diff --git a/hydradx/spec/Swap.ipynb b/hydradx/spec/Swap.ipynb
index 136110ce..44e1b466 100644
--- a/hydradx/spec/Swap.ipynb
+++ b/hydradx/spec/Swap.ipynb
@@ -66,8 +66,10 @@
     "$$\n",
     "\\begin{align}\n",
     "\\tilde{\\Delta Q_j} &= Q_j\\frac{- \\Delta R_j}{R_j(1 - f_A) + \\Delta R_j}\\\\\n",
-    "\\tilde{\\Delta Q_i} &= -\\frac{\\tilde{\\Delta Q_j}}{1 - f_P}\\\\\n",
-    "\\Delta R_i &= R_i\\frac{- \\tilde{\\Delta Q_i}}{Q_i + \\tilde{\\Delta Q_i}}\n",
+    "\\tilde{\\Delta Q_i} &= -\\frac{\\tilde{\\Delta Q_j}}{1 - f_Q}\\\\\n",
+    "\\Delta R_i &= R_i\\frac{- \\tilde{\\Delta Q_i}}{Q_i + \\tilde{\\Delta Q_i}}\\\\\n",
+    "\\Delta Q_j^m &= f_A\\frac{(Q_j + \\tilde{\\Delta Q_j})\\tilde{\\Delta Q_j}}{Q_j}\\\\\n",
+    "\\Delta Q_j &= \\tilde{\\Delta Q_j} + \\Delta Q_j^m\n",
     "\\end{align}\n",
     "$$\n",
     "\n",
@@ -91,9 +93,11 @@
     "$$\n",
     "\\begin{align}\n",
     "\\Delta Q_i &= Q_i\\frac{- \\Delta R_i}{R_i + \\Delta R_i}\\\\\n",
-    "\\Delta Q_j' &= -\\Delta Q_i (1 - f_P)\\\\\n",
+    "\\Delta Q_j' &= -\\Delta Q_i (1 - f_Q)\\\\\n",
     "\\Delta R_j' &= R_j\\frac{- \\Delta Q_j'}{Q_j + \\Delta Q_j'}\\\\\n",
-    "\\Delta R_j &= \\Delta R_j'(1 - f_A)\n",
+    "\\Delta R_j &= \\Delta R_j'(1 - f_A)\\\\\n",
+    "\\Delta Q_j^m &= f_A \\frac{R_j \\Delta Q_j'}{R_j + \\Delta R_j'}\\\\\n",
+    "\\Delta Q_j &= \\Delta Q_j' + \\Delta Q_j^m\n",
     "\\end{align}\n",
     "$$"
    ]
@@ -111,52 +115,18 @@
    "id": "826aedb1-dcd5-4b35-8f81-7cc5ca6faad2",
    "metadata": {},
    "source": [
-    "The LRNA fee amount is $-\\Delta Q_i f_P$, and it is burned until $L$ is brought up to $0$, at which point fees are moved to the LRNA side of the HDX sub-pool in Omnipool.\n",
+    "The total LRNA fee amount is $-\\Delta Q_i f_Q$, where $f_Q$ is the LRNA fee. We will burn some of it, and the rest will be used to buy HDX which will be sent to treasury.\n",
+    "Our dynamic LRNA fee will be $f_Q$ while $f_B$ will be some fixed fee corresponding to the LRNA that gets burned. Note that we require $f_B \\leq f_Q$.\n",
     "$$\n",
     "\\begin{align}\n",
-    "\\Delta L &= min(-\\Delta Q_i f_P, -L)\\\\\n",
-    "\\Delta Q_H &= -\\Delta Q_i f_P - \\Delta L\n",
+    "\\Delta Q_B &= -\\Delta Q_i f_B\\\\\n",
+    "\\Delta Q_H &= -\\Delta Q_i f_Q - \\Delta Q_B\n",
     "\\end{align}\n",
     "$$\n",
-    "where $Q_H$ is the LRNA side of the HDX sub-pool.\n",
+    "where $\\Delta Q_B$ is the amount of LRNA that gets burned, and $\\Delta Q_H$ is the amount of LRNA that is used to buy HDX.\n",
     "\n"
    ]
   },
-  {
-   "cell_type": "markdown",
-   "id": "e4ce191c-569b-424f-b494-1ffefd0b2116",
-   "metadata": {},
-   "source": [
-    "## Direct calculation (updating pool state)\n",
-    "### $\\Delta R_i > 0$ provided\n",
-    "$$\n",
-    "\\begin{align}\n",
-    "\\Delta Q_i &= Q_i \\frac{-\\Delta R_i}{R_i + \\Delta R_i}\\\\\n",
-    "\\Delta Q_j &= -\\Delta Q_i(1 - f_P)\\\\\n",
-    "\\Delta R_j &= R_j \\frac{-\\Delta Q_j}{Q_j + \\Delta Q_j}(1 - f_A)\\\\\n",
-    "\\Delta L &= min(-\\Delta Q_i f_P, -L)\\\\\n",
-    "\\Delta Q_H &= -\\Delta Q_i f_P - \\Delta L\n",
-    "\\end{align}\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "549adee6-8c99-4801-be86-10c77b584467",
-   "metadata": {},
-   "source": [
-    "### $\\Delta R_j < 0$ provided\n",
-    "$$\n",
