關於 mean math ，我們在網路上蒐集到這些相關的討論、資訊與評價

最好的自己不在未來，就在現在。

認真回想，以前我好像在每一段關係中都是在扮演某一個角色，不由自主的在為每個對象做調整。不只有在戀愛上，有時候可能跟朋友、工作夥伴都是，尤其是在前輩面前，常常我都會本能反應似的改變自己。聲音要甜一點嗎？笑話要低級一點嗎？從穿衣服就可以看出，我今天是要跟誰吃飯。雖然是微調，但透露出了一件事，我好像不夠喜歡真正的自己，所以才會擔心大家也不會喜歡真正的我。我告訴自己即使累，但如果持續努力，有一天就能變成大家都喜歡的Lara。

久而久之我和世界的相處模式只有「由外到內」，會依照別人對我的反應去斷定我的下一步。大家會覺得我很好相處，我卻越來越不快樂。現在我慢慢開始懂了，原來我把順序弄反了。我該優先處理的是自己內心的感受，其他人買不買單，其實跟你一點關係都沒有。就像找另外一半，如果我跟夢中情人在一起，但我根本無法在他面前做自己，最後還是會走不下去。

我現在33歲，So What? 終於懂，「當下的自己」才是人生時間軸上最重要的！

對了，有人好奇這次「{}」的意思嗎？簡單說是「大括號」，在數學運算式中稱作「空集合」。數學不好的人就想像一個清空的行李箱吧。「空」不是無，而是無限可能。括號還有另外一個更好懂的意思，就是「優先處理」。將過去整理過後的自己，回歸純粹，態度成熟自信！(失戀後重新出發的人應該很有感觸吧！🤣)

Your best self is not in the future, but in the Now.

Looking back, I seem to have been playing a role in almost every relationship I was ever in. I'm not just talking about romantic relationships. Whether it was with friends or colleagues (especially seniors), I would modify myself almost instinctively. Should my voice be sweeter? My jokes dirtier? If you wanted to know who I was eating with that day, all you had to do was look at the clothes I was wearing.

The changes were subtle, but they still reflected a truth: I didn’t feel like the real me was enough so I was worried that other people would also find her lacking. I told myself that if I just kept working at it, I would someday become a version of Lara that everyone accepted.

Eventually I only knew how to get my cues from the external world. I would always base my next step on feedback I got from others. Most people considered me easy to get along with, but I grew more and more unhappy.

I think I had it all wrong. My first priority should have been listening to my inner voice. Whether or not other people agree with it is not in my control nor of my concern. Take looking for a partner. Even if I found the person of my dreams, it would never last if I couldn’t be myself around them.

I am 33 years old. So What? I finally understand that "the present self" is the most important thing on the timeline of life! It’s time to prioritize me!

By the way, is anyone curious about the meaning of "{}" this time? Besides the more commonly known representation of parentheses as prioritization, these curly brackets are called “the empty set”. The empty what? If like me, math is not your strongest suit, think of it as an empty suitcase. "Empty" doesn’t mean nothing, but infinite possibilities! What will you fill your life with this time?

2021全新專輯《來者何人{}》
數位收聽：https://LaraLiang.lnk.to/DearYou2

<再也沒有你> 陳勢安 Andrew Tan
數位收聽：https://kkbox.fm/Iss26l

#Lara梁心頤2021全新專輯二部曲 #來者何人{} 全球發行
#再也沒有你 #NoMoreU #SoWhat30

📜 [專欄新文章] Uniswap v3 Features Explained in Depth

✍️ 田少谷 Shao

📥 歡迎投稿： https://medium.com/taipei-ethereum-meetup #徵技術分享文 #使用心得 #教學文 #medium

Once again the game-changing DEX 🦄 👑

Image source: https://uniswap.org/blog/uniswap-v3/

Outline

0. Intro1. Uniswap & AMM recap2. Ticks 3. Concentrated liquidity4. Range orders: reversible limit orders5. Impacts of v36. Conclusion

0. Intro

The announcement of Uniswap v3 is no doubt one of the most exciting news in the DeFi place recently 🔥🔥🔥

While most have talked about the impact v3 can potentially bring on the market, seldom explain the delicate implementation techniques to realize all those amazing features, such as concentrated liquidity, limit-order-like range orders, etc.

