Monday, June 30, 2025

Properties of supremum and infimum

The proofs are exercises from Zalduendo’s Calculus off the Beaten Path.

Write this in symbolic logic:

s=supA if and only if s is an upper bound of A and for each ε > 0 we may find a ε ∈ A such that s − ε < a ε ≤ s.

A compact way to put the sentence entirely in first-order logical symbols is

s=supA[x(xAxs)][ε(ε>0x(xAsε<xs))].


  • x(xAxs)” says that s is an upper bound of A.
  • ε>0xA(sε<xs)” formalizes that for every ε>0 we can find an element of A within ε of s from below.

Saturday, June 28, 2025

Proof that a|b ∧ b|c .→. a|c

Show that if a|b and b|c, then a|c.

To prove: If ab and bc, then ac.


Definitions:

  • ab means there exists an integer k such that b=ak.
  • bc means there exists an integer m such that c=bm.

Proof:

Given:

  • b=ak for some kZ (since ab)
  • c=bm for some mZ (since bc)

Substitute b=ak into the equation for c:

c=bm=(ak)m=a(km)

Since kmZ (product of integers is an integer), we have:

c=a(km)ac


Conclusion:

We have shown that c is a multiple of a, so ac. Thus, if ab and bc, then ac. □

Set intersection and multiplication

What makes set intersection algebraically similar to multiplication?

Mathematicians see an almost one-for-one match between the usual arithmetic operations “ + ” and “ · ” and the set operations “∪” (union) and “∩” (intersection). Intersection plays the rôle of “multiplication” for several independent reasons:

Feature Ordinary multiplication Set intersection
Binary operation on a universe • acts on numbers in ℝ, ℚ, ℤ, … ∩ acts on all subsets of a fixed universe U
Commutative a · b = b · a A ∩ B = B ∩ A
Associative (a · b) · c = a · (b · c) (A ∩ B) ∩ C = A ∩ (B ∩ C)
Identity element 1: a · 1 = a U: A ∩ U = A
Zero (annihilator) 0: a · 0 = 0 ∅: A ∩ ∅ = ∅
Distributive over “addition” a · (b + c) = a · b + a · c A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Absorption law a + a · b = a A ∪ (A ∩ B) = A
Complement gives a two-element algebra ±1 in {0, 1} give an idempotent semiring complements ¬A act like −a in the Boolean ring

These shared laws make the algebra of sets a semiring (and, if you swap union for symmetric difference, a Boolean ring). In other words, if you translate:

  • “numbers” ↔ “subsets of U”
  • “+” ↔ “∪”
  • “·” ↔ “∩”
  • “0” ↔ “∅”
  • “1” ↔ “U”

the familiar ring axioms turn into the identities that every Venn diagram respects.

A more structural viewpoint

Every subset AU can be identified with its indicator (characteristic) function

χA:U{0,1},χA(x)={1xA0xA.

Writing ordinary arithmetic on {0,1}:

  • pointwise multiplication of indicators corresponds to intersection: χAB(x)=χA(x)χB(x);
  • pointwise addition mod 2 corresponds to symmetric difference, and from that one can build union and complement.

Thus the algebra of all subsets of U is literally the ring ({0,1}U,,) with addition “⊕” = mod-2 sum and multiplication “·” = ∩. What we informally call “multiplying sets” is just multiplying 0-1 functions, which explains why the familiar distributive and absorption rules hold exactly as they do for numbers.

Intuition from counting and probability

  • Counting: if A and B are independent choices, |A||B| counts ordered pairs; intersecting constraints is the combinatorial analogue of requiring all conditions simultaneously, just as multiplying numbers combines factors.
  • Logic: ∩ is the logical AND of membership conditions; AND is the “multiplication” in every Boolean algebra.
  • Measure/probability: for independent events P(A ∩ B) = P(A)·P(B).

Because these perspectives all coincide, we say “intersection behaves like multiplication” not as a metaphor but as an exact algebraic fact: within the Boolean algebra of sets, ∩ is the multiplication operation.

Friday, June 20, 2025

What Patrick McKenzie means by a “Dangerous Professional”

The following are selected posts by Patrick McKenzie @patio11 which bear on his idea of the "Dangerous Professional". Read them and write a comprehensive summary of what his idea is about, with examples of things that such a professional does and says, with context for each.

