For example, the equation y = x corresponds to the set of all the points on the plane whose x-coordinate and y-coordinate are equal.

For example, using Cartesian coordinates on the plane, the distance between two points ( x < sub > 1 </ sub >, y < sub > 1 </ sub >) and ( x < sub > 2 </ sub >, y < sub > 2 </ sub >) is defined by the formula

For example, Barrett points out that the Pastoral Epistles are concerned that ministers of the generation of Timothy and Titus should pass on the doctrine they had received to the third generation.

For example, the graph contains the points ( 1, 1 ), ( 2, 0. 5 ), ( 5, 0. 2 ), ( 10, 0. 1 ), ... As the values of x become larger and larger, say 100, 1000, 10, 000 ..., putting them far to the right of the illustration, the corresponding values of y,. 01,. 001,. 0001, ..., become infinitesimal relative to the scale shown.

For example, if the player rolls a 6 and a 3 ( notated as " 6-3 "), that player must move one checker six points forward, and another or the same checker three points forward.

The " Deuteronomic " perspective ( that of the book of Deuteronomy ) is particularly evident in prayers and speeches spoken by key figures at major transition points: Solomon's speech at the dedication of the Temple is a key example.

The military tiles ( since there is only one each ) are also considered to be five mixed " pairs " ( for example, the 3-6 and 4-5 tiles " match " because they have same total points and both in the military suit ).

The given example sequence shows the importance of including the boundary points of the interval, since the limit points must be in the space itself: an open ( or half-open ) interval of the real numbers is not compact.

For example, the real line equipped with the discrete topology is closed and bounded but not compact, as the collection of all singleton points of the space is an open cover which admits no finite subcover.

For example, the file to which the link < tt >/ bin / ls </ tt > points in a typical Unix-like system probably has a defined size that seldom changed.

For example, a circle of radius 2 may be described as the set of all points whose coordinates x and y satisfy the equation x < sup > 2 </ sup > + y < sup > 2 </ sup > = 4.

For example, philosophy professor Robert Carroll points to Dianetics ' lack of empirical evidence:

In the transition between two positions the diver may for example bend their legs or curve at the waist, and points will not be deducted for doing so.

It has also been suggested that space and distance themselves are relational, and that an electron can appear to be in " two places at once " – for example, at both slits – because its spatial relations to particular points on the screen remain identical from both slit locations.

Other attributes also set each map apart, for example, restrictions on the placement of initial settlements, whether tiles are distributed randomly, the number of victory points needed to win, and special victory point awards, usually for building on islands across the sea.

For example, if u < sub > 1 </ sub > is an eigenvector of A, with a real eigenvalue smaller than one, then the straight lines given by the points along α u < sub > 1 </ sub >, with α ∈ R, is an invariant curve of the map.

An example PPF with illustrative points marked The production-possibility frontier ( PPF ) is an expository figure for representing scarcity, cost, and efficiency.

For example, Euclid assumed implicitly that any line contains at least two points, but this assumption cannot be proved from the other axioms, and therefore must be an axiom itself.

However, a particularly powerful piece of equipment — for example, a cloak that renders its wearer invisible for a brief period — may be treated as a Power that the hero must spend their initial element points on.

For example, in quantum field theory " locality " means that quantum fields at different points of space do not interact with one another.

For example, the Survey of Professional Forecasters contains forecasts for many forecasters ( cross-sectional observations ), at many points in time ( time series observations ), and at multiple forecast horizons ( a third dimension ).

For example, many key failure points associated with wire and connectors will be eliminated thus hours spent troubleshooting wires and connectors will be reduced.

For example many fundamental investors use technicals for deciding entry and exit points.

Falstaff and Henry meet at several points, and Falstaff is a familiar of Henry's, but his plot and Henry's do not mix.

To Serenissimus such tribes as the Cossacks of the Don or those ex-bandits the Zaporogian Cossacks ( in whose islands along the lower Dnieper the Polish novelist Sienkiewicz would one day place With Fire And Sword ) were just elements for enforced resettlement in, say, Bessarabia, where, as `` the faithful of the Black Sea borders '', he could use their presence as bargaining points in the Czarina's territorial claims against Turkey.

A general line L meets Q in two points, Af and Af, through each of which passes a unique generator of the regulus, Af, whose lines are simple secants of Aj.

) Hilton also claims a Roman Catholic monarch would therefore be unable to be crowned by the Archbishop of Canterbury and points to the examples of European states that have similar religious provisions for their monarchs: Denmark, Norway, and Sweden, whose constitutions compel their monarchs to be Lutherans, the Netherlands, the constitution of which insists its monarchs be members of the Protestant House of Orange, and Belgium, which has a constitution that provides for the succession to be through Roman Catholic houses.

The celebration of deeds of ancient Danish and Swedish heroes, the poem beginning with a tribute to the royal line of Danish kings, but written in the dominant literary dialect of Anglo-Saxon England, for a number of scholars points to the 11th century reign of Canute, the Danish king whose empire included all of these areas, and whose primary place of residence was in England, as the most likely time of the poem's creation, the poem being written as a celebration of the king's heroic royal ancestors, perhaps intended as a form of artistic flattery by one of his English courtiers.

* Natural Banach function algebra: A uniform algebra whose all characters are evaluations at points of X.

In the same way one defines a Cartesian space of any dimension n, whose points can be identified with the tuples ( lists ) of n real numbers, that is, with.

As a result, simple distillation is usually used only to separate liquids whose boiling points differ greatly ( rule of thumb is 25 ° C ), or to separate liquids from non-volatile solids or oils.

The most common geometric arrangement is where some convex polyhedron is in its canonical form, which is to say that the all its edges must be tangent to a certain sphere whose centre coincides with the centre of gravity ( average position ) of the tangent points.

Progress cards granting victory points are an exception, being played immediately ( without regards to whose turn it is ), while the Alchemist progress card, which allows a player to select the roll of the white and red dice, necessitates the card being played instead of rolling the numerical dice.

An ellipse is also the locus of all points of the plane whose distances to two fixed points add to the same constant.

The distance between antipodal points on the ellipse, or pairs of points whose midpoint is at the center of the ellipse, is maximum along the major axis or transverse diameter, and a minimum along the perpendicular minor axis or conjugate diameter.

Yet another equivalent definition of the ellipse is that it is the set of points that are equidistant from one point in the plane ( a focus ) and a particular circle, the directrix circle ( whose center is the other focus ).

If any part of the ball reaches any part of the imaginary vertical plane transected by this line while in-bounds and in possession of a player whose team is striving toward that end of the field, this is considered a touchdown and scores six points for the team whose player has advanced the ball to, or recovered the ball in, this position.

Given a trigonometric series f ( x ) with S as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that had S ' as its set of zeros, where S ' is the set of limit points of S. If p ( 1 ) is the set of limit points of S, then he could construct a trigonometric series whose zeros are p ( 1 ).

In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is that rate of increase.

* Moduli space, in mathematics a geometric space whose points represent algebro-geometric objects

One may say that Diophantus was studying rational points — i. e., points whose coordinates are rational — on curves and algebraic varieties ; however, unlike the Greeks of the Classical period, who did what we would now call basic algebra in geometrical terms, Diophantus did what we would now call basic algebraic geometry in purely algebraic terms.