An approximate introduction to how zk-SNARKs are possible (part 2)
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.
see part 3
(Edited)
Hackernoon
4 years ago
👏 Awesome post! When is part 3 coming?
Trent Lapinski
4 years ago
Very complex topic, great explanation
More on Web3 & Crypto
Coinbase
4 years ago
10 Predictions for Web3 and the Cryptoeconomy for 2022
By Surojit Chatterjee, Chief Product Officer
2021 proved to be a breakout year for crypto with BTC price gaining almost 70% yoy, Defi hitting $150B in value locked, and NFTs emerging as a new category. Here’s my view through the crystal ball into 2022 and what it holds for our industry:
1. Eth scalability will improve, but newer L1 chains will see substantial growth — As we welcome the next hundred million users to crypto and Web3, scalability challenges for Eth are likely to grow. I am optimistic about improvements in Eth scalability with the emergence of Eth2 and many L2 rollups. Traction of Solana, Avalanche and other L1 chains shows that we’ll live in a multi-chain world in the future. We’re also going to see newer L1 chains emerge that focus on specific use cases such as gaming or social media.
2. There will be significant usability improvements in L1-L2 bridges — As more L1 networks gain traction and L2s become bigger, our industry will desperately seek improvements in speed and usability of cross-L1 and L1-L2 bridges. We’re likely to see interesting developments in usability of bridges in the coming year.
3. Zero knowledge proof technology will get increased traction — 2021 saw protocols like ZkSync and Starknet beginning to get traction. As L1 chains get clogged with increased usage, ZK-rollup technology will attract both investor and user attention. We’ll see new privacy-centric use cases emerge, including privacy-safe applications, and gaming models that have privacy built into the core. This may also bring in more regulator attention to crypto as KYC/AML could be a real challenge in privacy centric networks.
4. Regulated Defi and emergence of on-chain KYC attestation — Many Defi protocols will embrace regulation and will create separate KYC user pools. Decentralized identity and on-chain KYC attestation services will play key roles in connecting users’ real identity with Defi wallet endpoints. We’ll see more acceptance of ENS type addresses, and new systems from cross chain name resolution will emerge.
5. Institutions will play a much bigger role in Defi participation — Institutions are increasingly interested in participating in Defi. For starters, institutions are attracted to higher than average interest-based returns compared to traditional financial products. Also, cost reduction in providing financial services using Defi opens up interesting opportunities for institutions. However, they are still hesitant to participate in Defi. Institutions want to confirm that they are only transacting with known counterparties that have completed a KYC process. Growth of regulated Defi and on-chain KYC attestation will help institutions gain confidence in Defi.
6. Defi insurance will emerge — As Defi proliferates, it also becomes the target of security hacks. According to London-based firm Elliptic, total value lost by Defi exploits in 2021 totaled over $10B. To protect users from hacks, viable insurance protocols guaranteeing users’ funds against security breaches will emerge in 2022.
7. NFT Based Communities will give material competition to Web 2.0 social networks — NFTs will continue to expand in how they are perceived. We’ll see creator tokens or fan tokens take more of a first class seat. NFTs will become the next evolution of users’ digital identity and passport to the metaverse. Users will come together in small and diverse communities based on types of NFTs they own. User created metaverses will be the future of social networks and will start threatening the advertising driven centralized versions of social networks of today.
8. Brands will start actively participating in the metaverse and NFTs — Many brands are realizing that NFTs are great vehicles for brand marketing and establishing brand loyalty. Coca-Cola, Campbell’s, Dolce & Gabbana and Charmin released NFT collectibles in 2021. Adidas recently launched a new metaverse project with Bored Ape Yacht Club. We’re likely to see more interesting brand marketing initiatives using NFTs. NFTs and the metaverse will become the new Instagram for brands. And just like on Instagram, many brands may start as NFT native. We’ll also see many more celebrities jumping in the bandwagon and using NFTs to enhance their personal brand.
9. Web2 companies will wake up and will try to get into Web3 — We’re already seeing this with Facebook trying to recast itself as a Web3 company. We’re likely to see other big Web2 companies dipping their toes into Web3 and metaverse in 2022. However, many of them are likely to create centralized and closed network versions of the metaverse.
