Improve knowledge base

16 files changed, 296 insertions, 266 deletions

diff --git a/source/know/concept/bernstein-vazirani-algorithm/index.md b/source/know/concept/bernstein-vazirani-algorithm/index.md
index 884cca3..4f36d3c 100644
--- a/source/know/concept/bernstein-vazirani-algorithm/index.md
+++ b/source/know/concept/bernstein-vazirani-algorithm/index.md
@@ -24,8 +24,8 @@ of $$x$$ with an unknown $$N$$-bit string $$s$$:
 
 $$\begin{aligned}
     f(x)
-    = s \cdot x \:\:(\bmod \: 2)
-    = (s_1 x_1 + s_2 x_2 + \:...\: + s_N x_N) \:\:(\bmod \: 2)
+    \equiv s \cdot x \:\bmod 2
+    = (s_1 x_1 + s_2 x_2 + \:...\: + s_N x_N) \:\bmod 2
 \end{aligned}$$
 
 The goal is to find $$s$$.
diff --git a/source/know/concept/deutsch-jozsa-algorithm/index.md b/source/know/concept/deutsch-jozsa-algorithm/index.md
index 44b06ad..223877a 100644
--- a/source/know/concept/deutsch-jozsa-algorithm/index.md
+++ b/source/know/concept/deutsch-jozsa-algorithm/index.md
@@ -72,8 +72,8 @@ $$\begin{aligned}
     + \frac{1}{2} \Ket{1} \Big( \Ket{0 \oplus f(1)} - \Ket{1 \oplus f(1)} \Big)
 \end{aligned}$$
 
-The parenthesized superpositions can be reduced.
-Assuming that $$f(b) = 0$$, we notice:
+The parenthesized superpositions can be reduced:
+let us suppose that $$f(b) = 0$$, then:
 
 $$\begin{aligned}
     \Ket{0 \oplus f(b)} - \Ket{1 \oplus f(b)}
@@ -91,7 +91,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 We can thus combine both cases, $$f(b) = 0$$ or $$f(b) = 1$$,
-into the following single expression:
+into the following expression:
 
 $$\begin{aligned}
     \Ket{0 \oplus f(b)} - \Ket{1 \oplus f(b)}
@@ -106,8 +106,8 @@ $$\begin{aligned}
     \frac{1}{2} \Big( (-1)^{f(0)} \Ket{0} + (-1)^{f(1)} \Ket{1} \Big) \Big( \Ket{0} - \Ket{1} \Big)
 \end{aligned}$$
 
-The second qubit in state $$\Ket{-}$$ is garbage; it is no longer of interest.
-The first qubit is given by:
+The second qubit in state $$\Ket{-}$$ is garbage (i.e. no longer of interest).
+The first qubit is:
 
 $$\begin{aligned}
     \frac{1}{\sqrt{2}} \Big( (-1)^{f(0)} \Ket{0} + (-1)^{f(1)} \Ket{1} \Big)
@@ -126,8 +126,8 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Depending on whether $$f$$ is constant or balanced,
-the mearurement outcome of this state will be $$\Ket{0}$$ or $$\Ket{1}$$
-with 100\% probability. We have solved the problem!
+the measurement outcome of this state will be $$\Ket{0}$$ or $$\Ket{1}$$
+with 100% probability. We have solved the problem!
 
 Note that we only consulted the oracle (i.e. applied $$U_f$$) once.
 A classical computer would need to query it twice,
@@ -146,7 +146,7 @@ This algorithm is then implemented by the following quantum circuit:
     alt="Deutsch-Jozsa circuit" %}
 
 There are $$N$$ qubits in initial state $$\Ket{0}$$, and one in $$\Ket{1}$$.
-For clarity, the oracle $$U_f$$ works like so:
+The oracle $$U_f$$ performs this action:
 
 $$\begin{aligned}
     \Ket{x_1} \Ket{x_2} \cdots \Ket{x_N} \Ket{y}
@@ -167,7 +167,7 @@ $$\begin{aligned}
 Where $$\Ket{x} = \Ket{x_1} \cdots \Ket{x_N}$$ denotes a classical binary state.
 For example, if $$x = 5 = 2^0 + 2^2$$ in the summation,
 then $$\Ket{x} = \Ket{1} \Ket{0} \Ket{1} \Ket{0}^{\otimes N-3}$$
-(from least to most significant).
+(from least to most significant digit).
 
 We give this state to the oracle,
 and, by the same logic as for the Deutsch algorithm,
@@ -217,8 +217,8 @@ we only need to measure the $$N$$ qubits once;
 $$f$$ is constant if and only if all are zero.
 
 The Deutsch-Jozsa algorithm needs only one oracle query to give an error-free result,
-whereas a classical computer needs $$2^{N-1} + 1$$ queries in the worst case;
-a revolutionary discovery.
+whereas a classical computer needs $$2^{N-1} + 1$$ queries in the worst case.
+A revolutionary discovery!
 