(The posts are omitted for brevity.)

What Patrick McKenzie means by a “Dangerous Professional”

A Dangerous Professional (DP) is not a hit-man in a suit; it is an ordinary-looking person who has mastered the practical magic of paperwork, procedure, and persuasion. They are “dangerous” only from the perspective of a large organization that hoped to keep things sloppy, verbal, or undocumented. To the victim of a fraud, the customer stuck in a phone-tree, or the junior employee facing “Um — please stop taking notes,” a DP is simply the person you want on your side.

McKenzie’s scattered Twitter threads form an informal field-manual. Taken together they describe a mindset, a toolkit, and a style of speech that turns asymmetric bureaucratic fights into winnable games.


1. Core attitudes

Attitude Explanation Representative quotes / situations
“Get it in writing.” Written records trigger statutes, shot-clocks and institutional accountability. Verbal promises can safely be ignored; signed PDFs and timestamped emails cannot. Gate-agent rebooking; Reg E dispute e-mails; post-meeting “Hi Bob, thanks for meeting — we discussed…” mail.
“Ask for citations.” Force the other side to say which policy, law or contract clause they are relying on. Often they can’t. “Please cite, in writing, your refusal to repay a fraud victim under Regulation E’s ‘no printed receipt’ clause.’”
Document first, argue later. A single contemporaneous diary entry, e-mail, or transcript will beat fuzzy recollections years later. Being told to stop taking notes → hand-wrote a transcript immediately after the meeting.
Use the system’s own rules. Know which regulator, statute, or SLA starts a clock and invoke it deliberately. FDCPA 30-day dispute window; 10-day FOIA appeal timeline; airline “irregular operations” options budget.
Polite, calm, relentless. No table-pounding unless you lack facts or law. A DP sounds boringly professional while making it clear they can escalate. Airline chat: “I’m sure you have options for me.”
Not necessarily a lawyer. Lawyers may be DPs, but anyone who learns the playbook can do the job. “Some lawyers are Dangerous Professionals and some Dangerous Professionals are lawyers, but not all DPs are lawyers.”

2. Characteristic moves & phrases

Move Typical wording Context & purpose
Memorialize the meeting. “Hi Bob, thanks for meeting today. We discussed … Looking forward to working with you.” Locks the oral conversation into an immutable timeline.
Force written authority. “Could you please set forth, in writing, the policy that governs X?” Either they pony up a real document or admit it doesn’t exist.
Invoke the regulator. “Under Regulation E you are required to complete your investigation within 10 business days.” Flips the risk onto the institution’s legal/compliance team.
Escalate by address line. Letters marked Office of the President / Chief Compliance Officer / Investor Relations or sent certified mail, return-receipt requested. Short-circuits the CSR tier and lands on someone who can say “yes.”
Apply the “What are my options?” incantation. Direct, calm request repeated until the agent presses the escalation button. Airline IRROPs, billing disputes, any “computer says no” moment.
Preserve the shot-clock. “I reiterate that you have not performed X. Your clock began on [date of first letter].” Prevents a bureaucracy from resetting its own deadline.
Professional presentation props. A notebook, neatly tabbed binder, diary app with timestamps, or even a FOIA appeal letterhead. Visual signals that you keep records and will use them.

3. Field-tested examples

Domain What the DP did Outcome / rationale
Bank fraud (tap-to-pay scam) Asked bank to cite in writing why Reg E didn’t apply; pointed out that newspapers are interested. Legal & PR risk flips, bank almost always refunds.
FOIA appeal First-time appellant writes a letter whose “load-bearing paragraph” threatens cheap, predictable litigation. Agency promises to respond within statutory timeframe.
Airline IRROPs Calmly repeats “What are my options?”; refuses to guess at alternatives; triggers supervisor override; re-booked on competitor at airline’s expense. Airlines have budgets for this; agent just needed permission.
Medical billing error Drafts letter to provider’s general counsel: invoice, receipt, card statement attached; notes FDCPA obligations. Billing department fixes error rather than escalate.
Payroll tax mismatch Letter “in a tone so polite the paint peels” warns bank that Ministry regulators may be consulted. Forces institution, not customer, to project-manage the clean-up.
Meeting where note-taking banned Continues note-taking, keeps physical control of notebook, immediately diaries that the request was “extraordinary.” Creates contemporaneous evidence and signals unwillingness to be intimidated.
Consumer loans / debt collection Cites FDCPA; demands validation in writing; records start date of dispute. Debt collector must stop until it can prove claim.
Customer-service dead end Buys bank’s “cheapest premium product” to gain elite-tier phone number, then internally transfers to the needed department. Small fee buys access to staff with actual decision authority.