10. Time for DAO 2.0 — We’ll see DAOs become more mature and mainstream. More people will join DAOs, prompting a change in definition of employment — never receiving a formal offer letter, accepting tokens instead of or along with fixed salaries, and working in multiple DAO projects at the same time. DAOs will also confront new challenges in terms of figuring out how to do M&A, run payroll and benefits, and coordinate activities in larger and larger organizations. We’ll see a plethora of tools emerge to help DAOs execute with efficiency. Many DAOs will also figure out how to interact with traditional Web2 companies. We’re likely to see regulators taking more interest in DAOs and make an attempt to educate themselves on how DAOs work.
Thanks to our customers and the ecosystem for an incredible 2021. Looking forward to another year of building the foundations for Web3. Wagmi.
Onchain Wizard
3 years ago
Three Arrows Capital & Celsius Updates
I read 1k+ page 3AC liquidation documentation so you don't have to. Also sharing revised Celsius recovery plans.
3AC's liquidation documents:
Someone disclosed 3AC liquidation records in the BVI courts recently. I'll discuss the leak's timeline and other highlights.
Three Arrows Capital began trading traditional currencies in emerging markets in 2012. They switched to equities and crypto, then purely crypto in 2018.
By 2020, the firm had $703mm in net assets and $1.8bn in loans (these guys really like debt).
The firm's net assets under control reached $3bn in April 2022, according to the filings. 3AC had $600mm of LUNA/UST exposure before May 9th 2022, which put them over.
LUNA and UST go to zero quickly (I wrote about the mechanics of the blowup here). Kyle Davies, 3AC co-founder, told Blockchain.com on May 13 that they have $2.4bn in assets and $2.3bn NAV vs. $2bn in borrowings. As BTC and ETH plunged 33% and 50%, the company became insolvent by mid-2022.
3AC sent $32mm to Tai Ping Shen, a Cayman Islands business owned by Su Zhu and Davies' partner, Kelly Kaili Chen (who knows what is going on here).
3AC had borrowed over $3.5bn in notional principle, with Genesis ($2.4bn) and Voyager ($650mm) having the most exposure.
Genesis demanded $355mm in further collateral in June.
Deribit (another 3AC investment) called for $80 million in mid-June.
Even in mid-June, the corporation was trying to borrow more money to stay afloat. They approached Genesis for another $125mm loan (to pay another lender) and HODLnauts for BTC & ETH loans.
Pretty crazy. 3AC founders used borrowed money to buy a $50 million boat, according to the leak.
Su requesting for $5m + Chen Kaili Kelly asserting they loaned $65m unsecured to 3AC are identified as creditors.
Celsius:
This bankruptcy presentation shows the Celsius breakdown from March to July 14, 2022. From $22bn to $4bn, crypto assets plummeted from $14.6bn to $1.8bn (ouch). $16.5bn in user liabilities dropped to $4.72bn.
In my recent post, I examined if "forced selling" is over, with Celsius' crypto assets being a major overhang. In this presentation, it looks that Chapter 11 will provide clients the opportunity to accept cash at a discount or remain long crypto. Provided that a fresh source of money is unlikely to enter the Celsius situation, cash at a discount or crypto given to customers will likely remain a near-term market risk - cash at a discount will likely come from selling crypto assets, while customers who receive crypto could sell at any time. I'll share any Celsius updates I find.
Conclusion
Only Celsius and the Mt Gox BTC unlock remain as forced selling catalysts. While everything went through a "relief" pump, with ETH up 75% from the bottom and numerous alts multiples higher, there are still macro dangers to equities + risk assets. There's a lot of wealth waiting to be deployed in crypto ($153bn in stables), but fund managers are risk apprehensive (lower than 2008 levels).
We're hopefully over crypto's "bottom," with peak anxiety and forced selling behind us, but we may chop around.
To see the full article, click here.
