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is and bounded
This theorem is similar to the theorem of Kakutani that there exists a circumscribing cube around any closed, bounded convex set in Af.
This is true of all components which have such a bounded support.
The tract is bounded by Island Ave., Dicks Ave., 61st St., and Eastwick Ave..
The Atlantic Ocean is bounded on the west by North and South America.
** A uniform space is compact if and only if it is complete and totally bounded.
The region is bounded by the Black Sea to the north, the Mediterranean Sea to the south and the Aegean Sea to the west.
The Anatolian peninsula, also called Asia Minor, is bounded by the Black Sea to the north, the Mediterranean Sea to the south, the Aegean Sea to the west, and the Sea of Marmara to the northwest, which separates Anatolia from Thrace in Europe.
Under this definition, Anatolia is bounded to the East by the Armenian Highland, and the Euphrates before that river bends to the southeast to enter Mesopotamia.
To the southeast, it is bounded by the ranges that separate it from the Orontes valley in Greater Syria and the Mesopotamian plain.
thus, as T decreases, ΔG and ΔH approach each other ( so long as ΔS is bounded ).
We now consider how the choice of description language affects the value of K, and show that the effect of changing the description language is bounded.
The South African plateau, as far as about 12 ° S, is bounded east, west and south by bands of high ground which fall steeply to the coasts.
It lies on Abadan Island ( long, 3 – 19 km or 2 – 12 miles wide, the island is bounded in the west by the Arvand waterway and to the east by the Bahmanshir outlet of the Karun River ), from the Persian Gulf, near the Iraqi-Iran border.
The ACT is bounded by the Goulburn-Cooma railway line in the east, the watershed of Naas Creek in the south, the watershed of the Cotter River in the west, and the watershed of the Molonglo River in the north-east.
The Arabian Sea ( Persian Sea ) is a region of the Indian Ocean bounded on on the north by Pakistan and Iran, on the south by northeastern Somalia, on the east by India and on the west by the Arabian Peninsula.
However, the recursion is bounded because in each recursive application either m decreases, or m remains the same and n decreases.
It is bounded by the Scandinavian Peninsula, the mainland of Europe, and the Danish islands.
The Black Sea is bounded by Europe, Anatolia and the Caucasus and is ultimately connected to the Atlantic Ocean via the Mediterranean and the Aegean Seas and various straits.
Since Bulgarian lev is anyway bounded to Euro Bulgaria is looking for more positive outcomes of the becoming a member of the Eurozone rather than risks.
In general, a linear mapping on a normed space is continuous if and only if it is bounded on the closed unit ball.
Note that the requirement that the maps be continuous is essential ; if X is infinite-dimensional, there exist linear maps which are not continuous, and therefore not bounded.

is and roughly
( Since the time-span of the nation-state coincides roughly with the separate existence of the United States as an independent entity, it is perhaps natural for Americans to think of the nation as representative of the highest form of order, something permanent and unchanging.
It was recognized years ago that the transition from daytime to nighttime propagation conditions, and vice versa, is not an instantaneous process, but takes place over periods of time from roughly 2 hours before sunset until about 2 hours after sunset, and again from roughly 2 hours before sunrise until some 2 hours after sunrise.
It is roughly shaped like a large pear, and when properly ripened, its dark green skin covers a meaty, melon-like pulp that has about the consistency of a ripe Bartlett pear, but oily.
Sometimes it is necessary to roughly calculate the square inch area of the opening but the calculation can usually be made with sufficient accuracy that it won't affect the final computation.
The average charge for this additional insurance coverage is roughly $1.00 a day.
Rates for American cars are somewhat higher, ranging from about $8.00 a day up to $14.00 a day for a Chevrolet Convertible, but the rate per kilometer driven is roughly the same as for the larger European models.
In the second sentence if drinking water is a gerundial clause and without drinking water is roughly equivalent in meaning to unless I drink water, there will be stronger stress on water than on drinking ; ;
( The P-20 input is of interest because it corresponds roughly to the light emission of conventional X-ray fluorescent screens ).
Each scene is shot straight through, as had been the universal custom, from a camera fixed in a single position, but in the outdoor scenes, especially in the capture and destruction of the outlaws, Porter's camera position breaks, necessarily, with the camera position standard until then, which had been, roughly, that of a spectator in a center orchestra seat at a play.
As a result, it is not easy to find a stock priced as the Morton issue was priced ( at roughly 10 times 1960 earnings, to yield a little over 5 per cent on the 64-cent anticipated dividend ).
According to a newspaper report of the 1961 statistics of the Church of England, the `` total of confirmed members is 9,748,000, but only 2,887,671 are registered on the parochial church rolls '', and `` over 27 million people in England are baptized into the Church of England, but roughly only a tenth of them continue ''.
The number of neutrons, N, is known as the neutron number of the atom ; thus, A = Z + N. Since protons and neutrons have approximately the same mass ( and the mass of the electrons is negligible for many purposes ), and the mass defect is usually very small compared to the mass, the atomic mass of an atom is roughly equal to A.
The mixture is roughly half sand ( 50 %), one-third clay ( 35 %), and one-sixth straw ( 15 %) by weight.
All other nuclides ( isotopes of hydrogen and all other elements ) have more nucleons than electrons, so the fraction of mass taken by the nucleus is closer to 100 % for all of these types of atoms, than for hydrogen-1 .</ ref > with protons and neutrons having roughly equal mass.
It is held roughly every ten years and invitation is by the Archbishop of Canterbury.
" The Dark " or " Misty Hel ") This realm is roughly analogous to Greek Tartarus.
The total energy of the recoil nucleus is readily calculable, and is roughly the weight of the alpha ( 4 u ) divided by the weight of the parent ( typically about 200 u ) times the total energy of the alpha.
For example, the prime number theorem states that the number of prime numbers less than or equal to N is asymptotically equal to N / ln N. Therefore the proportion of prime integers is roughly 1 / ln N, which tends to 0.

