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Translational Invariance
In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by . In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation. Discrete translational symmetry is invariant under discrete translation. Analogously an operator on functions is said to be translationally invariant with respect to a translation operator T_\delta if the result after applying doesn't change if the argument function is translated. More precisely it must hold that \forall \delta \ A f = A (T_\delta f). Laws of physics are translationally invariant under a spatial translation if they do not distinguish different points in space. According to Noether's theorem, space translational symmetry of a physical system is equivalent to the momentum conservation law. Translational symmetry of an object means that a particular translation does not change the object. For ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Translation Of A Set
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''translating'' (a written text) and '' interpreting'' (oral or signed communication between users of different languages); under this distinction, translation can begin only after the appearance of writing within a language community. A translator always risks inadvertently introducing source-language words, grammar, or syntax into the target-language rendering. On the other hand, such "spill-overs" have sometimes imported useful source-language calques and loanwords that have enriched target languages. Translators, including early translators of sacred texts, have helped shape the very languages into which they have translated. Because of the laboriousness of the translation process, since the 1940s efforts have been made, with varying degrees ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q'', there could be other scenarios where ''P'' is true and ''Q'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lebesgue Measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called ''n''-dimensional volume, ''n''-volume, or simply volume. It is used throughout real analysis, in particular to define Lebesgue integration. Sets that can be assigned a Lebesgue measure are called Lebesgue-measurable; the measure of the Lebesgue-measurable set ''A'' is here denoted by ''λ''(''A''). Henri Lebesgue described this measure in the year 1901, followed the next year by his description of the Lebesgue integral. Both were published as part of his dissertation in 1902. Definition For any interval I = ,b/math>, or I = (a, b), in the set \mathbb of real numbers, let \ell(I)= b - a denote its length. For any subset E\subseteq\mathbb, the Lebesgue oute ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Function
In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problem (mathematics education), word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic variety ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, which will output a function depending on temporal frequency or spatial frequency respectively. That process is also called ''analysis''. An example application would be decomposing the waveform of a musical chord into terms of the intensity of its constituent pitches. The term ''Fourier transform'' refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of space or time. The Fourier transform of a function is a complex-valued function representing the complex sinusoids that comprise the original function. For each frequency, the magnitude (absolute value) of the complex value represents the amplitude of a constituent complex sinusoid with that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frieze Pattern
In mathematics, a frieze or frieze pattern is a two-dimensional design that repeats in one direction. Such patterns occur frequently in architecture and decorative art. Frieze patterns can be classified into seven types according to their symmetries. The set of symmetries of a frieze pattern is called a frieze group. Frieze groups are two-dimensional line groups, having repetition in only one direction. They are related to the more complex wallpaper groups, which classify patterns that are repetitive in two directions, and crystallographic groups, which classify patterns that are repetitive in three directions. General Formally, a frieze group is a class of infinite discrete symmetry groups of patterns on a strip (infinitely wide rectangle), hence a class of groups of isometries of the plane, or of a strip. A symmetry group of a frieze group necessarily contains translations and may contain glide reflections, reflections along the long axis of the strip, reflections along the n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Translation Invariance Of Less-than-relation
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''translating'' (a written text) and '' interpreting'' (oral or signed communication between users of different languages); under this distinction, translation can begin only after the appearance of writing within a language community. A translator always risks inadvertently introducing source-language words, grammar, or syntax into the target-language rendering. On the other hand, such "spill-overs" have sometimes imported useful source-language calques and loanwords that have enriched target languages. Translators, including early translators of sacred texts, have helped shape the very languages into which they have translated. Because of the laboriousness of the translation process, since the 1940s efforts have been made, with varying degree ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cross-section (geometry)
In geometry and science, a cross section is the non-empty intersection of a solid body in three-dimensional space with a plane, or the analog in higher-dimensional spaces. Cutting an object into slices creates many parallel cross-sections. The boundary of a cross-section in three-dimensional space that is parallel to two of the axes, that is, parallel to the plane determined by these axes, is sometimes referred to as a contour line; for example, if a plane cuts through mountains of a raised-relief map parallel to the ground, the result is a contour line in two-dimensional space showing points on the surface of the mountains of equal elevation. In technical drawing a cross-section, being a projection of an object onto a plane that intersects it, is a common tool used to depict the internal arrangement of a 3-dimensional object in two dimensions. It is traditionally crosshatched with the style of crosshatching often indicating the types of materials being used. With computed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wallpaper Group
A wallpaper is a mathematical object covering a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. To a given wallpaper there corresponds a group of such congruent transformations, with function composition as the group operation. Thus, a wallpaper group (or plane symmetry group or plane crystallographic group) is in a mathematical classification of a two‑dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art, especially in textiles, tessellations and tiles as well as wallpaper. What this page calls pattern Any periodic tiling can be seen as a wallpaper. More particularly, we can consider as a wallpaper a tiling by identical tiles edge‑to‑edge, necessarily periodic, and conceive from it a wallpaper by decorating in the same manner every tiling element, and eventually erase partly or entirely the bou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Group
In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional linear transformations, and the name "modular group" comes from the relation to moduli spaces and not from modular arithmetic. Definition The modular group is the group of linear fractional transformations of the upper half of the complex plane, which have the form :z\mapsto\frac, where , , , are integers, and . The group operation is function composition. This group of transformations is isomorphic to the projective special linear group , which is the quotient of the 2-dimensional special linear group over the integers by its center . In other words, consists of all matrices :\begin a & b \\ c & d \end where , , , are integers, , and pairs of matrices and are considered to be identical. The group operation is the usual mult ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cross Product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is denoted by the symbol \times. Given two linearly independent vectors and , the cross product, (read "a cross b"), is a vector that is perpendicular to both and , and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product (projection product). If two vectors have the same direction or have the exact opposite direction from each other (that is, they are ''not'' linearly independent), or if either one has zero length, then their cross product is zero. More generally, the magnitude of the product equals the area of a parallelogram with the vectors for sides; in particular, the magnitude of the product of two perpendic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Region
Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each of these orbits. It serves as a geometric realization for the abstract set of representatives of the orbits. There are many ways to choose a fundamental domain. Typically, a fundamental domain is required to be a connected subset with some restrictions on its boundary, for example, smooth or polyhedral. The images of a chosen fundamental domain under the group action then tile the space. One general construction of fundamental domains uses Voronoi cells. Hints at a general definition Given an action of a group ''G'' on a topological space ''X'' by homeomorphisms, a fundamental domain for this action is a set ''D'' of representatives for the orbits. It is usually required to be a reasonably nice set topologically, in one of several preci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