-    "\\begin{align}\n",
-    "\\Delta Q_j &= Q_j\\frac{-\\Delta R_j}{R_j(1 - f_A) + \\Delta R_j}\\\\\n",
-    "\\Delta Q_i &= -\\frac{\\Delta Q_j}{1 - f_P}\\\\\n",
-    "\\Delta R_i &= R_i\\frac{-\\Delta Q_i}{Q_i + \\Delta Q_i}\\\\\n",
-    "\\Delta L &= min(-\\Delta Q_i f_P, -L)\\\\\n",
-    "\\Delta Q_H &= -\\Delta Q_i f_P - \\Delta L\n",
-    "\\end{align}\n",
-    "$$"
-   ]
-  },
   {
    "cell_type": "markdown",
    "id": "7ac04084-09e9-4e27-bdea-d4deed673ce3",
@@ -172,8 +142,10 @@
    "source": [
     "We have\n",
     "$$\n",
-    "\\Delta r^\\alpha_i = -\\Delta R_i\\\\\n",
-    "\\Delta r^\\alpha_j = -\\Delta R_j\n",
+    "\\begin{align}\n",
+    "\\Delta r^\\alpha_i &= -\\Delta R_i\\\\\n",
+    "\\Delta r^\\alpha_j &= -\\Delta R_j\n",
+    "\\end{align}\n",
     "$$"
    ]
   },
diff --git a/hydradx/spec/SwapLRNA.ipynb b/hydradx/spec/SwapLRNA.ipynb
index 6dc0b6b3..e8565966 100644
--- a/hydradx/spec/SwapLRNA.ipynb
+++ b/hydradx/spec/SwapLRNA.ipynb
@@ -78,9 +78,19 @@
    "source": [
     "### Case 1: LRNA sold, $\\Delta q^\\alpha < 0$ specified\n",
     "\n",
-    "If $-\\Delta q^\\alpha > q^\\alpha$, the user does not have enough LRNA to sell, and the transaction must fail."
+    "If $-\\Delta q^\\alpha > q^\\alpha$, the user does not have enough LRNA to sell, and the transaction must fail.\n",
+    "We first calculate the feeless amount $\\Delta R_i'$. We next calculate the amount of LRNA we will mint to match the asset fee $\\Delta Q_m$.\n",
+    "The total amount of asset $i$ coming out is $\\Delta R_i$, while the total change in LRNA in the pool (incorporating both the LRNA sold to the pool and the LRNA minted) is $\\Delta Q_i$."
    ]
   },
+  {
+   "metadata": {},
+   "cell_type": "code",
+   "outputs": [],
+   "execution_count": null,
+   "source": "",
+   "id": "653a53e961b30275"
+  },
   {
    "cell_type": "markdown",
    "id": "4554c820-71a3-4981-92ea-3442640e4537",
@@ -88,9 +98,10 @@
    "source": [
     "$$\n",
     "\\begin{align}\n",
-    "\\Delta Q_i &= -\\Delta q^\\alpha\\\\\n",
-    "\\Delta R_i &= R_i\\frac{- \\Delta Q_i}{Q_i + \\Delta Q_i}(1 - f_A)\\\\\n",
-    "\\Delta L &= -\\frac{\\Delta Q_i (Q + L)}{Q + \\Delta Q_i} - \\Delta Q_i\\\\\n",
+    "\\Delta R_i' &= R_i\\frac{\\Delta q^\\alpha}{Q_i -\\Delta q^\\alpha}\\\\\n",
+    "\\Delta Q_m &= -\\frac{f_A \\Delta q^\\alpha(Q_i -\\Delta q^\\alpha)}{Q_i}\\\\\n",
+    "\\Delta R_i &= \\Delta R_i'(1-f_A)\\\\\n",
+    "\\Delta Q_i &= -\\Delta q^\\alpha + \\Delta Q_m\\\\\n",
     "\\Delta r_i^\\alpha &= - \\Delta R_i\\\\\n",
     "\\end{align}\n",
     "$$"
@@ -132,9 +143,9 @@
     "$$\n",
     "\\begin{align}\n",
     "\\Delta R_i &= -\\Delta r_i^\\alpha\\\\\n",
-    "\\Delta Q_i &= Q_i\\frac{-\\Delta R_i}{R_i(1 - f_A) + \\Delta R_i}\\\\\n",
-    "\\Delta L &= -\\frac{\\Delta Q_i (Q + L)}{Q + \\Delta Q_i} - \\Delta Q_i\\\\\n",
-    "\\Delta q^\\alpha &= - \\Delta Q_i\\\\\n",
+    "\\Delta q^\\alpha &= Q_i\\frac{\\Delta R_i}{R_i(1 - f_A) + \\Delta R_i}\\\\\n",
+    "\\Delta Q_m &= -\\frac{f_A(Q_i -\\Delta q^\\alpha)\\Delta R_i}{R_i(1 - f_A) + \\Delta R_i}\\\\\n",
+    "\\Delta Q_i &= -\\Delta q^\\alpha + \\Delta Q_m\\\\\n",
     "\\end{align}\n",
     "$$\n",
     "\n",
@@ -199,7 +210,8 @@
    "source": [],
    "metadata": {
     "collapsed": false
-   }
+   },
+   "id": "c8ec5d4b56e4a375"
   }
  ],
  "metadata": {