Examples of scalar-flat hypersurfaces in $${\mathbb{R}^{n+1}}$$

Given a hypersurface M of null scalar curvature in the unit sphere , n ≥ 4, such that its second fundamental form has rank greater than 2, we construct a singular scalar-flat hypersurface in as a normal graph over a truncated cone generated by M. Furthermore, this graph is 1-stable if the cone is strictly 1-stable.     

Hypersurfaces in nearly Kaehler manifold $$\pmb {\mathbb {S}}^{\varvec{3}}\varvec{\times } \pmb {\mathbb {S}}^{\varvec{3}}$$

In this paper, we initiate the study of contact and minimal hypersurfaces in nearly Kaehler manifold \({\mathbb {S}}^3\times {\mathbb {S}}^3\) with a conformal vector field. There are three almost contact metric structures on a hypersurface of \({\mathbb {S}}^3\times {\mathbb {S}}^3\), and we will give some important properties of them. Besides, we study the influence of the conformal vector field on the almost contact metric structures and use it to characterize the hypersurfaces in \({\mathbb {S}}^3\times {\mathbb {S}}^3\).     

Closed characteristics on non-compact hypersurfaces in $${\mathbb {R}^{2n}}$$

Viterbo demonstrated that any (2n − 1)-dimensional compact hypersurface of contact type has at least one closed characteristic. This result proved the Weinstein conjecture for the standard symplectic space (, ω). Various extensions of this theorem have been obtained since, all for compact hypersurfaces. In this paper we consider non-compact hypersurfaces coming from mechanical Hamiltonians, and prove an analogue of Viterbo’s result. The main result provides a strong connection between the top half homology groups H _i (M), i = n, . . . , 2n − 1, and the existence of closed characteristics in the non-compact case (including the compact case). J. B. van den Berg is supported by NWO VENI grant 639.031.204. R. C. Vandervorst and F. Pasquotto are supported by NWO VIDI grant 639.032.202. This research is also partially supported by the RTN project ‘Fronts-Singularities’.     

On Linear Codes over $${\mathbb{Z}}_{2^{s}}$$

In an earlier paper the authors studied simplex codes of type α and β over and obtained some known binary linear and nonlinear codes as Gray images of these codes. In this correspondence, we study weight distributions of simplex codes of type α and β over The generalized Gray map is then used to construct binary codes. The linear codes meet the Griesmer bound and a few non-linear codes are obtained that meet the Plotkin/Johnson bound. We also give the weight hierarchies of the first order Reed-Muller codes over The above codes are also shown to satisfy the chain condition.A part of this paper is contained in his Ph.D. Thesis from IIT Kanpur, India     

Some aspects of shift-like automorphisms of $$\pmb {\mathbb {C}}^{\varvec{k}}$$

The goal of this article is two fold. First, using transcendental shift-like automorphisms of \(\mathbb C^k, ~k \ge 3\) we construct two examples of non-degenerate entire mappings with prescribed ranges. The first example exhibits an entire mapping of \(\mathbb C^k, ~k \ge 3\) whose range avoids a given polydisc but contains the complement of a slightly larger concentric polydisc. This generalizes a result of Dixon–Esterle in \(\mathbb C^2.\) The second example shows the existence of a Fatou–Bieberbach domain in \(\mathbb C^k,~k \ge 3\) that is constrained to lie in a prescribed region. This is motivated by similar results of Buzzard and Rosay–Rudin. In the second part we compute the order and type of entire mappings that parametrize one dimensional unstable manifolds for shift-like polynomial automorphisms and show how they can be used to prove a Yoccoz type inequality for this class of automorphisms.     

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.     

Invariant surfaces with constant mean curvature in $${\mathbb{H}}^2 \times {\mathbb{R}}$$

We classify the profile curves of all surfaces with constant mean curvature in the product space , which are invariant under the action of a 1-parameter subgroup of isometries. The author was supported by INdAM (Italy) and Fapesp (Brazil).     

Gluing constructions amongst constant mean curvature hypersurfaces in $${\mathbb {S}^{n+1}}$$

Four constructions of constant mean curvature (CMC) hypersurfaces in \mathbb Sⁿ⁺¹{\mathbb {S}^{n+1}} are given, which should be considered analogues of ‘classical’ constructions that are possible for CMC hypersurfaces in Euclidean space. First, Delaunay-like hypersurfaces, consisting roughly of a chain of hyperspheres winding multiple times around an equator, are shown to exist for all the values of the mean curvature. Second, a hypersurface is constructed which consists of two chains of spheres winding around a pair of orthogonal equators, showing that Delaunay-like hypersurfaces can be fused together in a symmetric manner. Third, a Delaunay-like handle can be attached to a generalized Clifford torus of the same mean curvature. Finally, two generalized Clifford tori of equal but opposite mean curvature of any magnitude can be attached to each other by symmetrically positioned Delaunay-like ‘arms’. This last result extends Butscher and Pacard’s doubling construction for generalized Clifford tori of small mean curvature.     

