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
3 years ago
👏 Awesome post! When is part 3 coming?

Trent Lapinski
3 years ago
Very complex topic, great explanation
More on Web3 & Crypto
Langston Thomas
3 years ago
A Simple Guide to NFT Blockchains
Ethereum's blockchain rules NFTs. Many consider it the one-stop shop for NFTs, and it's become the most talked-about and trafficked blockchain in existence.
Other blockchains are becoming popular in NFTs. Crypto-artists and NFT enthusiasts have sought new places to mint and trade NFTs due to Ethereum's high transaction costs and environmental impact.
When choosing a blockchain to mint on, there are several factors to consider. Size, creator costs, consumer spending habits, security, and community input are important. We've created a high-level summary of blockchains for NFTs to help clarify the fast-paced world of web3 tech.
Ethereum
Ethereum currently has the most NFTs. It's decentralized and provides financial and legal services without intermediaries. It houses popular NFT marketplaces (OpenSea), projects (CryptoPunks and the Bored Ape Yacht Club), and artists (Pak and Beeple).
It's also expensive and energy-intensive. This is because Ethereum works using a Proof-of-Work (PoW) mechanism. PoW requires computers to solve puzzles to add blocks and transactions to the blockchain. Solving these puzzles requires a lot of computer power, resulting in astronomical energy loss.
You should consider this blockchain first due to its popularity, security, decentralization, and ease of use.
Solana
Solana is a fast programmable blockchain. Its proof-of-history and proof-of-stake (PoS) consensus mechanisms eliminate complex puzzles. Reduced validation times and fees result.
PoS users stake their cryptocurrency to become a block validator. Validators get SOL. This encourages and rewards users to become stakers. PoH works with PoS to cryptographically verify time between events. Solana blockchain ensures transactions are in order and found by the correct leader (validator).
Solana's PoS and PoH mechanisms keep transaction fees and times low. Solana isn't as popular as Ethereum, so there are fewer NFT marketplaces and blockchain traders.
Tezos
Tezos is a greener blockchain. Tezos rose in 2021. Hic et Nunc was hailed as an economic alternative to Ethereum-centric marketplaces until Nov. 14, 2021.
Similar to Solana, Tezos uses a PoS consensus mechanism and only a PoS mechanism to reduce computational work. This blockchain uses two million times less energy than Ethereum. It's cheaper than Ethereum (but does cost more than Solana).
Tezos is a good place to start minting NFTs in bulk. Objkt is the largest Tezos marketplace.
Flow
Flow is a high-performance blockchain for NFTs, games, and decentralized apps (dApps). Flow is built with scalability in mind, so billions of people could interact with NFTs on the blockchain.
Flow became the NBA's blockchain partner in 2019. Flow, a product of Dapper labs (the team behind CryptoKitties), launched and hosts NBA Top Shot, making the blockchain integral to the popularity of non-fungible tokens.
Flow uses PoS to verify transactions, like Tezos. Developers are working on a model to handle 10,000 transactions per second on the blockchain. Low transaction fees.
Flow NFTs are tradeable on Blocktobay, OpenSea, Rarible, Foundation, and other platforms. NBA, NFL, UFC, and others have launched NFT marketplaces on Flow. Flow isn't as popular as Ethereum, resulting in fewer NFT marketplaces and blockchain traders.
Asset Exchange (WAX)
WAX is king of virtual collectibles. WAX is popular for digitalized versions of legacy collectibles like trading cards, figurines, memorabilia, etc.
Wax uses a PoS mechanism, but also creates carbon offset NFTs and partners with Climate Care. Like Flow, WAX transaction fees are low, and network fees are redistributed to the WAX community as an incentive to collectors.
WAX marketplaces host Topps, NASCAR, Hot Wheels, and cult classic film franchises like Godzilla, The Princess Bride, and Spiderman.
Binance Smart Chain
BSC is another good option for balancing fees and performance. High-speed transactions and low fees hurt decentralization. BSC is most centralized.
Binance Smart Chain uses Proof of Staked Authority (PoSA) to support a short block time and low fees. The 21 validators needed to run the exchange switch every 24 hours. 11 of the 21 validators are directly connected to the Binance Crypto Exchange, according to reports.
While many in the crypto and NFT ecosystems dislike centralization, the BSC NFT market picked up speed in 2021. OpenBiSea, AirNFTs, JuggerWorld, and others are gaining popularity despite not having as robust an ecosystem as Ethereum.

