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Fourth order differential equation
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aCa(y Acta Univ algebraic independence analogy applied assume B-class Banach space bilinear mapping conjugate context-free grammar context-sensitive grammar continuously differentiable coordinate space Corollary corresp defined denote differential equation differential system eigenfunction Euler-Maclaurin sum formula exists fc-norm four times continuously grammar G Green's function Hence hypotheses of Theorem inf f(xy interval a,b k-norm leftmost lemma lemma 18 linear linear subspace llvll llxll lt follows mapping Math Mathematics University matrix minimal radial solution MUSTONEN natural numbers non-empty nonterminals norm numbers obtain pN(M polynomials productions Professor SAKARI PIHA proved recursively enumerable languages scattered context grammars SCIENTIAE RERUM NATURALIUM second kind seminormal SEPPO sequence spaces SERIES A SCIENTIAE singular point subset subspace sup f(xy symbols Technica Series third kind Tienari total solution TRANSCENDENCE AND ALGEBRAIC trapezoidal rule unique radial solution University of Oulu X-free xeS(y yEpNW