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Topic: Hamel bases

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	Station Information - Hamel basis (Site not responding. Last check: 2007-10-15)
		In mathematics, a Hamel basis of a vector space is a set B of vectors in the space such that
		Every Hamel basis of this space is much bigger than this merely countably infinite set of functions.
		Hamel bases of spaces of this kind are of little if any interest; orthonormal bases of these spaces are important to Fourier analysis.
	www.stationinformation.com /encyclopedia/h/ha/hamel_basis.html (276 words)

	Orthonormal basis - Wikipedia, the free encyclopedia
		Then S will be an orthogonal basis of V; which may of course be smaller than H itself, being an incomplete orthogonal set, or be H, when it is a complete orthogonal set.
		Note that in the infinite-dimensional case, an orthonormal basis will not be a basis in the sense of linear algebra; to distinguish the two, the latter bases are also called Hamel bases.
		(Hamel bases are of little practical interest in inner product spaces, while orthonormal bases are of major importance - the distinction may though shed light on what an orthonormal basis is.)
	en.wikipedia.org /wiki/Orthonormal_basis (576 words)

	Hamel bases (Site not responding. Last check: 2007-10-15)
		Also, all bases of a vector space have thesame cardinality (number of elements), called the dimension of the vector space.
		To betterdistinguish these notions, vector space bases are also called Hamelbases and the vector space dimension is also known as Hamel dimension.
		Hamel basesof spaces of this kind are of little if any interest; orthonormal bases of these spaces are important to Fourier analysis.
	www.therfcc.org /hamel-bases-276093.html (753 words)

	Variety of uses in the cards if Navy air base closes - Boston.com (Site not responding. Last check: 2007-10-15)
		The commission said one reason to close the base rather than just cut back its operations was to give local and state officials a chance to redevelop the site.
		Former military bases are put to a variety of uses, said Tim Ford, executive director of the Association of Defense Communities, which works with communities around closed bases.
		Hamel, who once worked for the Pease Development Authority, said Portsmouth wanted to draw aviation companies to the property, while Newington wanted to prohibit them.
	www.boston.com /news/local/new_hampshire/articles/2005/08/26/variety_of_uses_in_the_cards_if_navy_air_base_closes (943 words)

	Basis (linear algebra) (Site not responding. Last check: 2007-10-15)
		The phrase Hamel basis is sometimes used to refer to a basis as defined above, in which the fact that all linear combinations are finite is crucial.
		Except in the finite-dimensional case, this concept is not purely algebraic, and is distinct from a Hamel basis; it is also more generally useful.
		To keep clear the distinction of bases using finite and infinite combination, vector space bases are called Hamel bases if the context requires it, and the vector space dimension is also known as Hamel dimension.
	www.arikah.net /encyclopedia/Basis_vector (859 words)

	[No title] (Site not responding. Last check: 2007-10-15)
		In that context, the "usual" basis is called a Hamel basis: it is a collection of vectors {v_i}, i in some index set, such that every vector in the space can be written uniquely as a linear combination of some finite set of v_i's.
		Hamel bases always exist, but can't be constructed without the Axiom of Choice.
		Concrete examples of Schauder bases, on the other hand, are known for most of the "natural" separable Banach spaces.
	www.math.niu.edu /~rusin/papers/known-math/96/schauder (766 words)

	Hamel Hawks Box Score (Nick Bell Classic)
		Hamel tied the game in the bottom of the inning on an unearned run, but the Grillers came right back with four runs in the top of the fourth.
		With two outs and the bases loaded, Putz was hit by a pitch that scored Aaron Rocheleau from third and then Holman hit a fly ball to right that was lost in the night and landed for a double to score Casey Haugen, Quinn, and Putz.
		Hamel got two runs back in the bottom of the fourth to cut the lead to 6-4, but with Holman on first base, Schloe put the game out of reach by smacking a 2-run home run in the top of the seventh.
	www.geocities.com /ultimategrillersbaseball/2005/boxscores/hamelboxNBC.htm (603 words)

	Heritage of Blue eyes after many generations
		One of the places where Hamel stayed during his years in Korea was at a military base for which Hamel's name was Saijsingh.
		SinsOng means "new fortress," and corresponds very well with Hamel's Saijsingh, which means the same thing but is written with the pure Korean word for "new." It appears that after the Imjin wars the Korean military used it as one of their own bases.
		Hamel's transcriptions of Korean words into the Dutch orthography of the 17th century are not always transparent, but in no case does he have a spelling as far-fetched as "Saijsingh" for "Chwasuyong." The -singh for K. sOng is found in other Hamel words and appears to reflect his hearing of the sounds.
	sln.fi.edu /tfi/units/life/forums/anatomy/0718000310.html (896 words)