Nabil Alouani
3 years ago
Why Cryptocurrency Is Not Dead Despite the FTX Scam
A fraud, free-market, antifragility tale
Crypto's only rival is public opinion.
In less than a week, mainstream media, bloggers, and TikTokers turned on FTX's founder.
While some were surprised, almost everyone with a keyboard and a Twitter account predicted the FTX collapse. These financial oracles should have warned the 1.2 million people Sam Bankman-Fried duped.
After happening, unexpected events seem obvious to our brains. It's a bug and a feature because it helps us cope with disasters and makes our reasoning suck.
Nobody predicted the FTX debacle. Bloomberg? Politicians. Non-famous. No cryptologists. Who?
When FTX imploded, taking billions of dollars with it, an outrage bomb went off, and the resulting shockwave threatens the crypto market's existence.
As someone who lost more than $78,000 in a crypto scam in 2020, I can only understand people’s reactions. When the dust settles and rationality returns, we'll realize this is a natural occurrence in every free market.
What specifically occurred with FTX? (Skip if you are aware.)
FTX is a cryptocurrency exchange where customers can trade with cash. It reached #3 in less than two years as the fastest-growing platform of its kind.
FTX's performance helped make SBF the crypto poster boy. Other reasons include his altruistic public image, his support for the Democrats, and his company Alameda Research.
Alameda Research made a fortune arbitraging Bitcoin.
Arbitrage trading uses small price differences between two markets to make money. Bitcoin costs $20k in Japan and $21k in the US. Alameda Research did that for months, making $1 million per day.
Later, as its capital grew, Alameda expanded its trading activities and began investing in other companies.
Let's now discuss FTX.
SBF's diabolic master plan began when he used FTX-created FTT coins to inflate his trading company's balance sheets. He used inflated Alameda numbers to secure bank loans.
SBF used money he printed himself as collateral to borrow billions for capital. Coindesk exposed him in a report.
One of FTX's early investors tweeted that he planned to sell his FTT coins over the next few months. This would be a minor event if the investor wasn't Binance CEO Changpeng Zhao (CZ).
The crypto space saw a red WARNING sign when CZ cut ties with FTX. Everyone with an FTX account and a brain withdrew money. Two events followed. FTT fell from $20 to $4 in less than 72 hours, and FTX couldn't meet withdrawal requests, spreading panic.
SBF reassured FTX users on Twitter. Good assets.
He lied.
SBF falsely claimed FTX had a liquidity crunch. At the time of his initial claims, FTX owed about $8 billion to its customers. Liquidity shortages are usually minor. To get cash, sell assets. In the case of FTX, the main asset was printed FTT coins.
Sam wouldn't get out of trouble even if he slashed the discount (from $20 to $4) and sold every FTT. He'd flood the crypto market with his homemade coins, causing the price to crash.
SBF was trapped. He approached Binance about a buyout, which seemed good until Binance looked at FTX's books.
Binance's tweet ended SBF, and he had to apologize, resign as CEO, and file for bankruptcy.
Bloomberg estimated Sam's net worth to be zero by the end of that week. 0!
But that's not all. Twitter investigations exposed fraud at FTX and Alameda Research. SBF used customer funds to trade and invest in other companies.
Thanks to the Twitter indie reporters who made the mainstream press look amateurish. Some Twitter detectives didn't sleep for 30 hours to find answers. Others added to existing threads. Memes were hilarious.
One question kept repeating in my bald head as I watched the Blue Bird. Sam, WTF?
Then I understood.
SBF wanted that FTX becomes a bank.
Think about this. FTX seems healthy a few weeks ago. You buy 2 bitcoins using FTX. You'd expect the platform to take your dollars and debit your wallet, right?
No. They give I-Owe-Yous.
FTX records owing you 2 bitcoins in its internal ledger but doesn't credit your account. Given SBF's tricks, I'd bet on nothing.
What happens if they don't credit my account with 2 bitcoins? Your money goes into FTX's capital, where SBF and his friends invest in marketing, political endorsements, and buying other companies.
Over its two-year existence, FTX invested in 130 companies. Once they make a profit on their purchases, they'll pay you and keep the rest.
One detail makes their strategy dumb. If all FTX customers withdraw at once, everything collapses.
