And What Have You've Been Doing During Lockdown?

Some folks, it would appear, may have a wee bit too much free time on their hands...:

'Arbitrary thresholding' and Statistical Practice

Black-and-white thinking and p-values... a recent post (and comments) from Andrew Gelman's blog:

https://statmodeling.stat.columbia.edu/2020/05/24/banishing-black-white-thinking-a-trio-of-teaching-tricks/

Sudoku Frenzy

OK, for all ye Sudoku buffs out there, 2 incredible Sudoku puzzles passed along by Ben Orlin on Twitter (no, really, incredible puzzles):

Siobhan Roberts Returns With More...

You probably think you've read enough about him by now, but yet another fantastic Siobhan Roberts' piece on John Conway in today's NY Times (ought not be missed):

https://www.nytimes.com/2020/05/16/science/john-conway-math.html

ADDENDUM:  ...and a day later, Jim Propp adds his own personal memories/experience to the mix:
https://mathenchant.wordpress.com/2020/05/16/confessions-of-a-conway-groupie/

Peeping Through a Keyhole

"Peeping through my keyhole I see within the range of only about thirty percent of the light that comes from the sun; the rest is infrared and some little ultraviolet, perfectly apparent to many animals, but invisible to me. A nightmare network of ganglia, charged and firing without my knowledge, cuts and splices what I do see, editing it for my brain. Donald E. Carr points out that the sense impressions of one-celled animals are not edited for the brain. 'This is philosophically interesting in a rather mournful way, since it means that only the simplest animals perceive the universe as it is.'"

                                             -- Annie Dillard ("Pilgrim At Tinker Creek")

"What is going on here?"

Today's re-run is a passage about the role of intuition in mathematics (from "How Mathematicians Think" by William Byers):

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"For the mathematician, the idea is everything.  Profound ideas are hard to come by, and when they surface they are milked for every possible consequence that one can squeeze out of them. Those who describe mathematics as an exercise in pure logic are blind to the living core of mathematics -- the mathematical idea -- that one could call the fundamental principle of mathematics. Everything else, logical structure included, is secondary.

"The mathematical idea is an answer to the question, 'What is going on here?' Now the mathematician can sense the presence of an idea even when the idea has not yet emerged. This happens mainly in a research situation, but it can also happen in a learning environment. It occurs when you are looking at a certain mathematical situation and it occurs to you that 'something is going on here.' The data that you are observing are not random, there is some coherence, some pattern, and some reason for the pattern. Something systematic is going on, but at the time you are not aware of what it might be…

"The feeling that 'something is going on here' can even be brought on by a single fact, a single number. A case in point happened in 1978, when my colleague John McKay noticed that 196884 = 196883+1. What, one might ask, is so important about the fact that some specific integer is one larger than its predecessor? The answer is that these are not just any two numbers. They are significant mathematical constants that are found in two different areas of mathematics. The first arises in the context of the mathematical theory of modular forms. The second arises in the context of the irreducible representations of a finite simple group called the Monster. McKay intuitively realized that the relationship between these two constants could not be a coincidence, and his observation started a line of mathematical inquiry that led to a series of conjectures that go by the name 'monstrous moonshine.' The main conjecture in this theory was finally proved by Fields Medalist winner Richard E. Borcherds. Thus the initial observation plus recognition that such an unusual coincidence must have some deep mathematical significance led to the development of a whole area of significant mathematical research….

"…But still it is possible to say 'we do not really understand what is going on.' Understanding what is going on is an ongoing process -- the very heart of mathematics."

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In essence, this seems very much what Eugene Wigner's famous notion of "the unreasonable effectiveness of mathematics" revolves around… how is it that the human brain is capable of such seemingly successful intuitions about the world around us….

Another Classic Puzzle

Re-running a fun, simple old golden-oldie from Alfred Posamentier's "Mathematical Amazements and Surprises":

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"You are seated at a table in a dark room. On the table there are twelve pennies, five of which are heads up and seven of which are  tails up. (You know where the coins are, so you can move or flip any coin, but because it is dark you will not know if the coin you are touching was originally heads up or tails up.) You are to separate the coins into two piles (possibly flipping some of them) so that when the lights are turned on there will be an equal number of heads in each pile."

