Mathematicians constantly need to learn and read about concepts with
which they are unfamiliar. Keeping mathematical notes in an
Obsidian.md vault can help with this learning
process as Obsidian.md makes it easy to edit notes, link notes to one
another, organize notes, read notes, and access notes.
This library currently includes functionalities to 1. Parse LaTeX
documents (e.g. those available on arxiv.org)
and divide them into reasonably lengthed parts/notes/excerpts as .md
files, especially for viewing and editing on Obsidian.md. 2. Use a
machine learning model to categorize the type of text each excerpt is
(e.g. introducing definitions/notations, presenting a concept,
presenting a proof). 3. Use a machine learning model to identify
notations introduced in each excerpt. 4. Create accompanying notes for
each notation as more .md files. 5. Use a machine learning model to
summarize what these notations denote in the created accompanying notes.
As some of these functionalities use machine learning models, they ultimately cannot run perfectly. Nevertheless, some of these models, particularly those described in 2 and 3, perform well enough that these functionalities are quite useful as tools to help reading mathematical papers.
Moreover, the results of the machine learning models are recorded in the
notes/.md files. One can very well correct these recorded results by
manually editing the affected .md files with any file editor.
At the time of this writing (04/03/2023), there is only one author/contributor of this library. Nevertheless, the author often refers to himself as the author, the authors, or the author/authors in writing this library. Moreover, the author often uses the editorial we in writing this library.
Use this library at your own risk as using this library can write or modify files in your computer and as the documentation of some components of this library may be inaccurate or outdated. Moreover, there are components of this library which use external API’s (including but possibly not limited to OpenAI’s API) that may not be free to use. By using this library, you agree that the author/authors of this library is/are not responsible for any damages or losses, including but not limited to data or monetary losses, from this library and related components.
This library is still somewhere in-between prototype and alpha. Moreover, the library itself may be unstable and subject to abrupt changes.
The author/authors of this library is/are also not affiliated with
Obsidian.md, fast.ai, Hugging Face, and OpenAI.
We recommend having at least 3GB of free space to install trouver and
related components.
pip install trouverYou may also have to manually install other libraries which are required
by the fast.ai and/or Hugging Face libraries.
We recommend installing Obsidian.md to view,
edit, and modify mathematical notes created by or which interact with
trouver.
See how_to.install_trouver for more details on installation.
Warning At the time of this writing,
trouverhas not been tested on MacOS extensively. We have also found that running the latest Python versions in Jupyter notebooks on modern MacOS machines (e.g. those using the M1 processor and in particular the arm64 architecture) lead to some issues. cf. stackexchange discussions such as- https://apple.stackexchange.com/questions/436801/m1-mac-mach-o-file-but-is-an-incompatible-architecture-have-x86-64-need-a - https://stackoverflow.com/questions/71502583/jupiter-wont-launch-from-terminal-on-m1-macbook.
In the following section, we present some of trouver’s functionalities
with some sample code. To run code, make sure to copy the contents of
the nbs/_tests folder of the trouver GitHub repository into the
working directory (as a folder names _tests).
See also tutorial.walkthrough for a tutorial to set up a basic
trouver workflow.
Trouver can parse LaTeX documents and split them up into parts which
are convenient to read in Obsidian.md and to take notes on. For
example, the following code splits up this
paper in creates a folder in an
Obsidian.md vault[^4].
import os
from pathlib import Path
import shutil
import tempfile
from trouver.helper import _test_directory, text_from_file
from trouver.latex.convert import (
divide_preamble, divide_latex_text, custom_commands,
setup_reference_from_latex_parts
)# This context manager is implemented to make sure that a temporary
# folder is created and copies contents from `test_vault_5` in `nbs/_tests`,
# only the contents of the temporary folder are modified, and
with (tempfile.TemporaryDirectory(prefix='temp_dir', dir=os.getcwd()) as temp_dir):
temp_vault = Path(temp_dir) / 'test_vault_5'
shutil.copytree(_test_directory() / 'test_vault_5', temp_vault)
sample_latex_file = _test_directory() / 'latex_examples' / 'kim_park_ga1dcmmc' / 'main.tex'
sample_latex_text = text_from_file(sample_latex_file)
preamble, _ = divide_preamble(sample_latex_text)
parts = divide_latex_text(sample_latex_text)
cust_comms = custom_commands(preamble)
vault = temp_vault
location = Path('') # The path relative to the vault of the directory in which to make the new folder containing the new notes.
