As a means of getting away from reviewing for a calculus exam, I'm going to spend a moment raving to myself, since I doubt anyone reads this drivel, about my opinions on the Madoff scandal.

Although there is little doubt that Madoff's behavior was immoral, the reason he was getting away with it was that he was giving people what they wanted. They wanted something that was easy. They wanted to be able to make millions without even trying, just by entrusting part of their income to the Great Miracle Man. I hope that at least some people have learned something from this. For example, don't invest any substantial amount of money in the fulfillment of a fantasy. If it sounds like it's too good to be true, it probably is.

In other news, people are still talking about Michael Jackson. LET THE MAN DIE!!!

Hey, on those taxes:

You know, we really should cut the corporate tax. We don't have the world's highest corporate tax, but we do have the highest corporate tax compared to income tax as far as I know. Frankly, I would like to know why. If there is a good reason for the difference, that's a little different.

You want to know what's going to sink America?

An inability to make hard choices. The government needs to make a decision whether it's going to cut away most of our government or raise taxes.

You know, I have had people accusing me of basing my views on being an impoverished student who doesn't have anything to pay taxes on. That really shows me how ignorant it is to cast judgment on people you don't know because I'm actually first in line to feel the consequences of economic trouble. I may be a penniless student myself, but my bread is buttered by a guy who lives on revenue from about a million dollars in investments, which is a lot less money than most people think it is, and whether we can afford certain luxuries or non-essential travel is determined 150% by how well his stocks and bonds are doing and how much interest he's making on his CDs. Like most people, I am not immune to the consequences of economic lag, and I don't consider myself to be so.

Here is a bold claim for you: I think that raising taxes on the country's wealthiest would probably benefit the country's wealthiest--and everyone else--in the long-run. My reason for thinking so comes down to a basic grasp on real-world economics.

When the government is going rapidly into debt, this causes feelings of uncertainty and insecurity in the world's investors. It makes the government look weak and unstable if it can't raise money to pay for anything. On the other hand, if the government were to show that it is still strong enough to pull sufficient revenue to pay down its debts, it would make investors feel a lot less like we're about to go into a full-sail debt crisis. For the economy to move, people need to be willing to put money into the system. When would-be investors are nervous, they sit on their money for as long as they can before parting with it.

This isn't about wanting to "punish the rich people for being wealthy." Try to imagine that there is someone in the world besides socialists, rock-red Republicans, and whatever is "in-between." The reality we all have to deal with is that our economy is going to go down the crapper unless a cork is put in the gaping hole in the federal government's pocket. I'm content for rich people to have as much surplus wealth as they want as long as they don't spend it on making shameful pricks of themselves, like Koch Industries.

Somehow, I am fearful that my countrymen are going to continue trying to repair the economy by going on doing the same stupid nonsense that messed us up in the first place. For example, the fools in our government are probably pondering widening the deregulation in the financial services industry that was started in 1999.

The idea behind deregulating financial services back in 1999 was to encourage investment by making it easier for people to borrow money. It sounded like a good idea, but it was this glamorously retarded borrowing spree that led to the 2008 Wall Street fiasco...you know, the one that we're still trying to bail ourselves out of. It doesn't work. How many times do you have to be kicked in the nuts to realize that it's going to hurt?

Unfortunately, the retarded, brain-dead conservatives in our government seem to think, "oh, if we just made it easier for people to borrow money to start a business, we'll have more investment and more jobs. Ingenious!" What that essentially means is kicking down all of the regulations that we put there in the first place because not having them there was threatening to leave us with the economic development of Western Somalia.

When conservatives complain about regulation, it is a lot like hearing some drivers complain about the speed limit. Now, imagine that a group of angry drivers were to see one too many irresponsible drivers zip by them during their daily commute and decided, "we're tired of feeling all pokey while these law-breakers are driving 20mph faster than us. If we abolished speed limits everywhere, we'd all be able to get to work faster, and it would be good for the economy." They're not taking into account the fact that speed limits are a tried and tested means of preventing deadly accidents. They feel stifled by the speed limits, and in their egocentric thinking the purpose is therefore to stifle them.

