Critical point (mathematics)Critical point is a wide term used in many branches of mathematics. When dealing with functions of a real variable, a critical point is a point in the domain of the function where the function is either not differentiable or the derivative is equal to zero. When dealing with complex variables, a critical point is, similarly, a point in the function's domain where it is either not holomorphic or the derivative is equal to zero. Likewise, for a function of several real variables, a critical point is a value in its domain where the gradient is undefined or is equal to zero.
Dirac equationIn particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac particles", such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics.
Orthonormal basisIn mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, the standard basis for a Euclidean space is an orthonormal basis, where the relevant inner product is the dot product of vectors. The of the standard basis under a rotation or reflection (or any orthogonal transformation) is also orthonormal, and every orthonormal basis for arises in this fashion.
Heat equationIn mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quantity such as heat diffuses through a given region. As the prototypical parabolic partial differential equation, the heat equation is among the most widely studied topics in pure mathematics, and its analysis is regarded as fundamental to the broader field of partial differential equations.
Projection (linear algebra)In linear algebra and functional analysis, a projection is a linear transformation from a vector space to itself (an endomorphism) such that . That is, whenever is applied twice to any vector, it gives the same result as if it were applied once (i.e. is idempotent). It leaves its unchanged. This definition of "projection" formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object.
Kronecker deltaIn mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: or with use of Iverson brackets: For example, because , whereas because . The Kronecker delta appears naturally in many areas of mathematics, physics, engineering and computer science, as a means of compactly expressing its definition above.
River deltaA river delta is a landform shaped like a triangle, created by the deposition of sediment that is carried by a river and enters slower-moving or stagnant water. This occurs when a river enters an ocean, sea, estuary, lake, reservoir, or (more rarely) another river that cannot carry away the supplied sediment. It is so named because its triangle shape resembles the Greek letter Delta. The size and shape of a delta are controlled by the balance between watershed processes that supply sediment, and receiving basin processes that redistribute, sequester, and export that sediment.
Prompt criticalityIn nuclear engineering, prompt criticality describes a nuclear fission event in which criticality (the threshold for an exponentially growing nuclear fission chain reaction) is achieved with prompt neutrons alone and does not rely on delayed neutrons. As a result, prompt supercriticality causes a much more rapid growth in the rate of energy release than other forms of criticality. Nuclear weapons are based on prompt criticality, while nuclear reactors rely on delayed neutrons or external neutrons to achieve criticality.
Morse theoryIn mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiable function on a manifold will reflect the topology quite directly. Morse theory allows one to find CW structures and handle decompositions on manifolds and to obtain substantial information about their homology.
Dirac delta functionIn mathematical physics, the Dirac delta distribution (δ distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one. The current understanding of the unit impulse is as a linear functional that maps every continuous function (e.g., ) to its value at zero of its domain (), or as the weak limit of a sequence of bump functions (e.g.
Rational unified processThe rational unified process (RUP) is an iterative software development process framework created by the Rational Software Corporation, a division of IBM since 2003. RUP is not a single concrete prescriptive process, but rather an adaptable process framework, intended to be tailored by the development organizations and software project teams that will select the elements of the process that are appropriate for their needs. RUP is a specific implementation of the Unified Process.
Discrete Laplace operatorIn mathematics, the discrete Laplace operator is an analog of the continuous Laplace operator, defined so that it has meaning on a graph or a discrete grid. For the case of a finite-dimensional graph (having a finite number of edges and vertices), the discrete Laplace operator is more commonly called the Laplacian matrix. The discrete Laplace operator occurs in physics problems such as the Ising model and loop quantum gravity, as well as in the study of discrete dynamical systems.
