scholarly journals Coordinate finite type rotational surfaces in Euclidean spaces

Filomat ◽  
2014 ◽  
Vol 28 (10) ◽  
pp. 2131-2140
Author(s):  
Bengü Bayram ◽  
Kadri Arslan ◽  
Nergiz Nen ◽  
Betü Bulca
Keyword(s):  
Euclidean Space ◽  
Finite Type ◽  
Euclidean Spaces ◽  
Coordinate Functions ◽  
Rotational Surfaces

Submanifolds of coordinate finite-type were introduced in [10]. A submanifold of a Euclidean space is called a coordinate finite-type submanifold if its coordinate functions are eigenfunctions of ?. In the present study we consider coordinate finite-type surfaces in E4. We give necessary and su_cient conditions for generalized rotation surfaces in E4 to become coordinate finite-type. We also give some special examples.

scholarly journals Complete hypersurfaces in cylinders

1989 ◽  
Vol 46 (2) ◽  
pp. 313-318
Author(s):  
Thomas Hasanis
Keyword(s):  
Euclidean Space ◽  
Scalar Curvature ◽  
Sectional Curvature ◽  
Positive Scalar Curvature ◽  
Positive Scalar ◽  
Coordinate Functions ◽  
Bounded Below ◽  
Complete Hypersurfaces

AbstractWe consider the extent of certain complete hypersurfaces of Euclidean space. We prove that every complete hypersurface in En+1 with sectional curvature bounded below and non-positive scalar curvature has at least (n − 1) unbounded coordinate functions.

scholarly journals Characterisation of upper gradients on the weighted Euclidean space and applications

2021 ◽  
Author(s):  
Danka Lučić ◽  
Enrico Pasqualetto ◽  
Tapio Rajala
Keyword(s):  
Euclidean Space ◽  
Sobolev Space ◽  
Radon Measure ◽  
Lipschitz Functions ◽  
Euclidean Spaces ◽  
Upper Gradient

AbstractIn the context of Euclidean spaces equipped with an arbitrary Radon measure, we prove the equivalence among several different notions of Sobolev space present in the literature and we characterise the minimal weak upper gradient of all Lipschitz functions.

scholarly journals Local and Nonlocal Singular Liouville Equations in Euclidean Spaces

2019 ◽  
Author(s):  
Ali Hyder ◽  
Gabriele Mancini ◽  
Luca Martinazzi
Keyword(s):  
Asymptotic Behavior ◽  
Euclidean Space ◽  
Finite Volume ◽  
Existence Of Solutions ◽  
Euclidean Spaces ◽  
Open Problems ◽  
Supercritical Regime

AbstractWe study the metrics of constant $Q$-curvature in the Euclidean space with a prescribed singularity at the origin, namely solutions to the equation \begin{equation*} (-\Delta)^{\frac{n}{2}}w=e^{nw}-c\delta_{0} \ \textrm{on}\ {\mathbb{R}}^n, \end{equation*}under a finite volume condition. We analyze the asymptotic behavior at infinity and the existence of solutions for every $n\ge 3$ also in a supercritical regime. Finally, we state some open problems.

SEQUENTIAL COLLISION-FREE OPTIMAL MOTION PLANNING ALGORITHMS IN PUNCTURED EUCLIDEAN SPACES

2020 ◽  
Vol 102 (3) ◽  
pp. 506-516
Author(s):  
CESAR A. IPANAQUE ZAPATA ◽  
JESÚS GONZÁLEZ
Keyword(s):  
Euclidean Space ◽  
Motion Planning ◽  
Euclidean Spaces ◽  
Optimal Motion ◽  
Topological Theory ◽  
Theory Of Motion ◽  
Optimal Motion Planning ◽  
Planning Algorithms

In robotics, a topological theory of motion planning was initiated by M. Farber. We present optimal motion planning algorithms which can be used in designing practical systems controlling objects moving in Euclidean space without collisions between them and avoiding obstacles. Furthermore, we present the multi-tasking version of the algorithms.

scholarly journals Chen’s Biharmonic Conjecture and Submanifolds with Parallel Normalized Mean Curvature Vector

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 710 ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Mean Curvature Vector ◽  
Biharmonic Submanifolds ◽  
Euclidean Spaces ◽  
Principal Curvatures ◽  
Chen’S Conjecture ◽  
Biharmonic Submanifold

The well known Chen’s conjecture on biharmonic submanifolds in Euclidean spaces states that every biharmonic submanifold in a Euclidean space is a minimal one. For hypersurfaces, we know from Chen and Jiang that the conjecture is true for biharmonic surfaces in E 3 . Also, Hasanis and Vlachos proved that biharmonic hypersurfaces in E 4 ; and Dimitric proved that biharmonic hypersurfaces in E m with at most two distinct principal curvatures. Chen and Munteanu showed that the conjecture is true for δ ( 2 ) -ideal and δ ( 3 ) -ideal hypersurfaces in E m . Further, Fu proved that the conjecture is true for hypersurfaces with three distinct principal curvatures in E m with arbitrary m. In this article, we provide another solution to the conjecture, namely, we prove that biharmonic surfaces do not exist in any Euclidean space with parallel normalized mean curvature vectors.

