Scientists figure out a bit about a toad’s brain (observation, hypothesis, experiment, model, predction)

First a bonus: Jerry Coyne’s website has a post about mayfly emergence showing up on radar!

Toad Brain Activity
A friend alerted me to this post, which is about how a toad reacts to stimuli which mimics prey in the wild. There was a bit of a “ha, ha, watch the stupid toad get “owned”” but the videos are quite interesting and illuminate how science works.

First, there is the observation (toad hunting a worm).


(photo: Heidi Carpenter)

Then some conjectures are made: “what type of stimuli elicits a “hunt” response”?
Then there is an series of “experiment followed by a refined conjecture”; here we see what “looks like” prey to the toad and what doesn’t, and what sort of response does the toad make? Then we look at the signals in the toad’s brain.

It turns out that there are a couple of receptors involved: one if the “predator” sensor is activated, it sends a signal which cancels the “hunt maneuver” response. How is this verified: one can disconnect the “canceling signal” pathway.

Then the whole lot is modeled by a neural network which elicits the predicted response. Yes, there is some mathematics that underlies this, which includes signal theory, neural networks, probability and possibly fuzzy set theory as the “predator/prey” sets appear to be fuzzy.

The videos total 30 minutes but are worth watching.

July 22, 2014 Posted by | frogs, mathematics, science, technology | , , , | Leave a comment

How even elementary math tricks our brain…

This simple example shows how a political campaign (or an advertising team) can use our intuitions to trick us, while remaining completely factual.

Suppose we ship 100 pounds of watermelons. At the start, each watermelon is 99 percent water (by weight).
The shipment arrives, and upon arrival, we find that some of the water has evaporated. Each watermelon is 98 percent water. There is no other change.

How much did the shipment weigh upon arrival?

Now our intuitions don’t handle things like percentages very well. Seriously.

So, what is the answer?

Start: the watermelons weigh 100 pounds: 99 pounds water, 1 pound not-water.
Finish: the 1 pound of “not water” is the same and comprises 2 percent of the arrival weight (arrival: 98 percent water).
So the finishing weight is: \frac{1}{.02} = 50 pounds.

Yes, going from 99 percent water to 98 percent water involves a loss of 50 pounds?

Don’t believe me?

99 pounds water + 1 pound not water = 100 pounds.
49 pounds water + 1 pound not water = 50 pounds, and \frac{49}{50} = .98 which is 98 percent.

Imagine how this could be used in a political ad.

(Here I describe where I got the answer from; no I did not give the answer on this blog as that blog’s audience should have no difficulty solving this problem).

July 15, 2014 Posted by | mathematics, social/political | | Leave a comment

Fooling yourself to think that you are smarter than you are….

I just had a case of this on a professional level.
Though I teach college mathematics for a living, I have a modest publication record. (Example) But my research is in pure mathematics, and I am far from being elite…

So don’t even think about hitting me up for a loan. There is a reason I vote Democrat. :-)

So, back to research….I am studying something called “wild knots” and “wild arcs“.

If you’ve had some calculus: roughly speaking, an arc can me thought of as the one to one image of a map f from [0,1] into 3 space. For example, think of a piece of string that isn’t allowed to intersect itself.

Arcs that come from differentiable functions are very well understood. What can be challenging to understand are arcs that are merely the image of continuous functions; for example:


This arc has no well defined tangent at the rightmost end point (where all the stuff curls up).

One known fact is: all arcs that have at most ONE point that doesn’t have a tangent (called a “wild point”) have a certain property called “being cellular”. It is a technical property; an implication of this property is that the set complement (everything in three space that is NOT on the arc) is “homeomorphic” to the complement of a smooth arc; that is, there exists a one to one, onto continuous map with continuous inverse, between the complement of this arc with a wild point to the complement of an arc that doesn’t have any wild points.

I was reviewing this fact and tried to produce a proof. I was able to and when I looked at my proof, it was the same proof that famous mathematician R. H. Bing produced in his book.

