Chaos in Prime Numbers

About a year ago I sat in on a seminar by Dr. Boris Kupershmidt on the topic of prime numbers. I'm not a mathematician and therefore most of what he said went right over my head. He did say something, however, that caught my attention: "The distribution of prime numbers is chaotic." If I remember right he was referring to "small" prime numbers. Of course in number theory 10^16 is "small." I had previously taken a course in nonlinear systems and had studied chaos. Mathematical chaos has a very interesting and subtle structure; there is order in chaos. My post entitled A Mathematical Perspective of Global Warming talked about chaos in the weather. Anyway, I am enthralled with chaos to the point that I see strange attractors everywhere. Every time I see a flag blowing in the wind I see a strange attractor. My work in hydrodynamic instability deals with turbulence and turbulence is very closely related to chaos so I get a personal satisfaction of coupling my work with my perspective of nature. I digress. I decided to see for myself if by "chaotic" he meant chaotic in the mathematical sense or chaotic in the nonmathematical vernacular.

I read a book by James Gleick called Chaos: Making a New ScienceISBN 0-140-09250-1. In it he discussed a group from the University of California at Santa Cruz called the Dynamical Systems Collective. They devised an experiment to find attractors in water dripping from a faucet. They saw that in general, the water dripped at a steady rate. If the faucet was disturbed they could force the drips to fall in groups. In order to visualize the attractor they plotted the time between one drip verses the time for the next. The plots showed "blobs" centered around two points for pairs of drips, three points for 3 drips and so forth. I got the idea to plot prime numbers in the same way: the "distance" on a number line between one appearance of a prime and the next. Low and behold some very interesting patterns emerged!

The first plot shows the result for the first 100000 primes. Many points actually plot on top of each other. The more points you plot the more complete the graph becomes and the farther it extends in both the x and y directions. However, this is extremely slow since so many plot directly on top of each other.

My next idea was to take the second difference and plotting it against the first difference

I took the 3^rd difference and plotted it against the 2nd difference

Beyond that, the pattern wasn't much different and the shape of the envelope only changed slightly.

As far as predicting prime numbers, I can't really comment on these results other than they seem to give an envelope in which many primes clearly fall and inside the envelope comes organized patterns so that one would not (so) blindly test for primes. As far as theory goes, I'm not worthy enough to even try. The most recent developments in Riemann's Conjecture and computational experiments in prime numbers suggest that this is likely to have been an exercise in futility. That's why I figured it was safe to post this to a blog rather than somewhere in the mathematical community. Either way, if somebody does find this important, I would appreciate some credit!

And now a deep thought…

Is our definition of mathematics universal?

Poverty Fallacy

These are excerpts from “Understanding Poverty in America”

“Each year, the U.S. Census Bureau counts the number of "poor" persons in the U.S. In 2005, the Bureau found 37 million "poor" Americans.”

“Today, the expenditures per person of the lowest-income one-fifth (or quintile) of households equal those of the median American household in the early 1970s, after adjusting for inflation.”

“The following are facts about persons defined as "poor" by the Census Bureau, taken from various government reports:

• Forty-six percent of all poor households actually own their own homes. The average home owned by persons classified as poor by the Census Bureau is a three-bedroom house with one-and-a-half baths, a garage, and a porch or patio.
• Seventy-six percent of poor households have air conditioning. By contrast, 30 years ago, only 36 percent of the entire U.S. population enjoyed air conditioning.
• Only 6 percent of poor households are overcrowded. More than two-thirds have more than two rooms per person.
• The average poor American has more living space than the average individual living in Paris, London, Vienna, Athens, and other cities throughout Europe. (These comparisons are to the average citizens in foreign countries, not to those classified as poor.)
• Nearly three-quarters of poor households own a car; 30 percent own two or more cars.
• Ninety-seven percent of poor households have a color television; over half own two or more color televisions.
• Seventy-eight percent have a VCR or DVD player; 62 percent have cable or satellite TV reception.
• Seventy-three percent own microwave ovens, more than half have a stereo, and a third have an automatic dishwasher.”

“As a group, America's poor are far from being chronically undernourished… Most poor children today are, in fact, supernourished and grow up to be, on average, one inch taller and 10 pounds heavier that the GIs who stormed the beaches of Normandy in World War II.”

“Eighty-nine percent of the poor report their families have "enough" food to eat, while only 2 percent say they "often" do not have enough to eat.”

