In mathematics, a transformation is a function that maps elements from one set to another, often changing their form or structure in some way. Transformations can be classified into various types depending on their properties and the context in which they are used. Here are a few key types of transformations: 1. **Geometric Transformations**: These are transformations that affect the position, size, and orientation of geometric figures.

The history of geometry is a fascinating journey that spans thousands of years, encompassing various cultures and developments that have shaped the field as we know it today. Here’s an overview of significant milestones in the history of geometry: ### Ancient Origins 1. **Prehistoric and Early Civilizations (circa 3000 BCE)**: - Geometry has its roots in ancient practices, particularly in surveying and land measurement, which were essential for agriculture.

Graph minor theory is a significant area of research within graph theory that deals with the concept of "minors." A graph \( H \) is said to be a minor of a graph \( G \) if \( H \) can be formed from \( G \) by a series of operations that include: 1. **Deleting vertices**: Removing a vertex and its associated edges. 2. **Deleting edges**: Removing edges between vertices.

Extremal graph theory is a branch of combinatorial mathematics that studies the extremal properties of graphs. Specifically, it focuses on questions related to the maximal or minimal number of edges in a graph that satisfies certain properties or conditions. The primary goal is often to determine the extremal (that is, maximum or minimum) values for specific parameters of graphs (like the number of edges, number of vertices, etc.) that meet certain constraints, such as containing a particular subgraph or avoiding certain configurations.

Graph databases are a type of database specifically designed to represent and store data in the form of graphs, which consist of nodes (entities) and edges (relationships). This model excels in scenarios where relationships and connections between data points are crucial and often complex. ### Key Characteristics of Graph Databases: 1. **Nodes and Edges**: - **Nodes**: Represent entities or objects, such as people, places, products, etc.

The history of the Internet is a complex tale of innovation, collaboration, and technological evolution that spans several decades. Here are the key milestones in its development: ### 1960s: The Foundations - **ARPANET**: The Advanced Research Projects Agency Network (ARPANET) was developed by the U.S. Department of Defense's ARPA (Advanced Research Projects Agency) in the late 1960s. It is often considered the precursor to the modern Internet.

The history of computer networks is a fascinating evolution that reflects the advancements in technology, communication theories, and computing power. Here's an overview of the significant milestones in the development of computer networks: ### 1960s: The Beginning - **Early Concepts**: The idea of networking devices for communication emerged alongside the development of computers. Theories about packet switching were proposed by researchers like Paul Baran and Donald Davies.

"Computing by year" typically refers to a historical overview or timeline of significant events, developments, and milestones in the field of computing and technology throughout the years. This can include the invention of key technologies, the creation of programming languages, advancements in hardware and software, and the rise of the internet, among other important developments.

A **powerful \( p \)-group** is a special type of \( p \)-group (a group where the order of every element is a power of a prime \( p \)) that satisfies certain conditions regarding its commutator structure.

The history of computer companies is a fascinating journey through technological innovation, commercialization, and the evolution of the computer industry over decades. Below is a brief overview highlighting key milestones and notable companies: ### 1940s - 1950s: The Birth of Computing - **ENIAC (1945)**: Often considered the first general-purpose electronic digital computer, designed by John W. Mauchly and J. Presper Eckert.