scholarly journals Implicit Equations of the Henneberg-Type Minimal Surface in the Four-Dimensional Euclidean Space

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 279
Author(s):  
Erhan Güler ◽  
Ömer Kişi ◽  
Christos Konaxis
Keyword(s):  
Euclidean Space ◽  
Minimal Surface ◽  
Parameter Family ◽  
Weierstrass Representation ◽  
Dimensional Euclidean Space ◽  
Algebraic Equations ◽  
Implicit Equations ◽  
Positive Integers ◽  
Two Parameter

Considering the Weierstrass data as ( ψ , f , g ) = ( 2 , 1 - z - m , z n ) , we introduce a two-parameter family of Henneberg-type minimal surface that we call H m , n for positive integers ( m , n ) by using the Weierstrass representation in the four-dimensional Euclidean space E 4 . We define H m , n in ( r , θ ) coordinates for positive integers ( m , n ) with m ≠ 1 , n ≠ - 1 , - m + n ≠ - 1 , and also in ( u , v ) coordinates, and then we obtain implicit algebraic equations of the Henneberg-type minimal surface of values ( 4 , 2 ) .

scholarly journals Family of Enneper Minimal Surfaces

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 281
Author(s):  
Erhan Güler
Keyword(s):  
Euclidean Space ◽  
Minimal Surface ◽  
Algebraic Equation ◽  
Minimal Surfaces ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
The Family ◽  
Free Representation ◽  
Positive Integers

We consider a family of higher degree Enneper minimal surface E m for positive integers m in the three-dimensional Euclidean space E 3 . We compute algebraic equation, degree and integral free representation of Enneper minimal surface for m = 1 , 2 , 3 . Finally, we give some results and relations for the family E m .

scholarly journals Bour’s theorem and helicoidal surfaces with constant mean curvature in the Bianchi–Cartan–Vranceanu spaces

2021 ◽  
Author(s):  
Renzo Caddeo ◽  
Irene I. Onnis ◽  
Paola Piu
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Classical Result ◽  
Constant Mean Curvature ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Helicoidal Surfaces ◽  
Helicoidal Surface ◽  
Groups Of Isometries ◽  
Two Parameter

AbstractIn this paper, we generalize a classical result of Bour concerning helicoidal surfaces in the three-dimensional Euclidean space $${\mathbb {R}}^3$$ R 3 to the case of helicoidal surfaces in the Bianchi–Cartan–Vranceanu (BCV) spaces, i.e., in the Riemannian 3-manifolds whose metrics have groups of isometries of dimension 4 or 6, except the hyperbolic one. In particular, we prove that in a BCV-space there exists a two-parameter family of helicoidal surfaces isometric to a given helicoidal surface; then, by making use of this two-parameter representation, we characterize helicoidal surfaces which have constant mean curvature, including the minimal ones.

scholarly journals SOME OBSERVATIONS ON THE DIOPHANTINE EQUATION y2=x!+A AND RELATED RESULTS

2012 ◽  
Vol 86 (3) ◽  
pp. 377-388 ◽  
Author(s):  
MACIEJ ULAS
Keyword(s):  
Diophantine Equation ◽  
Parameter Family ◽  
Computational Results ◽  
Three Solutions ◽  
Positive Integers ◽  
Two Parameter

AbstractWe consider the Brocard–Ramanujan type Diophantine equation y2=x!+A and ask about values of A∈ℤ for which there are at least three solutions in the positive integers. In particular, we prove that the set 𝒜 consisting of integers with this property is infinite. In fact we construct a two-parameter family of integers contained in 𝒜. We also give some computational results related to this equation.

Parallel Curves

1954 ◽  
Vol 6 ◽  
pp. 99-107 ◽  
Author(s):  
G. P. Henderson
Keyword(s):  
Euclidean Space ◽  
Euclidean Plane ◽  
Parameter Family ◽  
Dimensional Euclidean Space

In the Euclidean plane a curve C has a one-parameter family of parallel involutes and a unique evolute C* which coincides with the locus of the centres of the osculating circles of C. If is parallel to C, C* is also the evolute of . We will study parallel curves in n-dimensional Euclidean space and obtain generalizations of the properties given above.We will study parallel curves in n-dimensional Euclidean space and obtain generalizations of the properties given above.

On Vitali's Theorem for Groups of Motions of Euclidean Space

1999 ◽  
Vol 6 (4) ◽  
pp. 323-334
Author(s):  
A. Kharazishvili
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Finite Dimensional

Abstract We give a characterization of all those groups of isometric transformations of a finite-dimensional Euclidean space, for which an analogue of the classical Vitali theorem [Sul problema della misura dei gruppi di punti di una retta, 1905] holds true. This characterization is formulated in purely geometrical terms.

scholarly journals The holographic conformal manifold of 3d $$ \mathcal{N} $$ = 2 S-fold SCFTs

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Nikolay Bobev ◽  
Friðrik Freyr Gautason ◽  
Jesse van Muiden
Keyword(s):  
String Theory ◽  
Three Dimensional ◽  
Parameter Family ◽  
Global Symmetry ◽  
Maximal Supergravity ◽  
All Solutions ◽  
Operator Spectrum ◽  
Type Iib String Theory ◽  
Two Parameter

Abstract We employ a non-compact gauging of four-dimensional maximal supergravity to construct a two-parameter family of AdS4 J-fold solutions preserving $$ \mathcal{N} $$ N = 2 supersymmetry. All solutions preserve $$ \mathfrak{u} $$ u (1) × $$ \mathfrak{u} $$ u (1) global symmetry and in special limits we recover the previously known $$ \mathfrak{su} $$ su (2) × $$ \mathfrak{u} $$ u (1) invariant $$ \mathcal{N} $$ N = 2 and $$ \mathfrak{su} $$ su (2) × $$ \mathfrak{su} $$ su (2) invariant $$ \mathcal{N} $$ N = 4 J-fold solutions. This family of AdS4 backgrounds can be uplifted to type IIB string theory and is holographically dual to the conformal manifold of a class of three-dimensional S-fold SCFTs obtained from the $$ \mathcal{N} $$ N = 4 T [U(N)] theory of Gaiotto-Witten. We find the spectrum of supergravity excitations of the AdS4 solutions and use it to study how the operator spectrum of the three-dimensional SCFT depends on the exactly marginal couplings.

Some properties of Wigner coefficients and hyperspherical harmonics

1956 ◽  
Vol 52 (3) ◽  
pp. 424-430 ◽  
Author(s):  
A. P. Stone
Keyword(s):  
Angular Momentum ◽  
Euclidean Space ◽  
Rotation Group ◽  
Dimensional Euclidean Space ◽  
Hyperspherical Harmonics ◽  
Shift Operators ◽  
Analytical Relations

ABSTRACTGeneral shift operators for angular momentum are obtained and applied to find closed expressions for some Wigner coefficients occurring in a transformation between two equivalent representations of the four-dimensional rotation group. The transformation gives rise to analytical relations between hyperspherical harmonics in a four-dimensional Euclidean space.

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