reference_name = 'kim_park_ga1dcmmc'
author_names = ['Kim', 'Park']
setup_reference_from_latex_parts(
parts, cust_comms, vault, location,
reference_name,
author_names)
# os.startfile(os.getcwd()) # This open the current working directory; find the temporary folder in here.
# input() # There should be an input prompt; make an input here when you are done viewing the
While Obsidian.md is not strictly necessary to use trouver or to
read and write the files created by
setup_reference_from_latex_parts
(in fact, any traditional file reader/writer can be used for such
purposes), reading and writing the files on Obsidian.md can be
convenient. Moreover, even when you use Obsidian, your data is in a
local folder. In particular, even if Obsidian.md happens to get shut
down, get bought, or change privacy policy, you will (supposedly) not
lose access to your data.
We have trained a few ML models to detect/predict and provide
information about short (or at least not-too-long) mathematical text.
These ML models are available on
Hugging Face and as such, they can be
downloaded to and used from one’s local machines. Please note that ML
models can be large and the locations that the Hugging Face
Transformers library
downloads such models to can vary from machine to machine.
For each of these models, we may or may not have also written some instructions on how to train similar models given appropriately formatted data1.
Note that the data used to train these models contains mathematical text pertaining mostly to fields closely related to number theory and algebraic geometry.
One of these ML models predicts the type of a piece of mathematical writing. For example, this model may predict that
Let $L/K$ be an field extension. An element $\alpha \in L$ is said to be algebraic over $K$ if there exists some polynomial $f(x) \in K[x]$ such that $f(\alpha) = 0$.introduces a definition. For the purposes of trouver, an Obsidian.md
note containing ought to be labeled with the #_meta/definition tag by
adding the text _meta/definition to the tags field in the
frontmatter YAML metadata of the note:
See markdown.obsidian.personal.machine_learning.information_note_types
for more details.
This ML model is trained using the fast.ai
library with the ULMFiT
approach;
see how_to.train_ml_model.fastai for the steps taken to train this
model. This ML model is also available on Hugging
Face under the repository
hyunjongkimmath/information_note_type
The following code downloads the model into the local Hugging Face cache (if necessary) and loads the model.
import pathlib
from pathlib import WindowsPath
import platform
from huggingface_hub import from_pretrained_fastairepo_id = 'hyunjongkimmath/information_note_type'
# There is a PosixPath problem when trying to load
# the model on Windows; we get around this problem
# within the `if` statement.
if platform.system() == 'Windows':
temp = pathlib.PosixPath # See https://stackoverflow.com/questions/57286486/i-cant-load-my-model-because-i-cant-put-a-posixpath
pathlib.PosixPath = pathlib.WindowsPath
information_note_type_model = from_pretrained_fastai(repo_id)
pathlib.PosixPath = temp
else:
information_note_type_model = from_pretrained_fastai(repo_id)Fetching 4 files: 0%| | 0/4 [00:00<?, ?it/s]
sample_prediction_1 = information_note_type_model.predict(r'Let $L/K$ be an field extension. An element $\alpha \in L$ is said to be algebraic over $K$ if there exists some polynomial $f(x) \in K[x]$ such that $f(\alpha) = 0$.')
print(sample_prediction_1)
sample_prediction_2 = information_note_type_model.predict(r'Theorem. Let $q$ be a prime power. Up to isomorphism, there is exactly one field with $q$ elements.')