Financially savvy people think FTX's collapse resembles a bank run, and they're right. SBF designed FTX to operate like a bank.
You expect your bank to open a drawer with your name and put $1,000 in it when you deposit $1,000. They deposit $100 in your drawer and create an I-Owe-You for $900. What happens to $900?
Let's sum it up: It's boring and headache-inducing.
When you deposit money in a bank, they can keep 10% and lend the rest. Fractional Reserve Banking is a popular method. Fractional reserves operate within and across banks.
Fractional reserve banking generates $10,000 for every $1,000 deposited. People will pay off their debt plus interest.
As long as banks work together and the economy grows, their model works well.
SBF tried to replicate the system but forgot two details. First, traditional banks need verifiable collateral like real estate, jewelry, art, stocks, and bonds, not digital coupons. Traditional banks developed a liquidity buffer. The Federal Reserve (or Central Bank) injects massive cash into troubled banks.
Massive cash injections come from taxpayers. You and I pay for bankers' mistakes and annual bonuses. Yes, you may think banking is rigged. It's rigged, but it's the best financial game in 150 years. We accept its flaws, including bailouts for too-big-to-fail companies.
Anyway.
SBF wanted Binance's bailout. Binance said no, which was good for the crypto market.
Free markets are resilient.
Nassim Nicholas Taleb coined the term antifragility.
“Some things benefit from shocks; they thrive and grow when exposed to volatility, randomness, disorder, and stressors and love adventure, risk, and uncertainty. Yet, in spite of the ubiquity of the phenomenon, there is no word for the exact opposite of fragile. Let us call it antifragile. Antifragility is beyond resilience or robustness. The resilient resists shocks and stays the same; the antifragile gets better.”
The easiest way to understand how antifragile systems behave is to compare them with other types of systems.
Glass is like a fragile system. It snaps when shocked.
Similar to rubber, a resilient system. After a stressful episode, it bounces back.
A system that is antifragile is similar to a muscle. As it is torn in the gym, it gets stronger.
Time-changed things are antifragile. Culture, tech innovation, restaurants, revolutions, book sales, cuisine, economic success, and even muscle shape. These systems benefit from shocks and randomness in different ways, but they all pay a price for antifragility.
Same goes for the free market and financial institutions. Taleb's book uses restaurants as an example and ends with a reference to the 2008 crash.
“Restaurants are fragile. They compete with each other. But the collective of local restaurants is antifragile for that very reason. Had restaurants been individually robust, hence immortal, the overall business would be either stagnant or weak and would deliver nothing better than cafeteria food — and I mean Soviet-style cafeteria food. Further, it [the overall business] would be marred with systemic shortages, with once in a while a complete crisis and government bailout.”
Imagine the same thing with banks.
Independent banks would compete to offer the best services. If one of these banks fails, it will disappear. Customers and investors will suffer, but the market will recover from the dead banks' mistakes.
This idea underpins a free market. Bitcoin and other cryptocurrencies say this when criticizing traditional banking.
The traditional banking system's components never die. When a bank fails, the Federal Reserve steps in with a big taxpayer-funded check. This hinders bank evolution. If you don't let banking cells die and be replaced, your financial system won't be antifragile.
The interdependence of banks (centralization) means that one bank's mistake can sink the entire fleet, which brings us to SBF's ultimate travesty with FTX.
FTX has left the cryptocurrency gene pool.
FTX should be decentralized and independent. The super-star scammer invested in more than 130 crypto companies and linked them, creating a fragile banking-like structure. FTX seemed to say, "We exist because centralized banks are bad." But we'll be good, unlike the centralized banking system.
FTX saved several companies, including BlockFi and Voyager Digital.
FTX wanted to be a crypto bank conglomerate and Federal Reserve. SBF wanted to monopolize crypto markets. FTX wanted to be in bed with as many powerful people as possible, so SBF seduced politicians and celebrities.
Worst? People who saw SBF's plan flaws praised him. Experts, newspapers, and crypto fans praised FTX. When billions pour in, it's hard to realize FTX was acting against its nature.
Then, they act shocked when they realize FTX's fall triggered a domino effect. Some say the damage could wipe out the crypto market, but that's wrong.
Cell death is different from body death.
FTX is out of the game despite its size. Unfit, it fell victim to market natural selection.
Next?
The challengers keep coming. The crypto economy will improve with each failure.
