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My dear friends,

thank you ever so much for bearing with me. As I suspect is so often the case, it is the mathematician, and not the math, that has gone wrong. Thanks to all your feedback and encouragement, I think I can now call this case closed. For your visual satisfaction, I present below the final graphical solution to this little endeavour:



I don't know exactly where I went wrong with that last graph, but the solution as it stands now should graph as depicted above, and it looks like a set of solutions quite in agreement with what one would expect from the problem as I'd set it out.

Anyway, I really do appreciate not only the conversations we've had, but, more profoundly, that I have friends with whom I can have such conversations. Many thanks again for all your help.

Most fondly,
Paul

One Problem For Another...

All right. At last I know where I went wrong: a little sign change poked its ugly head in there at the last minute, and where it should have read, A=2x^3+x, it read A=2x^3-x. So that helps somewhat. However, that graph looks like this:




As you can see, I am now stuck with the opposite problem; You could give me an x, say, -.25, and my new better function will give you a positive number! I'll let the $20 stand...

A Mathematical Excursion (or, Where I Went Wrong)

o.k. I've taken to "amusing" myself lately by attempting to solve little geometrical problems I've set for myself. On the whole this has been successful, as measured both by the satisfactory completion of said problems (i.e. solving them), and by the "amusement" level associated with the process (e.g., not thinking about my foot, debts, work...).

HOWEVER, I've run into a stumbling block. I've come up with a solution to a problem which just doesn't seem like it could possibly be right. $20 to the first person who can show me where I went wrong. I'll lay out the problem, and then some basic things we'll need to agree on:

I thought it would be fun to see if, given some x, I could find some other number A, such that the line going between (A, 0) and (x, x^2) would be perpendicular to the function f(x)=x^2. A thousand words:

My thinking went something like this:

Axioms:
1. The slope of f(x) at x is equal to the derivative of f(x) evaluated at x.
2. if f(x)=x^n, f'(x)=nx^(n-1) [the "Power Rule"]
3. The slope of a line between two points is the quotient of the difference between y values and x values: Dy/Dx= (y2-y1)/(x2-x1)
4. If two lines are perpendicular, then their slopes are negative reciprocals of each other.

All of this should be, if not obvious, at least readily verifiable. Still with me? Then here's one argument, with justification for each step in brackets[]:

I. f(x)= x^2, therefore f'(x)=2x [stipulated, power rule]
II. The slope of our little green line, then, should be the negative reciprocal of the slope at x, or -1/2x [stipulated, 4]
III. The slope of our little green line should also equal (0-x^2)/(A-x) [stipulated, 3]
IV. Therefore, -1/2x= (0-x^2)/(A-x) [II., III.]

It's not terribly tricky to solve for A, but, for the sake of being explicit, here's how I did it:

1. -(A-x)/2x = (0-x^2) [mult. (A-x)]

2. -(A-x) = 2x(0-x^2) [mult. 2x]

3. -(A-x) = -2x^3 [dist.]

4. (A-x) = 2x^3 [mult. -1}

5. A = (2x^3) - x [sub. x]

Cool. So now we have a function that, when you put in some x, gives you back an A! Can you imagine the plot of this function? I did, and that was when I noticed that something fishy was going on. A thousand more words:


Problem: If (A, 0) is supposed to be the point where the line perpendicular to f(x) intersects the x axis, why would it ever be negative in the first quadrant? I imagine those lines getting closer and closer to vertical as one approaches the origin (f'(0)=0, and the negative reciprocal of this, while undefined, approaches -infinity, something like -1/0), but they would have to have a positive slope to get from anywhere on f(x) in quadrant one to the negative x axis.

So, there you have it. I can't see how to refute the math, but I can't see how it could be right either. For now there is just a hole where reason should be.

