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Truncated first moment of the parton distribution - a modified approach

We derive the LO DGLAP evolution equation for the full Mellin moments of the truncated at $x_0$ first moment of the nonsinglet parton distribution. This "moment of moment" approach allows to determine the small-$x_0$ behaviour of the truncated first moment. We compare our predictions to results obtained from $x-$space solutions for parton distributions with use of the Chebyshev polynomial method and to solutions of the evolution equations for the truncated moments proposed by other authors. The comparison is performed for different input parametrisations for $10^{-5}\leq x_0\leq 0.1$ and $1\leq Q^2\leq 100$ ${\rm GeV}^2$. We give an example of application to the determination of the contribution to the Bjorken Sum Rule.