Let $\psi : \mathbb {N} \to [0,1/2]$ be given. The Duffin–Schaeffer conjecture, recently resolved by Koukoulopoulos and Maynard, asserts that for almost all reals $\alpha$ there are infinitely many coprime solutions $(p,q)$ to the inequality $|\alpha - p/q| < \psi (q)/q$ , provided series $\sum _{q=1}^\infty \varphi (q) / q$ is divergent. In present paper, we establish a quantitative version...

Erdos and Rado defined a A-system, as a family in which every two members have the same intersection. Here we obtain a new upper bound on the maximum cardinality q ( n , q ) of an n-uniform family not containing any A-system of cardinality q. Namely, we prove that, for any a > 1 and q , there exists C = C(a, q ) such that, for any n ,

We study tne `q approximation of the sparsest solution of underdetermined linear systems. Mainly we present a condition on the matrix associated with an underdetermined linear system under which the solution of `q minimization is the sparsest solution of the system. Our condition generalizes a similar condition in [Candés, Romberg and Tao’06] ensuring that the solution of the `1 minimization is...

Let $p=(q^4+q^3+q^2+q+1)/(5,q-1)$ be a prime number, where $q$ is a prime power. In this paper, we will show $Gcong mathrm{PSL}(5,q)$ if and only if $|G|=|mathrm{PSL}(5,q)|$, and $G$ has a conjugacy class size $frac{| mathrm{PSL}(5,q)|}{p}$. Further, the validity of a conjecture of J. G. Thompson is generalized to the groups under consideration by a new way.

noindent We have two goals in this paper. First, we investigate and construct cofree coalgebras over $n$-representations of quivers, limits and colimits of $n$-representations of quivers, and limits and colimits of coalgebras in the monoidal categories of $n$-representations of quivers. Second, for any given quivers $mathit{Q}_1$,$mathit{Q}_2$,..., $mathit{Q}_n$, we construct a new quiver $math...

In this paper a comprehensive approach is presented to improve power swings damping based on direct Lyapunov method. The approach combines superconducting magnetic energy storage (SMES) system with static synchronous compensator (STATCOM). Considering the energy absorption/injection ability of SMES, in transient states the combination exchanges both active and reactive powers with power system....