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Balancing Act

I was going to start a sequence in physics this semester, but in the end I decided to switch to one in chemistry. There were a few motivating factors, but chief among them was the dread of having to sit through deriving Newton’s laws and equations again for the seemingly billionth time.

Also, there was the professor’s hat.

It was a hat that said “I’m certain about my status as an out-of-the-box thinker! So certain, in fact, that I must trumpet it like an over-caffienated swan!” Don’t get me wrong, everyone has a right to their own sartorial style and if I were taking a class called “Wacky Hats for the Urban Gentleman 231″, this hat would have been a normal, perhaps even a little understated choice. On a physics or math professor, though, such a thing is a robot-arm-waving, “DANGER! DANGER!” situation.

Anyway, one of the things I’ve run into in chemistry is balancing chemical equations. This is fairly simple stuff, but the text implies that the best way to solve balancing problems is to just use a hodge-podge of intuition and trial and error. The first one I did, I heard Steve Brule saying “Just use linear algebra, dummy!”.

So, here’s how to do it:

Take a reaction like the following with iron(III) hydroxide reacting with sulfuric acid to form iron(III) sulfate and water:

\text{Fe(OH)}_3+\text{H}_2\text{SO}_4 \rightarrow \text{Fe}_2\text{(SO}_4\text{)}+\text{H}_2\text{O}

From this reaction, you can form an augmented matrix with one row for each element and one column for each participating compound (see below). In each individual entry of the matrix, we put the number of atoms of that element found in that compound.

Here’s the matrix for the example reaction above:

\begin{array}{c}\text{Fe}\\\text{O}\\\text{H}\\\text{S}\end{array}\left[\begin{array}{ccc}1 & 0 & -2\\3 & 4 & -12\\3 & 2 & 0\\0 & 1 & -3\end{array}\right|\left.\begin{array}{c}0\\1\\2\\0\end{array}\right]

The entry in position (3,2) (value: 2) has the number of atoms of element 3 (H) in compound 2 (\text{H}_2\text{SO}_4).

Pick one of the compounds from the right hand side of the equation (it doesn’t matter which) and put that one in the augmented column. It’s usually easiest to just use the last compound in the reaction. When you put any other compounds from the right hand side in, make sure to change the sign of that entry to ensure that we won’t get any negative results.

Putting this matrix in reduced row echelon form yields:

\left[\begin{array}{ccc}1 & 0 & -2\\3 & 4 & -12\\3 & 2 & 0\\0 & 1 & -3\end{array}\right|\left.\begin{array}{c}0\\1\\2\\0\end{array}\right]\rightarrow\left[\begin{array}{ccc}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\\0 & 0 & 0\end{array}\right|\left.\begin{array}{c}\frac{1}{3}\\\frac{1}{2}\\\frac{1}{6}\\0\end{array}\right]

The last column of the RREF gives us a vector containing the coefficients of the chemical reaction: \left(\frac{1}{3},\frac{1}{2},\frac{1}{6},1\right)

First off, note that the last entry in the column does not mean that the last coefficient is 0. Instead since that row is all zeros, it’s a free variable and we can choose it to be 1 since that is the simplest and makes physical sense.

The second thing we see is that the rest of the coefficients are all fractions. This is clearly no good for chemical compounds since it doesn’t make physical sense to talk about one-sixth of a water molecule. However, this is easily remedied by multiplying the entire vector by the greatest denominator among them.

In this case, we multiply by 6:

6\times\left(\frac{1}{3},\frac{1}{2},\frac{1}{6},1\right)=\left(2,3,1,6\right)

From left to right, these are the coefficients of the balanced chemical reaction:

2\text{Fe(OH)}_3+3\text{H}_2\text{SO}_4 \rightarrow \text{Fe}_2\text{(SO}_4\text{)}+6\text{H}_2\text{O}

It may seem like quite a bit of work, especially if you’re unfamiliar with linear algebra, but this process always produces the correct answer and would seem to be quite useful for balancing large and/or strange reactions.