print(sample_prediction_2)(['#_meta/definition', '#_meta/notation'], tensor([False, False, False, False, False, False, True, False, False, False,
True, False, False, False]), tensor([1.9631e-03, 3.4931e-04, 1.7551e-02, 4.8163e-02, 5.7628e-06, 3.0610e-06,
9.6544e-01, 2.3179e-03, 2.4539e-03, 1.6170e-02, 5.8807e-01, 4.5185e-03,
2.5055e-04, 4.6183e-03]))
(['#_meta/concept', '#_meta/proof'], tensor([False, False, False, True, False, False, False, False, False, False,
False, True, False, False]), tensor([3.4701e-03, 6.6588e-05, 7.8861e-02, 9.7205e-01, 8.8357e-06, 6.1183e-06,
9.5552e-02, 4.0747e-03, 2.7043e-04, 2.7545e-02, 1.3064e-02, 5.6198e-01,
1.5603e-04, 5.5122e-03]))
At the time of this writing (01/18/2023), the model seems to incorrect
predict - in sample_prediction_1 that the text introduces a
notation. - in sample_prediction_2 that the text contains a proof.
# from trouver.markdown.obsidian.personal.machine_learning.information_note_types importWhile one can make use of the model’s predict method as is, trouver
also provides functions which predict the types of mathematical text
written in notes formatted in a specific way and record on these notes
the predictions made. This way, one can make the model predict once and
use these predictions for later, which can save computational resources.
from trouver.markdown.obsidian.vault import VaultNote
from trouver.markdown.obsidian.personal.notes import notes_linked_in_notes_linked_in_note
from trouver.markdown.obsidian.personal.machine_learning.information_note_types import automatically_add_note_type_tagswith (tempfile.TemporaryDirectory(prefix='temp_dir', dir=os.getcwd()) as temp_dir):
temp_vault = Path(temp_dir) / 'test_vault_8'
shutil.copytree(_test_directory() / 'test_vault_8', temp_vault)
reference = 'number_theory_reference_1'
index_note = VaultNote(temp_vault, name=f'_index_{reference}')
# `notes` below is a list of `VaultNote` objects.
# Also, the `notes_linked_in_note` function can be a useful
# alternative to the `notes_linked_in_notes_linked_in_note` function.
notes = notes_linked_in_notes_linked_in_note(index_note, as_dict=False)
print("This is what one of the notes looks like before predicting its note type:\n\n")
print(notes[0].text())
print("\n\nTagging notes\n\n")
# Note that `information_note_type_model` was loaded previously.
automatically_add_note_type_tags(information_note_type_model, temp_vault, notes)
print("This is what the same note looks like after predicting its note type:\n\n")
print(notes[0].text())
# os.startfile(os.getcwd()) # This opens the current working directory; find the temporary folder in here and explore it if desired.
# input() # There should be an input prompt; make an input here when you are done viewing theThis is what one of the notes looks like before predicting its note type:
---
cssclass: clean-embeds
aliases: [number_theory_reference_1_ring]
tags: [_meta/literature_note, _reference/number_theory_reference_1]
---
# Ring[^1]
A **(commutative) ring** is a set $R$, equipped with two binary operators, denoted $+$ and $\cdot$, such that the following hold:
1. $R$ is an abelian group under $+$ with identity element $0$.
2. $R$ is an commutative monoid under $\cdot$ with identity element $1$.
3. For all $a,b,c \in R$, we have $a \cdot (b+c) = a \cdot b + a \cdot c$.
# See Also
# Meta
## References
![[_reference_number_theory_reference_1]]
## Citations and Footnotes
[^1]: Kim, Definition 1.1, Page 1
Tagging notes
This is what the same note looks like after predicting its note type:
---
cssclass: clean-embeds
aliases: [number_theory_reference_1_ring]
tags: [_reference/number_theory_reference_1, _meta/literature_note, _auto/_meta/definition]
---
# Ring[^1]
A **(commutative) ring** is a set $R$, equipped with two binary operators, denoted $+$ and $\cdot$, such that the following hold:
1. $R$ is an abelian group under $+$ with identity element $0$.
2. $R$ is an commutative monoid under $\cdot$ with identity element $1$.
3. For all $a,b,c \in R$, we have $a \cdot (b+c) = a \cdot b + a \cdot c$.
# See Also
# Meta
## References
![[_reference_number_theory_reference_1]]
## Citations and Footnotes
[^1]: Kim, Definition 1.1, Page 1
Another ML model predicts locations of notations introduced in text.
