The yield spread between 10-year and 2-year US Treasury bonds has fallen below 0.2 percent, its lowest level since March 2020. A flattening or negative yield curve can be a bad sign for the economy.

What Is An Inverted Yield Curve? 

In the yield curve, bonds of equal credit quality but different maturities are plotted. The most commonly used yield curve for US investors is a plot of 2-year and 10-year Treasury yields, which have yet to invert.

A typical yield curve has higher interest rates for future maturities. In a flat yield curve, short-term and long-term yields are similar. Inverted yield curves occur when short-term yields exceed long-term yields. Inversions of yield curves have historically occurred during recessions.

Inverted yield curves have preceded each of the past eight US recessions. The good news is they're far leading indicators, meaning a recession is likely not imminent.

Every US recession since 1955 has occurred between six and 24 months after an inversion of the two-year and 10-year Treasury yield curves, according to the San Francisco Fed. So, six months before COVID-19, the yield curve inverted in August 2019.

Looking Ahead

The spread between two-year and 10-year Treasury yields was 0.18 percent on Tuesday, the smallest since before the last US recession. If the graph above continues, a two-year/10-year yield curve inversion could occur within the next few months.

According to Bank of America analyst Stephen Suttmeier, the S&P 500 typically peaks six to seven months after the 2s-10s yield curve inverts, and the US economy enters recession six to seven months later.

Investors appear unconcerned about the flattening yield curve. This is in contrast to the iShares 20+ Year Treasury Bond ETF TLT +2.19% which was down 1% on Tuesday.

So you want to get into sneakers? Buying a few sneakers and figuring it out seems simple. Then you miss out on the weekend's instant-sellout releases, so you head to eBay, Twitter, or your local  sneaker group to see what's available, since you're probably not ready to pay Flight Club prices just yet.

That's when you're bombarded with new nicknames, abbreviations, and general sneaker slang. It would take months to explain every word and sneaker, so here's a starter kit of ten simple terms to get you started. (Yeah, mostly Jordan. Does anyone really start with Kith or Nike SB?)

10. Colorways

Colorways are a common term in fashion, design, and other visual fields. It's just the product's color scheme. In the case of sneakers, the colorway is often as important as the actual model. Are this year's "Chicago" Air Jordan 1s more durable than last year's "Black/Gum" colorway? Because of their colorway and rarity, the Chicagos are worth roughly three pairs of the Black/Gum kicks.

9. Beaters

A “beater” is a well-worn, likely older model of shoe that has significant wear and tear on it. Rarely sold with the original box or extra laces, beaters rarely sell for much. Unlike most “worn” sneakers, beaters are used for rainy days and the gym. It's exactly what it sounds like, a box full of beaters, and they're a good place to start if you're looking for some cheap old kicks.

8. Retro

In the world of Jordan Brand, a “Retro” release is simply a release (or re-release) of a colorway after the shoe model's initial release. For example, the original Air Jordan 7 was released in 1992, but the Bordeaux colorway was re-released in 2011 and recently (2015). An Air Jordan model is released every year, and while half of them are unpopular and unlikely to be Retroed soon, any of them could be re-released whenever Nike and Jordan felt like it.

7. PP/Inv

In spite of the fact that eBay takes roughly 10% of the final price, many sneaker buyers and sellers prefer to work directly with PayPal. Selling sneakers for $100 via PayPal invoice or $100 via PayPal friends/family is common on social media. Because no one wants their eBay account suspended for promoting PayPal deals, many eBay sellers will simply state “Message me for a better price.”

6. Yeezy

Kanye West and his sneakers are known as Yeezys. The rapper's first two Yeezys were made by Nike before switching to Adidas. Everything Yeezy-related will be significantly more expensive (and therefore have significantly more fakes made). Not only is the Nike Air Yeezy 2 “Red October” one of the most sought-after sneakers, but the Yeezy influence can be seen everywhere.

5. GR/Limited

Regardless of how visually repulsive, uncomfortable, and/or impractical a sneaker is, if it’s rare enough, people will still want it. GR stands for General Release, which means they're usually available at retail. Reselling a “Limited Edition” release is costly. Supply and demand, but in this case, the limited supply drives up demand. If you want to get some of the colorways made for rappers, NBA players (Player Exclusive or PE models), and other celebrities, be prepared to pay a premium.

4. Grails

A “grail” is a pair of sneakers that someone desires above all others. To obtain their personal grails, people are willing to pay significantly more than the retail price. There doesn't have to be any rhyme or reason why someone chose a specific pair as their grails.

3. Bred

Anything released in “Bred” (black and red) will sell out quickly. Most resale Air Jordans (and other sneakers) come in the Bred colorway, which is a fan favorite. Bred is a good choice for a first colorway, especially on a solid sneaker silhouette.

2. DS

DS = Deadstock = New. That's it. If something has been worn or tried on, it is no longer DS. Very Near Deadstock (VNDS) Pass As Deadstock It's a cute way of saying your sneakers have been worn but are still in good shape. In the sneaker world, “worn” means they are no longer new, but not too old or beat up.

1. Fake/Unauthorized

The words “Unauthorized,” “Replica,” “B-grades,” and “Super Perfect” all mean the shoes are fake. It means they aren't made by the actual company, no matter how close or how good the quality. If that's what you want, go ahead and get them. Do not wear them if you do not want the rest of the sneaker world to mock them.

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.