[permalink] [id link]
This is a generalization of the Heine – Borel theorem, which states that any closed and bounded subspace S of R < sup > n </ sup > is compact and therefore complete.
from
Wikipedia
Some Related Sentences
is and generalization
Sir Henry Sumner Maine, a hundred years before Communism was a force to be reckoned with, wrote his brilliant legal generalization, that `` the progress of society is from status to contract ''.
A generalization of Artin's theorem states that whenever three elements in an alternative algebra associate ( i. e. ) the subalgebra generated by those elements is associative.
It is impossible to make a blanket generalization about how the blind were treated in literature beyond that point – they were marvelous, gifted, evil, malicious, ignorant, wise, helpless, innocent, or burdensome depending upon who wrote the story – except to say that blindness is perceived to be such a loss that it leaves an indelible mark on a person ’ s character.
The B-tree is a generalization of a binary search tree in that a node can have more than two children.
where Γ ( z ) is the gamma function, a generalization of the factorial function to non-integer values.
A generalization of the class of Horn formulae is that of renamable-Horn formulae, which is the set of formulae that can be placed in Horn form by replacing some variables with their respective negation.
Of course, this generalization increases the cost of such a program ; the process to establish a legally defensible assessment of an entire profession is very extensive.
* Signed measure, in mathematics, a generalization of the concept of a measure that is allowed to have negative values
As a generalization of the uniform circular motion case, suppose the angular rate of rotation is not constant.
If in the third identity we take H = G, we get that the set of commutators is stable under any endomorphism of G. This is in fact a generalization of the second identity, since we can take f to be the conjugation automorphism.
Directed sets are a generalization of nonempty totally ordered sets, that is, all totally ordered sets are directed sets ( contrast partially ordered sets which need not be directed ).
In mathematics, more specifically in abstract algebra and ring theory, a Euclidean domain ( also called a Euclidean ring ) is a ring that can be endowed with a certain structure – namely a Euclidean function, to be described in detail below – which allows a suitable generalization of the Euclidean division of the integers.
Also every ideal in a Euclidean domain is principal, which implies a suitable generalization of the Fundamental Theorem of Arithmetic: every Euclidean domain is a unique factorization domain.
is and Heine
That this form of compactness holds for closed and bounded subsets of Euclidean space is known as the Heine – Borel theorem.
An example of this phenomenon is Dirichlet's theorem, to which it was originally applied by Heine, that a continuous function on a compact interval is uniformly continuous: here continuity is a local property of the function, and uniform continuity the corresponding global property.
The Heine – Borel theorem, as the result is now known, is another special property possessed by closed and bounded sets of real numbers.
Again from the Heine – Borel theorem, the closed unit ball of any finite-dimensional normed vector space is compact.
This causal relationship is unidirectional ; an observable feature of an entity does not define the underlying essence ( Dar-Nimrod & Heine, 2011 ).
The history of what today is called the Heine – Borel theorem starts in the 19th century, with the search for solid foundations of real analysis.
Émile Borel in 1895 was the first to state and prove a form of what is now called the Heine – Borel theorem.
Indeed, the Heine – Borel definition of compactness — that every covering of a space by open sets admits a finite subcovering — is relatively recent.
There are clear echoes of the Wandering Jew in Wagner's The Flying Dutchman, whose plot line is adapted from a story by Heinrich Heine in which the Dutchman is referred to as ' the Wandering Jew of the ocean ', and his final opera Parsifal features a woman called Kundry who is in some ways a female version of the Wandering Jew.
This period in Heine's life is not very clear, but it seems that his father Samson's business deteriorated and Samson Heine effectively became the ward of his brother Salomon.
In the French edition of " Lutetia " Heine wrote, one year before he died: " This confession, that the future belongs to the Communists, I made with an undertone of the greatest fear and sorrow and, oh !, this undertone by no means is a mask!
The compactness of U ( n ) follows from the Heine – Borel theorem and the fact that it is a closed and bounded subset of M < sub > n </ sub >( C ).
