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Invariant
Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, a property of a program loop that is true before (and after) each iteration * A data type in method overriding that is neither covariant nor contravariant * Class invariant, an invariant used to constrain objects of a class Physics, mathematics, and statistics * Invariant (mathematics), a property of a mathematical object that is not changed by a specific operation or transformation ** Rotational invariance, the property of function whose value does not change when arbitrary rotations are applied to its argument ** Scale invariance, a property of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor ** Topological invariant * Invariant (physics), something does not change under a transformation, such as from one reference frame to another * Invari ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invariant (computer Science)
In mathematics, an invariant is a property of a mathematical object (or a class of mathematical objects) which remains unchanged after operations or transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used. For example, the area of a triangle is an invariant with respect to isometries of the Euclidean plane. The phrases "invariant under" and "invariant to" a transformation are both used. More generally, an invariant with respect to an equivalence relation is a property that is constant on each equivalence class. Invariants are used in diverse areas of mathematics such as geometry, topology, algebra and discrete mathematics. Some important classes of transformations are defined by an invariant they leave unchanged. For example, conformal maps are defined as transformations of the plane that preserve angles. The discovery of invariants is an important s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invariant (mathematics)
In mathematics, an invariant is a property of a mathematical object (or a class of mathematical objects) which remains unchanged after operations or transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used. For example, the area of a triangle is an invariant with respect to isometries of the Euclidean plane. The phrases "invariant under" and "invariant to" a transformation are both used. More generally, an invariant with respect to an equivalence relation is a property that is constant on each equivalence class. Invariants are used in diverse areas of mathematics such as geometry, topology, algebra and discrete mathematics. Some important classes of transformations are defined by an invariant they leave unchanged. For example, conformal maps are defined as transformations of the plane that preserve angles. The discovery of invariants is an important ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oxford University Invariant Society
The Oxford University Invariant Society, or 'The Invariants', is a university society open to members of the University of Oxford, dedicated to promotion of interest in mathematics. The society regularly hosts talks from professional mathematicians on topics both technical and more popular, from the mathematics of juggling to the history of mathematics. Many prominent British mathematicians were members of the society during their time at Oxford. History The Society was founded in 1936 by J. H. C. Whitehead together with two of his students at Balliol College, Graham Higman and Jack de Wet. The name of the society was chosen at random by Higman from the titles of the books on Whitehead's shelf; in this case, Oswald Veblen's ''Invariants of Quadratic Differential Forms''. The opening lecture was given by G. H. Hardy in Hilary Term 1936, with the title 'Round Numbers'. Though many members joined the armed forces during the war, meetings continued, including lectures by Dougl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loop Invariant
In computer science, a loop invariant is a property of a program loop that is true before (and after) each iteration. It is a logical assertion, sometimes checked within the code by an assertion call. Knowing its invariant(s) is essential in understanding the effect of a loop. In formal program verification, particularly the Floyd-Hoare approach, loop invariants are expressed by formal predicate logic and used to prove properties of loops and by extension algorithms that employ loops (usually correctness properties). The loop invariants will be true on entry into a loop and following each iteration, so that on exit from the loop both the loop invariants and the loop termination condition can be guaranteed. From a programming methodology viewpoint, the loop invariant can be viewed as a more abstract specification of the loop, which characterizes the deeper purpose of the loop beyond the details of this implementation. A survey article covers fundamental algorithms from many a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invariant (physics)
In theoretical physics, an invariant is an observable of a physical system which remains unchanged under some transformation. Invariance, as a broader term, also applies to the no change of form of physical laws under a transformation, and is closer in scope to the mathematical definition. Invariants of a system are deeply tied to the symmetries imposed by its environment. Invariance is an important concept in modern theoretical physics, and many theories are expressed in terms of their symmetries and invariants. Examples In classical and quantum mechanics, invariance of space under translation results in momentum being an invariant and the conservation of momentum, whereas invariance of the origin of time, i.e. translation in time, results in energy being an invariant and the conservation of energy. In general, by Noether's theorem, any invariance of a physical system under a continuous symmetry leads to a fundamental conservation law. In crystals, the electron density is peri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invariant (music)
The twelve-tone technique—also known as dodecaphony, twelve-tone serialism, and (in British usage) twelve-note composition—is a method of musical composition first devised by Austrian composer Josef Matthias Hauer, who published his "law of the twelve tones" in 1919. In 1923, Arnold Schoenberg (1874–1951) developed his own, better-known version of 12-tone technique, which became associated with the " Second Viennese School" composers, who were the primary users of the technique in the first decades of its existence. The technique is a means of ensuring that all 12 notes of the chromatic scale are sounded as often as one another in a piece of music while preventing the emphasis of any one notePerle 1977, 2. through the use of tone rows, orderings of the 12 pitch classes. All 12 notes are thus given more or less equal importance, and the music avoids being in a key. Over time, the technique increased greatly in popularity and eventually became widely influential on 20th-c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covariance And Contravariance (computer Science)
