ece124: look man i dunno either

docs/1b/ece124.md

@@ -462,6 +462,68 @@ An **equivalent state** is such that each input has the same output and an equiv
 2. For each state, if not all states transition to the same group, subgroup them such that they do
 3. Repeat as necessary
 
+## Asynchronous sequential circuits
+
+ASCs hae no clocks, relying on feedback from outputs for their memory effect.
+
+!!! warning
+    ASCs break down if any of these assumptions fail.
+    
+    - Only one input is allowed to change at a time
+    - Inputs change only after the circuit stabilises
+    - There is no propagation delay, although it may be compensated for with a delay element for the output / feedback
+
+### Analysis
+
+1. Determine logic expressions for next state and output in terms of current state and input
+2. Create transition and flow tables
+3. Circle stable states (will lead to itself)
+4. Replace bits with letters
+5. Assign bit variables to avoid changing more than one input at a time (as it is undefined)
+
+To create a circuit:
+
+1. Create a state diagram
+2. Flow table
+3. Minimise state
+4. Excitation table
+5. Circuit
+
+### Reducing state
+
+1. Partition per [#Minimising state](#minimising-state)
+  - *don't cares* are no longer equivalent unless both states have them in the same columns
+2. States are compatible if and only if, regardless of input:
+  - their output is the same
+  - their next state is the same, is stable, or are unspecified
+3. Merger diagram, identifying conflicts/compatible pairs
+4. Connect diagram, merging a subset (not all) of compatible states
+  - states can only be in at most one subset
+5. Repeat
+
+### Avoiding races
+
+!!! definition
+    - **Non-critical races** result in the same stable end state.
+    - **Critical races** cause *problems*.
+    - A **hazard** is unwanted switching due to unequal propagation delays.
+
+To avoid races: an $n$-dimensional cube with one vertex per state, ensuring that changes only move along one edge. If more states are needed to avoid this, they are automagically *unstable*.
+
+Alternatively, if $n\leq 4$, a state $A$ can be split into equivalent states $A1, A2$.
+
+1. Create a cube with $2n$ vertices
+2. Pairs must be adjacent
+3. Determine next states by following cube lines only
+
+Alternatively, each state can be assigned exactly one `1` bit, and transitions from one to another have `1`s at the states they transition between.
+
+### Hazards
+
+**Static-1/0 hazards** occur when output should stay constant, but suddenly flickers to the other. These can be fixed by covering minterms adjacent but not connected with another gate as an extra check.
+
+**Dynamic hazards** occur when outputs flip multiple times before stabilising. These can be avoided by switching everything to 2-term POS or SOP and fixing static hazards.
+
 ## VHDL
 
 VHDL is a hardware schematic language.

docs/1b/math119.md

@@ -635,3 +635,76 @@ The Jacobian is $r^2\sin\theta$.
     It is clear that $\tan\theta=\sqrt 3\implies\theta=\frac\pi 3,r=3$. Thus:
     
     $$\int^3_0\int^{\pi/3}_0,\int^{2\pi}_0 \frac{\rho}{\sqrt{3}}\rho\ d\phi\ d\theta\ d\rho=\frac{243\pi}{5}$$
+
+## Approximation and interpolation
+
+Each of these finds roots, so a rooted equation is needed.
+
+!!! example
+    To find an $x$ where $x=\sqrt 5$, the root of $x^2-5=0$ should be found.
+
+### Bisection
+
+1. Select two points that are guaranteed to enclose the point
+2. Select an arbitrary $x$ and check if it is greater than or less than zero
+3. Slice the remaining section in half in the correct direction
+
+### Newton's method
+
+The below formula can be repeated after plugging in an arbitrary value.
+
+$$x_1=x_0-\frac{f(x_0)}{f'(x_0}$$
+
+!!! warning
+    If Newton's method converges to the wrong root, bisection is necessary to brute force the result.
+    
+### Polynomial interpolation
+
+Where $\Delta^k y_0$ are the $k$th differences between the $y$ points:
+
+$$f(x)=y_0+x\Delta y_0+x(x-1)\frac{\Delta^2y_0}{2!}+x(x-1)(x-2)\frac{\Delta^3 y_0}{3!} ...$$
+
+### Taylor polynomials
+
+The $n$th order Taylor polynomial centred at $x_0$ is:
+
+$$\boxed{P_{n,x_0}(x)=\sum^n_{k=0}\frac{f^{(k)}(x_0)(x-x_0)^k}{k!}}$$
+
+**Maclaurin's theorem** states that if some function $P^{(k)}(x_0)=f^{(k)}(x_0)$ for all $k=0,...n$:
+
+$$P(x)=P_{n,x_0}(x)$$
+
+!!! example
+    If $P(x)=1+x^3+\frac{x^6}{2}$ and $f(x)=e^{x^5}}$, ... TODO
+
+The desired function $P(x)$ being the $n$th degree Maclaurin polynomial implies that $P(kx^m)$ is the $(mn)$th degree polynomial for $f(kx^m)$.
+
+Therefore, if you have the Maclaurin polynomial $P(x)$ where $P$ is the $n$th order Taylor polynomial:
+
+- $P'(x)=P_{n-1,x_0}(x)$ for $f'(x)$
+- $\int P(x)dx=P_{n+1,x_0}(x)$ for $\int f(x)dx$
+
+The integration constant $C$ can be found by substituting $x_0$ as $x$ and solving.
+
+For $m\in\mathbb Z\geq 0$, where $P(x)$ is the Maclaurin polynomial for $f(x)$ of order $n$, $x^mP(x)$ is the $(m+n)$th order polynomial for $x^mf(x)$.
+
+### Taylor inequalities
+
+The **triangle inequality** for integrals applies itself many times over the infinite sum.
+
+$$\left|\int^b_af(x)dx\right|\leq\int^b_a|f(x)|dx$$
+
+The **Taylor remainder** is the error between a Taylor polynomial and its actual value. Where $k$ is an arbitrary value chosen as the **upper bound** of the difference of the first derivative between $x_0$ and $x$: $k\geq |f^{(n+1)}(z)|$
+
+$$|R_n(x)|\leq\frac{k|x-x_0|^{n+1}}{(n+1)!}$$
+
+An approximation correct to $n$ decimal places requires that $|R_n(x)|<10^{-n}$.
+
+!!! warning
+    $k$ should be as small as possible. When rounding, round down for the lower bound, and round up for the upper bound.
+    
+### Integral approximation
+
+The upper and lower bounds of a Taylor polynomial are clearly $P(x)\pm R(x)$. Integrating them separately reveals creates bounds for the integral.
+
+$$\int P(x)dx-\int R(x)dx\leq\int P(x)\leq\int P(x)dx +\int R(x)dx$$