master > master: updated comments and user input

docs/examples_dilations.ipynb

@@ -30,9 +30,8 @@
     "import sys;\n",
     "\n",
     "# NOTE: need this to force jupyter to reload imports:\n",
-    "for key in list(sys.modules.keys()):\n",
-    "    if key.startswith('src.'):\n",
-    "        del sys.modules[key];\n",
+    "for key in filter(lambda key: key.startswith('src.'), list(sys.modules.keys())):\n",
+    "    del sys.modules[key];\n",
     "\n",
     "os.chdir(os.path.dirname(_dh[0]));\n",
     "sys.path.insert(0, os.getcwd());\n",
@@ -48,18 +47,12 @@
    "outputs": [],
    "source": [
     "# User input:\n",
-    "N = 4; # dimension of the Hilbert space.\n",
+    "N = 6; # dimension of the Hilbert space; must be divisible by 2 for this construction.\n",
     "d = 4;\n",
-    "\n",
-    "# If you ensure that the failure of S_{T,K} >= 0 only occurs for K = {1,2,...,d}\n",
-    "# + you want S_TK > 0 (strictly) for all K ≠ {1,2,...,d}:\n",
-    "alpha = 1/math.sqrt(d - 0.5);\n",
-    "\n",
-    "# If you ensure that the failure of S_{T,K} >= 0 only occurs for K = {1,2,...,d}\n",
-    "# alpha = 1/math.sqrt(d - 1);\n",
-    "\n",
-    "# Otherwise:\n",
-    "# alpha = 1;"
+    "# force k-th order dissipation operators to be strictly postive for k < k0\n",
+    "# force k-th order dissipation operators to be non-positive for k >= k0\n",
+    "# one can choose 2 <= k0 <= d\n",
+    "k0 = d;"
    ]
   },
   {
@@ -69,6 +62,7 @@
    "outputs": [],
    "source": [
     "# create the generators `A_i` of the marginal semigroups `T_i`:`\n",
+    "alpha = 1/math.sqrt(k0 - 0.5);\n",
     "A = [\n",
     "    generate_semigroup_generator(\n",
     "        shape    = [N, N],\n",

src/examples_dilations.py

@@ -46,7 +46,7 @@ def generate_semigroup_generator(
     - `base`     - <int>      If `rational = True`, fixes the denominator of the rational numbers.
     - `alpha`    - <float>    Additional parameter to scale the D_i operators.
 
-    NOTE: in the paper `α = 1` was chosen. However one can choose any value in `(1/√d, \infty)`.
+    NOTE: One can choose any value of `α ∈ (1/√d, \infty)`.
     By choosing any value `α ≥ 1/√(d-1)`, by the computations in Proposition 5.3
     one can force that the S_TK operators only fail to be positive when |K| > d-1.