HA! I figured I was hot stuff for having been able to do what Bing did. Then I remembered: “who did you have your first graduate topology class from”? (yep, it was him). Where did you learn your topology from? (yes, from the University of Texas, where he had a major influence). OF COURSE I’d approach the problem in a similar way…it would have been strange if I didn’t.

The real genius is to be able to come up with new techniques on my own…and I didn’t do that. :-)

Bottom line: those who we’ve studied and have talked to greatly influence what we are able to do; this is one reason that “closed” societies that didn’t have much contact with other societies didn’t progress nearly as much as those which did.

July 12, 2014 Posted by | mathematics, social/political | , | Leave a comment

Some humor (intentional and unintentional)

The ankle is still slightly sore, so I’ll do a run on the treadmill and walk outside. I’ll write more later in the day; I am closing in on finishing some revisions for a paper.


You need a bit of math background to get the joke; hint: the sigma in the picture doesn’t stand for “sigma”.


(photo taken from a “wedding photo fails” collection): My guess that the above is an inside joke of some sort; perhaps it relates to an incident in the bride’s life? Something like this happened to my 8′th grade English teacher in class while she was walking around, checking student’s work at the desks. And, unlike these women, she was wearing hose but no underwear.


Rihanna takes a stand against visible panty lines.


This is the type of thing that your crazy e-mail forwarding uncle is getting so worked up about:


Picky, picky, picky
A right wing US Senate candidate in South Dakota was charged with election fraud. Reason: she signed a statement that she personally witnessed people signing petitions to get her on the ballot. At the date of signing: she was actually overseas. Ooops.

Climate Change: that is Jesus getting angry with us:


Reading comprehension
I get a negative impression of conservatives because, well, the ones I encounter can’t understand what they read:


June 5, 2014 Posted by | butt, injury, mathematics, politics/social, social/political | , , , | Leave a comment

PC’ness, mathematics, quantum mechanics and other stuff….

Shameless fluff: I really like President Obama:

(yes, I know; it is probably edited)

Confession: when I am around a really accomplished mathematician or scientist, well, I end up acting a bit like Chester….. :-)

Quantum Mechanics
Little Boy Boo collapses Leghorn’s “position” wave function:

Of course quantum mechanics uses a lot of mathematics. And mathematics uses things like equations and formulas. But equations and formulas are a relatively recent invention in human history (relative to the time humans have been using writing). Prior to that, an equation such as 4x + 3 = 7 would have to be written as “a number, when multiplied by 4 and subsequently added to three yields 7″ or something like that. You can see how mathematical progress would have been glacially slow!

For more on this, read this Guardian article.

Academia: It is nice to see an accomplished liberal academic speaking out against smothering “political correctness. Jerry Coyne talks about some instances that were lampooned by the Wall Street Journal. He then notes:

The WSJ is, of course, a conservative organ, and goes on to decry the “loopiness” of the left wing and the ostracism of conservative professors, as well the tendency of universities to allow “the nuttiest professors to dumb down courses and even whole disciplines into tendentious gibberish.” That’s an exaggeration, but still, it’s disturbing that we see the left attacking, in effect, freedom of speech. If you don’t like Condaleeza Rice (and I sure don’t), that doesn’t mean you should mount such a protest against her that she has to withdraw. Are all speakers to be vetted for signs of cryptic conservatism? Are students that loath to hear views that might disagree with them?

I’m no conservative, but these Commencement Police frighten me, and paint students as self-entitled, fragile beings who can’t countenance dissent—unless it’s their own. At my own commencement at William and Mary in 1971, we had an undistinguished state legislator as speaker—and this after many of us wanted a more leftist person. But we didn’t shout him down, or pressure the university to withdraw his invitation. Instead, we organized a “counter commencement,” held at a different time and place, and our class invited and paid for Charles Evers, the older brother of slain civil rights worker Medgar Evers.

On one point the Journal has it right:

No one could possibly count the compromises of intellectual honesty made on American campuses to reach this point. It is fantastic that the liberal former head of Berkeley should have to sign a Maoist self-criticism to be able to speak at Haverford. Meet America’s Red Guards.

Indeed. The remedy for speech you don’t like and have rational arguments against, is this: more speech—counter speech.

