Mathematics with my son and daughter

For the past few years, I've had the pleasure of helping my children with homework. I think the best times are when we study mathematics. It's not just that I like math but I find that in helping each of them I learn more about math that I didn't learn in high school.

For example, not long ago my son, who's in the eighth grade, was learning about functions. So, in his homework there was a function, say, g(x) = x+5. Pretty simple.

One problem asked him to evaluate g(3). Now, most of us would say g(3) = 8. Simple. He didn't.

He said, well, g is a function, the parentheses without any operator imply multiplication. So, g(3) is not, as we might say "g of 3" but is simply "g times 3." "It's 3g, Dad, that's it."

I had to think about that because on the one hand one response is "No, that's wrong." On the other hand, we do write terms like g(3) to mean "g times 3." It came to me that here is an example of ambiguity within mathematics and how one has to know what the statement means at a level above the statement itself.

Here's another observation. Two nights ago I was helping one of daughters with math; she's in the 11th grade. She had to solve quadratics and one problem involved a term such as sqrt(x-3). So she manipulated the equation, put the square root on one side, with other terms on the other side of the equation. She squared both sides and solved for x. Simple. Not really.

After she determined the answer, I told her, "Let's try these answers in the original problem." As quadratics we found two solutions. So, she substituted her solutions, and lo and behold, with each answer she had two possible paths. For x=12, one of the solutions, the term sqrt(12-3) is 3 but it's also -3. As she explored each possibility, only the solution of 3 worked. Another solution was, I believe, 7. So the term sqrt(7-3) is 4 or -4. Well, x = 4 did not work but when she selected x = -4 this worked!

I told her, you're on to something deep here. You found two solutions but when you substitute them back in the original problem you then have to make choices as to how to use them to get the solution to work.

I asked her to talk with her teacher about this and I haven't heard if she did.

My takeaway message is this: When you have a chance to go back to elementary levels of work, you may find new insights worth your time. I know I have.

Papers: Rejection and publication

Never has there been a better time!

Rejecta Mathematica is a new, open access, online journal that publishes only papers that have been rejected from peer-reviewed journals (or conferences with comparable review standards) in the mathematical sciences. We are currently seeking submissions for our inaugural issue.

About Rejecta Mathematica

At Rejecta Mathematica, we believe that many previously rejected papers can nonetheless have a very real value to the academic community. This value may take many forms:

• "mapping the blind alleys of science": papers containing negative results can warn others against futile directions;
• "reinventing the wheel": papers accidentally rederiving a known result may contain new insight or ideas;
• "squaring the circle": papers discovered to contain a serious technical flaw may nevertheless contain information or ideas of interest;
• "applications of cold fusion": papers based on a controversial premise may contain ideas applicable in more traditional settings;
  
• "misunderstood genius": other papers may simply have no natural home among existing journals.

Edge: An interesting site

Here's an interesting site to peruse at one's leisure. I've bookmarked on my toolbar so I'll see it most days.

Enjoy.
(hat tip: Ilachina)

Brian Hayes on Complexity

Brian Hayes, a columnist for American Scientist magazine, gives a wonderful blog entry about the complexity and its confusing use of strange, often weird, terminology.

My concern here is not with the difficulty of the concepts—there’s not much we can do about that—but with the notation and terminology. Do locutions like P^#P and NISZK and (NP ∩ coNP)/poly roll trillingly off your tongue? How about EXP, EEXP, NEXP, PEXP and SUBEXP? And while we’re on the subject of EXP and friends, I’ve been wondering how to pronounce NEXPTIME. (I’m kind of hoping the “P” is silent, as in Pterodactyl.)

And it gets even better:

The sad truth is, the naming conventions for furniture at Ikea make for a more consistent language than those of complexity theory.

Read it all, just for fun.

Invisible tank

In secret trials last week, the Army said it had made a vehicle completely disappear and predicted that an invisible tank would be ready for service by 2012.

The new technology uses cameras and projectors to beam images of the surrounding landscape onto a tank.

A soldier, who was at the trials, said: "This technology is incredible. If I hadn't been present I wouldn't have believed it. I looked across the fields and just saw grass and trees - but in reality I was staring down the barrel of a tank gun."

I have believed for years that this was possible. Years ago I hypothesized such an invention with fiber optic cables where images from one side of an object would be "transmitted around" the object. That was fanciful and people I told about it where less than impressed. Still the idea nagged at me. But using cameras with projectors, if that's how this is actually, done, makes sense. The technological hurdles must be substantial.

Videos for Science: Journal of Visualized Experiments

Here are two links, here and here, that talk about scientists using video---cheap, easy, do-it-yourself videos---to describe their work and tell others how to do what they do or have done.

