions echoes the connection of the plactic monoid with the theory of symmetric functions. This quasi-crystal structure and the associated quasi-Kashiwara operators are shown to interact just as neatly with the combinatorics of quasi-ribbon tableaux and with the hypoplactic version of the Robinson–Schensted–Knuth correspondence. A study is then made of the interaction of the crystal graph for the plactic monoid and the quasi-crystal graph for the hypoplactic monoid. Finally, the quasi-crystal structure is applied to prove some new results about the hypoplactic monoid.     

Exceptional sequences of 8 line bundles on $$({\mathbb {P}}^1)^3$$ ( P 1 ) 3

Journal of Algebraic Combinatorics - We investigate maximal exceptional sequences of line bundles on $$({\mathbb {P}}^1)^r$$ , i.e., those consisting of $$2^r$$ elements. For $$r=3$$ we show that...     

Foliations and polynomial diffeomorphisms of $${\mathbb{R}^{3}}$$

Let be a C ² map and let Spec(Y) denote the set of eigenvalues of the derivative DY _p , when p varies in . We begin proving that if, for some ϵ > 0, then the foliation with made up by the level surfaces {k = constant}, consists just of planes. As a consequence, we prove a bijectivity result related to the three-dimensional case of Jelonek’s Jacobian Conjecture for polynomial maps of The first author was supported by CNPq-Brazil Grant 306992/2003-5. The first and second author were supported by FAPESP-Brazil Grant 03/03107-9.     

2-Arc-transitive cyclic covers of $$K_{n,n}$$

In this paper, a complete classification is achieved of all the regular covers of the complete bipartite graphs \(K_{n,n}\) with cyclic covering transformation group, whose fibre-preserving automorphism group acts 2-arc-transitively. All these covers consist of one threefold covers of \(K_{6,6}\), one twofold cover of \(K_{12, 12}\) and one infinite family X(r, p) of p-fold covers of \(K_{p^r,p^r}\) with p a prime and r an integer such that \(p^r\ge 3\). This infinite family X(r, p) can be derived by a very simple and nice voltage assignment f as follows: \(X(r, p)=K_{p^r, p^r}\times _f \mathbb {Z}_p\), where \(K_{p^r, p^r}\) is a complete bipartite graph with the bipartition \(V=\{ \alpha \bigm |\alpha \in V(r,p)\}\cup \{ \alpha '\bigm |\alpha \in V(r,p)\}\) for the r-dimensional vector space V(r, p) over the field of order p and \(f_{\alpha ,\beta '}=\sum _{i=1}^ra_ib_i,\,\, \mathrm{for\,\,all}\,\,\alpha =(a_i)_r, \beta =(b_i)_r\in V(r,p)\).     

Examples of factorial threefolds complete intersection in $${\mathbb{P}^5}$$ with many singularities

Denote by ν _m (d) the maximal integer for which there exists for d >> 0{d \gg 0} a threefold X ì \mathbbP⁵{X\subset \mathbb{P}^5} complete intersection of hypersurfaces of degree respectively d and d − 1 such that X has only ordinary singularities of order m and |Sing(X)| = ν _m (d). We prove that, n_m(d) 3 j(d){\nu_m(d)\ge \varphi(d)} where j(d) ~ d⁵{\varphi(d)\sim d^5} asymptotically. This result extends (Di Gennaro and Franco in Commun Contemp Math 10(5):745–764, 2008, Corollary 2.10).     

Global Stabilization of a Class of Bilinear Systems in $${\mathbb{R}^3}$$

In this paper, we consider a class of bilinear systems in dimension three which can be an extension of another one in \({\mathbbm{R}^{2}}\). We prove that there exists some homogeneous feedback of degree zero stabilizing the considered class if and only if these feedbacks are constants.     

Complete negatively curved immersed ends in $$\mathbb {R}^3$$

This paper extends, in a sharp way, the famous Efimov’s Theorem to immersed ends in \(\mathbb {R}^3\). More precisely, let M be a non-compact connected surface with compact boundary. Then there is no complete isometric immersion of M into \(\mathbb {R}^3\) satisfying that \(\int _M |K|=+\infty \) and \(K\le -\kappa <0\), where \(\kappa \) is a positive constant and K is the Gaussian curvature of M. In particular Efimov’s Theorem holds for complete Hadamard immersed surfaces, whose Gaussian curvature K is bounded away from zero outside a compact set.     

Geometric Separation in $$mathbb {R}^3$$ R 3

Journal of Fourier Analysis and Applications - The geometric separation problem, initially posed by Donoho and Kutyniok (Commun Pure Appl Math 66:1–47, 2013), aims to separate a distribution...     

Trajectory of an Observer Tracking Object Motion around Convex Obstacles in $${{\mathbb{R}}^{2}}$$ and $${{\mathbb{R}}^{3}}$$

Doklady Mathematics - An autonomous object moves at a constant speed along the shortest path, while bypassing an ordered collection of pairwise disjoint convex sets. The object is tracked by an...