Many believe Putin is simply sabre rattling and intimidating us. They see no threat of nuclear war. We can send NATO troops into Ukraine without risking a nuclear war.

I keep reading that Putin is just using nuclear blackmail and that a strong leader will call the bluff. That, in my opinion, misunderstands the danger of sending NATO into Ukraine.
It assumes that once NATO moves in, Putin can either push the red nuclear button or not.
Sure, Putin won't go nuclear if NATO invades Ukraine. So we're safe? Can't we just move NATO?

No, because history has taught us that wars often escalate far beyond our initial expectations. One domino falls, knocking down another. That's why having clear boundaries is vital. Crossing a seemingly harmless line can set off a chain of events that are unstoppable once started.
One example is WWI. The assassin of Archduke Franz Ferdinand could not have known that his actions would kill millions. They couldn't have known that invading Serbia to punish them for not handing over the accomplices would start a world war. Every action triggered a counter-action, plunging Europe into a brutal and bloody war. Each leader saw their actions as limited, not realizing how they kept the dominos falling.

Nobody can predict the future, but it's easy to imagine how NATO intervention could trigger a chain of events leading to a total war. Let me suggest some outcomes.
NATO creates a no-fly-zone. In retaliation, Russia bombs NATO airfields. Russia may see this as a limited counter-move that shouldn't cause further NATO escalation. They think it's a reasonable response to force NATO out of Ukraine. Nobody has yet thought to use the nuke.
Will NATO act? Polish airfields bombed, will they be stuck? Is this an article 5 event? If so, what should be done?

It could happen. Maybe NATO sends troops into Ukraine to punish Russia. Maybe NATO will bomb Russian airfields.

Putin's response Is bombing Russian airfields an invasion or an attack? Remember that Russia has always used nuclear weapons for defense, not offense. But let's not panic, let's assume Russia doesn't go nuclear.

Maybe Russia retaliates by attacking NATO military bases with planes. Maybe they use ships to attack military targets. How does NATO respond? Will they fight Russia in Ukraine or escalate? Will they invade Russia or attack more military installations there?
Seen the pattern? As each nation responds, smaller limited military operations can grow in scope.

So far, the Russian military has shown that they begin with less brutal methods. As losses and failures increase, brutal means are used. Syria had the same. Assad used chemical weapons and attacked hospitals, schools, residential areas, etc.
A NATO invasion of Ukraine would cost Russia dearly. “Oh, this isn't looking so good, better pull out and finish this war,” do you think? No way. Desperate, they will resort to more brutal tactics. If desperate, Russia has a huge arsenal of ugly weapons. They have nerve agents, chemical weapons, and other nasty stuff.

What happens if Russia uses chemical weapons? What if Russian nerve agents kill NATO soldiers horribly? West calls for retaliation will grow. Will we invade Russia? Will we bomb them?

We are angry and determined to punish war criminal Putin, so NATO tanks may be heading to Moscow. We want vengeance for his chemical attacks and bombing of our cities.
Do you think the distance between that red nuclear button and Putin's finger will be that far once NATO tanks are on their way to Moscow?

We might avoid a nuclear apocalypse. A NATO invasion force or even Western cities may be used by Putin. Not as destructive as ICBMs. Putin may think we won't respond to tactical nukes with a full nuclear counterattack. Why would we risk a nuclear Holocaust by launching ICBMs on Russia?

Maybe. My point is that at every stage of the escalation, one party may underestimate the other's response. This war is spiraling out of control and the chances of a nuclear exchange are increasing. Nobody really wants it.

Fear, anger, and resentment cause it. If Putin and his inner circle decide their time is up, they may no longer care about the rest of the world. We saw it with Hitler. Hitler, seeing the end of his empire, ordered the destruction of Germany. Nobody should win if he couldn't. He wanted to destroy everything, including Paris.

In other words, the danger isn't what happens after NATO intervenes The danger is the potential chain reaction. Gambling has a psychological equivalent. It's best to exit when you've lost less. We humans are willing to take small risks for big rewards. To avoid losses, we are willing to take high risks. Daniel Kahneman describes this behavior in his book Thinking, Fast and Slow.

