- Beta function: It is also called the Euler integral of the first kind,
is a special function defined by
for Re(x), Re(y) > 0.
- Calabi-Yau manifolds: They are a special class of manifolds used in some branches of mathematics (such as algebraic geometry) as well as in theoretical physics. For instance, in superstring theory the extra dimensions of spacetime are sometimes conjectured to take the form of a 6-dimensional Calabi-Yau manifold. The designation "Calabi-Yau space" for a member of this class was coined by physicists. Physical insights about Calabi-Yau manifolds, especially mirror symmetry, led to tremendous progress in pure mathematics.
- Chaos theory (in mathematics but also in physics): It describes the behaviour of certain nonlinear dynamical systems that may exhibit dynamics that are highly sensitive to initial conditions (popularly referred to as the butterfly effect). As a result of this sensitivity, which manifests itself as an exponential growth of perturbations in the initial conditions, the behaviour of chaotic systems appears to be random. This happens even though these systems are deterministic, meaning that their future dynamics are fully defined by their initial conditions, with no random elements involved.
- Difference equations: A difference equation is a specific type of recurrence
relation. In mathematics, a recurrence relation is an equation that defines
a sequence recursively: each term of the sequence is defined as a function
of the preceding terms. An example of a recurrence relation is the logistic
map:
Solving a recurrence relation means obtaining a closed-form solution: a non-recursive function of n.
- Dirac equation: It is a relativistic quantum mechanical wave equation that provides a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity. The equation demands the existence of antiparticles and actually predated their experimental discovery, making the discovery of the positron, the antiparticle of the electron, one of the greatest triumphs of modern theoretical physics.
- Drake equation: It is a result in the speculative fields of exobiology and the search for extraterrestrial intelligence (SETI).This equation was devised by Dr. Frank Drake in an attempt to estimate the number of extraterrestrial civilizations in our galaxy with which we might come in contact. The main purpose of the equation is to allow scientists to quantify the uncertainty of the factors which determine the number of such extraterrestrial civilizations.
- Einstein tensor: It is a mathematical entity expressing the curvature of spacetime in the Einstein field equations, which describes gravitation according to the theory of general relativity. It is also sometimes called the trace-reversed Ricci tensor.
In physics and differential geometry, the Einstein tensor is a tensor of
rank 2 defined over Riemannian manifolds. In index-free notation it is defined
as
where is the Ricci tensor, is the metric tensor and R is the scalar curvature.
In component form, the previous equation reads as
- Einstein-Hilbert action: It is a mathematical object (an action) that
is used to derive Einstein's field equations of general relativity. In 1913,
Albert Einstein realized the geometry would work if spacetime was curved,
and not flat as had always been the classical assumption. Einstein proposed
that mass and energy would warp spacetime and that gravitational force was
merely an expression of the spacetime curvature. Einstein initially found
an equation which was correct for vacuum spacetime (but in the presence
of matter was wrong as it omitted the term ?Rgmn/2) and finally found the
correct equations in 1915 by trial and error. Hilbert, who had been studying
Einstein's work, also found the missing term) at about the same time as
Einstein, by deriving Einstein's equation as the Euler-Lagrange equations
for the Einstein-Hilbert action with respect to variation of the metric.
The action S[g] which gives rise to the vacuum Einstein equations is given
by the following integral of the Lagrangian
where is the determinant of a spacetime Lorentz metric, R is the Ricci scalar,
k is the constant c4/16?G, the Lagrangian being , and the integral is taken
over a region of spacetime. The Einstein equations in the presence of matter
are given by adding the Lagrangian for the matter into the integral. Note
that is an invariant 4-volume element, so the action can be also written
in the following (somewhat more elegant) fashion:
- Euclidean geometry: It is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. The method consists of assuming a small set of intuitively appealing axioms, and then proving many other propositions (theorems) from those axioms. Although many of Euclid's results had been stated by earlier Greek mathematicians, Euclid was the first to show how these propositions could be fit together into a comprehensive deductive and logical system.
- Fourier series: It is a type of Fourier analysis, which is used on functions that might otherwise be difficult or impossible to analyze. The series decomposes a periodic function into a sum of simple functions, which may be sines and cosines or may be complex exponentials.
- Fractal: Generally it is a rough or fragmented geometric shape that can be subdivided into parts, each of which is (at least approximately) a reduced-size copy of the whole, a property called self-similarity.
- Green's function: It is a type of function used to solve inhomogeneous differential equations subject to boundary conditions. The term is used in physics, specifically in quantum field theory and statistical field theory, to refer to various types of correlation functions, even those that do not fit the mathematical definition.
- Group theory: In abstract algebra, it means the study the algebraic structures
known as groups. A group is a set G closed under a binary operation satisfying
three axioms:
1. The operation is associative.
2. The operation has an identity element.
3. Every element has an inverse element.
Groups are building blocks of more elaborate algebraic structures such as
rings, fields, and vector spaces, and recur throughout mathematics.
- Large numbers: They are numbers that are significantly larger than those ordinarily used in everyday life. The term typically refers to large positive integers, or more generally, large positive real numbers, but it may also be used in other contexts.
- Lie group: is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups represent the best developed theory of continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics. They provide a natural framework for analysing the continuous symmetries of differential equations (Differential Galois theory), much in the same way as permutation groups are used in Galois Theory for analysing the discrete symmetries of algebraic equations.
- Mirror symmetry: It is a surprising relation that can exist between two Calabi-Yau manifolds. It happens, usually for two such six-dimensional manifolds, that the shapes may look very different geometrically, but nevertheless they are equivalent if they are employed as hidden dimensions of string theory. The classical formulation of mirror symmetry relates two Calabi-Yau threefolds M and W whose Hodge numbers h1,1 and h1,2 are swapped; string theory compacted on these two manifolds can be proved to lead to identical physical phenomena.
- Perturbation theory: It comprises mathematical methods that are used
to find an approximate solution to a problem which cannot be solved exactly,
by starting from the exact solution of a related problem. Perturbation theory
is applicable if the problem at hand can be formulated by adding a "small"
term to the mathematical description of the exactly solvable problem. Perturbation
theory leads to an expression for the desired solution in terms of a power
series in some "small" parameter that quantifies the deviation
from the exactly solvable problem. The leading term in this power series
is the solution of the exactly solvable problem, while further terms describe
the deviation in the solution, due to the deviation from the initial problem.
Formally, we have for the approximation to the full solution A, a series
in the small parameter (here called ?), like the following:
In this example, A0 would be the known solution to the exactly solvable
initial problem and A1,A2,... represent the "higher orders" which
are found iteratively by some systematic procedure. For small ? these higher
orders become successively less important.
- Riemannian geometry: It is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives in particular local notions of angle, length of curves, surface area, and volume. From those some other global quantities can be derived by integrating local contributions.
- Schrödinger equation: It is an equation that describes how the quantum state of a physical system varies. According to the Copenhagen interpretation of quantum mechanics, the state vector is used to calculate the probability that a physical system is in a given quantum state. Schrödinger's equation is primarily applied to microscopic systems, such as electrons and atoms, but is sometimes applied to macroscopic systems (such as the whole universe).
- Wave equation: It is an important second-order linear partial differential
equation that describes the propagation of a variety of waves, such as sound
waves, light waves and water waves. The wave equation is the prototypical
example of a hyperbolic partial differential equation. In its simplest form,
the wave equation refers to a scalar function u that satisfies:
where is the Laplacian and where c is a fixed constant equal to the propagation
speed of the wave.