Deuber’s Theorem says that, given any m, p, c, r in N, there exist n, q, μ in N such that whenever an (n, q, c)-set is r-coloured, there is a monochrome (m, p, c)-set. This theorem has been used in conjunction with the algebraic structure of the StoneČech compactification βN of N to derive several strengthenings of itself. We present here an algebraic proof of the main results in βN and derive ...

The minimizers of the anisotropic fractional isoperimetric inequality with respect to a convex body $ K in \mathbb{R}^n are shown be equivalent star bodies whenever is strictly and unconditional. From this Pólya-Szeg? principle for seminorms derived by using symmetrization bodies.

let $g$ be a $p$-group of nilpotency class $k$ with finite exponent $exp(g)$ and let $m=lfloorlog_pk floor$. we show that $exp(m^{(c)}(g))$ divides $exp(g)p^{m(k-1)}$, for all $cgeq1$, where $m^{(c)}(g)$ denotes the c-nilpotent multiplier of $g$. this implies that $exp( m(g))$ divides $exp(g)$, for all finite $p$-groups of class at most $p-1$. moreover, we show that our result is an improvement...

It is established that a Q-homeomorphism in R, n ≥ 2, is absolute continuous on lines, furthermore, in W 1,1 loc and differentiable a.e. whenever Q ∈ Lloc.

We combine the H Coe cients technique and the Coupling technique to improve security bounds of balanced Feistel schemes. For q queries and round functions of n−bits to n−bits, we nd that the CCA Security of 4 + 2r rounds Feistel schemes is upperbounded by 2q r+3 ( 4q 2n ) r+1 2 + q(q−1) 2·22n . This divides by roughly 1.5 the number of needed rounds for a given CCA Security, compared to the pre...

Let E be an optimal elliptic curve over Q of conductor N having analytic rank one, i.e., such that the L-function LE(s) of E vanishes to order one at s = 1. Let K be a quadratic imaginary field in which all the primes dividing N split and such that the L-function of E over K vanishes to order one at s = 1. Suppose there is another optimal elliptic curve over Q of the same conductor N whose Mord...

For a positive integer $N$ and $\mathbb{A}$, subset of $\mathbb{Q}$, let $\mathbb{A}$-$\mathcal{KS}(N)$ denote the set $\alpha=\dfrac{\alpha_{1}}{\alpha_{2}}\in \mathbb{A\setminus} \{0,N\}$, where $\alpha_{2}r-\alpha_{1}$ divides $\alpha_{2}N-\alpha_{1}$ for every prime divisor $r$ $N$. The is called $N$-Korselt bases in $\mathbb{A}$. Let $p, q$ be two distinct numbers. In this paper, we prove ...

Let R be a commutative Noetherian ring, a an ideal of R, M and N be two finitely generated R-modules. Let t be a positive integer. We prove that if R is local with maximal ideal m and M ⊗R N is of finite length then H t m (M, N) is of finite length for all t ≥ 0 and lR(H t m (M, N)) ≤ ∑t i=0 lR(Ext i R (M, H m (N))). This yields, lR(H t m (M, N)) = lR(Ext t R(M, N)). Additionally, we show that ...

Let k ≤ n be two positive integers, and let F be a field with characteristic p. A sequence f : {1, . . . , n} → F is called k-constant, if the sum of the values of f is the same for every arithmetic progression of length k in {1, . . . , n}. Let V (n, k, F ) be the vector space of all kconstant sequences. The constant sequence is, trivially, k-constant, and thus dim V (n, k, F ) ≥ 1. Let m(k, F...

Chen et al. (Appl Math Lett 17:281–285, 2004) conjectured that for even m, $$R(T_n,W_m)=2n-1$$ if the maximum degree $$\varDelta (T_n)$$ is small. However, they did not state how small it is. Related to this conjecture, also interesting know which tree $$T_n$$ causes Ramsey number $$R(T_n,W_m)$$ be greater than $$2n-1$$ whenever m even. In paper, we determine $$R(T_n,W_8)$$ all trees with (T_n)...