* For a finite field of prime order p, the algebraic closure is a countably infinite field which contains a copy of the field of order p < sup > n </ sup > for each positive integer n ( and is in fact the union of these copies ).

For many geologic applications, isotope ratios are measured in minerals that give the amount of time that has passed since a rock passed through its particular closure temperature, the point at which different radiometric isotopes stop diffusing into and out of the crystal lattice.

For about the next 10 years, the country saw a steady process of secular Westernization through Atatürk's Reforms, which included the unification of education ; the discontinuation of religious and other titles ; the closure of Islamic courts and the replacement of Islamic canon law with a secular civil code modeled after Switzerland's and a penal code modeled after the Italian Penal Code ; recognition of the equality between the sexes and the granting of full political rights to women on 5 December 1934 ; the language reform initiated by the newly founded Turkish Language Association ; replacement of the Ottoman Turkish alphabet with the new Turkish alphabet derived from the Latin alphabet ; the dress law ( the wearing of a fez, is outlawed ); the law on family names ; and many others.

For S a subset of a Euclidean space, x is a point of closure of S if every open ball centered at x contains a point of S ( this point may be x itself ).

For example, the set of rational numbers, as a subset of R has the property that the interior has an empty closure, but it is not nowhere dense ; in fact it is dense in R.

For many, gacaca has been a vehicle for closure, and prisoners ' testimonies have helped many families locate victims.

For example, in the category of topological abelian groups, the image of a morphism actually corresponds to the inclusion of the closure of the range of the function.

For example, the figure depicting the law of closure portrays what we perceive to be a circle on the left side of the image and a square on the right side of the image.

For example, the universal closure of

The reflexive closure of a binary relation ~ on a set S is the smallest relation ~′ such that ~′ is a superset of ~ and ~′ is reflexive on S. This is equivalent to the union of ~ and the identity relation on S. For example, the reflexive closure of x < y is x ≤ y.

For example, in the topological space ω < sub > 1 </ sub >+ 1, the element ω < sub > 1 </ sub > is in the closure of the subset ω < sub > 1 </ sub > even though no sequence of elements in ω < sub > 1 </ sub > has the element ω < sub > 1 </ sub > as its limit: an element in ω < sub > 1 </ sub > is a countable set ; for any sequence of such sets, the union of these sets is the union of countably many countable sets, so still countable ; this union is an upper bound of the elements of the sequence, and therefore of the limit of the sequence, if it has one.

If, for example, we simply look at a curve in the real affine plane there might be singular P modulo the stalk, or alternatively as the sum of m ( m − 1 )/ 2, where m is the multiplicity, over all infinitely near singular points Q lying over the singular point P. Intuitively, a singular point with delta invariant δ concentrates δ ordinary double points at P. For an irreducible and reduced curve and a point P we can define δ algebraically as the length of where is the local ring at P and is its integral closure.

For a wound wider than 15 mm — especially a wound that comprehends the alar lobule and the lateral wall of the nose — primary closure is not indicated, because such a wound closure imposes excessive stresses upon the skin flap, thereby risking either blanching ( whitening ) or distortion, or both.

For example the algebraic closure, the separable closure, the cyclic closure et cetera.

For example if we take P ( K ) to be the property " every nonconstant polynomial f in K has a root in K ", then a P-closure of k is just an algebraic closure of k.

For example, if X is a set of airports and x R y means " there is a direct flight from airport x to airport y ", then the transitive closure of R on X is the relation R < sup >+</ sup >: “ it is possible to fly from x to y in one or more flights .”

For any relation R, the transitive closure of R always exists.

For example, the closure under subtraction of the set of natural numbers, viewed as a subset of the real numbers, is the set of integers.

For example, " every field has an algebraic closure " is not provable in ZF set theory, but the restricted form " every countable field has an algebraic closure " is provable in RCA < sub > 0 </ sub >, the weakest system typically employed in reverse mathematics.

For one thing, this is not a subject often discussed or analyzed.

For better or for worse, we all now live in welfare states, the organizing principle of which is collective responsibility for individual well-being.

For one thing, there is a natural belt of rock across the river bed ; ;

For the family is the simplest example of just such a unit, composed of people, which gives us both some immunity from, and a way of dealing with, other people.

Even so astute a commentator as Harold Clurman of The Nation has said that `` Waiting For Godot '' is `` the concentrate of the contemporary European mood of despair ''.

For one thing, the world that Beckett sees is already shattered.

Harold Clurman is right to say that `` Waiting For Godot '' is a reflection ( he calls it a distorted reflection ) `` of the impasse and disarray of Europe's present politics, ethic, and common way of life ''.

For the beatnik, like the hipster, is in opposition to a society that is based on the repression of the sex instinct.

For this reason, too, their language is more forthright and earthy.

For the present it is enough to note that in the grotesque figure of Jacoby, at the moment of his collapse, all these elements come together in prophetic parody.

