Stefan Catoiu

On the nth Quantum Derivative

The nth quantum derivative Dnf (x) of the real-valued function f is defined for each real non-zero x as

Area and Perimeter Bisecting Lines of a Triangle

Summary Any triangle has between one and three lines that bisect both the area and the perimeter. We determine the conditions for each possible number.

Linearizing Mile Run Times

In 1975, Steve Elliot ran the mile in T := 247.4 seconds, setting the record for Michigan high school runners [1]. By 1982, Michigan high schools had replaced the mile event with the 1600 meter race. A mile is approximately 1609 meters (actually, one mile = 5280 feet × 12 inches feet × .0254 meters inch = 1609.344 meters, but 1609 will do for our purposes); so when Earl Jones in 1982 ran 1600 meters in 247.2 seconds [1], a natural question arose. Since the former time linearizes to a lower time of 246.0 (= T × 160

The Centroid as a Nontrivial Area Bisecting Center of a Triangle

Summary We advertise a relatively new and little known subject, bisecting envelopes or deltoids, and illustrate its virtues by giving a short, simple proof to a classical theorem of convex geometry, the 1953 result of H. G. Eggleston that the centroid of a triangle is the unique point in the plane such that three lines through it divide the area of the triangle into six equal-area regions.

Nonuniqueness of Sixpartite Points

Abstract It is known that the area of any bounded, convex plane figure can be divided into equal sixths by three concurrent lines, and it is not hard to see that the same is true for perimeters. Calling the points of intersection of such lines area sixpartite points and perimeter sixpartite points, it is known that they are unique for triangles. We prove that they are not unique in general. Moreover, given any finite set of points in the plane we construct a convex polygon in which each of these points is an area sixpartite point, and a second polygon in which each is a perimeter sixpartite point.

Quantum symmetric ^{} derivatives

For 1 < p < ∞, a one-parameter family of symmetric quantum derivatives is defined for each order of differentiation as are two families of Riemann symmetric quantum derivatives. For 1 < p < ∞, symmetrization holds, that is, whenever the L P kth Peano derivative exists at a point, all of these derivatives of order k also exist at that point. The main result, desymmetrization, is that conversely, for 1 < p < ∞, each L p symmetric quantum derivative is a.e. equivalent to the L p Peano derivative of the same order. For k = 1 and 2, each kth L p symmetric quantum derivative coincides with both corresponding kth L p Riemann symmetric quantum derivatives, so, in particular, for k = 1 and 2, both kth L p Riemann symmetric quantum derivatives are a.e. equivalent to the L p Peano derivative.

Bisecting the Perimeter of a Triangle

It is well known that amedian of a triangle divides it into two triangles of equal area, and that the three medians meet at the centroid. Hence, every triangle has at least one point through which pass three area-bisecting lines. Are there always others? The answer is yes: In 1972, J. A. Dunn and J. E. Pretty proved in [10] that the area-bisecting lines of a triangle remain tangent to a three-cusped closed curve, their envelope; see Figure 1. The subject remained dormant for almost 30 years until it was tackled again by an internet newsgroup [14]. They named the bisecting envelope a deltoid, and it has many striking properties, which we now list as (AB1)–(AB6). The label (ABn) refers to the nth property of the (area-bisecting) deltoid, a known object that is included here only for motivational purpose. It is different from the perimeter-bisecting deltoid, which is introduced in this article, and the proof of whose properties listed as (PB1)–(PB10) will be the main focus of this article. Both of these deltoids are different from the classical deltoid or the hypocycloid with three cusps that was discovered by Steiner in 1856. For more on Steiner’s deltoid, see [13], [17], [19] or [20].

Two-Sided Ideals of Some Finite-Dimensional Algebras

We set up the framework for discussing general properties of the lattice and the semigroup of ideals of a finite-dimensional algebra. We work out the examples u(sl 2), uς(sl 2) and kSL 2 (F p ) explicitly. Algebras with distributive lattice and commutative semigroup of ideals are classified.