A Simple Introduction To Chaotic Systems

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Have you heard of the butterfly effect? In one of its many versions it stipulates, somewhat facetiously, that a butterfly flapping its wings in New Mexico could cause a hurricane in China. The serious underlying point is that small changes in the atmospheric conditions that determine weather phenomena can cause a large impact on the ensuing weather. Why? Because weather is a chaotic dynamic system.

Hands-on Tutorials

Search Algorithms In The Physical World

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Imagine you want to read (or re-read) your copy of Infinite Jest. You have thousands of books arrayed across bookcases around your house. How would you find that particular book?


You could walk around the house scanning each and every book on each and every shelf until you find Infinite Jest. This would work but would be quite tiresome and time-consuming. Such a strategy is known as a brute-force algorithm in Computer Science. It establishes that a solution exists and what the worst case is. There’s nowhere to go but up from the brute-force strategy.

Getting Started

A friendly introduction to the Friendship Paradox

An Unfriendly Truth

Your friends most likely have more friends than you do. You may find this humbling, hurtful, improbable, impossible, or simply quaint. But it’s the truth.

Hands-on Tutorials

A Markovian Method To Quantify Collective Preferences From Individuals’ Ranked Choices

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What’s For Dinner?

The other day my family of four had to make a critical decision: what to get for dinner. The options on the table were sushi, fried chicken, Indian, or pizza. We could have done what we’d always done before: each person votes for their top choice and the option with the highest votes wins. But, I had been thinking about how to more accurately model collective preferences, and we decided to give that a whirl. So each of us ranked the 4 options from the most preferred to the least. Here’s how we voted (preserving anonymity to protect the innocent):

You expect the physical world around you to behave a certain way. If you knock a glass off the table, it will fall to the floor and likely shatter. How do you know this? Primarily from everyday experience, which your brain distills into your intuition. Once you’ve seen enough things fall, you subconsciously generalize that to the intuition that everything falls. Then you encounter a helium balloon, see it rise instead of fall, and you layer on conscious knowledge over your subconscious intuition. You learn about gravity and buoyancy and how to reason about an imbalance between the two causing…

Getting Started

More Math, Less Money To Keep Your Data Safe

You have precious data. Be it videos of your baby’s first steps or documents you’ve scanned for a paperless life or information you store for users of your startup product. You want to keep this data safe, of course. Safe from permanent loss, temporary unavailability, and unauthorized access.

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A typical practice is to replicate your data and spread the replicas across storage domains that have uncorrelated failures. So, for example, you store a copy of your video in three different cloud storage providers. This is a perfectly reasonable way to keep your data safe. But it has two main drawbacks:

What is luck? It’s a measure of how much chance you need to achieve a desired outcome, after accounting for any agency you have in affecting the outcome. If you attain the outcome you want, you realize a positive amount of luck i.e. good luck. And the more chance you needed for the outcome, the luckier you were. On the contrary, if the event does not pan out as you would have liked, you realize a negative amount of luck i.e. bad luck. …

You may be familiar with real numbers and their subtypes (rationals, irrationals, integers, etc.). If you need a refresher see “Real Numbers, From The Ground Up”. You may also know that there is an infinite number of real numbers and also an infinite number of each subtype of real numbers. And some of those infinities are equal to others (e.g. # integers = #rationals) while some are larger than others (# irrationals > # rationals). See “Number Of Numbers: Infinite Weirdness” for a discussion of the cardinality of real numbers. What you may not be too familiar with is that…

Would you answer truthfully if you were asked in a survey whether you consume cannabis? Maybe you do like to wind down with an edible after a long day. But the legal contradictions between federal and state laws make you wary. Besides, what will your kids' friends' parents think? The glare of judgment is blinding. Or, what if someone asked you whether you voted for Donald Trump in 2016? What about in 2020? The systematic polling errors in each of those US presidential elections live in infamy. …

The 8-cell (Photo: Wikimedia Commons)

How many spatial dimensions do we live in? How many can we directly perceive? Are the two answers the same? If they are different, what effect, if any, do any higher dimensions have on us?

What Is a Dimension?

We exist in physical space. In this space, every point or location can be labeled by a set of coordinates in a coordinate system. There can be more than one coordinate system for the same space. But once we choose a system, two points have the same coordinates if and only if they are the same point in space. The number of independent coordinates needed…

Vishesh Khemani, Ph.D.

Mindful Thinker | Software Engineer (Google, Amazon) | Theoretical Physicist (MIT) | Husband, Dad, Dog Dad

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