Many programming language type systems support subtyping. For instance, if the type is a subtype of , then an expression of type should be substitutable wherever an expression of type is used. Variance refers to how subtyping between more complex types relates to subtyping between their components. For example, how should a list of s relate to a list of s? Or how should a function that returns relate to a function that returns ? Depending on the variance of the type constructor, the subtyping relation of the simple types may be either preserved, reversed, or ignored for the respective complex types. In the OCaml programming language, for example, "list of Cat" is a subtype of "list of Animal" because the list type constructor is covariant. This means that the subtyping relation of the simple types are preserved for the complex types. On the other hand, "function from Animal to String" is a subtype of "function from Cat to String" because the function type constructor is contr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Class Invariant
In computer programming, specifically object-oriented programming, a class invariant (or type invariant) is an invariant used for constraining objects of a class. Methods of the class should preserve the invariant. The class invariant constrains the state stored in the object. Class invariants are established during construction and constantly maintained between calls to public methods. Code within functions may break invariants as long as the invariants are restored before a public function ends. With concurrency, maintaining the invariant in methods typically requires a critical section to be established by locking the state using a mutex. An object invariant, or representation invariant, is a computer programming construct consisting of a set of invariant properties that remain uncompromised regardless of the state of the object. This ensures that the object will always meet predefined conditions, and that methods may, therefore, always reference the object without the risk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invariant Estimator
In statistics, the concept of being an invariant estimator is a criterion that can be used to compare the properties of different estimators for the same quantity. It is a way of formalising the idea that an estimator should have certain intuitively appealing qualities. Strictly speaking, "invariant" would mean that the estimates themselves are unchanged when both the measurements and the parameters are transformed in a compatible way, but the meaning has been extended to allow the estimates to change in appropriate ways with such transformations. The term equivariant estimator is used in formal mathematical contexts that include a precise description of the relation of the way the estimator changes in response to changes to the dataset and parameterisation: this corresponds to the use of "equivariance" in more general mathematics. General setting Background In statistical inference, there are several approaches to estimation theory that can be used to decide immediately what est ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invariance (magazine)
''Invariance'' is a French magazine edited by Jacques Camatte, published since 1968. It emerged from the Italian left-communist tradition associated with Amadeo Bordiga and it originally bore the subtitle "Invariance of the theory of the proletariat", indicating Bordiga's notion of the unchanging nature of communist theory. Invariance was instrumental in bringing unknown and obscure texts of the Communist Left (Bordiga and the Italian Left, Gorter and Pannekoek from the Dutch-German Left, as well as many others, including Lukacs and Sylvia Pankhurst to the attention of the '68 generation. It also played a major -albeit largely unknown- role in the 1970's crises and dissolution of councilist organisations such as ICO in France and Solidarity in the UK. However, around 1972-75 it broke with many of the tenets of Bordigism and Marxism per se, arguing that in the aftermath of May '68 there was no longer any potential for the working class to escape the domination of capital through re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotational Invariance
In mathematics, a function defined on an inner product space is said to have rotational invariance if its value does not change when arbitrary rotations are applied to its argument. Mathematics Functions For example, the function :f(x,y) = x^2 + y^2 is invariant under rotations of the plane around the origin, because for a rotated set of coordinates through any angle ''θ'' :x' = x \cos \theta - y \sin \theta :y' = x \sin \theta + y \cos \theta the function, after some cancellation of terms, takes exactly the same form :f(x',y') = ^2 + ^2 The rotation of coordinates can be expressed using matrix form using the rotation matrix, :\begin x' \\ y' \\ \end = \begin \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end\begin x \\ y \\ \end. or symbolically x′ = Rx. Symbolically, the rotation invariance of a real-valued function of two real variables is :f(\mathbf') = f(\mathbf) = f(\mathbf) In words, the function of the rotated coordinates takes exactly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scale Invariance
In physics, mathematics and statistics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality. The technical term for this transformation is a dilatation (also known as dilation), and the dilatations can also form part of a larger conformal symmetry. *In mathematics, scale invariance usually refers to an invariance of individual functions or curves. A closely related concept is self-similarity, where a function or curve is invariant under a discrete subset of the dilations. It is also possible for the probability distributions of random processes to display this kind of scale invariance or self-similarity. *In classical field theory, scale invariance most commonly applies to the invariance of a whole theory under dilatations. Such theories typically describe classical physical processes with no characteristic length scale. *In quantum field theory, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
