Hello all:

I started working at my first real job in mid-July. I am finally getting a life-schedule together, and I hope to soon have regular updates going. In the meantime, here is another old poem (humorously intended):

Simply Simplifying, or Solving Saliva

Once, while pondering an equation,

I happened upon a bedraggled man,

And in his saliva drippings accrued,

Was my equation, gleaming true!,

And then, as it was proven, something weird began to happen,

It reacted with the amylase,

And strange things began to happen at a quick pace,

The universe turned inside out,

The space-time continuum lost its clout,

And then what did I do?

I multiplied the equation

 by negative 2.

The Best Hallucinogen

A thought:

There are an infinite number of infinities, and some infinities are bigger than others. A simple statement. A mind-blowing statement. A true statement [for the proper definitions of each of those words, at least].

The procedure that mathematicians rely upon to compare the size of two collections of stuff (this striving to be both incredibly general and exactly specific can be a source of confusion to the uninitiated - but therein dwells the true power of mathematics) is as follows:

We match things, one by one. Yes, just like in Pre-School. But it works! Let's go on:

Think about counting. Suppose you need to get some apples from the store. If you are buying a single apple, say for a quick bite, you don't even need to think about counting – we are able to innately see the single object and see it as one thing.

Maybe you're buying a few apples, for an apple pie, let's say. Almost all people can do the same for 4 apples as you did for one. Beyond that, you need to count.

What are you doing as you count these apples? One Apple. Two Apples. Three Apples. What a mathematician would tell you if asked (and do we love to be asked!) is that you are matching, one by one, the counting numbers to the apples. You are making pairs: 'Green Apple and One' 'Red Apple and Two' 'Red-with-funny-shape Apple and Three'.

Generally speaking, two collections of things are the same size if I can pair all of them up with no bachelors left from either set.

In everyday life, infinity is generally used as more-or-less equivalent to “without end”. This is a very good definition, and it is a decent way to describe what is meant by the 'infinitely large' and 'infinitely small' one might first (and for all too many, also last) encounter in undergraduate Calculus.

For this, however, a different way of thinking about infinity will develop. It is important to note that neither view is true or false. In mathematics, the cake of the same general idea may be baked and sliced in many different ways, each appropriate to a different meal, each perfectly delicious in its own domain. Onward to the Cake!

Consider this: are there more fractions (or equivalently, all decimals that have after some point repeat the same exact sequence without end (this conveniently includes both 1/3 =.33333... and ½ = .500000....)) than there are counting numbers (1, 2, 3, 4, ...hahaha...)? The shocking answer is NO! There are just as many! Basically, you can construct a grid, kind of like a multiplication table, and at each box, instead of multiplying, you write the corresponding fraction, so that row 5, column 7 would contain the fraction 7/5 (or 3.5, if you prefer). Of course, there is no end to this grid. And here is the magic: the matching. Even though each row and each column go on forever, each diagonal (going from the bottom left to upper right) is of a finite length. These I can can match too! I can match each fraction to exactly one counting number by zig-zagging back and forth. And so, each counting number is matched up with its destined fraction, and they live happily ever after.

There are two hurdles to be wary of: One is that care must be taken too not double count – ½ is equivalent to 2/4, and both appear on this list. Second is that it is the process of matching itself that must be defined and shown here, as an actual implementation would be terminally lengthy.

Thus, you have counted to infinity, and shown that there are just as many fractions as there are counting number themselves. How astounding! What madness! – the first column (1/1, 2/1, 3/1, and so on) are the counting numbers themselves! The counting numbers are just as numerous as their extended family, when including themselves in the census. Mere chemical hallucinogens have nothing on mathematics.

But now, here is the final magic, the wonder! There is an infinity that we can't count by this matching method! The irrationals – the numbers that cannot be written as fractions (hence ir – ratio – nal). Part and parcel of this fact is that their decimal expansions (in other words, when I state as many subsequent digits as I can before I drive my audience to insanity) are rather wild – they never settle down into a pattern. This is not to say that they are random in the manner of thrown dice – the calculation of any digit you please is quite well-defined and orderly [although one can certainly ask for one that is... annoying to calculate – simple and well defined does not imply easy]. Rather, it is the case that instructions to build a real number are not going to be like 1/6, whereupon I might instruct you to write '.' and then '16' eternally, or dare I say 1/7 – a Sisyphean sentence most cruel of '.' then '142857' until Hades itself is overturned!