Since I’ve covered Uniswap v1 & v2 (if you happen to know Mandarin, here are v1 & v2), there’s no reason for me to not cover v3 as well ✅

Thus, this article aims to guide readers through Uniswap v3, based on their official whitepaper and examples made on the announcement page. However, one needs not to be an engineer, as not many codes are involved, nor a math major, as the math involved is definitely taught in your high school, to fully understand the following content 😊😊😊

If you really make it through but still don’t get shxt, feedbacks are welcomed! 🙏

There should be another article focusing on the codebase, so stay tuned and let’s get started with some background noise!

1. Uniswap & AMM recap

Before diving in, we have to first recap the uniqueness of Uniswap and compare it to traditional order book exchanges.

Uniswap v1 & v2 are a kind of AMMs (automated market marker) that follow the constant product equation x * y = k, with x & y stand for the amount of two tokens X and Y in a pool and k as a constant.

Comparing to order book exchanges, AMMs, such as the previous versions of Uniswap, offer quite a distinct user experience:

AMMs have pricing functions that offer the price for the two tokens, which make their users always price takers, while users of order book exchanges can be both makers or takers.

Uniswap as well as most AMMs have infinite liquidity¹, while order book exchanges don’t. The liquidity of Uniswap v1 & v2 is provided throughout the price range [0,∞]².

Uniswap as well as most AMMs have price slippage³ and it’s due to the pricing function, while there isn’t always price slippage on order book exchanges as long as an order is fulfilled within one tick.

In an order book, each price (whether in green or red) is a tick. Image source: https://ftx.com/trade/BTC-PERP

¹ though the price gets worse over time; AMM of constant sum such as mStable does not have infinite liquidity

² the range is in fact [-∞,∞], while a price in most cases won’t be negative

³ AMM of constant sum does not have price slippage

2. Tick

The whole innovation of Uniswap v3 starts from ticks.

For those unfamiliar with what is a tick:

Source: https://www.investopedia.com/terms/t/tick.asp

By slicing the price range [0,∞] into numerous granular ticks, trading on v3 is highly similar to trading on order book exchanges, with only three differences:

The price range of each tick is predefined by the system instead of being proposed by users.

Trades that happen within a tick still follows the pricing function of the AMM, while the equation has to be updated once the price crosses the tick.

Orders can be executed with any price within the price range, instead of being fulfilled at the same one price on order book exchanges.

With the tick design, Uniswap v3 possesses most of the merits of both AMM and an order book exchange! 💯💯💯

So, how is the price range of a tick decided?

This question is actually somewhat related to the tick explanation above: the minimum tick size for stocks trading above 1$ is one cent.

The underlying meaning of a tick size traditionally being one cent is that one cent (1% of 1$) is the basis point of price changes between ticks, ex: 1.02 — 1.01 = 0.1.

Uniswap v3 employs a similar idea: compared to the previous/next price, the price change should always be 0.01% = 1 basis point.

However, notice the difference is that in the traditional basis point, the price change is defined with subtraction, while here in Uniswap it’s division.

This is how price ranges of ticks are decided⁴:

Image source: https://uniswap.org/whitepaper-v3.pdf

With the above equation, the tick/price range can be recorded in the index form [i, i+1], instead of some crazy numbers such as 1.0001¹⁰⁰ = 1.0100496621.

As each price is the multiplication of 1.0001 of the previous price, the price change is always 1.0001 — 1 = 0.0001 = 0.01%.

For example, when i=1, p(1) = 1.0001; when i=2, p(2) = 1.00020001.

p(2) / p(1) = 1.00020001 / 1.0001 = 1.0001

See the connection between the traditional basis point 1 cent (=1% of 1$) and Uniswap v3’s basis point 0.01%?

Image source: https://tenor.com/view/coin-master-cool-gif-19748052

But sir, are prices really granular enough? There are many shitcoins with prices less than 0.000001$. Will such prices be covered as well?

Price range: max & min

To know if an extremely small price is covered or not, we have to figure out the max & min price range of v3 by looking into the spec: there is a int24 tick state variable in UniswapV3Pool.sol.

Image source: https://uniswap.org/whitepaper-v3.pdf

The reason for a signed integer int instead of an uint is that negative power represents prices less than 1 but greater than 0.

