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Interpreting of another proposition full of symbols

kalish

Member
Oct 7, 2013
99
Could someone help me interpret the following proposition full of symbols? I've been struggling to comprehend it as well. Thanks in advance.

Proposition: Suppose that $f:\mathbb{R^n} \rightarrow \mathbb{R^n}, g:\mathbb{R^n} \rightarrow \mathbb{R}$ is a positive function, and $\phi$ is the flow of the differential equation $\dot{x}=f(x)$. If the family of solutions of the family of initial value problems $$\dot{y} = g(\phi(y,\xi)),$$ $$y(0)=0,$$ with parameter $\xi \in \mathbb{R^n}$, is given by $\rho: \mathbb{R} \times \mathbb{R^n} \rightarrow \mathbb{R}$, then $\psi$, defined by $\psi(t,\xi)=\phi(\rho(t,\xi),\xi)$ is the flow of the differential equation $\dot{x}=g(x)f(x)$.

I have crossposted this here as well: differential equations - Interpreting another proposition full of symbols - Mathematics Stack Exchange
 

HallsofIvy

Well-known member
MHB Math Helper
Jan 29, 2012
1,151
Could someone help me interpret the following proposition full of symbols? I've been struggling to comprehend it as well. Thanks in advance.

Proposition: Suppose that $f:\mathbb{R^n} \rightarrow \mathbb{R^n}, g:\mathbb{R^n} \rightarrow \mathbb{R}$ is a positive function,
f is a function that maps n real variables to a value of n ordered numbers (OR f maps an n dimensional vector to an n dimensional vector). g is a function that maps n real variables to a single positive number (OR g maps an n dimensional vector to a single number).

and $\phi$ is the flow of the differential equation $\dot{x}=f(x)$.
[tex]\phi[/tex] is the general solution to the differential equation. Note that the equation can be thought of as a set of n interrelated differential equations.

If the family of solutions of the family of initial value problems $$\dot{y} = g(\phi(y,\xi)),$$ $$y(0)=0,$$ with parameter $\xi \in \mathbb{R^n}$, is given by $\rho: \mathbb{R} \times \mathbb{R^n} \rightarrow \mathbb{R}$, then $\psi$, defined by $\psi(t,\xi)=\phi(\rho(t,\xi),\xi)$ is the flow of the differential equation $\dot{x}=g(x)f(x)$.

I have crossposted this here as well: differential equations - Interpreting another proposition full of symbols - Mathematics Stack Exchange