Upper and lower bounds are given for the difference between the arithmetic and geometric means of n positive real numbers in terms of the variance of these numbers. In this note we prove a simple refinement of the arithmetic mean-geometric mean inequality. Our result solves a problem posed by Kenneth S. Williams in [5] and generalizes an inequality on p. 215 of [3]. Other estimates for the difference between the means are discussed in [2], [3] and [4]. THEOREM. Suppose that Xk E [a, b] and Pk > 0 for k = 1, ... , n, where a > 0, and suppose that ,-lPk = 1. Then, writing x = E4=ipkxk, we have 2 Pk(Xk x)2X I (Xkp) 0, and let yt = fb t dm(t) and a2 f= b(t -_ [)2 dm(t) be the mean and variance of m. Then 2b a2 < exp(( log(t) dm(t)) 2< a2 This follows from our theorem and the weak* density of the measures of the form En lPk6kx (where 6x denotes the probability measure which is concentrated at the point x) in the set of all probability measures on [a, b]. (See [1, p. 709].) Notice that the inequality /b expt log(t) dm (t) < tt Received by the editors August 15, 1977. AMS (MOS) subject classifications (1970). Primary 26A87.

https://www.ams.org/proc/1978-071-01/S0002-9939-1978-0476971-2/S0002-9939-1978-0476971-2.pdf

A refinement of the arithmetic mean-geometric mean inequality

This note gives a concise algorithm for computing a normal form for a real linear Hamiltonian differential equatin which has purely imaginary eigenvalues. This algorithm is then applied to the differential equation which comes from the quadratic terms of the Hamiltonian of the restricted three body problem at a Lagrange equilateral triangle equilibrium point.

Normal forms for real linear Hamiltonian systems with purely imaginary eigenvalues

We define the tensor product A 3 B for arbitrary semilattices A and B. The construction is analogous to one used in ring theory (see 4J, [71, 181) and different from one studied by A. Waterman [121, D. Mowat [9J, and Z. Shmuely 1101. We show that the semilattice A 0 B is a distributive lattice whenever A and B are distributive lattices, and we investigate the relationship between the Stone space of A 0 B and the Stone spaces of the factors A and B. We conclude with some results concerning tensor products that are projective in the category of distributive lattices. 1. Preliminaries. For terminology and basic results of lattice theory and universal algebra, consult Birkhoff [3] and Gratzer [5], [6]. The join and meet of elements al, . . ., an of a lattice are denoted by I.n 1 ai and H1' l ai respectively. All semilattices considered are join-semilattices. The smallest and largest elements of a lattice, if they exist, are denoted by 0 and 1 respectively. We denote by 2 the two element lattice consisting of 0 and 1. The category of distributive lattices is denoted by V. 2. Existence of the semilattice tensor product. DEFINITION 2.1. Let A, B and C be semilattices. A function f: A x B C is a bihomomorphism if the functions ga: B C defined by g,(b) = f(a, b) and hb :A -+ C defined by hb(a) = f(a, b) are homomorphisms for all a EA and b EB. DEFINITION 2.2. Let A and B be semilattices. A semilattice C is a tensor product of A and B if there is a bihomomorphism f: A x B C such that C is generated by f(A x B) and for any semilattice D and any bihomomorphism g: A x B -+D there is a homomorphism h: C D satisfying g = hf. Note that since f(A x B) generates C, the homomorphism h is necessarily unique. THEOREM 2.3. Let A and B be semilattices. Then a tensor product of A and B exists and is unique up to isomorphism. PROOF. Let K be the free semilattice on A x B and let w be the canonical inclusion map of A x B into K. Let p be the set of all ordered pairs of the Received by the editors November 4, 1974. AMS (MOS) subject classifications (1970). Primary 06A20, 06A35. 183 Copyright

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The semilattice tensor product of distributive lattices

Dilations are discussed from the point of view of a theory of non-commutative stationary Markov processes. We show that a theory of stationary Markov processes can be formulated in terms of certain dilations, and we put various results on dilations into the systematic context of such a theory. Finally, we discuss a class of non-commutative Poisson processes.

Survey on a theory of non-commutative stationary markov processes

Absolute continuity of the essential spectrum of\( - \frac{{d^2 }}{{dt^2 }} + q(t)\) without monotony ofq

J. v. Neumann, Mathematische Grundlagen der Quantenmechanik. (Die Grundlehren der math. Wissenschaften in Einzeldarstellungen, Band 38). VIII + 262 S. m. 4 Abb. Berlin/Heidelberg/New York 1968. Springer‐Verlag. Preis geb. DM 28, ‐ .

W. Gasser

Papers

J. v. Neumann, Mathematische Grundlagen der Quantenmechanik. (Die Grundlehren der math. Wissenschaften in Einzeldarstellungen, Band 38). VIII + 262 S. m. 4 Abb. Berlin/Heidelberg/New York 1968. Springer‐Verlag. Preis geb. DM 28, ‐ .