Symmetry groupIn group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object. A frequent notation for the symmetry group of an object X is G = Sym(X). For an object in a metric space, its symmetries form a subgroup of the isometry group of the ambient space.
Rotational symmetryRotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which it looks exactly the same for each rotation. Certain geometric objects are partially symmetrical when rotated at certain angles such as squares rotated 90°, however the only geometric objects that are fully rotationally symmetric at any angle are spheres, circles and other spheroids.
Staggered conformationIn organic chemistry, a staggered conformation is a chemical conformation of an ethane-like moiety abcX–Ydef in which the substituents a, b, and c are at the maximum distance from d, e, and f; this requires the torsion angles to be 60°. It is the opposite of an eclipsed conformation, in which those substituents are as close to each other as possible. Such a conformation exists in any open chain single chemical bond connecting two sp3-hybridised atoms, and is normally a conformational energy minimum.
Trigonometric integralIn mathematics, trigonometric integrals are a family of integrals involving trigonometric functions. The different sine integral definitions are Note that the integrand is the sinc function, and also the zeroth spherical Bessel function. Since sinc is an even entire function (holomorphic over the entire complex plane), Si is entire, odd, and the integral in its definition can be taken along any path connecting the endpoints. By definition, Si(x) is the antiderivative of sin x / x whose value is zero at x = 0, and si(x) is the antiderivative whose value is zero at x = ∞.
Reflection symmetryIn mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D there is a line/axis of symmetry, in 3D a plane of symmetry. An object or figure which is indistinguishable from its transformed image is called . In conclusion, a line of symmetry splits the shape in half and those halves should be identical.
Icosahedral symmetryIn mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of the icosahedron) and the rhombic triacontahedron. Every polyhedron with icosahedral symmetry has 60 rotational (or orientation-preserving) symmetries and 60 orientation-reversing symmetries (that combine a rotation and a reflection), for a total symmetry order of 120.
File descriptorIn Unix and Unix-like computer operating systems, a file descriptor (FD, less frequently fildes) is a process-unique identifier (handle) for a or other input/output resource, such as a pipe or network socket. File descriptors typically have non-negative integer values, with negative values being reserved to indicate "no value" or error conditions. File descriptors are a part of the POSIX API.
Algebraic expressionIn mathematics, an algebraic expression is an expression built up from constant algebraic numbers, variables, and the algebraic operations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number). For example, 3x2 − 2xy + c is an algebraic expression. Since taking the square root is the same as raising to the power 1/2, the following is also an algebraic expression: An algebraic equation is an equation involving only algebraic expressions.
Symmetry (geometry)In geometry, an object has symmetry if there is an operation or transformation (such as translation, scaling, rotation or reflection) that maps the figure/object onto itself (i.e., the object has an invariance under the transform). Thus, a symmetry can be thought of as an immunity to change. For instance, a circle rotated about its center will have the same shape and size as the original circle, as all points before and after the transform would be indistinguishable.
Sinc functionIn mathematics, physics and engineering, the sinc function, denoted by sinc(x), has two forms, normalized and unnormalized. In mathematics, the historical unnormalized sinc function is defined for x ≠ 0 by Alternatively, the unnormalized sinc function is often called the sampling function, indicated as Sa(x). In digital signal processing and information theory, the normalized sinc function is commonly defined for x ≠ 0 by In either case, the value at x = 0 is defined to be the limiting value for all real a ≠ 0 (the limit can be proven using the squeeze theorem).
Window functionIn signal processing and statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside of some chosen interval, normally symmetric around the middle of the interval, usually approaching a maximum in the middle, and usually tapering away from the middle. Mathematically, when another function or waveform/data-sequence is "multiplied" by a window function, the product is also zero-valued outside the interval: all that is left is the part where they overlap, the "view through the window".
Closed-form expressionIn mathematics, an expression is in closed form if it is formed with constants, variables and a finite set of basic functions connected by arithmetic operations (+, −, ×, ÷, and integer powers) and function composition. Commonly, the allowed functions are nth root, exponential function, logarithm, and trigonometric functions . However, the set of basic functions depends on the context.