Jonathan Vanian
4 years ago
What is Terra? Your guide to the hot cryptocurrency
With cryptocurrencies like Bitcoin, Ether, and Dogecoin gyrating in value over the past few months, many people are looking at so-called stablecoins like Terra to invest in because of their more predictable prices.
Terraform Labs, which oversees the Terra cryptocurrency project, has benefited from its rising popularity. The company said recently that investors like Arrington Capital, Lightspeed Venture Partners, and Pantera Capital have pledged $150 million to help it incubate various crypto projects that are connected to Terra.
Terraform Labs and its partners have built apps that operate on the company’s blockchain technology that helps keep a permanent and shared record of the firm’s crypto-related financial transactions.
Here’s what you need to know about Terra and the company behind it.
What is Terra?
Terra is a blockchain project developed by Terraform Labs that powers the startup’s cryptocurrencies and financial apps. These cryptocurrencies include the Terra U.S. Dollar, or UST, that is pegged to the U.S. dollar through an algorithm.
Terra is a stablecoin that is intended to reduce the volatility endemic to cryptocurrencies like Bitcoin. Some stablecoins, like Tether, are pegged to more conventional currencies, like the U.S. dollar, through cash and cash equivalents as opposed to an algorithm and associated reserve token.
To mint new UST tokens, a percentage of another digital token and reserve asset, Luna, is “burned.” If the demand for UST rises with more people using the currency, more Luna will be automatically burned and diverted to a community pool. That balancing act is supposed to help stabilize the price, to a degree.
“Luna directly benefits from the economic growth of the Terra economy, and it suffers from contractions of the Terra coin,” Terraform Labs CEO Do Kwon said.
Each time someone buys something—like an ice cream—using UST, that transaction generates a fee, similar to a credit card transaction. That fee is then distributed to people who own Luna tokens, similar to a stock dividend.
Who leads Terra?
The South Korean firm Terraform Labs was founded in 2018 by Daniel Shin and Kwon, who is now the company’s CEO. Kwon is a 29-year-old former Microsoft employee; Shin now heads the Chai online payment service, a Terra partner. Kwon said many Koreans have used the Chai service to buy goods like movie tickets using Terra cryptocurrency.
Terraform Labs does not make money from transactions using its crypto and instead relies on outside funding to operate, Kwon said. It has raised $57 million in funding from investors like HashKey Digital Asset Group, Divergence Digital Currency Fund, and Huobi Capital, according to deal-tracking service PitchBook. The amount raised is in addition to the latest $150 million funding commitment announced on July 16.
What are Terra’s plans?
Terraform Labs plans to use Terra’s blockchain and its associated cryptocurrencies—including one pegged to the Korean won—to create a digital financial system independent of major banks and fintech-app makers. So far, its main source of growth has been in Korea, where people have bought goods at stores, like coffee, using the Chai payment app that’s built on Terra’s blockchain. Kwon said the company’s associated Mirror trading app is experiencing growth in China and Thailand.
Meanwhile, Kwon said Terraform Labs would use its latest $150 million in funding to invest in groups that build financial apps on Terra’s blockchain. He likened the scouting and investing in other groups as akin to a “Y Combinator demo day type of situation,” a reference to the popular startup pitch event organized by early-stage investor Y Combinator.
The combination of all these Terra-specific financial apps shows that Terraform Labs is “almost creating a kind of bank,” said Ryan Watkins, a senior research analyst at cryptocurrency consultancy Messari.
In addition to cryptocurrencies, Terraform Labs has a number of other projects including the Anchor app, a high-yield savings account for holders of the group’s digital coins. Meanwhile, people can use the firm’s associated Mirror app to create synthetic financial assets that mimic more conventional ones, like “tokenized” representations of corporate stocks. These synthetic assets are supposed to be helpful to people like “a small retail trader in Thailand” who can more easily buy shares and “get some exposure to the upside” of stocks that they otherwise wouldn’t have been able to obtain, Kwon said. But some critics have said the U.S. Securities and Exchange Commission may eventually crack down on synthetic stocks, which are currently unregulated.
What do critics say?
Terra still has a long way to go to catch up to bigger cryptocurrency projects like Ethereum.