Combinatorics!

On Saturday, we visited my extended family for a semi-annual “soup day” event. Soup Day is where everyone brings a different kind of soup so you can try a variety or, more accurately, you can pig out and eat enough soup to drown in. Afterwards, we played Munchkin with a full crew and generally had a great time.

When I got home, I got a call from my sister. She and Brian had fallen to discussing the upcoming (Jan. 4)  Mega-millions lottery drawing for $290 million (now $355 million)  and were wondering how many tickets you would have to buy to guarantee a win. There was a little initial confusion about the rules, but my back-of-the-envelope estimate was at least a 100-200 million tickets.

After we hung up, I looked up the actual rules as well as the given chances of winning. The rules have you pick six numbers from 1-56 (of which you have to match 5), plus one additional number from 1-46 that also has to match to win. The chances of winning the jackpot are listed as 1 in 175,711,536 or \frac{1}{175,711,536}.

Where does this number come from?

First off, you need to find the number of possible winning combinations. That is, use combinatorics to count up all the possible ways to pick the 5 + 1 winning numbers. For the first part, you’re choosing 5 numbers from a pool of 56. There are \binom{56}{5}=\frac{56!}{5!(51!)}=3,819,816 different ways to choose those numbers. For the final number, there are obviously just 46 different ways to choose it. The number of ways the two can be chosen at the same time is then 3,819,816 \times 46 = 175,711,536 . So, each ticket has a \frac{1}{175,711,536} chance of winning. To guarantee a win you would need to buy 175,711,536 tickets at $1/ticket.

Why don’t people with $176 million laying about, just buy 176 million tickets to get the jackpot?

Because people with $176 million dollars have enough money to hire people to do their math for them.

When the jackpot was just $290 million, the one-time payout was something like $182 million dollars. This would only be a 3.5% return on investment. None too shabby provided, of course, that you were a senior citizen opening an account at your local credit union. Rich people don’t get richer  by making investments like that.

Now that the jackpot is up to $355 million, the payout is over $224 million. This is a much more respectable 27% return, but there’s a problem and it’s a doozy: both of these scenarios assume you’re the only winner. If just one other person wins you would stand to lose 3.6% of your principal. Of course, the more winners, the worse off you are. Again, rich people don’t get richer making investments like this.

My Dad

My Dad and MeMy dad has not been well since having a fall back in March.  I’ve been leery about writing about it since he was always a private person and for a long time he had no way of telling me whether it would be OK or not.

Unfortunately, though, he passed away in his sleep early on Sunday morning (9/5/2010). Although I’ve known since he fell that given the severity of his injury and his other complicating health issues this would always be a possibility, it was and remains quite a shock. He had been making so much progress and he was so hopeful that it just seems so incredibly unfair to have this be the outcome.

He was my radio buddy, my best friend, and most importantly, he was my dad. I’ll miss you. Goodbye John Eugene Tracy (8/10/1943 – 9/5/2010).

Harris RF-280

My dad, who has had a lifelong obsession with radios of all kinds, acquired three Harris RF-280 HF radios in his wheelings and dealings over the years. He’s charged me with trying to sell them so that we can buy him an HF rig that is more suited to his current situation.

I’ve been trying to dig up information about them and so far, I’m coming up fairly short. I know that these were mobile* radios  sold to the US military and that they are also known by the designation URC-94V.  I know a lot of technical details about the radios (since I have a manual and since my dad used to train people on how to repair them), but I have no idea about their value or what the best way would be to go about selling them.

(*) Mobile is a figurative term as anyone who ever picked one of these up can attest.

So, I’m putting this out there in hopes that the great Google will dump either someone with more information or someone looking for RF-280s to this post.

There are three radios in total. All are in need of some repair. None has the original cover/case.