Writing to Heinrich Heine in 1839, he offered the fatalistic view: I believe that richess is like love in the theatres and novels: no matter how often one encounters it ... it never misses its target if effectively wielded ... can grow back the foreskin of which we are robbed on the eighth day of life ; those who, on the ninth day, do not bleed from this operation shall continue to bleed an entire lifetime, even after death.
is and –
However implausible this is, it is certainly the case that Aristotle's rigid separation of action from production, and his justification of the subservience of slaves and others to the virtue – or arete – of a few justified the ideal of aristocracy.
Reference to them is made according to the organization of Immanuel Bekker's Royal Prussian Academy edition ( Aristotelis Opera edidit Academia Regia Borussica, Berlin, 1831 – 1870 ), which in turn is based on ancient classifications of these works.
In 2009, the American Anthropological Association's Commission on the Engagement of Anthropology with the US Security and Intelligence Communities released its final report concluding, in part, that, " When ethnographic investigation is determined by military missions, not subject to external review, where data collection occurs in the context of war, integrated into the goals of counterinsurgency, and in a potentially coercive environment – all characteristic factors of the HTS concept and its application – it can no longer be considered a legitimate professional exercise of anthropology.
Apollo ( Attic, Ionic, and Homeric Greek:, Apollōn ( gen .: ); Doric:, Apellōn ; Arcadocypriot:, Apeilōn ; Aeolic:, Aploun ; ) is one of the most important and complex of the Olympian deities in ancient Greek and Roman religion, Greek and Roman mythology, and Greco – Roman Neopaganism.
The last stage in the development of the Kouros type is the late archaic period ( 520 – 485 BC ), in which the Greek sculpture attained a full knowledge of human anatomy and used to create a relative harmonious whole.
These form thirteen established families ( plus perhaps Shompen, which is poorly attested, as a fourteenth ), which have traditionally been grouped into two, as Mon – Khmer and Munda.
However, Ancient Egyptian is highly divergent from Proto-Afroasiatic ( Trombetti 1905: 1 – 2 ), and considerable time must have elapsed in between them.
* Independence of observations – this is an assumption of the model that simplifies the statistical analysis.
The ANOVA F – test ( of the null-hypothesis that all treatments have exactly the same effect ) is recommended as a practical test, because of its robustness against many alternative distributions.
Under Parthian and Sassanian Iranian empires, scholars concentrated on exchanging knowledge and inventions by the countries around them – India, China, and the Roman Empire, when it is thought to be expanded over the other countries.
In the famous long scroll Along the River During the Qingming Festival painted by Zhang Zeduan ( 1085 – 1145 AD ) during the Song Dynasty ( 960 – 1297 AD ), a suanpan is clearly seen lying beside an account book and doctor's prescriptions on the counter of an apothecary's ( Feibao ).
Alphabets often come to be associated with a standard ordering of their letters, which can then be used for purposes of collation – namely for the listing of words and other items in what is called alphabetical order.
A distinction is sometimes made between the smaller adobes, which are about the size of ordinary baked bricks, and the larger adobines, some of which may be one to two yards ( 1 – 2 m ) long.
On average, the Atlantic is the saltiest major ocean ; surface water salinity in the open ocean ranges from 33 to 37 parts per thousand ( 3. 3 – 3. 7 %) by mass and varies with latitude and season.
It is named after André-Marie Ampère ( 1775 – 1836 ), French mathematician and physicist, considered the father of electrodynamics.
As the ship moves, momentum is not conserved – the ship eventually comes to a stop again when not thrusting.
The Asterales order probably originated in the Cretaceous ( 145 – 65 Mya ) on the supercontinent Gondwana which broke up from 184 – 80 Mya, forming the area that is now Australia, South America, Africa, India and Antarctica.
An Oligocene ( 34 – 23 Mya ) pollen is known for Asteraceae and Goodeniaceae, and seeds from Oligocene and Miocene ( 23 – 5. 3 Mya ) are known for Menyanthaceae and Campanulaceae respectively.
0.100 seconds.