4. Negative space: who the DP isn’t

  • Not a bully. Table-pounding is what lacks facts or law. A DP rarely needs theatrics.
  • Not always competent. McKenzie has met “Dangerous-to-Themselves-and-Others Professionals” whose titles outrun their skills.
  • Not secretive about methods. The “deep magick” works even when everybody knows the spell; the power lies in doing the boring work.

5. Learning the craft

Skill to acquire McKenzie’s tips
Reg-reading & citation. Practice Googling the primary source (“Fannie Mae Seller’s Guide §…”) until you can pull a paragraph in 30 sec.
Write like a DP. Keep templates; use clear subjects; state facts, request, desired timeline. LLMs can draft a passable DP letter now.
Note-taking & diaries. Day One journal, or 1-minute phone videos. Any timestamped medium becomes powerful evidence later.
Channel selection. Pick the communication path that makes their lawyer or compliance officer read your message.
Understand incentives. Know who actually wears financial or regulatory liability inside the org and aim the spotlight there.

6. Why the archetype matters now

  • Cheap generative AI means anyone can summon a respectable DP letter, so institutions that rely on friction (“write us a letter within 30 days”) will face a flood.
  • Regulatory shot-clocks (Reg E, FDCPA, FOIA, FDCPA, etc.) were designed when writing letters was hard. Automation breaks that assumption.
  • Professional veneer is democratizing. LLM-drafted correspondence turns former class-gated advantages (clarity, citations, formatting) into commodities, shifting power toward individual consumers.

In one sentence

A Dangerous Professional is a calm, paperwork-wielding adult who knows which letter, statute, or shibboleth makes a billion-dollar institution clear its throat and say, “Yes ma’am, right away.”

Thursday, June 19, 2025

Square trisection

use code to draw a picture that divides a unit square into three equal parts using two inscribed parabolas, one positive and one negative, such that the vertex of the concave-up parabola is the lower left corner of the square and the vertex of the concave-down parabola is the upper right corner of the square

Here’s why the picture you see cuts the unit square into three equal-area bands:

  • Concave-up parabola y1(x)=x2 has its vertex at (0,0).

    Area below y1=01x2,dx=13.

  • Concave-down parabola y2(x)=x2+2x has its vertex at (1,1).

    Area below y2=01(x2+2x),dx=23.

  • Three equal regions

    • Region A (bottom strip): area =13.
    • Region B (between the two parabolas): 2313=13.
    • Region C (top strip): 123=13.

Because each region occupies 13 of the square’s total area, the two parabolas achieve the desired trisection while meeting the specified vertices. The code above draws the square’s boundary and both parabolic arcs so you can verify the construction visually or adapt it further (e.g., change sampling density, export as SVG, add shading, etc.).

The code was as follows:

import numpy as np
import matplotlib.pyplot as plt

# x-coordinates inside the unit square
x = np.linspace(0, 1, 400)

# Two parabolas that split the square into three equal-area regions
y_up = x**2            # concave-up, vertex at (0, 0)
y_down = -x**2 + 2*x   # concave-down, vertex at (1, 1)

# Draw the unit square
plt.figure()
plt.plot([0, 1], [0, 0])   # bottom
plt.plot([1, 1], [0, 1])   # right
plt.plot([1, 0], [1, 1])   # top
plt.plot([0, 0], [1, 0])   # left

# Plot the parabolas
plt.plot(x, y_up, label=r"$y = x^2$")
plt.plot(x, y_down, label=r"$y = -x^2 + 2x$")

# Formatting
plt.xlim(0, 1)
plt.ylim(0, 1)
plt.gca().set_aspect("equal", adjustable="box")
plt.title("Parabolic trisection of a unit square")
plt.legend()
plt.show()