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For instance, an infinite geometric sum is implicit in Zeno's paradox of the dichotomy .< ref name =" Stillwell Infinite Series Early Results "> Later, Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids.
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For and instance
`` For instance, regarding the fact that the Gross Committee issued two interim announcements to the press during its investigation.
For instance, college-educated people consistently show up in study after study as more often than others supporters of the Bill of Rights and other democratic rights and liberties.
For instance, so-called `` conservative '' organizations, some of them secret, are sprouting in the garden of joining where `` liberal '' organizations once took root.
For instance -- what about all those people Harold Rhodes went toward unhesitatingly, as if this were the one moment they would ever have together, their one chance of knowing each other??
For instance, a site adjoining other publicly owned lands, such as a national forest or a public road, may be desirable, whereas a site next to an industrial plant might not.
( For instance, see Example 2 of Section 5-5, on red cards in hands of 5.
For instance, the following statement was rated low in compulsivity, `` She's naturally quite neat about things, but it doesn't bother her at all if her room gets messy.
For instance, in giving school grades or in making recommendations for the award of a college scholarship, does he consciously or unconsciously favor students of one or another social class??
For instance, there have been two Presidential Commissions on higher education since World War 2.
For instance, the dreamer sees himself seated behind neighbor Smith and, with photographic realism, sees Smith driving the car ; ;
For instance:
`` For instance, Djakarta, Indonesia, has three groups of dancers interested in coming here.
For instance, the Edwin Pauleys Jr., formerly of Chantilly Rd., are now at home on North Arden Dr. in Beverly Hills.
For instance, we cannot know whether even for church members the degree of conformity to Christian standards of morality increased or declined as the proportion of church members in the population rose.
`` For instance, Hesperus agreed to help me find my property, and I agreed to take him to Earth.
For instance, a C-Leg knee alone is not a prosthesis, but only a prosthetic part.
For instance, the word " bank " has several distinct lexical definitions, including " financial institution " and " edge of a river ".
For instance, a local society in the middle of a large city may have regular meetings with speakers, focusing less on observing the night sky if the membership is less able to observe due to factors such as light pollution.
For instance, according to Classical Arabic and Modern Standard Arabic, the Arabic root ( to sacrifice ) can be derived the forms ( he sacrificed ), ( you ( masculine singular ) sacrificed ), ( he slaughtered ), ( he slaughters ), and ( slaughterhouse ).
For instance, we read of Whiting, the last abbot of Glastonbury, judicially murdered by Henry VIII, that his house was a kind of well-ordered court, where as many as 300 sons of noblemen and gentlemen, who had been sent to him for virtuous education, had been brought up, besides others of a lesser rank, whom he fitted for the universities.
For instance, the n
For instance, in 51, Agrippina ordered the execution of Britannicus ’ tutor Sosibius because he had confronted her and was outraged by Claudius ’ adoption of Nero and his choice of Nero as successor, instead of choosing his own son Britannicus.
For instance, antibacterial resistance genes can be exchanged between different bacterial strains or species via plasmids that carry these resistance genes.
For instance, iron changes from a body-centered cubic structure ( ferrite ) to a face-centered cubic structure ( austenite ) above 906 ° C, and tin undergoes a transformation known as tin pest from a metallic phase to a semiconductor phase below 13. 2 ° C.
For instance member nations of the Commonwealth where English is not spoken natively, such as India, often closely follow British English forms, while many American English usages are followed in other countries which have been historically influenced by the United States, such as the Philippines.
For and infinite
For example for any ( even infinite ) collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate selection, but for an infinite collection of pairs of socks ( assumed to have no distinguishing features ), such a selection can be obtained only by invoking the axiom of choice.
** Tarski's theorem: For every infinite set A, there is a bijective map between the sets A and A × A.
For instance, the field of all algebraic numbers is an infinite algebraic extension of the rational numbers.
* 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 ).
Some sets are infinite ; these sets have more than n elements for any integer n. For example, the set of natural numbers, denotable by, has infinitely many elements, and we cannot use any normal number to give its size.
For an infinite group, the derived series need not terminate at a finite stage, and one can continue it to infinite ordinal numbers via transfinite recursion, thereby obtaining the transfinite derived series, which eventually terminates at the perfect core of the group.
For example, an equivalence relation with exactly two infinite equivalence classes is an easy example of a theory which is ω-categorical, but not categorical for any larger cardinal number.
For example, the set of all positive integers is infinite:
Thus intuitionists absolutely disallow the blanket assertion: " For all propositions P concerning infinite sets D: P or ~ P " ( Kleene 1952: 48 ).
For example, one could create a function that creates an infinite list ( often called a stream ) of Fibonacci numbers.
For any luminosity from a given distance L ( r ) N ( r ) proportional to r < sup > a </ sup >, is infinite for a ≥ − 1 but finite for a < − 1.
For point masses the gravitational energy decreases without limit as they approach zero separation, and it is convenient and conventional to take the potential energy as zero when they are an infinite distance apart, and then negative ( since it decreases from zero ) for smaller finite distances.
For an example of an infinite abelian p-group, see Prüfer group, and for an example of an infinite simple p-group, see Tarski monster group.
For instance, the normalizer N of a proper subgroup H of a finite p-group G properly contains H, because for any counterexample with H = N, the center Z is contained in N, and so also in H, but then there is a smaller example H / Z whose normalizer in G / Z is N / Z = H / Z, creating an infinite descent.
For a continuous random variable, the probability of any specific value is zero, whereas the probability of some infinite set of values ( such as an interval of non-zero length ) may be positive.
For an infinite set the order type determines the cardinality, but not conversely: well-ordered sets of a particular cardinality can have many different order types.
For a countably infinite set the set of possible order types is even uncountable.
For example, Stephen Kleene describes the notion of a Turing machine tape as " a linear tape, ( potentially ) infinite in both directions.
For the purposes of the Sprague – Grundy theorem, a game is a two-player game of perfect information satisfying the ending condition ( all games come to an end: there are no infinite lines of play ) and the normal play condition ( a player who cannot move loses ).
For example, the immortal monkey could randomly type G as its first letter, G as its second, and G as every single letter thereafter, producing an infinite string of Gs ; at no point must the monkey be " compelled " to type anything else.
For infinite sets, the supremum will always be 0 while the infimum can well be greater than 1.
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