Figures Circumscribing Circles
by Tom M. Apostol and Mamikon A. Mnatsakanian
The centroid of the interior of an arbitrary triangle need not be at the same point as the centroid of its boundary. But we have discovered that the two centroids are always collinear with the center of the inscribed circle, at distances in the ratio 2:3 from the center. This paper generalizes this elegant and surprising result to any polygon that circumscribes a circle.
Every triangle circumscribes a circle called the incircle, whose radius is called the inradius and whose center is called the incenter. A polygon with more than three edges may or may not circumscribe a circle. Those that do are examples of what we call circumgons. Each has an inradius and an incenter. Circumgons include all triangles, all regular polygons, some irregular polygons, some nonconvex polygons (such as star-shaped polygons), and other plane figures composed of line segments and circular arcs. This paper shows that all circumgons share common properties relating to area-perimeter ratios and centroids. For example, the ratio of the area of any region bounded by a circumgon to its semiperimeter is equal to its inradius (just as the ratio of the area of a circular disk to its semiperimeter is its radius). Also, the area centroid of any region bounded by a circumgon and the centroid of its boundary curve are collinear with the incenter, at distances in the ratio 2:3 from the incenter, as in the case of a triangle.
Corresponding results are derived for circumgonal rings, plane regions lying between two similar circumgons. These rings have constant width. The ratio of the area to the semiperimeter of such a ring is equal to this constant width. Relations connecting the area centroid of a circumgonal ring with the centroid of its boundary are also given.
Asymptotic Behaviour of Nonlinear Systems
by Hartmut Logemann and Eugene P. Ryan
This paper, which has a tutorial flavour, develops a self-contained, elementary, and unified approach to a variety of results (including LaSalle's Invariance Principle and generalizations thereof) pertaining to asymptotic behaviour and stability of solutions of (nonautonomous) ordinary differential equations and (autonomous) differential inclusions. Concepts of meagreness and weak meagreness of functions form the basis of the approach. These concepts, in conjunction with hypotheses of uniform continuity on particular subsets of [0, ∞) , capture certain asymptotic properties of functions defined on [0, ∞) and provide generalizations of Barbalat’s lemma (a simple observation that says that a uniformly continuous integrable function defined on [0, ∞) converges to 0 as the argument goes to ). Two illustrative examples are detailed.
Extreme Curvature of Polynomials
Stephanie Edwards and Russell A. Gordon
Since polynomials have been studied extensively for centuries, it is difficult to find new polynomial problems that are elementary to state, interesting to study, and do not require elaborate techniques to prove. We were fortunate enough to find such a problem in single-variable calculus: to determine the number of extreme points for the curvature of a polynomial. To be specific, we consider the following conjecture: if k? is the curvature of a real curvature of a polynomial of degree n, then k' has at most n – 1 real roots. We prove a special case of this conjecture, look at some counterintuitive examples, show that certain classes of polynomials have a limited number of real zeros, relate this problem to one posed by George Pólya, and mention a connection with the Schwarzian derivative.
Can One Drop L1-Boundedness in Komlós’s Subsequence Theorem?
by Heinrich v. Weizacker
A Three-point Characterization of Central Symmetry
by G. D. Chakerian and M. S. Klamkin
L’Hospital Rules for Monotonicity and the Wilker-Anglesio Inequality
by Iosif Pinelis
Infinitely Many Insolvable Diophantine Equations
by Noriaki Kimura and Kenneth S. Williams
Forward Shifts and Backward Shifts in a Rearrangement of a Conditionally Convergent Series
by Jón R. Stefánsson
Problems and Solutions
Mathematics for Finance: An Introduction to Financial Engineering
by Marek Capinski and Tomasz Zastawniak
Reviewed by Philip Protter
Index to Volume 111