關於 comparison pdf ，我們在網路上蒐集到這些相關的討論、資訊與評價

LỘ TRÌNH IELTS 6.5 KÈM LINK TẢI TÀI LIỆU [TỰ HỌC TẠI NHÀ TRONG MÙA DỊCH]
VIDEO CHI TIẾT: https://www.youtube.com/watch?v=fPL3-cODrVU&t=2s
- Tổng lộ trình: 8 tháng - 1 năm
- Gồm 4 giai đoạn
▪Giai đoạn 1: Củng cố nền tảng tiếng Anh cơ bản
▪Giai đoạn 2: IELTS - Luyện dạng bài
▪Giai đoạn 3: IELTS - Luyện đề từng phần
▪Giai đoạn 4: IELTS - Luyện Full Test

🎗🎗GIAI ĐOẠN 1: CỦNG CỐ NỀN TẢNG TIẾNG ANH CĂN BẢN
- Thời gian: Khoảng 1-2 tháng
- Việc cần làm: Củng cố nền tảng tiếng Anh cơ bản.
- Phần cần ôn: Phát âm - Ngữ pháp - Từ vựng

🎗PHÁT ÂM
Ôn lại Bảng phiên âm quốc tế IPA, đây là link ôn:
- Giọng Anh Anh: https://www.bbc.co.uk/learningenglish/english/features/pronunciation
- Giọng Anh Mỹ (vowels): https://www.youtube.com/watch?v=c97xwLdSsXU&t=91s
- Giọng Anh Mỹ (consonants): https://www.youtube.com/watch?v=4cU9fqpCqBA

App sử dụng để hỗ trợ:
- Duolingo
- Elsa

🎗NGỮ PHÁP
MẤT CĂN BẢN:
Nếu bạn đã mất căn bản ngữ pháp, bạn có thể chọn 2 sách này để ôn song song cùng với nhau:
- Giải thích ngữ pháp Tiếng Anh của tác giả Mai Lan Hương: https://shorten.asia/U4k3CwdF
- Ngữ pháp tiếng Anh bằng sơ đồ tư duy: https://shorten.asia/7gVj5atW

ĐÃ CÓ CĂN BẢN
Nếu bạn đã có căn bản và bạn muốn ôn lại/tìm hiểu sâu hơn về một số chủ điểm ngữ pháp nào đó bạn có thể tham khảo quyển:
- Grammar In Use (intermediate): https://shorten.asia/ATt4MJ53
- Grammar In Use (advanced): https://shorten.asia/dqCXQjfU

Nếu bạn chưa biết trình độ của mình đang ở đâu, bạn có thể làm bài test tại trang:
- Oxford Online English: https://www.oxfordonlineenglish.com/english-level-test/grammar
- examenglish: https://www.examenglish.com/leveltest/grammar_level_test.htm

🎗TỪ VỰNG
VỐN TỪ RẤT HẠN CHẾ
Nếu vốn từ vựng của bạn còn rất hạn chế, bạn có thể ôn các tài liệu sau để tích lũy vốn từ cơ bản:
- Tự học 2000 từ tiếng Anh theo chủ đề: https://shorten.asia/7ygbgpY1
- Từ vựng tiếng Anh bằng sơ đồ tư duy: https://shorten.asia/UEzGbSBt

ĐÃ CÓ VỐN TỪ CƠ BẢN
Nếu bạn đã có vốn từ vựng cơ bản và muốn nâng cấp thêm vốn từ của mình, bạn có thể tham khảo bộ sách Grammar In Use với các trình độ:
- Elementary: https://shorten.asia/qTvVfFJZ
- Pre-inter + intermediate: https://shorten.asia/9Wng394Y
- Upper-intermediate: https://shorten.asia/qW6q6ZDn
- Advanced: https://shorten.asia/q6BmA4pU

Nếu chưa biết trình độ từ vựng của bạn đang ở level nào, bạn có thể làm bài test tại đây nhé:
- oxfordonlineenglish: https://www.oxfordonlineenglish.com/english-level-test/vocabulary
- examenglish: https://www.examenglish.com/leveltest/grammar_level_test.htm