When someone tells them it's a bad idea, I hear them saying, "oh, you're just one of those liberals who can't afford a nice sport car, so you want to force everyone to drive slow like you have to anyway. Oh, boo-hoo. Poor you. If you worked hard enough, you could afford to get a good car that goes faster, and you wouldn't feel like you had to punish people who are more worthwhile than you are." Well, this is the same kind of tone that I keep hearing out of conservatives. It is laughable, but this really is how these people think.

If you're not poor, the argument reverses itself. "Oh, you rich liberals can afford to fly around in a private jet. Us honest working people have to drive on the roads, though." These are the kinds of mental gymnastics I see coming out of conservatives on a regular basis. I really think that being conservative should be considered to be a mental illness.

The point is that we can't keep going into debt, and that's all she wrote. I don't really care how the government does it, but somehow a cork needs to be shoved into this hole.

More Math

I've been stuck on trying to think through how to form a strategy based on numerical calculations that is better than just working out your strategy on a game board, but the well of creativity has run dry. I guess I'll just amuse myself for now by working out a calculus problem.

200 + 8x^3 + x^4 and find the interval of increase. To do this one, the first thing to do is to take the derivative.

Now, a derivative is nothing but the slope of a curved line. That means that, wherever the line is flat, the derivative is equal to 0. This also means that anything that ONLY affects on the position of the line on the y-axis, because slope is a proportion, doesn't change the value of the slope. Therefore, I can eliminate 200, giving me:

8x^3 +x^4

Now, on the face of it, you take a derivative in the same way that you would a slope. Now, the way that you would calculate a slope for a straight line would to simply make this little computation: y(#2) - y(#1) divided by x(#2) - x(#2). This would give us a slope equal to the difference in y divided by the difference in x.

Now, in calculus, we think of y as a "function of x." What this means is that when we say "y = x^2" then you can ALWAYS substitute x^2 in for y. This means that you could theoretically type x^2 - x^2 divided by x-x. The problem with this theory is that you can't divide anything by zero because it gives you an undefined number.

Therefore, we're going to use an algebraic trick to get around this problem. We're going to add another variable to x wherever it appears in the equation, giving us instead (x+h)^2 - x^2 divided by x+h -x. The denominator simplifies to h.

Simplifying further, we break down the binomial x+h squared into x^2 + 2xh + h^2 to give us a numerator of x^2 + 2xh + h^2 - x^2

x^2 eliminates, giving us 2xh + h^2 divided by h. Factor out the h, and you get 2x + h.

Now, by putting in this extra variable, I have simplified the expression considerably by getting the fraction out of the picture. However, now I have a problem: I didn't really want to have that h there!

That's okay. Since I only really intended h as an algebraic place-holder, I can just say "Oh, well it was always equal to 0 anyway," thereby leaving me with simply 2x. Therefore, I have taken the limit of my original function as the difference between x (#1) and x(#2) approached the value of 0.

Back to my equation: 8x^3 + x^4

Okay, gonna throw in a few rules.

1) if you add together two slopes, you get the slope of the two functions you got them from. This means that, because, when you add together 2x and 3x, you get the sum of their slopes times x, you can always assume that, when you add two functions together, you add up the slopes.

Therefore, let's find the slope of 8x^3 first. Now, we know that a slope is 0 when a function is flat. We also know that a slope is negative when a function is "going down from left to right." Well, we know that any function cubed increases exponentially on one side and decreases exponentially on the other, so we can assume that the function is flat in the middle and steep on the sides.

We can also assume that it is increasing from right to left on both sides, right?

Therefore, the slope will be positive on both sides...right?

Therefore, what we are looking for is an equation that is close to 0 in the middle and positive on both sides.

Let's say it's x^2, then.

Now, here is another thing that we know: the higher the exponent, the more steeply it rises. Well, doesn't this mean that a larger exponent means a larger slope?

Well, let's say we did some algebra and found out that the slope of x^3 is equal to 3x^2. Well, hell, that makes sense, doesn't it? That works with the theory that a larger exponent begets a larger slope.

