Demonstrates the logical relationship between percentage composition and an empirical formula using a technique that depends on determining a minimum molar mass for the compound based on the mass percent of each element. (DDR)

Multidimensional unfolding methods suffer from the degeneracy problem in almost all circumstances. Most degeneracies are easily recognized: the solutions are perfect but trivial, characterized by approximately equal distances between points from different sets. A definition of an absolutely degenerate solution is proposed, which makes clear that…

John Napier was born in 1550 in the Tower of Merchiston, near Edinburgh, Scotland. Napier's work on logarithms greatly influenced the work that was to be done in the future. The logarithm's ability to simplify calculations meant that Kepler and many others were able to find the relationships and formulas for motion of bodies. In turn, Kepler's…

Two new prophecy formulas for estimating item response theory (IRT)-based reliability of a shortened or lengthened test are proposed. Some of the relationships between the two formulas, one of which is identical to the well-known Spearman-Brown prophecy formula, are examined and illustrated. The major assumptions underlying these formulas are…

Within the general framework of a multifactor term structure model, the fundamental partial differential equation (PDE) satisfied by a default-free zero-coupon bond price is derived via a martingale-oriented approach. Using this PDE, a result characterizing a model belonging to an exponential affine class is established using only a system of…

We provide a geometric proof of the formula for the sine of the sum of two positive angles whose measures sum to less than [pi]/2. (Contains 1 figure.)

It has been widely acknowledged that there is some discrepancy in the teaching of vector calculus in mathematics courses and other applied fields. The curl of a vector field is one topic many students can calculate without understanding its significance. In this paper, we explain the origin of the curl after presenting the standard mathematical…

A formula in terms of a definite integral for the measure of a polygonal solid angle in a Euclidean space of arbitrary dimension is proved. The formula is applied to the study of the geometry of n-simplices.

The discrete dynamical system of absolute differences defined by the map [psi]( x[subscript 1] , x[subscript 2] , x[subscript 3] , x[subscript 4] ) = ([vertical line] x[subscript 2] - x[subscript 1] [vertical line], [vertical line] x[subscript 3] - x[subscript 2] [vertical line], [vertical line] x[subscript 4] - x[subscript 3] [vertical line],…

The method of least squares enables the determination of an estimate of the slope and intercept of a straight line relationship between two quantities or variables X and Y. Although a theoretical relationship may exist between X and Y of the form Y = mX + c, in practice experimental or measurement errors will occur, and the observed or measured…

This article shows how Microsoft Excel's ability to vary parameters with sliders allows students to "stand on the shoulders of giants" and discover characteristics of polynomial functions.

This article outlines an engineering problem requiring the use of a specialized trigonometric formula, and offers an answer to that age-old classroom question, "When are we gonna have to use this"?

This study reports on how students can be led to make meaningful connections between such structures on a set as a partition, the set of equivalence classes determined by an equivalence relation and the fiber structure of a function on that set (i.e., the set of preimages of all sets {b} for b in the range of the function). In this paper, I first…

This note introduces an interesting random walk on a straight path with cards of random numbers. The method of recurrent relations is used to obtain the convergent probability of the random walk with different initial positions.

A simple proof to some known results on the convergence of linear recursive sequences with nonnegative coefficients is given, using the technique of monotone convergence.