THỜI GIAN BIỂU
Trong giai đoạn này bạn có thể áp dụng thời gian biểu mẫu sau (bạn có thể điều chỉnh linh hoạt để phù hợp với lịch làm việc/học tập của bạn nhé:
- Thứ 2+4+6: Phát âm + Từ vựng
- Thứ 3+5+7: Phát âm + Ngữ pháp

Trong đó mình dành khoảng 30-45' cho Phát âm, còn 1h15-1h30' cho Ngữ pháp/Từ vựng.

🎗🎗GIAI ĐOẠN 2: IELTS - ÔN THEO DẠNG BÀI
- Thời gian: Khoảng 3 tháng
- Việc cần làm: Tìm hiểu 4 kỹ năng - Theo dạng bài

🎗TÀI LIỆU

Trong giai đoạn này bạn có thể chọn 1 trong những bộ tài liệu sau để ôn nhé (bạn có thể ra nhà sách xem hoặc đọc nhận xét trên mạng để xem bạn phù hợp với bộ nào nhé):

- Complete IELTS (Gồm 3 quyển tương ứng với 3 level): https://shorten.asia/YK6Fmh9C
- Mindset For IELTS (Gồm 4 quyển tương ứng với 4 level): https://shorten.asia/1GqAYcRz
- Get Ready for IELTS (Gồm 4 quyển, mỗi kỹ năng 1 quyển): https://shorten.asia/shPub8gC
- Hacker IELTS (Gồm 4 quyển, mỗi kỹ năng 1 quyển - có 2 bộ: basic và bình thường): https://shorten.asia/7ApMwrPb

🎗THỜI GIAN BIỂU MẪU
- Thứ 2: Listening + Reading
- Thứ 3: Speaking + Writing 1
- Thứ 4: Listening + Writing 2
- Thứ 5: Speaking + Reading
- Thứ 6: Listening + Writing 1
- Thứ 7: Speaking + Writing 2

🎗LISTENING + READING

Đối với Listening và Reading, chúng ta sẽ tìm hiểu về:
- Thang điểm: Với mục tiêu 6.5 thì chúng ta cần bao nhiêu câu đúng ở 2 kỹ năng này?
- Các dạng bài + Cách làm: Có bao nhiêu dạng bài chính và cách làm từng dạng như thế nào. Lưu ý: Chúng ta sẽ ôn dạng dễ → khó.
- Vocab + Keyword Table: Đây là phần quan trọng nhất để giúp chúng ta đạt được mục tiêu. Sau khi học/giải xong bài nào, chúng ta luôn cần phải phân tích lại bài, rút từ, rút bảng Keyword Table và học: https://drive.google.com/file/d/0B0MP2MIzbYmsczdrYllOZ2dzeEE/view?usp=sharing&resourcekey=0-R2f6SGJzKRXL54Yp8qX69g

🎗SPEAKING
- 4 tiêu chí chấm của IELTS Speaking
- Dạng câu hỏi chính + chủ đề phổ biến trong các phần.

🎗WRITING
4 tiêu chí chấm cho từng Task
Các dạng bài chính trong từng Task
PARAPHRASE ⇒ Đây là 1 trong những phần quan trọng nhất: https://ielts-nguyenhuyen.com/paraphrase-trong-ielts/

Writing Task 1
Đối với Writing Task 1, sau giai đoạn này chúng ta cần nắm được Từ vựng và Cấu trúc cho:
- Trend (Xu hướng tăng - giảm - dao động - giữ ổn định)
- Comparison (Thấp hơn - Cao hơn, Thấp nhất - Cao nhất, (gần) bằng nhau)
- Prediction (Cho các bài có năm tương lai)
- Process (gồm manmade và natural)
- Map

Writing Task 2
Đối với Writing Task 2, sau giai đoạn này chúng ta cần nắm được 1 số cấu trúc/từ vựng cho các chủ đề phổ biến thường ra thi:

Cấu trúc chỉ:
- Nguyên nhân
- Ảnh Hưởng
- Giải pháp
- ...

