scholarly journals Equilibrium of Surfaces in a Vertical Force Field

2021 ◽  
Vol 19 (1) ◽  
Author(s):  
Antonio Martínez ◽  
A. L. Martínez-Triviño
Keyword(s):  
Force Field ◽  
Minimal Surfaces ◽  
Quadratic Growth ◽  
Complete Graphs ◽  
Vertical Force ◽  
Full Classification

AbstractIn this paper, we study $$\varphi $$ φ -minimal surfaces in $$\mathbb {R}^3$$ R 3 when the function $$\varphi $$ φ is invariant under a two-parametric group of translations. Particularly those which are complete graphs over domains in $$\mathbb {R}^2$$ R 2 . We describe a full classification of complete flat-embedded $$\varphi $$ φ -minimal surfaces if $$\varphi $$ φ is strictly monotone and characterize rotational $$\varphi $$ φ -minimal surfaces by its behavior at infinity when $$\varphi $$ φ has a quadratic growth.

Powerfully nilpotent groups of rank 2 or small order

2020 ◽  
Vol 23 (4) ◽  
pp. 641-658
Author(s):  
Gunnar Traustason ◽  
James Williams
Keyword(s):  
Detailed Analysis ◽  
Special Kind ◽  
Nilpotent Groups ◽  
Nilpotency Class ◽  
Central Series ◽  
Rank 2 ◽  
Full Classification

AbstractIn this paper, we continue the study of powerfully nilpotent groups. These are powerful p-groups possessing a central series of a special kind. To each such group, one can attach a powerful nilpotency class that leads naturally to the notion of a powerful coclass and classification in terms of an ancestry tree. In this paper, we will give a full classification of powerfully nilpotent groups of rank 2. The classification will then be used to arrive at a precise formula for the number of powerfully nilpotent groups of rank 2 and order {p^{n}}. We will also give a detailed analysis of the ancestry tree for these groups. The second part of the paper is then devoted to a full classification of powerfully nilpotent groups of order up to {p^{6}}.

scholarly journals On the maximal Sobolev regularity of distributions supported by subsets of Euclidean space

2016 ◽  
Vol 15 (05) ◽  
pp. 731-770 ◽  
Author(s):  
D. P. Hewett ◽  
A. Moiola
Keyword(s):  
Maximal Regularity ◽  
Cantor Sets ◽  
Characteristic Functions ◽  
Bessel Potential ◽  
Empty Interior ◽  
Cartesian Products ◽  
Sobolev Regularity ◽  
Full Classification ◽  
Differential And Integral Equations

This paper concerns the following question: given a subset [Formula: see text] of [Formula: see text] with empty interior and an integrability parameter [Formula: see text], what is the maximal regularity [Formula: see text] for which there exists a non-zero distribution in the Bessel potential Sobolev space [Formula: see text] that is supported in [Formula: see text]? For sets of zero Lebesgue measure, we apply well-known results on set capacities from potential theory to characterize the maximal regularity in terms of the Hausdorff dimension of [Formula: see text], sharpening previous results. Furthermore, we provide a full classification of all possible maximal regularities, as functions of [Formula: see text], together with the sets of values of [Formula: see text] for which the maximal regularity is attained, and construct concrete examples for each case. Regarding sets with positive measure, for which the maximal regularity is non-negative, we present new lower bounds on the maximal Sobolev regularity supported by certain fat Cantor sets, which we obtain both by capacity-theoretic arguments, and by direct estimation of the Sobolev norms of characteristic functions. We collect several results characterizing the regularity that can be achieved on certain special classes of sets, such as [Formula: see text]-sets, boundaries of open sets, and Cartesian products, of relevance for applications in differential and integral equations.

scholarly journals Natural Paracontact Magnetic Trajectories on Unit Tangent Bundles

Axioms ◽  
2020 ◽  
Vol 9 (3) ◽  
pp. 72
Author(s):  
Mohamed Tahar Kadaoui Abbassi ◽  
Noura Amri
Keyword(s):  
Gaussian Curvature ◽  
Tangent Bundle ◽  
Two Dimensional ◽  
Constant Gaussian Curvature ◽  
Metric Structures ◽  
Unit Tangent Bundle ◽  
Tangent Bundles ◽  
Full Classification ◽  
Kaluza Klein

