By analogy with complex analysis, we can define a split-complex valued function to be pseudo-analytic at iff it locally depends only on , not . That is,
By the chain rule,
“” is the wave operator, or “d’Alembertian”. One of or “sort of” corresponds to time. There are many different conventions. In the usual complex analysis where , we get
All split-complex valued functions have unique real valued real and imaginary parts, and , respectively. Why unique? The short answer is “because I say so”. The long answer is that there are many possible representations in which the real and imaginary parts are unique, so “I’m allowed to say so”. One beautiful thing about complex/split-complex analysis is that the “numbers” can be interpreted in great generality. For example, in my opinion the simplest computational representation is given by
Aside: If you move your cursor over the equations, it will show you the code I use to create them. This is the best WordPress-LaTeX guide I’ve seen.
The above algebra is called split-quaternion. Four vector dimensions are often, but not always, necessary, if you try to add a third dimension, because an algebra requires multiplication. The split-quaternions give you , and anti-commutation together. For complex , & , it gives
where det is the matrix determinant, in that representation. You can also construct a 4D commuting system, in which case you must have the third . These are only representations, though, and there are infinitely many.
If pseudo-analytic, the parts of must satisfy the pseudo-Cauchy-Riemann equations:
where and are the real and imaginary parts of .
It is clear that all pseudo-analytic functions and all their parts must satisfy the wave equation in . Hence d’Alembert gives the solution over the entire split-complex plane, given is real on the whole real axis,
You might argue the concept of “pseudo-analytic” is therefore trivial, but actually it might be useful. A complex-analytic function at a point must converge to its Taylor series on a neighbourhood, and this formula I have derived tells you its value if instead you put into the Taylor series, assuming you know its value horizontal to the point.
Given it’s been more than a month since my last post, it seems keeping my “results” secret isn’t very productive. I worked this stuff out a few months ago. But recently SWIM ran out of energy. And then it turns out the bag of coke he had as back-up isn’t much good, because the drug is awful. It has much worse withdrawal, damages his nose much more, and causes more psychosis than meth. The increase in mathematical ability is negligible, compared to amphetamine. As Wiki clearly states, the drug is more dangerous than any amphetamine, and can cause sudden heart attack. I think the primary reason people buy cocaine is because they think it’s somehow safer than meth, because it’s natural. That’s not true. It probably would be useful in dentistry as a local anaesthetic, and was used that way back in the 19th century.
Another important point: MDMA = MDMeth (aka ecstasy) is not safer than meth. In my opinion, meth is safer. But an Australian drug dealer would never admit that to their customers because it would scare them away because of the meth stigma. MDMA was invented before WWI in Germany, while meth was not invented until after WWI, in the Japanese Empire. The Japs also invented our antennas, around about that time. The German inventor of MDMA died as a soldier in the War. Meth is the newer drug which hasn’t been around for as long. And it’s only semi-synthetic, unless it’s racemic in which case it looks ugly. The crystals are pretty while MDMA is ugly.
You know you want one. SWIM snorted 25mg before the final for this course. By the time he was writing the solution to the third question, he was doing the three problems in his head at once, because it was fun. It was the first and last maths exam of his life that he actually enjoyed, and the only one on which he was high. He still thinks measure theory is a joke. Unfortunately the crystals oxidise your neurons.
Don’t Do It. I have never, nor will ever, swallow a pill made by someone who refuses to show me how it was made. If you ever swallow a black-market pill, you need to see the ingredients that go into it with your own eyes! I strongly suspect there are many Australians who are taking an ecstasy pill every day, possibly as anti-depressants.
Aside: Please don’t use the new Silk Road. For all the good that people on the old one may have done, the guy running it was a psychopath who tried to murder a user. I believe most of the FBI indictment is true. If there is someone on there you want to do business with, you can ask them to move to another website. Otherwise you are supporting incompetent libertarians who think it’s okay to use your money to murder people. The new guy supports the old guy. Your name and address was being stored in a database by Ulbricht, and one of his employees was trying to blackmail him with user data, before he could blackmail you. It is impossible to simultaneously use encryption and a “shopping cart”. You can’t use two PGP keys from two drug dealers at the same time! In hindsight, the website was clearly designed to fool people into thinking drugs were safe by including a shopping cart, and designed to harvest your name and address. All your details and money are now seized by the FBI. Ulbricht has not been charged yet with soliciting murder, but he will be. At a conscious personal level, he is probably a sweet charismatic guy! Don’t trust what people say, especially stupid people. I should never have mentioned the website here. I’m sorry if any harm came about as a result of that – I should have known better. I suspect Ulbricht’s tolerance of scamming resulted in Australians receiving the “worst” batches of drugs like MDMA, as a form of retaliation against a few Australian drug dealers who kept scamming foreign dealers. There is a great deal of animosity towards Australians, because of some domestic dealers. The website needlessly encouraged scamming, in my opinion. No revenue was ever donated to anything. In future, you might want to pretend you’re British or American and handle mail redirection/smuggling yourself. Our customs are good, but I know an envelope with crystals sloshing around will get straight through. The envelope is a baggie. I would guess at least 90% of apparent failed deliveries were scams.
An isosceles trapezium of pain.