Surfaces in $${\mathbb {P}^{4}}$$ fibered in cubics

Any algebraic surface in which is fibered in cubics, so that the generic fibre is a twisted cubic, gives rise to a curve Γ in a suitable compactification X of the space of smooth rational cubics of In this paper the case n = 4 is addressed and the corresponding space X is studied. We apply our results to complete the classification of smooth, rational surfaces in ruled in cubics. This work is within the framework of the national research project “Geometry on Algebraic Varieties” Cofin 2006 of MIUR.     

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...     

Chow points of $${\mathbb{C}}$$-orbits

We consider free algebraic actions of the additive group of complex numbers on a complex vector space X embedded in the complex projective space. We find an explicit formula for the map p that assigns to a generic point x ? X the Chow point of the closure of the orbit through x. The properties Hausdorff quotient topology and proper action are equivalently characterized by the closure of the image of p in the closed Chow variety.     

3D Oriented Projective Geometry Through Versors of $${\mathbb{R}^{3,3}}$$

It is possible to set up a correspondence between 3D space and \({\mathbb{R}^{3,3}}\), interpretable as the space of oriented lines (and screws), such that special projective collineations of the 3D space become represented as rotors in the geometric algebra of \({\mathbb{R}^{3,3}}\). We show explicitly how various primitive projective transformations (translations, rotations, scalings, perspectivities, Lorentz transformations) are represented, in geometrically meaningful parameterizations of the rotors by their bivectors. Odd versors of this representation represent projective correlations, so (oriented) reflections can only be represented in a non-versor manner. Specifically, we show how a new and useful ‘oriented reflection’ can be defined directly on lines. We compare the resulting framework to the unoriented \({\mathbb{R}^{3,3}}\) approach of Klawitter (Adv Appl Clifford Algebra, 24:713–736, 2014), and the \({\mathbb{R}^{4,4}}\) rotor-based approach by Goldman et al. (Adv Appl Clifford Algebra, 25(1):113–149, 2015) in terms of expressiveness and efficiency.     

Heat Kernel Empirical Laws on $${\mathbb {U}}_N$$ and $${\mathbb {GL}}_N$$GLN

This paper studies the empirical laws of eigenvalues and singular values for random matrices drawn from the heat kernel measures on the unitary groups \({\mathbb {U}}_N\) and the general linear groups \({\mathbb {GL}}_N\), for \(N\in {\mathbb {N}}\). It establishes the strongest known convergence results for the empirical eigenvalues in the \({\mathbb {U}}_N\) case, and the first known almost sure convergence results for the eigenvalues and singular values in the \({\mathbb {GL}}_N\) case. The limit noncommutative distribution associated with the heat kernel measure on \({\mathbb {GL}}_N\) is identified as the projection of a flow on an infinite-dimensional polynomial space. These results are then strengthened from variance estimates to \(L^p\) estimates for even integers p.     

On MDS convolutional codes over $${\mathbb {Z}}_{p^{r}}$$

Maximum distance separable (MDS) convolutional codes are characterized through the property that the free distance meets the generalized Singleton bound. The existence of free MDS convolutional codes over \({\mathbb {Z}}_{p^{r}}\) was recently discovered in Oued and Sole (IEEE Trans Inf Theory 59(11):7305–7313, 2013) via the Hensel lift of a cyclic code. In this paper we further investigate this important class of convolutional codes over \({\mathbb {Z}}_{p^{r}}\) from a new perspective. We introduce the notions of p-standard form and r-optimal parameters to derive a novel upper bound of Singleton type on the free distance. Moreover, we present a constructive method for building general (non necessarily free) MDS convolutional codes over \({\mathbb {Z}}_{p^{r}}\) for any given set of parameters.     

Regular complete permutation polynomials over $${\mathbb {F}}_{2^{n}}$$ F 2 n

Designs, Codes and Cryptography - Applications of permutation polynomials in cryptography are closely related to their cycle structures. For example, many block ciphers use permutation polynomials...     

Maximal q-Plurisubharmonic Functions in {\mathbb{C}^{n}}

In this paper we will study maximal q-plurisubharmonic functions in ${\mathbb{C}^n}$ . At the same time, we define a notion above weakly q-plurisubharmonic functions and describe the relation between these functions and maximal q-plurisubharmonic functions.     

A new characterization of Sobolev spaces on $${\mathbb{R}^{n}}$$

In this paper we present a new characterization of Sobolev spaces on . Our characterizing condition is obtained via a quadratic multiscale expression which exploits the particular symmetry properties of Euclidean space. An interesting feature of our condition is that depends only on the metric of and the Lebesgue measure, so that one can define Sobolev spaces of any order of smoothness on any metric measure space.     

Rank two aCM bundles on general determinantal quartic surfaces in $${\mathbb {P}^{3}}$$

Let \(F\subseteq {\mathbb {P}^{3}}\) be a smooth determinantal quartic surface which is general in the Nöther–Lefschetz sense. In the present paper we give a complete classification of locally free sheaves \({\mathcal E}\) of rank 2 on F such that \(h^1(F,{\mathcal E}(th))=0\) for \(t\in \mathbb {Z}\).