Most financial transactions involving Terra-related cryptocurrencies have originated in Korea, where its founders are based. Although Terra is becoming more popular in Korea thanks to rising interest in its partner Chai, it’s too early to say whether Terra-related currencies will gain traction in other countries.
Terra’s blockchain runs on a “limited number of nodes,” said Messari’s Watkins, referring to the computers that help keep the system running. That helps reduce latency that may otherwise slow processing of financial transactions, he said.
But the tradeoff is that Terra is less “decentralized” than other blockchain platforms like Ethereum, which is powered by thousands of interconnected computing nodes worldwide. That could make Terra less appealing to some blockchain purists.
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Frank Andrade
3 years ago
I discovered a bug that allowed me to use ChatGPT to successfully web scrape. Here's how it operates.
This method scrapes websites with ChatGPT (demo with Amazon and Twitter)
In a recent article, I demonstrated how to scrape websites using ChatGPT prompts like scrape website X using Python.
But that doesn’t always work.
After scraping dozens of websites with ChatGPT, I realized that simple prompts rarely work for web scraping.
Using ChatGPT and basic HTML, we can scrape any website.
First things first, use ChatGPT's advanced version (Playground)
ChatGPT Playground's enhanced version is needed to scrape websites quickly. It generates code faster and has less constraints.
Here’s how it looks.
The classic ChatGPT display just allows you to type a prompt. Playground is more customizable and generates code faster than the basic version.
No more delays or restrictions.
In this tutorial, we'll write prompts underneath Playground.
How to Use ChatGPT to Scrape Any Website
To show you how to utilize ChatGPT to scrape any website, we'll use subslikescript, which lists movies.
Later in this post, I'll show you how to scrape Amazon and Twitter, but let's start easy.
If we try a basic prompt like “scrape the movie titles on this website: https://subslikescript.com/movies” it won’t scrape anything. Here’s when we need some HTML knowledge to build the right prompt for ChatGPT.Let's scrape the above movie titles. First, check the webpage. Right-click any element and choose Inspect.
Then see below.
We identify the element to scrape (those in the red box). The data elements are ul, a, and scripts-list.
Here's the prompt for movie titles.
Scrape this website: https://subslikescript.com/movies with Python and Beautiful Soup. Locate the element with tag “ul” and class “scripts-list”. Scrape all the “a” elements inside. Get the text attribute and print it.
Code generated:
import requests
from bs4 import BeautifulSoup
page = requests.get('https://subslikescript.com/movies/')
soup = BeautifulSoup(page.content, 'html.parser')
scripts_list = soup.find(class_="scripts-list")
all_a_elements = scripts_list.find_all('a')
for element in all_a_elements:
print(element.get_text())It extracts movie titles successfully.
Let's scrape Amazon and Twitter.
ChatGPT's Amazon scraping
Consider scraping Amazon for self-help books. First, copy the Amazon link for self-help books.
Here’s the link I got. Location-dependent connection. Use my link to replicate my results.
Now we'll check book titles. Here's our element.
If we want to extract the book titles, we need to use the tag name span, class attribute name and a-size-base-plus a-color-base a-text-normalattribute value.
This time I'll use Selenium. I'll add Selenium-specific commands like wait 5 seconds and generate an XPath.
Scrape this website https://www.amazon.com/s?k=self+help+books&sprefix=self+help+%2Caps%2C158&ref=nb_sb_ss_ts-doa-p_2_10 with Python and Selenium.
Wait 5 seconds and locate all the elements with the following xpath: “span” tag, “class” attribute name, and “a-size-base-plus a-color-base a-text-normal” attribute value. Get the text attribute and print them.
Code generated: (I only had to manually add the path where my chromedriver is located).
from selenium import webdriver
from selenium.webdriver.common.by import By
from time import sleep
#initialize webdriver
driver = webdriver.Chrome('<add path of your chromedriver>')
#navigate to the website
driver.get("https://www.amazon.com/s?k=self+help+books&sprefix=self+help+%2Caps%2C158&ref=nb_sb_ss_ts-doa-p_2_10")
#wait 5 seconds to let the page load
sleep(5)
#locate all the elements with the following xpath
elements = driver.find_elements(By.XPATH, '//span[@class="a-size-base-plus a-color-base a-text-normal"]')
#get the text attribute of each element and print it
for element in elements:
print(element.text)
#close the webdriver
driver.close()It pulls Amazon book titles.