This model is trained as a categorizer - given a piece of mathematical
text in LaTeX in which a single LaTeX math mode string (surrounded
either by the dollar sign $ or double dollar signs $$) is surrounded
by double asterisks **, the model should determine whether or not the
LaTeX math mode string contains a newly introduced notation.
For example, suppose that we want to find notations introduced in the following text:
Let $L/K$ be a Galois field extension. Its Galois group $\operatorname{Gal}(L/K)$ is defined as the group of automorphisms of $L$ fixing $K$ pointwise.Our approach is to consider each latex math mode strings in this text
(of which there are 4: ** are surround one of these math mode strings, and use the
model to predict whether that math mode string contains a newly
introduced notation. In particular, we pass through the model the
following pieces of text:
Let **$L/K$** be a Galois field extension. Its Galois group $\operatorname{Gal}(L/K)$ is defined as the group of automorphisms of $L$ fixing $K$ pointwise.Let $L/K$ be a Galois field extension. Its Galois group **$\operatorname{Gal}(L/K)$** is defined as the group of automorphisms of $L$ fixing $K$ pointwise.Let $L/K$ be a Galois field extension. Its Galois group $\operatorname{Gal}(L/K)$ is defined as the group of automorphisms of **$L$** fixing $K$ pointwise.Let $L/K$ be a Galois field extension. Its Galois group $\operatorname{Gal}(L/K)$ is defined as the group of automorphisms of $L$ fixing **$K$** pointwise.Ideally, the model should determine only the second version of text to contain a newly introduced notation
See markdown.obsidian.personal.machine_learning.notation_identifcation
for more details.
This ML model is also trained using the fast.ai library with the
ULMFiT
approach,
and is available on Hugging Face under the repository
hyunjongkimmath/notation_identification.
import pathlib
from pathlib import WindowsPath
import platform
from huggingface_hub import from_pretrained_fastairepo_id = 'hyunjongkimmath/notation_identification'
# There is a PosixPath problem when trying to load
# the model on Windows; we get around this problem
# within the `if` statement.
if platform.system() == 'Windows':
temp = pathlib.PosixPath # See https://stackoverflow.com/questions/57286486/i-cant-load-my-model-because-i-cant-put-a-posixpath
pathlib.PosixPath = pathlib.WindowsPath
notation_identification_model = from_pretrained_fastai(repo_id)
pathlib.PosixPath = temp
else:
notation_identification_model = from_pretrained_fastai(repo_id)Fetching 4 files: 0%| | 0/4 [00:00<?, ?it/s]
contains_a_notation = notation_identification_model.predict(r'Let $L/K$ be a Galois field extension. Its Galois group **$\operatorname{Gal}(L/K)$** is defined as the group of automorphisms of $L$ fixing $K$ pointwise.')
does_not_contain_a_notation = notation_identification_model.predict(r'Let **$L/K$** be a Galois field extension. Its Galois group $\operatorname{Gal}(L/K)$ is defined as the group of automorphisms of $L$ fixing $K$ pointwise.')
print(contains_a_notation)
print(does_not_contain_a_notation)('True', tensor(1), tensor([9.0574e-08, 1.0000e+00]))
('False', tensor(0), tensor([1.0000e+00, 4.8617e-06]))
# TODO: examples of using functions in markdown.obsidian.personal.machine_learning.notation_identifcation.Similarly as with the information_note_type model, trouver provides
functions (namely
automatically_mark_notations)
which locate within notes mathematical notations that are newly
introduced in the text of the notes and record on these notes locations
of such notations (by surrounding double asterisks ** to LaTeX math
mode strings). Note that this is done by applying the
notation_identification model’s predict method as many times on a
single piece of text as there are LaTeX math mode strings in the text.
As such, these predictions often take a long time.
To save time, it is recommended to apply
automatically_mark_notations
only on notes which have the _meta/definition or _meta/notation tags
(or _auto/_meta/definittion or _auto/_meta/notation) in their
frontmatter YAML metadata2.