24 bits can cover the range between 1.0001 ^ (2²³ — 1) and 1.0001 ^ -(2)²³. Even Google cannot calculate such numbers, so allow me to offer smaller values to have a rough idea of the whole price range:

1.0001 ^ (2¹⁸) = 242,214,459,604.341

1.0001 ^ -(2¹⁷) = 0.000002031888943

I think it’s safe to say that with a int24 the range can cover > 99.99% of the prices of all assets in the universe 👌

⁴ For implementation concern, however, a square root is added to both sides of the equation.

How about finding out which tick does a price belong to?

Tick index from price

The answer to this question is rather easy, as we know that p(i) = 1.0001^i, simply takes a log with base 1.0001 on both sides of the equation⁴:

Image source: https://www.codecogs.com/latex/eqneditor.php

Let’s try this out, say we wanna find out the tick index of 1000000.

Image source: https://ncalculators.com/number-conversion/log-logarithm-calculator.htm

Now, 1.0001¹³⁸¹⁶² = 999,998.678087146. Voila!

⁵ This formula is also slightly modified to fit the real implementation usage.

3. Concentrated liquidity

Now that we know how ticks and price ranges are decided, let’s talk about how orders are executed in a tick, what is concentrated liquidity and how it enables v3 to compete with stablecoin-specialized DEXs (decentralized exchange), such as Curve, by improving the capital efficiency.

Concentrated liquidity means LPs (liquidity providers) can provide liquidity to any price range/tick at their wish, which causes the liquidity to be imbalanced in ticks.

As each tick has a different liquidity depth, the corresponding pricing function x * y = k also won’t be the same!

Each tick has its own liquidity depth. Image source: https://uniswap.org/blog/uniswap-v3/

Mmm… examples are always helpful for abstract descriptions 😂

Say the original pricing function is 100(x) * 1000(y) = 100000(k), with the price of X token 1000 / 100 = 10 and we’re now in the price range [9.08, 11.08].

If the liquidity of the price range [11.08, 13.08] is the same as [9.08, 11.08], we don’t have to modify the pricing function if the price goes from 10 to 11.08, which is the boundary between two ticks.

The price of X is 1052.63 / 95 = 11.08 when the equation is 1052.63 * 95 = 100000.

However, if the liquidity of the price range [11.08, 13.08] is two times that of the current range [9.08, 11.08], balances of x and y should be doubled, which makes the equation become 2105.26 * 220 = 400000, which is (1052.63 * 2) * (110 * 2) = (100000 * 2 * 2).

We can observe the following two points from the above example:

Trades always follow the pricing function x * y = k, while once the price crosses the current price range/tick, the liquidity/equation has to be updated.

√(x * y) = √k = L is how we represent the liquidity, as I say the liquidity of x * y = 400000 is two times the liquidity of x * y = 100000, as √(400000 / 100000) = 2.

What’s more, compared to liquidity on v1 & v2 is always spread across [0,∞], liquidity on v3 can be concentrated within certain price ranges and thus results in higher capital efficiency from traders’ swapping fees!

Let’s say if I provide liquidity in the range [1200, 2800], the capital efficiency will then be 4.24x higher than v2 with the range [0,∞] 😮😮😮 There’s a capital efficiency comparison calculator, make sure to try it out!

Image source: https://uniswap.org/blog/uniswap-v3/

It’s worth noticing that the concept of concentrated liquidity was proposed and already implemented by Kyper, prior to Uniswap, which is called Automated Price Reserve in their case.⁵

⁶ Thanks to Yenwen Feng for the information.

4. Range orders: reversible limit orders

As explained in the above section, LPs of v3 can provide liquidity to any price range/tick at their wish. Depending on the current price and the targeted price range, there are three scenarios:

current price < the targeted price range

current price > the targeted price range

current price belongs to the targeted price range

The first two scenarios are called range orders. They have unique characteristics and are essentially fee-earning reversible limit orders, which will be explained later.

The last case is the exact same liquidity providing mechanism as the previous versions: LPs provide liquidity in both tokens of the same value (= amount * price).

There’s also an identical product to the case: grid trading, a very powerful investment tool for a time of consolidation. Dunno what’s grid trading? Check out Binance’s explanation on this, as this topic won’t be covered!

In fact, LPs of Uniswap v1 & v2 are grid trading with a range of [0,∞] and the entry price as the baseline.