Utilizing ChatGPT to scrape Twitter
Say you wish to scrape ChatGPT tweets. Search Twitter for ChatGPT and copy the URL.
Here’s the link I got. We must check every tweet. Here's our element.
To extract a tweet, use the div tag and lang attribute.
Again, Selenium.
Scrape this website: https://twitter.com/search?q=chatgpt&src=typed_query using Python, Selenium and chromedriver.
Maximize the window, wait 15 seconds and locate all the elements that have the following XPath: “div” tag, attribute name “lang”. Print the text inside these elements.
Code generated: (again, I had to add the path where my chromedriver is located)
from selenium import webdriver
import time
driver = webdriver.Chrome("/Users/frankandrade/Downloads/chromedriver")
driver.maximize_window()
driver.get("https://twitter.com/search?q=chatgpt&src=typed_query")
time.sleep(15)
elements = driver.find_elements_by_xpath("//div[@lang]")
for element in elements:
print(element.text)
driver.quit()You'll get the first 2 or 3 tweets from a search. To scrape additional tweets, click X times.
Congratulations! You scraped websites without coding by using ChatGPT.
Josh Chesler
3 years ago
10 Sneaker Terms Every Beginner Should Know
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.
Pro Tip: A colorway with a well-known nickname is almost always worth more than one without, and the same goes for collaborations.
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.
Pro Tip: Know which shoes clean up nicely. The shape of lower top sneakers with wider profiles, like SB Dunk Lows and Air Jordan 3s, tends to hold better over time than their higher and narrower cousins.
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.
Pro Tip: Now that the Air Jordan line has been around for so long, the model that tends to be heavily retroed in a year is whichever shoe came out 23 (Michael Jordan’s number during the prime of his career) years ago. The Air Jordan 6 (1991) got new colorways last year, the Air Jordan 7 this year, and more Air Jordan 8s will be released later this year and early next year (1993).
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.”
Pro Tip: PayPal invoices protect buyers well, but gifting or using Google Wallet does not. Unless you're certain the seller is legitimate, only use invoiced goods/services payments.
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.
Pro Tip: If you're going to buy Yeezys, make sure you buy them from a reputable retailer or reseller. With so many fakes out there, it's not worth spending a grand on something you're not 100% sure is real.
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.
Pro Tip: Limited edition sneakers, like the annual Doernbecher Freestyle sneakers Nike creates with kids from Portland's Doernbecher Children's Hospital, will always be more expensive and limited. Or, you can use automated sneaker-buying software.
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.
Pro Tip: For those who don't have them, the OG "Bred" or "Royal" Air Jordan 1s, the "Concord" Air Jordan 11s, etc., are all 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.
Pro Tip: Apart from satisfying the world's hypebeasts, Bred sneakers will probably match a lot of your closet.
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.
Pro Tip: Ask for photos of any marks or defects to see what you’re getting before you buy used shoes, also find out if they come with the original box and extra laces, because that can be a sign that they’re in better shape.
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.
Pro Tip: If you’re not sure if shoes are real or not, do a “Legit Check” on Twitter or Facebook. You'll get dozens of responses in no time.
Shalitha Suranga
3 years ago
The Top 5 Mathematical Concepts Every Programmer Needs to Know
Using math to write efficient code in any language
Programmers design, build, test, and maintain software. Employ cases and personal preferences determine the programming languages we use throughout development. Mobile app developers use JavaScript or Dart. Some programmers design performance-first software in C/C++.
A generic source code includes language-specific grammar, pre-implemented function calls, mathematical operators, and control statements. Some mathematical principles assist us enhance our programming and problem-solving skills.
We all use basic mathematical concepts like formulas and relational operators (aka comparison operators) in programming in our daily lives. Beyond these mathematical syntaxes, we'll see discrete math topics. This narrative explains key math topics programmers must know. Master these ideas to produce clean and efficient software code.