Từ vựng cho các chủ đề phổ biến:
- Health
- Work
- Environment
- ...

🎗🎗GIAI ĐOẠN 3: IELTS - LUYỆN ĐỀ TỪNG PHẦN
- Thời gian: Khoảng 2-3 tháng
- Việc cần làm: Luyện đề từng phần

🎗TÀI LIỆU
Trong giai đoạn này chúng ta có thể tham khảo các tài liệu sau:
- The Official Cambridge Guide to IELTS
- Bộ Cambridge 7 - 16: https://shorten.asia/Aae5k4jq

🎗THỜI GIAN BIỂU MẪU
- Thứ 2: Listening + Writing 1
- Thứ 3: Speaking + Writing 2
- Thứ 4: Reading
- Thứ 5: Listening + Writing 1
- Thứ 6: Speaking + Writing 2
- Thứ 7: Reading

🎗LISTENING + READING
Đối với 2 kỹ năng này chúng ta sẽ giải đề theo từng phần.
- Listening: Có 4 phần
- Reading: Có 3 bài văn

⇒ Chúng ta sẽ giải phần đầu cho tới khi nào số câu đúng ở phần này dao động theo đúng mục tiêu mà chúng ta đề ra.

Ví dụ: Chúng ta có mục tiêu là Part 1 của Listening phải đúng ít nhất 7 câu/10 câu → Chúng ta sẽ giải Part 1 cho tới khi nào số câu đúng dao động liên tục 7/10 → Lúc đó ta mới giải tới Part 2, tương tự như vậy với các Phần còn lại.

Chúng ta vẫn tiếp tục phân tích bài sau khi giải xong + từ vựng và bảng Keyword Table.

🎗SPEAKING
Ở giai đoạn này chúng ta tập trung chủ yếu ôn Part 1 + 2.
Đối với Part 3 chúng ta sẽ học cách kéo dài câu trả lời: https://drive.google.com/file/d/1BMnhpWE5URrjP1CI4V1U9XXFt_Gl_Yct/view?usp=sharing

🎗WRITING
Đối với Writing ở giai đoạn này chúng ta sẽ tập viết theo dạng bài, và đối với từng dạng chúng ta sẽ không viết nguyên bài mà tập viết theo từng phần.
- Note 1: https://drive.google.com/file/d/1UUqZv8VD0F0HYE85HCOvcnC_pLHj54xu/view?usp=sharing
- Note 2: https://drive.google.com/file/d/1gAJvtO-wTwPaas1wGidgzg_WHBTb8rm2/view?usp=sharing

🎗🎗GIAI ĐOẠN 4: FULL TEST
- Thời gian: Khoảng 2-3 tháng
- Việc cần làm: Giải Full test

🎗TÀI LIỆU
- Bộ Cambridge 7 - 16
- IELTS Trainer 1
- IELTS Trainer 2
- Actual Tests: https://www.facebook.com/ieltsfocusmode/posts/1697444003777215

🎗LISTENING & READING
- Đối với 2 kỹ năng này chúng ta sẽ giải Full Test và làm quen với Answer Sheet → Đây là phiếu trả lời đáp án trong phòng thi: https://www.ielts.org/-/media/pdfs/114189_ielts_listening_answer_sheet.ashx?la=en
- Quan trọng nhất vẫn là phân tích bài sau khi giải xong + rút từ và học.

🎗SPEAKING
- Tới giai đoạn này chúng ta sẽ ôn cả 3 phần - và khi thực hành chúng ta cũng sẽ thực hành cả 3 phần.
- Chúng ta có thể thực hành cùng bạn hoặc tham khảo Cambly - nơi chúng ta có thể thực hành cùng giáo viên bản xứ.

