Unsolvability of Quintic Equations

General formulas for roots of quadratic, cubic, and quartic equations can be written in closed form using the following algebraic operations: addition, subtraction, multiplication, division, raising to an integer power, and taking an integer root.

However, roots of quintic equations, \[ax^5 + bx^4 + cx^3 + dx^2 + e x + f = 0,\] cannot be written in closed form using these operations.

My PhD advisor, Ron Larson, had told me that this was one of the questions he was asked on his oral PhD qualifying exam. I knew the fact, but never understood the proof, since it involved math that I was not familiar with.

Fred Akalin presents a nice proof using plenty of interactive demos, visualizations, and not much advanced math.

David Brooks: Resume Virtues versus Eulogy Virtues

Last week, I heard an interview with David Brooks on Intelligence Squared. Even though I was a few years late to the party (the show was from 2015), I found the content riveting.

Here is a video of that interview:

I found his distinction of "resume virtues" and "eulogy virtues" helpful as a compass on how to lead the good life. Here is a relevant excerpt from an NYT article:

The résumé virtues are the skills you bring to the marketplace. The eulogy virtues are the ones that are talked about at your funeral — whether you were kind, brave, honest or faithful. Were you capable of deep love? 

We all know that the eulogy virtues are more important than the résumé ones. But our culture and our educational systems spend more time teaching the skills and strategies you need for career success than the qualities you need to radiate that sort of inner light. Many of us are clearer on how to build an external career than on how to build inner character. 

But if you live for external achievement, years pass and the deepest parts of you go unexplored and unstructured. You lack a moral vocabulary. It is easy to slip into a self-satisfied moral mediocrity. You grade yourself on a forgiving curve. You figure as long as you are not obviously hurting anybody and people seem to like you, you must be O.K. But you live with an unconscious boredom, separated from the deepest meaning of life and the highest moral joys. Gradually, a humiliating gap opens between your actual self and your desired self, between you and those incandescent souls you sometimes meet.

Multitasking Doesn't Work

Yesterday, I saw a YouTube video in which we are asked to complete two tasks in serial, and in parallel (multitasking). While I am not sure if the test is representative of multitasking in everyday life, it is obvious even from this simple exercise that multitasking is counterproductive.

Switching costs decrease efficiency, quality of experience, and accuracy, while raising stress levels. Multitasking on people degrades relationships.

[...] evidence suggests that the human "executive control" processes have two distinct, complementary stages. They call one stage "goal shifting" ("I want to do this now instead of that") and the other stage "rule activation" ("I'm turning off the rules for that and turning on the rules for this"). Both of these stages help people to, without awareness, switch between tasks.  

Although switch costs may be relatively small, sometimes just a few tenths of a second per switch, they can add up to large amounts when people switch repeatedly back and forth between tasks. Thus, multitasking may seem efficient on the surface but may actually take more time in the end and involve more error. Meyer has said that even brief mental blocks created by shifting between tasks can cost as much as 40 percent of someone's productive time.

It causes collateral damage beyond that inflicted on the multitasker. Maria Konnikova writes in the New Yorker,

When Strayer and his colleagues observed fifty-six thousand drivers approaching an intersection, they found that those on their cell phones were more than twice as likely to fail to heed the stop signs. In 2010, the National Safety Council estimated that twenty-eight per cent of all deaths and accidents on highways were the result of drivers on their phones.

The vast majority (~98%) of us cannot multitask well, and shouldn't delude ourselves.

I like the quote at the opening of Christine Rosen's essay,

In one of the many letters he wrote to his son in the 1740s, Lord Chesterfield offered the following advice: “There is time enough for everything in the course of the day, if you do but one thing at once, but there is not time enough in the year, if you will do two things at a time.” To Chesterfield, singular focus was not merely a practical way to structure one’s time; it was a mark of intelligence. “This steady and undissipated attention to one object, is a sure mark of a superior genius; as hurry, bustle, and agitation, are the never-failing symptoms of a weak and frivolous mind.”

The Ritual

The place was bustling with activity. The annual ritual had begun.

Every January, lots of people don their workout gear, and hit the gym. If the past is anything to go by, the crowds will thin out in a month or so. A handful of regulars will persist.

All points on a circle are equally important. Yet, some points like January 1st are more important than others!

I observe all this, not with judgment or condescension. The optimism of a new year is infectious. 

People around, through their actions, seem to say, "Forget and forgive the past, for this year, I resolve to get into shape." It is hard not to be inspired by that.

Even if most of them will fail.

It is a recognition that though we are flawed, we will strive!

Happy New Year.

Some Links

1. A nice portrait of Maryam Mirzakhani (NYT)

Three years ago, Mirzakhani, 37, became the first woman to win the Fields Medal, the Nobel Prize of mathematics. News of the award, and the obvious symbolism (first woman, first Iranian, an immigrant from a Muslim country) sat uneasily with her. She was puzzled when she discovered that some people thought mathematics was not for women — it was not an idea that she or her friends encountered growing up in Iran — but she was not inclined, by personality, to tell others what to think.

2. Similar operations using sed and awk

3. Wikipedia and Fake Claims (neurologica)