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Nearly 80 years ago, physicist Maria Goeppert Mayer showed that when the nucleus of an atom contains certain numbers of protons and neutrons, such as 50 or 82, it becomes exceptionally stable. In the years since, researchers amassed evidence of more such “magic numbers”, which are found in the most stable, and therefore most abundant, elements in our universe.

Goeppert Mayer and her contemporaries explained these numbers by proposing that protons and neutrons occupy discrete energy levels, or shells. This model, which is still used to interpret many nuclear physics experiments, treats each particle in the nucleus as independent, but our best quantum theories assert that particles within nuclei actually interact strongly.

Jiangming Yao at Sun Yat-sen University in China and his colleagues have now resolved this contradiction and, in the process, elucidated how magic numbers emerge from these interactions.

Yao says the shell model relies on input from experiments and doesn’t encode details of interactions between each particle. Instead, he and his team started their calculations from first principles, which means they mathematically described how particles interact with each other, how they stick together and how much energy is needed to move them apart in more detail.

The two descriptions are analogous to images taken at low and high resolution, respectively, says Yao. “Before, people directly modelled the system at low resolution, or they tried to understand nuclear structure at high resolution. We used modern methods to build the bridge between these descriptions.”

He and his colleagues started with the high-resolution description, gradually made it blurrier in each step of the calculation and tracked how the structure the particles formed changed.

As they moved across their mathematical bridge, the researchers saw the symmetry of the particle’s quantum states change – drawing a graph based on the equations for these states would produce shapes with different symmetries at different resolutions. This change resulted in a nuclear structure such that the nucleus achieved most stability when particles within it grouped themselves in magic numbers.

Jean-Paul Ebran at the French Alternative Energies and Atomic Energy Commission says this work offers a theoretical probe – a mathematical microscope of sorts – that mirrors how experiments work. “Nature looks different depending on the resolution at which you observe. This [study] really maps onto what we do experimentally.”

The change in symmetry that the researchers identified is related to effects described by Albert Einstein’s theory of special relativity, thus painting an even fuller picture of how magic numbers marry together different facets of nuclear theory, says Ebran.

So far, the researchers have tested their theoretical work on a type of tin that is doubly magic because its nuclei each contain 50 protons and 82 neutrons, as well as on several additional nuclei. Going forward, they want to extend their analysis to heavier atomic nuclei, which are typically unstable, and study processes by which heavy nuclei are created in exploding stars or merging neutron stars, says Yao. %!s()

Dawei Lu at the Southern University of Science and Technology in China and his colleagues have seemingly broken this law with a molecule of crotonic acid, which contains atoms of carbon, hydrogen and oxygen. The researchers used the nuclei of four of its carbon atoms as qubits, which are the basic building blocks of quantum computers and can store quantum information. When used in computation, researchers normally control the quantum states of the qubits with bursts of electromagnetic radiation, but in this case, the team leveraged this control to make heat flow from a colder, lower-temperature qubit towards a hotter one instead.

This would never spontaneously happen to something in our macroscopic world, like a cup of coffee, as it would require extra energy to fuel the backwards flow. But in the quantum setting, other forms of fuel are available – in this case, a form of quantum information called “coherence”. “By injecting and controlling this quantum information, we can reverse the direction of heat flow,” says Lu. “We were excited.”

The fact that the laws of thermodynamics break down in the quantum realm is perhaps unsurprising, as they were laid down in the 19th century, about 100 years before the formalisation of quantum physics. To fix this issue, Lu and his colleagues calculated each qubit’s “apparent temperature,” which is a modification of conventional temperature that accounts for some of an object’s quantum properties, like coherence, and saw the second law of thermodynamics become satisfied again and heat flow from a higher apparent temperature to a lower one.

Roberto Serra at the Federal University of ABC in Brazil says that quantum properties like coherence can be considered a type of thermodynamic resource analogous to how, for example, heat is a resource that is used to make a steam engine work. He says that when these quantum, microscopic resources are manipulated, thermodynamics laws can be apparently broken. “But the usual laws of thermodynamics were developed thinking that we do not have access to these microscopic states. This is just an apparent violation because we have to write new laws considering that we have this access,” says Serra.

The researchers now want to turn their heat-reversal experiment into a more practical protocol for controlling heat among qubits, says Lu. Beyond uncovering fundamental connections between quantum information and heat, finding new practical ways to cool qubits could improve quantum computers. This could be of great importance for the burgeoning quantum computing industry as, ultimately, even conventional computers can only work as well as they can avoid heating up, says Serra. %!s()

But that love became an obsession when, after Bordeaux, I stepped out of the Paris metro and onto the Champs Élysées. The lights, the energy, the mystique! There was magic in the air. I felt like I had known the city my entire life and I was simply returning home.

Since then, Paris has had a firm grip on my heart. Over the years, I spent countless visits exploring France and Paris in particular. I’ve run tours in the city and I spent a few months living there in 2019.

Last August, after yet another sojourn, I thought to myself, “What if I moved back?”