This is absolutely wonderful and a great way to communicate one's work.

To begin, a video let's the viewer see and hear what a scientist is thinking and wants to show you. You don't have to be limited to paper and text which are quite good but are static. What's more, videos can be archived and indexed so that you can search through them if you make a library for yourself, or go to a library on-line.

I think this idea is right up there with publishing on-line so that everyone can have access to each other's work. I was reading just a few months ago about how expensive journals are so that many researchers cannot afford them. What's worse, libraries (even university libraries) are starting to limit their subscriptions because of the tremendous cost.

If you go to ArXiv, you'll find a collection of pre-prints that researchers have written and posted there for others to read and for editors to check for publication. In fact, Grigory Perlman, who recently proved The Poincare Conjecture posted his work to that site. Anyone could get it; many did; and each could enjoy his wonderful research. (I looked at his papers and didn't have clue where to even begin. But that's not the point.)

Having these sites, like JOVE (see the link at the top) is a truly wonderful beginning to getting research out to others for review, comments, and general dissemination.

(Hat tip: Ilachina)

P.S. There is always the chance for people to abuse these sites and post junk. It happens but I think that generally others recognize the junk and filter it away.

Mapping the Internet by Function, Not just connnectivitiy

It's the first study to look at how the Internet is organized in terms of function, as well as how it's connected, says Shai Carmi, a physicist who took part in the research at the Bar Ilan University, in Israel. "This gives the most complete picture of the Internet available today," he says.

While efforts have been made previously to plot the topological structure in terms of the connections between Internet nodes--computer networks or Internet Service Providers that act as relay stations for carrying information about the Net--none have taken into account the role that these connections play. "Some nodes may not be as important as other nodes," says Carmi.

The researchers' results depict the Internet as consisting of a dense core of 80 or so critical nodes surrounded by an outer shell of 5,000 sparsely connected, isolated nodes that are very much dependent upon this core. Separating the core from the outer shell are approximately 15,000 peer-connected and self-sufficient nodes.

(Hat tip: Ilachina)

Art of sizes

This new series looks at contemporary American culture through the austere lens of statistics. Each image portrays a specific quantity of something: fifteen million sheets of office paper (five minutes of paper use); 106,000 aluminum cans (thirty seconds of can consumption) and so on. My hope is that images representing these quantities might have a different effect than the raw numbers alone, such as we find daily in articles and books. Statistics can feel abstract and anesthetizing, making it difficult to connect with and make meaning of 3.6 million SUV sales in one year, for example, or 2.3 million Americans in prison, or 426,000 cell phones retired every day. This project visually examines these vast and bizarre measures of our society, in large intricately detailed prints assembled from thousands of smaller photographs. My underlying desire is to emphasize the role of the individual in a society that is increasingly enormous, incomprehensible, and overwhelming.

My only caveat about this series is that the prints must be seen in person to be experienced the way they are intended. As with any large artwork, their scale carries a vital part of their substance which is lost in these little web images. Hopefully the JPEGs displayed here might be enough to arouse your curiosity to attend an exhibition, or to arrange one if you are in a position to do so. The series is a work in progress, and new images will be posted as they are completed, so please stay tuned.

~chris jordan, Seattle, 2007

When you view these beautifully expressive images note the zooms and how the zoomed out views relate to the zoomed in pieces or pixels, if you will. These are a wonderful example of how you can build images with other images.

Various measures: obscure to unknown

Some examples, from the site:

Attoparsec
Parsecs are used in astronomy to measure enormous interstellar distances. A parsec is approximately 3.26 light-years or about 3.085×10¹⁶ m. Combining it with the "atto-" prefix yields attoparsec, a conveniently human-scaled unit of about 3.085 centimeters (about 1-7/32 inches) that has no obvious practical use but does have a proper SI symbol, apc. Interestingly, 1 attoparsec/microfortnight is nearly 1 inch/second (the actual figure is about 1.0043 inches per second, or approximately 2.55 cm/s).

Siriometer
The siriometer is a rarely used astronomical measure equal to one million astronomical units, i.e., one million times the average distance between the Sun and Earth. This distance is equal to about 15.8 light-years, about twice the distance from Earth to the star Sirius.

Light-nanosecond
The light-nanosecond was popularized as a unit of distance by Grace Hopper as the distance which a photon could travel in one billionth of a second (roughly 30 cm or one foot): "The speed of light is one foot per nanosecond."

Computer agents: Testing theories of social interaction

The complex behaviour of primates can be understood using artificially-intelligent computer 'agents' that mimic their actions, shows new research published in a special edition of Philosophical Transactions of the Royal Society B and presented at the BA Festival of Science in York.

Scientists using agents programmed with simple instructions to work out why some primate groups are 'despotic' whilst others are 'egalitarian' - overturning previous theories developed by primatologists.

