Fourier integrals vanishing on large circles

When evaluating contour integrals, it is often of interest to prove that Fourier-type integrals vanish on large enough semicircles (see the figure). This holds under the following condition:

Theorem. Suppose that $$f(z)=O(|z|^{-a}), \quad a>0$$ for $$z$$ in the upper half-plane. Then for any $$\lambda > 0$$ we have $$\int_{\gamma_R} f(z)\mathrm{e}^{i\lambda z} \rightarrow 0, \quad R\to+\infty,$$ where $$\gamma_R$$ is the upper half-circle of radius $$R$$.

This result is stronger than other ways of developing vanishing integration contours in the upper half-plane, compare for instance with the MIT lecture notes by Jeremy Orloff​1​. The version above can be found in advanced books on Fourier transforms, for example​2​.

To prove that, parametrize the upper half-circle $$\gamma_R$$ by $$z=R\mathrm{e}^{i\theta} = R(\cos\theta + i\sin\theta)$$ where $$0<\theta<\pi$$. Under this parametrization, the Fourier multiplier becomes $$\mathrm{e}^{i\lambda z} = \mathrm{e}^{-\lambda R \sin \theta}\mathrm{e}^{i R \lambda \cos\theta}$$. Thus, the integral can be bounded by $$\left|\int_{\gamma_R} f(z)\mathrm{e}^{i\lambda z}\right|\leqslant \int_{0}^{\pi} |f(R\mathrm{e}^{i\theta})| R \mathrm{e}^{-R\lambda \sin\theta} \mbox{d}\theta \\ \leqslant C\int_{0}^{\pi} R^{1-a} \mathrm{e}^{-R\lambda \sin\theta} \mbox{d}\theta\\ = 2C\int_{0}^{\frac{\pi}{2}} R^{1-a} \mathrm{e}^{-R\lambda \sin\theta} \mbox{d}\theta \\ \leqslant 2C\int_{0}^{\frac{\pi}{2}} R^{1-a} \mathrm{e}^{-2 R\lambda \theta / \pi} \mbox{d}\theta \\ = C\cdot \frac{\pi R^{- a} \left(1 – e^{- R \lambda}\right)}{\lambda},$$
which tends to zero as long as $$a>0$$ and $$R\to \infty$$.

1. 1.
Orloff J. Definite integrals using the residue theorem. Lecture Notes. Accessed 2023. https://math.mit.edu/~jorloff/18.04/notes/topic9.pdf
2. 2.
Spiegel MR. Laplace Transforms. McGraw Hill; 1965.

Expanding Inverse Functions

The problem of inverting the implicit function $$y=f(x)$$ in the form of power-series $$x = a + \sum_{k=1}^{\infty} b_k (y-f(a))^k$$

around a point of interest $x=a$, has a long history. Lagrange obtained a theoretical inversion formula [1]https://mathworld.wolfram.com/LagrangeInversionTheorem.html, yet efficient implementations are relatively recent [2]Brent, Richard P., and Hsiang T. Kung. „Fast algorithms for manipulating formal power series.” Journal of the ACM (JACM) 25, no. 4 (1978): 581-595. [3]Johansson, F., 2015. A fast algorithm for reversion of power series. Mathematics of Computation84(291), pp.475-484..

I this note I am sharing a bit simpler algorithm, performing Newton-like updates:
$$x_{k} = x_k-f(x_{k-1})[(y-f(a))^{k}]\cdot \frac{(y-f(a))^{k}}{f'(a)},\quad x_1 = a+\frac{y-f(a)}{f'(a)}.$$
We can see that it gradually produces more and more accurate terms. More precisely, assuming w.l.o.g. $$0=a=f(a)$$, suppose $$f(x) =y + O(y^{k})$$, then $$f(x + h ) = f(x)+f'(0) h + O(h^2)$$ by Taylor’s expansion, and so the update $$h=-\frac{f(x) [y^k] }{f'(0)}$$ (where $$[z^k]$$ is the operation of extracting the coefficient with $$z^k$$) gives $$f(x+h)=y+O(y^{k+1})$$ as desired.
I implemented this a proposal for Sympy.

References

↑1 https://mathworld.wolfram.com/LagrangeInversionTheorem.html Brent, Richard P., and Hsiang T. Kung. „Fast algorithms for manipulating formal power series.” Journal of the ACM (JACM) 25, no. 4 (1978): 581-595. Johansson, F., 2015. A fast algorithm for reversion of power series. Mathematics of Computation, 84(291), pp.475-484.

Monitoring Azure Experiments

Azure Cloud is a popular work environment for many data scientists, yet many features remain poorly documented. This note shows how to monitor Azure experiments in a more handy and detailed way than through web or cl interface.

