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数学
数学,是研究數量、结构以及空间等概念及其变化的一門学科,从某种角度看屬於形式科學的一種。數學利用抽象化和邏輯推理,從計數、計算、量度、對物體形狀及運動的觀察發展而成。數學家們拓展這些概念,以公式化新的猜想,以及從選定的公理及定義出發,嚴謹地推導出一些定理。
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13#1
数学
對數學基本概念的完善,早在古埃及、美索不达米亚及古印度历史上的古代數學文本便可觀見,而在古希臘那裡有更為嚴謹的處理。從那時開始,數學的發展便持續不斷地小幅進展,至16世紀的文藝復興時期,因为新的科學發現和數學革新兩者的交互,致使數學的加速发展,直至今日。
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13#2
数学
今日,數學使用在一些不同的領域中,包括科學、工程、醫學、經濟學和金融學等。數學對這些領域的應用通常被稱為應用數學,有時亦會激起新的數學發現,並導致全新學科的發展,例如物理学的实质性发展中建立的某些理论激发数学家对于某些问题的不同角度的思考。數學家也研究純粹數學,就是數學本身的实质性內容,而不以任何實際應用為目標。許多研究雖然以純粹數學開始,但其过程中也發現許多應用之处。
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13#3
数学
西方语言中“數學”()一詞源自於古希臘語的(),其有“學習”、“學問”、“科學”,以及另外還有個較狹義且技術性的意思-「數學研究」,即使在其語源內。其形容詞(),意思為「和學習有關的」或「用功的」,亦會被用來指「數學的」。其在英语中表面上的複數形式,及在法语中的表面複數形式',可溯至拉丁文的中性複數',由西塞罗譯自希臘文複數(),此一希臘語被亚里士多德拿來指「萬物皆數」的概念。
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13#4
数学
汉字表示的「數學」一詞大約产生于中国宋元時期。多指象數之學,但有時也含有今天上的數學意義,例如,秦九韶的《數學九章》(《永樂大典》記,即《數書九章》也被宋代周密所著的《癸辛雜識》記爲《數學大略》)、《數學通軌》(明代柯尚遷著)、《数学钥》(清代杜知耕著)、《數學拾遺》(清代丁取忠撰)。直到1939年,經過中國數學名詞審查委員會研究“算學”與“數學”兩詞的使用狀況後,確認以“數學”表示今天意義上的數學含義。
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13#5
数学
數學有着久遠的歷史。它被認為起源於人類早期的生產活動:中國古代的六艺之一就有「數」,數學一詞在西方有希腊语詞源(mathematikós),意思是“学问的基础”,源于(máthema,“科学,知识,学问”)。
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13#6
数学
史前的人類就已嘗試用自然的法則來衡量物質的多少、時間的長短等抽象的數量關係,比如时间单位有日、季節和年等。算術(加減乘除)也自然而然地產生了。古代的石碑及泥版亦證實了當時已有幾何的知識。
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13#7
数学
更進一步則需要寫作或其他可記錄數字的系統,如符木或於印加帝國內用來儲存數據的奇普。歷史上曾有過許多不同的記數系統。
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13#8
数学
在最初有歷史記錄的時候,數學內的主要原理是為了做稅務和貿易等相關計算,為了解數字間的關係,為了測量土地,以及為了預測天文事件而形成的。这些需要可以简单地被概括为数学对數量、结构、空间及时间方面的研究。
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13#9
数学
到了16世纪,算术、初等代数以及三角学等初等数学已大体完备。17世纪变量概念的产生使人们开始研究变化中的量与量的互相关系和图形间的互相变换,微积分的概念也在此時形成。随着數學轉向形式化,为研究数学基础而产生的集合论和数理逻辑等也开始发展。数学的重心从求解实际问题转变到对一般形式上的思考。
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13#10
数学
從古至今,數學便一直不斷地延展,且與科學有豐富的相互作用,兩者的發展都受惠於彼此。在歷史上有著許多數學發現,並且直至今日都不斷地有新的發現。據Mikhail B. Sevryuk於2006年1月的期刊中所說,「存放於數學評論資料庫中論文和書籍的數量自1940年(數學評論的創刊年份)現已超過了一百九十萬份,而且每年還增加超過七萬五千份。此一學海的絕大部份為新的數學定理及其證明。」 旁邊的馬雅數字 一個黑點代表一 一條黑直線代表五
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13#11
数学
每當有涉及數量、結構、空間及變化等方面的困難問題時,通常就需要用到數學工具去解決問題,而這往往也拓展了數學的研究範疇。一開始,數學的運用可見於貿易、土地測量及之後的天文學。今日,所有的科學都存在著值得數學家研究的問題,且數學本身亦給出了許多的問題。牛頓和莱布尼兹是微積分的發明者,費曼發明了費曼路徑積分,這是推理及物理洞察二者的產物,而今日的弦理論亦引申出新的數學。一些數學只和生成它的領域有關,且用來解答此領域的更多問題。但一般被一領域生成的數學在其他許多領域內也十分有用,且可以成為一般的數學概念。即使是「最純的」數學通常亦有實際的用途,此一非比尋常的事實,被1963年諾貝爾物理獎得主維格納稱為「數學在自然科學中不可想像的有效性」。
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MIRACL (zh) embedded with cohere.ai multilingual-22-12 encoder

We encoded the MIRACL dataset using the cohere.ai multilingual-22-12 embedding model.

