The fact that stretching is equivalent to compression may seem somewhat counterintuitive, but it makes sense if rubber is viewed as a one-dimensional gas.

Consider gas in a one-dimensional container ( e. g., a long thin tube ).

In the case of a Tonks – Girardeau gas ( TG ), so many properties of this one-dimensional string of bosons would be sufficiently fermion-like that the situation is often referred to as the ' fermionization ' of bosons. Tonks – Girardeau gas coincide with quantum Nonlinear Schrödinger equation for infinite repulsion, which can be efficiently analyzed by Quantum inverse scattering method.

* artificial ' quantum wires ' ( one-dimensional strips of electrons ) defined by applying gate voltages to a two-dimensional electron gas, or by other means ( lithography, AFM, etc.

Because of the absence of a net force in the azimuthal direction, conservation of angular momentum allows the radial component of this equation to be expressed solely with respect to the radial coordinate, r, and the angular momentum, yielding the radial equation ( a " fictitious one-dimensional problem " with only an r dimension ):

In statistics, the Kolmogorov – Smirnov test ( K – S test ) is a nonparametric test for the equality of continuous, one-dimensional probability distributions that can be used to compare a sample with a reference probability distribution ( one-sample K – S test ), or to compare two samples ( two-sample K – S test ).

This means that if one studied supergravity on an eleven-dimensional spacetime that looks like the product of a ten-dimensional spacetime with another very small one-dimensional manifold, one gets the Type IIA supergravity theory.

Rays ( one-dimensional subspaces ) in H are associated with states of the system.

The walls of a one-dimensional box may be visualised as regions of space with an infinitely large potential energy.

One attempt to overcome these limitations is to replace ordinary quantum field theory, which is based on the classical concept of a point particle, with a quantum theory of one-dimensional extended objects: string theory.

For one-dimensional simple harmonic motion, the equation of motion, which is a second-order linear ordinary differential equation with constant coefficients, could be obtained by means of Newton's second law and Hooke's law.

* A closed interval b is a simple example of a one-dimensional manifold with boundary.

Consequently, the k-fold product of W ′ with itself, W ′< sup > k </ sup >, is one-dimensional.

Mathematically, the most basic wave is the ( spatially ) one-dimensional sine wave ( or harmonic wave or sinusoid ) with an amplitude described by the equation:

As the form of this Hamiltonian makes clear, the N-dimensional harmonic oscillator is exactly analogous to N independent one-dimensional harmonic oscillators with the same mass and spring constant.

Combined with other nano-and microfabrication technologies, cold welding is anticipated to have potential applications in the future bottom-up assembly of metallic one-dimensional nanostructures.

The non-sequential ionization can be readily understood with one-dimensional models of atoms which until recently were the only to handle numerically.

The " marvelous " one-dimensional world differs from the bidimensional world of magical realism, as in the latter, the supernatural realm blends with the natural, familiar world ( arriving at the combination of two layers of reality: bidimensional ).

A good example of this is one-dimensional heat transport with Dirichlet boundary conditions, the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies ; this makes solutions very flexible.

The word " symmetry " in the above statement refers more precisely to the covariance of the form that a physical law takes with respect to a one-dimensional Lie group of transformations satisfying certain technical criteria.

The one-dimensional Schrödinger operator with the potential, defined initially on smooth compactly supported functions, is essentially self-adjoint ( that is, has a self-adjoint closure ) for < math > 0 <

One degree of freedom occurs when one has an autonomous ordinary differential equation in a single variable, with the resulting one-dimensional system being called a phase line, and the qualitative behaviour of the system being immediately visible from the phase line.

The real numbers may be viewed as a one-dimensional vector space with a compatible multiplication, and hence a one-dimensional algebra over itself.

Menger showed, in the 1926 construction, that the sponge is a universal curve, in that any possible one-dimensional curve is homeomorphic to a subset of the Menger sponge, where here a curve means any compact metric space of Lebesgue covering dimension one ; this includes trees and graphs with an arbitrary countable number of edges, vertices and closed loops, connected in arbitrary ways.

