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The authors determine a function that relates 6 pairs of hyperbolic and trigonometric functions, and explain the resemblance and some coincidences from the corresponding graphs.


The author verifies that a scalar field associated to a vector field, satisfying spherical symmetry and Laplace's equation, implies Newton's inverse square law.


Using Haruki's lemma, the author provides an easy proof of the Double Butterfly Theorem in plane geometry regarding a circle and its chords.

The author presents a uniform way of studying the weighted \(r\)th power means and their relations, as well as a simple proof for the limits of the means discussed in previous articles....

The author provides, with derivations, a list of common groups (subgroups) associated to the \(2\times 2\) matrices over \(Z_{p}\) either additively or multiplicatively.

The author finds visually sums of triangular numbers.

The author finds visually sums of squares and sums of odd squares.
Given positive integers \(n\geq 3\), the author finds a single inequality condition for every \(r\) of them (\(3 \leq r \leq n \)) to be the lengths of sides of a \(r\)-gon.