Correction to Polynomial Lattice

In a previous posting, I created a “polynomial lattice” out of structures found within the Pascal Triangle.  I have found some errors and boundary gaps within the definition I created.

Rather than post differences between the two, I claim that this post has Precedence.  The previous posting is now superseded.

Given a polynomial p(x) specified by a vector _b_ in Z^n as p(x) = b_0 + b_1(x | 1) + b_2(x | 2) + … + b_n-2(x | n-2) + b_n-1(x | n-1), the integer polynomial lattice surrounding p(x) is simply specified as q(x; y) = p`(x) + p(y); r(x; z) = p(x) + p`(z).  This lattice is simple, covers all the necessary bases, and achieves exactly what is required of it:  it is a lattice interleaving one set of polynomials with another; the balance of a polynomial and its parallels and the lattice-pairing of each of these is perfect; no gaps are admitted within the given parameters of the lattice, including boundaries such as when _b_ is an almost-zero vector.

The previous lattice definition allowed for many possibilities of error, and was also incorrectly specified.  Under this definition, it is possible to have _b_ = {0, 0, …, 0, 0, 1}, and to also have a viable lattice useful for determining many properties of p(x).



What is it that we hear every day?  Do we hear the radio, the voices from a favorite TV serial, the sounds of a commute along extended stretches of road, the familiar voices of loved ones?

What happens when that changes?  Do we experience subtle shifts in perspective?  Does the world turn upside down in protest?  If the radio is off during a particularly long commute, would that change everything?

Having recently been introduced to the first couple chapters of Born to Run, a journalist’s novel covering his experiences with a native tribe of Tamahumara in Mexico, and later finding that the tribe is turning from its roots in running from our modern society and every other influence that may cause annihilation amongst its people, I find that I wish they had never been discovered.

There is no possible way that such a culture could have truly remained both alive and hidden from the rest of the world permanently.  As the aforementioned book chronicled, the Mexican government was already seeking ways to put roads through the best portions of the Tamahumara hiding places.  Drug cartels already laid claim to much of the territory.  Eventually, their fate was almost guaranteed to be the same as that of the bushmen, and many other aboriginal cultures.

What does our modern culture offer to them?  Sugar, iPods, jeans, a sense of ownership of things of this world…  Noise.

To a Tamahumaran, I would imagine that a typical day, more than five years ago, would have consisted of a very different set of noises.  Mountain air, desert creature sounds, the feel of simple homespun cloth, the taste of iskiate and corn in every meal.  The culture of intimately knowing your neighbors, despite the long trek required for a visit.

Now, hearing news that this culture has been turning modern, wearing of jeans, eating sugar as a dietary staple, the younger generations adopting the common language of Mexico, being more receptive to outside influence, I wonder if the modernization arrived too quickly.  I wonder if anyone else regrets the losses that are so intangible, yet so central to what was.

I turned off my car radio six months ago.  I have experienced six months’ worth of relatively silent hour-long or longer commutes to job sites.  I have never experienced a greater amount of progress, innovation, or clarity of mind than the past six months.  I have (re-)invented designs for the internal combustion engine, started my own mathematics/coding blog, progressed farther than I ever thought possible along a train of mathematical thought, and become a much better husband and father.  I have begun to set aside those distractions that weighed me down, including the most significant, which was gaming.

I am not saying that total silence, total Gnosticism, total avoidance of the things that make this world our home is the best way.  I claim that research and guru-ism and the previous way of life of the Tamahumara all have things to teach us about how we should live.

In our human history, certainly for at least ~6000 years of it, there has been a great deal of silence.  Silence in the form of lack of advertising, silence in the form of lack of alternative spices, silence in the form of lack of alternative materials for clothing, silence in the form of lack of common entertainment.

150 years ago, if you wanted to see a show, listen to a musician or band, or taste a foreign cuisine, you would have to go Somewhere Else.  When you got There, the object of your journey would be the entire fulfillment of your adventure.  There would be silence along the journey, and silence on the return.  You would return to your common portion of existence.  You would have your memories of the experience, and any discussion with friends or family as reminders.

Today, you wouldn’t have to go There.  You could Download a copy of the song, or watch the show on YouTube.  You could get a recipe Online, and find the ingredients at a local grocery store.  You wouldn’t have to feel a bumpy road, whether through shoes, moccasins, stage-coach, or horseback.

Our modern society is not bad.  But the society we left behind has good in it as well.  We often forget the silence we have been bred into under the vast majority of our human existence.  This modern age of noises is new and foreign to our DNA.  We must not forget that as we transition ourselves and our fellow cultures into it.