Recently,I have successfully applied a new quantum simulation method,called SCA approach,to FM and AFM single-walled and double-walled nanotubes,calculated their magnetic and thermodynamic properties,especially the magnetic structures.In the first case,the long distance dipole-dipole interaction among the spins are also considered.In the present work,we propose a theoretical quantum model for a pair of double-walled magnetic nanotubes,and utilize it to investigate their macroscopic properties and magnetic structures.Since only the Heisenberg exchange parameters between the two shells are assumed to be different in signs,our calculated spontaneous magnetic and thermodynamic properties for this pair of nanotubes are precisely identical,exhibiting fascinating duality of the nanosystems and demonstrating the correctness of our theoretical model.The two spin systems are all frustrated,so that a phase transition occurs at TM2 bellow another one at TM1.However,only the inner shell composed of smaller A-type spins has been obviously affected.In the temperature region TM2 < T < TM1,the inner shell becomes semi-antiferromagnetic and its magnetization is considerably suppressed,whereas as temperature drops down below TM2 the shell gradually restores its ferromagnetic nature.The longitudinal hysteresis behavior of such a double-waled nanotube is ferromagnetic-like below TM2,but antiferromagnetic-like in the temperature interval TM2< T < TM1.Moreover,we find that the diameter of the nanotube has seemly no effects on its physical properties,but its length does change the two transition temperatures slightly,and also its spin configuration at very low temperatures if the tube is sufficiently long.