They have also found support for an existing theory of how dominant macaques make it to the safer positions at the middle of their troop without seeming to be pre-occupied with getting there.

Using agents programmed with two rules -- stay in a group for safety and pester subordinates until they move away -- scientists found that their more dominant agents would make their way to the centre of the group.

This is one of the few times I've seen agents being used to test theories. Oftentimes, agents are criticized for not being predictive. That is, you can run agent models, see interesting and thought-provoking patterns, but then ask: How does that help me to set a course of action?

While that question is unanswered, for now, the idea of mimicking social behavior and comparing the agent model to actual creatures, is quite fascinating and a marvelous application of this science.

Quote of the Year

My first heresy says that all the fuss about global warming is grossly exaggerated. Here I am opposing the holy brotherhood of climate model experts and the crowd of deluded citizens who believe the numbers predicted by the computer models. Of course, they say, I have no degree in meteorology and I am therefore not qualified to speak. But I have studied the climate models and I know what they can do. The models solve the equations of fluid dynamics, and they do a very good job of describing the fluid motions of the atmosphere and the oceans. They do a very poor job of describing the clouds, the dust, the chemistry and the biology of fields and farms and forests. They do not begin to describe the real world that we live in. The real world is muddy and messy and full of things that we do not yet understand. It is much easier for a scientist to sit in an air-conditioned building and run computer models, than to put on winter clothes and measure what is really happening outside in the swamps and the clouds. That is why the climate model experts end up believing their own models.

Freeman Dyson, Physicist

Making fun of physics

This site pokes fun at theoretical physics and physicists. Frankly, there's a place for physics and place for fun but I think that mocking the serious endeavors of physicists is a childish and shows a degree of disrespect for what is a noble profession.

The Book Site

The link above takes you a lovely site where you can download parts of classic books. I just downloaded Edwin Shrodinger's chapter "What is Life?" and Henri Poincare's "The Future of Mathematics."

Take a look, you'll find something you like. Try this link of authors to see authors and titles easily.

A council has put up a sign warning lorry drivers to ignore their satellite navigation systems after faulty sat-nav directions caused traffic chaos in Wales.
Vale of Glamorgan Council in South Wales is the first in the UK to use visual signs warning drivers not to believe sat-nav advice after once peaceful villages were reduced to bedlam when heavy-goods lorries got stuck in tiny country lanes.

Now a sign aimed largely at foreign drivers has been put up on the outskirts of the village of St Hilary.

"The proliferation of satellite navigation aids used in heavy goods vehicles, and their over-reliance, especially by overseas drivers, has presented itself as a problem within the Vale of Glamorgan," a spokesman for the council's highways department said.

GPS navigation is a wonderful invention and usually, it's quite useful. However, one has to be careful to use it intelligently. Just because the GPS aid says to turn, one should check that there is a road there on which to turn.

GPS: Think when in use.

Math and girls

I had forgotten about this picture and then found it today. Tara C. Smith's blog is about women and science. In this post she talks to actress Danica McKellar about her book Math Doesn't Suck.

The book, as I read about it on Amazon.com, is pretty insignificant and so I won't link to it. What's interesting here is that Ms. Smith promotes women in science yet, if you look at this screen capture and the ad at the top, her advertisers see women as something less than scientific.

If one wants to support women in science, and that's a worthy goal as is supporting men in science, at least have the decency not to objectify them with sexually-based ads. C'mon.

Reading: Just do it

If you read more you exercise your brain and will be less likely to suffer mental impairment.

Workers at lead-smelting plants can suffer substantial neural damage from exposure to the toxic heavy metal. Workers who read well, however, experience comparatively less mental impairment, a new study finds.

It's not that the better readers were smarter, but that they have more "cognitive reserve," explains study leader Margit L. Bleecker, a neurologist at the Center for Occupational and Environmental Neurology in Baltimore. She says that people typically gain cognitive reserve-better or more resilient neural connections in the brain-through reading, puzzle solving, and other mentally challenging activities.

Her team recruited 112 men at a lead smelter to participate in a battery of neural assessments. After measuring the men's reading abilities-a rough gauge of cognitive reserve-the researchers split the volunteers into two groups of equal size, consisting of high or low scorers. In other respects-age, number of years worked, educational background-the two groups were similar. Most important, participants in each group exhibited the same range of blood-lead concentrations.

In the July 31 Neurology, the researchers report that in each group, men with higher blood-lead values scored more poorly on tests of hand-eye coordination. That's typical of lead poisoning. However, men in the better-reading group performed 2.5 times as well on tests of memory, attention, and concentration-tasks not necessarily related to reading.