	[No title] (Site not responding. Last check: 2007-10-15)
		Congress approved four rounds of base closures in 1988, 1991, 1993 and 1995 in a push to cut military costs after a massive arms buildup during the Reagan years.
		But lawmakers who fear base closings will hurt their districts’ businesses and tax bases have fought to keep more facilities from falling under the ax, said Lilly Goren, author of “The Politics of Military Base Closings: Not in My District.” In fact, the fight already has begun over the next round of closings.
		Base closings may actually improve national security by allowing the Defense Department to funnel that money to more pressing needs, said John Allen Williams, a military expert at Loyola University in Chicago.
	www.armytimes.com /print.php?f=1-292925-1955260.php (680 words)

	[No title]
		Although there exist Hamel bases having measure zero (Sierpinski, 1920; see my Sept. 25 post), the set of all integer linear combinations of elements in any Hamel basis is nonmeasurable.
		CH implies there exists a Hamel basis for which the set of all its nonnegative rational linear combinations is first category in the reals.
		CH implies there exists a Hamel basis for which the @ set of all its nonnegative rational linear combinations @ is first category in the reals.
	www.math.niu.edu /~rusin/known-math/99/hamel (3445 words)

	Station Information - Orthonormal basis (Site not responding. Last check: 2007-10-15)
		Nor is it possible to speak of an orthormal basis of any vector space unless it first has an inner product; Banach spaces do not generally have orthonormal bases.
		Using Zorn's lemma, one can show that every Hilbert space admits an orthonormal basis; furthermore, any two orthonormal bases of the same space have the same cardinality.
		Hamel bases are of little if any interest in inner product spaces, but realizing that a Hamel basis is something that an orthonormal basis is not, may shed some light on what an orthonormal basis is.
	www.stationinformation.com /encyclopedia/o/or/orthonormal_basis.html (383 words)

	THE OMEGA FILE:[Greys, Nazis, Underground Bases, and the New World Order] by Branton (Site not responding. Last check: 2007-10-15)
		If the Nazi bases still exist in Antarctica then they would no doubt still have secret contact with the Bavarian cults which sponsored and were an integral part of the Nazi party, like the Bavarian THULE society for instance.
		They all lead to the establishing of the Antarctic base 211 at the end of war by means of German submarines and flying saucers and to the [failed ?] attempt to destroy it by the US Navy in 1947.
		Based on the following very, very brief summary of a huge file on this subject, it seems apparent to me that you have each played a witting or unwitting role in treason.
	pdharris0.tripod.com /omega.htm (16364 words)

	The Journal of Hamel and Korea; Email exchange (Site not responding. Last check: 2007-10-15)
		Hamel's distances are all quoted in 'mijl (miles), it was Churchill's translation from the French that introduced the word 'leagues'.
		Hamel's letters are in no way comparable to any of our modern writings and hard to interpret, I will attach a few examples, especially about the words about which we are arguing now.
		Since Hamel was not amongst the chosen ones to be taken away in 1655 to Quelpaert to act as interpreter he was speaking about an assistant a sailor and the under officer in charge of the rigs.
	www.hendrick-hamel.henny-savenije.pe.kr /koreanstudies.htm (10429 words)

	Dimension : 3-D
		All, however, are ultimately based on the concept of the dimension of Euclidean n-space E
		The Hausdorff dimension is defined for all metric spaces and, unlike the Hamel dimension, can also attain non-integer real values.
		The Krull dimension of a commutative ring is defined to be the maximal length of a strictly increasing chain of prime ideals in the ring.
	www.fastload.org /3-/3-D.html (484 words)

	Basis (linear algebra)
		3.1 An "orthonormal basis" of an infinite-dimensional Hilbert space is not a Hamel basis
		To better distinguish these notions, vector space bases are also called Hamel bases and the vector space dimension is also known as Hamel dimension.
		An "orthonormal basis" of an infinite-dimensional Hilbert space is not a Hamel basis
	www.sciencedaily.com /encyclopedia/basis__linear_algebra_ (828 words)

	Redeveloping bases: The biggest factor is location, experts say - Boston.com (Site not responding. Last check: 2007-10-15)
		The vote to close Loring Air Force Base came in 1991, the same year the flag was lowered for the last time at New Hampshire's Pease Air Force Base, a victim of the previous round of military base closings.
		Now called Pease International Tradeport, the former base now boasts 5,500 jobs, five times the number of civilian jobs it had when it was an Air Force base.
		When the Base Realignment and Closure Commission begins casting final votes on Wednesday, supporters of the Portsmouth Naval Shipyard and Brunswick Naval Air Station can take heart in the knowledge that both oceanfront sites are considered desirable for redevelopment.
	www.boston.com /news/local/new_hampshire/articles/2005/08/21/redeveloping_bases_the_biggest_factor_is_location_experts_say?mode=PF (880 words)

	Orthonormal basis -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-15)
		An orthonormal basis of a vector space V makes no sense unless V is given an inner product; (Click link for more info and facts about Banach space) Banach spaces do not generally have orthonormal bases.
		Using (Click link for more info and facts about Zorn's lemma) Zorn's lemma, one can show that every Hilbert space admits a basis and thus an orthonormal basis; furthermore, any two orthonormal bases of the same space have the same (Click link for more info and facts about cardinality) cardinality.
		A Hilbert space is (Click link for more info and facts about separable) separable if and only if it admits a (Click link for more info and facts about countable) countable orthonormal basis.
	www.absoluteastronomy.com /encyclopedia/o/or/orthonormal_basis.htm (603 words)

