* Of the four Maxwell's equations, two — Faraday's law and Ampère's law — can be compactly expressed using curl.

Using the summation symbol, the same polynomial is expressed more compactly as follows:

On larger kites, clever hinges and latches allow the kite to be disassembled and compactly folded for storage or transport.

The Lorentz transform for a boost in one of the above directions can be compactly written as a single matrix equation:

Very commonly, these values are powers of 10 ; so for instance, if / stands for one, − for ten and + for 100, then the number 304 can be compactly represented as < tt >+++ ////</ tt > and the number 123 as < tt >+ − − ///</ tt > without any need for zero.

for all continuous compactly supported functions ƒ.

Clothes are folded to allow them to be stored compactly, to prevent creasing, to preserve creases or to present them in a more pleasing manner, for instance when they are put on sale in stores.

In America, the term is not used for this purpose because the bands are metal compactly sewn, while ' lace ' seems to imply cloth sewn into patterns with holes in them.

This construct displays a person's genealogy compactly, without the need for a diagram such as a family tree.

The transformation rules for polar vectors and pseudovectors can be compactly stated as

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 <

Let U be an open set in R < sup > n </ sup > and φ: U → R < sup > n </ sup > an injective differentiable function with continuous partial derivatives, the Jacobian of which is nonzero for every x in U. Then for any real-valued, compactly supported, continuous function f, with support contained in φ ( U ),

: Variational definition: A surface M ⊂ R < sup > 3 </ sup > is minimal if and only if it is a critical point of the area functional for all compactly supported variations.

: Energy definition: A conformal immersion X: M → R < sup > 3 </ sup > is minimal if and only if it is a critical point of the Dirichlet energy for all compactly supported variations, or equivalently if any point p ∈ M has a neighbourhood with least energy relative to its boundary.

One can prove directly from the Fourier slice theorem and change of variables for integration that for a compactly supported continuous function ƒ of two variables

Sadval campaigned for the redrawing of the Russian – Azerbaijani border to allow for the creation of a single Lezgin state encompassing areas in Russia and Azerbaijan where Lezgins were compactly settled.

To complete the buildup of measure theory for locally compact spaces from the functional-analytic viewpoint, it is necessary to extend measure ( integral ) from compactly supported continuous functions.

# Definition of the upper integral μ *( g ) of a lower semicontinuous positive ( real-valued ) function g as the supremum ( possibly infinite ) of the positive numbers μ ( h ) for compactly supported continuous functions h ≤ g

This property is interesting in the case of locally compact topological groups, since locally compact compactly generated topological groups can be approximated by locally compact, separable metric factor groups of G. More precisely, for a sequence

One of the primary motivations for studying compactly generated spaces comes from category theory.

A more well-behaved category commonly used in homotopy theory is the category of compactly generated Hausdorff spaces ( also called k-spaces ).

In mathematics, a homology theory in algebraic topology is compactly supported if, in every degree n, the relative homology group H < sub > n </ sub >( X, A ) of every pair of spaces

If one has defined a homology theory over compact pairs, it is possible to extend it into a compactly supported homology theory in the wider category of Hausdorff pairs ( X, A ) with A closed in X, by defining that the homology of a Hausdorff pair ( X, A ) is the direct limit over pairs ( Y, B ), where Y, B are compact, Y is a subset of X, and B is a subset of A.

The following theorem represents positive linear functionals on C < sub > c </ sub >( X ), the space of continuous compactly supported complex-valued functions on a locally compact Hausdorff space X.

The MFP measure can be compactly introduced with an accounting procedure in the following calculation.

Specifically, a topological space X is compactly generated if it satisfies the following condition:

More precisely, a direction v is in the complement of if there is a compactly supported smooth function φ with φ ( x ) ≠ 0 and an open cone Γ containing v such that the following estimate holds for each positive integer N:

The operational semantics of BCL, apart from eta-reduction ( which is not required for Turing completeness ), may be very compactly specified by the following rewriting rules for subterms of a given term, parsing from the left:

A survey prepared by the Russian imperial authorities in 1823, several years before the 1828 Armenian migration from Persia to the newly established Armenian Province, shows that all Armenians of Karabakh compactly resided in its highland portion, i. e. on the territory of the five traditional Armenian principalities in Nagorno Karabakh, and constituted an absolute demographic majority on those lands.

A survey prepared by the Russian imperial authorities in 1823, a year after and several years before the 1828 Armenian migration from Persia to the newly-established Armenian Province, shows that all Armenians of Karabakh compactly resided in its highland portion, i. e., on the territory of the five traditional Armenian principalities, and constituted an absolute demographic majority on those lands.

The Chebyshev polynomials of the first kind can be defined compactly in an orbit through the trigonometric

Suppose first that f is a compactly supported smooth function.

The pips on the Asian-style facets are arranged more compactly than the Western-style facets.

Let be an oriented smooth manifold of dimension n and let be an n-differential form that is compactly supported on.

One normally obtains spaces of distributions by forming the strong dual of a space of test functions ( such as the compactly supported smooth functions on R < sup > n </ sup >).

Anopheles eggs may on occasion cluster together on the water, too, but the clusters do not generally look much like compactly glued rafts of eggs.

If K ( x ) has an isolated singularity at the origin, but is an otherwise " nice " function, then the principal value distribution is defined on compactly supported smooth functions by

A yurt is designed to be dismantled and the parts carried compactly on camels or yaks to be rebuilt on another site.

Because inflatable craft can be stored compactly they can also be transported on midget submarines such as those operated by the Advanced SEAL Delivery System.

More generally, in the n-dimensional Euclidean space R < sup > n </ sup >, the Radon transform of a compactly supported continuous function ƒ is a function Rƒ on the space Σ < sub > n </ sub > of all hyperplanes in R < sup > n </ sup >.

It improves on conventional cryptosystems by representing some elements of large finite fields compactly and therefore transmitting fewer bits.

Every bounded positive-definite measure μ on G satisfies μ ( 1 ) ≥ 0. improved this criterion by showing that it is sufficient to ask that, for every continuous positive-definite compactly supported function f on G, the function Δ < sup >– ½ </ sup > f has non-negative integral with respect to Haar measure, where Δ denotes the modular function.

In mathematics, a bump function is a function on a Euclidean space which is both smooth ( in the sense of having continuous derivatives of all orders ) and compactly supported.

Given any topological space X we can define a ( possibly ) finer topology on X which is compactly generated.

His already mentioned influential 1993 paper on the technological singularity compactly summarizes the basic ideas.