“The typical American defined as "poor" by the government has a car, air conditioning, a refrigerator, a stove, a clothes washer and dryer, and a microwave. He has two color televisions, cable or satellite TV reception, a VCR or DVD player, and a stereo. He is able to obtain medical care. His home is in good repair and is not overcrowded. By his own report, his family is not hungry and he had sufficient funds in the past year to meet his family's essential needs.”

Read the article for more details.

Book Review: Advice to Rocket Scientists

Its been a little while since I read this book, but it is a damn good book, so I needed to say something about. Quite frankly the book is a must read for engineering students, especially those who want to get into academics. The book discusses how to get a job, how to deal with a boss, how to move up in industry, how to publish, how to manage students, how to negotiate a job offer and quite a bit more. To sum it up, it talks about all the stuff that no one ever talks about and it does it in a very simple straight forward way.

Just about everything is good about this book - except for one serious but forgivable flaw - THE TITLE. He states in the first chapter that a rocket scientist is anyone with an aerospace degree or someone that works at NASA, Boeing, etc. This is an absolute shame. This definition is fantastically wrong. A rocket scientist is simply someone who does scientific research on rockets. Simple. All those other aerospace engineers could be working on planes or God knows what, but not rockets, and they are certainly not rocket scientists. The advice in this book pertains to engineers and scientists as a whole - not just rocket scientists.

In my mind this error is a result of one of two possibilities. One, they picked a catchy title so that people would by the book. A very real possibility. Or two, the author (an astronautics - orbital sciences kinda guy) likes to think of himself as a rocket scientist. He is a mission designer. His work is intimately involved with rockets and depends on rockets. But alas, he is not a rocket scientist. He just works really near to them. His incorrect definition may be an attempt to include himself in the 'rocket scientist' crowd.

Before I get all crazy let me bring this article back down to earth and reveal the rocket scientists secret. What we do is hard, damn hard. Rockets are a difficult beast to handle. However, that is not to say that what we do is any harder than a lot of other noble fields. There are a lot of other really smart people out there who do amazing work. By bitching about this title I am not being a high horsed rocket scientist complaining that someone is trying to steal our thunder. Instead, it is simply that rocket science is a very small field with very few people and those of us who actually do rocket science should be acknowledged as rocket scientists, other aerospace engineers are clearly doing other work.

In summary, the book is fantastic, read it, but everywhere he writes 'rocket scientist' replace it with 'engineer' or 'scientist' or 'researcher.' That is who this book is for.

Robotic Nation

There is an article by the creator the website 'How Stuff Works' named 'Robotic Nation.' It discusses the interesting implications of an increased use of robotics, mainly in the workplace. In summary the idea goes like this:

As time progresses more and more robots will be used. Skilled labor jobs, low wage jobs and assembly line jobs will be among the first to go. Specifically, one could imagine robots replacing the jobs of many low wage workers. As computer AI and user interface is improved it is likely that fast food restaurants and many other 'simple' jobs could easily be replaced.

The question then is: Are other jobs going to be created to replace the jobs lost to robots? The answer is likely no. It takes very few people to make and repair the robots compared to the jobs they will displace. And remember, robots will be building the robots.

What do these people do then? That is the interesting part. Will this cause a large lower class? Those who are unable to do more complicated technical tasks are not going to have anything to fall back on.

Brain Marshall proposes a welfare system of sorts. That everyone will get some kind of grant ($25,000 in fact) to supplement their living. His concern is that with money flowing into the corporations via the consumer, with little money leaving to the general population via salaries, the economy would be starved for money to be spent on products and we would become less efficient. (Where does this money come from - More Taxes - Ick.)

Now, I would suggest that you read the articles yourself and form your own opinion. It is an interesting thing to think about. My personal thought is that a welfare ideology is crap. That it is bound to fail. We need to teach people to be self reliant. Instead, allow a free market to determine its fate. It is likely it will pick what is best for itself (which typically means that it is better for the economy as well.) If you can't get a job as a minimum wage slacker, well then, tough. I would imagine that forcing society to be highly educated is probably going to do more good than bad. In lieu of this 'impending disaster' which only places more reliance on education, we could use our money to, ah... educate ourselves and our children.

If the future is going to demand education, then so be it.