"Your first reaction is 'you must be kidding!' How can anyone do this task without seeing which coins are heads or tails up? This is where a most clever (yet incredibly simple) use of algebra will be the key to the solution."

Posamentier Continues:

"Let's 'cut to the chase' (You might actually want to try it with 12 coins.) Separate the coins into two piles, of 5 and 7 coins each. Then flip over the coins in the smaller pile. Now both piles will have the same number of heads! That's all! You will think this is magic. How did this happen. Well, this is where algebra helps us understand what was actually done."

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Explanation: ...At the start, there are 5 heads showing among 12 coins. After separating into piles, let's say the 7-pile now has "h" heads. The 5-pile then has "5 - h" heads, and "5 - (5 - h)" tails (or, just "h" tails). Once you flip the entire smaller pile, all the tails ("h" of them) become heads, and all the heads become tails. Thus you are left with "h" heads in the "5-pile," the same as the number in the "7-pile." Whaaa-laaahhhh! Math is a beautiful thang!!

Meta-thoughts....

I've used this quote from the irascible David Berlinski several times over the years... I'm sure it's not everyone's cup-o-tea, but it remains a personal favorite:

"Like any other mathematician, Euclid took a good deal for granted that he never noticed.  In order to say anything at all, we must suppose the world stable enough so that some things stay the same, even as other things change. This idea of general stability is self-referential. In order to express what it says, one must assume what it means.

"Euclid expressed himself in Greek; I am writing in English. Neither Euclid's Greek nor my English says of itself that it is Greek or English. It is hardly helpful to be told that a book is written in English if one must also be told that written in English is written in English. Whatever the language, its identification is a part of the background. This particular background must necessarily remain in the back, any effort to move it forward leading to an infinite regress, assurances requiring assurances in turn.

"These examples suggest what is at work in any attempt to describe once and for all the beliefs 'on which all men base their proofs.' It suggests something about the ever-receding landscape of demonstration and so ratifies the fact that even the most impeccable of proofs is an artifact."

-- David Berlinski (from "The King of Infinite Space")

ASMR Sunday

For April, and a diversion from everything going on around us, time to insert another example of "fast and aggressive" ASMR from one of its practitioners:

The Dismal Science...

Today's golden-oldie re-run I first saw on Greg Ross's delightful "Futility Closet" site:

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....why rational choices ain't always so rational:

Another lovely puzzle/paradox today from Greg Ross's "Futility Closet" volume. It's known as the "dollar auction" paradox created by economist Martin Schubik. The setup (I've adapted from Wikipedia):

An auctioneer is to auction off a single dollar bill with the following rule: the bill goes to the highest bidder, AND the second-highest bidder LOSES the amount that they bid (to the auctioneer). The winner could gain a dollar for say 20 cents, for example, but only if no one else bids higher. The second-highest bidder is the biggest loser since they pay out their bid and get nothing in return.

The opening, minimum bid is 5 cents (with 5-cent increments thereafter) from one player, who would make a 95-cent profit if no one else bid. But it's sensible for another player to bid, say 10 cents, and still make a 90-cent profit. Then similarly, another bidder may now bid 15 cents, making 85-cents profit.

Whoever is the second-highest bidder at any point in time will wish to convert his potential loss to a gain by bidding higher than the highest-bidder, and so on. Obviously, if this keeps up, at some point, the dollar will COST someone a dollar to purchase -- but at least they will suffer no loss, while the 2nd highest bidder will lose 95 cents, giving them an incentive to bid $1.05 and thus decrease their loss to a nickel... at which point, the other bidder loses a whole dollar... and on and on. Bids beyond $1.00 mean that both top bidders lose money, thus minimizing the amount of loss then becomes the focus. A series of rational bids will reach and ultimately surpass the one dollar point, as the bidders seek to minimize their losses. Thus, "rational" bidding leads inevitably to both the two highest bidders losing money (while the auctioneer makes out well).