 
 ## References
diff --git a/source/know/concept/dirac-notation/index.md b/source/know/concept/dirac-notation/index.md
index 2830a33..bbf31e5 100644
--- a/source/know/concept/dirac-notation/index.md
+++ b/source/know/concept/dirac-notation/index.md
@@ -27,7 +27,8 @@ that maps kets $$\ket{V}$$ to other kets $$\ket{V'}$$.
 Recall that by definition the Hilbert inner product must satisfy:
 
 $$\begin{aligned}
-    \inprod{V}{W} = \inprod{W}{V}^*
+    \inprod{V}{W}
+    = \inprod{W}{V}^*
 \end{aligned}$$
 
 So far, nothing has been said about the actual representation of bras or kets.
@@ -36,12 +37,14 @@ the corresponding bras are given by the kets' adjoints,
 i.e. their transpose conjugates:
 
 $$\begin{aligned}
-    \ket{V} =
+    \ket{V}
+    =
     \begin{bmatrix}
         v_1 \\ \vdots \\ v_N
     \end{bmatrix}
-    \quad \implies \quad
-    \bra{V} =
+    \qquad \implies \qquad
+    \bra{V}
+    =
     \begin{bmatrix}
         v_1^* & \cdots & v_N^*
     \end{bmatrix}
@@ -88,8 +91,9 @@ then the bras are *functionals* $$F[u(x)]$$
 that take an arbitrary function $$u(x)$$ as an argument and return a scalar:
 
 $$\begin{aligned}
-    \ket{f} = f(x)
-    \quad \implies \quad
+    \ket{f}
+    = f(x)
+    \qquad \implies \qquad
     \bra{f}
     = F[u(x)]
     = \int_a^b f^*(x) \: u(x) \dd{x}
diff --git a/source/know/concept/ito-integral/index.md b/source/know/concept/ito-integral/index.md
index 4a725e1..9b092d6 100644
--- a/source/know/concept/ito-integral/index.md
+++ b/source/know/concept/ito-integral/index.md
@@ -10,8 +10,7 @@ layout: "concept"
 
 The **Itō integral** offers a way to integrate
 a given [stochastic process](/know/concept/stochastic-process/) $$G_t$$
-with respect to a [Wiener process](/know/concept/wiener-process/) $$B_t$$,
-which is also a stochastic process.
+with respect to a [Wiener process](/know/concept/wiener-process/) $$B_t$$.
 The Itō integral $$I_t$$ of $$G_t$$ is defined as follows:
 
 $$\begin{aligned}
@@ -47,21 +46,21 @@ which can be applied recursively, leading to:
 $$\begin{aligned}
     X_{t+h}
     \approx X_{t} + f(X_t) \: h
-    \quad \implies \quad
+    \qquad \implies \qquad
     X_t
     \approx X_0 + \sum_{s = 0}^{s = t} f(X_s) \: h
 \end{aligned}$$
 
-In the limit $$h \to 0$$, this leads to the following unsurprising integral for $$X_t$$:
+In the limit $$h \to 0$$, this unsurprisingly leads to the following integral for $$X_t$$:
 
 $$\begin{aligned}
-    \int_0^t f(X_s) \dd{s}
-    = \lim_{h \to 0} \sum_{s = 0}^{s = t} f(X_s) \: h
+    \lim_{h \to 0} \sum_{s = 0}^{s = t} f(X_s) \: h
+    = \int_0^t f(X_s) \dd{s}
 \end{aligned}$$
 
 In contrast, consider the *stochastic differential equation* below,
 where $$\xi_t$$ represents white noise,
-which is informally the $$t$$-derivative
+which is informally defined as the $$t$$-derivative
 of the Wiener process $$\xi_t = \idv{B_t}{t}$$:
 
 $$\begin{aligned}
@@ -89,9 +88,9 @@ $$\begin{aligned}
     = X_0 + \int_0^t g(X_s) \dd{B_s}
 \end{aligned}$$
 
-This integral is *defined* as below,
-analogously to the first, but with $$h$$ replaced by
-the increment $$B_{t+h} \!-\! B_t$$ of a Wiener process.
+The meaning of such an integral is *defined* below.
+It is analogous to the deterministic case,
+but $$h$$ is replaced by the increment $$B_{t+h} \!-\! B_t$$ of a Wiener process.
 This is an Itō integral:
 
 $$\begin{aligned}
@@ -100,7 +99,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 For more information about applying the Itō integral in this way,
-see the [Itō calculus](/know/concept/ito-process/).
+see [Itō calculus](/know/concept/ito-process/).
 
 
 
@@ -131,7 +130,7 @@ $$\begin{aligned}
 A more interesting property is the **Itō isometry**,
 which expresses the expectation of the square of an Itō integral of $$G_t$$
 as a simpler "ordinary" integral of the expectation of $$G_t^2$$
-(which exists by the definition of Itō-integrability):
+(which exists due to the definition of Itō-integrability):
 
 $$\begin{aligned}
     \boxed{
@@ -172,24 +171,16 @@ $$\begin{aligned}
 
 However, $$\mathcal{F}_t$$ says nothing about
 the increment $$(B_{t + h} \!-\! B_t) \sim \mathcal{N}(0, h)$$,
-meaning that the conditional expectation is zero:
+meaning that the conditional expectation is zero for $$t \ge s + h$$:
 
 $$\begin{aligned}
     \mathbf{E} \Big[ G_t G_s (B_{t + h} \!-\! B_t) (B_{s + h} \!-\! B_s) \Big]
     = 0
-    \qquad \mathrm{for}\; t \ge s + h
 \end{aligned}$$
 