Warning The
automatically_mark_notationsfunction not only adds double asterisks**to LaTeX math mode strings, but also removes components such as links and footnotes from the text of the note. It is recommended to only apply this function to notes whose text has not been embellished with such components3. Moreover, theautomatically_mark_notationsis currently buggy and should not be applied to the same note twice
The test vault used in the below example contains a single note which
has already been marked with the _meta/definition and _meta/notation
notes. The following example in particular locates notations in that
note at the very least.
from trouver.markdown.markdown.file import MarkdownFile
from trouver.markdown.obsidian.vault import VaultNote
from trouver.markdown.obsidian.personal.notes import notes_linked_in_notes_linked_in_note
from trouver.markdown.obsidian.personal.machine_learning.notation_identification import automatically_mark_notationswith (tempfile.TemporaryDirectory(prefix='temp_dir', dir=os.getcwd()) as temp_dir):
temp_vault = Path(temp_dir) / 'test_vault_8'
shutil.copytree(_test_directory() / 'test_vault_8', temp_vault)
reference = 'number_theory_reference_1'
index_note = VaultNote(temp_vault, name=f'_index_{reference}')
# `notes` below is a list of `VaultNote` objects.
# Also, the `notes_linked_in_note` function can be a useful
# alternative to the `notes_linked_in_notes_linked_in_note` function.
notes = notes_linked_in_notes_linked_in_note(index_note, as_dict=False)
one_note_with_notation_tag = VaultNote(temp_vault, name='number_theory_reference_1_Definition 1.7')
print("This is what one of the notes looks like before locating notations introduced:\n\n")
print(one_note_with_notation_tag.text())
print("\n\nFinding notations\n\n")
# Note that `information_note_type_model` was loaded previously.
automatically_add_note_type_tags(notation_identification_model, temp_vault, notes)
note_mfs = [MarkdownFile.from_vault_note(note) for note in notes]
# The below code ensures that the model searches for notations only in
# notes marked with a `_meta/definition` or a `_meta/notation`tag or
# their `_auto` versions.
notation_introducing_notes = [
note for note, mf in zip(notes, note_mfs)
if mf.has_tag('_auto/_meta/definition') or mf.has_tag('_auto/_meta/notation')
or mf.has_tag('_meta/definition') or mf.has_tag('_meta/notation')]
for note in notation_introducing_notes:
automatically_mark_notations(note, notation_identification_model, reference_name=reference)
print("This is what the same note looks like after locating notations introduced:\n\n")
print(one_note_with_notation_tag.text())
# os.startfile(os.getcwd()) # This opens the current working directory; find the temporary folder in here and explore it if desired.
# input() # There should be an input prompt; make an input here when you are done viewing theThis is what one of the notes looks like before locating notations introduced:
---
cssclass: clean-embeds
aliases: [number_theory_reference_1_ring_of_integers_modulo_n]
tags: [_meta/literature_note, _reference/number_theory_reference_1, _meta/definition, _meta/notation]
---
# Ring of integers modulo $n$[^1]
The ring of integers modulo $n$, denoted $\mathbb{Z}/n\mathbb{Z}$ has the elements $[m]$ for each integer $m$ where $[m_1] = [m_2]$ if and only if $m_1-m_2$ is divisible by $n$. As a ring, it has the following structure:
1. $[m_1] + [m_2] = [m_1+m_2]$
2. $[m_1] \cdot [m_2] = [m_1 \cdot m_2]$.
# See Also
# Meta
## References
![[_reference_number_theory_reference_1]]
## Citations and Footnotes
[^1]: Kim, Definition 1.7, Page 3
Finding notations
This is what the same note looks like after locating notations introduced:
---
cssclass: clean-embeds
aliases: [number_theory_reference_1_ring_of_integers_modulo_n]
tags: [_meta/literature_note, _auto/l, _auto/e, _auto/F, _meta/notation, _auto/a, _meta/definition, _reference/number_theory_reference_1, _auto/s]
---
# Topic[^1]
The ring of integers modulo $n$, denoted **$\mathbb{Z}/n\mathbb{Z}$** has the elements $[m]$ for each integer $m$ where $[m_1] = [m_2]$ if and only if $m_1-m_2$ is divisible by $n$. As a ring, it has the following structure:
1. $[m_1] + [m_2] = [m_1+m_2]$
2. $[m_1] \cdot [m_2] = [m_1 \cdot m_2]$.