The brain is like a muscle, Bleecker concludes: Exercising it strengthens it and makes it better able to counter the ravages of disease and poisoning.

To turn a phrase: Reading, just do it!

Out of your body experiences

The study participant sits in a chair wearing a pair of head-mounted video displays. These have two small screens over each eye, which show a live film recorded by two video cameras placed beside each other two metres behind the participant's head. The image from the left video camera is presented on the left-eye display and the image from the right camera on the right-eye display. The participant sees these as one 'stereoscopic' (3D) image, so they see their own back displayed from the perspective of someone sitting behind them. (Credit: Image courtesy of University College London) (Reference: Here.)

The above link is a fascinating article on inducing out of body experiences. The part that caught my attention is this:

A separate set of experiments was carried out by Henrik Ehrsson, an assistant professor of neuroscience at the Karolinska Institute in Stockholm.

Last year, when Dr. Ehrsson was “a bored medical student at University College London,” he wondered, he said, “what would happen if you ‘took’ your eyes and moved them to a different part of a room.”

“Would you see yourself where your eyes were placed?” he said. “Or from where your body was placed?”

To find out, he asked people to sit in a chair and wear goggles connected to two video cameras placed six feet behind them. The left camera projected to the left eye, the right camera to the right eye. As a result, people saw their own backs from the perspective of a virtual person sitting behind them.

Using two sticks, Dr. Ehrsson stroked each person’s chest for two minutes with one stick while moving the second stick just under the camera lenses, as if it were touching the virtual body.

Again, when the stroking was synchronous, people reported the sense of being outside their own bodies, in this case looking at themselves from a distance where their “eyes” were situated.

Then Dr. Ehrsson grabbed a hammer. While people were experiencing the illusion, he pretended to smash the virtual body by waving the hammer just below the cameras. Immediately, the subjects registered a threat response as measured by sensors on their skin. They sweated, and their pulses raced. They also reacted emotionally, as if they were watching themselves get hurt.

Years ago, my friend Andy mentioned just such a experiment. He put as: What would you see if you could take your eyeballs out of your head and the point them at yourself? I think we have an answer here.

Also, here's a thought: If people feel that these experiments are actually happening to them, that is, the threat of a hammer induces feeling of being hit by hammer, then we have a very effective means for torture. If a captive can be led to think wrongly, then torture would be physically harmless (or somewhat, given that one responds physically the effect is not truly harmless) yet highly effective.

Hat tip: Ilachina

Plagiarism in physics

From Ars Mathematica I learned about an article at Ars Technica describing a scandal involving plagiarism of theoretical physics papers by about 20 different people, some of them students at the Middle East Technical University in Ankara. Many of the papers were refereed and published in well-known journals, and one made it into what is now perhaps the most well-known particle theory journal, the Journal of High Energy Physics.

The rest of the post gives more detail and quotes.

Frankly, I think this just shows the greed in people.

According to Dr. Sarioglu, [faculty member at METU] two of the authors of this paper were graduate students with a prodigious track record of publication: over 40 papers in a 22-month span.

Think about those numbers. That's almost two papers a month. Incredible. To do that kind of work as graduate students, who generally work for themselves without students under them, is too good to be true; and it was, obviously.

My question is: Where was the faculty advisor?

If you go to the link above, check out the comments for more insights.

Teenageres in love, Madly in love

Adolescents who claim they are "madly in love" might not be too far off the mark: a new study suggests that they show almost manic behaviours.

Serge Brand of the Psychiatric University Clinics in Basel, Switzerland, and his colleagues surveyed 113 teenagers at around 17 years of age, asking them to complete questionnaires about their conduct and mood and to keep a log of their sleep patterns. Of those, 65 indicated they had recently fallen in love and experienced intense romantic emotions.

Thelovestruck teenagers showed many behaviours resembling "hypomania" – a less intense form of mania. For example, they required about an hour less sleep each night than teens who didn't have a sweetheart. They were also more likely to report acting compulsively, with 60% saying they spent too much money compared with fewer than 30% of teenagers who were not in love.

Moreover, the lovestruck teens were more than twice as likely to say they had lots of ideas and creative energy. Worryingly, they were also more likely to say they drove fast and took risks on the road.

Yep, that about explains it Speaking from experience, once as a teenager, now father to three teenagers, one who has a boyfriend, I can testify that teenagers do some wacky things. It's nice to see that scientists have confirmed what most parents already know. The article doesn't say how long this behavior will last but, as most parents also know, there's always an end.

Is this real?

My last post was about image editing. So, I went to the Astronomy Picture of the Day for a real image. Alas, it's not, from their caption:

Colors in the gorgeous image were picked to emphasize light emitted by specific elements in the nebula excited by the energetic starlight.

So, what picture does capture reality?