-By swapping $$s$$ and $$t$$, the exact same result can be obtained for $$s \ge t \!+\! h$$:
-
-$$\begin{aligned}
-    \mathbf{E} \Big[ G_t G_s (B_{t + h} \!-\! B_t) (B_{s + h} \!-\! B_s) \Big]
-    = 0
-    \qquad \mathrm{for}\; s \ge t + h
-\end{aligned}$$
-
-This leaves only one case which can be nonzero: $$[t, t\!+\!h] = [s, s\!+\!h]$$.
-Applying the law of total expectation again yields:
+By swapping $$s$$ and $$t$$, the exact same result can be obtained for $$s \ge t \!+\! h$$.
+This leaves only one possibly nonzero case: $$[t, t\!+\!h] = [s, s\!+\!h]$$.
+Applying the law of total expectation again:
 
 $$\begin{aligned}
     \mathbf{E} \bigg[ \sum_{t = a}^{t = b} G_t (B_{t + h} \!-\! B_t) \bigg]^2
@@ -198,15 +189,15 @@ $$\begin{aligned}
     &= \sum_{t = a}^{t = b} \mathbf{E} \bigg[ \mathbf{E} \Big[ G_t^2 (B_{t + h} \!-\! B_t)^2 \Big| \mathcal{F}_t \Big] \bigg]
 \end{aligned}$$
 
-We know $$G_t$$, and the expectation value of $$(B_{t+h} \!-\! B_t)^2$$,
-since the increment is normally distributed, is simply the variance $$h$$:
+We know $$G_t$$,
+and the expectation value of $$(B_{t+h} \!-\! B_t)^2$$ is simply the variance $$h$$:
 
 $$\begin{aligned}
     \mathbf{E} \bigg[ \sum_{t = a}^{t = b} G_t (B_{t + h} \!-\! B_t) \bigg]^2
     &= \sum_{t = a}^{t = b} \mathbf{E} \big[ G_t^2 \big] h
-    \longrightarrow
-    \int_a^b \mathbf{E} \big[ G_t^2 \big] \dd{t}
 \end{aligned}$$
+
+Taking the limit $$h \to 0$$ then yields the desired result.
 {% include proof/end.html id="proof-isometry" %}
 
 
@@ -239,7 +230,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 We now have everything we need to calculate $$\mathbf{E} [ I_t | \mathcal{F_s} ]$$,
-giving the martingale property:
+leading to the martingale property:
 
 $$\begin{aligned}
     \mathbf{E} \big[ I_t | \mathcal{F}_s \big]
@@ -250,10 +241,10 @@ $$\begin{aligned}
 
 For the existence of $$I_t$$,
 we need $$\mathbf{E}[G_t^2]$$ to be integrable over the target interval,
-so from the Itō isometry we have $$\mathbf{E}[I]^2 < \infty$$,
-and therefore $$\mathbf{E}[I] < \infty$$,
-so $$I_t$$ has all the properties of a Martingale,
-since it is trivially $$\mathcal{F}_t$$-adapted.
+which implies via the Itō isometry that $$\mathbf{E}[I]^2$$ is finite.
+Therefore $$\mathbf{E}[I]$$ is also finite,
+so $$I_t$$ has all the properties of a Martingale
+(since it is trivially $$\mathcal{F}_t$$-adapted).
 {% include proof/end.html id="proof-martingale" %}
 
 
diff --git a/source/know/concept/korteweg-de-vries-equation/index.md b/source/know/concept/korteweg-de-vries-equation/index.md
index 2857e23..e8035d1 100644
--- a/source/know/concept/korteweg-de-vries-equation/index.md
+++ b/source/know/concept/korteweg-de-vries-equation/index.md
@@ -162,11 +162,11 @@ $$\begin{aligned}
     = q_0 - \frac{g}{q_0} \Big( \eta(x, t) + \alpha + \gamma(x, t) \Big)
 \end{aligned}$$
 
-Where $$\alpha$$ is a constant parameter
-(which we will use to handle velocity discrepancies
-between the linear and nonlinear theories).
+Where $$\alpha$$ is a constant parameter,
+which we will use to handle velocity discrepancies
+between the linear and nonlinear theories.
 The correction represented by $$\gamma$$ is much smaller,
-i.e. $$\eta \sim \alpha \gg \gamma$$.
+i.e. $$\eta \gg \alpha \gg \gamma$$.
 We insert this ansatz into the above equations, yielding:
 
 $$\begin{aligned}
@@ -265,14 +265,15 @@ $$\begin{aligned}
     \equiv \frac{h^3}{3} - \frac{h T}{g \rho}
 \end{aligned}$$
 
-What about $$\alpha$$?
+But what about $$\alpha$$?
 Looking at the ansatz for $$f$$, we see that
-the body of water is already assumed to be moving at $$q_0$$,
-minus $$g \alpha / q_0$$, so by varying $$\alpha$$
-we are modifying the water's velocity.
-The term in the KdV equation simply corrects for our chosen value of $$\alpha$$.
-It has no deeper meaning than that: for any value of $$\alpha$$,
-the full range of KdV solutions can still be obtained.
+the body of water is assumed to be moving at $$q_0 - g \alpha / q_0$$,
+and $$q_0$$ is set to $$\pm \sqrt{g h}$$ by almost all authors,
+so $$\alpha$$ controls the velocity of our reference frame.
+Nonlinear waves do not travel at the same speed as linear waves,
+so we can choose $$\alpha$$ to make the wave stationary
+without breaking the $$q_0$$ "tradition".
+That term in the KdV equation simply corrects for our chosen value of $$\alpha$$.
 
