In mathematics, particularly in the field of abstract algebra, a **semimodule** is a generalization of the concept of a module, specifically over a semiring instead of a ring. ### Definitions 1. **Semiring**: A semiring is an algebraic structure consisting of a set equipped with two binary operations: addition (+) and multiplication (×). These operations must satisfy certain properties: - The set is closed under addition and multiplication.

The term "CSA Trust" can refer to different concepts depending on the context. One common interpretation is related to "Community Supported Agriculture" (CSA), where a trust might be set up to support local farms and agricultural initiatives. In some contexts, it could also refer to specific trusts that are established for a particular community or social cause.

The "Hockey-stick identity" is a mathematical identity in combinatorics that describes a certain relationship involving binomial coefficients. It gets its name from the hockey stick shape that graphs of the identity can resemble.

A **torsion-free abelian group** is an important concept in group theory, a branch of abstract algebra.

A trivial semigroup is a specific type of algebraic structure in the field of abstract algebra, particularly in the study of semigroups. A semigroup is defined as a set equipped with an associative binary operation. The trivial semigroup is the simplest form of a semigroup, consisting of a single element.

A **variety of finite semigroups** is a class of semigroups that can be defined using certain algebraic properties or operations. More specifically, a variety is generated by a set of finite semigroups and is characterized by the types of identities they satisfy. In algebra, varieties are often used to study structures that share common defining properties, much like varieties in other algebraic contexts (such as groups or rings). ### Key Concepts 1.

Wilkinson's polynomial is a polynomial that is specifically constructed to demonstrate the phenomenon of numerical instability in polynomial root-finding algorithms. It is named after the mathematician James H. Wilkinson.

The Laurent series is a representation of a complex function as a series, which can include both positive and negative powers of the variable. It is particularly useful for analyzing functions that have singularities (points at which they are not defined or fail to be analytic).

Electrophysiology is a branch of physiology that studies the electrical properties of biological cells and tissues. It primarily focuses on how cells generate and respond to electrical signals, which are crucial for various physiological processes, including the functioning of the nervous system and the heart. Key aspects of electrophysiology include: 1. **Membrane Potential**: Electrophysiologists investigate how the differences in ion concentrations inside and outside cells produce a membrane potential, which is critical for the initiation and propagation of electrical signals.

In mathematics, particularly in topology and algebraic geometry, the term "genus" has several related but distinct meanings depending on the context. Here are some of the most common interpretations: 1. **Genus in Topology**: The genus of a topological surface refers to the number of "holes" or "handles" in the surface.

Genus theory, particularly in the context of algebraic geometry and topology, deals with the concept of genus, which is a topological invariant that characterizes surfaces and, more generally, algebraic varieties. In simpler terms, the genus of a surface refers to the number of "holes" it has. For example: - A sphere has a genus of 0 (no holes). - A torus has a genus of 1 (one hole).

Schlömilch's series is a series associated with a particular type of mathematical expansion, often related to functions in mathematical analysis, particularly in the study of special functions and series expansions. Specifically, it typically refers to a series that arises in the context of approximating certain kinds of functions or in the solution of differential equations. One notable example of a Schlömilch series is related to the expansion of the logarithm function or other functions in terms of powers of certain variables.

Biographical films about mathematicians explore the lives, struggles, and achievements of notable figures in the field of mathematics. These films often delve into the personal and professional challenges faced by mathematicians, highlighting their contributions to the discipline and society at large. They typically blend historical accuracy with dramatic storytelling to engage audiences.

Bisection in software engineering typically refers to a debugging technique used to identify the source of a problem in code by systematically narrowing down the range of possibilities. The basic idea is to perform a "binary search" through the versions of the codebase to determine which specific change or commit introduced a bug or issue. ### How Bisection Works 1. **Identify the Range**: The developer begins with a known working version of the code and a version where the bug is present.

The Siberian High, also known as the Siberian Anticyclone, is a vast area of high atmospheric pressure that forms over Siberia during the winter months. This meteorological phenomenon is characterized by cold, dense air that accumulates due to extremely low temperatures in the region. The Siberian High typically develops in late fall and persists through the winter, influencing weather patterns not only in Siberia but also in surrounding areas, including parts of East Asia and the Arctic.

A symmetric function is a type of function in mathematics that remains unchanged when its variables are permuted.

The term "grade," in the context of slope, generally refers to the steepness or incline of a surface, such as a road, hill, or ramp. It is often expressed as a percentage or ratio, indicating how much vertical rise occurs over a horizontal distance. ### Here are key points about grade: 1. **Percentage**: Grade can be expressed as a percentage, which is calculated by dividing the vertical rise by the horizontal run and then multiplying by 100.