Nikolaj-K

Video where this script is discussed: https://youtu.be/Lsf4eAGvODs

Consider a non-strictly ordered space $X$ with an (absolutely homogenous) absolute value function $|\cdot|: X\to X$ If $B:(X\times X)\to X$ is a symmetric map, then

$B(\max(x, y)-x, \max(x, y)-y) = B(0, |x-y|)$

Proof

Nikolaj-K

#!/usr/bin/env python3
"""
Minimal Wikipedia semantic search wrapper for web integration.

Install hints (CPU):
  - python, duh
  - pip install -U sentence-transformers faiss-cpu numpy

Expected index artifacts (see zip):
  - path/to/pages_embeddings.index (main data item, 200 MB)

Nikolaj-K

This file starts with Exercises, which ends at exercise 10 and is followed by "Notes: Applied Optimization".

Search for "Exercise 9" for the Regression cost function minimization exercises.

Greetings Nikolaj Kuntner

---

Exercises - Applied Optimization

Nikolaj-K

Script for the video

https://youtu.be/LfiIiSPg3Ms

$f$ ... Lebesgue-integrable over (all of) ${\mathbb R}$,

$\int_{-\infty}^\infty f\big(x + d\big),{\mathrm d}x = \int_{-\infty}^\infty f\big(x\big),{\mathrm d}x$

$\int_{-\infty}^\infty f\big(x - \dfrac{1}{x}\big),{\mathrm d}x = \int_{-\infty}^\infty f\big(x\big),{\mathrm d}x$

Nikolaj-K

Probability Theory 1 – Script skeleton

• 2024W-Prob1__script.pdf: sections 1–5 (up to and including 5. Lebesgue-Integral)
• 2025W-Prob1-Collection.pdf: sections 0–3 (up to and including 3. Das Lebesgue-Integral)

Notes:

• Warning: I dropped \mathcal, since it doesn't render in gist
• Likewise, _\# became _H
• The items have rough importance rankings
• roughly "important" vs "neutral" and "random"; I'll derank a few later

Nikolaj-K

The relevant content starts at page 17 in the script. The chapter ends on page 25. Note that the exercises are different ones from the listing in the script.

==== Exercise's context ====

We consider the general minimization problem $p^{\ast} := \inf_{x \in M \subseteq X} f(x),$ where $X \subseteq \mathbb{R}^n$, $f:X\to\mathbb{R}$ is continuous,

Nikolaj-K

"""
Code and prompt used in the video
https://youtu.be/xcE_0azawvM
"""

"""You'll get two tasks
First part:
Consider a joint distirbution of two random variables X_k with k in {a, b} (i.e. two random variables X_a and X_b),
over a finite outcome sets of size n_a and n_b.
Explain what the marginal distributions are.

Nikolaj-K

""" Script to the video at https://youtu.be/8w4jHN1LsnY """

===== Moving quantile series ===== === Preliminary - Exponential distribution example === Density: $p_\mu(x) := F'(x) = \frac{1}{\mu} {\mathrm e}^{-\frac{x}{\mu}}$

Nikolaj-K

"""
Proof prsented in the video

https://youtu.be/Ydyhe7KdRsk
"""

Reminder:

=== Theorem 1, the geometric sum and series formulae ===
$\sum_{k=0}^n a^k = \dfrac{a^{n+1}-1}{a-1}$

Nikolaj-K

"""
Code for the video at
https://youtu.be/CquQe7Y05VA

This script will be easy to generalize, also.
"""

from datetime import datetime
import imageio.v2 as imageio
from io import BytesIO