🎗WRITING
- Ở giai đoạn này chúng ta sẽ viết và gửi sửa. Huyền đã review 4 dịch vụ sửa bài mà Huyền đã và đang sử dụng tại bài viết này, mọi người có thể tham khảo nhé.
- Chúng ta cố gắng phân tích thêm bài mẫu, đọc các bài báo để tích lũy thêm từ vựng và ý tưởng cho các chủ đề. Huyền đã chia sẻ các nguồn báo mà Huyền thường đọc để lấy ý tưởng tại bài viết này.

🎗ÔN DƯỚI ÁP LỰC THI

Trước thi ít nhất khoảng 1 tháng, khi giải đề chúng ta hãy căn thời gian làm bài để tập làm quen với áp lực thi nhé.

Với các bạn có nguyên ngày để ôn, bạn hãy cố gắng giải đề TRONG KHUNG THỜI GIAN THI THỰC TẾ nhé.

- 9h: Giải đề Listening
- 9h45/10h: Giải đề Reading
- 10h45/11h: Writing
- 13h-13h30: Speaking

Chúc page mình học tốt <3
https://ielts-nguyenhuyen.com/lo-trinh-hoc-ielts-6-5/

🎉🎉!!ยู่ยี่เปิดสอนภาษาอังกฤษออนไลน์แล้วนะ!!🎉🎉

ตอนนี้ทางเพจเปิดอยู่ 4 คอร์สนะคะ (ทุกคอร์สเป็นออนไลน์ค่ะ)
(ผู้ที่อยู่ต่างประเทศสามารถลงเรียนได้ทุกคอร์สนะคะ)

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1. คอร์ส YooYee's Essentials 1

👉🏻คอร์สนี้เหมาะสำหรับ?

✅ คนที่ต้องการจะเรียนภาษาอังกฤษด้วยตัวเอง แต่ไม่รุ้จะเริ่มตรงไหน
✅ คนที่อยากทบทวนปูพื้นฐาน-ระดับกลางของแกรมม่า คำศัพท์ การออกเสียงที่ต้องรู้ให้แน่นเป๊ะ
✅ คนที่ต้องการอยากจะเพิ่มหรือพัฒนาคะแนนสอบ ไม่ว่าจะสนามไหนก็ตาม

👉🏻คอร์สนี้จะสอนอะไร?

✅ เป็นคอร์สแกรมม่า+คำศัพท์ภาษาอังกฤษนะคะ.เน้นการนำไปต่อยอดใช้ในการสื่อสารและสอบในชีวิตจริงได้ (แกรมม่า + คำศัพท์เน้นๆ + เรียนแกรมม่ากับศัพท์จากหนังจากเพลงตามฉบับยู่ยี่) อะไรที่ไม่จำเป็นไม่เอามาสอนให้ปวดหัวนะ ยู่ยี่รับรอง

👉🏻คอร์สนี้สอนอะไรบ้าง
-Noun
-Article
-Quantifier
-Pronoun
-Verb
-Finite verb
-Present tense
-Past tense
-Future tense
-Passive voice
-Non-Finite verb
-Adjective
-Adverb
-Comparison
-Final Test
-Reading Practice

👉🏻คอร์สนี้เรียนยังไง?

- เรียนออนไลน์เป็นวิดิโอ สามารถเรียนได้ทุกที่ ทุกเวลา ได้ในทุกอุปกรณ์ที่ต่ออินเตอร์เนท และดูได้กี่ครั้งก็ได้ในระยะเวลา 6 เดือน และสามารถส่งคำถามมาถามยู่ยี่ได้ตลอดเวลา!!

👉🏻ใครสมควรลงคอร์สนี้?

✅ เหมาะสำหรับเด็กตั้งแต่มอต้น-วัยทำงาน เพราะยู่ยี่จะปูพื้นฐานให้ใหม่ทั้งหมดเพื่อที่ว่าถ้าเรียนจบคอร์สนี้แล้วไม่อยากต่อคอร์สก็สามารถนำความรู้ที่ได้ไปต่อยอดไม่ว่าจะเป็นการเรียน การสอบ หรือการสนทนาได้แน่นอน เพราะยู่ยี่ได้รวบรวมแกรมม่า คำศัพท์ การออกเสียงที่สำคัญ ตัวแม่ๆในการเรียน การพูดภาษาอังกฤษมาให้แล้ว และที่สำคัญคือยู่ยี่อัดแบบฝึกหัดแน่นมาก และเฉลยทุกข้อ!!