# See Also
# Meta
## References
![[_reference_number_theory_reference_1]]
## Citations and Footnotes
[^1]: Kim, Definition 1.7, Page 3
Now that we have found notations introduced in text and created notation
notes for them in our Obisidian.md vault, we now generate summaries
for these notations.
The ML model in question fine-tuned from a T5
model
This ML model is available on Hugging Face under the repository
hyunjongkimmath/notation_summarizations_model.
from transformers import AutoModelForSeq2SeqLM, AutoTokenizer, pipelinemodel = AutoModelForSeq2SeqLM.from_pretrained('hyunjongkimmath/notation_summarizations_model')
tokenizer = AutoTokenizer.from_pretrained('hyunjongkimmath/notation_summarizations_model')
summarizer = pipeline('summarization', model=model, tokenizer=tokenizer)The summarizer pipeline can be used to summarize notations newly introduced in a piece of mathematical text. The text needs to be formatted as follows:
summarize: <mathematical_text_goes_here>
latex_in_original: $<notation_to_summarize>$type(summarizer)transformers.pipelines.text2text_generation.SummarizationPipeline
summarizer("summarize:Let us now define the upper half plane $\mathbb{H}$ as the set of all complex numbers of real part greater than $1$.\n\n\nlatex_in_original: $\mathbb{H}$")Your max_length is set to 200, but you input_length is only 54. You might consider decreasing max_length manually, e.g. summarizer('...', max_length=27)
[{'summary_text': 'the upper half plane of the real part greater than $1$. It is defined as the set of all complex numbers of real parts greater than $$.'}]
In the above example, the summarizer determines that the notation
$\mathbb{H}$ introduced in the text
Let us now define the upper half plane $\mathbb{H}$ as the set of all complex numbers of real part greater than $1$.denotes
'the upper half plane of the complex plane $\\ mathbb{ H} $. It is defined as the set of all complex numbers of real part greater than $1$.'.
Once we mark notations introduced in information notes by surrounding
LaTeX math mode strings with double asterisks ** (manually and/or by
using the notation_identification model, see the section about the
notation_identification
model above), we
can use the
make_notation_notes_from_double_asts
function to make notation notes dedicated to those introduced notations
and to link these newly created notation notes to the information notes.
After making these notation notes, we can use the
append_summary_to_notation_note
function to predict what each notation is supposed to denote and add
these predicted summaries to the notation notes themselves.
For the example below, there is at least one information note with
notations already marked with double asterisks **.
from trouver.markdown.obsidian.personal.notation import make_notation_notes_from_double_asts, notation_notes_linked_in_see_also_section
from trouver.markdown.obsidian.personal.machine_learning.notation_summarization import append_summary_to_notation_notewith (tempfile.TemporaryDirectory(prefix='temp_dir', dir=os.getcwd()) as temp_dir):
temp_vault = Path(temp_dir) / 'test_vault_8'
shutil.copytree(_test_directory() / 'test_vault_8', temp_vault)
reference = 'number_theory_reference_1'
index_note = VaultNote(temp_vault, name=f'_index_{reference}')