Free markets are antifragile because their fragile parts compete, fostering evolution. With constructive feedback, evolution benefits customers and investors.
FTX shows that customers don't like being scammed, so the crypto market's health depends on them. Charlatans and con artists are eliminated quickly or slowly.
Crypto isn't immune to collapse. Cryptocurrencies can go extinct like biological species. Antifragility isn't immortality. A few more decades of evolution may be enough for humans to figure out how to best handle money, whether it's bitcoin, traditional banking, gold, or something else.
Keep your BS detector on. Start by being skeptical of this article's finance-related claims. Even if you think you understand finance, join the conversation.
We build a better future through dialogue. So listen, ask, and share. When you think you can't find common ground with the opposing view, remember:
Sam Bankman-Fried lied.

Elnaz Sarraf
3 years ago
Why Bitcoin's Crash Could Be Good for Investors

The crypto market crashed in June 2022. Bitcoin and other cryptocurrencies hit their lowest prices in over a year, causing market panic. Some believe this crash will benefit future investors.
Before I discuss how this crash might help investors, let's examine why it happened. Inflation in the U.S. reached a 30-year high in 2022 after Russia invaded Ukraine. In response, the U.S. Federal Reserve raised interest rates by 0.5%, the most in almost 20 years. This hurts cryptocurrencies like Bitcoin. Higher interest rates make people less likely to invest in volatile assets like crypto, so many investors sold quickly.

The crypto market collapsed. Bitcoin, Ethereum, and Binance dropped 40%. Other cryptos crashed so hard they were delisted from almost every exchange. Bitcoin peaked in April 2022 at $41,000, but after the May interest rate hike, it crashed to $28,000. Bitcoin investors were worried. Even in bad times, this crash is unprecedented.
Bitcoin wasn't "doomed." Before the crash, LUNA was one of the top 5 cryptos by market cap. LUNA was trading around $80 at the start of May 2022, but after the rate hike?
Less than 1 cent. LUNA lost 99.99% of its value in days and was removed from every crypto exchange. Bitcoin's "crash" isn't as devastating when compared to LUNA.
Many people said Bitcoin is "due" for a LUNA-like crash and that the only reason it hasn't crashed is because it's bigger. Still false. If so, Bitcoin should be worth zero by now. We didn't. Instead, Bitcoin reached 28,000, then 29k, 30k, and 31k before falling to 18k. That's not the world's greatest recovery, but it shows Bitcoin's safety.
Bitcoin isn't falling constantly. It fell because of the initial shock of interest rates, but not further. Now, Bitcoin's value is more likely to rise than fall. Bitcoin's low price also attracts investors. They know what prices Bitcoin can reach with enough hype, and they want to capitalize on low prices before it's too late.

Bitcoin's crash was bad, but in a way it wasn't. To understand, consider 2021. In March 2021, Bitcoin surpassed $60k for the first time. Elon Musk's announcement in May that he would no longer support Bitcoin caused a massive crash in the crypto market. In May 2017, Bitcoin's price hit $29,000. Elon Musk's statement isn't worth more than the Fed raising rates. Many expected this big announcement to kill Bitcoin.

Not so. Bitcoin crashed from $58k to $31k in 2021. Bitcoin fell from $41k to $28k in 2022. This crash is smaller. Bitcoin's price held up despite tensions and stress, proving investors still believe in it. What happened after the initial crash in the past?
Bitcoin fell until mid-July. This is also something we’re not seeing today. After a week, Bitcoin began to improve daily. Bitcoin's price rose after mid-July. Bitcoin's price fluctuated throughout the rest of 2021, but it topped $67k in November. Despite no major changes, the peak occurred after the crash. Elon Musk seemed uninterested in crypto and wasn't likely to change his mind soon. What triggered this peak? Nothing, really. What really happened is that people got over the initial statement. They forgot.
Internet users have goldfish-like attention spans. People quickly forgot the crash's cause and were back investing in crypto months later. Despite the market's setbacks, more crypto investors emerged by the end of 2017. Who gained from these peaks? Bitcoin investors who bought low. Bitcoin not only recovered but also doubled its ROI. It was like a movie, and it shows us what to expect from Bitcoin in the coming months.