 
 
@@ -383,14 +384,16 @@ These are the final scale parameter values,
 leading to the desired dimensionless form:
 
 $$\begin{aligned}
-    0
-    &= \tilde{\eta}_{\tilde{t}} - 6 \tilde{\eta} \tilde{\eta}_{\tilde{x}} + \tilde{\eta}_{\tilde{x} \tilde{x} \tilde{x}}
+    \boxed{
+        0
+        = \tilde{\eta}_{\tilde{t}} - 6 \tilde{\eta} \tilde{\eta}_{\tilde{x}} + \tilde{\eta}_{\tilde{x} \tilde{x} \tilde{x}}
+    }
 \end{aligned}$$
 
 Recall that $$\alpha$$ sets the background fluid velocity,
 and $$v_c$$ controls the coordinate system's motion:
 our choice of $$v_c$$ simply cancels out the effect of $$\alpha$$.
-This reveals the point of $$\alpha$$:
+This demonstrates the purpose of $$\alpha$$:
 the KdV equation has solutions moving at various speeds,
 so, for a given $$\eta$$, we can always choose $$\alpha$$ (and hence $$v_c$$)
 such that the wave appears stationary.
diff --git a/source/know/concept/kramers-kronig-relations/index.md b/source/know/concept/kramers-kronig-relations/index.md
index 711023e..68e27dc 100644
--- a/source/know/concept/kramers-kronig-relations/index.md
+++ b/source/know/concept/kramers-kronig-relations/index.md
@@ -10,124 +10,145 @@ categories:
 layout: "concept"
 ---
 
-Let $$\chi(t)$$ be a complex function describing
-the response of a system to an impulse $$f(t)$$ starting at $$t = 0$$.
-The **Kramers-Kronig relations** connect the real and imaginary parts of $$\chi(t)$$,
-such that one can be reconstructed from the other.
-Suppose we can only measure $$\chi_r(t)$$ or $$\chi_i(t)$$:
+Let $$\chi(t)$$ be the response function of a system
+to an external impulse $$f(t)$$, which starts at $$t = 0$$.
+Assuming initial equilibrium, the principle of causality
+states that there is no response before the impulse,
+so $$\chi(t) = 0$$ for $$t < 0$$.
+To enforce this, we demand that $$\chi(t)$$ satisfies a **causality test**,
+where $$\Theta(t)$$ is the [Heaviside step function](/know/concept/heaviside-step-function/):
 
 $$\begin{aligned}
-    \chi(t) = \chi_r(t) + i \chi_i(t)
+    \chi(t)
+    = \chi(t) \: \Theta(t)
 \end{aligned}$$
 
-Assuming that the system was at rest until $$t = 0$$,
-the response $$\chi(t)$$ cannot depend on anything from $$t < 0$$,
-since the known impulse $$f(t)$$ had not started yet,
-This principle is called **causality**, and to enforce it,
-we use the [Heaviside step function](/know/concept/heaviside-step-function/)
-$$\Theta(t)$$ to create a **causality test** for $$\chi(t)$$:
-
-$$\begin{aligned}
-    \chi(t) = \chi(t) \: \Theta(t)
-\end{aligned}$$
-
-If we [Fourier transform](/know/concept/fourier-transform/) this equation,
-then it will become a convolution in the frequency domain
+If we take the [Fourier transform](/know/concept/fourier-transform/) (FT)
+$$\chi(t) \!\to\! \tilde{\chi}(\omega)$$ of this equation,
+the right-hand side becomes a convolution in the frequency domain
 thanks to the [convolution theorem](/know/concept/convolution-theorem/),
-where $$A$$, $$B$$ and $$s$$ are constants from the FT definition:
+where $$A$$, $$B$$ and $$s$$ are constants determined by
+how we choose to define our FT:
 
 $$\begin{aligned}
     \tilde{\chi}(\omega)
-    = (\tilde{\chi} * \tilde{\Theta})(\omega)
-    = B \int_{-\infty}^\infty \tilde{\chi}(\omega') \: \tilde{\Theta}(\omega - \omega') \dd{\omega'}
+    &= (\tilde{\chi} * \tilde{\Theta})(\omega)
+    \\
+    &= B \int_{-\infty}^\infty \tilde{\chi}(\omega') \: \tilde{\Theta}(\omega - \omega') \dd{\omega'}
 \end{aligned}$$
 
