Robert Kantrowitz

A Principle of Countability

1. M. A. Armstrong, Groups and Symmetry, Springer-Verlag, New York, NY, 1988. 2. M. Brennan, A note on the converse to Lagranges theorem, The Math. Gazette, 82 (494), July 1998, 286-288. 3. J. D. Dixon, Problems in Group Theory, Dover, New York, NY, 1973. 4. J. Gallian, On the converse to Lagranges theorem, tills MAGAZINE, 66 (1993), 23. 5. I. N. Herstein, Ahstmct Algebra, 2nd ed., Macmillan, New York, NY, 1990. 6. G. T. Hogan, More on the converse to Lagranges theorem, this MAGAZINE 69 (1996), 375-376. 7. T. W. Hungerford, Algebra, Springer-Verlag, New York, NY, 1974. 8. G. Mackiw, The linear group SL(2,3) as a source of examples, The Math. Gazette, March 1997, 64-67. 9. J. J. Rotman, An Introduction to the Theon) of Groups, 3rd ed., Wm. C. Brown, Dubuque, lA, 1988. 10. K. Spindler, Abstract Algebra with Applications, Dekker, New York, NY, 1994.

Another Face of the Archimedean Property

Summary We show that the Archimedean property for an abstract ordered field is equivalent to several convergence conditions from calculus, most notably the validity of the geometric series test.

Series that Converge Absolutely but Don't Converge

Summary If a series of real numbers converges absolutely, then it converges. The usual proof requires completeness in the form of the Cauchy criterion. Failing completeness, the result is false. We provide examples of rational series that illustrate this point. The Cantor set appears in connection with one of the examples.

More of Dedekind: His Series Test in Normed Spaces

Dedekind’s test for infinite series has a canonical interpretation in the context of normed spaces. It is shown that his test holds in a normed space precisely when the space is complete.

Optimization for products of concave functions

Ifh denotes the product of finitely many concave non-negative functions on a compact interval [a, b], then it is shown that there exist pointsα andβ witha≤α≤β≤b such thath is strictly increasing on [α, α), constant on (α, β), and strictly decreasing on (β, b]. This structure theorem leads to an extension of several classical optimization results for concave functions on convex sets to the case of products of concave functions.

MATHEMATICS IN A FOREIGN LANGUAGE

ABSTRACT A major obstacle students encounter in their study of mathematics is the language of the discipline. Apart from the mathematical concepts themselves, the words which are used to convey these concepts are foreign and conceal the notions which they are intended to reveal. Confusion arises from the spoken and written language, the very vehicle we use to clearly communicate the ideas. In this paper, we discuss this problem and present some suggestions to help make students more sensitive to and aware of the precision required to effectively communicate mathematics.

Completeness of Ordered Fields and a Trio of Classical Series Tests

This article explores the fate of the infinite series tests of Dirichlet, Dedekind, and Abel in the context of an arbitrary ordered field. It is shown that each of these three tests characterizes the Dedekind completeness of an Archimedean ordered field; specifically, none of the three is valid in any proper subfield of . The argument hinges on a contractive-type property for sequences in Archimedean ordered fields that are bounded and strictly increasing. For an arbitrary ordered field, it turns out that each of the tests of Dirichlet and Dedekind is equivalent to the sequential completeness of the field.

APPROXIMATION BY WEIGHTED COMPOSITION OPERATORS ON C ( X )

Given a pair of compact Hausdorff spaces X and Y, this article centers around the approximation of arbitrary continuous linear mappings from C(X) into C{Y) by weighted composition operators. Optimal results are obtained for compact operators and also for positive weakly compact operators.

A fixed point approach to the steady state for stochastic matrices

We provide two conditions, both in the spirit of classical regularity, that are equivalent to the existence of the steady state for a stochastic matrix. Our development of these characterizations sidesteps Perron-Frobenius theory for non-negative matrices, hinging instead on an elementary fixed point result that complements Banach’s contraction mapping theorem.

Yet Another Proof of Minkowski's Inequality

verges to a nice stellated polyhedron consisting of 18 right-triangular faces. The vol ume of the latter is < 0.95, as the reader can deduce from the calculations above. Thus at some point the volume of P? stops increasing and maximizes at about 1.1820. In [3], Bleecker presented a different isometric deformation of a cube, where the vol ume maximizes at about 1.2187. This value was further improved to 1.2567 in [4]. The largest value obtained by a bending of a cube remains an open problem (see [7]). The construction in this paper is a special case of a general approach by the au thor [7]. There, we prove that every (not necessarily convex) polyhedron in R3 has an isometric embedding of larger volume.