No wonder some call economics "the dismal science." ;-)

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Hahhh! I've pre-timed these re-run posts to run on forthcoming days, and on Monday (4/13) noticed that Kyle Evans entered The Aperiodical's "Big Math-Off" contest with a British version of this very same puzzle:
https://aperiodical.com/2020/04/the-big-lockdown-math-off-match-2/

Contemplation versus Dynamite

Today, just re-posting this old math meditation from Jordan Ellenberg:

"Outsiders sometimes have an impression that mathematics consists of applying more and more powerful tools to dig deeper and deeper into the unknown, like tunnelers blasting through the rock with ever more powerful explosives. And that's one way to do it. But Grothendieck, who remade much of pure mathematics in his own image in the 1960s and '70s, had a different view: 'The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration… the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it… yet it finally surrounds the resistant substance.'

"The unknown is a stone in the sea, which obstructs our progress. We can try to pack dynamite in the crevices of rock, detonate it, and repeat until the rock breaks apart, as Buffon did with his complicated computations in calculus. Or you can take a more contemplative approach, allowing your level of understanding gradually and gently to rise, until after a time what appeared as an obstacle is overtopped by the calm water, and is gone.

"Mathematics as currently practiced is a delicate interplay between monastic contemplation and blowing stuff up with dynamite."

-- Jordan Ellenberg in "How Not To Be Wrong"

Old Richard Wiseman Classic

With so many current distractions, am lacking the energy for posting new material, but will fill time re-posting some old Math-Frolic posts for any who missed them the first go-around, starting with this old Richard Wiseman classic puzzler:

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 Verbatim from Richard Wiseman's blog: 

"Imagine there is a country with a lot of people. These people do not die, the people consists of monogamous families only, and there is no limit to the maximum amount of children each family can have. With every birth there is a 50% chance its a boy and a 50% chance it is a girl.  Every family wants to have one son: they get children until they give birth to a son, then they stop having children. This means that every family eventually has one father, one mother, one son and a variable number of daughters.  What percent of the children in that country are male?"

Wiseman's Friday puzzles are frequently devious… but, often once the answer is given and explained, one feels impelled to slap one's forehead and exclaim "DOH!, well, of course!." So perhaps his best offerings are those that, even once explained, are still not totally clear, and generate a lot of ongoing discussion/debate...

The one from last Friday (above) is such an effort, once again proving how tricky and misleading, probabilities can be. I confess to requiring extra time to convince myself that 50% was the correct answer, and it stiiiill rankles my intuition (…reminds me of Cantor's "I see it but I don't believe it" reaction! ;-)). This seems to be one of those quirky puzzles that is patently obvious to many, yet hugely thorny for others (one of the keys, I think, is to remain tightly focused on strict statistical probability, and not let your brain get distracted by what could theoretically happen). Read all the discord for yourself:

http://richardwiseman.wordpress.com/2012/01/16/answer-to-the-friday-puzzle-139/

(...peruse as many of the 270+ comments as you care to -- in fact you have to read some to get the answer, since Richard neglected to post an answer or explanation in his own post).

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Classic

Haven't had much time for math or blogging lately so will just re-post a link to an old fun favorite, lest any have never seen it... George Boolos' "Gödel's Second Incompleteness Theorem Explained in Words of One Syllable":

https://www2.kenyon.edu/Depts/Math/Milnikel/boolos-godel.pdf

Stayin' Alive, Oh Yeah....

Hey, we all just wanna keep ’Stayin’ Alive,’ right:
(sorry, couldn't get these to embed within blogpost, but just follow the links)

https://www.youtube.com/watch?v=eHycxiW6XOQ

https://www.youtube.com/watch?v=LAO6-rkpYCY

https://www.youtube.com/watch?v=CyBVBNI-uSA

https://www.youtube.com/watch?v=HPh_Gi7PCqs

...admit it, you don't get content like this from Terry Tao!  ;)

Good Reads...

