"Automorphism" Pronounce,Meaning And Examples
"Automorphism" Natural Recordings by Native Speakers
"Automorphism" Meaning
An automorphism is an isomorphism from a mathematical object to itself, preserving its structure and properties. In other words, it's a self-map of the object that maintains all the fundamental relationships within the object. Automorphisms are often studied in various branches of mathematics, such as group theory, ring theory, and topology.
"Automorphism" Examples
1. In mathematics, an automorphism is a self-map of a structure that preserves all its properties. For example, an automorphism of a group G is a bijective function from G to itself that preserves the group operation.
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Example: The map f: Z → Z defined by f(n) = -n is an automorphism of the additive group of integers because it is a bijection and respects the addition operation: f(m+n) = -m-n = f(m) + f(n).
2. In algebraic geometry, an automorphism of a curve is a birational map from the curve onto itself.
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Example: The function g: xy → 1/x + 1/y is an automorphism of the projective plane, as it is a rational map that sends points on the curve yx = 1 to other points on the same curve.
3. In computer science, an automorphism of a graph is a permutation of its vertices that preserves adjacency.
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Example: For a simple graph with vertices A, B, C, and D connected as A-B, A-C, B-D, an automorphism could be the permutation (A B C D) → (B A D C), as it maintains the edge connections.
4. In field theory, an automorphism of a field F is a bijective function from F to itself that preserves the field operations.
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Example: Let F = Q(√2), the field of rational numbers adjoined with √2. The map h: a + b√2 → a - b√2 is an automorphism of F, since h(a + b√2) * h(c + d√2) = (a - b√2)(c - d√2) = ac - (bd)2, which is in F.
5. In linguistics, an automorphism of a language is a transformation of its grammar that produces an equivalent grammar.
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Example: In the context of Chomsky's hierarchy, a context-free grammar G can have an automorphism where non-terminal symbols are replaced with new non-terminals while preserving the production rules, resulting in a grammatically equivalent grammar G'.
"Automorphism" Similar Words
Automization
Automation refers to the use of technology, machines, or systems to perform tasks automatically, reducing or eliminating the need for human intervention. It involves the integration of equipment, software, and processes to streamline operations, increase efficiency, and minimize errors or inconsistencies. Automation can be found in various industries, from manufacturing and transportation to information technology and customer service.
Automobile
"Automobile" refers to a motor vehicle with four wheels, designed for the transport of passengers, typically powered by an internal combustion engine, although electric and hybrid models are also common now. It is commonly known as a car, and it is used for personal transportation, commuting, and recreational purposes.
Automobiles
"Automobiles" refers to motor vehicles designed for transportation on roads, typically having four wheels and powered by an internal combustion engine, electricity, or other means. They are also commonly known as cars, vehicles, or passenger vehicles and are used for personal, commercial, or recreational purposes.
Automobilism
Automobilism refers to the culture, industry, and activity surrounding automobiles, including the design, manufacture, use, and enthusiasts of cars. It encompasses aspects such as automotive engineering, racing, car clubs, and the societal impact of automobiles on transportation and daily life.
Automobilist
An "automobilist" is a person who owns, drives, or is enthusiastic about automobiles, particularly cars. It refers to someone who is interested in or frequently uses motor vehicles for transportation or leisure.
Automobilistic
"Automobilistic" refers to relating to or involving automobiles, especially in terms of their use, design, or industry. It pertains to the world of cars and motor vehicles.
Automonic
The word "autonomic" refers to a part of the nervous system that controls involuntary functions of the body, such as heart rate, digestion, and breathing. It is also often used to describe actions or processes that occur automatically or without conscious control.
Automorphic
"Automorphic" is an adjective that has different meanings depending on the context:<br><br>1. In mathematics, especially number theory, automorphic refers to a property or function that transforms in a specific way under certain operations, such as modular arithmetic. It often relates to forms or functions that "come back to themselves" after being transformed.<br><br>2. In linguistics, automorphic stems from the word "auto-" (self) and "-morphic" (shape or form). It describes a word or morpheme that changes its form by adding affixes to itself, without changing its basic meaning. For example, the English word "play" becomes "player" when a suffix is added, but the core meaning of "engaging in play" remains the same.<br><br>3. In computer science and software engineering, automorphic can refer to a program or algorithm that generates output similar to its input, often with some transformation or self-reference.<br><br>Overall, the term "automorphic" generally conveys the idea of something that transforms or self-modifies while retaining its fundamental nature.
Automotive
Automutilation
Autonoe
Autonomasy
Autonomic
Autonomical
Autonomies
Autonomism