In the case of Russia, the United States stated that it intended to discuss a bilateral reduction in the numbers of nuclear warheads, which would allow Russia to reduce its spending on missiles without decrease of comparative strength.

It is simpler to see the notational equivalences between ordinary notation and bra-ket notation, so for now ; consider a vector A as an element of 3-d Euclidean space using the field of real numbers, symbolically stated as.

Generalizing further, consider a vector A in an N dimensional vector space over the field of complex numbers, symbolically stated as.

However, in view of the difficulty in finding accurate counts in the first place, Caesar's propagandistic purposes, and the common exaggeration of numbers in ancient texts, the stated totals of enemy combatants are likely to be too high.

" Davis stated that men in the Confederacy had successfully fought for their own rights with inferior numbers during the Civil War and that the northern historians ignored this view.

Larger numbers are often stated in English or numbers instead of Greek.

However, while the results in real analysis are stated for real numbers, many of these results can be generalized to other mathematical objects.

Gödel's theorem, informally stated, asserts that any formal theory expressive enough for elementary arithmetical facts to be expressed and strong enough for them to be proved is either inconsistent ( both a statement and its denial can be derived from its axioms ) or incomplete, in the sense that there is a true statement about natural numbers that can't be derived in the formal theory.

In an interview for the DVD compilation Reel Talent, which included Lucas's original 4EB short, Lucas stated that he chose the letters and numbers for their aesthetic qualities, especially their symmetry.

In a resolution passed on June 29, 2005, they stated that " Section 215 of the USA PATRIOT Act allows the government to secretly request and obtain library records for large numbers of individuals without any reason to believe they are involved in illegal activity.

These numbers allow the compass to show azimuths or bearings, which are commonly stated in this notation.

When discussing sets of real numbers, the Lebesgue measure is assumed unless otherwise stated.

* The Goldbach conjecture for practical numbers, a prime-like sequence of integers, was stated by Margenstern in 1984, and proved by Melfi in 1996: every even number is a sum of two practical numbers.

Gauss discovered that the law of biquadratic reciprocity and its supplements were more easily stated and proved as statements about " whole complex numbers " ( i. e. the Gaussian integers ) than they are as statements about ordinary whole numbers ( i. e. the integers ).

Although General Marshall stated that he knew of no evidence that the CPC were being supplied by the Soviet Union, the CPC were able to capture a large number of weapons abandoned by the Japanese, including some tanks but it was not until large numbers of well trained KMT troops surrendered and joined the communist forces that the CPC were finally able to master the hardware.

The formula is actually true in a more general setting than stated above: if z and w are complex numbers, then

This is mostly of technical interest, since all true formal theories of arithmetic ( theories whose axioms are all true statements about natural numbers ) are ω-consistent, and thus Gödel's theorem as originally stated applies to them.

A precise ( homology ) version can be stated using winding numbers.

U. N. Special Representative for Children and Armed Conflict Radhika Coomaraswamy stated in January 2010 that " large numbers " of teenage boys are being recruited in tribal Yemeni fighting.

In the first book to include SPECTRE, " Thunderball ", it is stated that the numbers of each member changes periodically ( it " advances round a rota by two digits at midnight on the first of every month ") to avoid detection and Blofeld is in fact number 2.

For example, if P ( x ) is the propositional function " x is between 0 and 1 ", then, for a domain of discourse X of all natural numbers, the existential quantification " There exists a natural number x which is between 0 and 1 " is symbolically stated:

Eventually, an Act of Parliament was obtained in 1713, which appointed large numbers of Commissioners, but stated that work could only proceed if any nine of them could find someone to make the entire river navigable.

Had the situation been reversed, had, for instance, England been the enemy in 1898 because of issues of concern chiefly to New England, there is little doubt that large numbers of Southerners would have happily put on their old Confederate uniforms to fight as allies of Britain.

It will be noted that point f has seven nearest neighbors, h and e have six, and p has only one, while the remaining points have intermediate numbers.

During intercrisis periods the educational facilities of the community have the possibility of remolding the perspectives and altering the behavior of vast numbers of human beings of every age and condition.

Had Krim gone farther from New York than Chapel Hill, he might have discovered that large numbers of American Jews do not find his New York version of the Jews' lot remotely recognizable.

If he can't play with Mommy's magazines, he should have some old numbers of his own.

Large numbers of potential builders have been investigated, but none have been found to be as effective as the polyphosphates over the relatively wide range of conditions met in practice.

For example, in accordance with the fashion of the times, most transitional societies have announced economic development plans of varying numbers of years ; ;

For one thing, although considerable numbers of men have been trained, bureaucracies are still deficient in many respects ; ;

Elaborate studies have been made in labor surplus areas in order to identify sufficient numbers of local job vacancies and future replacement needs for certain skills to justify training programs for those skills.

The luminosity of the water is now believed to have been caused by the stimulation of vast numbers of the luminescent organism Noctiluca miliaris by the turbulence of the sea.

the fish have swallowed vast numbers of bottom-living diatoms, raised to the surface by the disturbance.

Whereas the primary meanings of the Lo Shu diagram seemed to have been based on its inner mathematical properties -- and we shall see that even its secondary meanings rested on some mathematical bases -- the urgent desire to place everything into categories of fives led to other groupings based on other numbers, until an exaggerated emphasis on mere numerology pervaded Chinese thought.

We have already seen that odd numbers were considered as being Yang, while the even numbers were Yin, so that the eight outer numbers of the Lo Shu represented these two principles in balanced equilibrium around the axial center.

Studies of membership trends, even in some areas where population is expanding, show that numbers of churches have had little net increase, though many new members were received.

In some of the numbers the instrumental parts have even been recorded at different times and then later combined on the master tape to produce special effects.

* If numbers have mean X, then.

Any decline in amphibian numbers will have an impact on the patterns of predation.

As of 2010, elements with atomic numbers 1 to 118 have been observed.

But some subsets of the real numbers do not have least elements.

After describing the manifestation of the Gospel in the Ogdoad and Hebdomad, he adds that the Basilidians have a long account of the innumerable creations and powers in the several ' stages ' of the upper world ( diastemata ), in which they speak of 365 heavens and say that " their great archon " is Abrasax, because his name contains the number 365, the number of the days in the year ; i. e. the sum of the numbers denoted by the Greek letters in ΑΒΡΑΣΑΞ according to the rules of isopsephy is 365:

As widespread inefficiencies in numbering ( such as the assignment of entire blocks of 10000 numbers to every competing carrier in every small village to support local number portability schemes ) have created shortages of available numbers, these prefixes are often " reclaimed " and issued as standard exchanges, moving the handful of numbers in them to one standard test exchange ( usually 958 ).