Those stuck at home now may have time for a lot more reading, so passing along two of my favorite long posting-links from the almost 10 years of doing this blog... in case you missed these (or worth re-reading if you saw them the first go-around).  Amazingly, both of them come from computational biologist Lior Pachter:

2014 piece on the “cultures” of math and biology:

https://liorpachter.wordpress.com/2014/12/30/the-two-cultures-of-mathematics-and-biology/

and fantastic 2015 piece on problems related to Common Core (for teachers especially):

https://liorpachter.wordpress.com/2015/09/20/unsolved-problems-with-the-common-core/

…finally, slightly similar to the above, here’s a very long, wonderful 2012 Tim Gowers’ piece, also on problems for math teachers:

https://gowers.wordpress.com/2012/06/08/how-should-mathematics-be-taught-to-non-mathematicians/#more-4305

Happy reading (and sheltering-in-place)… and stay well, warm, safe everyone.

Distraction

Maybe time for a little distraction from the approaching apocalypse and accompanying probabilities of the Trump virus, er, uhhh, I mean Corona virus, and get back to some math. So here's Alex Bellos' latest puzzle, a probability one from Peter Winkler, for The Guardian:
https://www.theguardian.com/science/2020/mar/23/can-you-solve-it-meet-the-puzzle-king

By the way, the above link to Winkler's brief Wikipedia page offers this chuckle-worthy story of Paul Erdos' attendance at the Bar Mitzvah celebration for Winklers' twin boys (the story is quoted from Hoffman's "The Man Who Loved Only Numbers"):

"Erdös came to my twins' bar mitzvah, notebook in hand," said Peter Winkler, a colleague of Graham's at AT&T. "He also brought gifts for my children--he loved kids--and behaved himself very well. But my mother-in-law tried to throw him out. She thought he was some guy who wandered in off the street, in a rumpled suit, carrying a pad under his arm. It is entirely possible that he proved a theorem or two during the ceremony."

Meanwhile, happy handwashing everyone... and in the immortal words of Yogi Berra (supposedly) keep in mind, "It's tough to make predictions, especially about the future."

Surviving ... [FWIW, more Covid-19]

There are so many sites with great coverage of the Covid-19 situation I don’t wish to spend too much time on it here (as I already devoted the previous post to it), but have to say the more extensively I read, the less consensus I see among various doctors, epidemiologists, and statisticians on approaches to the crisis. 

I mainly want to use this post to state below the regimen I use for winter cold/flu-like symptoms (in case it’s helpful for anyone else), but before getting to that feel compelled to say a little about all this emphasis on “flattening the curve” and “mitigation.”

1)  Virtually EVERYone will be exposed to this virus. Social distancing, isolation, home-sequester, will NOT prevent that. It may well reduce the frequency and intensity of exposure (especially any transmission from you to others), but you will still be exposed -- there is a difference between exposure and contraction (or infection) of the actual disease. As with most flu varieties, most people will naturally fight off the virus with their own immune system, and even if they contract it most will survive after some (and perhaps serious) unpleasantries — I realize many of you already know all this, but from the fear that has been stoked it’s also clear many do not. 

Frankly (for so many reasons), I think it almost certain that our medical system will inevitably be overwhelmed by this situation — the whole ‘flattening the curve’ argument just says that instead of being overrun, early-on, say in the next month, we will be less intensely overrun across several months later on (a worthy goal, but no panacea). The virus also may re-hit in separate waves further complicating matters.

2)  Since we all will be exposed to the virus (repeatedly) I’m disappointed that more emphasis hasn’t been placed on proactively bolstering our own individual health and immune system (for those who already have compromised health or underlying conditions, what I say may not apply, but I’m speaking to the bulk of average, reasonably healthy adults). And that means a simple emphasis on the basics:  good diet, regular exercise, hydration, adequate sleep/warmth, fresh air & sunshine, perhaps vitamins/supplements. Things you should be concentrating on all the time anyway (but of course most of us don’t), and even more especially NOW!

