A Fuzzy Theorem or two

An interval number A consists of a closed, bounded subset of the real numbers denoted here as A=[a_l, a_u] where a_l denotes the lower bound for the interval number A, and a_u denotes the upper bound for the interval number A. Define A as A={x: a_l<=x<=a_u}=[a_l, a_u]. Define addition of interval numbers as A+B=[a_l, a_u]+[b_l, b_u]=[a_l+b_l, a_u+b_u] Define the image of an interval number -A=-[a_l, -a_u] as [-a_u, -a_l], e. g. -[1, 2]=[-2, -1], -[-.1, .2]=[-.2, .1]. Define {0}=[0, 0] as the neutral for interval numbers, in the sense that for any interval number A, A+{0}={0}+A=A. Consider the subset N of interval numbers which lie close to {0} in the sense that n_l lies close to 0 and n_u lies close to 0.

Fuzzy Theorem: For interval numbers N which lie close to 0 in the sense that both n_l and n_u lie close to 0, if A+B approximately equals A+C, then B more or less approximately equals C.

First we'll need some lemmas.

Lemma 1: For all interval numbers A+B=B+A

Proof: A+B=[a_l, a_u]+[b_l+b_u]=[a_l+b_l, a_u+b_u]=[b_l+a_l, b_u+a_u]=[b_l, b_u]+[b_l+a_u]=B+A.

Lemma 2: For all interval numbers A+(B+C)=(A+B)+C

Proof: A+(B+C)=[a_l, a_u]+([b_l, b_u]+[c_l, c_u])=[a_l+a_u]+([b_l+c_l, b_u+c_u])
=[a_l+b_l_c_l,a_u+b_u+c_u]=[(a_l+b_l)+c_l, (a_u+b_u)+c_u]=([a_l+b_l, a_u+b_u])+[c_l, c_u]
=([a_l, a_u]+[b_l+b_u])+[c_l, c_u]=(A+B)+C

Fuzzy Lemma 3: N+(-N)=(-N)+N approximately lies close to {0}, or to put it another way, N+(-N)=(-N)+N=*{0} where =* indicates "approximately equals".

Demonstration: By lemma 1 we have N+(-N)=-N+N. N lies close to {0} in the sense that n_l and n_u lie close to 0. Consequently, -n_u and -n_l lies close to 0 also. N+(-N)=[n_l-n_u, n_u-n_l], which then approximately lies close to 0. Notice the addition of the hedge 'approxmiately' since B=[-.01, .01] may get said to lie close to {0}, while B+(-B)=[-.02, .02] has numbers a bit farther away from 0 than the numbers of [-.01, .01] and thus [-.01, .01] no longer merely lies close to {0}, but rather lies *approximately* close to {0}. But, consider C=[.01, .1]. [.01, .1]-[.01, .1]=[.01-.1, .1-.01]=[-.99, .99] which has some members which lie closer to 0 than that of [.01, .1], but also has some members which lie farther away. In this sense also, C+(-C) lies *approximately* close to {0}.

We may now proceed to demonstrate the fuzzy theorem:

We have that A+B approximately equals A+C, which I'll denote as A+B=*A+C. I'll denote more or less approximately as =**. We can then add -A to both sides to yield -A+(A+B)=*-A+(A+C). By lemma 2 we can rewrite this equation as (-A+A)+B=*(-A+A)+C. By fuzzy lemma 3 we have that (-A+A)+B=*{0}+B=B and (-A+A)+C=*{0}+C=C. So, we can more or less rewrite (-A+A)+B=*(-A+A)+C as B=*C, in other words, we have B=**C from the above.

One could also develop a similar fuzzy theorem for multiplication on intervals near {1}. Taken together these fuzzy theorems suggest an algebra which we may call an *approximate* commutative group (X, @) with the following axioms:
x_1@x_2=x_2@x_1
(x_1@x_2)@x_3=x_1@(x_2@x_3)
x_1@{0}={0}@x_1=x_1
x_1@(-x_1)=(-x_1)@(x_1)=*{0}

FF IV humor

Heal the Undead

I can't say I ever use to like the idea of magic in computer RPGs. It just seemed silly. Supernatural spells? That doesn't even make enough sense for a suspension of disbelief (not that I would have used such a term at the time). As I've started playing computer RPGs again and having recently played a good bit of the strategy game Civ III, I tend to think of magical spells , summons, or what have you in computer RPGs as a sort of way of enhancing strategy. Perhaps that's an understatement. Magic in computer RPGs makes RPGs into strategy games of a sort. Healing the undead seems to support this idea. After all, healing the living actually heals them. But, when you realize that the undead aren't living (i. e. you analyze their characteristics), healing them makes sense. It's just like knowing your enemy (opponent) in a strategy game and then pro-acting accordingly.

