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**A Strong Version of Liouville's Theorem**

By: Wolfhard Hansen

hansen@math.uni-bielefeld.de

An elementary proof for a statement of the following type is presented. Given a continuous bounded complex function on the complex plane, it is constant provided its value at each point is equal to the average of the values on some circle centered at this point. The only condition imposed on the radii of these circles is that, for each point which is far away from the origin, the corresponding radius may exceed its distance to the origin by at most a constant (not depending on the point). Since entire functions have this mean value property for any choice of the radii, such a result clearly implies Liouville's theorem. It is shown that the result fails if, at each point, a radius is admitted which may be up to approximately four times its distance to the origin.

**Conjugate Phase Portraits of Linear Systems**

By: Patrick D. McSwiggen and Kenneth R. Meyer

pat.mcswiggen@uc.edu, ken.meyer@uc.edu

We study the geometry of the solution curves, the phase portraits, of linear differential equations, and what aspects of the geometry are preserved under different types of change of variable. This is a study in the interplay between geometry and smoothness. Different smoothness conditions detect different geometric features of the portraits. In particular, the three intuitive components of a linear transformation, the rates of contraction, the rates of rotation, and the sheer from off-diagonal Jordan blocks, are seen to be not all equally robust.

**A Polynomial Analogue of the 3N + 1 Problem**?

By: Kenneth Hicks, Gary L. Mullen, Joseph L. Yucas, and Ryan Zavislak

hicks@ohio.edu, mullen@math.psu.edu, jyucas@math.siu.edu, ryanzavislak@hotmail.com

We provide a polynomial analogue of the well-known, but still open, integer 3n+1 problem. In our setting over the binary field, and in fact more generally over any field, we are able to show that this polynomial algorithm does, after some finite number of iterations, always end at 1.

**The Wedge Product and Analytic Geometry**

By: Mehrdad Khosravi and Michael D. Taylor

mehrkhos@gmail.com, mtaylor@pegasus.cc.ucf.edu

Many ideas of a geometric nature— the theorem of Pythagoras, the addition of forces, the law of cosines, etc.— are expeditiously handled by the machinery of vectors and linear algebra. It does not seem widely recognized that such ideas and results often generalize to statements about *k*-dimensional objects in *n*-dimensional Euclidean space. For example, the Pythagorean theorem becomes the statement that for an orthogonal *k*-dimensional simplex, the "volume'' squared of its oblique face is the sum of the ``volumes'' squared of the other faces. There is a beautiful idea standing behind such generalizations: the wedge product.

Though the wedge product is a familiar concept, it is usually thought of algebraically rather than geometrically. We take a different tack here: Simple *k*-vectors in —those of the form where each *a _{i}* is a vector — are introduced as equivalence classes of oriented parallelepipeds. This approach leads, via elementary matrix algebra, to the standard properties of exterior algebra. As geometric applications, we show what the law of vector addition becomes for

**Notes**

**Venn Symmetry and Prime Numbers: A Seductive Proof Revisited**

By: Stan Wagon and Peter Webb

wagon@macalester.edu, webb@math.umn.edu

**On Goursat's Proof of Cauchy's Integral Theorem**

By: Harald Hanche-Olsen

hanche@math.ntnu.no

**Number of Irreducible Polynomials in Several Variables Over Finite Fields**

By: Arnaud Bodin

**Baer's Result: The Infinite Product of the Integers Has No Basis**

By: Stefan Schroër

**Reviews**

*The Three Body Problem.*

By: Mauri Valtonen and Hannu Karrunen

Reviewed by: Donald G. Saari

dsaari@uci.edu