The current Bitcoin crash isn't as bad as the last one. LUNA is causing market panic. LUNA and Bitcoin are different cryptocurrencies. LUNA crashed because Terra wasn’t able to keep its peg with the USD. Bitcoin is unanchored. It's one of the most decentralized investments available. LUNA's distrust affected crypto prices, including Bitcoin, but it won't last forever.
This is why Bitcoin will likely rebound in the coming months. In 2022, people will get over the rise in interest rates and the crash of LUNA, just as they did with Elon Musk's crypto stance in 2021. When the world moves on to the next big controversy, Bitcoin's price will soar.
Bitcoin may recover for another reason. Like controversy, interest rates fluctuate. The Russian invasion caused this inflation. World markets will stabilize, prices will fall, and interest rates will drop.
Next, lower interest rates could boost Bitcoin's price. Eventually, it will happen. The U.S. economy can't sustain such high interest rates. Investors will put every last dollar into Bitcoin if interest rates fall again.
Bitcoin has proven to be a stable investment. This boosts its investment reputation. Even if Ethereum dethrones Bitcoin as crypto king one day (or any other crypto, for that matter). Bitcoin may stay on top of the crypto ladder for a while. We'll have to wait a few months to see if any of this is true.
This post is a summary. Read the full article here.
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Startup Journal
3 years ago
The Top 14 Software Business Ideas That Are Sure To Succeed in 2023
Software can change any company.
Software is becoming essential. Everyone should consider how software affects their lives and others'.
Software on your phone, tablet, or computer offers many new options. We're experts in enough ways now.
Software Business Ideas will be popular by 2023.
ERP Programs
ERP software meets rising demand.
ERP solutions automate and monitor tasks that large organizations, businesses, and even schools would struggle to do manually.
ERP software could reach $49 billion by 2024.
CRM Program
CRM software is a must-have for any customer-focused business.
Having an open mind about your business services and products allows you to change platforms.
Another company may only want your CRM service.
Medical software
Healthcare facilities need reliable, easy-to-use software.
EHRs, MDDBs, E-Prescribing, and more are software options.
The global medical software market could reach $11 billion by 2025, and mobile medical apps may follow.
Presentation Software in the Cloud
SaaS presentation tools are great.
They're easy to use, comprehensive, and full of traditional Software features.
In today's cloud-based world, these solutions make life easier for people. We don't know about you, but we like it.
Software for Project Management
People began working remotely without signs or warnings before the 2020 COVID-19 pandemic.
Many organizations found it difficult to track projects and set deadlines.
With PMP software tools, teams can manage remote units and collaborate effectively.
App for Blockchain-Based Invoicing
This advanced billing and invoicing solution is for businesses and freelancers.
These blockchain-based apps can calculate taxes, manage debts, and manage transactions.
Intelligent contracts help blockchain track transactions more efficiently. It speeds up and improves invoice generation.
Software for Business Communications
Internal business messaging is tricky.
Top business software tools for communication can share files, collaborate on documents, host video conferences, and more.
Payroll Automation System
Software development also includes developing an automated payroll system.
These software systems reduce manual tasks for timely employee payments.
These tools help enterprise clients calculate total wages quickly, simplify tax calculations, improve record-keeping, and support better financial planning.
System for Detecting Data Leaks
Both businesses and individuals value data highly. Yahoo's data breach is dangerous because of this.
This area of software development can help people protect their data.
You can design an advanced data loss prevention system.
AI-based Retail System
AI-powered shopping systems are popular. The systems analyze customers' search and purchase patterns and store history and are equipped with a keyword database.
These systems offer many customers pre-loaded products.
AI-based shopping algorithms also help users make purchases.
Software for Detecting Plagiarism
Software can help ensure your projects are original and not plagiarized.
These tools detect plagiarized content that Google, media, and educational institutions don't like.
Software for Converting Audio to Text
Machine Learning converts speech to text automatically.
These programs can quickly transcribe cloud-based files.
Software for daily horoscopes
Daily and monthly horoscopes will continue to be popular.
Software platforms that can predict forecasts, calculate birth charts, and other astrology resources are good business ideas.
E-learning Programs
Traditional study methods are losing popularity as virtual schools proliferate and physical space shrinks.
Khan Academy online courses are the best way to keep learning.
Online education portals can boost your learning. If you want to start a tech startup, consider creating an e-learning program.
Conclusion
Software is booming. There's never been a better time to start a software development business, with so many people using computers and smartphones. This article lists eight business ideas for 2023. Consider these ideas if you're just starting out or looking to expand.