# Also, the `notes_linked_in_note` function can be a useful
# alternative to the `notes_linked_in_notes_linked_in_note` function.
notes = notes_linked_in_notes_linked_in_note(index_note, as_dict=False)
one_note_with_notations_marked = VaultNote(temp_vault, name='number_theory_reference_1_Definition 2.3')
print("This is what the information note looks like before we add the links to the notation notes:\n\n")
print(one_note_with_notations_marked.text())
for note in notes:
new_notation_notes = make_notation_notes_from_double_asts(note, temp_vault, reference_name=reference)
print("\n\nThis is what the information note looks like after we add the links to the notation notes:\n\n")
print(one_note_with_notations_marked.text())
for note in notes:
notation_notes_linked_in_note = notation_notes_linked_in_see_also_section(note, temp_vault)
for notation_note in notation_notes_linked_in_note:
append_summary_to_notation_note(notation_note, temp_vault, summarizer)
print("\n\nThis is what the newly created notation notes look like after we add the predicted summaries:\n\n")
notation_notes_linked_in_the_one_note = notation_notes_linked_in_see_also_section(
one_note_with_notations_marked, temp_vault)
for notation_note in notation_notes_linked_in_note:
print(notation_note.text(), '\n')Your max_length is set to 200, but you input_length is only 166. You might consider decreasing max_length manually, e.g. summarizer('...', max_length=83)
This is what the information note looks like before we add the links to the notation notes:
---
cssclass: clean-embeds
aliases: []
tags: [_meta/literature_note, _reference/number_theory_reference_1, _meta/definition, _meta/notation]
---
# Quotient ring of a ring by an ideal[^1]
Let $R$ be a ring and let $I$ be an ideal. The quotient ring **$R/I$** is the ring whose elements are the equivalence classes of elements of $R$ with respect to the equivalence relation **$\sim$** given by $x \sim y$ if $x-y \in I$ and whose ring structure is given by
$$\begin{align*}
[x]+[y] &= [x+y] \\
[x] \cdot [y] &= [x \cdot y].
\end{align*}$$
# See Also
# Meta
## References
![[_reference_number_theory_reference_1]]
## Citations and Footnotes
[^1]: Kim,
This is what the information note looks like after we add the links to the notation notes:
---
cssclass: clean-embeds
aliases: []
tags: [_meta/literature_note, _reference/number_theory_reference_1, _meta/definition, _meta/notation]
---
# Quotient ring of a ring by an ideal[^1]
Let $R$ be a ring and let $I$ be an ideal. The quotient ring **$R/I$** is the ring whose elements are the equivalence classes of elements of $R$ with respect to the equivalence relation **$\sim$** given by $x \sim y$ if $x-y \in I$ and whose ring structure is given by
$$\begin{align*}
[x]+[y] &= [x+y] \\
[x] \cdot [y] &= [x \cdot y].
\end{align*}$$
# See Also
- [[number_theory_reference_1_notation_R_I]]
- [[number_theory_reference_1_notation_sim]]
# Meta
## References
![[_reference_number_theory_reference_1]]
## Citations and Footnotes
[^1]: Kim,
Your max_length is set to 200, but you input_length is only 166. You might consider decreasing max_length manually, e.g. summarizer('...', max_length=83)
This is what the newly created notation notes look like after we add the predicted summaries:
---
detect_regex: []
latex_in_original: [R/I]
tags: [_auto/notation_summary]
---
$R/I$ [[number_theory_reference_1_Definition 2.3|denotes]] the quotient ring of the ideal $R/I$. It is the ring whose elements are the equivalence classes of elements of $R$.
---
detect_regex: []
latex_in_original: ["\\sim"]
tags: [_auto/notation_summary]
---
$\sim$ [[number_theory_reference_1_Definition 2.3|denotes]] the quotient ring of the ideal $R/I$. It is the ring whose elements are the equivalence classes of elements of a ring $R$.
At the time of this writing (1/30/2023), the author of trouver
believes that this summarization model could be improved upon with more
data; thus far, this model was trained on less than 1700 data points.
Many of the functions and methods in this library are accompanied by examples demonstrating how one might use them.
These examples are usually also tests of the functions/methods; the
developer of this library can use nbdev’s
nbdev_test
command-line command to automatically run these tests56. Moreover,
there is a GitHub workflow in the repository for this library (see the
.github/workflows/test.yaml) which automatically runs these
examples/tests on GitHub Actions when changes to are committed to the
GitHub repository7.
