In a magazine article [1] on problems and progress in quantum field theory, Wood writes of Feynman path integrals, “No known mathematical procedure can meaningfully average an infinite number of objects covering an infinite expanse of space in general. The path integral is more of a physics philosophy than an exact mathematical recipe.”

This article [2] provides a method for averaging an arbitrary collection of objects; however, the average can be any number in the extension of the range of these objects. (Note, an arbitrary collection of these objects is a function.)

Question: Suppose anything meaningful has an application in quantum field theory. Is there a way of meaningfully choosing a unique, finite average of a function whose graph matches the description in Wood's quote?

For more info, see this post [3].

[1]: https://www.quantamagazine.org/mathematicians-prove-2d-version-of-quantum-gravity-really-works-20210617/

[2]: https://arxiv.org/pdf/2004.09103

[3]: https://math.stackexchange.com/q/5052005/125918

I finally know what I want.

Let $n\in \mathbb{N}$ and suppose function $f:A\subseteq {\mathbb{R}}^n\to \mathbb{R}$, where $A$ and $f$ are Borel. Let ${\text{dim}}_{\text{H}}(\cdot )$ be the Hausdorff dimension, where ${\mathcal{H}}^{{\text{dim}}_{\text{H}}(\cdot )}(\cdot )$ is the Hausdorff measure in its dimension on the Borel $\sigma$-algebra.

§1. Motivation

Suppose, we define everywhere surjective $f$:

Let $(A,\mathrm{T})$ be a standard topology. A function $f:A\subseteq {\mathbb{R}}^n\to \mathbb{R}$ is everywhere surjective from $A$ to $\mathbb{R}$, if $f[V]=\mathbb{R}$ for every $V\in \mathrm{T}$.

If f is everywhere surjective, whose graph has zero Hausdorff measure in its dimension (e.g., [1]), we want a unique, satisfying [2] average of $f$, taking finite values only. However, the expected value of $f$:

$$\mathbb{E}[f]=\frac{1}{{\mathcal{H}}^{{\text{dim}}_{\text{H}}(A)}(A)}{\int }_{A}fd{\mathcal{H}}^{{\text{dim}}_{\text{H}}(A)}$$

is undefined since the integral of $f$ is undefined: i.e., the graph of $f$ has Hausdorff dimension $n+1$ with zero $(n+1)$-dimensional Hausdorff measure. Thus, w.r.t a reference point $C\in {\mathbb{R}}^{n+1}$, choose any sequence of bounded functions converging to $f$ [2, §2.1] with the same satisfying [2, §4] and finite expected value [2, §2.2].

[1]: https://mathoverflow.net/questions/476471/is-there-an-explicit-everywhere-surjective-f-mathbbr-to-mathbbr-whose-gr

[2]: https://www.researchgate.net/publication/389499633_Defining_a_Unique_Satisfying_Expected_Value_From_Chosen_Sequences_of_Bounded_Functions_Converging_to_an_Everywhere_Surjective_Function/stats

#HausdorffMeasure #HausdorffDimension
#EverywhereSurjectiveFunction
#ExpectedValue
#Average
#research

Important interview with #privileged #treasurysec #bessent on #BloombergTV
#federalworkforce #dogees #tusk #ukraine #zelensky

#Protip #media interviews: Never call anyone an “ #averageAmerican”. Nor their kids #average. You have no experience of life outside your bubble of #privilege. Working for the gummint was steady work with steady benefits. Your idea is to make everyone grovel.

And learn to stop smirking on camera. Save it for #SiliconValley @bloomberg

Shoot for Good Enough doesn't fit as well on motivational influencers' t-shirts though. I once read "Perfectionism is the ultimate form of self-abuse", which was probably in Chicken Soup for the Soul or some such dreck, but it does strike a certain chord of truth too.

#mediocrity #average #perfectionism #goodenough #happiness

https://getpocket.com/explore/item/it-s-okay-to-be-good-and-not-great

#average : any charge in addition to the regular charge for freight of goods shipped

- French: moyen

- German: durchschnittlich

- Italian: media

- Portuguese: médio / média

- Spanish: mediano

------------

Report an incorrect translation @ https://wordofthehour.org/r/translations

>> So, there you have it. The interiors of our homes, coffee shops and restaurants all look the same. The buildings where we live and work all look the same. The cars we drive, their colours and their logos all look the same. The way we look and the way we dress all looks the same. Our movies, books and video games all look the same. And the brands we buy, their adverts, identities and taglines all look the same.