And so bettors who have lost a lot begin taking bigger risks to make up for it. We get a snowball effect. NATO involvement in the Ukraine conflict is akin to entering a casino and placing a bet. We'll start taking bigger risks as we start losing to Russian retaliation. That's the game's psychology.

It's impossible to stop. So will politicians and citizens from both Russia and the West, until we risk the end of human civilization.

You can avoid spiraling into ever larger bets in the Casino by drawing a hard line and declaring “I will not enter that Casino.” We're doing it now. We supply Ukraine. We send money and intelligence but don't cross that crucial line.

It's difficult to watch what happened in Bucha without demanding NATO involvement. What should we do? Of course, I'm not in charge. I'm a writer. My hope is that people will think about the consequences of the actions we demand. My hope is that you think ahead not just one step but multiple dominos.

More and more, we are driven by our emotions. We cannot act solely on emotion in matters of life and death. If we make the wrong choice, more people will die.

Read the original post here.

If tasked with the problem of coming up with a zk-SNARK protocol, many people would make their way to this point and then get stuck and give up. How can a verifier possibly check every single piece of the computation, without looking at each piece of the computation individually? But it turns out that there is a clever solution.

Polynomials

Polynomials are a special class of algebraic expressions of the form:

• x+5
• x^4
• x^3+3x^2+3x+1
• 628x^{271}+318x^{270}+530x^{269}+…+69x+381

i.e. they are a sum of any (finite!) number of terms of the form cx^k

There are many things that are fascinating about polynomials. But here we are going to zoom in on a particular one: polynomials are a single mathematical object that can contain an unbounded amount of information (think of them as a list of integers and this is obvious). The fourth example above contained 816 digits of tau, and one can easily imagine a polynomial that contains far more.

Furthermore, a single equation between polynomials can represent an unbounded number of equations between numbers. For example, consider the equation A(x)+ B(x) = C(x). If this equation is true, then it's also true that:

• A(0)+B(0)=C(0)
• A(1)+B(1)=C(1)
• A(2)+B(2)=C(2)
• A(3)+B(3)=C(3)

And so on for every possible coordinate. You can even construct polynomials to deliberately represent sets of numbers so you can check many equations all at once. For example, suppose that you wanted to check:

• 12+1=13
• 10+8=18
• 15+8=23
• 15+13=28

You can use a procedure called Lagrange interpolation to construct polynomials A(x) that give (12,10,15,15) as outputs at some specific set of coordinates (eg. (0,1,2,3)), B(x) the outputs (1,8,8,13) on thos same coordinates, and so forth. In fact, here are the polynomials:

• A(x)=-2x^3+\frac{19}{2}x^2-\frac{19}{2}x+12
• B(x)=2x^3-\frac{19}{2}x^2+\frac{29}{2}x+1
• C(x)=5x+13

Checking the equation A(x)+B(x)=C(x) with these polynomials checks all four above equations at the same time.

Comparing a polynomial to itself

You can even check relationships between a large number of adjacent evaluations of the same polynomial using a simple polynomial equation. This is slightly more advanced. Suppose that you want to check that, for a given polynomial F, F(x+2)=F(x)+F(x+1) with the integer range {0,1…89} (so if you also check F(0)=F(1)=1, then F(100) would be the 100th Fibonacci number)

As polynomials, F(x+2)-F(x+1)-F(x) would not be exactly zero, as it could give arbitrary answers outside the range x={0,1…98}. But we can do something clever. In general, there is a rule that if a polynomial P is zero across some set S=\{x_1,x_2…x_n\} then it can be expressed as P(x)=Z(x)*H(x), where Z(x)=(x-x_1)*(x-x_2)*…*(x-x_n) and H(x) is also a polynomial. In other words, any polynomial that equals zero across some set is a (polynomial) multiple of the simplest (lowest-degree) polynomial that equals zero across that same set.

Why is this the case? It is a nice corollary of polynomial long division: the factor theorem. We know that, when dividing P(x) by Z(x), we will get a quotient Q(x) and a remainder R(x) is strictly less than that of Z(x). Since we know that P is zero on all of S, it means that R has to be zero on all of S as well. So we can simply compute R(x) via polynomial interpolation, since it's a polynomial of degree at most n-1 and we know n values (the zeros at S). Interpolating a polynomial with all zeroes gives the zero polynomial, thus R(x)=0 and H(x)=Q(x).