Range orders

To understand range orders, we’d have to first revisit how price is discovered on Uniswap with the equation x * y = k, for x & y stand for the amount of two tokens X and Y and k as a constant.

The price of X compared to Y is y / x, which means how many Y one can get for 1 unit of X, and vice versa the price of Y compared to X is x / y.

For the price of X to go up, y has to increase and x decrease.

With this pricing mechanism in mind, it’s example time!

Say an LP plans to place liquidity in the price range [15.625, 17.313], higher than the current price of X 10, when 100(x) * 1000(y) = 100000(k).

The price of X is 1250 / 80 = 15.625 when the equation is 80 * 1250 = 100000.

The price of X is 1315.789 / 76 = 17.313 when the equation is 76 * 1315.789 = 100000.

If now the price of X reaches 15.625, the only way for the price of X to go even higher is to further increase y and decrease x, which means exchanging a certain amount of X for Y.

Thus, to provide liquidity in the range [15.625, 17.313], an LP needs only to prepare 80 — 76 = 4 of X. If the price exceeds 17.313, all 4 X of the LP is swapped into 1315.789 — 1250 = 65.798 Y, and then the LP has nothing more to do with the pool, as his/her liquidity is drained.

What if the price stays in the range? It’s exactly what LPs would love to see, as they can earn swapping fees for all transactions in the range! Also, the balance of X will swing between [76, 80] and the balance of Y between [1250, 1315.789].

This might not be obvious, but the example above shows an interesting insight: if the liquidity of one token is provided, only when the token becomes more valuable will it be exchanged for the less valuable one.

…wut? 🤔

Remember that if 4 X is provided within [15.625, 17.313], only when the price of X goes up from 15.625 to 17.313 is 4 X gradually swapped into Y, the less valuable one!

What if the price of X drops back immediately after reaching 17.313? As X becomes less valuable, others are going to exchange Y for X.

The below image illustrates the scenario of DAI/USDC pair with a price range of [1.001, 1.002] well: the pool is always composed entirely of one token on both sides of the tick, while in the middle 1.001499⁶ is of both tokens.

Image source: https://uniswap.org/blog/uniswap-v3/

Similarly, to provide liquidity in a price range < current price, an LP has to prepare a certain amount of Y for others to exchange Y for X within the range.

To wrap up such an interesting feature, we know that:

Only one token is required for range orders.

Only when the current price is within the range of the range order can LP earn trading fees. This is the main reason why most people believe LPs of v3 have to monitor the price more actively to maximize their income, which also means that LPs of v3 have become arbitrageurs 🤯

I will be discussing more the impacts of v3 in 5. Impacts of v3.

⁷ 1.001499988 = √(1.0001 * 1.0002) is the geometric mean of 1.0001 and 1.0002. The implication is that the geometric mean of two prices is the average execution price within the range of the two prices.

Reversible limit orders

As the example in the last section demonstrates, if there is 4 X in range [15.625, 17.313], the 4 X will be completely converted into 65.798 Y when the price goes over 17.313.

We all know that a price can stay in a wide range such as [10, 11] for quite some time, while it’s unlikely so in a narrow range such as [15.625, 15.626].

Thus, if an LP provides liquidity in [15.625, 15.626], we can expect that once the price of X goes over 15.625 and immediately also 15.626, and does not drop back, all X are then forever converted into Y.

The concept of having a targeted price and the order will be executed after the price is crossed is exactly the concept of limit orders! The only difference is that if the range of a range order is not narrow enough, it’s highly possible that the conversion of tokens will be reverted once the price falls back to the range.

As price ranges follow the equation p(i) = 1.0001 ^ i, the range can be quite narrow and a range order can thus effectively serve as a limit order:

When i = 27490, 1.0001²⁷⁴⁹⁰ = 15.6248.⁸

When i = 27491, 1.0001²⁷⁴⁹¹ = 15.6264.⁸

A range of 0.0016 is not THAT narrow but can certainly satisfy most limit order use cases!

⁸ As mentioned previously in note #4, there is a square root in the equation of the price and index, thus the numbers here are for explantion only.

