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It occurs in the proofs of several theorems of crucial importance, for instance the Hahn – Banach theorem in functional analysis, the theorem that every vector space has a basis, Tychonoff's theorem in topology stating that every product of compact spaces is compact, and the theorems in abstract algebra that every nonzero ring has a maximal ideal and that every field has an algebraic closure.
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occurs and proofs
The idea of multiplying all known primes occurs in some proofs of the infinitude of the prime numbers, where it is used to derive the existence of another prime.
# Proofs of other theorems are examined to see if the newly found lemma or the new proof-generated concept occurs in them: this concept may be found lying at cross-roads of different proofs, and thus emerge as of basic importance.
occurs and several
It occurs in several allomorphs depending on its phonological environment, assimilating voicing of the previous segment or inserting a schwa when following an alveolar stop:
This case is also used for nouns in several other senses, e. g., where the action occurs " because of " or " without " a certain noun ; nouns indicating distance or direction.
Viral induction of apoptosis occurs when one or several cells of a living organism is infected with a virus leading to cell death.
It occurs several times in the Shatapatha Brahmana in the sense of " non-injury " without a moral connotation.
This occurs when two surfaces converge rapidly and obliquely at a small angle, and high-temperature highly shocked material is expelled from the convergence zone with velocities that may be several times larger than the impact velocity.
The only other location where this occurs is Fort Erie, Ontario, lying south of several cities in Niagara County, New York.
It rapidly oxidizes in air, so that bulk oxidation of a centimeter-sized sample occurs within several days.
Electricity occurs due to several types of physics:
The lateral fricative occurs as the ll of Welsh, as in Lloyd, Llewelyn, and the town of Machynlleth (), as the unvoiced ' hl ' and voiced ' dl ' or ' dhl ' in the several languages of Southern Africa ( such as Xhosa and Zulu ), and in Mongolian.
The expansion of trade occurs after the Portuguese reach this region in 1446, bringing great wealth to several local slave trading tribes.
When hypoglycemia occurs repeatedly, a record or " diary " of the spells over several months, noting the circumstances of each spell ( time of day, relation to last meal, nature of last meal, response to carbohydrate, and so forth ) may be useful in recognizing the nature and cause of the hypoglycemia.
It may here be requisite for me to explain, that by the term Hypnotism, or Nervous Sleep, which frequently occurs in the following pages, I mean a peculiar condition of the nervous system, into which it may be thrown by artificial contrivance, and which differs, in several respects, from common sleep or the waking condition.
* 2006 – 2006 São Paulo violence: a major rebellion occurs in several prisons in Brazil.
The format of the ceremony for taking the precepts occurs several times in the canon in slightly different forms, and each temple or tradition has slightly different ordination ceremonies.
In Buddhist tradition the process occurs across five or six realms of existence, including the human, any kind of animal and several types of supernatural being.
Samarium occurs with concentration up to 2. 8 % in several minerals including cerite, gadolinite, samarskite, monazite and bastnäsite, the last two being the most common commercial sources of the element.
An experienced hunter, Peckinpah was fascinated with firearms and was known to shoot the mirrors in his house while abusing alcohol, an image which occurs several times in his films.
The burial trenches have been evaluated to be secure for at least 600 years before the first pinhole penetration of some lead containment areas of the reactor compartment packages occurs, and several thousand years before leakage becomes possible.
From an astronomical view, the equinoxes and solstices would be the middle of the respective seasons, but a variable seasonal lag means that the meteorological start of the season, which is based on average temperature patterns, occurs several weeks later than the start of the astronomical season.
The notion of weighted mean plays a role in descriptive statistics and also occurs in a more general form in several other areas of mathematics.
* September 2 – The Rock Springs Massacre occurs in Rock Springs, Wyoming ; 150 white miners attack their Chinese coworkers, killing 28, wounding 15, and forcing several hundred more out of town.
If the winner of the seat is determined purely by the candidate with the most votes, as occurs in the plurality vote and the alternative vote, the plurality system has several characteristics that can serve to discourage the development of third parties and reward the two major parties.