One of them works as-is. It has a custom-fabricated replacement cover:

The RF-280 that works. Note the thickness of the manual.

That’s the manual atop it. Note its thickness.

The second radio powers on and receives, but does not transmit:

Receive, no transmit.

The third radio does not work at all and has been scavenged heavily for parts for the other two:

Good for parts!

There are also various modules and parts for the radios as well as a manual which would be indispensable to someone bent on repairing them:

A necessary evil

Finally, there’s also the RF-281 automatic antenna tuner which is paired with the RF-280 and according to my dad would allow you transmit at full power on a bent paperclip:

I might actually try hooking it up to a paperclip.

So, if anyone stumbles across this and wants to add their two cents or ask any questions, please feel free to leave a comment.

The line between genius and madness

I just saw a motorcycle with a sidecar. This in itself would be a novelty, but additionally there was a giant, person-sized stuffed animal on the motorcycle behind the rider and the sidecar was empty.

So … why wouldn’t you put your ape-shaped carnival prize in your motorcycle’s sidecar? Is it because you realized that the only thing more absurd than the sight of that would be the sight of it riding behind you, grinning and fake fur whipping in the wind? Are you a genius or are you mad?

Say hello to my little(r) friend

I meant to do this post a few weeks ago when this was fresh news, but tempest fuckit as the saying goes, so:

Some physicists have recently used some ingenious techniques to show that the diameter of the proton is roughly 4% smaller than previously thought. For most people, I’m sure, this falls into the “I’m sorry, I wasn’t listening” category, but it really is a quite shocking development for a number of reasons.

The first is just, how could we not have known? Yes, the proton is exceedingly small and hard to corral, but it’s the proton, not some exotic meson that only exists in a particle collider for 10^{-10} seconds. It’s been studied extensively, both theoretically and experimentally since before my grandmother was born. This means that mother nature has got some serious ‘splainin  to do.  There are simply going to be some completely out-of-left-field corrections to the fundamentals of theoretical particle physics. Here’s your chance to shine string theorists ;-) .

Second, although a 4% reduction of something so tiny would seem to be insignificant, it actually has some rather enormous implications, due in part to Heisenberg’s uncertainty principle*.  Just for illustrative (i.e. decidedly non-rigorous) purposes, here’s why something like this might matter so much.

The uncertainty principle in terms of position and momentum has the form:

\Delta p\Delta x\ge\frac{\hbar}{2}

Where \Delta p represents a particle’s uncertainty in momentum, \Delta x  a particle’s uncertainty in position (space), and \hbar is Planck’s constant divided by 2\pi .

So, if we imagine we’re dealing with one of these new slimmer, sexier protons, we can (sort of) replace the size, or spread in space, with the reduced size:

\Delta p (\Delta x - 0.04 \times \Delta x) \ge\frac{\hbar}{2}

\Delta p \cdot 0.96 \Delta x\ge\frac{\hbar}{2}

With a little re-jiggering, we then have:

\Delta p \ge\frac{\hbar}{2\cdot0.96 \Delta x}

Now, \hbar is a constant, so the this means that as \Delta x gets smaller and smaller, \Delta p gets larger and larger. Since we’re talking in such general terms, it really doesn’t matter what \hbar actually is, but rather the overall form of how \Delta p varies with either \Delta x or 0.96\Delta x. What better way to compare this relationship than with a graph:

Delta p as a function of Delta x

Here, the blue line represents momentum (\Delta p) as an inverse function of spatial spread (\Delta x^{-1}) and the red line represents momentum as an inverse function of ((0.96\Delta x)^{-1}). This shows the somewhat obvious result that a smaller overall \Delta x is going to have more momentum to go around.

So we come to the question: what difference does that make?

Well, the other major uncertainty relation of quantum physics has to do with energy:

\Delta E \Delta t \ge \frac{\hbar}{2}

Which (trust me) can be related to above equation like this:

\Delta E\ge v \frac{\hbar}{2\Delta x}

This looks a lot like our relation above involving momentum, so with similar reasoning, we can conclude that a reduction in the size means that there should be a lot more energy to play with during interactions involving protons.