-We look up the FT of the step function $$\tilde{\Theta}(\omega)$$,
+We look up the full expression for $$\tilde{\Theta}(\omega)$$,
 which involves the signum function $$\mathrm{sgn}(t)$$,
 the [Dirac delta function](/know/concept/dirac-delta-function/) $$\delta$$,
-and the Cauchy principal value $$\pv{}$$.
-We arrive at:
+and the [Cauchy principal value](/know/concept/cauchy-principal-value/) $$\pv{}$$.
+Inserting that, we arrive at:
 
 $$\begin{aligned}
     \tilde{\chi}(\omega)
     &= \frac{A B}{|s|} \pv{\int_{-\infty}^\infty \tilde{\chi}(\omega')
-    \Big( \pi \delta(\omega - \omega') + i \:\mathrm{sgn} \frac{1}{\omega - \omega'} \Big) \dd{\omega'}}
+    \bigg( \pi \delta(\omega - \omega') + i \frac{\mathrm{sgn}(s)}{\omega - \omega'} \bigg) \dd{\omega'}}
     \\
-    &= \Big( \frac{1}{2} \frac{2 \pi A B}{|s|} \Big) \tilde{\chi}(\omega)
-    + i \Big( \frac{\mathrm{sgn}(s)}{2 \pi} \frac{2 \pi A B}{|s|} \Big)
+    &= \bigg( \frac{2}{2} \frac{\pi A B}{|s|} \bigg) \tilde{\chi}(\omega)
+    + i \: \mathrm{sgn}(s) \bigg( \frac{2 \pi}{2 \pi} \frac{A B}{|s|} \bigg)
     \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}(\omega')}{\omega - \omega'} \dd{\omega'}}
 \end{aligned}$$
 
-From the definition of the Fourier transform we know that
-$$2 \pi A B / |s| = 1$$:
+From the definition of the FT we know that
+$$2 \pi A B / |s| = 1$$, so this reduces to:
 
 $$\begin{aligned}
     \tilde{\chi}(\omega)
     &= \frac{1}{2} \tilde{\chi}(\omega)
-    + \mathrm{sgn}(s) \frac{i}{2 \pi} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}(\omega')}{\omega - \omega'} \dd{\omega'}}
+    + i \: \mathrm{sgn}(s) \frac{1}{2 \pi} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}(\omega')}{\omega - \omega'} \dd{\omega'}}
 \end{aligned}$$
 
-We isolate this equation for $$\tilde{\chi}(\omega)$$
-to get the final version of the causality test:
+We rearrange this equation a bit to get the final version of the causality test:
 
 $$\begin{aligned}
     \boxed{
         \tilde{\chi}(\omega)
-        = - \mathrm{sgn}(s) \frac{i}{\pi} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}(\omega')}{\omega - \omega'} \dd{\omega'}}
+        = i \: \mathrm{sgn}(s) \frac{1}{\pi}
+        \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}(\omega')}{\omega - \omega'} \dd{\omega'}}
     }
 \end{aligned}$$
 
-By inserting $$\tilde{\chi}(\omega) = \tilde{\chi}_r(\omega) + i \tilde{\chi}_i(\omega)$$
-and splitting the equation into real and imaginary parts,
-we get the Kramers-Kronig relations:
+Next, we split $$\tilde{\chi}(\omega)$$
+into its real and imaginary parts,
+i.e. $$\tilde{\chi}(\omega) = \tilde{\chi}_r(\omega) + i \tilde{\chi}_i(\omega)$$:
+
+$$\begin{aligned}
+    \tilde{\chi}_r(\omega) + i \tilde{\chi}_i(\omega)
+    = i \: \mathrm{sgn}(s) \frac{1}{\pi} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}_r(\omega')}{\omega - \omega'} \dd{\omega'}}
+    - \mathrm{sgn}(s) \frac{1}{\pi} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}_i(\omega')}{\omega - \omega'} \dd{\omega'}}
+\end{aligned}$$
+
+This equation can likewise be split into real and imaginary parts,
+leading to the **Kramers-Kronig relations**,
+which enable us to reconstruct $$\tilde{\chi}_r(\omega)$$
+from $$\tilde{\chi}_i(\omega)$$ and vice versa:
 
 $$\begin{aligned}
     \boxed{
         \begin{aligned}
             \tilde{\chi}_r(\omega)
-            &= \mathrm{sgn}(s) \frac{1}{\pi} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}_i(\omega')}{\omega' - \omega} \dd{\omega'}}
+            &= - \mathrm{sgn}(s) \frac{1}{\pi} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}_i(\omega')}{\omega - \omega'} \dd{\omega'}}
             \\
             \tilde{\chi}_i(\omega)
-            &= - \mathrm{sgn}(s) \frac{1}{\pi} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}_r(\omega')}{\omega' - \omega} \dd{\omega'}}
+            &= \mathrm{sgn}(s) \frac{1}{\pi} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}_r(\omega')}{\omega - \omega'} \dd{\omega'}}
         \end{aligned}
     }
 \end{aligned}$$
 
-If the time-domain response function $$\chi(t)$$ is real
-(so far we have assumed it to be complex),
-then we can take advantage of the fact that
-the FT of a real function satisfies
-$$\tilde{\chi}(-\omega) = \tilde{\chi}^*(\omega)$$, i.e. $$\tilde{\chi}_r(\omega)$$
-is even and $$\tilde{\chi}_i(\omega)$$ is odd. We multiply the fractions by
-$$(\omega' + \omega)$$ above and below:
+The sign of these expressions deserves special attention:
+it depends on an author's choice of FT definition via $$\mathrm{sgn}(s)$$,
+and, to make matters even more confusing,
+many also choose to use the opposite sign in the denominator,
+i.e. they write $$\omega' - \omega$$ instead of $$\omega - \omega'$$.
+
+In the special case where $$\chi(t)$$ is real,
+we can take advantage of the property that
+the FT of a real function always satisfies
+$$\tilde{\chi}(-\omega) = \tilde{\chi}^*(\omega)$$.
+Here, this means that $$\tilde{\chi}_r(\omega)$$ is even
+and $$\tilde{\chi}_i(\omega)$$ is odd.
+To use this fact, we simultaneously
+multiply and divide the integrands by $$\omega + \omega'$$:
 
