Two slits are illuminated by a plane waveGeometry for far field fringesMuch of the behaviour of light can be modelled using classical wave theory.

* Catoptrics, which concerns the mathematical theory of mirrors, particularly the images formed in plane and spherical concave mirrors.

Since the fundamental group is a homotopy invariant, the theory of the winding number for the complex plane minus one point is the same as for the circle.

One of the most famous and productive problems of graph theory is the four color problem: " Is it true that any map drawn in the plane may have its regions colored with four colors, in such a way that any two regions having a common border have different colors?

The roots of the theory go back to the result of Émile Picard in 1879, showing that a non-constant complex-valued function which is analytic in the entire complex plane assumes all complex values save at most one.

In the early 1920s Rolf Nevanlinna, partly in collaboration with his brother Frithiof, extended the theory to cover meromorphic functions, i. e. functions analytic in the plane except for isolated points in which the Laurent series of the function has a finite number of terms with a negative power of the variable.

Jakobson's universalizing structural-functional theory of phonology, based on a markedness hierarchy of distinctive features, was the first successful solution of a plane of linguistic analysis according to the Saussurean hypotheses.

Pythagoras emphasized subduing emotions and bodily desires in order to enable the intellect to function at the higher plane of theory.

However, the complete theory of the Abbe sine condition shows that if a lens is corrected for coma and spherical aberration, as all good photographic objectives must be, the second principal plane becomes a portion of a sphere of radius f centered about the focal point, ..." In this sense, the traditional thin-lens definition and illustration of f-number is misleading, and defining it in terms of numerical aperture may be more meaningful.

In electromagnetic theory, the phase constant, also called phase change constant, parameter or coefficient is the imaginary component of the propagation constant for a plane wave.

Due to his background working with Triffids, Masen has developed a theory that they were bioengineered in the USSR and then accidentally released into the wild when a plane smuggling their seeds was shot down.

Many fundamental tasks in mathematics, geometry, trigonometry, graph theory and graphing are performed in a two-dimensional space, or in other words, in the plane.

The topological plane is the natural context for the branch of graph theory that deals with planar graphs, and results such as the four color theorem.

In the eighteenth century came the beginnings of the theory of plane algebraic curves, in general.

Specific examples of mathematics, statistics, and physics applied to music composition are the use of the statistical mechanics of gases in Pithoprakta, statistical distribution of points on a plane in Diamorphoses, minimal constraints in Achorripsis, the normal distribution in ST / 10 and Atrées, Markov chains in Analogiques, game theory in Duel and Stratégie, group theory in Nomos Alpha ( for Siegfried Palm ), set theory in Herma and Eonta, and Brownian motion in N ' Shima.

In Planetary waves on beta planes, he developed a beta plane approximation for simplifying the equations of classical tidal theory, whilst at the same time developing planetary wave relations.

By virtue of these interpretive connections, the network can function as a scientific theory: From certain observational data, we may ascend, via an interpretive string, to some point in the theoretical network, thence proceed, via definitions and hypotheses, to other points, from which another interpretive string permits a descent to the plane of observation.

In 1849 he published a long paper on the dynamical theory of diffraction, in which he showed that the plane of polarisation must be perpendicular to the direction of propagation.

For example, in the case of the bosonic open string theory in 26-dimensional flat spacetime, a general element of the Fock-space of the BRST quantized string takes the form ( in radial quantization in the upper half plane ),

For example, the treatment of the projective plane starting from axioms of incidence is actually a broader theory ( with more models ) than is found by starting with a vector space of dimension three.

This does not mean that the ' spiritual ' theory of race is acceptable, but it had at least the merit of not totally failing to see certain values, to refuse the German aberrations and the ones modeled after them and to try to keep racism on a plane of cultural problems worthy of the name " ( Storia degli Ebrei Italiani sotto il Fascismo, or The History of Italian Jews under Fascism 1977 ; 465 ).

Specific examples of mathematics, statistics, and physics applied to music composition are the use of the statistical mechanics of gases in Pithoprakta, statistical distribution of points on a plane in Diamorphoses, minimal constraints in Achorripsis, the normal distribution in ST / 10 and Atrées, Markov chains in Analogiques, game theory in Duel and Stratégie, group theory in Nomos Alpha ( for Siegfried Palm ), set theory in Herma and Eonta, and Brownian motion in N ' Shima.

-- On the basis of a differentiability assumption in function space, it is possible to prove that, for materials having the property that the stress is given by a functional of the history of the deformation gradients, the classical theory of infinitesimal viscoelasticity is valid when the deformation has been infinitesimal for all times in the past.

He supports the steady-state theory which holds that matter is continually being created in space.

The elementary theory of optical systems leads to the theorem: Rays of light proceeding from any object point unite in an image point ; and therefore an object space is reproduced in an image space.

In mathematics, specifically in measure theory, a Borel measure is defined as follows: let X be a locally compact Hausdorff space, and let be the smallest σ-algebra that contains the open sets of X ; this is known as the σ-algebra of Borel sets.