How? Here is How! Here is Cantor's magic! Imagine, dear reader, that some shady salesman offers you a once-in-a-lifetime deal: the magical list of all Irrational Numbers [in no particular order] – and they are bulleted, on by one:

1.    √48 = 6.92820323028.....

2.    √2 = 1.4142135623.....

3.    π = 3.14159265.....

4.    √2 = 2.6457513....

5.    and so on...

Now, say you suspect that he might be selling you adder oil – so you consider for a couple millenia and say (probably to the FTC):

“Your claim is false! There is an irrational not on your list!”

First, for brevity, let's say that this novel number is between 0 and 1. This way we only have to deal with one side of the decimal point. Then:

Irrational # 1's 1rst digit after the decimal is 9: my number is now 0.0

Irrational # 2's 2nd digit after the decimal is 1: my number is now 0.02

Irrational # 3s 3rd digit after the decimal is 1: my number is now 0.022

Irrational # 4's 4th digit after the decimal is 7: my number is now 0.0228

.

.

.

and so on.

Now! Your number doesn't match any number on this cheat's list! It can't be the first, the first digit mismatches, and on and on and on. These – they are more numerous than those! And so, we named the small infinity ℵ₀ [Aleph Null] and this larger infinity ℵ_C. There are infinitely many infinities, but that is a story for another day.

And so: We, these transient specks on a speck huddling by a spark in the vast darkness - Yet we have seen beyond even ℵ₀ - and so we journey onward forevermore, light into the darkness.

If you're interested in learning more on this topic, I highly recommend Everything and More, by David Foster Wallace.

Off-the-cuff attempted description of Tensor Fields that came up in a conversation.

   Basically - imagine kind of an arrow at every point in space that describes some force at that point (maybe a bathtub draining, and each arrow is the direction and strength that water is flowing at that point). This is called a vector field. a tensor field is that, but the arrows indicate some other property of space at that point, for example, one could talk about a tensor field (the tensor field is the term for the collection of all of these arrows) that describes the stretching forces on a rubber band as you stretch it. They can also describe curvature of space itself - when Riemann first published about them they were considered very abstract and useless to any real world thing. When Einstein was formulating the theory of General Relativity he realized that this is the very language with which to describe the very structure of the space and time of the Universe itself! It would be hard to conceive of something more real than that.

   [Einstein was himself not so familiar with this branch of mathematics, and recruited assistance for this from among his colleagues (contrary to popular belief, he was not 'bad at math', however, he had received poor math grades in school)].

   For books on this subject for the man on the street - multitudes have been written - almost any public library will likely have at least a few books on this exact topic in particular, probably under Dewey Decimal codes 500,521, and 539.

I have decided that I will post my mathematical poetry here as well. I will stagger the posting of those already written over time, starting with the one whose inception is temporally most distant.

This is the first:

The Wellspring of Imagination

You are the world to all humanity

Yet misled masses despise you greatly.

You are physics, which is totality,

Yet people think of you abhorrently!

You are perfect, you are most absolute.

You are Nature’s bedrock, its graceful heart,

Timeless, infinite, profoundly acute,

You are the mother of all human art!

You are hard logic, an eternal guide,

You are chaos, who romps in atom’s hearts,

You are concrete, harmony you provide,

You are pure muse, in poets’ teeming hearts.

You, most beautiful in all creation,

The Eternal Wellspring of Imagination.

  I often amuse myself by considering odd and interesting questions, and thinking to a reasonable estimate. This is an excellent way to pass the time while say, delayed by the MTA.

  One morning in this past fall, the subway train I was on was opening and shutting it doors in a futile attempt to mimic a politician flip-flopping. Of course, this was due to various persons holding the doors open for stragglers.

  Irritating as this is for all commuters (unless that commuter is oneself, of course) I found myself considering what cost - and benefit - is entailed by this action:

  First, how many people are on this entire train? Hmmmm. The car I am in has about 25 people. We are split about 50-50 between college students and people who are (presumably) traveling to a workplace or to an errand-place. There are a few children, who I will discount for the purposes of this calculation.

  There are 10 cars on this particular train (NYC MTA lines vary between 8 and 11 cars (according to Wikipedia)). Therefore, assume a total of say 120 students, 120 [working] adults, and 30 children. (For the purpose of this thought, a student is one (presumably) in college, an adult is one who is at least that old but not in a college, and a child is anyone else (there were no elderly or retired people in my car, so I assume that at this time of day on this line there is a negligible number)).

  Now, I will estimate the smallest value of time (in dollars per second) of all students and adults on the train. [Value of time with respect to work is far easier to define and calculate. Simply, it is the amount of money that one could earn at one's regular job in that amount of time (in theory).]