	Bambooweb: Core competency
		To take an example from the automotive industry, it has been claimed that Volvo?s core competence is safety.
		Ever since Prahalad and Hamel introduced the term in the 1990?s many researchers have tried to highlight and further illuminate the meaning of core competence.
		Coyne, Hall, and Clifford (1997) proposed that "a core competence is a combination of complementary skills and knowledge bases embedded in a group or team that results in the ability to execute one or more critical processes to a world class standard." Two ideas are especially important here.
	www.bambooweb.com /articles/c/o/Core_competency.html (610 words)

	Atlas: Marczewski measurable Vitali sets and Hamel bases by Strashimir G. Popvassilev (Site not responding. Last check: 2007-10-15)
		Under CH there is a Hamel basis which is a Lusin set (and, hence, Marczewski null set).
		In ZFC there is a Marczewski null Hamel basis for the plane as a vector space over the rationals.
		There exists a Hamel basis for the reals which is a Marczewski null set, though the proof is a little messier than the one for the plane.
	atlas-conferences.com /cgi-bin/abstract/cacl-50 (251 words)

	[No title]
		We also show there is a Marczewski null Hamel basis for the reals, although a Vitali set cannot be Marczewski null.
		The proof of the existence of a Marczewski null Hamel basis for the plane is easier than for the reals and we give it first.
		We show that there is no easy way to get a Marczewski null Hamel basis for the reals from one for the plane by showing that there is no one-to-one additive Borel map from the plane to the reals.
	www.math.wisc.edu /~miller/res/hamel.tex (1010 words)

	afrol News - Religious knowledge set found to check Africa's development
		This is one of the conclusions in a draft working paper by Jacques Hamel, a scientific affairs officer at the Addis Ababa-based UN Economic Commission for Africa (UNECA).
		Mr Hamel told afrol News that African nations may turn out to be losers of the global knowledge race "in the absence of adequate knowledge and governance strategies."
		This was said to be the kind of "sterile knowledge bases that pervades and keeps big chunks of African Knowledge Societies from developing." There was a need to introduce development efforts aimed at deliberately "deleting, subtracting, abandoning or transforming obsolete or non-development knowledge," Mr Hamel urges.
	www.afrol.com /articles/15368 (630 words)

	Dictionary of Meaning www.mauspfeil.net
		In the infinite-dimensional case the distinction matters: the definition given above requires only that the span of an orthonormal basis be ''dense in'' the vector space, not that it equal the entire space.
		Note that in the infinite-dimensional case, an orthonormal basis will not be a basis in the sense of linear algebra; to distinguish the two, the latter bases are also called basis (linear algebra) Hamel bases.
		(Hamel bases are of little practical interest in inner product spaces, while orthonormal bases are of major importance - the distinction may though shed light on what an orthonormal basis is.) See also Hamel basis.
	www.mauspfeil.net /Orthonormal_basis.html (950 words)

	The cardinality of Hamel bases of Banach spaces (ResearchIndex)
		Abstract: We show that in an in nite dimensional Banach space, every Hamel base has the cardinality of the Banach space, which is at least the cardinality of the continuum.
		1 Facts concerning Hamel bases In this article we investigate by set theoretical methods the cardinality of Hamel bases (also called \algebraic bases") of Banach spaces.
		0.3: On The Complexity Of Hamel Bases Of Infinite Dimensional Banach..
	citeseer.ist.psu.edu /292209.html (393 words)

	Alchemy and Alchemical Practices
		They base their atomic system on Ether, the universal fluid (the astral light of the Kabbala), and on the principle of evolution.
		According to them matter is composed of molecules divided into infinitesimal particles, called atoms, which again are but one degree removed from Ether; hence matter is compact energy which in the last resort is able to dissolve itself into free energy, into force.
		The Alchemist - even in modern days - is always a priest who bases his physical or moral work on the Primordial Genesis which is none other than Tarot.
	www.tarot-decks.com /alchemy.htm (2946 words)

	Re: General relativity vs flat Minkowski spacetime
		A subset $B\subseteq V$ of a vector space is called a Hamel base, if $B$ is linear independent (Hamel) and if every $v\in V$ can be represented as finite linear combination of elements contained in $B$.
		If $B$ is a Hamel base, the coefficients $a_i$ in $v=\sum_{i=1}^n a_i x_i$ with $v\in V$ are unique.
		However, for Hamel bases the summation in the representation $t=t^{j_1 \ldots j_p}_{k_1 \ldots k_q}e_{j_1} \otimes\cdots \otimes e_{j_p} \otimes \theta^{k_1} \otimes\cdots \otimes \theta^{k_q}$ of a $(p,q)$-tensor $t$ is not finite.
	www.lns.cornell.edu /spr/1999-12/msg0020402.html (516 words)

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