For Plato, `` imitation '' is twice removed from reality, being a poor copy of physical appearance, which in itself is a poor copy of ideal essence.

For both Plato and Aristotle artistic mimesis, in contrast to the power of dialectic, is relatively incapable of expressing the character of fundamental reality.

For Hammer, nothing is forbidden.

For example, suppose a man wearing a $200 watch, driving a 1959 Rolls Royce, stops to ask a man on the sidewalk, `` What time is it ''??

For the occasion on which everyone already knows everyone else and the host wishes them to meet one or a few honored newcomers, then the `` open house '' system is advantageous because the honored guests are fixed connective points and the drifting guests make and break connections at the door.

For this change is not a change from one positive position to another, but a change from order and truth to disorder and negation.

For paradigmatic history `` breaks '' rather than unfolds precisely when the movement is from order to disorder, and not from one order to a new order.

For this love of the boy for his mother is a hopeless and forbidden love, doomed by its nature.

For innocence, of all the graces of the spirit, is I believe the one most to be prayed for.

For what we propose, however, a psychoanalyst is not necessary, even though one aim is to enable the reader to get beneath his own defenses -- his defenses of himself to himself.

For this purpose a degree of intellectual and emotional involvement is necessary ; ;

For example, in modern Western cultures, there are alternative styles of clothing that characterized older and younger generations.

For Epicurus, the purpose of philosophy was to attain the happy, tranquil life, characterized by ataraxia — peace and freedom from fear — and aponia — the absence of pain — and by living a self-sufficient life surrounded by friends.

For example, in 1994, Irving Kirsch characterized hypnosis as a " nondeceptive placebo ," i. e., a method that openly makes use of suggestion and employs methods to amplify its effects.

For optical applications, this is characterized by the index of refraction.

For most of human history, people have lived in stateless societies, characterized by a lack of concentrated authority, and the absence of large inequalities in economic and political power.

For example, classical thermodynamics is characterized by its study of materials that have equations of state or characteristic equations.

For optical fibers, a power-law index profile is an index of refraction profile characterized by

For an optical fiber, a step-index profile is a refractive index profile characterized by a uniform refractive index within the core and a sharp decrease in refractive index at the core-cladding interface so that the cladding is of a lower refractive index.

U interface: For basic-rate access in an Integrated Services Digital Network ( ISDN ) environment, a user-to-network interface reference point that is characterized by the use of a 2-wire-loop transmission system that

For two centuries New Netherland Dutch culture characterized the region ( today's Capital District around Albany, the Hudson Valley, western Long Island, northeastern New Jersey, and New York City ).

For example, the Finale is a combined fugue and sonata form movement: the first theme ( characterized by the downward leap of an octave ) appears in the exposition as a four-part fugue in the strings and the concluding theme of the exposition is presented first as a chorale in the brass, then as a four-part fugue in the development, and culminating in a double fugue with the first theme at the recapitulation ; additionally, the coda combines not only these two themes but also the main theme of the first movement.

For instance, Charles Rosen has characterized Mozart's various works in imitation of Baroque style as pastiche, and Edvard Grieg's Holberg Suite was written as a conscious homage to the music of an earlier age.

For example, ADHD is characterized by an increase of antisaccade errors and an increase in delays for visually guided saccade.

For the Reformers, the culmination of this gradual corruption was typified, in a concentrated way, in the office of the Pope who took on ancient titles such as Pontifex Maximus and supreme power in the church, which they characterized in its final form as being an usurpatious throne of Satanic authority set up in pretense of ruling over the Kingdom of God.

For example, multiple endocrine neoplasia 1 ( MEN 1 ), a hereditary disorder characterized by parathyroid hyperplasia and pancreatic islet-cell and pituitary adenomas, is caused by a mutation in the menin gene on human chromosome 11q13.

For bosonic particles that exhibit rest mass when they are free and " real ," virtual interactions are characterized by the relatively short range of the force interaction produced by particle exchange.

For example, a person born into a wealthy family characterized by traits such as popularity, talents and high values will have many expectations growing up.

For example, Iwata poses the question: "... is a change in temperature more accurately characterized by the presentation of cold ( heat ) or the removal of heat ( cold )?

For instance, the melting of ice is characterized by a temperature, known as the melting point at which solid and liquid phases are in equilibrium with each other.

For his part, Bennett characterized the On-to-Ottawa Trek as " not a mere uprising against law and order but a definite revolutionary effort on the part of a group of men to usurp authority and destroy government.

For liquid-packed vessels, thermal relief valves are generally characterized by the relatively small size of the valve necessary to provide protection from excess pressure caused by thermal expansion.

Steinberg characterized the change as follows: “ For a long time, the World Jewish Congress was meant to be the greatest secret of Jewish life, because the nature of diplomacy after the war was quiet diplomacy.

For the rest of their lives together, their relationship was, as biographer Benita Eisler characterized it,

For given polynomials and the classical orthogonal polynomials are characterized by being solutions of the differential equation