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Dec
6
revised Periodic solution of nonlinear differential equation
added 2 characters in body
Dec
6
asked Periodic solution of nonlinear differential equation
Dec
5
answered Periodic parametric curve on cylinder
Dec
5
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.
Dec
5
revised Parametric curve on cylinder surface
added 411 characters in body
Dec
5
answered Parametric curve on cylinder surface
Dec
5
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
Dec
5
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
Dec
4
accepted Space of bounded functions is reflexive if the domain is finite
Dec
4
accepted Showing that $(X^*)^{**}=(X^{**})^{*}$, where $X$ is a Banach space
Dec
3
awarded  Citizen Patrol
Dec
3
asked Space of bounded functions is reflexive if the domain is finite
Dec
3
comment Showing that $(X^*)^{**}=(X^{**})^{*}$, where $X$ is a Banach space
No. I find the question from exercise of conway book
Dec
3
asked Showing that $(X^*)^{**}=(X^{**})^{*}$, where $X$ is a Banach space
Nov
30
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?
Nov
30
asked Strictly convex Inequality in $l^p$
Nov
24
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.
Nov
24
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 $t>0$)
Nov
24
asked Differential equation, Stability , Lyapunov function
Nov
19
accepted Surjective map from quotient space of Banach Space that satisfy certain condition