scholarly journals Reflection-Like Maps in High-Dimensional Euclidean Space

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 872
Author(s):  
Zhiyong Huang ◽  
Baokui Li
Keyword(s):  
Euclidean Space ◽  
Convex Domain ◽  
Orthogonal Transformation ◽  
High Dimensional ◽  
Dimensional Euclidean Space ◽  
Local Domain ◽  
Euclidean Spaces ◽  
Line Cross ◽  
Klein Model

In this paper, we introduce reflection-like maps in n-dimensional Euclidean spaces, which are affinely conjugated to θ : ( x 1 , x 2 , … , x n ) → 1 x 1 , x 2 x 1 , … , x n x 1 . We shall prove that reflection-like maps are line-to-line, cross ratios preserving on lines and quadrics preserving. The goal of this article was to consider the rigidity of line-to-line maps on the local domain of R n by using reflection-like maps. We mainly prove that a line-to-line map η on any convex domain satisfying η ∘ 2 = i d and fixing any points in a super-plane is a reflection or a reflection-like map. By considering the hyperbolic isometry in the Klein Model, we also prove that any line-to-line bijection f : D n ↦ D n is either an orthogonal transformation, or a composition of an orthogonal transformation and a reflection-like map, from which we can find that reflection-like maps are important elements and instruments to consider the rigidity of line-to-line maps.

scholarly journals Bott–Taubes–Vassiliev cohomology classes by cut-and-paste topology

2019 ◽  
Vol 30 (10) ◽  
pp. 1950047
Author(s):  
Robin Koytcheff
Keyword(s):  
Finite Type ◽  
Integer Lattice ◽  
Configuration Spaces ◽  
Euclidean Spaces ◽  
The Real ◽  
Knot Invariants ◽  
Trivalent Graph ◽  
Higher Dimensional ◽  
Valued Graph ◽  
Knots And Links

Bott and Taubes used integrals over configuration spaces to produce finite-type a.k.a. Vassiliev knot invariants. Cattaneo, Cotta-Ramusino and Longoni then used these methods together with graph cohomology to construct “Vassiliev classes” in the real cohomology of spaces of knots in higher-dimensional Euclidean spaces, as first promised by Kontsevich. Here we construct integer-valued cohomology classes in spaces of knots and links in [Formula: see text] for [Formula: see text]. We construct such a class for any integer-valued graph cocycle, by the method of gluing compactified configuration spaces. Our classes form the integer lattice among the previously discovered real cohomology classes. Thus we obtain nontrivial classes from trivalent graph cocycles. Our methods generalize to yield mod-[Formula: see text] classes out of mod-[Formula: see text] graph cocycles, which need not be reductions of classes over the integers.

scholarly journals Quotient manifolds of flows

2017 ◽  
Vol 26 (02) ◽  
pp. 1740005 ◽  
Author(s):  
Robert E. Gompf
Keyword(s):  
Euclidean Space ◽  
Vector Fields ◽  
Topological Manifold ◽  
Fundamental Groups ◽  
Smooth Manifolds ◽  
Euclidean Spaces ◽  
Orbit Spaces ◽  
Quotient Maps ◽  
Fixed Dimension

This paper investigates which smooth manifolds arise as quotients (orbit spaces) of flows of vector fields. Such quotient maps were already known to be surjective on fundamental groups, but this paper shows that every epimorphism of countably presented groups is induced by the quotient map of some flow, and that higher homology can also be controlled. Manifolds of fixed dimension arising as quotients of flows on Euclidean space realize all even (and some odd) intersection pairings, and all homotopy spheres of dimension at least two arise in this manner. Most Euclidean spaces of dimensions five and higher have families of topologically equivalent but smoothly inequivalent flows with quotient homeomorphic to a manifold with flexibly chosen homology. For [Formula: see text], there is a topological flow on (ℝ2r+1 − 8 points) × ℝm that is unsmoothable, although smoothable near each orbit, with quotient an unsmoothable topological manifold.

scholarly journals Finite, primitive and euclidean spaces

1988 ◽  
Vol 1 (3) ◽  
pp. 177-196 ◽  
Author(s):  
Efim Khalimsky
Keyword(s):  
Image Processing ◽  
Euclidean Space ◽  
Computer Tomography ◽  
Quotient Space ◽  
Robot Vision ◽  
Euclidean Spaces ◽  
Finite Spaces ◽  
Path Connected ◽  
Connected Spaces ◽  
Digital Spaces

Integer and digital spaces are playing a significant role in digital image processing, computer graphics, computer tomography, robot vision, and many other fields dealing with finitely or countable many objects. It is proven here that every finite T0-space is a quotient space of a subspace of some simplex, i.e. of some subspace of a Euclidean space. Thus finite and digital spaces can be considered as abstract simplicial structures of subspaces of Euclidean spaces. Primitive subspaces of finite, digital, and integer spaces are introduced. They prove to be useful in the investigation of connectedness structure, which can be represented as a poset, and also in consideration of the dimension of finite spaces. Essentially T0-spaces and finitely connected and primitively path connected spaces are discussed.

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