So anyway I want to offer the regimen I’ve worked out over decades that helps me avoid or recover from typical cold-and-flu-like symptoms (I don’t claim these specific ones will work for everyone, but that something like them will benefit most individuals):

These are steps I take at the FIRST SIGN of a sore throat, congestion/stuffy-nose, fever, headache, etc. (IF you wait more than 24 hrs. the chance of success diminishes):

1)  WARMTH — THE FIRST AND BY FAR MOST IMPORTANT step is to throw on layers of clothing… to the point almost of sweating; i.e. raise your body temperature — a fever of course is the body’s attempt to fight an infection, and if you can proactively raise your own temperature early on I often find that alone stops an infection in its tracks. This includes covering the extremities (hat, toboggan, or ski cap for head, gloves for hands, and thick socks or a double-pair of socks — I especially like latex gloves that fit tight but allow dexterous finger movement, but winter gloves OK too; also cover neck, turtleneck attire is good).  I spend the day so-clothed AND sleep in a similar bundle! -- if you awake in the middle of the night soaked in sweat, that's a GOOD thing, not a bad thing; and you can always remove layers, as needed. I'll also wear a dust mask at night (including sleep) to breathe in warm air. Depending on time of year, a room humidifier at night, may be a good idea as well.

BTW, before putting on the layered attire a hot shower and all-over application of body lotion or baby oil is in order; for me lotion on the back of the neck seems especially crucial (though I don’t even know why!).

2)  DIET — surprise, surprise, eat well! Plenty of veggies/fruits/salads; fewer starchy carbs, pasta, bread, pastries, dairy products, and definitely much less sugar (…and I’m a sweets-and-carb-addict).  When I was younger the ol' staple chicken noodle soup (and even Ramen noodles) helped, but my body no longer does well with high sodium, so I avoid these (but if you can tolerate the sodium by all means) — of course if you have Fawn Nguyen’s recipe for “pho” on-hand that just might be a more powerful cure-all than all the aforementioned ;))))  Also, at least one apple-a-day — seriously, helps me!

Plenty of water and liquids; especially hot green tea (preferably Jasmine), citrus fruits, and my one surrender to sugar is (I hate to admit it) Mountain Dew or Ginger Ale (preferably with real sugar, no fructose or artificial sweetener; 8-16 oz. a day at most) — otherwise, I avoid sodas and other sugary drinks, and decrease alcohol (...I also try to decrease coffee, but never succeed). p.s... I like to add lemon juice to plain bottled or tap water. And sucking on ice cubes is yet another small tidbit I've found helpful. 

3)  EXERCISE  — DO NOT BECOME SEDENTARY! — I fear the folks in the grocery stores right now filling their carts to the hilt with chips and crackers, ice cream and jelly beans, and going home, possibly to play video games and eat junk food… and then act surprised when they come down with the virus! The body was made to move. Even without special equipment there are plenty of exercises you can do right at home (look up YouTube exercise videos if you need to), but better yet get outside and walk, jog, hike, bicycle, etc. (aerobic exercise best; alternatively, weight training) — avoid other people, but move, move, move around outdoors… in your layered clothing, and perhaps even sipping a thermos of Jasmine tea ;)

Sunshine is a great sterilizer, and indoor air is often horrid! Try to spend at least an hour, preferably more, in the sun and fresh air (even on a cloudy day) every day; even if it’s just sitting in a lawn chair working crossword puzzles (unless it is simply too cold/windy to be outside).

In addition to exercise, one certainly might add meditation, yoga, tai chi, and the like to a daily routine of otherwise active movement.

4)  Try for a good night’s sleep each evening. (I can't do it, but maybe you can.)

5)  VITAMINS/SUPPLEMENTS — these vary tremendously from one individual to the next. I’m partial to vitamin C, garlic tablets, vitamin D, fish oil, and a Chinese herbal product a friend turned me on to, called “Cold Snap.” But I certainly know people who swear by the benefits of zinc, turmeric, ginger, honey, echinacea, bee pollen, ginseng, mushrooms, and on and on — you just have to play around and see what, if anything, benefits your body and genetics.

