The General Properties of Nanoparticles Distributions
There are many possible variants of nanoparticles distributions. But, The general properties of nanoparticles distribution may be found with the aid of generalization of Pafnutii L. Tchebysheff’s (Chebyshev’s) inequality and corrections to Andrei N. Kolmogoroff’s (Kolmogorov’s) estimation. Author has found for all possible variants the universal expressions for nanoparticles distributions with the aid of his generalizations and corrections. These new inequalities are following
P{M≥nm}≥{MS^α-[(n-1)m]^α}/(kmax*m)^α-[(n-1)m]^α }
and P{M≥nm}≤{MS^α-(kmin*m)^α }/(n *m)^α-(kmin*m)^α}
where P{M≥nm} is share of nanoparticles with mass more or equal nm, n=kmin+1, kmin+2, …, kmax, kmax and kmin are integers, α is any positive or negative exponent, MS^α- is the average mean of raised mass particles with exponent α, m is minimize difference in mass of nanoparticles, kmin*m is minimize value of nanoparticle mass, kmax*m is maximize value of nanoparticle mass. MS^α can be determined by the experiment of by the theoretical calculation. The algorithm for point estimation of nanoparticle distribution has been created with the aid of stated inequalities. It was made an application of the new algorithm to nanoparticles distributions. All experimental data are fully correspondent the theoretical stated estimation of nanoparticle distributions.