Let G be a transitive permutation group on a set ? and let m be a positive integer. If no element of G moves any subset of ? by more than m points, then |? | [2mp I (p-1)] wherep is the least odd primedividing |G |. When the bound is attained, we show that | ? | = 2 p q ….. q where ? is a non-negative integer with 2 < p, r 1 and q is a prime satisfying p < q < 2p, ? = 0 or 1, I i n....

‎let $lambda_1,dots,lambda_n$ be positive real numbers such that‎‎$sum_{k=1}^n lambda_k=1$‎. ‎we prove that for‎‎any positive operators $a_1,a_2,cdots‎, ‎a_n$ in semifinite von‎‎neumann algebra $m$ with faithful normal trace that $t(1)

Let M be the Doob maximal operator on a filtered measure space and let v an Ap weight with 1&lt;p&lt;+∞. We try proving that ∥Mf∥Lp(v)≤p′[v]Ap1p−1∥f∥Lp(v), where 1/p+1/p′=1. Although we do not find approach which gives constant p′, obtain ∥Mf∥Lp(v)≤p1p−1p′[v]Ap1p−1∥f∥Lp(v), limp→+∞p1p−1=1.

MicroRNAs (miRNAs) post-transcriptionally repress gene expression via the miRNA-induced silencing complex (miRISC), which includes miRNA, Argonaute and a GW182 family member. Here we show that in Caenorhabditis elegans, miRNA-mediated gene silencing is modulated by macroautophagy, a lysosome-mediated degradation process. Loss of autophagy activity suppresses developmental defects caused by part...

Let n be a positive integer, and let ρ =(n,n-1,n-2,...,1) the “staircase” partition of size N=n+1 2. The Saxl conjecture asserts that every irreducible representation S λ symmetric group N appears as subrepresentation tensor square ⊗S . In this short note we give two proofs in cube

Let G be a simple connected graph with n ≤ 2 vertices and m edges, and let μ1 ≥ μ2 ≥...≥μn-1 >μn=0 be its Laplacian eigenvalues. The Kirchhoff index and Laplacian-energy-like invariant (LEL) of graph G are defined as Kf(G)=nΣi=1n-1<...

Let A be an alphabet and let F a set of words with letters in A. We show that the sum all no consecutive subwords F, as formal power series noncommuting variables, is reciprocal coefficients 0, 1 or -1. also explain how this result related to Curtis Greene on lattices M\"obius function 1,

let g be a (p, q) graph. let f : v (g) → {1, 2, . . . , k} be a map. for each edge uv, assign the label gcd (f(u), f(v)). f is called k-prime cordial labeling of g if |vf (i) − vf (j)| ≤ 1, i, j ∈ {1, 2, . . . , k} and |ef (0) − ef (1)| ≤ 1 where vf (x) denotes the number of vertices labeled with x, ef (1) and ef (0) respectively denote the number of edges labeled with 1 and not labeled with 1....

a graph is called textit{circulant} if it is a cayley graph on a cyclic group, i.e. its adjacency matrix is circulant. let $d$ be a set of positive, proper divisors of the integer $n>1$. the integral circulant graph $icg_{n}(d)$ has the vertex set $mathbb{z}_{n}$ and the edge set e$(icg_{n}(d))= {{a,b}; gcd(a-b,n)in d }$. let $n=p_{1}p_{2}cdots p_{k}m$, where $p_{1},p_{2},cdots,p_{k}$ are disti...

a polynomial   1 2 ( , , , ) n f x x x is called multilinear if it is homogeneous and linear in every one of its variables. in the present paper our objective is to prove the following result: let   r be a prime k-algebra over a commutative ring   k with unity and let 1 2 ( , , , ) n f x x x be a multilinear polynomial over k. suppose that   d is a nonzero derivation on r such that ...