 $$\begin{aligned}
     \tilde{\chi}_r(\omega)
-    &= \mathrm{sgn}(s) \bigg( \frac{1}{\pi} \pv{\int_{-\infty}^\infty \frac{\omega' \tilde{\chi}_i(\omega')}{ {\omega'}^2 - \omega^2} \dd{\omega'}}
-    + \frac{\omega}{\pi} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}_i(\omega')}{ {\omega'}^2 - \omega^2} \dd{\omega'}} \bigg)
+    &= - \mathrm{sgn}(s) \frac{1}{\pi}
+    \bigg( \!\pv{\int_{-\infty}^\infty \frac{\omega \tilde{\chi}_i(\omega')}{\omega^2 - {\omega'}^2} \dd{\omega'}}
+    + \pv{\int_{-\infty}^\infty \frac{\omega' \tilde{\chi}_i(\omega')}{\omega^2 - {\omega'}^2} \dd{\omega'}} \bigg)
     \\
     \tilde{\chi}_i(\omega)
-    &= - \mathrm{sgn}(s) \bigg( \frac{1}{\pi} \pv{\int_{-\infty}^\infty \frac{\omega' \tilde{\chi}_r(\omega')}{ {\omega'}^2 - \omega^2} \dd{\omega'}}
-    + \frac{\omega}{\pi} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}_r(\omega')}{ {\omega'}^2 - \omega^2} \dd{\omega'}} \bigg)
+    &= \mathrm{sgn}(s) \frac{1}{\pi}
+    \bigg( \!\pv{\int_{-\infty}^\infty \frac{\omega \tilde{\chi}_r(\omega')}{\omega^2 - {\omega'}^2} \dd{\omega'}}
+    + \pv{\int_{-\infty}^\infty \frac{\omega' \tilde{\chi}_r(\omega')}{\omega^2 - {\omega'}^2} \dd{\omega'}} \bigg)
 \end{aligned}$$
 
-For $$\tilde{\chi}_r(\omega)$$, the second integrand is odd, so we can drop it.
-Similarly, for $$\tilde{\chi}_i(\omega)$$, the first integrand is odd.
-We therefore find the following variant of the Kramers-Kronig relations:
+In $$\tilde{\chi}_r(\omega)$$'s equation, the first integrand is odd,
+so the integral's value is zero.
+Similarly, for $$\tilde{\chi}_i(\omega)$$, the second integrand is odd, so we drop it too.
+We thus arrive at the following common variant of the Kramers-Kronig relations,
+only valid for real $$\chi(t)$$:
 
 $$\begin{aligned}
     \boxed{
         \begin{aligned}
             \tilde{\chi}_r(\omega)
-            &= \mathrm{sgn}(s) \frac{2}{\pi} \pv{\int_0^\infty \frac{\omega' \tilde{\chi}_i(\omega')}{ {\omega'}^2 - \omega^2} \dd{\omega'}}
+            &= - \mathrm{sgn}(s) \frac{2}{\pi}
+            \pv{\int_0^\infty \frac{\omega' \tilde{\chi}_i(\omega')}{\omega^2 - {\omega'}^2} \dd{\omega'}}
             \\
             \tilde{\chi}_i(\omega)
-            &= - \mathrm{sgn}(s) \frac{2 \omega}{\pi} \pv{\int_0^\infty \frac{\tilde{\chi}_r(\omega')}{ {\omega'}^2 - \omega^2} \dd{\omega'}}
+            &= \mathrm{sgn}(s) \frac{2}{\pi}
+            \pv{\int_0^\infty \frac{\omega \tilde{\chi}_r(\omega')}{\omega^2 - {\omega'}^2} \dd{\omega'}}
         \end{aligned}
     }
 \end{aligned}$$
 
-To reiterate: this version is only valid if $$\chi(t)$$ is real in the time domain.
+Note that we have modified the integration limits
+using the fact that the integrands are even,
+leading to an extra factor of $$2$$.
 
 
 
diff --git a/source/know/concept/lagrange-multiplier/index.md b/source/know/concept/lagrange-multiplier/index.md
index 6b5e3fc..4c2e957 100644
--- a/source/know/concept/lagrange-multiplier/index.md
+++ b/source/know/concept/lagrange-multiplier/index.md
@@ -117,7 +117,7 @@ We often assign $$\lambda$$ an algebraic expression rather than a value,
 usually without even bothering to calculate its final actual value.
 In fact, in some cases, $$\lambda$$'s only function is to help us reason
 about the interdependence of a system of equations
-(see [example 3](https://en.wikipedia.org/wiki/Lagrange_multiplier#Example_3:_Entropy) on Wikipedia);
+(see Wikipedia's [entropy example](https://en.wikipedia.org/wiki/Lagrange_multiplier#Examples));
 then $$\lambda$$ is not even given an expression!
 Hence it is sometimes also called an *undetermined multiplier*.
 
diff --git a/source/know/concept/laser-rate-equations/index.md b/source/know/concept/laser-rate-equations/index.md
index c81f02b..feec168 100644
--- a/source/know/concept/laser-rate-equations/index.md
+++ b/source/know/concept/laser-rate-equations/index.md
@@ -30,7 +30,7 @@ $$\begin{aligned}
 
