Penguicon 2015: Scheduling

I’ll be at Penguicon this weekend, Michigan’s largest sci-fi/open source convention! I’ll be giving a talk about my upcoming research paper “Scheduling a conference to minimize RSVP conflicts” which is currently undergoing peer review.

Here are the slides for my talk: Scheduling: The first academic paper about Penguicon

A pre-print of the actual paper: Scheduling a conference to minimize RSVPs

Foundations of the golden ratio base

Positional numeration systems have come to dominate mathematics, with the ubiquitous base-ten number system used nearly universally. In addition to base-ten, other bases such as base-two and base-sixteen have found widespread usage (for example in computer engineering). We review a particularly novel take on the positional numeration system: the golden ratio base, first introduced by George Bergman in 1957, who was a 12 year old junior high student at the time. We shall prove that the number system is correct, starting with basic properties of the golden ratio up to proofs of the existence and uniqueness of representations for certain classes of numbers, which rely on algebraic number theory. In addition we will introduce simple algorithms for performing arithmetic in the system.

Download PDF: Foundations of the golden ratio base

Convergence of arithmetic and geometric means of the n-th root of a sequence of certain binomial coefficients

Introduction

Blaise Pascal first introduced the triangle that would later come to hold his name [1], although in modern notations the so-called binomial coefficient denoted by $n \choose k$ may be more familiar to the reader. We shall prove a few interesting results regarding a sequence of the $n$-th root of means of the set of binomial coefficients
\begin{equation}\label{binomial}
{n \choose 0}, {n \choose 1}, {n \choose 2}, {\cdots}, {n \choose n}.
\end{equation}
In particular, if $A_n$ is the arithmetic mean of (\ref{binomial}) and $G_n$ is the geometric mean of (\ref{binomial}), we will show that the infinite sequences
\begin{equation*}
S_A = A_1, \sqrt{A_2}, \sqrt[3]{A_3}, \sqrt[4]{A_4}, \cdots, \quad S_G = G_1, \sqrt{G_2}, \sqrt[3]{G_3}, \sqrt[4]{G_4}, \cdots
\end{equation*}
converge to $2$ and $\sqrt{e}$ respectively.

Continue reading “Convergence of arithmetic and geometric means of the n-th root of a sequence of certain binomial coefficients”

Real number approximations in finite space with IEEE 754

Jeffrey Quesnelle

Computer systems often need to represent real numbers as a part of their operation, but how to encode these numbers in fixed, finite space is non-trivial. If the size of the variable in question is bounded then so-called fixed point arithmetic can be used, treating both sides of the decimal point as integer values. In general however it would be useful to have a more flexible method of representing values that can hold both $10$ and $10^{-9}$ in a small, fixed amount of memory.

The current base ten number system is a relatively new invention [1]. We may take for granted that there is a well-defined way of writing down numbers (with only terminating reals having non-unique representations) but the problem of representing an arbitrary real number in fixed space (say, in computer memory) raises several interesting tradeoffs between precision and accuracy; we give an extremely abbreviated overview of the most popular method of representing reals in computers: IEEE 754.

IEEE Standard for Floating-Point Arithmetic (IEEE 754-2008 or simply 754) is the internationally accepted method for performing operations on and transmitting approximations of reals on computer systems. The key property of 754 is that the decimal point “floats”, i.e. if a number is very large or very small the decimal point can be “moved around” so that most bits are used to represent the significant digits of the number. Compare this method to fixed point arithmetic which has a bias towards numbers closer to zero; in 8.8 fixed point math (8 bits for whole part, 8 bits for fractional part) the number $2^{7}$ is represented as $1000 \; 0000.0000 \; 0000$ and $2^8$ cannot be represented at all, even though both numbers contain only one “significant digit”!

Continue reading “Real number approximations in finite space with IEEE 754”

Modeling weight loss with differential equations

This comes from a talk I’m giving at Penguicon 2012. The talk in general is about weight loss and getting healthy but part of it involves doing some predictive modeling of future weights based on calorie counting.

Consider the question “if I eat X number of calories a day, will I gain or lose weight?”. It’s a supremely practical question but without doing some research it isn’t immediately apparent how to answer this. The best place to start is one of the formulas for calculating basal metabolic rate. The obligatory Wikipedia link provides some more background but in short it gives us a formula for the “baseline” number of calories our body needs. Mifflin-St. Jeor is considered by many [citation needed] to be the most accurate. It is of the form:

In this formula, m is mass in kilograms, h is height in centimeters, a is age in years, and s is a gender offset which is either +5 for men or -161 for women. This gives us P, our basal metabolic rate in calories. An interesting thing about BMR is that it’s a measure of your caloric needs at more or less complete rest. A normal person’s caloric needs to maintain weight will usually be strictly greater than their BMR. Because of this, P is usually scaled by an “activity factor” which factors in daily life. This can range from ~1.2 for someone with a desk job who gets no exercise to ~1.9 for someone who engages in some serious workouts. Now that we know our caloric needs, how can use this to model where our weight will be in the future? Continue reading “Modeling weight loss with differential equations”