By Frank William Sinden
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Extra info for An Oscillation Theorem for Algebraic Eigenvalue Problems and its Applications
Et M. Krejn, "Surles matrices completement non-n~gatives et oscillatoires", Compositio Math. 4 (1937), 445-476:- 3 ) Schoenberg, I. , "Ueber variationsvermindernde lineare Transformationen", Math. , 32 (1930), p. 321. , Basel (1933). , and G. Szego, "Aufgaben und Lehrslltze aus der Analysis", (Springer, Berlin 1925), V. , Kap. 1, Aufg. 7, 13, 15. 6 ) Haupt, 0. , Wtirzburg (1911). 7 ) Janczewsky, S. , 2. , 29, (1928), pp. , 3. , 31 (1930), ·pp. 663-680. 8 ) Davis, Amer. Journ. , "Einschliessungssatz ftir die charakteristischen Zahlen von Matrizen", Math.
From ~ and the transformed have the variation number V (N\,) vector we form the quotients q. S.. 2 > · · · > :An > 0 lies between Max (qi) and Min (qi). , satisfies the hypotheses of the corollary, then the (n - k + 1) -st eigenvalue lies between Max (qi) and Min (q 1). or Proof. Let be the (positive) diagonal matrix whose diagonal elements are the quotients qr The trial vector M satisfies the equation According to the theorem of Part I the problem (1 ') has an inversely oscillatory eigensolution (as well as problem (1) ).
43 Appendix 9. VIBRATING ROD WITH INTERMEDIATE SUPPORTS. Very often in the vibrating rod problem there are one or more supports or bearings between the end -points. These are usually idealized as sharp fulcra which completely obstruct transverse motion, but through which the bending moment is freely transmitted. ) (20) =0 Yo y yl y' continuous y2 py" continuous y3 (py")' (in general) discontinuous The first two conditions are geometrically plausible. y 2 is essentially the bending moment and y 3 is essentially the transversal shearing force.
An Oscillation Theorem for Algebraic Eigenvalue Problems and its Applications by Frank William Sinden