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The theory of the equation f(x)=x has produced some of the most generally useful results in mathematics. Banach's fixed point theorem and Brouwer's fixed point theorem are two pillars of the theory that every student will learn about, and they are in turn the main ingredients for fundamental applications ranging from biology, numerical computations, economics, hydrodynamics, differential equations to game theory.

INTO JAPANESE

方程式f（x）= xの理論は、数学で最も一般的に有用な結果をもたらしました。 Banachの不動点定理とBrouwerの不動点定理は、すべての生徒が学ぶ理論の2つの柱であり、生物学、数値計算に至るまでの基本的なアプリケーションの主要な要素です。

BACK INTO ENGLISH

The theory of the equation f (x) = x has yielded the most generally useful results in mathematics. Banach's fixed point theorem and Brouwer's fixed point theorem are the two pillars of theory that all students learn, and are key elements of basic applications ranging from biology to numerical computation.

INTO JAPANESE

方程式f（x）= xの理論は、数学で最も一般的に有用な結果をもたらしました。 Banachの不動点定理とBrouwerの不動点定理は、すべての学生が学ぶ理論の2つの柱であり、生物学から数値計算までの基本的なアプリケーションの重要な要素です。

BACK INTO ENGLISH

The theory of the equation f (x) = x has yielded the most generally useful results in mathematics. Banach's fixed-point theorem and Brouwer's fixed-point theorem are two pillars of theory that all students learn, and are important elements of basic applications from biology to numerical computation.

INTO JAPANESE

方程式f（x）= xの理論は、数学で最も一般的に有用な結果をもたらしました。 Banachの固定小数点定理とBrouwerの固定小数点定理は、すべての学生が学ぶ理論の2つの柱であり、生物学から数値計算までの基本的なアプリケーションの重要な要素です。

BACK INTO ENGLISH

The theory of the equation f (x) = x has yielded the most generally useful results in mathematics. Banach's fixed-point theorem and Brouwer's fixed-point theorem are two pillars of theory that all students learn, and are important elements of basic applications from biology to numerical computation.