Matrix (mathematics)In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a " matrix", or a matrix of dimension . Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra.
Squeeze mappingIn linear algebra, a squeeze mapping, also called a squeeze transformation, is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is not a rotation or shear mapping. For a fixed positive real number a, the mapping is the squeeze mapping with parameter a. Since is a hyperbola, if u = ax and v = y/a, then uv = xy and the points of the image of the squeeze mapping are on the same hyperbola as (x,y) is.
Mathematical formulation of quantum mechanicsThe mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. This mathematical formalism uses mainly a part of functional analysis, especially Hilbert spaces, which are a kind of linear space. Such are distinguished from mathematical formalisms for physics theories developed prior to the early 1900s by the use of abstract mathematical structures, such as infinite-dimensional Hilbert spaces (L2 space mainly), and operators on these spaces.
SymmetrySymmetry () in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations, such as translation, reflection, rotation, or scaling. Although these two meanings of the word can sometimes be told apart, they are intricately related, and hence are discussed together in this article.
Cardinal point (optics)In Gaussian optics, the cardinal points consist of three pairs of points located on the optical axis of a rotationally symmetric, focal, optical system. These are the focal points, the principal points, and the nodal points. For ideal systems, the basic imaging properties such as image size, location, and orientation are completely determined by the locations of the cardinal points; in fact only four points are necessary: the focal points and either the principal or nodal points.
Symmetry in biologySymmetry in biology refers to the symmetry observed in organisms, including plants, animals, fungi, and bacteria. External symmetry can be easily seen by just looking at an organism. For example, the face of a human being has a plane of symmetry down its centre, or a pine cone displays a clear symmetrical spiral pattern. Internal features can also show symmetry, for example the tubes in the human body (responsible for transporting gases, nutrients, and waste products) which are cylindrical and have several planes of symmetry.
Infrared spectroscopyInfrared spectroscopy (IR spectroscopy or vibrational spectroscopy) is the measurement of the interaction of infrared radiation with matter by absorption, emission, or reflection. It is used to study and identify chemical substances or functional groups in solid, liquid, or gaseous forms. It can be used to characterize new materials or identify and verify known and unknown samples. The method or technique of infrared spectroscopy is conducted with an instrument called an infrared spectrometer (or spectrophotometer) which produces an infrared spectrum.
Radius of curvature (optics)Radius of curvature (ROC) has specific meaning and sign convention in optical design. A spherical lens or mirror surface has a center of curvature located either along or decentered from the system local optical axis. The vertex of the lens surface is located on the local optical axis. The distance from the vertex to the center of curvature is the radius of curvature of the surface. The sign convention for the optical radius of curvature is as follows: If the vertex lies to the left of the center of curvature, the radius of curvature is positive.
Fourier-transform infrared spectroscopyFourier-transform infrared spectroscopy (FTIR) is a technique used to obtain an infrared spectrum of absorption or emission of a solid, liquid, or gas. An FTIR spectrometer simultaneously collects high-resolution spectral data over a wide spectral range. This confers a significant advantage over a dispersive spectrometer, which measures intensity over a narrow range of wavelengths at a time. The term Fourier-transform infrared spectroscopy originates from the fact that a Fourier transform (a mathematical process) is required to convert the raw data into the actual spectrum.
Molecular orbitalIn chemistry, a molecular orbital (ɒrbədl) is a mathematical function describing the location and wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of finding an electron in any specific region. The terms atomic orbital and molecular orbital were introduced by Robert S. Mulliken in 1932 to mean one-electron orbital wave functions. At an elementary level, they are used to describe the region of space in which a function has a significant amplitude.