However, it's all well and good to say that the slope of our function will look something like the slope of x^3, but that doesn't tell us what to do with the constant, 8, in the slope of the expression 8x^3. I know how to solve that little problem, though.

Let's act out an example, here, using a linear equation:

We know the slope of 5x is 5. Well, let me use this as an example. The slope of x is 1, and we know that the constant 5 gives us a straight line therefore a slope of 0.

Solve this little equation: (0 * x) + (5 * 1) That is "the slope of 5 times x" plus "5 times the slope of x."

(0 * x) + (5 * 1) = 5. Now let's try it with the equation we're trying to work with.

The slope of 8 is 0.

The slope of x^3 is 3x^2

(0 * x^3) + (8 * 3x^2) = 24x^2

Therefore, the derivative of 8x^2 is the coefficient 8 times the exponent times x^(exponent minus 1)

So let's just take it on faith that the same rules would work with x^4

Derivative of (that is the "slope of the function") 8x^3 + x^4 is 24x^2 + 4x^3

+++++++++++++++++++++++

Now, to find the interval of increase, I just work out where my function is greater than 0.

24x^2 + 4x^3 = 0

x^2 * (24 + 4x) = 0

0 * any number is equal to zero, so the functions can be divided into x^2 = 0, 24 + 4x = 0

x = -6, 0

x = -5 means 24*(-5)^2 + 4*(-5)^3=100 positive

x = -7 gives -196

x = 1 gives 28

This is as expected because, with this type of equation where you have the highest exponent odd and the other exponent one less, it sort of "kisses" the x-axis at 0. Graph as many of these as you want to, but it's always going to be positive on both sides of 0. The other x value is less than 0, and the coefficient of the other function in the equation greater than 0. Below the line it goes.

Intervals of increase are (-6,0) and (0,ifni)

++++++++++++++++++++++

I want the maximum, so I have to find where the slope is 0 and negative on both sides. That isn't the case anywhere, though, because I'm looking at a negative, a positive, and then a positive. That means that the shape of my line goes down, stops, rises, stops, and then rises again. There really isn't a maximum val there.

On the other hand, I do have a minimum because I have a place on the line where the slope is 0 and positive on both sides, which is at x = -6. Well, this is easy because, in my original equation, 200 + 8x^3 + x^4, plugging -6 in for x gives me -232.

++++++++++++++++++++++

Okay, concavity, concavity. That means I want to know where the slope stops increasing and starts decreasing or the reverse of that. I can determine that with a second derivative.

Derivative of 24x^2 + 4x^3 = 48x + 12x^2

Stops increasing or decreasing at 48x + 12x^2 = 0

x(48 + 12x)

x = 0

x = -4

Curve's shaped generally like a quadratic, crosses through the x-axis twice, positive on both sides, irregularity at -4 that I could probably take a third derivative and say stuff about, I'm pretty sure it's concave downward between -4 and 0.

What this of course means is that the slope stops increasing and starts decreasing at -4, which means that it starts to go in the direction of flattening out from being negative. Then at 0 it switches again and starts increasing.

Essentially, this means that our base equation has a sort of wiggle or "s-shape" between -6 and 0, where it's sort of WANTING to go down again but doesn't quite manage to move any in that direction before it crosses over point 0 and starts getting steeper again.

++++++++++++++++++++++

200 + 8x^3 + x^4

Okay, to get inflection points, I plug in my 0s for the second derivative (into my original equation), and I of course get 200 for point 0 giving me (0, 200) and for -4, let's see, 4 cubed is equal to 8 squared, and we get 8 cube after multiplying with the coefficient, getting 2 cubed cubed or 2^9 and negative because the coefficient is negative and the exponent odd, and then 4^4 is 2 squared^4 or 2^(2*4) getting 2^8. 0 minus the difference is -2^8 which is -256. That subtracted from 200 is -56. 

Chess by Numbers

A chessboard is an 8*8 square, giving it a total of 64 positions. If we were to think of the movements of the pieces in terms of numerical operations and treat the board as a linear grid, consider the motions of the knight.