Better luck to you,
Paulman

howdyhowdyhowdy

What the heck?! It's been, like, well, way too long since I last posted, and all kinds of things have been going on. I wrecked my heel and have been hobbling around since last Wednesday, but the doctor's say I haven't broken anything, so it's just going to be painful and suck for a few weeks; I miss Kenpo :( I'm also bummed because this means that I can't practise riding the 5' giraffe unicycle I recently acquired. I purchased it for quite a reasonable sum from Joe Lind of Compulsion Cycles. It teases me from my living room. Anyway, hopefully I'll be back to my more physical self soon; in the meantime, I've been laying down the beats for OCEANOVER's upcoming full length album. That's been moving along swimmingly, though there's really never quite enough time. In part because I'm soon to begin my lab work at the U of M. It's all been squared away with the powers that be, and it looks like I'll start Wednesday next. 'Til then I've been trying to learn as much about MRI and DTI (diffusion tensor imaging) specifically as I can. There's a lot to learn though, and it would be easy to spend years on any particular facet. More on this as it comes...
In other news I finished the latest (and, need I say, final) installment of the Harry Potter series. Can I just say, DAMN! I was wrecked for days after I finished it. I haven't yet been to a support group or anything, but that is some series J.K. put together.
I've also been working on some LED lamps.. something's not quite right...

I am STARVING!! What am I doing writing? Dinner, then maybe more. Anyway, I'm still out here, sort of. LOVE!!!!

-p

Concerning Reason

As is often the case, I've been meaning to write for some time, but for one reason or another, I've failed to make space for it. In any event, I have recently noticed a confluence of themes sprouting up into conversations I've had with different people, and, as it's nearly spring (nevermind the flurries- whoever invented Minnesota should be shot), I thought I'd re-plant them here.

I had originally intended to put to paper, as it were, a stimulating conversation I was fortunate enough to take part in with Christopher Beckett over some delicious coffee and a game of Othello. I believe it was back in February, when we had one of those welcome respites from the cold which last only a day or so, but which help remind us of just what warmth is. Chris and I had parked it outside on a bench, and he mentioned that he knew someone who was something of Go freak. A Go addict? Anyway, it seems that those who become interested in the game rarely take only a passing interest in it, and ultimately learn the names of particular strategies and famous players, much like serious Chess players. Othello, as it happens, is something like Go, though I don't imagine there being any professional Othello players. As Chris and I sat playing and talking, the discussion turned towards the kind of skill employed by game players, and what its relationship was to other kinds of skills and knowledge, or even life in general. While we certainly had to concede that many players use a kind of analytic approach similar to that used in science, there also seemed to be an artfulness employed, a kind of schizophrenic winging it that was often just as useful. Anyone who's played these games will surely agree that such improvisation only bears fruit with any regularity once one has gained some level of accomplishment, but I find it a little curious nevertheless that so un-thinking an approach should bear fruit with any frequency greater than chance at all. The question also arose as to what sorts of knowledge could be gained through this kind of strategy, and that reminded me of a book I believe I have mentioned here before: Jonathan Strange and Mr. Norrell, by Susanna Clarke. It is the story of two nineteenth century magicians, among other things, and is quite an excellent read in many respects. Of particular interest to me is the way in which one of the characters goes about learning magic: He discovers that faeries have a particular affinity for the insane, and he manages to obtain an elixir that renders him temporarily insane for the purposes of communicating with them, and seeing as they do. Again, what is the relationship between knowledge and irrational thinking? Perhaps it is not quite "thinking" properly, but it seems to be some kind of mental activity, akin perhaps to what Pierce called "abduction", or the hypothesizing of Sherlock Holmes, musement. The topic came up again at a concert at First Avenue, where some friends of mine and I met to hear Subtle ("Mind blowing" is the best way to describe them, so by all means check them out, but I digress). Susan, a painter among other things, and I were discussing art. It occurred to us that it is, on the one hand, perfectly natural that art should sometimes convey an emotion, idea, or meaning, more efficaciously than words which might more literally represent those same things, but also that, on the other hand, what could be stranger? That is, why should the words, "I miss you" not convey that emotion at least as well as some melancholy piece of music, for example? Perhaps it is just that words are abstract enough that they are sometimes more ambiguous than the particular thing communicated by a piece of art, but it strikes me as odd. Perhaps, too, this is actually all a part of what reason is. That is, perhaps I have construed what it is to do the mental activity we call thinking too narrowly. But there is surely some limit to what we will call rational thought? How else can we account for reason going wrong, as in the case of religious metaphysics, or racism? There are clearly some instances when "reason" is not reason at all, and leaves us in a state of un-knowing. Maybe that is the price we pay for being able to synthesize new ideas. It is a method more prone towards error, but also one perhaps more capable of radically clarifying our perception of the world and ourselves.

