** The Baire category theorem about complete metric spaces, and its consequences, such as the open mapping theorem and the closed graph theorem.

Considerable confusion exists about the meanings of the SI ( or metric ) prefixes used with the unit byte, especially concerning prefixes such as kilo ( k or K ) and mega ( M ) as shown in the chart Prefixes for bit and byte.

Because the FLRW metric assumes a uniform distribution of mass and energy, it applies to our Universe only on large scales — local concentrations of matter such as our galaxy are gravitationally bound and as such do not experience the large-scale expansion of space.

It is still used in metric English-speaking countries ( such as Canada ), and remains the standard unit of classification for air conditioning units manufactured and sold in many non-English-speaking metric countries.

Various equivalent notions of compactness, such as sequential compactness and limit point compactness, can be developed in general metric spaces.

* If the metric space X is compact and an open cover of X is given, then there exists a number such that every subset of X of diameter < δ is contained in some member of the cover.

The utility of Cauchy sequences lies in the fact that in a complete metric space ( one where all such sequences are known to converge to a limit ), the criterion for convergence depends only on the terms of the sequence itself.

In mathematics, a contraction mapping, or contraction, on a metric space ( M, d ) is a function f from M to itself, with the property that there is some nonnegative real number < math > k < 1 </ math > such that for all x and y in M,

Thus, if ( M, d ) and ( N, d ) are two metric spaces, and, then there is a constant k such that

If S is an arbitrary set, then the set S < sup > N </ sup > of all sequences in S becomes a complete metric space if we define the distance between the sequences ( x < sub > n </ sub >) and ( y < sub > n </ sub >) to be, where N is the smallest index for which x < sub > N </ sub > is distinct from y < sub > N </ sub >, or 0 if there is no such index.

The fixed point theorem is often used to prove the inverse function theorem on complete metric spaces such as Banach spaces.

The expansion constant of a metric space is the infimum of all constants such that whenever the family intersects pairwise, the intersection

Thus differential geometry may study differentiable manifolds equipped with a connection, a metric ( which may be Riemannian, pseudo-Riemannian, or Finsler ), a special sort of distribution ( such as a CR structure ), and so on.

In such a situation, a company's lifespan was measured by its burn rate: that is, the rate at which a non-profitable company lacking a viable business model ran through its capital served as the metric.

A relatively weak gravitational field, such as the Earth's or the sun's, is represented by a metric that is approximately, but not exactly, Euclidean.

Given both Einstein's equations and suitable equations for the properties of matter, such a solution consists of a specific semi-Riemannian manifold ( usually defined by giving the metric in specific coordinates ), and specific matter fields defined on that manifold.

The Kaluza – Klein action for such a metric is given by

However, some properties of the metric, such as completeness, cannot be said to be inherited.

Visiting advocates of metrication also claim that they use metric units for many things internally with exceptions such as old petrol pumps in Myanmar, calibrated in British Imperial gallons.

Some countries such as Guyana, for example, have officially adopted the metric system, but have had some trouble over time implementing it.

Other Caribbean countries such as Saint Lucia are officially metric but are still in the process toward full conversion .< ref >

This turns the seminormed space into a pseudometric space ( notice this is weaker than a metric ) and allows the definition of notions such as continuity and convergence.

Therefore, 5 ml of such material is, or 5 500 000 000 metric tons.

If this connection is the Levi-Civita connection induced by a Riemannian metric, then the geodesics are ( locally ) the shortest path between points in the space.

In general, a metric space may have no geodesics, except constant curves.

If the covariant derivative is the Levi-Civita connection of a certain metric, then the geodesics for the connection are precisely those geodesics of the metric that are parametrised proportionally to their arc length.

If ∇ is a metric connection, then the affine geodesics are the usual geodesics of Riemannian geometry and are the locally distance minimizing curves.

If the covariant derivative is the Levi-Civita connection of a certain metric then the geodesics for the connection are precisely the geodesics of the metric that are parametrised by arc length.

The straight lines in the hyperbolic plane ( geodesics for this metric tensor, i. e. curves which minimize the distance ) are represented in this model by circular arcs perpendicular to the x-axis ( half-circles whose origin is on the x-axis ) and straight vertical lines ending on the x-axis.