However the “Red Guards” snark is an exaggeration; after all, these people can be stood up to; no one is going to shoot you.

Personal life
Where I was wrong: there was a time when I was part of “a calorie is a calorie” crowd. I was “sort of” right. After all, one cannot get fat if one doesn’t ingest calories; the fat has to come from somewhere.

But though the energy balance is still true, some foods have no available energy for us at all (at least for humans: think “grass”). Some foods are put to work making energy and some foods are more prone to get stored as fat and NOT be immediately used. Hence the new conjecture as to why fat people might be hungry all of the time. Note: I am no longer morbidly obese but I not only cut back how much I eat, I changed what I eat (drastically).

Personal note

Do we sometimes benefit from doing what we don’t want to do? This essay argues “yes”. This is similar to the line in John Denver’s song “Thank God I am a Country Boy”: “fiddle when I can, work when I should”. This essay has an interesting paragraph:

Dr. King taught that every life is marked by dimensions of length, breadth and height. Length refers to self-love, breadth to the community and care of others, and height to the transcendent, to something larger than oneself. Most would agree with Dr. King’s prescription that self-fulfillment requires being able to relate yourself to something higher than the self. Traditionally, that something “higher” was code for God, but whatever the transcendent is, it demands obedience and the willingness to submerge and remold our desires.

Perhaps you relish running marathons. Perhaps you even think of your exercise regimen as a form of self-improvement. But if your “something higher” is, say, justice and equality, those ideals might behoove you to delegate some of the many hours spent pounding the track on tutoring kids at the youth center. Our desires should not be the ultimate arbiters of vocation. Sometimes we should do what we hate, or what most needs doing, and do it as best we can.

He also mentions the situation in which a skilled doctor saved up so much money he could retire and roller blade full time (his true passion) which, while it is what he wanted to do, ended up depriving patients of his life saving skills. I really can’t weigh in as, well, I really don’t have “essential skills”. But I can do things like volunteer (as I do to help new runners build to a healthy life style) and give blood (and I hate that, but will keep doing it…and complaining about doing it).

Note: My body can no longer can stand training for hours on end so the “marathon” paragraph really doesn’t apply to me.

May 23, 2014 Posted by | Barack Obama, mathematics, obesity, physics, political/social, science, social/political | , , , , , | Leave a comment

A bit of mathematics: discrete log problem

This is an interesting post about mathematics (“applied number theory”) and applications to cryptography.

Here is the discrete log problem: if one wants to solve, say, a^x = b where a is a positive real number and b is a positive real number, then we know x = log_a(b) or, equivalently, x = \frac{ln(b)}{ln(a)} where ln(x) is the natural (base e ) logarithm.

Now a field is a system of objects that can be added, multiplied where the usual distribution properties hold and each object, save the additive identity (the “zero”) has a multiplicative inverse; the usual examples are the real numbers, the rational numbers, the complex numbers, and the integers modulo a prime number).

A finite field is a field that has only a finite number of elements (say, the integers mod a prime number). Finite fields always have characteristic p^n where p is a prime number.

The discrete log problem is to solve x^n = b where x, b are known elements of the field. This problem doesn’t always has a solution even if the field is of characteristic p .

Example: if we work “integers mod 7 ” and we wish to solve 2^n = 5 the solution does not exist. Why? 2^2 = 4, 2^3 = 1 therefore 2^n \in \{1, 2, 4\} mod(7). The problem: the set of units mod(7) with multiplication from a group of order 6, and 2 generates a subgroup of order 3. Note that “4″ generates a subgroup of order 2.

So, this is an interesting and difficult problem to solve, in general, at least in a “reasonable mount of time”.

Note: this is one branch of mathematics where it makes sense to do some experimentation, even though mathematical results require proof; an experiment with lots of data doesn’t cut it.

For more:

May 20, 2014 Posted by | mathematics, science | , | Leave a comment

Blogging…superficial here but

I have been busy here.

May 4, 2014 Posted by | mathematics | | Leave a comment

Mathematics and God….

Mano Singham talks about how there might be religious objections to…set theory?

