The extreme version of this intrinsic approach is that associated with Walter Pater, Oscar Wilde, and the French symbolists and summarized in the slogan "art for art's sake". Such thinkers and writers believe that art is not only an end in itself but also a sufficient justification of itself.
For the sake of contradiction, suppose that there exists a complete substring π i π i +1… π k such that k< j.
(b) | L| ≠ |ℒ(T b )|: We claim that there does not exist any edge (pa(w), w) ∈ E(T v ) such that ℒ(T w ) is either ℒ(S u ) or L. Let us suppose, for the sake of contradiction, that such an edge exists.
In the 20th century, duels still took place occasionally in France though often only for form's sake, with precautions such that neither sword nor pistol could prove fatal, or even for publicity, the last recorded duel occurring in 1967.
For any, by the former claim, there exists a subsequence of, denoted for the sake of simplicity, such that (3.12).
Thus, for any, by the former conclusion and Lemma 2.3, there exists a subsequence of, denoted by for the sake of simplicity, such that (3.19).
We assume that there exist a positive number (bar{alpha} >0) and a subsequence (we denote the subsequence by the sequence itself for the sake of notational simplicity) such that (| r_{i}^{k}| geqbar{alpha}) for sufficiently large k.
He argues that a person does desire his own happiness for its own sake and that, therefore, happiness as such is desired by and desirable for its own sake for humanity as a whole ("The aggregate of all persons") (IV 3).
For the sake of simplicity, the geometry is such that the problem considered is two-dimensional.
Let Γ : [ a, b ] → S be the required Δ-integral curve, and, for the sake of simplicity, choose a parameter such that Γ ( 0 ) = P.