The query embeddings can be found in Cohere/miracl-zh-queries-22-12 and the corpus embeddings can be found in Cohere/miracl-zh-corpus-22-12.

For the orginal datasets, see miracl/miracl and miracl/miracl-corpus.

Dataset info:

MIRACL 🌍🙌🌏 (Multilingual Information Retrieval Across a Continuum of Languages) is a multilingual retrieval dataset that focuses on search across 18 different languages, which collectively encompass over three billion native speakers around the world.

The corpus for each language is prepared from a Wikipedia dump, where we keep only the plain text and discard images, tables, etc. Each article is segmented into multiple passages using WikiExtractor based on natural discourse units (e.g., \n\n in the wiki markup). Each of these passages comprises a "document" or unit of retrieval. We preserve the Wikipedia article title of each passage.

Embeddings

We compute for title+" "+text the embeddings using our multilingual-22-12 embedding model, a state-of-the-art model that works for semantic search in 100 languages. If you want to learn more about this model, have a look at cohere.ai multilingual embedding model.

Loading the dataset

In miracl-zh-corpus-22-12 we provide the corpus embeddings. Note, depending on the selected split, the respective files can be quite large.

You can either load the dataset like this:

from datasets import load_dataset
docs = load_dataset(f"Cohere/miracl-zh-corpus-22-12", split="train")

Or you can also stream it without downloading it before:

from datasets import load_dataset
docs = load_dataset(f"Cohere/miracl-zh-corpus-22-12", split="train", streaming=True)

for doc in docs:
    docid = doc['docid']
    title = doc['title']
    text = doc['text']
    emb = doc['emb']

Search

Have a look at miracl-zh-queries-22-12 where we provide the query embeddings for the MIRACL dataset.

To search in the documents, you must use dot-product.

And then compare this query embeddings either with a vector database (recommended) or directly computing the dot product.

A full search example:

# Attention! For large datasets, this requires a lot of memory to store
# all document embeddings and to compute the dot product scores.
# Only use this for smaller datasets. For large datasets, use a vector DB

from datasets import load_dataset
import torch

#Load documents + embeddings
docs = load_dataset(f"Cohere/miracl-zh-corpus-22-12", split="train")
doc_embeddings = torch.tensor(docs['emb'])

# Load queries 
queries = load_dataset(f"Cohere/miracl-zh-queries-22-12", split="dev")

# Select the first query as example
qid = 0
query = queries[qid]
query_embedding = torch.tensor(queries['emb'])

# Compute dot score between query embedding and document embeddings
dot_scores = torch.mm(query_embedding, doc_embeddings.transpose(0, 1))
top_k = torch.topk(dot_scores, k=3)

# Print results
print("Query:", query['query'])
for doc_id in top_k.indices[0].tolist():
    print(docs[doc_id]['title'])
    print(docs[doc_id]['text'])

You can get embeddings for new queries using our API:

#Run: pip install cohere
import cohere
co = cohere.Client(f"{api_key}")  # You should add your cohere API Key here :))
texts = ['my search query']
response = co.embed(texts=texts, model='multilingual-22-12')
query_embedding = response.embeddings[0] # Get the embedding for the first text

Performance

In the following table we compare the cohere multilingual-22-12 model with Elasticsearch version 8.6.0 lexical search (title and passage indexed as independent fields). Note that Elasticsearch doesn't support all languages that are part of the MIRACL dataset.

We compute nDCG@10 (a ranking based loss), as well as hit@3: Is at least one relevant document in the top-3 results. We find that hit@3 is easier to interpret, as it presents the number of queries for which a relevant document is found among the top-3 results.

Note: MIRACL only annotated a small fraction of passages (10 per query) for relevancy. Especially for larger Wikipedias (like English), we often found many more relevant passages. This is know as annotation holes. Real nDCG@10 and hit@3 performance is likely higher than depicted.

Model cohere multilingual-22-12 nDCG@10 cohere multilingual-22-12 hit@3 ES 8.6.0 nDCG@10 ES 8.6.0 acc@3
miracl-ar 64.2 75.2 46.8 56.2
miracl-bn 61.5 75.7 49.2 60.1
miracl-de 44.4 60.7 19.6 29.8
miracl-en 44.6 62.2 30.2 43.2
miracl-es 47.0 74.1 27.0 47.2
miracl-fi 63.7 76.2 51.4 61.6
miracl-fr 46.8 57.1 17.0 21.6
miracl-hi 50.7 62.9 41.0 48.9
miracl-id 44.8 63.8 39.2 54.7
miracl-ru 49.2 66.9 25.4 36.7
Avg 51.7 67.5 34.7 46.0

Further languages (not supported by Elasticsearch):

Model cohere multilingual-22-12 nDCG@10 cohere multilingual-22-12 hit@3
miracl-fa 44.8 53.6
miracl-ja 49.0 61.0
miracl-ko 50.9 64.8
miracl-sw 61.4 74.5
miracl-te 67.8 72.3
miracl-th 60.2 71.9
miracl-yo 56.4 62.2
miracl-zh 43.8 56.5
Avg 54.3 64.6
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