(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.
- 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
- 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.
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.
It is better to look up the Mathworld. The reason I put this word
here is the difficulty of English. In Japanese,
I always confuse which is Zensha and which is Tansha.
- 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.
bijection has inverse map. There is another saying of bijection
``equipollent.'' But it seems this is used in logic.
- 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).
- trapezoid: Daikei in Japanese (Dai means
support, kei means shape)
Copyright (C) 2002-2007 YAMAUCHI Hitoshi