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The Hidden Math of Ocean Waves

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The Hidden Math of Ocean Waves


In 2011, Deconinck and Oliveras simulated different disturbances with higher and higher frequencies and watched what happened to the Stokes waves. As they expected, for disturbances above a certain frequency, the waves persevered.

But as the pair continued to dial up the frequency, they suddenly began to see destruction again. At first, Oliveras worried that there was a bug in the computer program. “Part of me was like, this can’t be right,” she said. “But the more I dug, the more it persisted.”

In fact, as the frequency of the disturbance increased, an alternating pattern emerged. First there was an interval of frequencies where the waves became unstable. This was followed by an interval of stability, which was followed by yet another interval of instability, and so on.

Deconinck and Oliveras published their finding as a counterintuitive conjecture: that this archipelago of instabilities stretches off to infinity. They called all the unstable intervals “isole”—the Italian word for “islands.”

It was strange. The pair had no explanation for why instabilities would appear again, let alone infinitely many times. They at least wanted a proof that their startling observation was correct.

Bernard Deconinck and Katie Oliveras uncovered a strange pattern in computational studies of wave stability.

Photograph: Courtesy of Bernard Deconinck

The Hidden Math of Ocean Waves

Photograph: Courtesy of Katie Oliveras

For years, no one could make any progress. Then, at the 2019 workshop, Deconinck approached Maspero and his team. He knew they had a lot of experience studying the math of wavelike phenomena in quantum physics. Perhaps they could figure out a way to prove that these striking patterns arise from the Euler equations.

The Italian group got to work immediately. They started with the lowest set of frequencies that seemed to cause waves to die. First, they applied techniques from physics to represent each of these low-frequency instabilities as arrays, or matrices, of 16 numbers. These numbers encoded how the instability would grow and distort the Stokes waves over time. The mathematicians realized that if one of the numbers in the matrix was always zero, the instability would not grow, and the waves would live on. If the number was positive, the instability would grow and eventually destroy the waves.

To show that this number was positive for the first batch of instabilities, the mathematicians had to compute a gigantic sum. It took 45 pages and nearly a year of work to solve it. Once they’d done so, they turned their attention to the infinitely many intervals of higher-frequency wave-killing disturbances—the isole.

First, they figured out a general formula—another complicated sum—that would give them the number they needed for each isola. Then they used a computer program to solve the formula for the first 21 isole. (After that, the calculations got too complicated for the computer to handle.) The numbers were all positive, as expected—and they also seemed to follow a simple pattern that implied they would be positive for all the other isole as well.



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Do Waterproof Sneakers Keep the Slosh In or Out? Let WIRED Explain

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Do Waterproof Sneakers Keep the Slosh In or Out? Let WIRED Explain


Running with wet feet, in wet socks, in wet shoes is the perfect recipe for blisters. It’s also a fast track to low morale. Nothing dampens spirits quicker than soaked socks. On ultra runs, I always carry spares. And when faced with wet, or even snowy, mid-winter miles, the lure of weatherproof shoes is strong. Anything that can stem the soggy tide is worth a go, right?

This isn’t as simple an answer as it sounds. In the past, a lot of runners—that includes me—felt waterproof shoes came with too many trade-offs, like thicker, heavier uppers that change the feel of your shoes or a tendency to run hot and sweaty. In general, weatherproof shoes are less comfortable.

But waterproofing technology has evolved, and it might be time for a rethink. Winterized shoes can now be as light as the regular models, breathability is better, and the comfort levels have improved. Brands are also starting to add extra puddle protection to some of the most popular shoes. So it’s time to ask the questions again: Just how much difference does a bit of Gore-Tex really make? Are there still trade-offs for that extra protection? And is it really worth paying the premium?

I spoke to the waterproofing pros, an elite ultra runner who has braved brutal conditions, and some expert running shoe testers. Here’s everything you need to know about waterproof running shoes in 2026. Need more information? Check out our guide to the Best Running Shoes, our guide to weatherproof fabrics, and our guide to the Best Rain Jackets.

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How Do Waterproof Running Shoes Work?

On a basic level, waterproof shoes add extra barriers between your nice dry socks and the wet world outside. If you’re running through puddles deep enough to breach your heel collars, you’re still going to get wet feet. But waterproof shoes can protect against rain, wet grass, snow, and smaller puddles.

Gore-Tex is probably the most common waterproofing tech in footwear, but it’s not the only solution in town. Some brands have proprietary tech, or you might come across alternative systems like eVent and Sympatex. That GTX stamp is definitely the one you’re most likely to encounter, so here’s how GTX works.

The water resistance comes from a layered system that is composed of a durable water repellent (DWR) coating to the uppers with an internal membrane, along with other details like taped seams, more sealed uppers with tighter woven mesh, gusseted tongues, and higher, gaiter-style heel collars.



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I’ve Been Waiting Months for This Gorgeous Laptop to Drop in Price. It Finally Happened

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I’ve Been Waiting Months for This Gorgeous Laptop to Drop in Price. It Finally Happened


After a long time of resisting significant price drops, the Asus Zenbook S 16 has finally dropped down to $1,000, which is $500 off its retail price.