#average #culture #bored

https://www.alexmurrell.co.uk/articles/the-age-of-average

#CosmicLatte: The #Average #Colour of the #Universe
#Astronomy #Picture of the Day

https://apod.nasa.gov/apod/ap241201.html

It’s #InternationalCoffeeDay! While you’re waking up with your first cup, why not take a moment to find out about the early history of coffee – and how it’s connected to insurance, of all things! https://www.youtube.com/watch?v=4ODk5nL3JX8

10 Ways to Stand Out from the Crowd

https://www.sahilbloom.com/newsletter/10-ways-to-stand-out-from-the-crowd?ref=refind

Wow – it’s yet another insurance-related day on the calendar! This time it’s #InsuranceNerdDay, which definitely calls for our detailed and very nerdy video about the surprisingly interesting history of insurance! https://youtu.be/g977bQJUeFs

Who knew there were so many insurance-related days in the ‘holiday’ calendar? Turns out today is #InsuranceAwarenessDay, so why not make yourself aware of the surprisingly interesting history of insurance, with our video on Coffee, Insurance, and Mathematics! https://youtu.be/g977bQJUeFs

I've seen another "life expectancy (at birth) used to be so low, so we dramatically improved everything" argument. First: Yes, life expectancy by all means is going up, but not the way you might think.

The best, unbutchered source is here: https://ourworldindata.org/life-expectancy-how-is-it-calculated-and-how-should-it-be-interpreted

If you want my tl;dr: [Life expectancy at birth](https://ourworldindata.org/grapher/life-expectancy?time=1950..latest) vs [life expectancy at 15](https://ourworldindata.org/grapher/life-expectancy-at-age-15?tab=chart&facet=none&country=CHN~BRA~SWE~ITA~GBR~OWID_WRL~OWID_EUR) shows the huge gap that infant & young adolescent mortality makes in the stats. For example in 1950 the world average (at birth) is **46.5** years, whereas for those that survive to the puberty it's **62.0** years. If my math is mathing correctly that's +33%. And since the gap is closing towards the modern age, it's not unreasonable to expect that it widened towards past times.

Even [wikipedia has notes on this](https://en.wikipedia.org/wiki/Life_expectancy) for English nobles who _survived_ to the age of 21 could expect to live:

- 1200–1300: to age 64
- 1300–1400: to age 45 (because of the bubonic plague)
- 1400–1500: to age 69
- 1500–1550: to age 71

And even more interesting for Classical Greece (480 - 323 BC):

> Most Greeks and Romans died young. About half of all children died before adolescence. Those who survived to the age of 30 had a reasonable chance of reaching 50 or 60. The truly elderly, however, were rare. Because so many died in childhood, life expectancy at birth was probably between 20 and 30 years.

That's +100% gap between "life expectancy at birth" and "life expectancy at the age of 30".

So please stop saying skewed things like "you would not live up to this age in (ancient times)". Yes, you would quite likely not, but mainly because you'd die before you could do anything about it.

Japan's Nikkei index crossed the 40,000 threshold for the first time early Monday on robust technology issues following the advance of their U.S. counterparts late last week.
The Nikkei benchmark quickly topped the 40,000 line after the market opening. In the first 15 minutes of trading, the 225-issue #Nikkei #Stock #Average rose 237.73 points, or 0.60 percent, from Friday to 40,148.55. The broader Topix index was down 1.55 points, or 0.06 percent, at 2,707.87.

https://english.kyodonews.net/news/2024/03/82e7324f0c19-urgent-nikkei-tops-40000-for-1st-time-after-upbeat-wall-st.html

“‘AI’ is pretty much just shorthand for mediocre”

Just read through “You sound like a bot” by Adi Robertson in the Verge. I hadn’t really put my finger on the right word for my feelings about AI until reading that article but that’s it: it feels very mediocre.
If you want to get a rough overview of how the average frontend engineer might feel about a JavaScript framework, ChatGPT is useful enough. If […]

https://piperhaywood.com/ai-is-pretty-much-just-shorthand-for-mediocre/

NORTHLAND FORECAST FOR JANUARY 17, 2024:

Well #below #average #temperatures continue across the #Northland with #lows below zero Tuesday night and #highs in the single digits Wednesday. #Wind #chills could drop as low as 30-below.

#Lake #effect #snow #showers will continue along the #Bayfield #Peninsula and #Upper Peninsula. The rest of the region remains #dry with a bit of #sunshine.

VIDEO: https://www.fox21online.com/2024/01/16/tuesday-evening-northland-forecast-1-16-2024/