The term "covariance group" can refer to different contexts in mathematics and physics, often related to how certain structures behave under transformations. However, it is not a widely used or standardized term like "group theory" or "covariance" in statistics or relativity. In general, covariance is a measure of how two variables change together.

A physical object is anything that has a tangible presence and occupies space. This means that it has specific dimensions (length, width, height), mass, and is made of matter, which can be solid, liquid, or gas. Physical objects can be perceived through our senses, particularly sight and touch.

Quantum non-equilibrium refers to the state of a quantum system that is not in thermodynamic equilibrium. In thermodynamics, systems at equilibrium exhibit well-defined macroscopic properties, such as temperature and pressure, and statistical distributions of their internal states (like the Boltzmann distribution). In contrast, non-equilibrium systems display time-dependent behavior and can have spatial gradients in quantities such as temperature, chemical potential, and density.

Analytic philosophy is a tradition in Western philosophy that emphasizes clarity of expression, logical reasoning, and the use of formal logic to analyze philosophical problems. This approach emerged in the early 20th century, primarily in the English-speaking world, and is often contrasted with continental philosophy, which may focus more on historical context, existential themes, and subjective experience.

The Analytical Society was a group formed in the early 19th century, primarily in Britain, that aimed to promote the use and understanding of analytical methods in mathematics, particularly calculus. Founded in 1813, it was a response to the predominance of the traditional calculus taught in British universities, which was often based on the work of Newton rather than the more rigorous methods developed by mathematicians like Joseph-Louis Lagrange and Augustin-Louis Cauchy.

A tensor is a mathematical object that generalizes scalars, vectors, and matrices to higher dimensions. Tensors are used in various fields such as physics, engineering, and machine learning to represent data and relationships in a structured manner. ### Basic Definitions: 1. **Scalar**: A tensor of rank 0, which is a single number (e.g., temperature, mass).

Geometry is a branch of mathematics that deals with the properties, measurements, and relationships of points, lines, shapes, and spaces. It encompasses various aspects, including: 1. **Shapes and Figures**: Geometry examines both two-dimensional shapes (like triangles, circles, and rectangles) and three-dimensional objects (like spheres, cubes, and cylinders). 2. **Properties**: It studies properties of these shapes, such as area, perimeter, volume, angles, and symmetry.

Graph theory is a branch of mathematics and computer science that studies the properties and applications of graphs. A graph is a collection of nodes (or vertices) connected by edges (or arcs). Graph theory provides a framework for modeling and analyzing relationships and interactions in various systems. Key concepts in graph theory include: 1. **Vertices and Edges**: The basic building blocks of a graph. Vertices represent entities, while edges represent the connections or relationships between them.

Mathematical analysis is a branch of mathematics that deals with the properties and behaviors of real and complex numbers, functions, sequences, and series. It provides the rigorous foundation for calculus and focuses on concepts such as limits, continuity, differentiation, integration, and sequences and series convergence. Key topics within mathematical analysis include: 1. **Limits**: Exploring how functions behave as they approach a specific point or infinity.

**Probability and Statistics** are two related but distinct branches of mathematics that deal with uncertainty and data analysis. ### Probability Probability is the branch of mathematics that deals with the likelihood or chance of different outcomes occurring. It provides a framework for quantifying uncertainty and making predictions based on known information. Some key concepts in probability include: - **Experiment**: A procedure that yields one of a possible set of outcomes (e.g., rolling a die).

Historians of mathematics are scholars who study the development, context, and impact of mathematical ideas throughout history. This field, often referred to as the history of mathematics, involves examining ancient texts, manuscripts, and artifacts to understand how mathematical concepts, techniques, and practices evolved over time and how they influenced various cultures and societies.

Quaternions are a number system that extends complex numbers and was first introduced by the Irish mathematician William Rowan Hamilton in 1843. The historical treatment of quaternions encompasses their discovery, development, and applications, as well as the controversies and advancements in mathematical theory associated with them. ### Discovery and Development 1. **Early Concepts**: Before quaternions were formally defined, mathematicians used various forms of complex numbers.

The historiography of mathematics is the study of the history of mathematics and how it has been interpreted, understood, and communicated over time. This field focuses not only on the historical development of mathematical concepts, theories, and practices, but also on how these developments have been recorded and analyzed by historians, scholars, and mathematicians themselves.

In the context of Wikipedia and other collaborative encyclopedia projects, a "stub" is a short article or entry that provides limited information on a topic and is often marked for expansion. The "History of mathematics" stubs would refer to short articles related to various aspects of the historical development of mathematics that need further elaboration. These stubs can cover a wide range of topics, such as: - Key mathematicians and their contributions throughout history. - Important mathematical discoveries and theories.

Mathematical problems are questions or challenges that require the application of mathematical concepts, principles, and techniques to find solutions or answers. These problems can arise in various fields, including pure mathematics, applied mathematics, engineering, science, economics, and beyond. Mathematical problems can be categorized in several ways: 1. **Type of Mathematics**: - **Arithmetic Problems**: Involving basic operations like addition, subtraction, multiplication, and division.

"Mathematics by culture" refers to the idea that mathematical practices, concepts, and understanding are influenced by the cultural context in which they are developed and used. It emphasizes that mathematics is not a universal language in a vacuum but is shaped by social, historical, philosophical, and cultural factors. Here are some key aspects to consider: 1. **Cultural Context**: Different cultures have developed unique mathematical ideas, systems, and tools that reflect their specific needs, environments, and philosophies.

The Antikythera mechanism is an ancient Greek analog device, believed to be one of the earliest known mechanical computers. It was discovered in a shipwreck off the coast of the Greek island Antikythera in 1901 and dates to around 150-100 BCE. The device is made up of a complex system of gears and is thought to have been used to calculate astronomical positions and predict celestial events, such as eclipses and the positions of the sun and moon.

Eudemus of Rhodes was an ancient Greek philosopher and a significant figure in the Peripatetic school, which was founded by Aristotle. He is generally thought to have lived during the 4th century BCE and is most commonly recognized for his contributions to ethics and the study of logic, as well as for his work on the history of philosophy, particularly his study of previous philosophical doctrines. Eudemus is often noted for his efforts in systematizing and clarifying Aristotle's teachings.