For along with his steady diet of giggles and unconditional love, Barney offers our children a one-dimensional world where everyone must be happy and everything must be resolved right away.

After two seasons, Reeves was dissatisfied with the one-dimensional role and low salary.

When applied to a function defined on a one-dimensional domain, it denotes its standard derivative as defined in calculus.

A one-dimensional line or curve can be represented by the coordinates as a function of one parameter.

and may yield greater accuracy for the same number of function evaluations than repeated integrations using one-dimensional methods.

The Artin reciprocity law applies to a Galois extension of algebraic number fields whose Galois group is abelian, assigns L-functions to the one-dimensional representations of this Galois group ; and states that these L-functions are identical to certain Dirichlet L-series or more general series ( that is, certain analogues of the Riemann zeta function ) constructed from Hecke characters.

Fourier's Law provides a simplified one-dimensional description of the variation in heat flow Q as a function of thermal conductivity k and thermal gradient dT / dz:

* The wave function of the ground state of a particle in a one-dimensional well is a half-period sine wave which goes to zero at the two edges of the well.

These two operators do not commute as may be seen by considering the effect of their products x ( d / dx ) and ( d / dx ) x on a one-dimensional wave function ψ ( x ):

Scalar quantization, typically denoted as, is the process of using a quantization function ( ) to map a scalar ( one-dimensional ) input value to a scalar output value.

The Airy function is the solution to Schrödinger's equation for a particle confined within a triangular potential well and for a particle in a one-dimensional constant force field.

The Fourier transform ( a one-dimensional function ) of the resulting signal is taken as the window is slid along the time axis, resulting in a two-dimensional representation of the signal.

The Jacobi theta function is the unique solution to the one-dimensional heat equation with periodic boundary conditions at time zero.

For one-dimensional integration, quadrature methods such as the trapezoidal rule, Simpson's rule, or Newton – Cotes formulas are known to be efficient if the function is smooth.

The < tt > findgen </ tt > function in the above example returns a one-dimensional array of floating point numbers, with values equal to a series of integers starting at 0.

More generally, a typical energy function of a one-dimensional system has a Taylor expansion in the extension q:

This makes the Schwarzian derivative an important tool in one-dimensional dynamics since it implies that all iterates of a function with negative Schwarzian will also have negative Schwarzian.

Schematic one-dimensional test function ( black ) and STUN effective potential ( red & blue ), where the minimum indicated by the arrows is the best minimum found so far.

For a real-valued continuous function f, defined on an interval b ⊂ ℝ, its total variation on the interval of definition is a measure of the one-dimensional arclength of the curve with parametric equation x ↦ f ( x ), for x ∈ b.

For the graph of a one-dimensional function, this corresponds to a point on the graph where the tangent is parallel to the x-axis.

However, this condition is purely arithmetic and cannot be considered for a general one-dimensional function field in a straightforward way.

In practice, this is usually the maximally-localized set, in which the Wannier function is localized around the point R and rapidly goes to zero away from R. For the one-dimensional case, it has been proved by Kohn that there is always a unique choice that gives these properties ( subject to certain symmetries ).

ambiguity function consists of examining the one-dimensional

* Take a two-dimensional function f ( r ), project it onto a ( one-dimensional ) line, and do a Fourier transform of that projection.

where A < sub > 1 </ sub > represents the Abel transform operator, projecting a two-dimensional circularly symmetric function onto a one-dimensional line, F < sub > 1 </ sub > represents the 1-D Fourier transform

In mathematics, the Lehmer – Schur algorithm ( named after Derrick Henry Lehmer and Issai Schur ) is a root-finding algorithm extending the one-dimensional bracketing used by the bisection method to find the roots of a function of one complex variable inside any rectangular region of the function's holomorphicity ( i. e., analyticity ).