Lx=d: SundayResearch

Japanese version It has been while. We thought about a machine which generates the next number of the input last time. Here, we have seen a procedure of ''increment one.'' We coined this procedure as ''increment one,'' like there is something meaning. Or for human, this procedure has a meaning, ''increment one.'' But, the machine SUCC 1 just moves some symbols around. I think SUCC 1 did not understand the numbers. Here is too much to think about what is ''meaning'' or what is ''understanding.'' There is a hypothesis called ''Society of Mind'' by Marv in Minsky (this is a nice book and I recommend this.) which said a complex combination of simple functionalities creates intelligence, or you can not distinguish such complex thing from an intelligent thing. But SUCC 1 is such a simple machine and I believe I could safely say it has no intelligence. (By the way, are there any relationship betwe...

Japanese version Several weeks ago, one of my friends visited me. We told about lambda calculus and the meaning of computation. Lambda calculus tried to define the computation only in a formal way without thinking its meaning. On the other hand, it is usually important what I am computing in every day life. Interestingly, I started to think about the meaning of computation more deeply when I think about how-to-not-to-think-about-the-meaning. I think something I start to understand when it was missing. There is a film called `` Paper moon ''. In the film, there are several cheating methods which are good examples of using misunderstanding of meaning of the computation. If I remember correctly, a classic Japanese Rakugo, ``Tubozan'' has the same tricks. There is also a good example of how a physicist connects the meaning and computation in a book, ``Surely You're Joking, Mr. Feynman!.'' In this article, ''three guests in a restaurant''is a fa...

Japanese version Today, I read a paper, ``A signal processing approach to fair surface design'' (G. Taubin), but I did not understand it. It said that the eigenvalues of a Laplacian matrix like Equation 1 is Equation 2. Eq 1 Eq 2 Today, my friend CR taught me why this is. Actually, it is not so difficult. Here, we can back to the basics of eigenvalues, a characteristics function: Also we know a Laplacian matrix is a second order differentiation. Therefore, it is One of the solution of this differential equation is exponential function (since the second derivative is back to the original one except a constant factor). If you remember the Euler's equation, you can get Equation 2. Additionally, Equation 1 is symmetric. Therefore, its eigenvalues are real and eigenvectors are orthogonal. Moreover, this is a diffusion equation and you can see the relationship with Fourier basis. Now, I can see why the paper's title is ``signal processing approach.'' But, one who is ...

Japanese version If I remember correctly, when I was a high school student, there was a exhibition near a station. The theme was Children's dream and it was provided by a Kindergarten. Each child draws a picture with a title "I wish it was ..." Some already drew a picture to please their parents. Some others were totally free from anything. One of my friend G is a designer and once told me that "A picture by a child is very strong and I sometimes feel I can never overcome their drawing." He also said, "They never consider an object must have some three dimensional constraint. They draw as they feel when they saw a thing, which is totally irrelevant what they saw. Their drawings have overwhelming power." When a child feel "The mouth is big!", then she/he draws a mouth that is larger than the face without any doubt. He even feels some kind of fear in their picture with incredible freedom. I remember one of them, "Flexible sized television....

Yesterday I lost a friend. I don't know why. I feel something wrong in this world, but I wish he never left. I liked him. The world became a bit less attractive. All my wish was just him alive. Disappointed. Sad. I miss him. May rest in peace. Sorry for our loss.

Japanese version Last week, I visited to my old friends in Edinburgh and Durham. Walking in the old city with my friends, that was fun holidays. The history of the train system started in the United Kingdom. Therefore, I thought the train system is also advanced, however, it seems not so. For example, you can see the hand-written cargo numbers. Our train was missing one coach J, and the announce was "Today, we are missing one coach J. If you have a reservation in the coach J, you can find your seat in coach B. If you have a reservation in the coach B and can not find your seat, you can find it in coach F. Sorry for the inconvenience." We thought, wow, in England, the system is so difficult. How the people can manager that? But soon we found out many people were walking around and asked "Do you know where the coach x's seat?" We told a woman about the announce and she said, "What a beautiful organization!" I see why the English comedy is so good.

Japanese version Blaise Pascal made a calculator for helping his father's job (tax calculation). Usually it is painful to calculate a huge amount of accounting. I am not good at calculation. So, I thought if I have a computer, I do not need to compute anything myself. That's one of the motivation I took a computer science course in my University. Some people totally misunderstand that a computer scientist is good at arithmetic. No. If someone is good at arithmetic, why does she/he need to learn that? If a human can fly faster than sound, maybe we do not need to use a plane. If people can communicate without speaking between thousand kilometers away, why we need a telephone? I can not do arithmetic, therefore I learned computer science. So, let's make a computer. Figure 4 is a computer SUCC mark 1 by Sirius Cybernetics corp. This computer gets one Church number as an input, and outputs another Church number. This computer does not understand what the number is, but just exec...

