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 .

In 1981 Walter Rudin exhibited a totally real embedding of the Klein bottle into C2. We show that the polynomially convex hull of Rudin’s Klein bottle contains an open subset of C2. We also describe another totally real Klein bottle in C2 whose hull has topological dimension equal to three.

a subgroup $h$ is said to be $nc$-supplemented in a group $g$ if there exists a subgroup $kleq g$ such that $hklhd g$ and $hcap k$ is contained in $h_{g}$, the core of $h$ in $g$. we characterize the supersolubility of finite groups $g$ with that every maximal subgroup of the sylow subgroups is $nc$-supplemented in $g$.

suppose that $h$ is a subgroup of $g$‎, ‎then $h$ is said to be‎ ‎$s$-permutable in $g$‎, ‎if $h$ permutes with every sylow subgroup of‎ ‎$g$‎. ‎if $hp=ph$ hold for every sylow subgroup $p$ of $g$ with $(|p|‎, ‎|h|)=1$)‎, ‎then $h$ is called an $s$-semipermutable subgroup of $g$‎. ‎in this paper‎, ‎we say that $h$ is partially $s$-embedded in $g$ if‎ ‎$g$ has a normal subgroup $t$ such that $ht...

Gromov and Thurston in [10] constructed, for each n 4, examples of compact n– manifolds which admit metrics of negative curvature, with arbitrarily small pinching constants, but do not admit metrics of constant curvature. We review these examples in Section 3. The main goal of this paper is to put convex projective structures on Gromov– Thurston examples. Suppose that  RP is an open subset and...

The answer is yes, if ... This note attempts to give amplification to the above statement, while at the same time arriving at a reasonable description of this lattice. The main theorem of the paper is no doubt the assertion that the lattice of torsion classes of lattice-ordered groups is completely distributive. The proof of this theorem depends on the notion of a value selector, and should not...

let $g=(v,e)$ be a simple graph. a set $dsubseteq v$ is adominating set of $g$ if every vertex in $vsetminus d$ has atleast one neighbor in $d$. the distance $d_g(u,v)$ between twovertices $u$ and $v$ is the length of a shortest $(u,v)$-path in$g$. an $(u,v)$-path of length $d_g(u,v)$ is called an$(u,v)$-geodesic. a set $xsubseteq v$ is convex in $g$ ifvertices from all $(a, b)$-geodesics belon...

let $g=(v,e)$ be a simple graph. a set $dsubseteq v$ is adominating set of $g$ if every vertex in $vsetminus d$ has atleast one neighbor in $d$. the distance $d_g(u,v)$ between twovertices $u$ and $v$ is the length of a shortest $(u,v)$-path in$g$. an $(u,v)$-path of length $d_g(u,v)$ is called an$(u,v)$-geodesic. a set $xsubseteq v$ is convex in $g$ ifvertices from all $(a, b)$-geodesics belon...

Abstract In this paper, we consider the conjugacy stability property of subgroups and provide effective procedures to solve problem in several classes groups. particular, start with free groups, that is, give an procedure find out if a finitely generated subgroup group is stable. Then further generalize result quasi-convex torsion-free hyperbolic groups limit