When the Banach algebra A is the algebra L ( X ) of bounded linear operators on a complex Banach space X ( e. g., the algebra of square matrices ), the notion of the spectrum in A coincides with the usual one in the operator theory.

In 1916, Albert Einstein published his theory of general relativity, which provided a unified description of gravity as a geometric property of space and time.

The Big Bang theory states that it is the point in which all dimensions came into existence, the start of both space and time.

In contrast, modern control theory is carried out in the state space, and can deal with multi-input and multi-output ( MIMO ) systems.

In contrast to the frequency domain analysis of the classical control theory, modern control theory utilizes the time-domain state space representation, a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations.

In it, he puts forth a new theory about the nature of space and describes how this theory influences thinking about architecture, building, planning, and the way in which we view the world in general.

Sequences, like patterns, promise to be tools of wider scope than building ( just as his theory of space goes beyond architecture ).

His work was a key aspect of Hermann Weyl and John von Neumann's work on the mathematical equivalence of Werner Heisenberg's matrix mechanics and Erwin Schrödinger's wave equation and his namesake Hilbert space plays an important part in quantum theory.

In 1926 von Neuman showed that if atomic states were understood as vectors in Hilbert space, then they would correspond with both Schrödinger's wave function theory and Heisenberg's matrices.

The related concept of " standard " numbers, which can only be defined within a finite time and space, is used to motivate axiomatic internal set theory, and provide a workable formulation for illimited and infinitesimal number.

Bifurcation theory considers a structure in phase space ( typically a fixed point, a periodic orbit, or an invariant torus ) and studies its behavior as a function of the parameter μ.

An implication of Einstein's theory of general relativity is that Euclidean space is a good approximation to the properties of physical space only where the gravitational field is weak.

* Relativity and Quantum field theory ( physics ), η ( with two subscripts ) represents the metric tensor of Minkowski space ( flat spacetime ).

Perhaps their most famous application is the theory of relativity, where empty spacetime with no matter is represented by the flat pseudo-Euclidean space called Minkowski space, spacetimes with matter in them form other pseudo-Riemannian manifolds, and gravity corresponds to the curvature of such a manifold.

In string theory, Eric G. Gimon and Petr Hořava have argued that in a supersymmetric five-dimensional Gödel universe, quantum corrections to general relativity effectively cut off regions of spacetime with causality-violating closed timelike curves.

In particular, in the quantum theory a smeared supertube is present that cuts the spacetime in such a way that, although in the full spacetime a closed timelike curve passed through every point, no complete curves exist on the interior region bounded by the tube.

Many areas of mathematics and computer science have been brought to bear on the problem, including elliptic curves, algebraic number theory, and quantum computing.

As the perturbation increases and the smooth curves disintegrate we move from KAM theory to

For example, profit maximization lies behind the neoclassical theory of the firm, while the derivation of demand curves leads to an understanding of consumer goods, and the supply curve allows an analysis of the factors of production.

Utility maximization is the source for the neoclassical theory of consumption, the derivation of demand curves for consumer goods, and the derivation of labor supply curves and reservation demand.

These developments were accompanied by the introduction of new tools, such as indifference curves and the theory of ordinal utility.

* Complex multiplication, a theory of elliptic curves.

In 1991 David Deutsch showed that quantum theory is fully consistent ( in the sense that the so-called density matrix can be made free of discontinuities ) in spacetimes with closed timelike curves.

Physicists have long been aware that there are solutions to the theory of general relativity which contain closed timelike curves, or CTCs — see for example the Gödel metric.

:* Congruence ( manifolds ), in the theory of smooth manifolds, the set of integral curves defined by a nonvanishing vector field defined on the manifold

The theory of indifference curves was developed by Francis Ysidro Edgeworth, who explained in his book " Mathematical Psychics: an Essay on the Application of Mathematics to the Moral Sciences ”, 1881, the mathematics needed for its drawing ; later on, Vilfredo Pareto was the first author to actually draw these curves, in his book " Manual of Political Economy ", 1906 ; and others in the first part of the 20th century.

Consumer theory uses indifference curves and budget constraints to generate consumer demand curves.

The different shapes of the curves imply different responses to a change in price as shown from demand analysis in consumer theory.

Choice theory formally represents consumers by a preference relation, and use this representation to derive indifference curves showing combinations of equal preference to the consumer.

He became well known for his general theory of curvilinear coordinates and his notation and study of classes of ellipse-like curves, now known as Lamé curves, and defined by the equation:

He made decisive contributions to meromorphic curves, value distribution theory, Riemann surfaces, conformal geometry, quasiconformal mappings and other areas during his career.

Monge's paper gives the ordinary differential equation of the curves of curvature, and establishes the general theory in a very satisfactory manner ; the application to the interesting particular case of the ellipsoid was first made by him in a later paper in 1795.

A fundamental advance in theory of curves was the advent of analytic geometry in the seventeenth century.