It’s normal for laptops to dip in price toward the end of their lifespan, close to when an update comes out. But the Asus Zenbook S 16 has held on. To be fair, it’s an extremely high-end Windows laptop, one of the prettiest to come out last year. It’s sleek, portable, and has a striking design. It even gets fantastic battery life, on par with a MacBook. Speaking of MacBooks, this Zenbook is the laptop I saw tech journalists traveling with more than anything else. Given how much tech they review, that’s quite an endorsement.

But the S 16 has always been hard for me to recommend when the cheapest model available was $1,500. I was always on the lookout for a more significant price cut, but it never dropped more than a couple hundred bucks. And even though it always came with 24 GB of RAM and a terabyte of storage, the price was a hard pill to swallow. Well, the day has finally come. It’s now down to $1,000 over at Best Buy as part of the store’s Presidents’ Day sale. That’s an incredible price for this much laptop.

Photograph: Christopher Null

Asus

Zenbook S 16 (UM5606)

The previously mentioned memory and storage still apply here, along with the 2880 x 1800 OLED display with a 120-Hz refresh rate. This laptop basically has every high-end feature you could imagine, but one of my favorite aspects is the ports. Despite the thin profile, the S 16 keeps all the legacy ports you might want, including HDMI, USB-A, and even a full-size SD card slot.

There is also a smaller, 14-inch model, but its discount is not as strong as the 16-inch model. It comes in at $1,300 right now, which is still a solid price for this configuration.

I should say that Asus has an update in the works for 2026 with the latest Intel chips, but it’s only coming to the 14-inch model. I won’t lie: Based on my testing, these CPUs will make a significant difference in performance—especially on the graphics front. But I have a feeling Asus will be selling this device for an even higher price for much longer, especially with the recent development around memory shortage.

While the Zenbook S 16 is certainly the best deal at Best Buy for its Presidents’ Day sale, I would also recommend the Asus Zephyrus G14, which is also $500 off. This configuration comes with a powerful RTX 5070 Ti graphics card and is one of our favorite gaming laptops.



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‘Uncanny Valley’: ICE’s Secret Expansion Plans, Palantir Workers’ Ethical Concerns, and AI Assistants

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‘Uncanny Valley’: ICE’s Secret Expansion Plans, Palantir Workers’ Ethical Concerns, and AI Assistants


Brian Barrett: They’ve got 80 billion or so to spend 75 billion of that I think they have to spend in the next four years. So yeah, they’re going to keep expanding. And when you think of how much of an impact 3000 agents officers had in Minneapolis alone, that’s like an eighth of the, they can repeat some version of that in a lot of different spots.

Leah Feiger: And I’ve been fielding, honestly, shout out to the many local reporters around the country who’ve been contacting me in the last day or so, just to ask questions about the locations that we named that are near them or in their states or cities. And the thing to me that keeps coming up is that in addition to new buildings, they’re getting put into preexisting government buildings, preexisting leases, or that that appears to be the plan. And then we’ve also found that a bunch of these ICE offices are being located near plans for giant immigration detention warehouses, and we’re looking at offices being set up, say 20 minutes, an hour and 20 minutes away for these. Yeah. So we’re looking at different, the triangulation of this around you have to have your lawyers, your agents, have a place to get their orders and put their computers and do in some ways very mundane things that are required of an operation like this one.

Brian Barrett: Well, Leah, that’s a good point. I think when people hear ICE offices or when I do just instinctively, I think of ICE as guys with guns and masks and all that, but that’s not exactly what we’re saying here. Do you mind talking through what these offices seem to be queued up to be used for and by whom? Because ICE is not just the masked guys with bad tattoos.

Leah Feiger: Yes, absolutely. So what we reported in this story as well was some of the specific parts of ICE that actually reached out to GSA and asked them to expedite the process of getting new leases, et cetera, included in that, for example, where representatives from Ola, Ola is ICE’s office of the principal legal advisor. So that’s the lawyers, those are the ICE lawyers that are working with the courts and arguing back or deportation orders saying yes, no, et cetera, signing the documents, putting everything in front of judges. This is a really important part of this entire operation that we’re not talking about a ton. There’s a lot of focus on the DOJ. There’s a lot of focus. There was an excellent article this week in Politico talking about all of these federal judges that are really, really upset that DHS and ICE are ignoring their requests for immigrants to not be detained anymore.

The missing level of that is the lawyers that are part of this that are representing ICE to the US government here, and that’s ola. So they’ve reached out to GSA extensively as we report to get these leasing locations, specifically with the OLA legal request. I just want to get across how big this is. How massive is this ICE repeatedly outlined its expansion to cities around the us And this one piece of memorandum that we got from Ola stated that ICE will be expanding its legal operations into Birmingham, Alabama, Fort Lauderdale, Fort Myers, Jacksonville, and Tampa, Des Moines, Iowa, Boise, Idaho, Louisville, Kentucky, Baton Rouge, Louisiana, grand Rapids, Michigan, St. Louis, Missouri, rally, North Carolina, long Island, New York, Columbus, Ohio, Oklahoma City, Oklahoma, Pittsburgh, Pennsylvania, Charleston and Columbia, South Carolina, Nashville, Tennessee, Richmond, Virginia, Spokane, Washington and Cord Delaine, Idaho and Milwaukee, Wisconsin. We have other locations as well throughout the rest of the article, but those are the requests from OLA.



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