Starting from position 1, the knight can move to positions 18 or 11. Therefore, the motions of the piece can be thought of as increasing or decreasing in increments of 8+/-2 or 16+/-1. In this manner, the piece can be moved as far as it will go to the opposite end of the board, "upward," by making the moves 1+17, (1+17*2), and 1+17*3, where it can go no farther without surpassing 64. Moving it in one direction "to the right," 1+10, 1+10*2 and 1+10*3=31, where it stops as it approaches point 8*4.

To move in the other direction generally upward from position 31, the piece will have to move in increments of 6. Therefore, 31+6, 31+6*2, 31+6*3=49, stopping at point 8*6+1. Therefore, let's think of the game set-up as such:

Pawns:

pb1-pb8@9-16

pw1-pw8@49-56

MOVES: +8&-8, respectively; non-continuous

KILL: +1 or -1 after move

SPECIAL: secondary move IF @origin

SPECIAL: If @>56 replaced by any missing piece

Queens:

qb@4

qw@60

MOVES:

+/-1 between multiples of 8; continuous between self

+/-8 between 1 and 64; continuous

+/-8+/-1; between 1 and 64; continuous

KILL: stops after kill, and may not kill same side

Kings:

kb@5

kw@61

MOVES:

+/-1 between multiples of 8; non-continuous

+/-8 between 1 and 64; non-continuous

+/-8+/-1 between 1 and 64; non-continuous

KILL: simple, and may not kill same side

Bishops:

bba@3

bbb@6

bwa@59

bwb@62

Moves:

+/-8+/-1 between multiples of 8; continuous

Knights

kba@2

kbb@7

kwa@58

kwb@63

Moves:

+/-8+/-2 but not crossing multiples of 8; non-continuous

+/-16+/-1 but not crossing multiples of 8; non-continuous

Rooks

rba@1

rbb@8

rwa@57

rwb@64

+/-8; continuous between 1 and 64 but not over obstructions

+/-1; continuous between multiples of 8 but not over obstructions

SPECIAL: a+4 or b-2 AND k+/-1 but not over obstructions

I think that pretty much sums up the rules of chess except the weird stuff about the king. However, to make things a lot easier, let's just list all of these pieces in accordance with their starting positions and then alter the numbers in our set according to their positions. Furthermore, let's divide them up into 8 sets, four of them for either side.

{1,2,3,4} for rook through queen on the black side and {5, 6, 7, 8} for king through rook. The others will follow the same pattern. Therefore:

a{01,02,03,04}, b{05,06,07,08}

a{9,10,11,12}, b{13,14,15,16} for the pawns. We can call them "queen's pawns" and "king's pawns."

a{49,50,51,51}, b{53,54,55,56} PAWNS!!! lol

a{57,58,59,60}, b{61,62,63,64}

Now, at the outset of the game, we know that certain pieces can't move. The way we'll think of this is to say, for example, we wanted to move the rook at position 01. We couldn't perform the maneuver +1 because that would create a duplicate at position 02. Same for the maneuver +8.

However, the knight at 02 wouldn't create a duplicate by moving 16+/-1, and any of the pawns could move either +8 or +16 from the origin. Now, I'm going to show right now how I'm going to start figuring out how to move these babies!

kba up and in: 01020304, 02+16+01=16+3=19, 01190304 ((Here, I eliminated the commas)

kwb down and in: 61626364, 63-16-01=63-17=46, 61624664

Okay, maybe that's comprehensible, maybe it isn't. However, I'm going to make a shorthand especially for representing moves by the knight. Therefore, for the same move, consider this:

kba++ means that for the set represented by 01020304, I calculate the addition of (8*2+1*1)*10^4.

