let $g $ be a finite group and let $gamma(g)$ be the prime graph‎ ‎of g‎. ‎assume $2 < q = p^{alpha} < 100$‎. ‎we determine finite groups‎ ‎g such that $gamma(g) = gamma(u_3(q))$ and prove that if $q neq‎ ‎3‎, ‎5‎, ‎9‎, ‎17$‎, ‎then $u_3(q)$ is quasirecognizable by prime graph‎, ‎i.e‎. ‎if $g$ is a finite group with the same prime graph as the‎ ‎finite simple group $u_3(q)$‎, ‎then $g$ has a un...

let l := u3(11). in this article, we classify groups with the same order and degree pattern as an almost simple group related to l. in fact, we prove that l, l:2 and l:3 are od-characterizable, and l:s3 is 5-fold od-characterizable.

Let $k$ be a totally real number field and $p$ prime. We show that the ``complexity'' of Greenberg's conjecture ($\lambda = \mu 0$) is $p$-adic nature governed (under Leopoldt's conjecture) by finite torsion group ${\mathcal T}_k$ Galois maximal abelian $p$-ramified pro-$p$-extension $k$, means images in ideal norms from layers $k_n$ cyclotomic tower (Theorem (5.2)). These are obtained via form...

Let $X_{1,n}\le\cdots\le X_{n,n}$ be the order statistics of $n$ independent random variables with a common distribution function $F$ having right heavy tail index $\gamma$. Given known constants $d_{i,n}$, $1\le i\le n$, consider weighted power sums $\sum^{k_n}_{i=1}d_{n+1-i,n}\log^pX_{n+1-i,n}$, where $p>0$ and $k_n$ are positive integers such that $k_n\to\infty$ $k_n/n\to0$ as $n\to\infty$. ...

We investigate criteria for circle packing(CP) types of disk triangulation graphs embedded into simply connected domains in $ \mathbb{C}$. In particular, by studying combinatorial curvature and the Gauss-Bonnet theorem involving boundary turns, we show that a graph is CP parabolic if \[ \sum_{n=1}^\infty \frac{1}{\sum_{j=0}^{n-1} (k_j +6)} = \infty, \] where $k_n$ degree excess sequence defined...

Simple drawings are of graphs in the plane or on sphere such that vertices distinct points, edges Jordan arcs connecting their endpoints, and intersect at most once (either a proper crossing shared endpoint). generalized twisted if there is point O every ray emanating from crosses edge drawing which exactly once. We show all $$K_n$$ contain $$2n-4$$ empty triangles, by this making substantial s...

after the classification of the flag-transitive linear spaces, the attention has been turned to line-transitive linear spaces. in this article, we present a partial classification of the finite linear spaces $mathcal s$ on which an almost simple group $g$ with the socle $g_2(q)$ acts line-transitively.

Hill's Conjecture states that the crossing number $\text{cr}(K_n)$ of complete graph $K_n$ in plane (equivalently, sphere) is $\frac{1}{4}\lfloor\frac{n}{2}\rfloor\lfloor\frac{n-1}{2}\rfloor\lfloor\frac{n-2}{2}\rfloor\lfloor\frac{n-3}{2}\rfloor=n^4/64 + O(n^3)$. Moon proved expected crossings a spherical drawing which points are randomly distributed and joined by geodesics precisely $n^4/64+O(n...

let g be a finite group and let $gk(g)$ be the prime graph of g. we assume that $n$ is an odd number. in this paper, we show that if $gk(g)=gk(b_n(p))$, where $ngeq 9$ and $pin {3,5,7}$, then g has a unique nonabelian composition factor isomorphic to $b_n(p)$ or $c_n(p)$ . as consequences of our result, $b_n(p)$ is quasirecognizable by its spectrum and also by a new proof, the validity of a con...