Curious.

-P

Happy Birthdays!

Well, it is a bit past my birthday, but it was a good one, and seems to be going strong still. Chad, Charles, and I got together last night to celebrate each other's existence, and I think we did a fine job. Before I get to that however, I should like to send out a most sincere note of thanks to me mum for, well, being me mum, but also for the generous Amazon.com gift card she gave me. It, in turn, helped me to acquire a couple of new books and CDs. Somehow this blog has turned into my ramblings about neuroscience and reviewing movies, books and CDs, so, if you're reading this, this should come as no surprise.

Anyway, while I've not tucked into it yet, the first of my purchases was of a recently published biography of Donald Coxeter, and American mathematician and geometer, written by Siobhan Roberts. I've always had a thing for geometry, despite how abstracted, how un-empirical it is, if you will. Quite looking forward to this read. Next I scooped up a book by Gyorgy Buzsaki titled Rhythms of the Brain, in which may be found the culmination of his work on neural oscillators, and populations of neurons firing synchronously. Naturally, any connection between neuroscience and rhythm was bound to catch my attention. I'm not too far in yet, but it seems to be written well enough, and there are heaps of notes and citations from which to hurtle myself should the need arise. I'll keep you posted, but so far so good.

It just wouldn't be a birthday without some new tune-age, so I treated myself to a disc I've long been hoping to add to my collection: Money Jungle, by Duke Ellington, Charles Mingus, and Max Roach. If you dig jazz at all, you'll know that these are some fantastic musicians; not just excellent on their respective instruments, but adept composers as well. Upon my fourth listen it continues to exceed my expectations, and I suspect it will be one of those albums I'll return to with some regularity. Top-notch.

As if all this weren't enough, Caralyn made me PIE and got me a fabulous Lego tee shirt (no, not one made out of Legos, rather, featuring a nerdy little Lego rocker-dude with a half-stack). Thanks! Mad thanks go out to Elliot for Blade Runner, Charles for the X-Men and the sweet tunes, to Chad for dooming me to innumerable days of electronic joy via a Gamecube, and to Chris for the delicious teas (I have been savouring each). It was a motherlode year, and while that's pretty sweet, it's been greater still to have spent these past few weeks with my friends. The band went out for dinner before our last show, and it was just cool to be together; Mom and Jim came to town that same weekend, and Chad, Elise, Eleanor, and Caralyn and I all got to spend some time over dinner together as well; I certainly value my time to myself, but friends are good, and I'm really glad I got a chance just to be with you all.

With greatest affection and thanks,

Paul

In Paul's Brain

Well, it's been a busy week. My meeting with Dan Franc at the Center for Magnetic Resonance Research at the U. was great fun. I got a little tour of the place, and then I got put into the machine. This was a 3 Tesla MRI I believe- about twice as strong as what you'd find in a hospital. We did a full scan of around 20 minutes, and then went over how one goes about analyzing the retrieved data. It's really quite amazing what it's possible to know about the structure and chemical composition of the brain without actually opening up the skull. The software we were using, for example, can discriminate between a number of different organic compounds, so one can determine their relative amounts. And it can be done with a fair degree of localization. This is, perhaps, not news to those familiar with MRI, but it's clever and amazing nevertheless. Here are some samples:

One through the eyes














One through the ears- I think that vertical white streak is one of my carotids














And, of course, my nose. Who would ever get an MRI of their nose?







I should get back to work I suppose. Just a quick book/author recommendation first though. Eric, a fellow Kenpo dude, suggested Neal Stephenson's Quicksilver to me last week, and though I haven't snooped into that just yet, I did pick up a copy of another of Stephenson's books titled Cryptonomicon, and it's pretty decent so far. It deals, not too surprisingly, with cryptography, both during the second world war and the present. No mention of Shannon yet (if you don't know who Claude Shannon is, a quick Google search will reveal just how central his work is to the field), but hopefully he'll make an appearance. Thanks, Eric!

Also, SEE SWEETLAND. Caralyn and I went to see it at the Grandview last week, and it is just amazing. It was produced, in part, by Alan Cummings, who also plays a role in it, and it gets all those words great works deserve: charming, haunting, poignant, sad, funny, joyful...a must see.

o.k. I go.

Take it easy,

Paul