Going back to our example, if we have a polynomial F that encodes Fibonacci numbers (so F(x+2)=F(x)+F(x+1) across x=\{0,1…98\}), then I can convince you that F actually satisfies this condition by proving that the polynomial P(x)=F(x+2)-F(x+1)-F(x) is zero over that range, by giving you the quotient:
H(x)=\frac{F(x+2)-F(x+1)-F(x)}{Z(x)}
Where Z(x) = (x-0)*(x-1)*…*(x-98).
You can calculate Z(x) yourself (ideally you would have it precomputed), check the equation, and if the check passes then F(x) satisfies the condition!

Now, step back and notice what we did here. We converted a 100-step-long computation into a single equation with polynomials. Of course, proving the N'th Fibonacci number is not an especially useful task, especially since Fibonacci numbers have a closed form. But you can use exactly the same basic technique, just with some extra polynomials and some more complicated equations, to encode arbitrary computations with an arbitrarily large number of steps.

WWDC 2022 began early Tuesday morning. WWDC brought a ton of new features (which went for just shy of two hours).

With so many announcements, we thought we'd compile them. And now...

WWDC?

WWDC is Apple's developer conference. This includes iOS, macOS, watchOS, and iPadOS (all of its iPads). It's where Apple announces new features for developers to use. It's also where Apple previews new software.

Virtual WWDC runs June 6-10.  You can rewatch the stream on Apple's website.

WWDC 2022 news:

Completely everything. Really. iOS 16 first.

iOS 16.

iOS 16 is a major iPhone update. iOS 16 adds the ability to customize the Lock Screen's color/theme. And widgets. It also organizes notifications and pairs Lock Screen with Focus themes. Edit or recall recently sent messages, recover recently deleted messages, and mark conversations as unread. Apple gives us yet another reason to stay in its walled garden with iMessage.

New iOS includes family sharing. Parents can set up a child's account with parental controls to restrict apps, movies, books, and music. iOS 16 lets large families and friend pods share iCloud photos. Up to six people can contribute photos to a separate iCloud library.

Live Text is getting creepier. Users can interact with text in any video frame. Touch and hold an image's subject to remove it from its background and place it in apps like messages. Dictation offers a new on-device voice-and-touch experience. Siri can run app shortcuts without setup in iOS 16. Apple also unveiled a new iOS 16 feature to help people break up with abusive partners who track their locations or read their messages. Safety Check.

Apple Pay Later allows iPhone users to buy products and pay for them later. iOS 16 pushes Mail. Users can schedule emails and cancel delivery before it reaches a recipient's inbox (be quick!). Mail now detects if you forgot an attachment, as Gmail has for years. iOS 16's Maps app gets "Multi-Stop Routing," .

Apple News also gets an iOS 16 update. Apple News adds My Sports. With iOS 16, the Apple Watch's Fitness app is also coming to iOS and the iPhone, using motion-sensing tech to track metrics and performance (as long as an athlete is wearing or carrying the device on their person). 

iOS 16 includes accessibility updates like Door Detection.

watchOS9

Many of Apple's software updates are designed to take advantage of the larger screens in recent models, but they also improve health and fitness tracking.

The most obvious reason to upgrade watchOS every year is to get new watch faces from Apple. WatchOS 9 will add four new faces.

Runners' workout metrics improve.
Apple quickly realized that fitness tracking would be the Apple Watch's main feature, even though it's been the killer app for wearables since their debut. For watchOS 9, the Apple Watch will use its accelerometer and gyroscope to track a runner's form, stride length, and ground contact time. It also introduces the ability to specify heart rate zones, distance, and time intervals, with vibrating haptic feedback and voice alerts.

The Apple Watch's Fitness app is coming to iOS and the iPhone, using the smartphone's motion-sensing tech to track metrics and performance (as long as an athlete is wearing or carrying the device on their person).