James White
3 years ago
Three Books That Can Change Your Life in a Day
I've summarized each.
Anne Lamott said books are important. Books help us understand ourselves and our behavior. They teach us about community, friendship, and death.
I read. One of my few life-changing habits. 100+ books a year improve my life. I'll list life-changing books you can read in a day. I hope you like them too.
Let's get started!
1) Seneca's Letters from a Stoic
One of my favorite philosophy books. Ryan Holiday, Naval Ravikant, and other prolific readers recommend it.
Seneca wrote 124 letters at the end of his life after working for Nero. Death, friendship, and virtue are discussed.
It's worth rereading. When I'm in trouble, I consult Seneca.
It's brief. The book could be read in one day. However, use it for guidance during difficult times.
My favorite book quotes:
Many men find that becoming wealthy only alters their problems rather than solving them.
You will never be poor if you live in harmony with nature; you will never be wealthy if you live according to what other people think.
We suffer more frequently in our imagination than in reality; there are more things that are likely to frighten us than to crush us.
2) Steven Pressfield's book The War of Art
I’ve read this book twice. I'll likely reread it before 2022 is over.
The War Of Art is the best productivity book. Steven offers procrastination-fighting tips.
Writers, musicians, and creative types will love The War of Art. Workplace procrastinators should also read this book.
My favorite book quotes:
The act of creation is what matters most in art. Other than sitting down and making an effort every day, nothing else matters.
Working creatively is not a selfish endeavor or an attempt by the actor to gain attention. It serves as a gift for all living things in the world. Don't steal your contribution from us. Give us everything you have.
Fear is healthy. Fear is a signal, just like self-doubt. Fear instructs us on what to do. The more terrified we are of a task or calling, the more certain we can be that we must complete it.
3) Darren Hardy's The Compound Effect
The Compound Effect offers practical tips to boost productivity by 10x.
The author believes each choice shapes your future. Pizza may seem harmless. However, daily use increases heart disease risk.
Positive outcomes too. Daily gym visits improve fitness. Reading an hour each night can help you learn. Writing 1,000 words per day would allow you to write a novel in under a year.
Your daily choices affect compound interest and your future. Thus, better habits can improve your life.
My favorite book quotes:
Until you alter a daily habit, you cannot change your life. The key to your success can be found in the actions you take each day.
The hundreds, thousands, or millions of little things are what distinguish the ordinary from the extraordinary; it is not the big things that add up in the end.
Don't worry about willpower. Time to use why-power. Only when you relate your decisions to your aspirations and dreams will they have any real meaning. The decisions that are in line with what you define as your purpose, your core self, and your highest values are the wisest and most inspiring ones. To avoid giving up too easily, you must want something and understand why you want it.

Thomas Huault
3 years ago
A Mean Reversion Trading Indicator Inspired by Classical Mechanics Is The Kinetic Detrender
DATA MINING WITH SUPERALGORES
Old pots produce the best soup.
Science has always inspired indicator design. From physics to signal processing, many indicators use concepts from mechanical engineering, electronics, and probability. In Superalgos' Data Mining section, we've explored using thermodynamics and information theory to construct indicators and using statistical and probabilistic techniques like reduced normal law to take advantage of low probability events.
An asset's price is like a mechanical object revolving around its moving average. Using this approach, we could design an indicator using the oscillator's Total Energy. An oscillator's energy is finite and constant. Since we don't expect the price to follow the harmonic oscillator, this energy should deviate from the perfect situation, and the maximum of divergence may provide us valuable information on the price's moving average.
Definition of the Harmonic Oscillator in Few Words
Sinusoidal function describes a harmonic oscillator. The time-constant energy equation for a harmonic oscillator is:
With
Time saves energy.
In a mechanical harmonic oscillator, total energy equals kinetic energy plus potential energy. The formula for energy is the same for every kind of harmonic oscillator; only the terms of total energy must be adapted to fit the relevant units. Each oscillator has a velocity component (kinetic energy) and a position to equilibrium component (potential energy).
The Price Oscillator and the Energy Formula
Considering the harmonic oscillator definition, we must specify kinetic and potential components for our price oscillator. We define oscillator velocity as the rate of change and equilibrium position as the price's distance from its moving average.
