‎‎for any group $g$‎, ‎we define an equivalence relation $thicksim$ as below‎: ‎[forall g‎, ‎h in g gthicksim h longleftrightarrow |g|=|h|]‎ ‎the set of sizes of equivalence classes with respect to this relation is called the same-order type of $g$ and denote by $alpha{(g)}$‎. ‎in this paper‎, ‎we give a partial answer to a conjecture raised by shen‎. ‎in fact‎, ‎we show that if $g$ is a nilpot...

We conjecture that if G is a graph of order sk, where s ~3 and k 2: 1 are integers, and d(x)+d(y) ~ 2(s-1)k for every pair of non-adjacent vertices x and y of G, then G contains k vertex-disjoint complete subgraphs of order s. This is true when s = 3, [6]. Here we prove this conjecture for k ~ 6.

A set S ⊆ V is called an q-set (q−-set, respectively) if S has at least two vertices and, for every u ∈ S, there exists v ∈ S, v 6= u such that N(u) ∩ N(v) 6= ∅ (N−(u)∩N−(v) 6= ∅, respectively). A digraph D is called s-quadrangular if, for every q-set S, we have | ∪ {N+(u) ∩ N(v) : u 6= v, u, v ∈ S}| ≥ |S| and, for every q−set S, we have | ∪ {N−(u) ∩ N−(v) : u, v ∈ S)} ≥ |S|. We conjecture that...

Abstract. We compute the asymptotical growth rate of a large family of Uq(sl2) 6j-symbols and we interpret our results in geometric terms by relating them to the volumes of suitable hyperbolic objects. We propose an extension of S. Gukov’s generalized volume conjecture to cover the case of hyperbolic links in S or #k . We prove this conjecture for the infinite family of universal hyperbo...

‎‎we investigate graham higman's paper emph{enumerating }$p$emph{-groups}‎, ‎ii‎, ‎in which he formulated his famous porc conjecture‎. ‎we are able to simplify some of the theory‎. ‎in particular‎, ‎higman's paper contains five pages of homological algebra which he uses in‎ ‎his proof that the number of solutions in a finite field to a finite set of‎ ‎emph{monomial} equations is porc‎. ‎it turn...

let $gamma(s_n)$ be the minimum number of proper subgroups‎ ‎$h_i, i=1‎, ‎dots‎, ‎l $ of the symmetric group $s_n$ such that each element in $s_n$‎ ‎lies in some conjugate of one of the $h_i.$ in this paper we‎ ‎conjecture that $$gamma(s_n)=frac{n}{2}left(1-frac{1}{p_1}right)‎ ‎left(1-frac{1}{p_2}right)+2,$$ where $p_1,p_2$ are the two smallest primes‎ ‎in the factorization of $ninmathbb{n}$ an...

In this article we consider the distribution of N points on the unit sphere S in R interacting via logarithmic potential. A characterization theorem of the stationary configurations is derived when N = d + 2 and two new log-optimal configurations minimizing the logarithmic energy are obtained for six points on S and seven points on S. A conjecture on the log-optimal configurations of d + 2 poin...

Preface Jules Henri Poincaré may rightly be considered the father of modern topology (Leonhard Euler and the Königsberg bridges notwithstanding). It is fitting, therefore, that many of the questions explored in this thesis originated with Poincaré. Poincaré's famous conjecture – which has driven so much of twentieth-century topology – arose, in fact, as the successor of an earlier conjecture. P...

rankE(Q) ? = ords=1 L(E/Q, s). This rank conjecture is known when ords=1 L(E/Q, s) ≤ 1 (Gross-Zagier, Kolyvagin, ...). The recent breakthrough of Bhargava-Skinner-W. Zhang shows that the rank conjecture holds for at least 66% of all elliptic curves over Q. Even more interestingly, BSD further predicts a refined BSD formula for the leading term of L(E/Q, s) at s = 1 in terms of various important...

 In this paper, we prove Frankl's Conjecture for an upper semimodular lattice $L$ such that $|J(L)setminus A(L)| leq 3$, where $J(L)$ and $A(L)$ are the set of join-irreducible elements and the set of atoms respectively. It is known that the class of planar lattices is contained in the class of dismantlable lattices and the class of dismantlable lattices is contained in the class of lattices ha...