Nucleus (mathematics) - Write

Japanese: 核（数学） - かく

…However, it was Volterra V. (1860-1940) and Fredholm E. Fredholm (1866-1927) who systematically discussed integral equations. In general, when f ( x ) and K ( x , y ) are known functions and φ( x ) is an unknown function, they are called Fredholm-type integral equation and Volterra-type integral equation, respectively, and K ( x , y ) is called the kernel of these equations. ^Let K1 ( x , y )＝ K ( x , y ), and Kn ⁽ x , y ) defined for n ＞1 is called the iteration kernel. …

From [Linear Mapping]

...For a linear mapping f : V → W , Ker( f ) = { a ∈ V | f ( a ) = 0} and Im( f ) = { f ( a ) | a ∈ V } are linear subspaces of V and W , respectively. Ker( f ) is called the kernel of f , Im( f ) is called the image of f , and the dimension of Im( f ) is called the rank of f . If V and W are both finite-dimensional, _and e1 , ..., en are _the basis of V , and e1 _' , ..., em ' are the basis of W _, then an element a of V can be written as follows. ...

*Some of the terminology explanations that mention "nucleus (mathematics)" are listed below.

Source | Heibonsha World Encyclopedia 2nd Edition | Information

Japanese:

…しかし積分方程式を系統的に論じたのは，ボルテラV.Volterra(1860‐1940)とフレドホルムE.I.Fredholm(1866‐1927)である。　一般的な形として，f(x)，K(x,y)を既知関数，φ(x)を未知関数とするとき，はそれぞれフレドホルム型積分方程式，ボルテラ型積分方程式と呼ばれ，K(x,y)をこれらの方程式の核という。K¹(x,y)＝K(x,y)とおき，n＞1で定義したKⁿ(x,y)を反復核という。…

【線形写像】より

…線形写像f：V→Wについて，Ker(f)＝｛a∈V｜f(a)＝0｝，Im(f)＝｛f(a)｜a∈V｝はそれぞれV,Wの線形部分空間になる。Ker(f)をfの核，Im(f)をfの像と呼び，Im(f)の次元をfの階数という。V,Wがともに有限次元であるとして，e₁，……，e_nがVの，e₁′，……，e_m′がWの基底とすると，Vの元aはと書ける。…