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.     

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.     

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.     

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...     

Resolution of the Piecewise Smooth Visible–Invisible Two-Fold Singularity in $$\mathbb {R}^3$$ R 3 Using Regularization and Blowup

Journal of Nonlinear Science - Two-fold singularities in a piecewise smooth (PWS) dynamical system in $$\mathbb {R}^3$$ have long been the subject of intensive investigation. The interest stems...     

Parallelisms of $${{\rm PG}(3, {\mathbb R})}$$ composed of non-regular spreads

Any continuous strictly monotonic function \({F : {{\mathbb R}^{{\geq}0} \to {\mathbb R}}}\) with F(0) = 0 and F(t) → ∞ for t → ∞ gives rise to a topological rotational spread of \({{\rm PG}\,(3,{\mathbb R})}\); this spread is non-regular, if F is not linear. The action of the group \({SO_3({\mathbb R})}\) on this spread yields a topological parallelism of \({{\rm PG}\,(3,{\mathbb R})}\). The article also contains a short investigation on rotational spreads. Moreover, we construct a parallelism P ₇₂ of \({{\rm PG}\,(3,{\mathbb R})}\) which is composed of piecewise regular spreads each consisting of two segments which are tacked together along a common regulus. Using Klein’s correspondence of line geometry and the Thas–Walker construction we represent every parallel class of P ₇₂ via two parallel half-lines being non-interior to a given sphere in \({{\mathbb R}^3}\). The parallelism P ₇₂ contains exactly one regular spread, all other members of P ₇₂ are piecewise regular spreads with two segments. However, P ₇₂ is not topological.     

Explicit div-curl inequalities in bounded and unbounded domains of $${{\mathbb {R}}}^3$$

We give a constructive proof of some functional inequalities related to the div and curl operators in bounded and unbounded domains of \({{\mathbb {R}}}^3\). Our new innovation consists in giving explicit constants in several geometric configurations. These inequalities are of a first use in solving div-curl systems and vector potential problems arising in physics.     

Syzygies of Multi-Variable Higher Spin Dirac Operators on $$\mathbb{R}^{3}$$

This paper is a short report on the generalization of some results of our previous paper [12] to the case of spin j/2 Dirac operators in real dimension three for arbitrary odd integer j. We use an explicit formula for the local expression of such operators to study their algebraic properties, construct the compatibility conditions of the overdetermined system associated to the operator in several spatial variables, and we prove that its associated algebraic complex, dual do the BGG sequence coming from representation theory, has substantially the same pattern as the Cauchy-Fueter complex. The author is a member of the Eduard Čech Center and his research is supported by the relative grants.     

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.     

Shape preserving interpolation by cubic G1 splines in $${\mathbb{R}^3}$$

In this paper, G ¹ continuous cubic spline interpolation of data points in , based on a discrete approximation of the strain energy, is studied. Simple geometric conditions on data are presented that guarantee the existence of the interpolant. The interpolating spline is regular, loop-, cusp- and fold-free.      

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.     

On the structure of level sets of uniform and Lipschitz quotient mappings from $$ \mathbb{R}^n $$ to $$ \mathbb{R} $$

We study two questions posed by Johnson, Lindenstrauss, Preiss, and Schechtman, concerning the structure of level sets of uniform and Lipschitz quotient mappings from . We show that if , is a uniform quotient mapping then for every has a bounded number of components, each component of separates and the upper bound of the number of components depends only on and the moduli of co-uniform and uniform continuity of .Next we prove that all level sets of any co-Lipschitz uniformly continuous mapping from to are locally connected, and we show that for every pair of a constant and a function with , there exists a natural number , so that for every co-Lipschitz uniformly continuous map with a co-Lipschitz constant and a modulus of uniform continuity , there exists a natural number and a finite set with card so that for all has exactly components, has exactly components and each component of is homeomorphic with the real line and separates the plane into exactly 2 components. The number and form of components of for are also described - they have a finite tree structure.     

On pairs of regular foliations in $${\mathbb{R}^3}$$ and singularities of map-germs

We study germs of pairs of codimension one regular foliations in \mathbbR³{\mathbb{R}^3} . We show that the discriminant of the pair determines the topological type of the pair. We also consider various classifications of the singularities of the discriminant.     

On the Exponential Dichotomy on $$\mathbb{R}$$ of Linear Differential Equations in $$\mathbb{R}^n$$

We present new results on the exponential dichotomy on the entire axis of linear differential equations in .