Japanese version Last time, a circle was a symbol to represent a number. But, there is no such thing (a symbol to represent a number) in the Peano's axiom. Peano's axiom only define a Zero and a successor. we employed a square to represent Zero. But when we tell two numbers to a machine, we can not distinguish two numbers if we have only Zeros. See Figure 2. Therefore, we use a circle as a delimiter. One could say, we can use a space, but we also need to tell a space to a machine, otherwise any machine can not know a space exist. We need something like a number 0. 0 means ``there is nothing.'' If we write down nothing, how we could know something is missing. If we put 0, then we know nothing actively exists. 0 can represent ``existence of nothing.'' This is an excellent invention of human being. By the way, speaking about space, there is no space character in Japanese. I think also Korean and Chinese do not have space character. Therefore, a processing of Asia...

Last Saturday, I walked around the city with my friend. DDR buildings are a bit chilly... Or it is already winter here.

Japanese version Last time we were talking about how Peano defined the natural number. Because Lambda calculus defines the numbers based on that. Mathematical formulation makes the discussion (proof) more exact, this ``exact'' is important for mathematician. But, the formulation causes increasing the exactness, which means, there are no such thing, like ``You know about the numbers, just do something like calculation in appropriate way.'' Because even every single obvious issue should be defined in formulation. As the side effect, we could execute these rules on a machine --- we can make a computer! That's the interesting point for me. There are many ways to how to implement a computer as a machine. Pascaline and Charles Babbage's differential engine are gear based. Here our base is Peano's axiom. Before Marvin points out, this explanation is Masahiko Sato and Takafumi Sakurai's ``Basic theory of programming.'' (By the way, It is not so conven...

Japanese version This is continued from the last article. Please refer the Peano's axiom in the last article. The second definition means that we can create a next natural number from the current one. This function creates a successor number from the current number, therefore it called ``successor'' function. This defines ``plus one'' function. We have already the first natural number, zero, then we can make a successor number from zero. This successor number is ``something'' of zero. It is usually called one, but not necessary. This definition just said, it is something different from zero. We have now: 1. there is the first number, 2. we can make a successor number from a number. Out of these two definitions/rules, it seems we can make the whole natural numbers, but this is not enough for that. The third definition said that there is no loop of successor function, i.e., if you repeat the successor function staring with x, the result of them will never retu...

Japanese version A few weeks ago, I visited Dresden and saw a bridge called Blauses Wunder. The formal name of the bridge is Loschwitzer Bruecke, but even this name is on a map. The literal meaning of this word is ``Blue Wonder,'' but in German, there is another meaning -- very surprising in a bad way. This is bit strange... The originally this bridge was blue. Dresden is a beautiful city. I have friends there, so I visited there several times. I first know the word, Blauses Wunder in the book from Preussler, ``Der Raueber Hotzenploz.'' (Am Ende der Spur sollte jeder von beiden sein blaues Wunder erleben, dafuer hatte Hotzenploz vorgesorgt. in Chapter 6) Größere Kartenansicht

Japanese version Peano defined natural numbers. He actually described properties of natural number, not seems to try to define the natural numbers. But these are somehow the same. The following five definitions are called Peano's axiom which defines the natural numbers. If you are not familiar with mathematical notation, it might be hard to get what they said. But, the basics are not so difficult. These are copied from Mathworld. Zero is a number. If a is a number, the successor of a is a number. zero is not the successor of a number. Two numbers of which the successors are equal are themselves equal. (induction axiom.) If a set S of numbers contains zero and also the successor of every number in S, then every number is in S. The first definition said, there is the first number called Zero. Here it said Zero, but it does not matter which number is. It should be a ``something.'' However, you may ask ``What is something?'' It is really just ``something.'...

One of my friends is an artist. Recently, he made an article, ``Stereo Shadow.'' He used two light sources, one is blue, the other is red. Audiences see their shadow with wearing a red-blue filter glasses. Then they can observe their own ``3D'' shadow in the wall. There were several exhibitions of this in Japan. I heard he has prepared a video of this, and I am looking forward to seeing that. The black dots in the picture is for audience's safety. He found some children try to touch their shadow and hit the wall without noticing. So the dot shows ``Here, there is a wall'' sign.

