(note) line is my line. Sometimes it is not sure. I am not a mathematician. I am just a wanabee of mathematician. So, please care what I wrote.

### A

• Analytic: (holomorphic) is a regular function in Complex Analysis. (note) The regular means there are derivatives. In the Garrity's book, analytic and holomorphic seems the same. But it is not clear for me that holomorphic is always analytic or not.
• Algebraic Multiplicities: In a equation, how many roots are the same is algebraic multiplicities. For example, the root of (x-1)^2 = 0 is `1' and multiplicities is 2.
In Japanese this is Jyukon or Jyukai. Jyu means multiple or overlapped, kon and kai are both root. So, it is overlapped roots. Somehow it is easy for me.
(note) Algebraic multiplicities are important for algebra, however, for me, it is more familiar with eigenvalue problem of linear algebra.
Eigenvalues are the root of the characteristic equation of linear operator T. The algebraic multiplicities are related with the basis of eigenspace. But in practical, most of the linear operator's eigenvalues usually have not algebraic multiplicities (or, the same thing, it is 1). For me, theoretically this is interesting, but practically there is not so much.
Maybe it is easy to imagine a n-degree equation to feel the reason that usual eigenvalues has no multiplicities. If n-degree equation has algebraic multiplicities, such point is quite special. For example, assume 2nd degree equation, y = (x-a)^2 + b = 0. Think about the graph of this. The case Algebraic multiplicities = 2 is only b = 0 case. If b is not 0, there are two roots all the time (including complex roots). If you move the graph a little bit, you can easy to remove the case of multiplicities = 2. Eigenvalues are the roots of characteristic polynomial = 0. So, just put some disturbance to your linear operator T, your eigenvalues have no more multiplicities >= 2. Or, more concrete, think about one representation of matrix A of T, just add epsilon to some of the element of A breaks eigenvalues multiplicities >= 2.

### B

• Bijection: It is better to look up the Mathworld. The reason I put this word here is the difficulty of English. In Japanese,
• Surjection: Zensha (Zen means all, sha means map), So all the source element goes to somewhere on the destination.
• Injection: Tansha (Tan means one, sha means map), So all the source element has one correcpondance on the destination.
• Bijection: Zentansha, so, Surjection and Injection is Bijection in Japanese.
I always confuse which is Zensha and which is Tansha.
Note only bijection has inverse map. There is another saying of bijection ``equipollent.'' But it seems this is used in logic.

### R

• Reductio ad Absurdum (Reduction of absurdity) : Hairihou or Kibyuuhou in Japanese
• Rhombus: Diamond shape. Hishigata in Japanese. Hishi means Rhombus, Kata is shape (type). There is a company called Mitubishi which means Mitu (= three) and bishi (= shi, the sound is a bit changed after vowel in Japanese.) So, I think you may know the Mitubishi's company logo. Also there is a company Yotubishi (Yotu means four).

### T

• trapezoid: Daikei in Japanese (Dai means support, kei means shape)