 Where $$n$$ is the background medium's refractive index,
 $$\omega_0$$ the two-level system's gap resonance frequency,
-$$|g| \equiv |\matrixel{e}{\vu{x}}{g}|$$ the transition dipole moment,
+$$|g| \equiv |\!\matrixel{e}{\vu{x}}{g}\!|$$ the transition dipole moment,
 $$\gamma_\perp$$ and $$\gamma_\parallel$$ empirical decay rates,
 and $$D_0$$ the equilibrium inversion.
 Note that $$\vb{E}^{-} = (\vb{E}^{+})^*$$.
@@ -110,7 +110,7 @@ $$\begin{aligned}
 
 Where the Lorentzian gain curve $$\gamma(\omega)$$
 (which also appears in the [SALT equation](/know/concept/salt-equation/))
-represents a laser's preferred spectrum for amplification,
+represents the laser's preferred spectrum for amplification,
 and is defined like so:
 
 $$\begin{aligned}
@@ -139,7 +139,7 @@ $$\begin{aligned}
 
 Next, we insert our ansatz for $$\vb{E}^{+}$$ and $$\vb{P}^{+}$$
 into the third MBE, and rewrite $$\vb{P}_0^{+}$$ as above.
-Using our identity for $$\gamma(\omega)$$,
+Using the aforementioned identity for $$\gamma(\omega)$$
 and the fact that $$\vb{E}_0^{+} \cdot \vb{E}_0^{-} = |\vb{E}|^2$$, we find:
 
 $$\begin{aligned}
@@ -218,8 +218,8 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Where $$\gamma_e$$ is a redefinition of $$\gamma_\parallel$$
-depending on the electron decay processes,
-and the photon loss rate $$\gamma_p$$, the gain $$G$$,
+depending on the electron decay processes.
+The photon loss rate $$\gamma_p$$, the gain $$G$$,
 and the carrier supply rate $$R_\mathrm{pump}$$
 are defined like so:
 
diff --git a/source/know/concept/lyddane-sachs-teller-relation/index.md b/source/know/concept/lyddane-sachs-teller-relation/index.md
index e80bf00..9cec9dc 100644
--- a/source/know/concept/lyddane-sachs-teller-relation/index.md
+++ b/source/know/concept/lyddane-sachs-teller-relation/index.md
@@ -219,8 +219,8 @@ i.e. the material becomes a perfect reflector:
 
 $$\begin{aligned}
     R
-    = \bigg| \frac{i \sqrt{|\varepsilon_r|} - 1}{i \sqrt{|\varepsilon_r|} + 1} \bigg|^2
-    = \frac{|\varepsilon_r|^2 + 1^2}{|\varepsilon_r|^2 + 1^2}
+    = \bigg| \frac{i \sqrt{-\varepsilon_r} - 1}{i \sqrt{-\varepsilon_r} + 1} \bigg|^2
+    = \frac{\varepsilon_r^2 + 1^2}{\varepsilon_r^2 + 1^2}
     = 1
 \end{aligned}$$
 
diff --git a/source/know/concept/magnetohydrodynamics/index.md b/source/know/concept/magnetohydrodynamics/index.md
index bcc23f3..4431dfa 100644
--- a/source/know/concept/magnetohydrodynamics/index.md
+++ b/source/know/concept/magnetohydrodynamics/index.md
@@ -24,24 +24,23 @@ and electric current density $$\vb{J}$$ are:
 
 $$\begin{aligned}
     p
-    = p_i + p_e
-    \qquad \quad
+    &= p_i + p_e
+    \\
     \vb{J}
-    = q_i n_i \vb{u}_i + q_e n_e \vb{u}_e
+    &= q_i n_i \vb{u}_i + q_e n_e \vb{u}_e
 \end{aligned}$$
 
 Meanwhile, the macroscopic mass density $$\rho$$
-and center-of-mass flow velocity $$\vb{u}$$
-are as follows, although the ions dominate due to their large mass:
+and center-of-mass flow velocity $$\vb{u}$$ are as follows,
+although the ions dominate both due to their large mass,
+so $$\rho \approx m_i n_i$$ and $$\vb{u} \approx \vb{u}_i$$:
 
 $$\begin{aligned}
     \rho
-    = m_i n_i + m_e n_e
-    \approx m_i n_i
-    \qquad \quad
+    &= m_i n_i + m_e n_e
+    \\
     \vb{u}
-    = \frac{1}{\rho} \Big( m_i n_i \vb{u}_i + m_e n_e \vb{u}_e \Big)
-    \approx \vb{u}_i
+    &= \frac{1}{\rho} \Big( m_i n_i \vb{u}_i + m_e n_e \vb{u}_e \Big)
 \end{aligned}$$
 
 With these quantities in mind,
@@ -75,9 +74,9 @@ $$\begin{aligned}
 \end{aligned}$$
 
