master > master: updated comments and user input

README.md

@@ -1,12 +1,57 @@
 # Notebooks #
 
-This repository is for presenting purely supplementary material to research papers.
+This repository is for presenting supplementary material to research papers.
-These intended for the purposes of presentation only.
 
-All scripts have been written by the owner of this repository.
+## Scope ##
+
+- All scripts have been written by the owner of this repository for research conducted by the owner.
+- The code in this repository is _not_ intended to be part of research submissions, but rather **exclusively for presentation purposes**.
 
 ## Presentations ##
 
 See the [docs/*.ipynb](docs) files.
-To run these, users require python v3.10.x and the jupyter module.
+
-To install the package requirements, call `just build` or `python3 -m pip install -r requirements.txt`.
+| Paper/Research | Notebook |
+| :------------- | :------- |
+| _Dilations of commuting $C\_{0}$-semigroups with bounded generators and the von Neumann polynomial inequality_ | [examples_dilations.ipynb](docs/examples_dilations.ipynb) |
+
+**Note:** The Git provider _Gitea_ does not have a display format for ipynb-files.
+To properly view and run the notebooks, users require python v3.10.x and the jupyter module.
+To install the package requirements, call `just build`
+or `python3 -m pip install -r requirements.txt`.#
+
+## Usage ##
+
+To run the presentations:
+
+- Ensure you have a working copy of [**python 3.10.x**](https://www.python.org/downloads) on your system with rights to install packages.
+- Clone the repository.
+- Install the requirements. Navigate to the root of the repository and carry out:
+  ```bash
+  python3 -m pip install --disable-pip-version-check -r requirements.txt
+  # for Windows users:
+  py -3 -m pip install --disable-pip-version-check -r requirements.txt
+  ```
+- Start the desired notebooks using the [jupyter](https://jupyter.org) UI or run
+  ```bash
+  python3 -m jupyter notebook notebooks/xyz.ipynb
+  # for Windows users:
+  py -3 -m jupyter notebook notebooks/xyz.ipynb
+  ```
+  to run the notebook `docs/xyz.ipynb`.
+
+Alternatively we recommend the following:
+
+- Ensure you have a working copy of [**python 3.10.x**](https://www.python.org/downloads) on your system with rights to install packages.
+- Ensure you have **bash** (for Windows users see the [Git SCM](https://gitforwindows.org))
+  and the [**justfile**](https://github.com/casey/just#installation) tool
+  (Windows users may wish to install [Chocolatey](https://chocolatey.org/install) beforehand).
+- Clone the repository.
+- Navigate in a bash terminal to the root of the repository and make use of the following commands:
+  ```bash
+  # for setup:
+  just build
+  # to run the notebook docs/xyz.ipynb:
+  just run xyz
+  ```
+  (ensure that the extension is no included in the command!).