5. Impacts of v3

Higher capital efficiency, LPs become arbitrageurs… as v3 has made tons of radical changes, I’d like to summarize my personal takes of the impacts of v3:

Higher capital efficiency makes one of the most frequently considered indices in DeFi: TVL, total value locked, becomes less meaningful, as 1$ on Uniswap v3 might have the same effect as 100$ or even 2000$ on v2.

The ease of spot exchanging between spot exchanges used to be a huge advantage of spot markets over derivative markets. As LPs will take up the role of arbitrageurs and arbitraging is more likely to happen on v3 itself other than between DEXs, this gap is narrowed … to what extent? No idea though.

LP strategies and the aggregation of NFT of Uniswap v3 liquidity token are becoming the blue ocean for new DeFi startups: see Visor and Lixir. In fact, this might be the turning point for both DeFi and NFT: the two main reasons of blockchain going mainstream now come to the alignment of interest: solving the $$ problem 😏😏😏

In the right venue, which means a place where transaction fees are low enough, such as Optimism, we might see Algo trading firms coming in to share the market of designing LP strategies on Uniswap v3, as I believe Algo trading is way stronger than on-chain strategies or DAO voting to add liquidity that sort of thing.

After reading this article by Parsec.finance: The Dex to Rule Them All, I cannot help but wonder: maybe there is going to be centralized crypto exchanges adopting v3’s approach. The reason is that since orders of LPs in the same tick are executed pro-rata, the endless front-running speeding-competition issue in the Algo trading world, to some degree, is… solved? 🤔

Anyway, personal opinions can be biased and seriously wrong 🙈 I’m merely throwing out a sprat to catch a whale. Having a different voice? Leave your comment down below!

6. Conclusion

That was kinda tough, isn’t it? Glad you make it through here 🥂🥂🥂

There are actually many more details and also a huge section of Oracle yet to be covered. However, since this article is more about features and targeting normal DeFi users, I’ll leave those to the next one; hope there is one 😅

If you have any doubt or find any mistake, please feel free to reach out to me and I’d try to reply AFAP!

Stay tuned and in the meantime let’s wait and see how Uniswap v3 is again pioneering the innovation of DeFi 🌟

Uniswap v3 Features Explained in Depth was originally published in Taipei Ethereum Meetup on Medium, where people are continuing the conversation by highlighting and responding to this story.

👏 歡迎轉載分享鼓掌

The voice of Shaun King, a Black activist.

Hate crimes, violence, and harassment against Asian-Americans have dramatically skyrocketed in this country over the past year. This is not fake news. This is not a conspiracy theory. This is not propaganda. This is not Asian-Americans trying to steal the spotlight from African Americans. It’s math. And it matters. Let me explain why.

“Injustice anywhere is a threat to justice everywhere.” --Dr. Martin Luther King, Jr.

I have tried to live by those words from Dr. King for the past 25 years. And I actually believe them. It’s really easy to think that what happens to one group of people that you aren’t a part of has NOTHING to do with you and won’t affect or impact you in any way at all, but that’s just not how this country works. Injustice against one group of people absolutely impacts justice for all of us.

Now, as a principle, I try to fight back against the mistreatment of people whether I think it impacts my world or not, but what I want to teach you today is that being permissive or dismissive of the mistreatment of people will ALWAYS make us all less safe. These words from Martin Niemöller come to mind…

"First they came for the socialists, and I did not speak out—because I was not a socialist.
Then they came for the trade unionists, and I did not speak out— because I was not a trade unionist.
Then they came for the Jews, and I did not speak out—because I was not a Jew.
Then they came for me—and there was no one left to speak for me."

When we allow one group of people to be spit on, called racist names, punched, slapped, pushed, and mauled - it lowers the bar of what’s acceptable in this country for all people - period. If you are OK with Asian American children or senior citizens being spit on and called “China virus” then just understand that it’s only a matter of time until that ugliness spreads to children and senior citizens that you know and love. We have to reject all of it. We have to fight back against all of it. We have to take all of it personally.

I posted about this on Instagram and Facebook earlier this morning - and I want to address a few recurring themes that I saw in the comments.

A lot of my Black followers said something to the effect of, “But when do they ever speak up or fight for us?”
Every day. I see Asian Americans fighting and advocating and speaking out for Black folk every single day. I see them marching. I see them posting about injustices against Black folk on social media. I see them in planning meetings. Everywhere - from coast to coast.

