Dear Mathletes: Looking for Algebraic Expressions of Concepts


#1

Strangely I developed an anti-math attitude in high school despite initially being good at it. Maybe my biggest regret, LOL. Anyways, I’ve been trying flex my mathematical thinking skills more often and I’ve really come to love simple algebraic expressions to describe things. Example, the motivation equation from Motivation Hacker:

Would love to make a list of more of these. Does anyone know of a resource that has a lot of them? Or perhaps you have some you’d like to share?


#2

I cannot give a persuasive explanation, but I would advise you to not attach with this idea much. If you like, better take a look at a textbook of cognitive psychology and find some times to actually concentrate on it. Reading articles cannot help you gain an insight about psychology.


#3

Fun question! I guess there are practically an unbounded number of them: F=m*a, money = P*e^(r*t), d=r*t, hyperbolically_discounted_utility=1/(1+r*t), …

As for the procrastination equation, the creator of Boss As a Service, @manasvinik, turned that into an interactive quiz:

Repeating myself from email with Manasvini:


i’m actually skeptical of the procrastination equation. i suspect that the equation itself is kind of a gimmick and really it’s just a statement of the four facts, {expecting to fail increases procrastination, enjoyment of the task decreases procrastination, getting distracted increases procrastination, more distant time horizons increase procrastination}.

for example, i’m not sure that M=E*V/I/D is any better than, say, M=E+V-I-D or anything else where the partial derivatives have the right signs.

but i haven’t read the actual book so i might be being super unfair…


I see @nick also hedges a bit in The Motivation Hacker but suggests that there’s more justification (plus an extra constant term in the denominator) in Piers Steel’s original research on the procrastination equation. But his use of it depends only on the signs of the partial derivatives (ie, motivation goes up/down when the components like expectancy, delay, etc go up/down).


#4

Two of my favorites are the Universal Scalability Law and Little’s Law, which are both in the territory of queueing theory and apply to modeling the throughput of software systems… and people systems. There’s a good explanation of both (applied to social contexts) in Adrian Coyler’s notes on the paper “Applying the Universal Scalability Law to Organizations”.

Little’s Law describes the average length of a queue, given the average rate at which new work items arrive and the average time to handle a single item.

 people waiting in line = arrival rate * time to complete one transaction

The Universal Scalability Law describes how the throughput of work items scales with the number of workers handling items. This one is a little more complicated, but I’ll break it down.

C(N) = N / ( 1 + alpha * (N-1) + beta * N * (N-1) )

C is the capacity (maximum possible throughput of the system), N is the number of workers, alpha is a measure of contention (how much workers are blocked trying to get access on shared resources), and beta is a measure of coherence (how often workers have to communicate in order to reach agreement on what to do). Both alpha and beta range from 0 to 1.

When both alpha and beta are 0, each worker can just crank through the items in their own line without any slow-down. Perfect scalability!

When you have contention (alpha > 0), workers have to wait on access to a central resource before they can complete a work item. Imagine a series of grocery store checkouts, with one manager supervising. When two clerks need help from the manager at the same time, that’s contention. If every transaction requires the manager, then that’s alpha = 1 and there’s no benefit from having multiple clerks.

Coherence costs (beta > 0) happen when the workers have to come to consensus about something in order to proceed. To understand coherence, imagine that none of the clerks in the grocery store can recognize which vegetable is which. Whenever a shopper is buying a vegetable, the clerk handling that transaction calls all the other clerks over to vote on whether or not the vegetable in question is broccoli.

Coherence costs can hurt worse than contention because they can make multi-worker performance worse than the single worker case.


#5

Another entry: