Growth of Drops By Condensation
Growth of Drops By Condensation
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' . - 13 - Such functions will be said to belong to class f\, • T heorem 1, Let u, u, v, v £ Jt, and let furthermore uu 1 < uu', vv' > vv', for all x > 0, Then K(u, v) < K(u, v), x > 0. Proof ; 1) z(u, 0) < z(u, 0) since u < u, pz{u, 0), z(v, 0) hence / ^(1)6. 1 z(v, 0) since v > v, hence z (v, 0) > z(v, 0), since u > 0, v > 0; con sequent ly Vi(z(v, 0)) < >i(z(v, 0)) ?wr - - (r) furthermore ^L£i < vy < V2, V(T) - V' j hence < £(u, v) < €(u, v) < — / x " a < ~ . But zm(z) "•V / 2 VXS "Y2 / z 2 1 = — exp(-z ) is an increasing function of z for < z < — ; yw " "V2 hence, £(u, v) n(^(u, v) ) < £(u, v)ci(£;(u, v) ) . Since also u(s)u'(s) < u(s)u' (s), it follows that f X u( p )u ' ( p } «(u, v)»(5(u f v))da < f * {r 2 s) " ( ^ ) 5(u, v)w(5(u, v)Jl8. 0x-s ' "Ox^-s / The proof is completed on adding the three inequalities. . ■ t - 111. - Theorem 2 8 Let the sequences u, v be recursively de- ^ n n fined by u (0) = v (0) = 1 n n u n u A = K(u n-l^ v n-l^ v n v A = K(v n-l' U n-l ) ' Then: 1) u is a monotonically increasing sequence and con- verges to a function u, 2) v is a monotonically decreasing sequence and con- verges to a function v, 3) uu' < vv' for all x > 0, l\.
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To quickly assess the difficulty of the text, read a short excerpt:
' . - 13 - Such functions will be said to belong to class f\, • T heorem 1, Let u, u, v, v £ Jt, and let furthermore uu 1 < uu', vv' > vv', for all x > 0, Then K(u, v) < K(u, v), x > 0. Proof ; 1) z(u, 0) < z(u, 0) since u < u, pz{u, 0), z(v, 0) hence / ^(1)6. 1 z(v, 0) since v > v, hence z (v, 0) > z(v, 0), since u > 0, v > 0; con sequent ly Vi(z(v, 0)) < >i(z(v, 0)) ?wr - - (r) furthermore ^L£i < vy < V2, V(T) - V' j hence < £(u, v) < €(u, v) < — / x " a < ~ . But zm(z) "•V / 2 VXS "Y2 / z 2 1 = — exp(-z ) is an increasing function of z for < z < — ; yw " "V2 hence, £(u, v) n(^(u, v) ) < £(u, v)ci(£;(u, v) ) . Since also u(s)u'(s) < u(s)u' (s), it follows that f X u( p )u ' ( p } «(u, v)»(5(u f v))da < f * {r 2 s) " ( ^ ) 5(u, v)w(5(u, v)Jl8. 0x-s ' "Ox^-s / The proof is completed on adding the three inequalities. . ■ t - 111. - Theorem 2 8 Let the sequences u, v be recursively de- ^ n n fined by u (0) = v (0) = 1 n n u n u A = K(u n-l^ v n-l^ v n v A = K(v n-l' U n-l ) ' Then: 1) u is a monotonically increasing sequence and con- verges to a function u, 2) v is a monotonically decreasing sequence and con- verges to a function v, 3) uu' < vv' for all x > 0, l\.
What to read after Growth of Drops By Condensation?
You can find similar books in the "Read Also" column, or choose other free books by Ignace I Kolodner to read online
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