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Two lower bounds of Sperb for the first eigenvalue of an elastically attached membrane are proved in an elementary way.

For every separable Banach spaceX there is a Banach spaceY with a separable dual such thatY ⊕X* ≈Y**. There is also a separable spaceZ so thatZ**/JZ is isomorphic toX.

Let H be a real or complex Hilbert space and let L p(1≦p

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This is a short exposition and slight reorgnization of Hahn’s work concerning discrete real time flows with quasi-discrete spectra.

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As usual we denote the number of integers not exceeding n and relatively prime to n by Euler's ϕ function ϕ(n). Lehmer calls the ϕ(n) integersthe totatives of n.Denote by ϕ(k,l,n) the number of a's satisfyingIf nl ≡ 0 (mod k) or n(l + 1) ≡ 0 (mod k) then, since n > k, (n - l/k, n) > 1 and (n(l + 1)/k, n)>1 respectively. Thus ϕ(k, l, n) is the number of totatives of n satisfying

A ring R is called, according to [2], a left p.p. ring if any principal left ideal of R is projective. A ring which is left and right p.p. is called a p.p. ring.In this short note we shall give some additional remarks to A. Hattori [2]. In Proposition 1 we shall give a characterization of commutative p.p. rings, and in Proposition 3 we shall give a generalization of Proposition 17 and 18 in [2], which shows also that the modified torsion theory over commutative p.p. rings coincides with the usual torsion theory.