Bidiagonal Factorizations of Totally Nonnegative Matrices
by Shaun Fallat
Motivated by types of positivity that are closed under matrix multiplication, we introduce the class of totally nonnegative matrices. A matrix is totally nonnegative if the determinants of all of its square submatrices are nonnegative. Beginning with Gantmacher and Krein in the thirties and continuing with Karlin, totally nonnegative matrices have had an illustrious history and arise in numerous applications. There has been a recent surge of papers studying this class, most of which are motivated by a surprising matrix factorization. Loosely stated, any totally nonnegative matrix can be factored into a product of matrices in which each factor has nonnegative main diagonal entries and at most one nonzero entry off the main diagonal, which must be positive and occur on the super- or sub-diagonal. We develop this factorization result, survey the relevant history along the way, and explore several applications that demonstrate its usefulness for investigating properties of totally nonnegative matrices.
Sequential Searches: Proofreading, Russian Roulette, and the Incomplete q-Eulerian Polynomials Revisited
by Don Rawlings
There are many natural contexts in which a sequence of searches is conducted for lost objects. Ranging from the commonplace to the exhilarating to the risky, they include proofreading, treasure hunts, and the clearing of dangerously littered live munitions from a region following a war.
A single fundamental question arises in all such scenarios: What is the expected number of searches needed to find all, or some acceptable percentage, of the lost objects? Time and resources are only finite after all!
Beyond resolving the fundamental question, our approach has some amusing sidelights. First, the relevant distributions lead to a probabilistic proof of an identity for the q-Eulerian polynomials studied by Carlitz and to the discovery of a new formula for the incomplete q-Eulerian polynomials introduced by Herbranson and Rawlings. Also, the machismo factor is computed for Sandells fair Russian roulette with several players.
Pictures of Ultrametric Spaces, the p-adic Numbers, and Valued Fields
by Jan Holly
Ultrametric spaces can be pictured as trees, which give an intuitive feel for distance. The tree picture easily displays properties such as the strong triangle inequality and the fact that every point in a given disc is a center of that disc. This paper presents the tree picture and gives an introduction to the field of p-adic numbers in the process. In addition, the tree picture serves for valued fields in general.
Jacobi Elliptic Functions from a Dynamical Systems Point of View
by Kenneth R. Meyer
We present a differential equation definition of the Jacobi elliptic functions and use it to obtain many of their elementary properties. This presentation illustrates the power of the dynamical systems approach to the theory of differential equations. The basic properties of these functions are an immediate applications of the fundamental theorems on existence, uniqueness, and continuous dependence of solutions on initial conditions, and thus this presentation gives some excellent examples for a course in differential equations. These functions are used to solve the pendulum equation and the undamped Duffing equation.
Large Torsional Oscillations in Suspension Bridges Visited Again: Vertical Forcing Creates Torsional Response
by P.J. McKenna and Cilliam OTuama
This paper continues an investigation into the large oscillations seen in the Tacoma Narrows suspension bridge before its famous collapse in 1940. Our previous paper studied the coupling of the torsional and vertical modes of oscillation, using a piecewise linear restoring force to model the resistance of the cable to expansion from the unloaded state. This paper explores the effect of having a smoother version of the same resistance. We urge the reader to get involved in the experiment. Download a Windows version of our visualization software from http://euclid.ucc.ie/applmath/projects/bridge/ and try some experiments for yourself. Let the authors know of any interesting results.
Absolutely Abnormal Numbers
by Greg Martin
Despite the fact that almost all real numbers are absolutely normal - that is, the digits in their expansion to any base occur in all possible configurations with the expected frequency - not one specific example of an absolutely normal number is known. In this paper we investigate the opposite extreme: absolutely abnormal numbers - numbers that are normal to no base whatsoever. We give a simple construction that yields such absolutely abnormal numbers in abundance.
Limiting Distributions for Derangments
by Christopher Stuart
The Finite Field Kakeya Problem
by Keith Rogers
The Fundamental Theorem of Algebra Revisted
by Airton Von Sohsten de Medeiros
Other Versions of the Steiner-Lehmus Theorem
by Mowaffaq Hajja
A Simpler Proof of sin p z = p z ? (1-z2/k2)
by Wladimir de Azevedo Pribitkin
A Quick Proof for the Volume of n-Balls
by Jean B. Lasserre
Niels Henrik Abel and His Times: Called Too Soon
By Arild Stubhaug
Reviewed by O. A. Laudal
A First Course in Fourier Analysis
By David W. Kammler
Reviewed by O. Carruth McGehee