The trick is to create a dashborad of experiments and their respective runs, up to a desired level of detail, from Python. The workhorse is the following handy utility function:

from collections import namedtuple

def get_runs_summary(ws):
"""Summarise all runs under a given workspace, with experiment name, run id and run status
Args:
ws (azureml.core.Workspace): Azure workspace to look into
"""
# NOTE: extend the scope of run details if needed
record = namedtuple('Run_Description',['job_name','run_id','run_status'])
for exp_name,exp_obj in ws.experiments.items():
for run_obj in exp_obj.get_runs():
yield(record(exp_name,run_obj.id,run_obj.status))

Now it’s time to see it in action 😎

# get the default workspace
from azureml.core import Workspace
import pandas as pd

ws = Workspace.from_config()

# generate the job dashboard and inspect
runs = get_runs_summary(ws)
summary_df = pd.DataFrame(runs)
# count jobs by status
summary_df.groupby('run_status').size()

Use the dashboard for to automatically manage experiments. For example, to kill running jobs:

from azureml.core import Experiment, Run

for exp_name,run_id in summary_df.loc[summary_df.run_status=='Running',['job_name','run_id']].values:
exp = Experiment(ws,exp_name)
run = Run(exp,run_id)
run.cancel()

Check the jupyter notebook in my repository for a one-click demo.

Poetry, a popular Python package manager, prefers to use keyring to manage passwords for private code repositories. Storing passwords in plain text is a secondary option, but may be needed in case of either issues in poetry itself or with keyring configuration (may not be properly installed, be locked etc). To disable the use of system keyring by poetry, set the null backend in the environmental variable PYTHON_KEYRING_BACKEND:

(.venv) azureuser@sense-mskorski:~/projects/test_project$export PYTHON_KEYRING_BACKEND=keyring.backends.null.Keyring (.venv) azureuser@sense-mskorski:~/projects/test_project$ poetry config http-basic.private_repo *** '' -vvv
Adding repository test_repo (https://***) and setting it as secondary
No suitable keyring backend found
No suitable keyring backends were found
Keyring is not available, credentials will be stored and retrieved from configuration files as plaintext.
Using a plaintext file to store credentials

Analogously, one of working backends can be enabled (make sure it works correctly!) . This is how to list available backends:

(.venv) azureuser@sense-mskorski:~/projects/test_project$keyring --list-backends keyring.backends.chainer.ChainerBackend (priority: -1) keyring.backends.fail.Keyring (priority: 0) keyring.backends.SecretService.Keyring (priority: 5) Marking Python Tests as Optional Often code tests are to be run on special demand, rather than in a CI/CD manner: for instance, they may be slow or work only in a local mode with protected data. This note shows how to declare code tests optional in pytest, the leading testing framework for Python. The article is inspired by the pytest documentation on test markers. The trick is to mark extra tests with a decorator and couple it with a runtime flag. The first step is to mark the relevant code blocks and is illustrated below: import pytest @pytest.mark.optional def test_addition(): assert 1+1==2 The second step is to instruct the config flag on how to interpret the decorator; it is also worth describing this action in the helper. The code below shows how to set this up: import pytest def pytest_addoption(parser): """Introduce a flag for optional tests""" parser.addoption( "--optional", action="store_true", default=False, help="run extra tests" ) def pytest_collection_modifyitems(config, items): """Instruct the testing framework to obey the decorator for optional tests""" if config.getoption("--optional"): # --extras given in cli: do not skip extra tests return # otherwise, execute skip_extras = pytest.mark.skip(reason="need --optional flag to run") for item in items: if "optional" in item.keywords: item.add_marker(skip_extras) def pytest_configure(config): """Make the decorator for optional tests visible in help: pytest --markers""" config.addinivalue_line("markers", "optional: mark test as optional to run") To see it in action, run pytest -k battery --optional 😎 test_battery.py::test_addition PASSED Debug CI/CD with SSH CircleCI is a popular platform Continuous integration (CI) and continuous delivery (CD). While its job status reports are already useful, one can do much more insights by debugging it in real time. Here I am sharing a real use-case of debugging a failing job deploing an app 🙂 Failing job are reported in red and the (most of) errors caught during the execution appear in the terminal. In this case, the environment is unable to locate Python: The error is not very informative. 😮 For more insights, let’s re-run the job in SSH mode: You will be welcomed with instructions on how to connect via SSH: Use this instruction to connect to inspect the environment at its failure stage: mskorski@SHPLC-L0JH Documents % ssh -p 64535 aa.bbb.cc.dd The authenticity of host '[aa.bbb.cc.dd]:64535 ([aa.bbb.cc.dd]:64535)' can't be established. ED25519 key fingerprint is SHA256:LsMhHb5fUPLHI9dFdyig4VKw44GTqrA2dkEWT0sZx4k. Are you sure you want to continue connecting (yes/no)? yes circleci@bc95bb40fff3:~$ ls project/venv/bin -l
total 300
...
lrwxrwxrwx 1 circleci circleci    7 Aug  1 12:37 python -> python3
lrwxrwxrwx 1 circleci circleci   49 Aug  1 12:37 python3 -> /home/circleci/.pyenv/versions/3.8.13/bin/python3
lrwxrwxrwx 1 circleci circleci    7 Aug  1 12:38 python3.9 -> python3