Therefore, kba++ is code to perform EXACTLY that operation for the set that's in! When I write kba++, I WITHOUT VARIATION perform that operation on that set. So here is a cause and effect picture in simple language:

cause: kba++

effect: black queen back-row {01020304} => 01190304

For the black queen, here is how a move three spaces down and to the right would look: qb3+-

qb3+-=01020344

44: next multiple of 8 is 48, and 48-44=4, which is greater than 3, which gives me space

44+3-8*3=23

0<23<64 therefore there is no vertical inhibition

However, now I have to complete it as a SERIES of operations. Therefore, if anything in my SERIES of one-step operations creates a duplicate, I must either make a kill or stop, depending on whether it's in the black set or the white set. The easiest way to do this would be to consider the number of spaces it would move with each duplication and just remember that.

Therefore, for the +- operation, I would ALWAYS be subtracting by 9. ALWAYS

You can see where this is going. Now, I'm going to use indicators h and v to indicate horizontal v. vertical movement, so I'll have qbh+3 for three spaces right, qbv+3 for three spaces down, etc.

How about I make up the rules for moving the other ones as I go along. Let's play!

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

a{01,02,03,04}, b{05,06,07,08}

a{09,10,11,12}, b{13,14,15,16} pawns

a{49,50,51,51}, b{53,54,55,56} pawns

a{57,58,59,60}, b{61,62,63,64}

kbb+-05060708--05062208

kwb--61626364--61624664

kba++01020304--01190304

kwa-+57585960--57425960

Updated board:

a{01,19,03,04}, b{05,06,22,08}

a{09,10,11,12}, b{13,14,15,16} pawns

a{49,50,51,52}, b{53,54,55,56} pawns

a{57,42,59,60}, b{61,62,46,64}

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

Now I'll do that sequence of moves using the method of treating the whole board as just one number.

0119030405062208091011121314151649505151535455565742596061624664

Operations:

kbb+-: (05060708+16*10^2-1*10^2)*10^(8*6)

0000000005062208000000000000000000000000000000000000000000000000

kwb--: (61626364-16*10^2-1*10^2)*10^(8*0)

0000000000000000000000000000000000000000000000000000000061624664

kba++: (01020304+16*10^4+1*10^4)*10^(8*7)

0119030400000000000000000000000000000000000000000000000000000000

kwa-+: (57585960-16*10^4+1*10^4)*10^(8*1)

0000000000000000000000000000000000000000000000005742596000000000

Therefore, adding them together

+0119030400000000000000000000000000000000000000000000000000000000

+0000000005062208000000000000000000000000000000000000000000000000

+0000000000000000000000000000000000000000000000005742596000000000

+0000000000000000000000000000000000000000000000000000000061624664

Plus add in the pawn rows:

+0000000000000000091011121314151649505151535455560000000000000000

=0119030405062208091011121314151649505151535455565742596061624664

And, as tedious as that was, it is clearly a waste of time to try using the method of treating the board as one number. However, the point has been proven that it can be done. It is very well possible to input a series of computations on one number and consider that series of computations to be the game. Anyway, let's get back to the game.

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

a{01,19,03,04}, b{05,06,22,08}

a{09,10,11,12}, b{13,14,15,16} pawns

a{49,50,51,52}, b{53,54,55,56} pawns

a{57,42,59,60}, b{61,62,46,64}

pb3+09101112--09101812 pawn move 

pw8--53545556--53545540 pawn move, this time double

Updated board:

a{01,19,03,04}, b{05,06,22,08}

a{09,10,18,12}, b{13,14,15,16} pawns

a{49,50,51,52}, b{53,54,55,56} pawns

a{57,42,59,60}, b{61,62,46,40} 

This is turning out to be an interesting project, and my real interest is in trying to see how I can determine how the math involved in chess actually works. However, I might at some point construct a full length game based on this method, using a spreadsheet program. This would allow me to study the mathematics of the game more easily, and perhaps I can begin developing a concept of strategy that goes beyond just instinct and gut reaction.

A political conversation with my father

When I visited home for my sister's birthday, my father and I ended up having a discussion. I was standing out on the porch in front of his farm house, and he stormed out after watching something on the news saying something about the Occupy Wall St. movement.

Now, I didn't give a happy damn at first about Occupy, but I thought his criticisms of Occupy were silly. He said "they're college kids, living on the parents' money, probably cutting class." Wait, if I understand correctly, unemployment is one of the things they are angry about. So the only people who are allowed to complain about unemployment now are people who have secure jobs? That's pretty cheap if you ask me.

