vasil'ev posed problem 16.26 in [the kourovka notebook: unsolved problems in group theory, 16th ed.,sobolev inst. math., novosibirsk (2006).] as follows:does there exist a positive integer $k$ such that there are no $k$ pairwise nonisomorphicnonabelian finite simple groups with the same graphs of primes? conjecture: $k = 5$.in [zvezdina, on nonabelian simple groups having the same prime gr...

Let $q$ be a prime with $q \equiv 7 \mod 8$, and let $K=\mathbb{Q}(\sqrt{-q})$. Then $2$ splits in $K$, we write $\mathfrak{p}$ for either of the primes $K$ above $2$. $K_\infty$ unique $\mathbb{Z}_2$-extension unramified outside $\mathfrak{p}$. For certain quadratic biquadratic extensions $\mathfrak{F}/K$, prove simple exact formula $\lambda$-invariant Galois group maximal abelian 2-extension ...

Given a partial edge coloring of complete graph $K_n$ and lists allowed colors for the non-colored edges $K_n$, can we extend to proper using only from lists? We prove that this question has positive answer in case when both color satisfy certain sparsity conditions.

The notion of weak saturation was introduced by Bollob\'as in 1968. Let $F$ and $H$ be graphs. A spanning subgraph $G \subseteq F$ is weakly $(F,H)$-saturated if it contains no copy but there exists an ordering $e_1,\ldots,e_t$ $E(F)\setminus E(G)$ such that for each $i \in [t]$, the graph \cup \{e_1,\ldots,e_i\}$ a $H'$ $e_i H'$. Define $wsat(F,H)$ to minimum number edges graph. In this paper,...

Consider a sequence of positive integers $$\{k_n,n\ge 1\}$$ , and an array nonnegative real numbers $$\{a_{n,i},1\le i\le k_n,n\ge satisfying $$\sup _{n\ge 1}\sum _{i=1}^{k_n}a_{n,i}=C_0\in (0,\infty ).$$ This paper introduces the concept $$\{a_{n,i}\}$$ -stochastic domination. We develop some techniques concerning this apply them to remove assumption in strong law large Chandra Ghosal (Acta Ma...

Let $E$ be an elliptic curve with positive rank over a number field $K$ and let $p$ odd prime number. $K_{cyc}$ the cyclotomic $\mathbb{Z}_p$-extension of $K_n$ denote its $n$-th layer. The Mordell--Weil is said to constant in tower if for all $n$, $E(K_n)$ equal $E(K)$. We apply techniques Iwasawa theory obtain explicit conditions above sense. then indicate potential applications Hilbert's ten...