![PI010101's tweet photo. Here is one surprisingly elegant equivalent reformulation of the Riemann Hypothesis: Let B be the pointwise maximal function on [0,1] satisfying B(0) = B(1) = 0 and 2B((x+y)/2) ≤ B(x) + B(y) + |x - y| for all x and y in [0,1]. Extend it periodically to all the real line.
As Grok correctly notes, this is Takagi function (or blancmange curve), with a fractal-like graph. It is continuous, nowhere differentiable, and less well-known than the Weierstrass function.
This function also emerged recently (and independently) in a joint work https://t.co/pgJw9MaEA1 with my student, Natanael Alpay, as a sharp lower bound for the L^1 norm of the square function applied to indicator functions of sets. I was unaware at the time that it is known as the Takagi function or of its connection to the Riemann Hypothesis. Large language models will undoubtedly accelerate scientific discoveries by forging connections across diverse fields of science.](https://pbs.twimg.com/media/G86cuY-WMAAM4o0.jpg)
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“It would seem, that Babylon must be rebuilt, and become again the center of the world, and the glory of kingdoms, as we have it represented in the 18th chapter of the Apocalypse… the wonder is that the restoration of Babylon has never yet been attempted; but the jealousy of the Great Powers, and the difficulty of dealing with Turkey, have, probably, caused the delay. As soon, however, as Christendom is united in the form of the Ten Confederate Kingdoms, all jealousy will be at an end, and the great prize may then be seized for the common good. No doubt commerce will be the exciting motive: the civilized world will, perhaps, combine to build a great central emporium, which by their united exertions will quickly surpass all other cities, and finally become the capital of the Antichrist.”
George Pember, Great Prophecies Concerning the Gentiles, the Jews and the Church of God p. 170-171
This will boost the late-model used car market in 2027...
https://t.co/9GwFzw2K2W
The Trump weak dollar strategy doesn’t exist - @ericwallerstein
He says dollar policy fears are overstated and FX will track macro fundamentals, not some hidden plan from Washington.
“There’s no weak dollar policy. There’s no overarching strategy.”
The chart below shows the national average retail gasoline price from an AAA Motor Club survey. Yesterday was day 11 since the war started on February 28.
The middle panel shows the 11-day dollar rise. The current 61-cent rise was exceeded only at the beginning of the Russia/Ukraine war in March 2022.
The bottom panel shows the 11-day percentage rise. Gas is up 20.6%, the largest 11-day percentage increase in the 22+-year history of this measure.
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https://t.co/dVduU2YIW1
As I'm a software engineer and I know what’s actually happening...The easy phase is over simply. slapping a subscription on a basic tool and calling it a startup doesn’t work anymore. users have too many options, switching cost is low, and AI is making basic features a commodity.
investors are not avoiding software, they are avoiding replaceable software. if ur product can be rebuilt in a weekend with open source + an API, it is not defensible. but if software is tied to real workflows, data moats, infra, hardware, regulation, or deep domain knowledge, it is still insanely valuable.
look at where money is going: dev tools, security, infra, vertical SaaS, AI applied to real industries, fintech rails, healthcare systems. that is all software, just not the copy paste kind.
software is not un-investable. lazy software is.
Pure software is rapidly becoming un-investable.
https://t.co/IdA57FqMI2
Trump Was Not Just a Friend of Epstein but an Integral Part of the Operation
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Solari Report Money & Markets: February 12, 2025 with Catherine Austin Fitts & John Titus
Full Report: https://t.co/7spWAlEwxt
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For all those who went hysterical that Spirit pf Alaska was dead... It's now haunting the Great Resetters - bigly!
https://t.co/S8emXX43eX
@grok what hotel in new york is this video from
https://t.co/p3X3rX4W3K
https://t.co/p3X3rX4W3K
https://t.co/p3X3rX4W3K
Here is one surprisingly elegant equivalent reformulation of the Riemann Hypothesis: Let B be the pointwise maximal function on [0,1] satisfying B(0) = B(1) = 0 and 2B((x+y)/2) ≤ B(x) + B(y) + |x - y| for all x and y in [0,1]. Extend it periodically to all the real line.
As Grok correctly notes, this is Takagi function (or blancmange curve), with a fractal-like graph. It is continuous, nowhere differentiable, and less well-known than the Weierstrass function.
This function also emerged recently (and independently) in a joint work https://t.co/pgJw9MaEA1 with my student, Natanael Alpay, as a sharp lower bound for the L^1 norm of the square function applied to indicator functions of sets. I was unaware at the time that it is known as the Takagi function or of its connection to the Riemann Hypothesis. Large language models will undoubtedly accelerate scientific discoveries by forging connections across diverse fields of science.
![PI010101's tweet photo. Here is one surprisingly elegant equivalent reformulation of the Riemann Hypothesis: Let B be the pointwise maximal function on [0,1] satisfying B(0) = B(1) = 0 and 2B((x+y)/2) ≤ B(x) + B(y) + |x - y| for all x and y in [0,1]. Extend it periodically to all the real line.
As Grok correctly notes, this is Takagi function (or blancmange curve), with a fractal-like graph. It is continuous, nowhere differentiable, and less well-known than the Weierstrass function.
This function also emerged recently (and independently) in a joint work https://t.co/pgJw9MaEA1 with my student, Natanael Alpay, as a sharp lower bound for the L^1 norm of the square function applied to indicator functions of sets. I was unaware at the time that it is known as the Takagi function or of its connection to the Riemann Hypothesis. Large language models will undoubtedly accelerate scientific discoveries by forging connections across diverse fields of science.](https://pbs.twimg.com/media/G86cwEeW0AAKGga.png)
Truth
Funny thing. Historically speaking, as in every single instance throughout all of human history. The women can and have been replaced by women from the outside. Never once has it ever worked when the women tried to replace the men of the land with men from outside. Never once!
>go to an Ivy League school
>be a corporate slave
>work 60-80hrs per week
>sacrifice social life for years
>get replaced by Indians
The American Dream in 2025
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What's in Hannahs closet?
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