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When does $\min_x \max_yf(x,y) = \min_y \max_x f(x,y)$ hold for a real function $f(x,y)$?

Let $f(x,y)$ be a real function of the variables $x,y$ (which can be real vectors). Under what conditions do we have the following equality: $$\min_x \max_yf(x,y) = \min_y \max_x f(x,y)$$ For example, this equality is true if $f(x,y) = xy$ and $x,y$ are real scalars. Note that this is not the same as Von Neumann’s […]

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$\min_x \max_yf(x,y) = \min_y \max_x f(x,y)$

Let $f(x,y)$ be a real function of the variables $x,y$ (which can be real vectors). Under what conditions do we have the following equality: $$\min_x \max_yf(x,y) = \min_y \max_x f(x,y)$$ For example, this equality is true if $f(x,y) = xy$ and $x,y$ are real scalars. Note that this is not the same as Von Neumann’s […]

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Conditions for $\min_x \max_yf(x,y) = \min_y \max_x f(x,y)$?

Let $f(x,y)$ be a real function of the variables $x,y$ (which can be real vectors). Under what conditions do we have the following equality: $$\min_x \max_yf(x,y) = \min_y \max_x f(x,y)$$ For example, this equality is true if $f(x,y) = xy$ and $x,y$ are real scalars. Note that this is not the same as Von Neumann’s […]

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Conditions for $\min_x \max_yf(x,y) = \min_y \max_x f(x,y)$?

Let $f(x,y)$ be a real function of the variables $x,y$ (which can be real vectors). Under what conditions do we have the following equality: $$\min_x \max_yf(x,y) = \min_y \max_x f(x,y)$$ For example, this equality is true if $f(x,y) = xy$ and $x,y$ are real scalars. Note that this is not the same as Von Neumann’s […]

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How to find an inverse of this type of functions?

Let $F(x,y)$ and $G(x,y)$ Be functions from where $x,y$ are whole numbers. ($Z^2 \to Z^2$) $F(x,y) = (x+3y,x+5y)$ $G(x,y) = (2x+3y,3x+5y)$ The question: One of these functions has an inverse, Prove it and find its inverse functions. My question: I don’t really know how to decide which one is inversible. I tried to show that […]

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Prove that $Ker (\phi ) = (X^2 – Y^3, Y^2-Z^3)$ where $\phi : k[X,Y,Z] \rightarrow k[T]$ s.t. $X \mapsto t^9, \; Y \mapsto t^6, \; Z \mapsto t^4$.

I want to prove that for a ring homomorphism $\phi : k[X,Y,Z] \rightarrow k[T]$ s.t. $X \mapsto t^9, \; Y \mapsto t^6, \; Z \mapsto t^4$, we have $Ker (\phi ) = (X^2 – Y^3, Y^2-Z^3)$. My attempt: It is easy to prove that $ I = (X^2 – Y^3, Y^2-Z^3) \subset Ker (\phi ) […]