docs/examples_dilations.ipynb

@@ -22,7 +22,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 12,
+   "execution_count": null,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -30,60 +30,39 @@
     "import sys;\n",
     "\n",
     "# NOTE: need this to force jupyter to reload imports:\n",
-    "for key in list(sys.modules.keys()):\n",
+    "for key in filter(lambda key: key.startswith('src.'), list(sys.modules.keys())):\n",
-    "    if key.startswith('src.'):\n",
+    "    del sys.modules[key];\n",
-    "        del sys.modules[key];\n",
     "\n",
     "os.chdir(os.path.dirname(_dh[0]));\n",
     "sys.path.insert(0, os.getcwd());\n",
     "\n",
-    "from src.examples_dilations import *;"
+    "from src.examples_dilations import *;\n",
+    "np.random.seed(7098123); # for repeatability"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 13,
+   "execution_count": null,
    "metadata": {},
    "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",
+    "# force k-th order dissipation operators to be strictly postive for k < k0\n",
-    "# If you ensure that the failure of S_{T,K} >= 0 only occurs for K = {1,2,...,d}\n",
+    "# force k-th order dissipation operators to be non-positive for k >= k0\n",
-    "# + you want S_TK > 0 (strictly) for all K ≠ {1,2,...,d}:\n",
+    "# one can choose 2 <= k0 <= d\n",
-    "alpha = 1/math.sqrt(d - 0.5);\n",
+    "k0 = d;"
-    "\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;"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 14,
+   "execution_count": null,
    "metadata": {},
-   "outputs": [
+   "outputs": [],
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "\n",
-      "The marginal semigroups T_i and their generators A_i:\n",
-      "\n",
-      " i    spec bound of A_i    A_i dissipative (<==> T_i contractive)?\n",
-      "---  -------------------  -----------------------------------------\n",
-      " 1        -0.465478                         True\n",
-      " 2        -0.465478                         True\n",
-      " 3        -0.465478                         True\n",
-      " 4        -0.465478                         True\n"
-     ]
-    }
-   ],
    "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",
@@ -98,7 +77,7 @@
     "for i, A_i in enumerate(A):\n",
     "    omega_Re = spec_bounds((1/2)*(A_i + A_i.T.conj()));\n",
     "    omega = spec_bounds(A_i);\n",
-    "    data.append((i+1, omega_Re, True if (omega_Re <= 0) else False));\n",
+    "    data.append((i+1, omega, True if (omega_Re <= 0) else False));\n",
     "\n",
     "repr = tabulate(\n",
     "    tabular_data = data,\n",
@@ -113,47 +92,18 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 15,
+   "execution_count": null,
    "metadata": {},
-   "outputs": [
+   "outputs": [],
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "\n",
-      "Dissipation operators:\n",
-      "\n",
-      "     K         min σ(S_{T,K})    S_{T,K} >= 0?\n",
-      "------------  ----------------  ---------------\n",
-      "     []           1.000000           True\n",
-      "    [3]           0.465478           True\n",
-      "    [2]           0.465478           True\n",
-      "    [1]           0.465478           True\n",
-      "    [0]           0.465478           True\n",
-      "   [2, 3]         0.212675           True\n",
-      "   [1, 3]         0.183738           True\n",
-      "   [1, 2]         0.217453           True\n",
-      "   [0, 3]         0.200206           True\n",
-      "   [0, 2]         0.301332           True\n",
-      "   [0, 1]         0.215681           True\n",
-      " [1, 2, 3]        0.058427           True\n",
-      " [0, 2, 3]        0.075037           True\n",
-      " [0, 1, 3]        0.056030           True\n",
-      " [0, 1, 2]        0.077350           True\n",
-      "[0, 1, 2, 3]     -0.082403           False\n"
-     ]
-    }
-   ],
    "source": [
     "# compute the dissipation operators `S_TK` for each `K ⊆ {1,2,...,d}``:\n",
     "S, beta_T = dissipation_operators(shape=[N, N], A=A);\n",
     "\n",
-    "data = [];\n",
-    "for K, S_TK, b in sorted(S, key=lambda x: len(x[0])):\n",
-    "    data.append((K, b, True if b >= -MACHINE_EPS else False))\n",
-    "\n",
     "repr = tabulate(\n",
-    "    tabular_data = data,\n",
+    "    tabular_data = [\n",
+    "        (K, b, True if b >= -MACHINE_EPS else False)\n",
+    "        for K, S_TK, b in sorted(S, key=lambda x: (len(x[0]), x[0]) )\n",
+    "    ],\n",
     "    headers      = ['K', 'min σ(S_{T,K})', 'S_{T,K} >= 0?'],\n",
     "    showindex    = False,\n",
     "    floatfmt = '.6f',\n",
@@ -165,20 +115,9 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 16,
+   "execution_count": null,
    "metadata": {},
-   "outputs": [
+   "outputs": [],
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "\n",
-      "β_T = min_K min σ(S_{T,K}) = -0.082403\n",
-      "⟹ T is not compeletely dissipative.\n",
-      "⟹ (by Thm 1.1) T does not have a regular unitary dilation.\n"
-     ]
-    }
-   ],
    "source": [
     "# Display summary:\n",
     "print('')\n",

justfile

@@ -89,7 +89,7 @@ clean:
     @just clean-notebooks
 clean-notebooks:
     @echo "Clean python notebooks."
-    @#{{PYTHON}} -m jupyter nbconvert --clear-output --inplace docs/*.ipynb
+    @{{PYTHON}} -m jupyter nbconvert --clear-output --inplace docs/*.ipynb
     @- {{PYTHON}} -m jupytext --update-metadata '{"vscode":""}' docs/*.ipynb 2> /dev/null
     @- {{PYTHON}} -m jupytext --update-metadata '{"vscode":null}' docs/*.ipynb 2> /dev/null
 clean-basic:

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.