We'll get sleep tracking, medication reminders, and drug interaction alerts. Your watch can create calendar events. A new Week view shows what meetings or responsibilities stand between you and the weekend.

iPadOS16

WWDC 2022 introduced iPad updates. iPadOS 16 is similar to iOS for the iPhone, but has features for larger screens and tablet accessories. The software update gives it many iPhone-like features.

iPadOS 16's Home app, like iOS 16, will have a new design language. iPad users who want to blame it on the rain finally have a Weather app. iPadOS 16 will have iCloud's Shared Photo Library, Live Text and Visual Look Up upgrades, and FaceTime Handoff, so you can switch between devices during a call.

Apple highlighted iPadOS 16's multitasking at WWDC 2022. iPad's Stage Manager sounds like a community theater app. It's a powerful multitasking tool for tablets and brings them closer to emulating laptops. Apple's iPadOS 16 supports multi-user collaboration. You can share content from Files, Keynote, Numbers, Pages, Notes, Reminders, Safari, and other third-party apps in Apple Messages.

M2-chip

WWDC 2022 revealed Apple's M2 chip. Apple has started the next generation of Apple Silicon for the Mac with M2. Apple says this device improves M1's performance.

M2's second-generation 5nm chip has 25% more transistors than M1's. 100GB/s memory bandwidth (50 per cent more than M1). M2 has 24GB of unified memory, up from 16GB but less than some ultraportable PCs' 32GB. The M2 chip has 10% better multi-core CPU performance than the M2, and it's nearly twice as fast as the latest 10-core PC laptop chip at the same power level (CPU performance is 18 per cent greater than M1).

New MacBooks

Apple introduced the M2-powered MacBook Air. Apple's entry-level laptop has a larger display, a new processor, new colors, and a notch.

M2 also powers the 13-inch MacBook Pro. The 13-inch MacBook Pro has 24GB of unified memory and 50% more memory bandwidth. New MacBook Pro batteries last 20 hours. As I type on the 2021 MacBook Pro, I can only imagine how much power the M2 will add.

macOS 13.0 (or, macOS Ventura)

macOS Ventura will take full advantage of M2 with new features like Stage Manager and Continuity Camera and Handoff for FaceTime. Safari, Mail, Messages, Spotlight, and more get updates in macOS Ventura.

Apple hasn't run out of California landmarks to name its OS after yet. macOS 13 will be called Ventura when it's released in a few months, but it's more than a name change and new wallpapers. 

Stage Manager organizes windows

Stage Manager is a new macOS tool that organizes open windows and applications so they're still visible while focusing on a specific task. The main app sits in the middle of the desktop, while other apps and documents are organized and piled up to the side.

Improved Searching

Spotlight is one of macOS's least appreciated features, but with Ventura, it's becoming even more useful. Live Text lets you extract text from Spotlight results without leaving the window, including images from the photo library and the web.

Mail lets you schedule or unsend emails.

We've all sent an email we regret, whether it contained regrettable words or was sent at the wrong time. In macOS Ventura, Mail users can cancel or reschedule a message after sending it. Mail will now intelligently determine if a person was forgotten from a CC list or if a promised attachment wasn't included. Procrastinators can set a reminder to read a message later.

Safari adds tab sharing and password passkeys

Apple is updating Safari to make it more user-friendly... mostly. Users can share a group of tabs with friends or family, a useful feature when researching a topic with too many tabs. Passkeys will replace passwords in Safari's next version. Instead of entering random gibberish when creating a new account, macOS users can use TouchID to create an on-device passkey. Using an iPhone's camera and a QR system, Passkey syncs and works across all Apple devices and Windows computers.

Continuity adds Facetime device switching and iPhone webcam.

With macOS Ventura, iPhone users can transfer a FaceTime call from their phone to their desktop or laptop using Handoff, or vice versa if they started a call at their desk and need to continue it elsewhere. Apple finally admits its laptop and monitor webcams aren't the best. Continuity makes the iPhone a webcam. Apple demonstrated a feature where the wide-angle lens could provide a live stream of the desk below, while the standard zoom lens could focus on the speaker's face. New iPhone laptop mounts are coming.

System Preferences

System Preferences is Now System Settings and Looks Like iOS
Ventura's System Preferences has been renamed System Settings and is much more similar in appearance to iOS and iPadOS. As the iPhone and iPad are gateway devices into Apple's hardware ecosystem, new Mac users should find it easier to adjust.

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