So much of what I’m saying here is little more than what your grandmother or great grandmother would have recommended, without any medical degree (except no one listens to grandma anymore with Dr. Oz a click away on the Web). It is not intended to sound like know-it-all stuff, but intended to simply say act proactively on behalf of your own health; don’t just sit around waiting for the Government to tell you what to do in lockstep, and then run off at the first sniffles or sore throat to clog up your medical facilities asking for a screening test (though I realize others will argue that IF the tests were readily available then mass screening should be done).

Stay warm, stay well, stay safe, move… and this too shall pass.

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ADDENDUM:  To lighten things up a bit, this seems to be making the rounds on the internet right now… and so timely (despite being 40 years old). From the incomparable Tom Lehrer, who I wrote about back HERE:

ADDENDUM II -- This is one of the better, and more nuanced, Twitter threads (in my view) of many good ones on the Covid-19 situation:

https://twitter.com/BallouxFrancois/status/1240624849879535618

...and another ADDENDUM:  After writing all the above, a number of smaller things occurred to me that I do regularly in the winter, or especially at the slightest hint of illness, so will pass them along as well, though not as key as the general actions above:

1)  boil water in a teakettle (or just a sauce pan) and hold face, obviously not to closely, over the steam (feels good, not sure it has any real illness-fighting ability)

2)  similarly, soak a wash cloth in hot water (as hot as I can stand to hold) and hold over nose, mouth, eyes, massaging face with it

3)  brush body (circular motions), especially aching areas, with a hard bristle brush (can also use self-massager, but I especially like a brush)

4)  keep fingernails trimmed short (to reduce ‘hiding’ places for virus)

5)  use saline nose spray as needed, but, NOT too often

6)  gargling with hydrogen peroxide solution or an herbal product (hard to find) called Alkalol

7)  use slow, calming breathing… inhaling slowly through nose, exhaling slowly through mouth (there are several variations practiced)

8)  Something I do anyway (because of a bad back) and now do even more, limit prolonged sitting; stay on your feet as much as possible.

Again my point in all this is that I wish there was at least a little more emphasis from the Government on individuals trying to maintain health and a strong immune system to successfully fight the virus, instead of only emphasis on trying to avoid exposure to the virus (which isn't likely possible). With all that said, DO wash hands, DON'T touch face, and stay warm and well-hydrated.

The End of Education As We’ve Known It ??? [The Covid-19 Edition]

I’m not involved with or in the loop of education, but many readers here are (at various levels). Especially at university level we are now seeing colleges all across the nation close down brick-and-mortar classes in favor of online education. This will likely last for the remainder of the current semester, and one can easily imagine it carrying on through the summer and next fall semester.

I can’t help but wonder how easy it will be, if this alteration continues very long, to return to traditional classrooms and lecture halls (for some courses of course, much easier than for others). Once young people get fully accustomed to an online format will they have tolerance for returning to buildings/desks/chairs/etc. I’ve watched, over time, how many friends have gotten ‘hooked’ on online shopping to the point that actual stores/malls/shopping seem like drudgery or a nuisance to avoid.

Where I live, for some decades now, the main state university has been planning a major expansion of its grounds to meet future needs (I believe it started as a 50+ year plan… except that we are now I think at least 25 years into it and little has been done, except pay for a lot of expensive independent studies that were then trashed). It involves transforming a lot of beautiful woodland and wild habitat into another separate university campus to meet future demands. From the beginning I wondered if this expansion was even necessary. In 50 or 100 years I could imagine the university requiring LESS, not more land/space and brick-and-mortar buildings — because students may not need classrooms, lecture halls, and other physical spaces, IF most learning takes place online. In the future students may not even matriculate at a single university, but instead take a smorgasbord of courses, over 4 years, from a variety of schools: physics 101 from Princeton, English Lit. from Yale, Economics from University of Michigan, American History from Stanford, and on and on. Of course there will always be some courses and labs that require student presence in a special building, but a lot of courses can indeed be covered digitally. Not only that, but in such a world you don’t need 100s of professors teaching calculus 101; perhaps 10 (or less) of the best could instruct ALL students throughout the nation, with plenty of assistants available more locally. 