As our conversation continued, I began to form the notion that the Occupy movement really isn't much of a different thing from the Tea Party movement. Some of the participants are different, but their concerns are pretty much the same. In response to this, my father blasted, "well, they were different. They actually had jobs. They worked for a living. Their issue was the government wasting their money. You know what TEA stands for, by the way? 'Taxed Enough Already.'" Okay, so apparently he thinks it's legitimate to complain about being taxed on money you earn at a job but not so legitimate to complain about unemployment making it impossible to find a job. Apparently, he hasn't been unemployed recently, and he has forgotten how completely it sucks.

Well, I finally cornered him on this one point. I asked him, "Okay, what would you do? What reforms would you propose if you could impose any reform?"

After explaining an example of one of Obama's lobbyists getting a fat check from the government and then going belly-up, he said essentially that he would want to cut down on lobbyists controlling where the money goes and how the laws are made.

Well, I gestured at the television, and I said to him, "okay, now what were the protesters advocating again?"

He declined to answer.

Morning Daydreams

I've been daydreaming about chess. I like to try to hold a vision of it in my mind and keep track of the movements of the pieces on it. In a way, it's sort of a memory exercise. How long can I keep an imagined game going before I lose track of which piece is where?

Lately, I've been dividing it up into a handful of basic games. In some, I start with the classic opening maneuver of putting my knights out front. In others, I might start out by moving a pawn forward to liberate a Bishop. In the same game, perhaps I'll start a "pawn assault" on the opposite side of the board. Therefore, it sounds like it might be hard to remember where all of the pieces are on the board, but you eventually start to crystallize how a particular game might go, at least at the outset.

However, I still haven't managed to play through a game yet before drifting off into other topics. Yesterday morning, I started thinking about the number seven. Did you know that, for any multiple of seven, you can play with the numbers in each place and come up with other multiples of seven?

7: 7-0=7 and 7-7=0. From 7, 0: 70. 70/7=10

14: 7-1=6 and 7-4=3. From 6, 3: 63. 63/7=9

21: 7-2=5 and 7-1=6. From 5, 6: 56. 56/7=8

28: 7-2=5 and 8-7=1. From 5, 1: 51. NON-DIVISIBLE BY 7 and PRIME

35: 7-3=4 and 7-5=2. From 4, 2: 42. 42/7=6

42: 7-4=3 and 7-2=5. From 3, 5: 35. 35/7=5

49: 7-4=3 and 9-7=2. From 3, 2: 32. NON-DIVISIBLE BY 7 and =2^5

56: 7-5=2 and 7-6=1. From 2, 1: 21. 21/7=3

63: 7-6=1 and 7-3=4. From 1, 4: 14. 14/7=2

70: 7-7=0 and 7-0=7. From 0, 7: 7. 7/7=1

Therefore, I have a clear progression except in two cases. Doing the operation on the other ones gave me the pattern you see above, from 10 to 1. I were to apply the technique used on the others, I would get the numbers 51 for the first and 31 for the other. That just wouldn't work. However, let me try something weird real quick.

12-7=5 and 18-7=11. From 5, 11: 61. Now let's do a decimal reflection:

61: 10-6=4 and 10-1=9. From 4, 9: 49. 49/7=7

Hmm...

49: 14-7=7 and 19-7=12. From 7, 12: 82 Gonna have to do something a little different here

82: 10-8=2 and 20-12=8. From 2, 8: 28. 28/7=3...HAD TO GO UP

Now that's interesting. Look at how that fits right in with my progression. However, I wonder if that's a valid method for all of the other combinations. Let's try.

21: 12-7=5 and 11-7=4. From 5, 4: 54

54: 10-5=5 and 10-4=6. From 5, 6: 56. SAME RESULT as by ORIGINAL METHOD

So it would seem that this roundabout method is the same in effect except that it circumvents that roadblock. Now that's interesting. I haven't ventured beyond 7*10 yet, but perhaps I will at some point.