  Since I am looking for the least amount, I will assume the minimum wage. At the time of this occurrence, it is $7.25 per hour (smartphones are awesome!). It is scheduled to increase (in NYC) to $9 per hour in stages, completing on the last day of 2015. For the adults on the train, this comes to (I will write the computation as it was done initially):
($7.25 per adult per hour) * (120 adults) =  $(72.5 * 12) per hour = $(725 + 145) per hour = $870 per hour.

  The students are more difficult. Some are heading to CCNY, and some to Columbia. The split is about 75-25 (rough guess, and there are likely some who are heading to neither (the split is based on how many left at those respective stops)) in my car. Extending this, there are 90 CCNY students and 30 Columbia students.

  Columbia estimates tuition at $48,846 per academic year (two semesters), excluding travel costs, food, etc. and $64,144 excluding only transportation. Assuming that time traveling has no college-value, and that everyone is spending 15 hours per week on campus utilizing the academic expenditure, and that one semester is 15 weeks:
$48,846 per academic year  = $24,423 per semester = $1628.20 per week ≈ $108.55 per hour.

30 Columbia Students * $108.55 per hour per student = (3 * 1085.50) = $(3000 + 240 + 15 + 1.5) per hour = $3256.5 per hour

  CCNY has a semesterly tuition of about $2300:
$2300 per semester ≈ $153 per week = $10.2 per hour.

90 CCNY Students * $10.2 per hour per student = $9*102 per hour = $918 per hour.


So the total time value per hour of this entire train is:

$(918 + 3256.5 + 870) = $5044.50 per hour

≈ $84.08 per minute

≈ $1.40 per second


  The average delay I measured on this train ride from someone holding the door was about 10 seconds. This is counted from the time at which the door stops closing and begins to open again through the time at which it fully closes. The amount of time per delay, as well as the probability of a delay occurring, presumably varies greatly with respect to time of day, and location.

The average time cost to of all of the passengers per holding event is therefore $14 (for this exact train demographic, of course). This implies that in order to have some sort of moral or financial equilibrium between the one(s) who the door is being held for and all the current passengers, the time cost in waiting for the next train for the former party would have to be equal to this $14. As the average time between 1 trains is about 5 minutes or so, the hourly wage for this individual would be:

$14 per 5 minutes * 60 minutes per hour = $(14 * 12) per hour = $168 per hour (about 23 times the then-current minimum wage). That hourly wage yields:

$168 per hour * 8 hours per day = $1344 per day
* 5 days per week = $6720 per week
* 46 weeks per year = $309,120 (assuming 8 weeks of vacation per year)
Generally one living in NYC (at least in Manhattan) seems to be considered "Doing OK" at $200,000 (gross) per year. One making half  that again might prefer not to utilize the subway at all...


  Therefore, taking into consideration all of this, as well as that a train traveling uptown in late morning is hardly crowded, I conclude that with the exception of quite empty trains (i.e. end of line, very late or early, etc.) the time cost to everyone else far outweighs any individual's gain. However, bear in mind that there are many top-of-the-head scenarios where this calculation will not apply, such as where social costs and gains are involved (being late to an anniversary dinner can be hazardous ;)) or relatively rare financial events - such as a job interview.


...and I missed my stop.

Well, that's why I math.

-Aryeh

Edit: Thanks to my friend S. Klein for pointing out a computational error in the original version of this post.

Hello World!

In this blog, I intend to talk about current events in mathematics and the sciences, as well as any interesting thoughts that cross my mind. To start, here is a blurb about how I first found mathematics in the first place:

            Why do I math? Why would I do a thing so dry and boring? For me, it is neither of those.

I first truly encountered mathematics when I was seven years old. It was wintertime. I had received from my parents my first video game (if you're curious). I quickly learned that there were many features of the game based on chance. Intrigued, I checked a book about probability out of the library.

            What wonder! What beauty! How purely laid out before me were the principles embodied in my beloved game! The world was ordered. It might be crazy, it might be chaos, but that too, possesses its own order, its own eternal beauty.

That which my elementary-school teachers taught, I quickly grasped firmly and built upon. Instead, I filled my notebooks with my own thoughts, with my own knowledge, beginning with creating vast tables expressing every possible tactical situation within that game.

From that beginning, I have continued growing in my pursuit of knowledge. The beauty of the world is multiplied endlessly by the prism of mathematics into the wondrous rainbow that is our understanding of the Universe.

To this day, my favorite hobby is analyzing other hobbies.