* The Jacobi fields correspond to the geodesics on the tangent bundle ( with respect to the metric on induced by the metric on ).

The " straightness " of a line, interpreted as the property that it minimizes distances between its points, can be generalized and leads to the concept of geodesics in metric spaces.

In general relativity, the geodesics of the Schwarzschild metric describe the motion of particles of infinitesimal mass in the gravitational field of a central fixed mass M. The Schwarzschild geodesics have been pivotal in the validation of the Einstein's theory of general relativity.

Because test particles follow geodesics in a fixed metric, the orbits of those particles may be determined using the calculus of variations, also called the Lagrangian approach.

The geodesics for this metric tensor are circular arcs perpendicular to the real axis ( half-circles whose origin is on the real axis ) and straight vertical lines ending on the real axis.

The geodesics of the metric ( obtained where is extremised ) must, in some limit ( e. g., toward infinite speed of light ), agree with the solutions of Newtonian motion ( e. g., obtained by Lagrange equations ).

If we had added the standard Euclidean metric instead, our curves would have become circles, but not geodesics.

IGRP supports multiple metrics for each route, including bandwidth, delay, load, MTU, and reliability ; to compare two routes these metrics are combined together into a single metric, using a formula which can be adjusted through the use of pre-set constants.

However, they all have adopted metric measures to some degree through international trade and standardisation for example, Sierra Leone switched to selling fuel by the litre in May 2011.

The Congress of 1866 legalized the use of the metric system through the passage of US code 1952 Ed., Title 15, Ch 6, section 204 and 205.

The spindle of an ordinary metric micrometer has 2 threads per millimetre, and thus one complete revolution moves the spindle through a distance of 0. 5 millimeter.

James Clerk Maxwell and Elihu Thomson ( through the British Association for the Advancement of Science-BAAS ) introduced the Centimetre gram second system of units ( cgs ) in 1874, in order to derive electric and magnetic metric units, following the recommendation of Carl Friedrich Gauss in 1832.

# cost: i. e. the cost or metric of the path through which the packet is to be sent

There are five ( 5 ) K values used in the Composite metric calculation-K1 through K5.

A funge is an esoteric programming language which models its programs as metric spaces with coordinate systems ( often, but not necessarily, Cartesian ) and which execute instructions located at points in their program space by moving an instruction pointer ( a position vector which indicates the currently executing instruction ) through that space.

" His inventive playing helped redefine the role of jazz rhythm section through the use of polyrhythms and metric modulation ( transitioning between mathematically related tempos and / or time signatures ).

More simply, R-values may be converted from SI to US units through the following, where RSI is the given unit in metric units:

This is not obvious from the right-hand side of Kosmann's local expression, as the right-hand side seems to depend on the metric through the spin connection ( covariant derivative ), the dualisation of vector fields ( lowering of the indices ) and the Clifford multiplication on the spinor bundle.

Unique to universes described by the FLRW metric, a de Sitter universe has a Hubble Law which is not only consistent through all space, but also through all time ( since the deceleration parameter is equal to ), thus satisfying the perfect cosmological principle that assumes isotropy and homogeneity throughout space and time.

Winold recommends that, " metric structure is best described through detailed analysis of pulse groupings on various levels rather than through attempts to represent the organization with a single term " ( Winold 1975, 217 )

The formal definition of proper time involves describing the path through spacetime that represents a clock, observer, or test particle, and the metric structure of that spacetime.

The Q Score is a metric developed by Marketing Evaluations, Inc. that determines a " quotient " (" Q ") factor or score through mail and online panelists who make up representative samples of the United States.

Under the Mendenhall Order in 1893, metric standards, developed through international cooperation under the auspices of BIPM, were adopted as the fundamental standards for length and mass in the United States.

U. S. citizens are frequently exposed to metric units through coverage of international sporting events, particularly the Olympic Games.

A typical large refinery processing 40, 000 metric tonnes of crude oil per day ( per day ) circulates about 80, 000 cubic metres of water per hour through its cooling tower system.

For comparison, in 1920-1921 the peasants surrendered to the state 367 million poods ( 6, 010, 000 metric tons ) of cereals through prodrazvyorstka, as opposed to 240 million poods ( 3, 930, 000 metric tons ) in 1921-1922 through prodnalog.