is and by
It is possible, although highly doubtful, that he killed none at all but merely let his reputation work for him by privately claiming every unsolved murder in the state.
The place is inhabited by several hundred warlike women who are anachronisms of the Twentieth Century -- stone age amazons who live in an all-female, matriarchal society which is self-sufficient ''.
since Bourbon whiskey, though of Kentucky origin, is at least as much favored by liberals in the North as by conservatives in the South.
In fact it has caused us to give serious thought to moving our residence south, because it is not easy for the most objective Southerner to sit calmly by when his host is telling a roomful of people that the only way to deal with Southerners who oppose integration is to send in troops and shoot the bastards down.
But apart from racial problems, the old unreconstructed South -- to use the moderate words favored by Mr. Thomas Griffith -- finds itself unsympathetic to most of what is different about the civilization of the North.
The two main charges levelled against the Bourbons by liberals is that they are racists and social reactionaries.
It became the sole `` subject '' of `` international law '' ( a term which, it is pertinent to remember, was coined by Bentham ), a body of legal principle which by and large was made up of what Western nations could do in the world arena.
Ratified in the Republican Party victory in 1952, the Positive State is now evidenced by political campaigns being waged not on whether but on how much social legislation there should be.
He was, and is, with the RAND Corporation, a nonprofit pool of thinkers financed by the U.S. Air Force.
They are huge areas which have been swept by winds for so many centuries that there is no soil left, but only deep bare ridges fifty or sixty yards apart with ravines between them thirty or forty feet deep and the only thing that moves is a scuttling layer of sand.
It is softened by the saltbush and the bluebush, has a peaceful quality, the hills roll softly.
On Fridays, the day when many Persians relax with poetry, talk, and a samovar, people do not, it is true, stream into Chehel Sotun -- a pavilion and garden built by Shah Abbas 2, in the seventeenth century -- but they do retire into hundreds of pavilions throughout the city and up the river valley, which are smaller, more humble copies of the former.
Poetry in Persian life is far more than a common ground on which -- in a society deeply fissured by antagonisms -- all may stand.
Nostalgic Yankee readers of Erskine Caldwell are today informed by proud Georgians that Tobacco Road is buried beneath a four-lane super highway, over which travel each day suburbanite businessmen more concerned with the Dow-Jones average than with the cotton crop.
All but the most rabid of Confederate flag wavers admit that the Old Southern tradition is defunct in actuality and sigh that its passing was accompanied by the disappearance of many genteel and aristocratic traditions of the reputedly languid ante-bellum way of life.
Westbrook further bemoans the Southern writers' creation of an unreal image of their homeland, which is too readily assimilated by both foreign readers and visiting Yankees: `` Our northerner is suspicious of all this crass evidence ( of urbanization ) presented to his senses.
As his disciples boast, even though his emphasis is elsewhere, Faulkner does show his awareness of the changing order of the South quite keenly, as can be proven by a quick recalling of his Sartoris and Snopes families.
The unit of form is determined subjectively: `` the Heart, by the way of the Breath, to the Line ''.

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