Harlan J. Brothers

Brothers Technology · Research & Development

The Bourr´ee Part I from Johann Sebastian Bach’s Cello Suite No. 3 provides a clear example of structural scaling. The recursive form of this structure can be visualized in the manner of a well known fractal construction — the Cantor set.

Benoit Mandelbrot always had a strong feeling that music could be viewed from a fractal perspective. However, without our eyes to guide us, how do we gain this perspective? Here we discuss precisely what it means to say that a piece of music is fractal and present new findings with respect to duration scaling.

Beginning with Newton's Maclaurin series expansion for e, we demonstrate how to derive new and elegant series that converge far more rapidly.

Nightingale is a fast, scalable, and secure hybrid cipher. Using a 64-bit block size and 1 multiple independent keys, it runs faster than hardware-accelerated AES-128 in CBC mode. Its output 2 under TestU01's BigCrush test battery is indistinguishable from random data. Because it uses only 3 scalable components, we believe Nightingale's expandable...

Iterated function systems (IFS) can be a surprisingly useful tool for studying structure in data. Here we present results stemming from a 2013 computational study by the author using IFS. The results include fractal patterns that reveal "repulsive" phenomena among primes in a wide range of classes, having specified arithmetic or congruence properti...

Benoit Mandelbrot had a childlike exuberance for learning and sharing this learning with others. He was especially interested in the education of students and wanted very much to spark the joy of learning in them. Thanks to Benoit’s vision, countless young minds around the globe have been introduced to the magic of mathematics through their exposur...

A system and method for encryption of data is disclosed. At least one block of the data is received. The at least one block of data is modified to cause each unique data element within the at least one block to appear with a respective predetermined frequency ratio. The block of data is encrypted into ciphertext based at least on an encryption key.

There is a large body of written materials, available online, that are easily accessed and which recount aspects of Benoît’s life, times, research, quotations, and opinions. But here we try to capture afresh the fact that he was one of us, a mathematician, and to give a glimpse and feeling, for the time that you read this, of the real and amazing m...

Pascal's triangle is well known for its numerous connections to probability theory [1], combinatorics, Euclidean geometry, fractal geometry, and many number sequences including the Fibonacci series [2,3,4]. It also has a deep connection to the base of natural logarithms, e [5]. This link to e can be used as a springboard for generating a family of...

A 3-dimensional generalization of Pascal's triangle, derived using the limit definition of e, unites a wide range of combinatoric number sequences. This is the supplementary material referenced in the paper “Pascal’s Prism,” from The Mathematical Gazette, Vol. 96, July 2012. It provides additional details regarding the construction of the object, a...

Pascal’s triangle has fascinated mathematicians for centuries. Beneath its many interesting properties lies the base of the natural logarithm, e.

If s_n is the product of the entries in row n of Pascal’s triangle then (s_(n + 1)/s_n)/(s_n/s_(n - 1)) = (1 + 1/n)^n, which has the limiting value e.

More than two centuries separate the writing of The Art of Fugue, by Johann Sebastian Bach, and the printing of the first graphical representation of the Mandelbrot set. Here we examine Bach’s Contrapunctus IX from the perspective of fractal geometry and listen as it accompanies a zoom into the Mandelbrot set.

The cello suites of Johann Sebastian Bach exhibit several types of power-law scaling, the best examples of which can be considered fractal in nature. This article examines scaling with respect to the characteristics of melodic interval and its derivative, melodic moment. A new and effective method for pitch-related analysis is described and then ap...

This article demonstrates how a simple restatement of the limit definition of e can lead to the derivation of a fascinating family of functions that can be used to convert the digits of pi to those of e.

Further applications of a fascinating family of functions that can be used to convert the digits of pi to those of e.

Based on the techniques demonstrated in the paper "Improving the convergence of Newton's series approximation for e," this is extensive list of new and elegant series that converge far more rapidly than Newton's original Maclaurin series expansion.

In this paper we take one of the oldest dogs in the college calculus curriculum - Taylor series - and teach you how to do new tricks with it that Newton, Euler and their successors do not seem to have discovered.

In this paper we dare to take one of the oldest dogs in the college calculus curriculum—the Taylor series—and teach *you* how to do new tricks with it that Newton, Euler and their successors do not seem to have discovered.

For e, there exists a straight-forward Maclaurin series summation that is quite accurate. In this article, we demonstrate that there exist alternative closed-form approximations to e that are also very accurate.

EVS provides a tamper-proof means for establishing the authenticity, time, and place of a digital recording along with a means for communicating this information to a remote location.