I was growing a bit tired of the NYC dating scene, the rising cost of living, and felt like creatively, I was in a rut. With AI coming for creators like me and the industry changing, I was a little lost on what, career wise, would come next.

In short, I needed a change.

And Paris seemed like the best place to go.

It’s cheaper than NYC, I had friends there already, I’ve always wanted to learn French, it would make a great base for exploring Europe, and I could start writing my next book there. There’s nothing like a change in scenery to get the creativity flowing!

But the question remained: how do you move to France?

After all, they don’t have a digital nomad visa, and you’re limited to three months if you visit on your regular Schengen tourist visa.

Well, it turns out that it’s not actually that hard. (And while I can only speak for Americans, I suspect the criteria are similar for Canadians, Aussies, Kiwis, and other “developed” nations.)

If you want to move to France, you have four main options:

• Student visa
• Long-term visitor visa
• Entrepreneur
• Talent visa

The student visa is pretty straightforward. You need to enroll in a full-time university program and show you have enough funds to support yourself (around 600 Euros per month). A friend is currently there on this visa, as she is doing a nine-month intensive French program to become fluent. You still have to apply for the visa, but if you’re enrolled in an accredited school, you’ll likely get approved. Another benefit to this visa is that you can work part-time on it!

But this visa is contingent on being in school, so if you leave or quit, it’s voided. However, when you are done, you can change this to a “job seekers” visa, which can give you up to another six months in France (or more depending on how long your program was).

I thought about applying for the entrepreneur visa, but the application process is pretty complex and can take months to be approved. You have to set up your business in France — and that is a lot of paperwork. You have to show that your business makes money, has clients (at least one of them being French), and can support you full time (you have to make at least the French minimum wage). They will scrutinize your assets and financials a lot and you’ll need to register your business in France and pay French taxes.

Additionally, France offers a talent visa. If you’re an expert in your field (with verifiable accreditations and accolades), this could be a good visa to apply for, especially if you plan to stay in France for a long time and want to physically work there. But, again, you’ll need to have some source of income and plan to do physical business in the France. If you’re not an academic but someone in the arts, you have to show how you are going to add to the “culture of France” in some way.

While I would meet the requirements for both those visas, the process for either would have been time consuming, and, since I’m not sure France is going to be my forever home, I decided not to go through that process.

So I went with the long-term visitor visa, officially called VLS-TS visiteur. This allows me to stay up to 12 months in France and is renewable in the country. It does come with a lot of restrictions, however: I can’t physically work here and I am not allowed into the French social welfare system.

It’s also the visa a lot of Americans (and most retirees) are coming in on. If you have passive income or retirement savings and just want to live in France, this is a good option for you. It allows you to set up a bank account and it can be renewed pretty much indefinitely.

The paperwork for this visa (which I’ll get into in a bit) essentially comes down to whether or not you can support yourself. French authorities want to make sure you won’t be a burden on the system.

My biggest question centered around whether or not I could “work” on this visa. A lot creators and digital nomads are applying for — and getting — this visa.

But remember: you aren’t allowed to work on this visa so how are you going to work if you can’t work?

So let me take a moment to talk about “work” from a legal standpoint.

As I mentioned, there’s no digital nomad visa that allows you to work in France. And the French tax office said last July that remote work is considered taxable — but there are no official laws that. That’s just their opinion and there’s been no progress in codifying that opinion. In fact, my visa doesn’t even come with a tax number, so it’s impossible for me to be taxed. I couldn’t pay taxes even if I wanted to.

The visa office asks how you’ll support yourself when you are in France and one of the sources of income they accept is “income from work.” I was very open that my means of support would be the income from this blog and my books.

The system there really hasn’t caught up to remote work. It still defines “work” as something you do physically that could take away a job from a French person.

So could I run walking tours in Paris? Nope. Could you work at a café? Nope.

But does writing this blog post count as work? After all, no one is paying for it. It’s a free resource. Does going to a café to work on the next great American novel count? What about my weekly newsletter that has affiliate links (that generate income) in it? Is writing that considered work?

The gray area is considerable, so the best thing to do is to talk to a lawyer. I recommend Daniel Tostado (yes, that’s his real name), one of the biggest American-French immigration lawyers in Paris. He has a huge law firm and regularly meets with government officials to clarify the rules on this. He gave me some solid advice.

For this long-term visitor visa, the application process is pretty easy. You go to the French government website, fill out your application, take the application number to the TLScontact website, and make an appointment. (France outsources the appointment process to this company, which then sends everything to the French consulate for approval.)

