Differential Equation Lecture n+2

Let’s review some stuff.

Wronskian is determinant of and their up to -th derivatives. If the set of solutions is linearly dependent, then Wronskian will vanish everywhere. Therefore, by testing that Wronskian of a set of solutions is nonzero at least at one point, a set of solution is shown to be linearly independent.

When testing for Wronskian, one often takes a trivial-looking value.

Abel’s identity can be used to compute Wronskian without taking a set of test solutions.

What have we learned so far?

- First-order : separation / integrating factor

- h2oODE : damped oscillation and stuff; “beat” behavior.

- nh2oODE : wronskians, variation of parameters, power series method.

General solution for where are analytic and are linearly independent, then

Then breakdown and stuff and stuff and… I forgot.

 

2

Quotes

Learn everything of something, and something of everything.

 

No idea is bad idea. The only bad idea is no idea.

 

Leadership is overrated. It was the first few followers that transformed a lone nut into a leader. (Derek Sivers)

 

If there’s a pensive fear, a wasted year, a man must learn to cope. (Dream Theater)

 

Language is the heart’s lament to circumvent the loneliness inherent in search for permanence. (Protest the Hero)

 

It’s a joke. It’s all a joke. (William Shakespear / Alan Moore)

 

I’m not interested in money or fame, I don’t want to be on display like an animal in a zoo. I’m not a hero of mathematics. (Grigori Perelman)

 

I’d hate to die twice. It’s so boring. (Richard Feynman)

 

Gather ye Rosebuds while ye may.

 

We can learn from the past, but those days are gone. We can hope for the future, but there may not be one. (Mike Portnoy)

 

Now we are all sons of bitches. (Charles Bainbridge)

 

She picked him up and hugged him: “No, baby,” she said. He was shaking. She followed his gaze toward treats sitting on pillows behind the glass: the chocolate bar and the magnetic monopole, the It-From-Bit and the Ethical Calculus; and so many other things, deeper inside. ”Maybe when you’re older, baby,” she whispered, setting him back on his feet and leading him home, “Maybe when you’re older.” (Jonathan Blow)

 

 

 

HiT Lecture n

What is “design”?

- 만들기 아름다움 명품 돈 가치 감각 편의

- 혁신과 innovation을 끌어내기 위해 기술력 등을 재배열하는 것.

Designing/mechanizing human emotion.

- Systematically induce certain emotions.

The basis of human enhancement technology.

- 인간이 “평범한 인간”으로 정의된 상태를 바꿀수 있느냐? 어떻게 바꿀 수 있느냐?

- Why not augment&enhance your body when there’s no problem with it? (in future lol)

Human Enhancement Technology

- cybernetic body

- artificial red blood cells

- AI, computer-aided human decision

- mobile computing, smart clothes

- augmented reality, smart drugs, neural implants

- genetics, immunology, synthetic biology and bio-engineering

“Stronger, Longer, Smarter, Faster”

Super-intelligent mouse, Planet of the Apes. Intelligence enhancement.

Minority Report: What if brain function can be mapped so thoroughly that one’s neural firing etc. can be predicted and hence predict crimes?

Super-soldiers / Soldier 2.0; enhanced battling ability

 

시험관 아기들. Genetic Super Baby

Differential Equations Lecture n+1

Ahem. a small remark about my university results… it does feel like shit to have all my months of efforts wasted like hell, but I guess that’s ok… considering that this CITE option is actually much better than what I felt it to be like in its early days.

 

Airy Differential Equation

Assume analyticity of and then substitute in the power series to obtain a recurrence relation with gap 3: with . This yields a power series with two degrees of freedom: . It’s a linear combination of two fixed power series which are

 

Meanwhile let’s try solving for polynomial … lolwhat.

 

 

Continuous optical density plot

Given a block of glass with varying optical densities given by , suppose a light ray passes through it. What would be the equation of the light ray, which would curve inside the medium?

 

for some constant because of the “” law. Rearranging, we have . By definition of optical density, and therefore . The previous expression then becomes  . I have no idea how to proceed here onwards. My friend sez it’s probably nonlinear PDE that can’t be solved (tbh I even forgot what “linear PDE” means) but meh whatever.

 

Initially I tried this out because of this idea: is it possible to set a material’s optical density plot such that an array of incident light (probably going in through a hole or something) will spiral inside it indefinitely such that the light, no matter how fast, will converge to a certain point?

Real Base Expansions

Credits to Herng Yi. Though it’s probably considered before.

Consider real and . Then one can recursively subtract from until it’s right before reaching the negative. Then one moves on to , and so on. This would generate a -base expansion of .

Instead of going generalized all the way like that, we can ask some smaller interesting questions like . In other words, if for positive integers , then can we easily express (computable in polynomial time or so) ?

Revelations

When I close my eyes while I’m very sleepy, weird images and videos race through my head, morphing extremely fast and uncontrollably. Just like how one knows how a scene looks like but needs to take effort to accurately describe it, those scenes are quite hard to describe, especially because they morph from one to another extremely rapidly. One I just had (I’m quite sleepy now) is about some machine shooting thick blue laser beam thing from the ceiling and rotating, with two other arms shooting smaller beams. The machine changed to have two ends that shot laser beams and rotating and stuff. If I sound retarded, that’s ok. I am.

I also try often to get inspirations from looking at random stuff and apply meanings to them. For example, I once saw a drawer with rectangular handle and a rectangular hole inside, which made me think of generalizing my current research task with an additional circle. There are a lot more; I’m going to pull off one now. I just thought about a possible generalization to Apollonius’ circle research; instead of two points rotating on two circles, I might consider those two circles rotating in turn with respect to another pair of circles. Guh, this one isn’t so good.

Mathematical Notations.

Mathematical notations should be revamped so as to maximize, or at least improve, the communication of ideas. Current system is most likely heavily limited by some arbitrary restrictions and rules such as using Greek/English characters, writing linearly, using special symbols for squareroots and division, etc. etc. etc.

This would require an interdisciplinary effort between linguistic and mathematics (and possibly physics, computer science, etc. etc.) departments. Or…. maybe some awesome guy like Donald Knuth would rise out of nowhere to change the history forever.

Experts would probably too lazy to change, or even participate, until there are some substantial contents to the change (developed by some enthusiastic group of people), and solid evidences of superiority of the new system.

Why blog

Pros: It’s an online archive that can be opened everywhere and be shared with everyone.

Cons: depends on internet connection and the server, which are not in my hands. Also doesn’t have a lot of freedom in terms of document design and stuff.

Digit Density

Decimal number system is retarded. Really. What meaning does the number ten hold apart from being number of our digits? It doesn’t have enough intrinsic reasons to be chosen as the basis to which real numbers be tabulated. Well enough with obvious ramblings.

In binary expansion of a real, compare density of 0 and 1. Which is more? More precisely, we can put it like this:

Problem. Given real , let -digit expansion of be . (here, to avoid ambiguities like 0.999… and 1.000… representing the same number, are determined recursively… I hope you see what I mean.) Then define where iff and otherwise . Now, consider limits for .

1. For what do exist?

2. How do different values of relate and compare?

I wonder if this can be easily computed for