The Case for Space

Eric and I were invited by the Citizens for Space Exploration to address members of Congress about the benefits of continued funding for NASA and NASA's endeavors (endeavors = projects…not more shuttlesJ). The CSE is a citizens group supporting the U.S.'s continuing support for space exploration. Once a year they invite students and community leaders to have closed room discussions lobbying for a 1% NASA budget. Most of the time we actually met with the technical advisor, not the congressmen. Apparently the technical advisor actually required to have a degree in a non-technical field like basket weaving. That was rude and I'm sorry, but I think most of the people I talked to would agree that it would be beneficial to have a technical person in the technical position. Oh well, they were all pleasant people and seemed quite responsive to the topic. Really I think the number one criterion for those jobs is to be personable enough to pacify an angry or aggressive lobbyist.

The organization of the trip was OK, but not great. I asked if we should have a game plan or a unified theme for each group of five that traveled together. The organizers basically said "wing it." I thought that was crap. These people are busy people trying to balance the problems of millions of constituents. If I know one thing, someone like that will want to hear a coordinated and well thought out summary of the purpose, benefit, and request/solution. But whatever, I did my best to convey a theme at our orientation and everyone seemed to be agreeable with it so we went with it.

I commented on the fact that my American education, and specifically my Indiana education, left be behind the international students when I entered graduate school. I spent about a year where I felt like I was playing catch up. Since we were meeting primarily with Indiana delegates and Indiana doesn't have a center for NASA like Ohio, Florida, Texas,… I guessed that education would be an easy sell. It turned out I was right on the money. My argument illustrated the momentum that President Kennedy built when we went to the moon and how there was a surge in engineers afterward because they were inspired. We haven't had that since and even now with the initiative to go to the moon and Mars we don't have it because there hasn't been the effort to inform and inspire the masses. If we make it to Mars and it is highly publicized, the kids would be self-motivated to be engineers and scientists. From that point I emphasized two things. The first being that if we can't get that public appeal for Mars now, once we get there we probably won't get farther for hundreds of years, and therefore we risk losing the ability for scientific inspiration as a form of motivation. The secondary point is that U.S. educational legislation lives (and will die) with the "no child left behind" idealism. By forcing the brightest students to be slowed down by the lesser we will create a situation where each generation of teachers possesses less knowledge than the previous. Therefore it becomes the responsibility of the child to motivate themselves. Again, without a goal the ability and the self awareness required for self motivation becomes diminished. Countries we consider "third-world" are investing in space exploration and launch capabilities. They are showing serious progress and competition and will eventually overtake us if we do not act.

One staff member we spoke with was military so I mentioned that the first "Space Lawyer" just graduated from a U.S. school. It will become a matter of international interest once other countries establish a significant presence on the moon and in space in general. Lunar resources, especially if water and power become scarce on earth, are an obvious area for international incident. Right now it's kind of like the old west - lawless. But unlike the old west, it's not guaranteed to be the property of the United States. Ideally space would be a unifying point for all the world's nations, but it is not human nature to play well with others. "There is hopeful symbolism in the fact that flags do not wave in a vacuum - Arthur C. Clarke." I think he knows better.

There are many more arguments in favor of space exploration. In the spirit of brevity I've only given the stripped down version of my group's focus. I feel those arguments are viable and pertinent and I should hope the U.S. government has the foresight to push for space now instead of scrambling to keep up later.

And now a deep thought…

"The meek shall inherit the Earth -- the rest of us will go to the stars."

1978 - from Notebooks of Lazarus Long

Blind Faith

Somehow I got into a discussion a while back with my physical therapist about the cold or flu that was "going around." I don't remember the details exactly, but I'm going to say that she wondered why there wasn't a cure or vaccine for the common cold. I replied along the lines of "the cold virus evolves frequently to counteract the medication and therefore it's difficult to design a drug or vaccine for." She replied "I don't believe in that." I didn't get it immediately. You don't believe in what? If it had been "I don't believe it," then there is no question. She doesn't believe the virus has the ability to adapt. Ok. I guess. Suspecting that wasn't what she meant, I asked "You don't believe in what, evolution?" She replied "No." End of conversation. Apparently evolution is a touchy topic for her for religious reasons.

Religion is a central point in the lives of billions worldwide. Most major religions are devoted to the teachings, if not a direct following of an omnipotent or exceptionally enlightened individual. We follow without truly, unarguable manifested proof. That is faith. Religious doctrine forms the basis for many laws. It lays a foundation for moral life, regardless of religion. I feel organized religion has many benefits for humanity. Unfortunately, truly blind faith harbors ignorance in certain circumstances. Evolution is a terrible topic for Christians (and probably many other religions). The notion that God didn't "just say go" and presto, the universe as we know it, is uncomfortable for those with blind faith. Even harder is the possibility that modern life was founded on the building blocks of the more primitive.

