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American Mathematical Monthly - February 2003



Complex Analysis: A Brief Tour into Higher Dimensions
by R. Michael Range

Join on this easy tour into the fascinating world of complex analysis in several variables. We will explore familiar concepts and results of the classical one variable theory from a multivariable perspective. While keeping the common roots in sight, we will discover some of the novel features, unexpected phenomena, and new difficulties. Along the way we will encounter central higher dimensional concepts such as domains of holomorphy, pseudoconvexity, and the “patching by decomposition” technique that is at the core of abstract analytic sheaf cohomology. At the end we will briefly look at newer routes to the high peaks, which make the territory more accessible to analysts without requiring major additional technical training.


Responding to Calls for Change in High School Mathematics: Implications for Collegiate Mathematics
by Harold L. Schoen and Christian R. Hirsch,

In 1983 the Conference Board of the Mathematical Sciences recommended that the high school mathematics curriculum be streamlined to make room for new topics and techniques from discrete mathematics, statistics, and probability and that the content, emphases, and approaches in algebra, geometry, and precalculus be re-examined in the light of emerging computer technologies. This paper provides an overview of a new four-year high school mathematics curriculum developed by the Core-Plus Mathematics Project (CPMP) that embodies these and other professional recommendations for change, including those in the National Council of Teachers of Mathematics Standards. We also describe patterns of mathematical learning, particularly as they relate to preparedness for college mathematics, of students who studied the CPMP curriculum and examine how those patterns differ from the learning of students in more traditional high school curricula. Strengths of students completing Standards-oriented curricula that college mathematics departments could potentially capitalize on are identified.


How Rare Is Symmetry in Musical 12-Tone Rows?
by David J. Hunter and Paul T. von Hippel,

Between 1914 and 1928, the Viennese composer Arnold Schoenberg developed a method for “12-tone” musical composition. In a 12-tone composition, all harmonies and melodies are based on a “12-tone row,” i.e., a permutation of the twelve chromatic pitches. A tone row can be used in its original form, or it can be transformed by four operations--transposition, retrogression, inversion, or cyclic shift. If we regard a tone row as an element of the symmetric group on twelve letters, then these four musical transformations can be represented as multiplication by fixed elements of this group. This perspective provides a convenient model for enumerating certain types of rows; in particular, we determine that symmetric rows (that is, rows that are invariant under some transformation) are quite rare in the universe of all 12-tone rows.

The music theory literature contains claims that certain composers prefer symmetric rows; our model provides a mathematical justification for this claim. By applying our model to the works of Schoenberg and two of his students, Anton Webern and Alban Berg, we find that these composers used symmetric rows far more often than might be attributed to chance.


Munshi’s Proof of the Nullstellensatz
by J. Peter May

Hilbert's Nullstellensatz is a cornerstone of the foundations of algebraic geometry. It is the essential starting point of the dictionary that translates concepts in algebraic geometry to concepts in commutative algebra. The usual textbook proofs focus on the relationship with a third subject, field theory, and specifically the study of algebraic extensions. Pedagogically, this obscures the essential simplicity of this fundamental theorem. The paper at hand gives a complete and rigorous treatment that never so much as mentions integrality or algebraic extensions. The theorem is deduced directly from a pair of simple and easily understandable results about prime ideals in integral domains and their polynomial algebras. These results are given complete elementary proofs, with a clear summary of the very little background material that is necessary.



Problems and Solutions


Euclid’s Argument on the Infinitude of Primes
by J.M. Aldaz and A. Bravo

Smooth Interpolation, Hölder Continuity, and the Takagi—van der Waerden Function
by Jack B. Brown and George Kozlowski

When Does the Position Vector of a Space Curve Always Lie in its Rectifying Plane?
by Bang-Yen Chen

The Sharp Rado Theorem for Majorizations
by Xingzhi Zhan

On the Fundamental Theorem of Finite Abelian Groups
by Gabriel Navarro



Ripples in Mathematics: The Discrete Wavelet Transform.
by Arne Jensen and Anders la Cour-Harbo

A First Course in Wavelets with Fourier Analysis.
by Albert Boggess and Francis J. Narcowich
Reviewed by M. Victor Wickerhauser

From Holomorphic Functions to Complex Manifolds.
by Karl Fritzsche and Hans Grauert
Reviewed by Steven G. Krantz