; Intermodulation distortion: This form of distortion occurs when two sine waves of frequencies X and Y are present at the input, resulting in the creation of several other frequency components, whose frequencies include ( X + Y ), ( X-Y ), ( 2X-Y ), ( 2Y-X ), and generally ( mX ± nY ) for integer m and n. Generally the size of the unwanted output falls rapidly as m and n increase.
This device occurs in the One Thousand and One Nights, which binds several tales in a story cycle.
In " The Herball, or Generall Historie of Plantes " ( 1597 ), John Gerard describes it under the name of raphanus rusticanus, stating that it occurs wild in several parts of England.
occurs and theorems
The discrepancy between the two " theorems " occurs because of the presence of more than one opponent.
occurs and crucial
There is much less consensus on the crucial question of how much knowledge acquired in school transfers to tasks encountered outside formal educational settings, and how such transfer occurs.
Such usages typically obscure the crucial aspect of a leitmotif, as opposed to the plain musical motif or theme-that it is transformable and recurs in different guises throughout the piece in which it occurs.
However when the event occurs, events happen slightly differently than in Desmond's vision and it is suggested that Charlie may have been able to save himself without jeopardizing the hopes of rescue, if he had not believed his death was crucial in the rescue of the other castaways.
Little is known why this occurs but research shows the buffer acting component to be crucial acid rain.
It is a crucial reaction that occurs in phagocytes to degrade internalized particles and bacteria.
Studies of various projectile types have been crucial in overturning a prior theory that spallation occurs at a threshold of pressure, independent of time.
occurs and importance
Mayr emphasized the importance of allopatric speciation, where geographically isolated sub-populations diverge so far that reproductive isolation occurs.
The " illusion " occurs when people consider the impact of one specific factor on their overall happiness, they tend to greatly exaggerate the importance of that factor, while overlooking the numerous other factors that would in most cases have a greater impact.
" It " is a multivalent term of great importance in later Indian Buddhism and Tibetan Buddhism " which " also occurs occasionally in Hindu and East Asian Buddhist esotericism.
Under the Achaemenid dynasty Merv is mentioned as being a place of some importance: under the name of Margu it occurs as part of one of the satrapies in the Behistun inscriptions ( ca 515 BC ) of the Persian monarch Darius Hystaspis.
The focusing effect ( or focusing illusion ) is a cognitive bias that occurs when people place too much importance on one aspect of an event, causing an error in accurately predicting the utility of a future outcome.
Despite the enormous ecological importance of the myrtle family in Australia ( e. g. Eucalyptus, Corymbia, Angophora, Melaleuca, Callistemon, Rhodamnia, Gossia ), only one species of Eugenia, E. reinwardtiana, occurs on that continent.
Mutation of APC also occurs commonly in incident cases of colorectal carcinoma, emphasizing its importance in this form of cancer.
Her role has varying degrees of importance, in that being a lady, she is required to remain at her own estate while Jim is away ; to oversee all that occurs in it ; and to maintain custom and courtesy in all situations.
One of the most important events of the year is the Janmashthami procession which starts from Dhakeshwari temple and then proceeds through the streets of Old Dhaka ; this occurs on the day of the Lord Krishna's birthday, which is also a national holiday in Bangladesh and second only to Durga Puja in importance in the Bengali Hindu calendar.
Burford abstention is closely related to Thibodaux abstention, derived from Louisiana Power & Light Co. v. City of Thibodaux, 360 U. S. 25 ( 1959 ), which occurs when a federal court sitting in diversity jurisdiction chooses to allow a state to decide issues of state law that are of great public importance to that state, to the extent that a federal determination would infringe on state sovereignty.
It may also describe a similar phenomenon that occurs when one may immediately detect words of importance originating from unattended stimuli, for instance hearing one's name in another conversation.
Meloidogyne occurs in 23 of 43 crops listed as having plant-parasitic nematodes of major importance, ranging from field crops, through pasture and grasses, to horticultural, ornamental and vegetable crops.
The Landau damping occurs on both electrons and ions, with the relative importance depending on parameters.
A major change brought about by habitat fragmentation is an increase in the proportion of edge exposed to other habitats, and the importance of this change depends to a degree on the contrast between the fragmented habitat and the matrix in which it occurs.
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