The point being, who knows what education will be like (or its needs be) 50-100 years from now; few of us can foresee or predict all the changes ahead. And what I’m wondering is whether or not this current disease-driven crisis is simply going to hasten the day where brick-and-mortar universities, like brick-and-mortar malls/stores will be a dying breed. A lot of smaller, less well-endowed colleges are already struggling or going out-of-business in the very competitive world of higher education. How many will be left standing in decades to come, and how big of a physical footprint will they require from generations raised living their lives so largely online?

Anyway, for now (and sorry to sound like such a doomsayer) it’s difficult for me to see any likely near-term scenario for this crisis that doesn’t result in a worldwide depression and potentially huge societal transformations. I’m amused at folks using the tepid word “recession” when surely they must realize the far worse implications/ripples for economies, business, employment, and daily lives that the current situation seems to augur — but hey, ‘necessity is the mother of invention’  so who knows what lies ahead, what creative solutions may evolve and how quickly society will adapt — if you see a realistic scenario that avoids widespread financial depression feel free to offer it in the comments. I'm not perceptive enough to imagine one and I need some opening for optimism!

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ADDENDUM (3/12):  With the Dow Jones Average tanking another 2300+ points today, the answer to how to attempt to avoid economic depression may have come in the form of the Fed announcing a plan for dropping a tidy 1.5 trillion dollars into the economy (I know some debt-ridden students who would like a chunk of that). They don’t have a lot of tools left in their arsenal, but they seem to be rapidly employing those they do have (...or maybe we can all just start printing up $20 bills at home, LOL).

https://news.yahoo.com/feds-1-5-trillion-intervention-224344024.html

Meanwhile, I’ve told friends for over a year that I thought the actual fair value for the DJA was somewhere between 18,000 and 22,000 and we’ve now reached that range with an awful lot of Covid-19 news still to come. Gonna be a wild ride.

ADDENDUM II (3/14):  In my town (like many), closings/cancellations are now happening fast and furious. Most of these closings are talking of a 30-day action to be re-evaluated at that time. But we are still far closer to the early stages of this situation than to the end. I can’t imagine that at the end of 30 days, many of the closings won’t be extended for another 30-60 days (…and then after that perhaps again) with devastating results for individuals’ economic lives, especially the MANY who live month-to-month. Still possible with spring around the corner that this novel virus might prove to be seasonal, and much of the danger dissipate soon, in a best-case scenario,** but still the ripples/ramifications outward won’t suddenly end.  Whenever this ends, how much of society and human behavior will have changed for the long-term?

   ** Addendum to an Addendum ;) ... for a take on why 'seasonality' might NOT be a good thing see:
https://fivethirtyeight.com/features/why-we-shouldnt-hope-covid-19-is-seasonal-like-the-flu/

ASMR by the Beach

A new month of crazy politics so who doesn't need a li'l dose of ASMR, before perhaps all Heck erupts.... 
Another long-standing ASMR producer is "ASMR Barber," early on often doing 'shaving' ASMR videos. Here's an example by the beach:



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side-note and RFI: ASMR is being heavily-studied by researchers to understand the wide variety of stimuli that generate the sensation, and why some folks are susceptible to it (to varying degrees) and others not at all. Anyway, it all makes me wonder a bit about an opposite sensation:  the screeching sound of fingernails down a blackboard or tennis shoes on a playing court is intensely irritating to many people, yet has no effect on others. Does anyone happen to know of research or explanations for these wide individual differences?

Whoooaaa

Here are a couple of wonderful recent optical motion illusions from @jagarikin on Twitter (there is NO motion; nothing is moving):

https://twitter.com/jagarikin/status/1228122662368448512

https://twitter.com/jagarikin/status/1233780266780389378

(....is it any wonder that eyewitness testimony is so unreliable!)