🏷 ราคา 3500 บาท (หนังสือเรียนเป็นไฟล์ pdf)
🏷 ความยาวคอร์ส 40 ชม. สามารถเรียนได้ 6 เดือนไม่จำกัด!

*ไม่ใช่คอร์สติวนะคะ แต่เป็นคอร์สสอนพื้นฐานให้แน่นเพื่อให้สามารถใช้ภาษาอังกฤษได้หยั่งยืนในทุกอย่างนะคะ จะสอนทุกอย่างอย่างละเอียดค่ะ*

ตัวอย่างบทเรียนในคอร์ส
https://www.facebook.com/YooYeesChannel/videos/312129119718583/
https://www.facebook.com/YooYeesChannel/videos/1627573717374460/
https://www.facebook.com/YooYeesChannel/videos/431929700718399/

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2. คอร์ส YooYee's Essentials 2 (คอร์สต่อจาก YooYee's Essentials1)

👉🏻เป็นคอร์สสำหรับผู้เรียนภาษาอังกฤษระดับกลาง-สูง ซึ่งคอร์สนี้นอกจากแกรมม่าแล้ว ยู่ยี่จะเน้นการเรียนศัพท์จากราก, Prefix/suffix ซึ่งจะทำให้จำศัพท์และเดาศัพท์ได้มากขึ้นหลายร้อยคำ พร้อมกับเน้นการฝึกอ่านบทความตั้งแต่ระดับกลาง จนถึงการอ่าน IELTS (จบคอร์สนี้คือจบแกรมม่าภาษาอังกฤษที่ต้องรู้หมดแล้ว ไม่ต้องหาเรียนต่อแล้ว)

👉🏻คอร์สนี้สอนอะไรบ้าง
-Part of speech (การสร้างคำและเดาคำศัพท์จาก prefix/suffix)
-Conditional sentence (If-Clause)
-Wish
-Question Tag
-Preposition
-Conjunction
-Noun Clause
-Adjective Clause
-Inversion
-Essential Phrasal verbs and idioms
-Reading Practices
-เรียนคำศํพท์จากรากศัพท์ท้ายทุกบท

*คอร์สนี้ไม่ใช่คอร์สติวนะคะ สอนทุกบทอย่างละเอียด มีที่มาที่ไป มีแบบฝึกหัดให้ฝึกมากมาย และเฉลยทุกข้อ จึงเหมาะกับคนที่ต้องการจริงจังกับภาษาอังกฤษจริงๆ*

🏷 ราคาค่าเรียน 3000 บาท (นักเรียนเก่า 2700)
🏷 ความยาวคอร์ส 25 ชั่วโมง ++ สามารถเรียนได้ 6 เดือนไม่จำกัด!

ตัวอย่างบทเรียนในคอร์ส
https://www.facebook.com/317990941658488/videos/898791184302818
https://www.facebook.com/317990941658488/videos/806652056896681
https://www.facebook.com/317990941658488/videos/185193010170674

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3. คอร์ส YooYee Daily life (แนะนำมาก!!)

👉🏻คอร์สนี้เหมาะสำหรับ?

✅ คนที่คิดว่าพื้นฐานแกรมม่าตัวเองดีพอควรแล้ว แต่เน้นการเอาไปใช้! ไปพูด!
✅ คนที่รู้แกรมม่านะแต่ก็ยัง.....
- ดูหนัง ฟังเพลง ไม่ค่อยจะรู้เรื่อง?? >>> เหมาะมาก
- คำศัพท์น้อย ไม่รู้จะเริ่มจำศัพท์อะไรดี >> เหมาะสุดๆ
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📜 [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.

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