Religious objections to mathematics is nothing new. Underwood Dudley talks about some instances of this in his book Mathematical Cranks. For example, there was a Priest who wrote a tome “attacking” non-Euclidean geometry and purporting to prove that the parallel axiom is really a theorem that follows from the previous axioms.

His objection to non-Euclidean geometry was religious.

May 2, 2014 Posted by | mathematics, religion, social/political | | Leave a comment

Mooching, coloring and Paradigming….

Mathematics: this 3-quarks daily post is about a “simple to state but hard to solve” mathematics problem::

From this book I learned of an irresistible 60+ year old question called the Hadwiger-Nelson problem. It’s easy to state:

If you want to color the points of the Euclidean plane in such a way as to guarantee that there are never two points of the same color which are exactly one unit apart, how many colors do you need?

By the Euclidean plane we just mean the usual xy-plane you use in geometry when you draw circles, parabolas, etc. Since the plane goes off to infinity in every direction there are infinitely many points on the plane. A reasonable first guess is that you’ll need infinitely many colors [3].

With a little sideways thinking, it might occur to you to tile the plane with squares sized so that the distance from corner to corner is a little less than one unit. All of the points within that square are then less than one unit apart so can be colored a single color without causing us any trouble. We then can color the points on the plane by instead coloring each square a single color. This (in retrospect) easy observation quickly gives a striking breakthrough to what seemed previously to be an unsolvable problem.

For example here is a coloring of the squares using nine colors:


And of course, one can do better. But no one knows, as of this time, how much better. We need more than 3, but less than 8.

Note: this is NOT the famous (solved) 4-color problem for a map on a plane. Here, if you can construct ONE map in which every “country” had diameter less than one unit but no two countries of the same color were within one unit of each other, you’d solve the problem.

Now of course, professional mathematicians often deal with cranks who think that they’ve solved such a famous problem. Of course, they haven’t. But it appears to be a human tendency to think that YOU, a non-specialist, are smarter (or more insightful) than the established specialists of another area. You (probably) aren’t. If you can’t outdo the best in your own area, what makes you think that you will outdo the best in areas that aren’t your own?

Now that it appears that Obamacare won’t implode on its own, Republicans are scrambling. Why? Obamacare is designed the way that it is because it has to be that way. That is: there can be no coverage for preexisting conditions if there aren’t healthy people in the pool. Sure, many will complain that they can’t afford it….while risking ruinous debt if they get a major illness. We can’t afford that either.

Fact: the Republicans don’t have an alternative because this was the conservative option to begin with.

The alternative is: the liberal option (single payer) which is what I support.

Interestingly enough, Republicans seem to be making a habit of…supporting moochers?

t is, in a way, too bad that Cliven Bundy — the rancher who became a right-wing hero after refusing to pay fees for grazing his animals on federal land, and bringing in armed men to support his defiance — has turned out to be a crude racist. Why? Because his ranting has given conservatives an easy out, a way to dissociate themselves from his actions without facing up to the terrible wrong turn their movement has taken.

For at the heart of the standoff was a perversion of the concept of freedom, which for too much of the right has come to mean the freedom of the wealthy to do whatever they want, without regard to the consequences for others.

Start with the narrow issue of land use. For historical reasons, the federal government owns a lot of land in the West; some of that land is open to ranching, mining and so on. Like any landowner, the Bureau of Land Management charges fees for the use of its property. The only difference from private ownership is that by all accounts the government charges too little — that is, it doesn’t collect as much money as it could, and in many cases doesn’t even charge enough to cover the costs that these private activities impose. In effect, the government is using its ownership of land to subsidize ranchers and mining companies at taxpayers’ expense.

It’s true that some of the people profiting from implicit taxpayer subsidies manage, all the same, to convince themselves and others that they are rugged individualists. But they’re actually welfare queens of the purple sage.

Mr. Bundy is a moocher, period. I don’t think that the story is over.

April 29, 2014 Posted by | economics, economy, mathematics, social/political | Leave a comment

A Practical Application of Game Theory…


I love Rat!

(Game theory, in part, deals with the value of cooperation versus acting selfishly.)

April 28, 2014 Posted by | humor, mathematics, social/political | | Leave a comment


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