In this paper, we study natural paracontact magnetic trajectories in the unit tangent bundle, i.e., those that are associated to g-natural paracontact metric structures. We characterize slant natural paracontact magnetic trajectories as those satisfying a certain conservation law. Restricting to two-dimensional base manifolds of constant Gaussian curvature and to Kaluza–Klein type metrics on their unit tangent bundles, we give a full classification of natural paracontact slant magnetic trajectories (and geodesics).

scholarly journals On the symmetries of three-dimensional generalized symmetric spaces

2021 ◽  
Vol 13(62) (2) ◽  
pp. 451-462
Author(s):  
Lakehal Belarbi
Keyword(s):  
Symmetric Spaces ◽  
Vector Fields ◽  
Killing Vector ◽  
Three Dimensional ◽  
Riemannian Metric ◽  
Killing Vector Fields ◽  
Generalized Symmetric Space ◽  
Generalized Symmetric Spaces ◽  
Full Classification

In this work we consider the three-dimensional generalized symmetric space, equipped with the left-invariant pseudo-Riemannian metric. We determine Killing vector fields and affine vectors fields. Also we obtain a full classification of Ricci, curvature and matter collineations

scholarly journals A Pair of Rigid Surfaces with pg= q = 2 and K2= 8 Whose Universal Cover is Not the Bidisk

2018 ◽  
Vol 2020 (11) ◽  
pp. 3453-3493
Author(s):  
Francesco Polizzi ◽  
Carlos Rito ◽  
Xavier Roulleau
Keyword(s):  
Minimal Surfaces ◽  
Universal Cover ◽  
The Family ◽  
Albanese Map ◽  
Rigid Surfaces

Abstract We construct two complex-conjugated rigid minimal surfaces with $p_g\!=q=2$ and $K^2\!=8$ whose universal cover is not biholomorphic to the bidisk $\mathbb{H} \times \mathbb{H}$. We show that these are the unique surfaces with these invariants and Albanese map of degree 2, apart from the family of product-quotient surfaces given in [33]. This completes the classification of surfaces with $p_g=q=2, K^2=8$, and Albanese map of degree 2.

Super-Penrose process

2020 ◽  
Vol 35 (02n03) ◽  
pp. 2040058
Author(s):  
O. B. Zaslavskii
Keyword(s):  
Black Hole ◽  
Flat Space ◽  
Space Time ◽  
Centre Of Mass ◽  
Rotating Black Hole ◽  
Mass Frame ◽  
Penrose Process ◽  
Energy Formula ◽  
Full Classification

If two particles collide near a rotating black hole, their energy in the centre of mass frame can become unbounded under certain conditions. In doing so, the Killing energy [Formula: see text] of debris at infinity is, in general, remain restricted. If [Formula: see text] is also unbounded, this is called the super-Penrose process. We elucidate when such a process is possible and give full classification of corresponding relativistic objects for rotating space-times. We also discuss the case of a pure electric super-Penrose process that is valid even in the flat space-time. The key role in consideration is played by the Wald inequalities.

scholarly journals Embedded minimal surfaces of finite topology

2019 ◽  
Vol 2019 (753) ◽  
pp. 159-191 ◽  
Author(s):  
William H. Meeks III ◽  
Joaquín Pérez
Keyword(s):  
Asymptotic Behavior ◽  
Riemann Surface ◽  
Finite Number ◽  
Minimal Surface ◽  
Minimal Surfaces ◽  
Compact Riemann Surface ◽  
Finite Topology ◽  
Compact Boundary ◽  
Interior Points

AbstractIn this paper we prove that a complete, embedded minimal surface M in {\mathbb{R}^{3}} with finite topology and compact boundary (possibly empty) is conformally a compact Riemann surface {\overline{M}} with boundary punctured in a finite number of interior points and that M can be represented in terms of meromorphic data on its conformal completion {\overline{M}}. In particular, we demonstrate that M is a minimal surface of finite type and describe how this property permits a classification of the asymptotic behavior of M.

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