Again: who cares?

Well, in relativistic particle physics, energy and mass are related (E=mc^2 **) so this means that the reduction in size and subsequent increase in energy will result in more mass and therefore new exotic particles that not only has no one seen, but no one has even predicted.

This ultimately translates to job security for everyone working at the Tevatron or the LHC since even after they find the Higgs boson they can set to work squeezing these new particles out of their colliders. Now we’ll never have to see an Ebay auction like this:

L@@K — Mesons Cheap!!!!

Auction ending: 5 \times 10^{-24}s Hurry!

Current Bid: $550,000 (0 bids)

Item Description: You are bidding on a lot of  \Delta^{++} and \Lambda^0 mesons …



(*) Craig Olivas, a friend of mine, says that if he wants to sound smart at a party he just adds “… vis-a-vis Heisenberg’s uncertainty principle” to the end of some random statement. This is however, not my intent here, it really is necessary.

(**) Now I’ve got E=mc^2 in here. Maybe I am just trying to sound smart. Crap.

Beginning with APRS

I’m sure there was something much more useful I could be doing, but I started playing around with APRS on my laptop and Yaesu VX-7R.

I used soundmodem for setting up the packet interface and xastir for doing the APRS munging.

I had to use my Thinkpad T61 for this since I discovered that the new Thinkpad T410s lacks discrete headphone and microphone jacks, supplying instead a single mongrel “headset” alternative. I’ll have to find a Y-cable somewhere with one of those dastardly 4-conductor 3.5mm audio connectors…

Anyway, here’s part of my setup with soundmodemconfig running:

Laptop running soundmodem and VX-7R

And here’s my setup with an added inline meow-plexer (of questionable utility):

Meow-plexer of questionable merit

After running xastir for about 20 minutes, I’ve picked up a surprising number of stations. Here they are dumped to KML and beautified by Google Maps:

An auspicious beginning with APRS

That’s me in the middle. But, before you come to rob us blind, Internet-ne’er-do-wells, please note that our house is not made of gold as this picture suggests.

-= UPDATE =-

I let the rig run for most of the day and night and so far I’ve received 293 stations. Woohoo!  The furthest is an apparently mobile rig from Lockhart, TX with a close second in Halifax, NS. You can check out the map, or just look at the pretty picture:

Some more APRS data

Buffalo “Buffalo”

Jody and I went to visit our friends Mike, Leslie and their son Alex in Buffalo over the weekend for a cookout. We had a great time with them as well as all our “Buffalo” friends (Jhen, Jane, and Martin).

As we were headed past the sign that sort-of welcomes you to Buffalo (it’s strangely worded, “Buffalo An All America City”, and strategically placed next to an enormous open pit mine), I was trying to remember a notorious sentence that consisted of nothing but repetitions of the word “Buffalo”. I knew the construction was grammatically/syntactically correct and the “trick”, if there was one, centered around the fact that the word “Buffalo” has numerous homophones and homonyms.

I decided the sentence had to be something like “Buffalo buffalo buffalo Buffalo buffalo”. The word “Buffalo” being used as the name for the animal, the name of the city, and as a verb, “to buffalo”, which I remembered as meaning “to confuse or bewilder”.

To see how the sentence works, it’s easiest to build it up from its most basic parts:

First, we’ll begin with the noun “buffalo”, which refers to an animal (actually animals), also known as the bison. Next we’ll add “Buffalo” as the name of the city, which will modify the animals, giving “The buffalo from Buffalo” or, as it’s used in the sentence, “Buffalo buffalo”. Then, adding the plural form of the verb with “Buffalo buffalo” as the subject, we have “Buffalo buffalo buffalo something”, meaning “The buffalo from Buffalo bewilder something”.  Finally, adding “Buffalo buffalo” as the object of the verb as well, we get “Buffalo buffalo buffalo Buffalo buffalo”, meaning “The buffalo from Buffalo bewilder the (other) buffalo from Buffalo”. A perfectly nonsensical bit of doggerel, but a valid bit of doggerel nonetheless.

