I need to determine whether the fixed point $x^{*}=0$ of the system $$\dot{x} = 0 \\ \dot{y}=x $$
is attracting, Liapunov stable, asymptotically stable, or none of the above.
For reference, a fixed point $x^{*}$ is said to be attracting if any trajectory that starts within a distance $\delta$ ...
In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output. Otherwise, a function is said to be a discontinuous function. A continuous function with a continuous inverse function is called a homeomorphism.
Continuity of functions is one of the core concepts of topology, which is treated in full generality below. The introductory portion of this article focuses on the special case where the inputs and outputs of functions are real numbers. A stronger form of continuity is uniform continuity. In addition...
I'm reading a class note of CS231n, and I'm wondering the dimension of the Jacobian matrices $\Large\frac{\partial Y}{\partial X}$. In the given example he provided a formula
$$\frac{\partial L}{\partial X}=\frac{\partial L}{\partial Y}\frac{\partial Y}{\partial X},$$
use the formula my calcula...
Consider an improper integral with a pole on the integration contour at say $z=1$,
$$
\tag{1} I = \int_{-\infty}^\infty \mathrm{d}z\ \frac{e^{-z^2}}{z-1+i\epsilon},~~~~~\epsilon>0.
$$
Let $$f(z) = \frac{e^{-z^2}}{z-1+i\epsilon}$$
then
$$
\sum_{residues~inside~\Gamma} = 0 = \oint_\Gamma f(z) ...