 We will assume that electrons' inertia
-is negligible compared to the [Lorentz force](/know/concept/lorentz-force/).
-Let $$\tau_\mathrm{char}$$ be the characteristic timescale of the plasma's dynamics,
-i.e. nothing noticable happens in times shorter than $$\tau_\mathrm{char}$$,
+is negligible compared to the Lorentz force.
+Let $$\tau_\mathrm{char}$$ be the characteristic timescale of the plasma's dynamics
+(i.e. nothing notable happens in times shorter than $$\tau_\mathrm{char}$$),
 then this assumption can be written as:
 
 $$\begin{aligned}
@@ -86,15 +85,14 @@ $$\begin{aligned}
     \sim \frac{m_e n_e |\vb{u}_e| / \tau_\mathrm{char}}{q_e n_e |\vb{u}_e| |\vb{B}|}
     = \frac{m_e}{q_e |\vb{B}| \tau_\mathrm{char}}
     = \frac{1}{\omega_{ce} \tau_\mathrm{char}}
-    \ll 1
 \end{aligned}$$
 
-Where we have recognized the cyclotron frequency $$\omega_c$$ (see Lorentz force article).
+Where we have recognized the cyclotron frequency $$\omega_c$$
+(see [Lorentz force](/know/concept/lorentz-force/)).
 In other words, our assumption is equivalent to
 the electron gyration period $$2 \pi / \omega_{ce}$$
-being small compared to the macroscopic dynamics' timescale $$\tau_\mathrm{char}$$.
-By construction, we can thus ignore the left-hand side
-of the electron momentum equation, leaving:
+being small compared to the macroscopic timescale $$\tau_\mathrm{char}$$.
+We can thus ignore the left-hand side of the electron momentum equation, leaving:
 
 $$\begin{aligned}
     m_i n_i \frac{\mathrm{D} \vb{u}_i}{\mathrm{D} t}
@@ -138,8 +136,8 @@ $$\begin{aligned}
 
 However, we found this by combining two equations into one,
 so some information was implicitly lost;
-we need a second momentum equation.
-Therefore, we return to the electrons' momentum equation,
+we need a second one to keep our system of equations complete.
+Therefore we return to the electrons' momentum equation,
 after a bit of rearranging:
 
 $$\begin{aligned}
@@ -154,14 +152,14 @@ so:
 $$\begin{aligned}
     \vb{E} + \vb{u}_e \cross \vb{B} - \frac{\nabla p_e}{q_e n_e}
     = \eta \vb{J}
-    \qquad \quad
+    \qquad \qquad
     \eta
     \equiv \frac{f_{ei} m_e}{n_e q_e^2}
 \end{aligned}$$
 
 Where $$\eta$$ is the electrical resistivity of the plasma,
 see [Spitzer resistivity](/know/concept/spitzer-resistivity/)
-for more information, and a rough estimate of this quantity for a plasma.
+for more information and a rough estimate of its value in a plasma.
 
 Now, using that $$\vb{u} \approx \vb{u}_i$$,
 we add $$(\vb{u} \!-\! \vb{u}_i) \cross \vb{B} \approx 0$$ to the equation,
@@ -183,34 +181,37 @@ $$\begin{aligned}
     - \nabla \cross \frac{\nabla p_e}{q_e n_e}
 \end{aligned}$$
 
-Where we have used Faraday's law.
+Where we have used [Faraday's law](/know/concept/maxwells-equations/).
 This is the **induction equation**,
 and is used to compute $$\vb{B}$$.
 The pressure term can be rewritten using the ideal gas law $$p_e = k_B T_e n_e$$:
 
 $$\begin{aligned}
     \nabla \cross \frac{\nabla p_e}{q_e n_e}
-    = \frac{k_B}{q_e} \nabla \cross \frac{\nabla (n_e T_e)}{n_e}
-    = \frac{k_B}{q_e} \nabla \cross \Big( \nabla T_e + T_e \frac{\nabla n_e}{n_e} \Big)
+    &= \frac{k_B}{q_e} \nabla \cross \frac{\nabla (n_e T_e)}{n_e}
+    \\
+    &= \frac{k_B}{q_e} \nabla \cross \Big( \nabla T_e + T_e \frac{\nabla n_e}{n_e} \Big)
 \end{aligned}$$
 
 The curl of a gradient is always zero,
 and we notice that $$\nabla n_e / n_e = \nabla\! \ln(n_e)$$.
-Then we use the vector identity $$\nabla \cross (f \nabla g) = \nabla f \cross \nabla g$$,
-leading to:
+Then we use the vector identity $$\nabla \cross (f \nabla g) = \nabla f \cross \nabla g$$ to get:
 
 $$\begin{aligned}
     \nabla \cross \frac{\nabla p_e}{q_e n_e}
-    = \frac{k_B}{q_e} \nabla \cross \big( T_e \: \nabla\! \ln(n_e) \big)
-    = \frac{k_B}{q_e} \big( \nabla T_e \cross \nabla\! \ln(n_e) \big)
-    = \frac{k_B}{q_e n_e} \big( \nabla T_e \cross \nabla n_e \big)
+    &= \frac{k_B}{q_e} \nabla \cross \big( T_e \: \nabla\! \ln(n_e) \big)
+    \\
+    &= \frac{k_B}{q_e} \big( \nabla T_e \cross \nabla\! \ln(n_e) \big)
+    \\
+    &= \frac{k_B}{q_e n_e} \big( \nabla T_e \cross \nabla n_e \big)
 \end{aligned}$$
 
 It is reasonable to assume that $$\nabla T_e$$ and $$\nabla n_e$$
 point in roughly the same direction,
 in which case the pressure term can be negle