  Let $A \in \Omega_n$ be doubly-stochastic $n \times n$ matrix. Alexander
Schrijver proved in 1998 the following remarkable inequality per(\widetilde{A})
\geq \prod_{1 \leq i,j \leq n} (1- A(i,j)); \widetilde{A}(i,j) =:
A(i,j)(1-A(i,j)), 1 \leq i,j \leq n.
  We use the above Shrijver's inequality to prove the following lower bound:
  \frac{per(A)}{F(A)} \geq 1; F(A) =: \prod_{1 \leq i,j \leq n} (1- A(i,j))^{1-
A(i,j)}.
  We use this new lower bound to prove S.Friedland's Asymptotic Lower Matching
Conjecture(LAMC) on monomer-dimer problem.
  We use some ideas of our proof of (LAMC) to disprove [Lu,Mohr,Szekely]
positive correlation conjecture.
  We present explicit doubly-stochastic $n \times n$ matrices $A$ with the
ratio $\frac{per(A)}{F(A)} = \sqrt{2}^{n}$; conjecture that
  \max_{A \in \Omega_n}\frac{per(A)}{F(A)} \approx (\sqrt{2})^{n} and give some
examples supporting the conjecture.
  If true, the conjecture (and other ones stated in the paper) would imply a
deterministic poly-time algorithm to approximate the permanent of $n \times n$
nonnegative matrices within the relative factor $(\sqrt{2})^{n}$. The best
current such factor is $e^n$.