The actual sentence is more complex than my half-remembered construction, but operates on the exact same principles. It was created by William J. Rappaport , an associate professor in computer science at (unsurprisingly) UB. His sentence is:

Buffalo buffalo Buffalo buffalo buffalo buffalo Buffalo buffalo.

Which parses as:

The buffalo from Buffalo that are bewildered by (other) buffalo from Buffalo also, in turn, bewilder buffalo from Buffalo.

I think I enjoy this sentence so much because its difficulty in parsing/interpretation does not stem from any deliberate removal of punctuation or other “trickery” to obfuscate the types or the levels of language being used.

Take another similarly notorious sentence: “James while John had had had had had had had had had had had a better effect on the teacher”. Without going into it too deeply, this confusing construction relies on the removal of both the punctuation that makes it easier to read, like commas, as well as quotation marks which indicate the shift from language to metalanguage (e.g. while John had had ‘had had’ …). This strikes me as a cheap trick and we all know what tricks are:

Also, part of the night’s drunk-scussion touched on what could be considered Buffalo’s quintessence, i.e. the most “Buffalo” of Buffalo and to what degree something or someone could be from Buffalo, but not be “Buffalo” Buffalo. This suggests a new kind of confused sentence like Rappaport’s, but unfortunately my mind is entirely buffaloed at its construction.

Burn up the old… bring in the new

Sad to say, but my old Thinkpad T61, which has served me so faithfully over the years, which has been there through fat and lean, which has thanklessly compiled line of code after line of code and rendered LaTex like a champ, has finally up and… kept working just as I would have hoped. Damn!

It is, however, my main computer for school and no longer under warranty. Although it has served me well, it’s unlikely that it will last another 2-3 years without major incident. Plus, I cannot rightly walk on to campus at RIT with my aging behemoth of a laptop and expect to be taken seriously. “What, your laptop has spinning disks with little magnets in it? How quaint. Does it also do TCP/IP over carrier pigeon?”

So I’m getting a new one: a Thinkpad T410s. It’s basically somewhat of an update to my existing T61 with nothing too flashy. I guess I’m most excited about the SSD and again having a graphics card that can play at least a few games (TF2, woohoo!).

As I’m sure everyone will be wearing out their F5 keys to find out more about this topic I’ll do my best to keep it up-to-date.

Stockeu-mageddon

So, on the descent of the Col de Stockeu on only the 2nd stage of the TdF, roughly 1.2 x 10^6 riders went down, including my faves from the firm of Schleck, Schleck, and Cancellara. Riders placed the blame on “some kind of oil slick or something.”

Several BP officials immediately materialized to deny that it was an oil slick and even if it was it wasn’t dangerous and anyway they have like 20 doctors who swear that crude oil is the recommended treatment for road rash.

Even Eddie Merckx went down and he hasn’t been in the TdF for 30+ years. Merckx had this to say about the crash: “That was fuckin’ weird, yo”.

Anyway, I’m totally with the peloton on their decision to hold back on the run into Spa to show both their solidarity with the riders who fell as well as against what were obviously dangerous conditions.

To the blood and guts, cutthroat types who were disappointed with the conclusion, I have this to say: “Shut up. No one likes you. That’s right, not even Meemaw.”

What I mean to say is that the truth about the TdF is that it is a strange mix of both competition an cooperation. No one rider or team could finish the race without some degree of support from the peloton.

This means that occasionally you have to act in the best interest of the entire group, of which the run into Spa was a shining example.

Good for them and especially good for Cancellara, who showed that he deserved to be in the maillot jaune a lot longer than he got to be.,