A subset X in the k-dimensional Euclidean space R that contains n points (elements) is called an n-point isosceles set if every triplet of points selected from them forms an isosceles triangle. In this paper, we show that there exist exactly two 11-point isosceles sets up to isomorphism and that the maximum cardinality of isosceles sets in R is 11 .

The paper formulates Maxwell’s equations in 4-dimensional Euclidean space by embedding the electromagnetic vector potential in the frame vector g0. Relativistic electrodynamics is the first problem tackled; in spite of using a geometry radically different from that of special relativity, the paper derives relativistic electrodynamics from space curvature. Maxwell’s equations are then formulated...

let (m,g ) be a compact immersed hypersurface of (rn+1,) , λ1 the first nonzeroeigenvalue, α the mean curvature, ρ the support function, a the shape operator, vol (m ) the volume of m,and s the scalar curvature of m. in this paper, we established some eigenvalue inequalities and proved theabove.1) 1 2 2 2 2m ma dv dvn∫ ρ ≥ ∫ α ρ ,2)( )2 2 1 2m 1 mdv s dvn nα ρ ≥ ρ∫ − ∫ ,3) if the scalar curvatu...

Equivalences between conformal foliations on Euclidean 3-space, Hermitian structures on Euclidean 4-space, shear-free ray congruences on Minkowski 4-space, and holomorphic foliations on complex 4-space are explained geometrically and twistorially; these are used to show that 1) any real-analytic complex-valued harmonic morphism without critical points defined on an open subset of Minkowski spac...

In 3-dimensional Euclidean space Scherk second surfaces are singly periodic embedded minimal surfaces with 4 planar ends. In this paper, we obtain a natural generalization of Scherk’s second surfaces in higher dimensional Euclidean spaces. In particular we show that, in higher dimensional Euclidean spaces R, for n ≥ 3, there exists n−1-periodic embedded minimal hypersurfaces with 4 hyperplanar ...

in this article we review an algebraic definition of the gyrogroup and a simplified version of the gyrovector space with two fundamental examples on the open ball of finite-dimensional euclidean space, which are the einstein and möbius gyrovector spaces. we introduce the structure of gyrovector space and the gyroline on the open convex cone of positive definite matrices and see its interest...