The College Mathematics Journal
January 1997
Featured Articles
Kevin Kirby, Of Memory, Neurons and Rank-One Corrections
The classical art of memory dealt with techniques for associatiing images with places. The rules for implementing these associations can be viewed as operating in a vector space, with memory itself being a linear transformation taking input patterns to output patterns. Memorization is revealed as the gradual modification of this linear transformation by many small rank-one corrections. The Dirac notation from quantum mechanics allows these simple rules to be formalized. When they are written down in a fixed basis, the rules yield a popular neural network architecture. In this architecture, the rank one correction rule re-appears as a steepest descent algorithm. In this unusual perspective on neural networks several ideas from linear algebra such as outer products and the pseudoinverse of a matrix arise very naturally.
Chris Bernhardt and Thomas Yuster, Periodic Points of the Difference Operator
The difference operator acts as a linear map on the vector space of sequences of real numbers, and it is natural to ask for the periodic points under iteration of this map. The points with period n under the difference operator form a subspace, and using a natural conjugacy between the difference operator and the shift map we determine this subspace explicitly, both for the space of ordinary infinite sequences and for bi-infinite sequences. An easy corollary is that for ordinary sequences there are periodic points of period n for every n, but for bi-infinite sequences there exist periodic points of all periods with the exception of period 2.
Garret Sobczyk: The Generalized Spectral Decomposition of a Linear Operator
The spectral basis for the commutative algebra generated by a linear operator t on a finite dimensional space consists of orthogonal idempotents and related nilpotent elements that can be determined quickly from the minimal polynomial of the operator. The generalized spectral decomposition expresses t in terms of this spectral basis--it is a more detailed form of the so-called Jordan decomposition of t into a sum of commuting semisimple and nilpotent operators. From this spectral decomposition one can express any function of t that is analytic on the spectrum, such as, 
, as an explicit polynomial in t. Numerical examples illustrating the theory are provided, and it is shown how the generalized spectral decomposition makes accessible several relatively advanced topics such as the Jordan form, the polar factorization, and the classical spectral decomposition for self-adjoint operators.
Ji Gao, An Application of Elementary Geometry in Functional Analysis
It is well known that a norm in a vector space comes from an inner product space in the standard way if and only if the so-called parallelogram law is satisfied. Recently J. Borlein and L. Keener asked whether a weaker condition is sufficient: if every rhombus inscribed in the unit circle relative to a given norm has sides of length , does the norm come from an inner product? After setting the stage, counterexamples are constructed by using only elementary vector algebra and properties of convex regions in the Euclidean plane.
Classroom Capsules
C. W. Groetsch, Halley's Gunnery Rule
In 1686 Edmond Halley presented a paper to the Royal Society in which he used calculus to determine how gunners could aim a cannon and find the proper powder charge to send a projectile to a given target, using the least expenditure of powder. Halley's argument, which is accessible to calculus students with a rudimentary understanding of kinetic and potential energy, is presented in modern notation and terminology, but set in its historical context.
Allen J. Schwenk, Introduction to Limits, or Why Can't We Just Trust The Table?
By investigating the values of 
on several sequences, each of which converges to zero, students are led to be wary when guessing 
from a table of values.
Martha E. Dasef and Steven M. Kautz, Some Sums of Some Significance
The sums 
and 
, for r > 0, have interesting combinatorial interpretations. The sums are determined by finding recurrence relations they satisfy. Intuitive counting arguments are used to show the connection between 
and the Bell numbers 
, and between 
(2) and the number 
of preferential arrangements of n objects.
Melissa Shepard, A Rose Is a Rose Is a Rose . . .
In this computer exploration and writing project for second-semester calculus, students examine how changing a parameter in the equation of a family of polar curves affects the shape and area.
Jingcheng Tong, Area and Perimeter, Volume and Surface Area
Most calculus teachers have observed that the derivative of the area of a circle with respect to its radius is the circumference, and the derivative of the volume of a sphere with respect to its radius is the surface area. This turns out to be a very general phenomenon, when the derivative is taken with respect to the proper linear dimension.
Computer Corner: Classroom Computer Capsule
Peter Zizler and Holly Fraser, Eigenpictures and Singular Values of a Matrix
The eigenpicture of an n x n matrix A shows a vector field on the unit sphere in 
, with the vectors Au attached at typical points u on this sphere. The locus of the tips of the vectors U + Au is shown, by the singular value decomposition of I + A, to be a (possibly degenerate) ellipsoid in . Examples are given to show that for estimating the curl or divergence of a plane vector field F(x,y) at a given point, the eigenpicture of the differential (Jacobian matrix) can be more useful than a plot of the vector field itself in a neighborhood of the point. Thus the eigenpicture of the differential plays a role in visualizing the rate of change of a vector field that is analogous to the role played by the tangent line in single variable differential calculus.