A set $S$ plus any number of functions $f_i:S\times S\to S$, such that each $f_i$ satisfies some properties of choice.

Key examples:

The order of a algebraic structure is just its cardinality.

Sometimes, especially in the case of structures with an infinite number of elements, it is often more convenient to talk in terms of some parameter that characterizes the structure, and that parameter is usually called the degree.

The degree of some algebraic structure is some parameter that describes the structure. There is no universal definition valid for all structures, it is a per structure type thing.

This is particularly useful when talking about structures with an infinite number of elements, but it is sometimes also used for finite structures.

Examples:

Examples:

A linear map is a function $f:V_1(F)\to V_2(F)$ where $V_1(F)$ and $V_2(F)$ are two vector spaces over underlying fields $F$ such that:

A common case is $F=\mathbb{R}$, $V_1={\mathbb{R}}_m$ and $V_2={\mathbb{R}}_n$.

One thing that makes such functions particularly simple is that they can be fully specified by specifyin how they act on all possible combinations of input basis vectors: they are therefore specified by only a finite number of elements of $F$.

Every linear map in finite dimension can be represented by a matrix, the points of the domain being represented as vectors.

As such, when we say "linear map", we can think of a generalization of matrix multiplication that makes sense in infinite dimensional spaces like Hilbert spaces, since calling such infinite dimensional maps "matrices" is stretching it a bit, since we would need to specify infinitely many rows and columns.

The prototypical building block of infinite dimensional linear map is the derivative. In that case, the vectors being operated upon are functions, which cannot therefore be specified by a finite number of parameters, e.g.

For example, the left side of the time-independent Schrödinger equation is a linear map. And the time-independent Schrödinger equation can be seen as a eigenvalue problem.

A form is a function from a vector space to elements of the underlying field of the vector space.

Examples:

A Linear map where the image is the underlying field of the vector space, e.g. ${\mathbb{R}}^n\to \mathbb{R}$.

The set of all linear forms over a vector space forms another vector space called the dual space.

For the typical case of a linear form over ${\mathbb{R}}^n$, the form can be seen just as a row vector with n elements, the full form being specified by the value of each of the basis vectors.

Packet Clearing House (PCH) is a non-profit organization that focuses on improving the performance and reliability of the Internet through the development and deployment of various network infrastructure solutions. PCH is known for providing services such as Internet exchange points, route servers, and DNS infrastructure to enhance connectivity among networks and facilitate the efficient exchange of Internet traffic. PCH plays a key role in increasing the resilience of global Internet infrastructure, particularly in regions that may be underserved or have less-developed telecommunications networks.

The term "public opinion brigades" is not a widely recognized or standard concept, so its meaning can vary based on context. However, it seems to refer to organized groups or initiatives aimed at influencing public opinion or collecting and analyzing public sentiment on various issues. Such organizations may work in the fields of politics, marketing, social movements, or research.

ZA Central Registry (ZACR) is the organization responsible for managing the registration of domain names under the .za (South Africa) country code top-level domain (ccTLD). Founded in 1995, ZACR oversees the infrastructure and policies related to domain registrations in South Africa, ensuring the stability and security of these domains. ZACR offers various services related to domain name registration, including support for second-level domains like .co.za, .net.za, .org.

French Internet celebrities, often referred to as "influenceurs" or "influenceuses" in French, are individuals who have gained significant popularity and a large following on various online platforms, such as YouTube, Instagram, TikTok, and Twitter. These personalities can span across different niches, including fashion, beauty, gaming, travel, food, and lifestyle.

The Service for French Internet Exchange (SFINX) refers to a network exchange point in France designed to facilitate the exchange of internet traffic between different networks, also known as Internet Service Providers (ISPs) and other organizations. SFINX is operated by the French National Institute for Research in Computer Science and Automation (Inria) and is located in Paris.

The World Internet Conference (WIC) is an annual event held in China that focuses on discussing the development and governance of the internet. The conference was first convened in 2014 and is organized by the Cyberspace Administration of China. It brings together leaders from government, business, academia, and civil society to share insights, exchange ideas, and address global internet-related issues.

The Conseil national du numérique (CNNum) is a French advisory body dedicated to digital issues and transformation. Established in 2011, it aims to provide recommendations and insights to the French government and other stakeholders on various aspects of the digital economy, technology, and digital policy. The CNNum's mission includes analyzing the impact of digital technology on society, the economy, and public policy, as well as promoting the development of digital skills, enhancing innovation, and fostering inclusivity in the digital realm.

France is home to a variety of internet technology companies that operate in different sectors, including software development, e-commerce, digital services, and more. Here are some notable internet technology companies based in France: 1. **Orange S.A.** - One of the largest telecommunications operators in the world, providing a range of internet and mobile services. 2. **Atos** - A multinational IT services company that offers consulting, digital services, and technology solutions to various industries.

Internet censorship in the United Kingdom refers to the regulation and restriction of online content and activities by the government, internet service providers (ISPs), or other entities. While the UK generally upholds free speech and access to information, various measures are in place that can restrict certain types of content.

As of my last knowledge update in October 2023, I cannot provide real-time data or the latest rankings for YouTube videos, including the most-viewed French music videos. However, some French music videos that have historically received a high number of views and could be included in such a list are: 1. **"Despacito" (Remix) - Luis Fonsi ft.