Science has shown glaring evidence that evolution is real, that it still exists, and that it will continue after the time of humans comes to an end. Who are we to state as fact that God's methods don't include evolution? 7 days in the bible could be a metaphor for 7 billion years. It seems reasonable since science shows records predating the timeframe of the bible as calculated with standard time references. For you literalists out there, let me ask you: "What is a day?" A day is based on the time it takes Earth to rotate on its axis. Earth! A day is subjective as to what planet you're on. In an infinite universe, the probability of life in some form on another planet is 100%. Therefore, our "day" is completely meaningless beyond our own planet. Maybe God created the birds in the sky and the fish in the sea from the primordial goo, as according to His plan. And maybe when Genesis was transcribed by Moses some of the facts became inspirational metaphor. After all, God the omnipotent is a bit more inspirational and poetic than God the chemist or God the physicist. Who knows? Does it really matter since a literal believer and a metaphorical believer follow the same moral code? If every word of the bible is absolutely true, then Alabama's math curriculum would have certainly been enlightened (see http://www.snopes.com/religion/pi.asp for the joke and http://www.straightdope.com/classics/a3_341.html for a nonreligious laugh at ignorance followed by a reply that is a perfect example that blind faith begets ignorance).

After all, I'm not arguing God's existence, His influence, or the importance of religion in general. I'm not really even talking about evolution in the sense that it becomes controversial in schools. I'm talking about the ability for a life form to foster selective mutation as a mechanism to adapt. It is basic science. It is not controversial. It is verifiable and repeatable over a very short time. There should be no opposing religious stance. So many of the differences in humans, too often described as beautiful, are a good example of adaptation on a genetic level. We are all "unique snowflakes" but snowflakes with a purpose. Dark skin comes from sun exposure. Thick hair is beneficial in cold climates. Baldness is actually considered as a modern example of evolution because the need to stop heat loss through the top of your head is lessened with the comfort of sophisticated shelter and clothing. We must take an objective stance when observing our existence. We must weigh the proven facts and the non-secular interpretations and find a balance that puts perspective in both worlds.

It is a responsibility of all humans to make decisions, both moral and common, on logic and reason. Rational and informed thinking is the only way to understand fundamental perspectives. So often, religious zealots put their fingers in their ears when a scientific finding (not to mention a religious difference) contradicts their one-dimensional perception of the world. They are hindering human social and intellectual progress. It is their fault because it is their choice.

And now a deep thought…

Extreme ignorance and extreme knowledge of the world and the universe are the only two places that can truly sustain blind faith.

The Cost of...

In this world nothing can be said to be certain, except death and taxes.
-Ben Franklin

I began writing this as a discussion of military spending vs. space exploration costs. I would have liked that to be a central theme, but my anger grew as I gathered more and more numbers until I decided to instead basically list a few interesting numbers to give a perspective on spending.

The general misconception is that NASA gets a bunch of money. It is then sometimes attacked as a waste. "We can spend that money on feeding the poor... etc," they exclaim. One thing I have personally come to dislike is military spending. Now, I enjoy the fact that our country has a large military, the largest. We account for close to half of the entire worlds military spending. What I think is that it is too large. Lets look at some numbers.

Now, to date, the Iraq war has cost approximately $515.75 billion. That is around 341 million dollars a day. The 2008 budget for NASA was $17.318 billion, or about 50 Iraq days (ID's). The highly successful Mars rovers cost around $820 million, 3 ID's. The Cassini Probe, another fantastic adventure, cost $3.26 billion, 10 ID's. The entire budget of NASA to date, adjusted for inflation totals to $618.412 billion. The cost of a meaningless war equals sending man into space, landing on the moon, building all the shuttles, a space station, sending countless probes and satellites and robots into space, and all the other stuff NASA does, all of which have added immensely to our base of knowledge.

I guess it just makes me a little sick.

Other numbers:
Yearly cost of nuclear weapons: $15.1 billion
Money spent on gambling in US, 2005: $84.65 billion
2008 Social Security: $608 billion
Money spent on pets in the US a year: around $40 billion
Nation Science Foundation 2008 budget: $5.9 billion

We need our priorities, but math and science and technology are taking a back seat to many other programs. Which of these programs are going to help our nation in the end? Which are going to keep us competitive in the global market? Which will inspire new technology and innovation, thus leading to a strong economy and more jobs and better education? I think it is easy to see.

Good Links: *(other data was collected easily using the Force =>Google)
http://www.nationalpriorities.org/
http://www.whitehouse.gov/omb/
Death and Taxes Poster