Expressions in mathematics and built-in mathematical functions
A source code can only contain a mathematical algorithm or prebuilt API functions. We develop source code between these two ends. If you create code to fetch JSON data from a RESTful service, you'll invoke an HTTP client and won't conduct any math. If you write a function to compute the circle's area, you conduct the math there.
When your source code gets more mathematical, you'll need to use mathematical functions. Every programming language has a math module and syntactical operators. Good programmers always consider code readability, so we should learn to write readable mathematical expressions.
Linux utilizes clear math expressions.
Inbuilt max and min functions can minimize verbose if statements.
How can we compute the number of pages needed to display known data? In such instances, the ceil function is often utilized.
import math as m
results = 102
items_per_page = 10
pages = m.ceil(results / items_per_page)
print(pages)Learn to write clear, concise math expressions.
Combinatorics in Algorithm Design
Combinatorics theory counts, selects, and arranges numbers or objects. First, consider these programming-related questions. Four-digit PIN security? what options exist? What if the PIN has a prefix? How to locate all decimal number pairs?
Combinatorics questions. Software engineering jobs often require counting items. Combinatorics counts elements without counting them one by one or through other verbose approaches, therefore it enables us to offer minimum and efficient solutions to real-world situations. Combinatorics helps us make reliable decision tests without missing edge cases. Write a program to see if three inputs form a triangle. This is a question I commonly ask in software engineering interviews.
Graph theory is a subfield of combinatorics. Graph theory is used in computerized road maps and social media apps.
Logarithms and Geometry Understanding
Geometry studies shapes, angles, and sizes. Cartesian geometry involves representing geometric objects in multidimensional planes. Geometry is useful for programming. Cartesian geometry is useful for vector graphics, game development, and low-level computer graphics. We can simply work with 2D and 3D arrays as plane axes.
GetWindowRect is a Windows GUI SDK geometric object.
High-level GUI SDKs and libraries use geometric notions like coordinates, dimensions, and forms, therefore knowing geometry speeds up work with computer graphics APIs.
How does exponentiation's inverse function work? Logarithm is exponentiation's inverse function. Logarithm helps programmers find efficient algorithms and solve calculations. Writing efficient code involves finding algorithms with logarithmic temporal complexity. Programmers prefer binary search (O(log n)) over linear search (O(n)). Git source specifies O(log n):
Logarithms aid with programming math. Metas Watchman uses a logarithmic utility function to find the next power of two.
Employing Mathematical Data Structures
Programmers must know data structures to develop clean, efficient code. Stack, queue, and hashmap are computer science basics. Sets and graphs are discrete arithmetic data structures. Most computer languages include a set structure to hold distinct data entries. In most computer languages, graphs can be represented using neighboring lists or objects.
Using sets as deduped lists is powerful because set implementations allow iterators. Instead of a list (or array), store WebSocket connections in a set.
Most interviewers ask graph theory questions, yet current software engineers don't practice algorithms. Graph theory challenges become obligatory in IT firm interviews.
Recognizing Applications of Recursion
A function in programming isolates input(s) and output(s) (s). Programming functions may have originated from mathematical function theories. Programming and math functions are different but similar. Both function types accept input and return value.
Recursion involves calling the same function inside another function. In its implementation, you'll call the Fibonacci sequence. Recursion solves divide-and-conquer software engineering difficulties and avoids code repetition. I recently built the following recursive Dart code to render a Flutter multi-depth expanding list UI:
Recursion is not the natural linear way to solve problems, hence thinking recursively is difficult. Everything becomes clear when a mathematical function definition includes a base case and recursive call.
Conclusion
Every codebase uses arithmetic operators, relational operators, and expressions. To build mathematical expressions, we typically employ log, ceil, floor, min, max, etc. Combinatorics, geometry, data structures, and recursion help implement algorithms. Unless you operate in a pure mathematical domain, you may not use calculus, limits, and other complex math in daily programming (i.e., a game engine). These principles are fundamental for daily programming activities.
Master the above math fundamentals to build clean, efficient code.