These examples may use a combination of the following:
- Mock patching via Python’s
unittest.mocklibrary. - The
fastcore.testmodule as assertion statements. - example/test files in the
nbs/_testsfolder in the repository8.-
The
_test_directory()function in thehelpermodule obtains this folder. -
Many of these examples also use the
tempfile.TemporaryDirectoryclass along with theshutil.copytreeto create a Python context manager of a temporary directory with contents copied from thenbs/_testsfolder. The temporary directory is automatically deleted once the context manager ends. We do this to run tests/examples which modify files/folders without modifying the files/folders in thenbs/_testsdirectory themselves.- For example, the code
with tempfile.TemporaryDirectory(prefix='temp_dir', dir=os.getcwd()) as temp_dir: temp_vault = Path(temp_dir) / 'test_vault_1' shutil.copytree(_test_directory() / 'test_vault_1', temp_vault) # run the rest of the example here # Uncomment the below lines of code to view the end-results of the example; # os.startfile(os.getcwd()) # os.input() # this line pauses the process until the user makes an input so the deletion of the temporary directory is delayed.
first creates a temporary directory starting
temp_dirin the current working directory and copies into this temporary directory the contents oftest_vault_1in thenbs/_testsfolder. One the example/test has finished running, the temporary directory is removed whether or not the test succeeds.
-
This repository is still in its preliminary stages and much of the code
and documentation may be faulty or not well formatted. The author
greatly appreciates reports of these issues, notifications of typos, and
suggestions on edits; please feel free to report them on the Issues
section of the GitHub repository for this library. The
author of this
repository, who is primarily a mathematician (a PhD student at the time
of this writing), does not guarantee quick responses or resolutions to
such issues, but will do his best to address them.
This repository is based on the nbdev
template. As such, code for the packages as well as the documentation
for the repository are written in jupyter notebooks (the .ipynb files
in the nbs folder) and the Python modules are auto-generated via the
command-line command
nbdev_export
(or
nbdev_prepare,
which among other things runs nbdev_export.).
- In the
nbs/_testsfolder, make sure that the folders that you want to test are not empty; since git does not track empty folders, empty folders will not be pushed in GitHub and the tests in GitHub Actions may yield different results than in a local computer.
Copyright © 2023 onward Hyun Jong Kim. Licensed under the Apache License, Version 2.0 (the “License”); you may not use this project’s files except in compliance with the License. A copy of the License is provided in the LICENSE file in this repository.
The author of trouver thanks Sun Woo
Park for agreeing to allow
their coauthored paper, Global $\mathbb{A}^1$-degrees covering maps
between modular curves, along with
some of Park’s expository writings, to be used in examples in this
library. The author of trouver further thanks Sun Woo Park for his
help in testing trouver on a MacOS computer and for reviewing the
tutorial.walkthrough page.
trouver was built using nbdev as a
template.
See release_notes.
Footnotes
-
Given time, the author of
trouvereventually plans on writing instructions on training each of the models. ↩ -
At the time of this writing (1/30/2023), the
information_note_typemodel is fairly good at telling when a note introduces a definition or a notation, but will often conflate the two. In other words, the model may predict that a note ought to have the_meta/definitiontag assigned to it when the_meta/notationtag should be assigned to it and vice versa, but the model will fairly usually assign at least one of the tags when the note introduces a definition or a notation and will assign neither of the tags when the note does not introduce a definition or a notation. ↩ -
More precisely,
automatically_mark_notationsfirst appliesprocess_standard_information_noteto aMarkdownFileobject constructed from theVaultNoteobject to roughly obtain the raw text of the note, uses that raw text to locate notations, marks the notations in the raw text, and then replaces the text from the note with the raw text with notations marked. In the process of obtaining the raw text, theprocess_standard_information_notefunction removes components such as links and footnotes from the text. ↩ -
There seems to be a bug in the above example where inexplicable tags (e.g.
_auto/s,_auto/a) are added to the note along with the double asterisks**. This issue is reported as Issue #33. ↩ -
cf. nbdev’s End-To-End Walkthrough to see how to use
nbdev_test↩ -
There are also tests which are hidden from the documentation website; one can find these tests in the jupyter notebook files in the
nbsfolder in the repository for this library as notebook cells marked with the#| hideflag, cf. nbdev’s End-to-End Walkthrough to see what the#| hideflag does. ↩ -
The
.github/workflows/test.yamlGitHub workflow file is set up in such a way that that allows GitHub Actions to access/use the contents of thenbs/_testsdirectory upon running the tests/examples. ↩ -
The
.github/workflows/test.yamlGitHub workflow file is set up in such a way that that allows GitHub Actions to access/use the contents of thenbs/_testsdirectory upon running the tests/examples. ↩