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### Browsed byMonth: September 2017

Commentary: Returning to Fundamentals in Tech

## Commentary: Returning to Fundamentals in Tech

I once heard a great analogy about the difference between mathematicians and engineers in their problem-solving approaches. If an engineer and a mathematician are tasked with crossing a river, the engineer will create a set of stepping stones and get you across quickly, and safely in most cases. The mathematician will spend a month examining the problem, and more months or years carefully constructing a very sturdy stone bridge that will last for lifetimes.

Reduce the Problem: Permutations and Modulo Arithmetic

## Reduce the Problem: Permutations and Modulo Arithmetic

We’ve all seen permutations before. If you have ten distinct items, and rearrange them on a shelf, you’ve just performed a permutation. A permutation is actually a function that is performing the arrangement on a set of labeled objects. For simplicity, we can just number the objects and work with permuting the numbers.

The Gravity of Mathematics: Summary of Tech Field Day at SDC 2017

## The Gravity of Mathematics: Summary of Tech Field Day at SDC 2017

Fair warning – this will likely be one of the least technical posts I write. On September 14, I gave a presentation at Tech Field Day that wasn’t actually storage related, but rather a call to rekindle the relationship between pure math and industry. Here I’ll post the slides from that talk and summarize some of the discussion that carried on around this topic.

The Central Limit Theorem isn’t a Statistical Silver Bullet

## The Central Limit Theorem isn’t a Statistical Silver Bullet

Chances are, if you took anything away from that high school or college statistics class you were dragged into, you remember some vague notion about the Central Limit Theorem. It’s likely the most famous theorem in statistics, and the most widely used. Most introductory statistics textbooks state the theorem in broad terms, that as the sample size increases, the sample distribution of the sum of the sample elements will be approximately normally distributed, regardless of the underlying distribution. Many things used in statistical inference as justification in a broad variety of fields, such as the classical z-test, rely on this theorem. Many conclusions in science, economics, public policy, and social studies have been drawn with tests that rely on the Central Limit Theorem as justification. We’re going to dive into this theorem a bit more formally, and discuss some counterexamples to this theorem. Not every sequence of random variables will obey the conditions of theorem, and the assumptions are a bit more strict than are used in practice.

Cauchy Sequences: the Importance of Getting Close

## Cauchy Sequences: the Importance of Getting Close

I am an analyst at heart, despite my recent set of algebra posts. Augustin Louis Cauchy can be argued as one of the most influential mathematicians in history, pioneering rigor in the study of calculus, almost singlehandedly inventing complex analysis and real analysis, though he also made contributions to number theory, algebra, and physics.

One of the fundamental areas he studied was sequences and their notion of convergence. Suppose I give you a sequence of numbers, and ask you what happens to this sequence if I kept appending terms forever? Would the path created by the sequence lead somewhere?