Wednesday, December 25, 2024

The Cantor Set

 Rummaging around, I found this in ChatGPT.

 The Cantor set is a famous example of a fractal and an important concept in mathematics, particularly in topology, measure theory, and real analysis. It is constructed by repeatedly removing the middle third of a line segment and has some remarkable and counterintuitive properties.

Construction of the Cantor Set

  1. Start with a line segment: Begin with the interval [0,1] on the real number line.
  2. Remove the middle third: Remove the open interval (1/3,2/3), leaving two closed intervals: [0,1/3] and [2/3,1].
  3. Repeat the process: For each remaining interval, remove its middle third. For example:
    • In the next step, remove (1/9,2/9) and (7/9,8/9).
    • This process continues indefinitely.

The Cantor set consists of all points that are never removed during this process.

Properties of the Cantor Set

  1. Uncountable: Despite appearing "small," the Cantor set is uncountably infinite. It contains as many points as the real numbers in [0,1].
  2. Zero measure: The total length of the intervals removed adds up to 1, so the Cantor set has Lebesgue measure zero. It is "small" in this sense.
  3. Nowhere dense: The Cantor set is nowhere dense in [0,1], meaning it does not contain any intervals of positive length.
  4. Self-similar: The Cantor set is a fractal. It looks "the same" at all scales and can be described by recursive patterns.
  5. Perfect and closed: The Cantor set is a perfect set (every point is a limit point of the set) and a closed set (it contains all its limit points).
  6. Ternary representation: A point in [0,1] belongs to the Cantor set if and only if it can be written in base 3 (ternary) without using the digit 1.

The Cantor set is a fundamental example in mathematics for illustrating ideas about infinity, measure, and topology. It is also used to demonstrate the concept of a set that is uncountably infinite yet has no "length" in the traditional sense.

 WOW!!!!!!!

Jim Simons (1938-2024)

 From “Notices of the Mathematical Society”.

 His investment fund averaged about 60% annually for many years.

 An excerpt:

 Jim Simons’s guiding principles.

Do something new—don’t run with the pack.

I am not such a fast runner. If I am one of 𝑁

people all working on the same problem, there is

very little chance I will win. If I can think of a new

problem in a new area, that will give me a chance.

Surround yourself with the smartest people you

can find.

When you see such a person, do all you can to

get them on board.

Be guided by beauty.

This is obviously true in doing mathematics or

writing poetry, but it is also true in fashioning an

organization that is running extremely well and

accomplishing its mission with excellence.

Don’t give up easily.

Some things take much longer than one initially

expects. If the goal is worth achieving just

stick with it.

Hope for Good Luck!

 Here is the link.

 https://www.ams.org/journals/notices/202501/202501FullIssue-optimized.pdf?adat=January%202025&trk=&cat=none&type=.pdf

 

Tuesday, December 24, 2024

Your Government at work

An unjustified, excessively aggressive raid by the ATF on a law-abiding family.

 Here is the link.

 https://youtu.be/mJMCeWLYzts 

Friday, December 20, 2024

Beware the Boomer Christmas

 From Jeff “Tank” Hoover at the American Handgunner.

 It’s remarkable that Us Old Guys managed to make it.

 Here is the link.

 https://americanhandgunner.com/discover/beware-the-boomer-christmas/

 

Wednesday, December 18, 2024

On the maturity of macroeconomics

From John Cochrane.

Forget the math. The English conveys the message.

The moral of the story is that you cannot trust what you read and hear in the media and from many economists to be coherent.

 Here is the link.

 https://www.grumpy-economist.com/p/bob-hall-and-consumption

Tuesday, December 10, 2024

2024 Top Story in Cardiology: The ESPRIT Trial

 Here is the link.

 https://www.practiceupdate.com/c/173221/2/2/?elsca1=emc_enews_daily-digest&elsca2=email&elsca3=practiceupdate_cardio&elsca4=cardiology&elsca5=newsletter&rid=MzEwODYyNjA4MjA0S0&lid=20845073

 Here is an excerpt.

 Hypertension remains a common and undertreated cardiovascular risk factor throughout the world. The recommended target for the treatment of high blood pressure (BP) has long been a systolic BP of less than 140 mm Hg. The SPRINT trial1 did show that targeting a systolic BP of less than 120 mm Hg was better than 140 mm Hg in patients at high cardiovascular risk but without diabetes or stroke. The ESPRIT trial2 has now validated and extended the SPRINT findings to include patients with diabetes or stroke.

Monday, December 09, 2024

Mystery Illness Kills Dozens in Congo

 Here is the link.

 https://www.practiceupdate.com/c/173373/2/6/?elsca1=emc_enews_daily-digest&elsca2=email&elsca3=practiceupdate_primary&elsca4=primary-care&elsca5=newsletter&rid=MzEwODYyNjA4MjA0S0&lid=20849334

 Here is an excerpt.

 Health officials in the Democratic Republic of Congo are racing to try to identify the cause of a mysterious, flu-like illness that has sickened 376 people and left 79 dead in that country.

Friday, December 06, 2024

Wind and Solar Can’t Support the Grid

 

Here is the link.

 https://judithcurry.com/2024/12/05/wind-and-solar-cant-support-the-grid/#more-31718

 Here is an excerpt.

 In October of 2025, the isolated small city of  Broken Hill in New South Wales, Australia with a 36 MW load (including the large nearby mines) could not be reliably served by 200 MW of wind, a 53 MW solar array, significant residential solar, and a large 50 MW battery all supplemented by diesel generators.

 Many people falsely believe that wind, solar and batteries have been demonstrated to provide grid support and deliver energy independently in large real word applications. Few people realize that we are a long way away from having wind, solar and batteries support a large power system without significant amounts of conventional spinning generation (nuclear, gas, coal, hydro, geothermal) on-line to support the grid.