beginner
Reputation
Next privilege 125 Rep.
Vote down
4 17
Impact
~27k people reached

# 255 Actions

 Dec6 asked Periodic solution of nonlinear differential equation Dec5 answered Periodic parametric curve on cylinder Dec5 comment Differential equation, Stability , Lyapunov function How about this argument: We can reparameterize $S=\{(\sqrt{C}\cos u,\frac{\sqrt{C}}{\sqrt{2}}\sin u, v): u,v\in \mathbb{R}\}$. Let $r(t)= (x(t),y(t),z(t))$ and $r(0)=(x_0,y_0,z_0)$, then $x(t)=\sqrt{C}\cos (t-t_0)$ and $y(t)=\frac{\sqrt{C}}{\sqrt{2}}\sin (t-t_0)$ with $t_0$ satisfying $x_0=\sqrt{C}\cos t_0$ and $y_0=-\frac{\sqrt{C}}{\sqrt{2}}\sin t_0$. Since $z'=xy$ then $z'(t)=\frac{C}{2\sqrt{2}}\sin(2t-t_0)$, hence $z(t)=-\frac{C}{4\sqrt{2}}\cos(2t-t_0)$. Since $r(2\pi)=(x_0,y_0,z_0)=r(0)$ then $r(t)$ is periodic. Dec5 revised Parametric curve on cylinder surface added 411 characters in body Dec5 answered Parametric curve on cylinder surface Dec5 comment Differential equation, Stability , Lyapunov function How about the possiblity if the solution only trace the first octan? In this octan, z'=xy>0 so z always increasing. What make you sure that the solution trace along all quadrant of domain Dec5 comment Differential equation, Stability , Lyapunov function How about the possiblity if the solution only trace the first octan? In this octan, z'=xy>0 so z always increasing. What make you sure that the solution trace along all quadrant of domain Dec4 accepted Space of bounded functions is reflexive if the domain is finite Dec4 accepted Showing that $(X^*)^{**}=(X^{**})^{*}$, where $X$ is a Banach space Dec3 awarded Citizen Patrol Dec3 asked Space of bounded functions is reflexive if the domain is finite Dec3 comment Showing that $(X^*)^{**}=(X^{**})^{*}$, where $X$ is a Banach space No. I find the question from exercise of conway book Dec3 asked Showing that $(X^*)^{**}=(X^{**})^{*}$, where $X$ is a Banach space Nov30 comment Strictly convex Inequality in $l^p$ If the vectors is multiple of the other by scalar implies the equality is trivial. How about the converse? Nov30 asked Strictly convex Inequality in $l^p$ Nov24 comment Differential equation, Stability , Lyapunov function I just took the definition from Hirsch and Smale's Book. And it say like that. $dV/dt<0$ for asymptotically stable and $dV/dt\leq 0$ for stable one. You can check the book now. Nov24 comment Differential equation, Stability , Lyapunov function In Lyapunov theorem, if $dV/dt<0$ then the equilibrium is asymptotically stable and $dV/dt\leq 0$ then the equilibrium is stable (An equilibrium point $x_0$ is stable iff for every neighboorhood $U$ of $x_0$ there is a neighboorhood $U_1$ of $x_0$ in $U$ such that every solution $r(t)$ with $r(0)$ in $U_1$ is defined and in $U$ for all $t>0$) Nov24 asked Differential equation, Stability , Lyapunov function Nov19 accepted Surjective map from quotient space of Banach Space that satisfy certain condition Nov18 comment Surjective map from quotient space of Banach Space that satisfy certain condition @martini: Is that projection surjective? Could you give me the surjective one?