Price kinetic energy:
It's like:
With
and
L is the number of periods for the rate of change calculation and P for the close price EMA calculation.
Total price oscillator energy =
Given that an asset's price can theoretically vary at a limitless speed and be endlessly far from its moving average, we don't expect this formula's outcome to be constrained. We'll normalize it using Z-Score for convenience of usage and readability, which also allows probabilistic interpretation.
Over 20 periods, we'll calculate E's moving average and standard deviation.
We calculated Z on BTC/USDT with L = 10 and P = 21 using Knime Analytics.
The graph is detrended. We added two horizontal lines at +/- 1.6 to construct a 94.5% probability zone based on reduced normal law tables. Price cycles to its moving average oscillate clearly. Red and green arrows illustrate where the oscillator crosses the top and lower limits, corresponding to the maximum/minimum price oscillation. Since the results seem noisy, we may apply a non-lagging low-pass or multipole filter like Butterworth or Laguerre filters and employ dynamic bands at a multiple of Z's standard deviation instead of fixed levels.
Kinetic Detrender Implementation in Superalgos
The Superalgos Kinetic detrender features fixed upper and lower levels and dynamic volatility bands.
The code is pretty basic and does not require a huge amount of code lines.
It starts with the standard definitions of the candle pointer and the constant declaration :
let candle = record.current
let len = 10
let P = 21
let T = 20
let up = 1.6
let low = 1.6Upper and lower dynamic volatility band constants are up and low.
We proceed to the initialization of the previous value for EMA :
if (variable.prevEMA === undefined) {
variable.prevEMA = candle.close
}And the calculation of EMA with a function (it is worth noticing the function is declared at the end of the code snippet in Superalgos) :
variable.ema = calculateEMA(P, candle.close, variable.prevEMA)
//EMA calculation
function calculateEMA(periods, price, previousEMA) {
let k = 2 / (periods + 1)
return price * k + previousEMA * (1 - k)
}The rate of change is calculated by first storing the right amount of close price values and proceeding to the calculation by dividing the current close price by the first member of the close price array:
variable.allClose.push(candle.close)
if (variable.allClose.length > len) {
variable.allClose.splice(0, 1)
}
if (variable.allClose.length === len) {
variable.roc = candle.close / variable.allClose[0]
} else {
variable.roc = 1
}Finally, we get energy with a single line:
variable.E = 1 / 2 * len * variable.roc + 1 / 2 * P * candle.close / variable.emaThe Z calculation reuses code from Z-Normalization-based indicators:
variable.allE.push(variable.E)
if (variable.allE.length > T) {
variable.allE.splice(0, 1)
}
variable.sum = 0
variable.SQ = 0
if (variable.allE.length === T) {
for (var i = 0; i < T; i++) {
variable.sum += variable.allE[i]
}
variable.MA = variable.sum / T
for (var i = 0; i < T; i++) {
variable.SQ += Math.pow(variable.allE[i] - variable.MA, 2)
}
variable.sigma = Math.sqrt(variable.SQ / T)
variable.Z = (variable.E - variable.MA) / variable.sigma
} else {
variable.Z = 0
}
variable.allZ.push(variable.Z)
if (variable.allZ.length > T) {
variable.allZ.splice(0, 1)
}
variable.sum = 0
variable.SQ = 0
if (variable.allZ.length === T) {
for (var i = 0; i < T; i++) {
variable.sum += variable.allZ[i]
}
variable.MAZ = variable.sum / T
for (var i = 0; i < T; i++) {
variable.SQ += Math.pow(variable.allZ[i] - variable.MAZ, 2)
}
variable.sigZ = Math.sqrt(variable.SQ / T)
} else {
variable.MAZ = variable.Z
variable.sigZ = variable.MAZ * 0.02
}
variable.upper = variable.MAZ + up * variable.sigZ
variable.lower = variable.MAZ - low * variable.sigZWe also update the EMA value.
variable.prevEMA = variable.EMAConclusion
We showed how to build a detrended oscillator using simple harmonic oscillator theory. Kinetic detrender's main line oscillates between 2 fixed levels framing 95% of the values and 2 dynamic levels, leading to auto-adaptive mean reversion zones.
Superalgos' Normalized Momentum data mine has the Kinetic detrender indication.
All the material here can be reused and integrated freely by linking to this article and Superalgos.
This post is informative and not financial advice. Seek expert counsel before trading. Risk using this material.