The solution in this case was to update cache. The issues may be far more complex than that, but being able to debug them live comes to the rescue. 😎

Prototype in Jupyter on Multiple Kernels

For data scientists, it is a must to prototype in multiple virtual environments which isolate different (and often very divergent) sets of Python packages. This can be achieved by linking one Jupyter installation with multiple Python environments.

Use the command <code>which jupyter</code> to show the Jupyter location and <jupyter kernelspec list> to show available kernels, as shown below:

ubuntu@ip-172-31-36-77:~/projects/eye-processing$which jupyter /usr/local/bin/jupyter ubuntu@ip-172-31-36-77:~/projects/eye-processing$ jupyter kernelspec list
Available kernels:
eye-processing    /home/ubuntu/.local/share/jupyter/kernels/eye-processing
pypupilenv        /home/ubuntu/.local/share/jupyter/kernels/pypupilenv
python3           /usr/local/share/jupyter/kernels/python3

To make an environment a Jupyter kernel, first activate it and install ipykernel inside.

ubuntu@ip-172-31-36-77:~/projects/eye-processing$source .venv/bin/activate (.venv) ubuntu@ip-172-31-36-77:~/projects/eye-processing$ pip install ipykernel
Collecting ipykernel
...
Successfully installed...

Then use ipykernel to register the active environment as a jupyter kernel (choose a name and the destination, e.g. user space; consult python -m ipykernel install –help for more options).

(.venv) ubuntu@ip-172-31-36-77:~/projects/eye-processing\$ python -m ipykernel install --name eye-processing --user
Installed kernelspec eye-processing in /home/ubuntu/.local/share/jupyter/kernels/eye-processing

The kernel should appear in the jupyter kernel lists. Notebooks may need restarting to notice it.

Open kernel.json file under kernel’s path to inspect config details:

{
"argv": [
"/home/ubuntu/projects/eye-processing/.venv/bin/python",
"-m",
"ipykernel_launcher",
"-f",
"{connection_file}"
],
"display_name": "eye-processing",
"language": "python",
"debugger": true
}
}
"~/.local/share/jupyter/kernels/eye-processing/kernel.json" [noeol] 14L, 234B

Modern Bibliography Management

A solid bibliography database is vital for every research project, yet building it is considered an ugly manual task by many – particularly by old-school researchers. But this does not have to be painful if we use modern toolkit. The following features appear particularly important:

• collaborative work (sharing etc)
• extraction magic (online search, automated record population)
• tag-annotation support (organize records with keywords or comments)
• multi-format support (export and import using BibTeX or other formats)
• plugins to popular editors (Office, GoogleDocs)

I have had a particularly nice experience with Zotero (many thanks to my work colleagues from SensyneHealth for recommending this!). Let the pictures below demonstrate it in action!

tag and search…

organize notes…

automatically extract from databases…

format and export…

National Bank of Poland leads on helping Ukraine’s currency

Ukrainian refugees see their currency hardly convertible at fair rates or even accepted. The lack of liquidity will also hit Ukraine’s government urgent expenses (medical supplies or weapons).

The National Bank of Poland (NBP) offered a comprehensive financial package to address these problems. Firstly, it enabled the refugees to exchange the currency at a nearly official rate, through the agreement with the National Bank of Ukraine (NBU) signed on March 18.

Secondly, in a follow-up agreement the NBP offered a FX Swap for USD 1 billion which will provide the Ukraine central bank with more liquidity.

As Reuters announced on March 24, EU countries are close to follow and agree on a scheme enabling exchanging Ukraine’s cash. The rules disclosed in the draft are close to the NBP scheme (exchange capped at around 10,000 UAH per individual)

Evidence-based quick wins in reducing oil demand

Following the Russia invasion on Ukraine an oil supply shock is expected and what is worse there may be no supply increase from OPEC+ . A way out is to cut the demand as advised by IEA. The proposed steps are quick-wins: easy to implement and reverse while having measurable and significant impact. The overall impact should balance the supply gap.

Quantitative estimates can be backed with empirical or even statistical evidence. A good example is the speed reduction, as recent research advances demonstrate quantitatively what drivers learn by experience: fuel consumption considerably increases with speed beyond a turning point. Turning this statement around: you can likely save lots of petrol driving at a lower speed. The study of He at al. finds cubic approximations fit well and enables several optimisation considerations for various groups of car users.