If you aren’t seeing what I’m seeing, this is because you likely aren’t following who I’m following and guess what? You see who and what you follow. You don’t see who and what you don’t follow. But I follow hundreds of Asian American activists and leaders because I have hundreds of Asian American friends - many are as close to me as family. But if you literally don’t have a single Asian American friend, it’s probably very much the case that you don’t see them speaking up for us - but that says more about you than it does to them.

A lot of my Black followers said something to the effect of, “But what about those horrible videos of Asian people mistreating Black people around the world.”

I’ve seen those videos. They are horrible. Do you know why? Some Asian people are horrible and bigoted. Do you know why? Because every ethnic group has people that are good, bad, ugly and everything in between. Period. I’ve also seen multiple videos of Black people horribly mistreating Asian people. Does that mean all Black people are bad? Of course it doesn’t. This world has BILLIONS of African and Asian people around the world. We can’t judge BILLIONS of people off of the actions of a few. And here’s the thing - I have consistently seen Asian-American people speak out against the injustices of Asians against Black people, but again, I see that, because of who I follow, know, and trust.

But let me be even more specific - if an Asian man in Kenya does something ugly to an African, that should never mean that we then turn a blind eye to an Asian-American child being spit on in New York City. That’s foolishness. And let’s be honest, we’d never want the reverse done to us.

Race and racism are both complex and painful. Our nation, and this whole world, honestly, are so deeply mired in a ping pong of racism and violence, that people’s nerves are rightly frayed. I understand. Mine are frayed as well, but I am choosing to interrupt the status quo by speaking out. And personally, I do this without the expectation of anything in return. I am speaking out because the mistreatment of people because of their race or ethnicity or national origin is always wrong.

?賭Sir 幫你急救 DSE 數學： http://www.HermanToMath.com
----------
?️賭Sir是杜氏數學Herman To Math的始創人
?全港唯一「完爆」【DSE Core+M1+M2】、【IAL 12科Maths】、【AL Pure+Applied】、【CE Maths+A.Maths】、【IGCSE Math+F.Pure】的數學導師
?全港第一最多訂閱粉絲的數學教育YouTuber
?YouTube觀看次數超越700萬、訂閱粉絲超過70000人
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? Mensa Club member
? 中文大學 數學碩士畢業（Big Data stream）
? 中文大學 風險管理學士畢業
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?流行文學作家，出版著作：
《賭馬男人嫁得過》?（2020）
《YouTuber新手到網紅》?（2019、2020再版增訂本）
《5**數學男人嫁得過》?（2019）
《碌葛男人嫁得過》?（2018）
《賭波男人嫁得過》?（2018、同年再版）
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?YouTuber Go網絡課程 全港最平+獨家 報讀優惠：
http://hermantomath.blogspot.com/2019/03/youtuber-go-link.html
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賭Sir考試戰績:

新制中六DSE: (2016 M2 + 2017 M1)
?數學必修 (Mathematics) 一take過 奪5**
?數學延伸M1 (Calculus and Statistics) 一take過 奪5**
?數學延伸M2 (Algebra and Calculus) 一take過 奪5**

國際高考International Advanced Level: (2017 + 2018)
?Core Math 1 2 一take過 奪A
?Core Math 3 4 一take過 奪A
?Further Pure Math 1 一take過 奪A
?Further Pure Math 2 一take過 奪A
?Further Pure Math 3 一take過 奪A
?Mechanics 1 一take過 奪A
?Mechanics 2 一take過 奪A
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?Statistics 1 一take過 奪A
?Statistics 2 一take過 奪A
?Statistics 3 一take過 奪A
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IGCSE: (2020)
?Mathematics 一take過 奪9
?Further Pure Mathematics 一take過 奪9

舊制中七高考: (2011)
?純粹數學 (Pure Mathematics) 一take過 奪A
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舊制中五會考: (2009)
?數學 (Mathematics) 一take過 奪A
?附加數學 (Additional Mathematics) 一take過 奪A
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#箱形圖 #DSE數學 #立即訂閱

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This video is about 7 Tips To Get Started As A Marketer.

https://youtu.be/4M61Uf1UI68
https://youtu.be/4M61Uf1UI68

Back with another Googled questions series, we brought Jasmine over to share more about herself and answer your dying questions. Is she truly a mean girl? Is she rich?!

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