  Event-by-event fluctuations of multiplicity and transverse momentum of
charged hadrons produced in heavy-ion collisions at FAIR energies, 10A, 20A,
30A and 40A GeV are studied in the framework of relativistic transport model,
URQMD. Dependence of two families of strongly intensive measures of
multiplicity($N$) and transverse momentum($p_{\rm T}$) fluctuations,
$\Delta[p_{\rm T},N]$ and $\Sigma[p_{\rm T},N]$, on collision centrality,
centrality bin-widths and pseudorapidity windows are examined. Attempts are
also made to study $NN$, $N$$p_{\rm T}$ and $p_{\rm T}$$p_{\rm T}$ fluctuations
using two window analysis method. The findings suggest that the measure,
$\Delta[p_{\rm T},N]$ be dealt with proper selection of centrality intervals.
This measure also exhibits a strong dependence on the widths of $\eta$ windows.
The variable $\Sigma[p_{\rm T},N]$, however, is observed to be insensitive to
the centrality bin-widths and shows a variation of $< 5\%$ with the widths of
$\eta$ windows. The analysis of data after event mixing gives $\Delta[p_{\rm
T},N]$ and $\Sigma[p_{\rm T},N]$ values as $\sim 1$ irrespective of the widths
of $\eta$ windows and collision centrality, as predicted by model of
independent particle emission, IPM. The study of joint fluctuations of the two
quantities on two $\eta$ windows separated in $\eta$ space, reveals that
$\Sigma[N_{\rm F},N_{\rm B}]$ values are $\sim 1$ irrespective of the position
of $\eta$ windows whereas, the values of $\Sigma[N_{\rm F},p_{\rm T_B}]$ and
$\Sigma[p_{\rm T_F},p_{\rm T_B}]$ firstly increase with $\eta_{sep}$ and later
acquire saturations. The observed trend of centrality dependence of
$\Sigma[N_{\rm F},N_{\rm B}], \Sigma[N_{\rm F},p_{\rm T_B}]$ and $\Sigma[p_{\rm
T_F},p_{\rm T_B}]$ agrees fairly well with those observed in MC simulated
studies carried out for AA collisions at LHC energies in the framework model of
string fusion.