With wide eyes I look
Simply spotting a short book.
Ink appears on a page
And I begin to read
But as soon as I see the last word
The page turns.

I try to go back a page
To better absorb what I read
But the pages have glued together
Thus I cannot turn back a leaf.

I see yet more words
Magically embedding into the page.
Once again as I finish scanning
The page turns and adheres to the rest.

I feel as if a story must lie
Within those pages
Which disapper so quickly.

I dare not try to turn a page forward
As the present page lies blank.

Marks appear again.
The page turns at tempo.
This time I did not see
What lie on that page.

I feel this book as one
With the pages to the left already filled
Surely the ones to the right
Must lie blank.

But something in me implores that
I sketch on them
Almost as if
I doodled upon a sketchpad
As a painter might doodle
For some work yet to come.

I someone feel
This book as one.
But I could not tell you how
It all fits together.

The books seems to grow ever larger in size
At least on the bottom left binding
Why can't I see the right?

Even though I still see writing appear
The book seems short,
With more blank pages
Soon to get filled.

It's probably either pretty clear or rather obscure about what I've talked about above. Or something in between. Anyone want to guess or comment?

I must feel like I did when I was 11 or 12. I picked up the Nintendo DSi the other day and Civilization Revolution and the Final Fantasy IV remake. I did beat the original back in the day, but I never owned it... I just rented it and played like crazy when I did. Now I play at a more relaxed pace. The story feels fresh, since I haven't played the game since sometime in the 90s, and the last time I remember seeing anyone play part of the game was something like 2002 when I lived in a college dorm. The music still sounds great!

However, I did feel disappointed that they made it so the main title screen would cut to the intro. cut-scene before the arpeggio runs segue into the "main theme". A few hours later I guessed out I could just set up like I would load it in a moment, and wait until the "main theme" came on, and sure enough I could hear the "main theme". Beautiful. I believe here's how the same prelude originally sounded on the SNES.
The piece bears the name 'Prelude'. Some of the other versions of Prelude that appear in later FF games (I've only played I, IV, VI, and VII, but I've listened to other versions via YouTube) don't quite match up to the Prelude in FF IV. The FF IV Prelude feels like it has more energy and I would call it edgy.

I've felt amazed several times by the graphics in the remake. I can still remember the original game's graphics, even though I haven't seen it in years. And the DSi is what like 1/4 the size of an SNES and it has its own monitor? Moore's Law and all I guess.

Having just played one game of CivRev it seems cool and all, but for a Civ game I can't help but feeling "WHERE ARE THE WORKERS?"

I did play a game of M.U.L.E. (via an emulator on my computer) the other night vs. 3 computers on Tournament level as a Bonzoid. I was second in the 11th turn and pulled ahead to 1st in the last round and the colony did survive. I laughed seeing my offworld investments in "artifical dumbness" pay off. I forgot about in-game jokes like that. I did make a sort of grid to keep track of the quality (no/low/medium/high) of crystite deposits, as I remember doing when I played the game back in college. But, I didn't do it like how I remember my brother doing it well before that. I used a piece of paper. I remember my brother actually using a felt-tipped pen and writing on the television screen! He's an actuary now... go figure.

Oh... and since I've talked about games so much now, I may as well say that I play Tetris all day at work. O. K., not exactly. I set up a truck with boxes on pallets and then load donations into boxes. I have to figure out what can fit where, how to rearrange the truck when I get it, so I can put the boxes down on pallets, and then I figure out how donations can fit into the boxes I have set up. In other words, it's all about spatial relationships, or Tetris. Which I could play once I have the truck set up if I don't have any donors at that time.

How much is Smithore really worth?

I have looked at some of the videos of old games on YouTube today. Fun games that never made any sense like BoulderDash. Hard games like Karateka where I actually remember kicking the princess one time after beating the final boss. Games that I watch to this day and still feel nerve-wracked, like Bruce Lee.

One of my favorite games I remember one of my brothers playing a lot. It's actually a sort of hidden educational game in very basic supply-demand economics. You land on a planet called iratA, develop land to produce food, energy, and other resources, and try to make more money than 3 other players (human or computer or both). I think the game almost deserves a remake, or at least some sort of way people could play against each other on the net (unless there exists one already, probably some emulator does it, I would guess). Edit, well here it is! Oh, I'm talking about M.U.L.E. For those that don't know, here's an explanation of the game.

So what do I do now?

After months of playing Hall of Fame type games over at CivFanatics http://www.civfanatics.com/ I've started to feel burned out on civ III. I still have some interest in Sid level (the hardest) and I keep dreaming up ideas of games to play. I could do a Huge histographic run or something, but that seems like a lot of work for just more points. I've thought about going back and playing old video games instead. I started watching Farscape and Stargate on library DVDs a few months, but haven't done much of that lately, so maybe I'll do more of that.