Japanese version According to the Wikipedia's page , Lambda calculus was introduced by Church and Kleene in the 1930s as part of an investigation into the foundations of mathematics. By the way, I am an amateur Sunday mathematician, therefore, please do not believe this blog without check by yourself. This is just I think I understand these stuffs. So I think there must be many errors. I try to avoid errors, but, this is not my profession. I am also not confident about my English. Welcome the comments. The origin of this blog is Wikipedia . I could say in cool way, I was inspired by the Wikipedia's page. If you understand the Wikipedia's entry, I don't recommend to waste of time by reading this blog. There are bunch of interesting text around the world. When you read ``Lambda calculus was introduced by Church and Kleene in the 1930s as part of an investigation into the foundations of mathematics.'', and if you think ``I see, that's the reason of why lambda ...

Japanese version Standard mathematics books explain mathematical stuffs as definition, theorem, proof, definition, theorem, proof, .... This is quite simple and enough abstracted. Therefore, we can also explain lambda calculus in the standard way. But Marvin will sure complain that is so depressed. More abstracted theory could be more applicable to many things. It becomes less unnecessary stuffs, then, it becomes simpler and also more beautiful in a sense. The theory is to the point when more abstract. Japanese sword seeks for the beauty in the sword itself, it never decorates with some kind of jewels. Because a sword maker/master thinks the beauty comes from the sword itself. They shamed if they need to cover a sword with non-sword component. We can find many swords, staffs, ... are decorated with gold or some jewels. I can also see some kind of gorgeousness in that, however, I prefer beauty in these kind of simpleness. A French pilot said ``A designer knows he has achieved perfectio...

I have a Japanese version of this page . There was a good question on my last article. ``What happen if the vending machine (a function) is broken?'' First of all, we need to define what broken means. Some of you would say, ``Broken is broken, what else?'' But, this answer does not make any sense for mathematicians. Are there no common sense in the head of mathematician? Maybe, yes. But, there is a reason. I am a Sunday mathematician/programmer. Mainly I program a code to solve some of my problems. To tell my computer to solve my problem, I need to interpret my problem to a code which my computer can execute/understand. Many of mathematics formulation is really formulated, which means you do not need to understand what it is, they are just a procedure. Then my computer can execute to solve my problem. I formulate some problem since after that is done, I do not need to think about that. Rest of the problem is solved automatically. This is fun for me. ``Computer, search s...

Japanese version My colleague told me this realtime raytracing demo with photon mapping (caustics). My machine has not enough power to run the demo. But it is quite impressive that just 4k bytes demo program can do this in realtime. Unfortunately, the code is not published. The rendering time in a movie sometimes took 8 hours to render a frame. Which means it takes more than 8 days for one second movie. But these people usually has a render farm that contains 1000 or more computers. So, after they start render images, 1000 frames will be ready. For this demo we can take video directory. When you visit to YouTube, there are higher quality movie. http://www.youtube.com/watch?v=LWx2HjvFzHw http://www.pouet.net/prod.php?which=51443

( Japanese version ) The word lambda calculus itself has ``calculus,'' so it can also calculate numbers. But, when I started to learn the lambda calculus, I want to know that what is ``calculate'' means. Then I also think about ``what is the number?'' If I want to teach ``what is the number'' to small children, I do not know how to teach it. Also I did not recall how to learn that. But, I think I know what numbers are. I can tell that there are some properties about numbers. First, it does not matter how to read it. In the hitchhiker's guide, there was a planet called Earth. The people on the planet have a lot of languages. For example, English, German, Latin, Japanese, and so forth. Interestingly enough, every language seems to have a concept about numbers. There are many representations for numbers, for example, (1, 2, 3, ...), (one, two, three, ...), (ein, zwei, drei, ...), (ichi, ni, san, ...), and so on. But no matter how you read them, there s...

lambda calculus and Hitchhiker's guide One day, I hit on to want to know about lambda calculus, and looked up Wikipedia. There are nice entry about that, but I could not understand some examples of calculation. It took a week to figure out. Maybe this is trivial, but I would like to brag myself about that. There is a common saying, ``they brag most who can do least.'' I am not familiar with lambda calculus. If I encounter something not familiar, I usually lookup some kind of guide book. I went to some city, I usually have a book called ``Chikyu no aruki kata (How to walk on the earth (in Japanese)),'' or Hitchhiker's guide to the Galaxy. This article could be a tiny version of a Hitchhiker's guide to lambda calculus. According to the Hitchhiker's guide to the Galaxy, the Hitchhiker's guide to the Galaxy is the most successful book in the universe. However, it seems no one knows the book is written in which language. If that is the most successful boo...