The main things you’ll want to demonstrate are that you have accommodation for 90 days on your application and enough income to support yourself. They just want to make sure you aren’t going to be a burden on the system. I went overboard. Here’s what I brought to my appointment:

• The application (you need to bring a printout)
• A confirmation of the appointment
• Extra passport photos
• Proof of accommodation (this can be Airbnbs, hotels, or someone’s home — if it’s the latter, make sure that they are the owner!)
• A letter attesting I won’t work
• A financial summary letter explaining why I am moving there and my source of income
• Health insurance that meets French legal requirements (I used Mondassur.)
• Three months of bank statements showing regular deposits
• A second bank account, my tax returns, and my W-2

I didn’t need the second bank account, tax returns, or W-2 (they weren’t required) but the woman at the appointment said the more proof you had the better and it wouldn’t hurt. And another friend who got this visa didn’t bring beyond what was asked in the application form. But I wanted to take no chances!

At the appointment, the staff double-checks your paperwork, takes your biometric data, collects the fees, and then sends your passport to the embassy for processing. Officially, it takes up to 15 days to process your visa, but everyone I know seems to get it back within a week.

The worst part was not knowing if you are approved until your passport is returned. While you can track the progress of your application, you don’t know the verdict until your passport is back in your hands. While I was waiting (and anxiously spiraling), I looked on Reddit and found that most denials were for lack of income or issues with their accommodation, which seem to be the two most important things to have perfect on your application.

Once you get your visa back, you can enter France any time after the start date. After you enter, you have 90 days to register your visa with the state (they give you a little QR code with instructions). This allows you to “officially” be on this long-term visa. (It’s a more paperwork kind of thing.)

Once that is done, there’s nothing else to do and you can stay in France (and Europe) until your visa expires!

So, if you’re considering moving to France, these are your options. But, as always, I’m not a lawyer and this is not legal advance. In regard to work and other issues or questions, it’s important that you seek legal counsel and get their expert opinion!

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That player is Usman Tariq, the Pakistan spinner.

He might have only played four international matches but, right now, his bowling action is the most discussed anywhere in the world.

"He doesn't care about these things," said Pakistan captain Salman Agha.

"Since the time he has started cricket, these talks are going on about him.

"He doesn't care about these things."

So what is the history, what are the rules and what has been said about him?

Get in touch

Send us your questions

Who is Usman Tariq and why has his bowling action been controversial?

Tariq is a 28-year-old off-spinner from Pakistan. He has played just two first-class matches, instead concentrating on T20s.

After an initial season in the Pakistan Super League in 2023-24, he has become more prominent over the past year with a successful PSL and Caribbean Premier League in 2025, plus two matches in this winter's ILT20 in the United Arab Emirates.

But it is Tariq's bowling action that has made him so discussed.

He trots to the crease and pauses in his delivery stride, also coming to a complete stop, before slinging the ball down with a low-armed action.

During the warm-up series between Australia and Pakistan before this World Cup, Australia batter Cameron Green mimicked a side-arm throwing action after being dismissed by Tariq. He later apologised.

During the ILT20, England batter Tom Banton also appeared to suggest to the umpire that Tariq was throwing.

Tariq's action has twice been reported by on-field umpires during the Pakistan Super League, in 2024 and again in 2025, but he was cleared on both occasions.

What are the laws?

The regulations around whether a bowler is 'throwing' are simple.

Bowlers are not allowed to straighten their arm by more than 15 degrees from the time the arm reaches the shoulder in the bowling action.

"An illegal bowling action is a bowling action where the player's elbow extension exceeds 15 degrees between their arm reaching the horizontal and the ball being released," the International Cricket Council's playing conditions say.

Bowlers with suspected actions are first reported by the on-field umpires or match referee and then must undergo testing at an ICC-approved testing centre.

Tariq's action was cleared at the centre in Lahore, Pakistan.

The laws and regulations around Tariq's pause are less clearcut, with the issue falling under the "unfair play" section of the laws and playing conditions – rather than being explicitly defined by a law.

The on-field umpires can interpret the unfair play law, possibly using Law 41.5 which covers the "deliberate distraction or deception of batter" or the more broader laws around unfair play, to ensure the match goes ahead fairly.

"If the pause is obviously longer or shorter, or if he bowls without a pause, that will be considered deception" was one interpretation offered by former ICC umpire Anil Chaudhary this week.

"Everything seems OK up to now. I am not seeing anything wrong."

The issue came up in 2014 when former India spinner Ravichandran Ashwin paused in his delivery stride, resulting in a tightening of how umpires interpreted these laws.

The umpires are left to judge whether the length of a pause in a bowler's action is fair or unfair.

What has been said?

Tariq has said any flex in his action comes because he has "two elbows" in the joint.

Pakistan captain Salman said the spinner is his "trump card".

"He has been bowling very well for the past few months," Salman said.

"And he has also bowled very well in the league cricket he was playing before playing for Pakistan.

"You can say that he is a trump card for us."

India captain Suryakumar Yadav was seen mimicking Tariq's action in the nets before the encounter and backed his batters to succeed.

"He is a different character when he comes to bowl, but at the same time we cannot just surrender to him," Suryakumar said.

"We practise with similar type of bowlers and similar actions and we will try to execute what we are practising in net sessions."

This article is the latest from BBC Sport's Ask Me Anything team. %!s()