This reduction has been accomplished by the general methods of linear algebra, i.e., by the primary decomposition theorem.

Fifteen primary schools were founded and the university was replaced by an ’ Ecole centrale ’ in which there were eight chairs, all very scientific in outlook: notably, arithmetic and stereometry, algebra and stereotomy, geometry and astronomy, mechanics and physics, navigation, chemistry, etc.

The commutative ring theory, commonly known as commutative algebra, primary concerns itself with problems and naturally-occurring ideas in algebraic number theory and algebraic geometry.

In linear algebra, the main diagonal ( sometimes leading diagonal or major diagonal or primary diagonal or principal diagonal ) of a matrix is the collection of entries where is equal to.

Lasker was also a research mathematician who was known for his contributions to commutative algebra, which included proving the primary decomposition of the ideals of polynomial rings.

However, the infinitesimal conformal transformations form an infinite-dimensional algebra, called the Witt algebra and only the primary fields ( or chiral fields ) are invariant with respect to the full infinitesimal conformal group.

In particular, it is possible to relate the spectrum of primary operators in a theory to the value of the central charge, c. The Hilbert space of physical states is a unitary module of the Virasoro algebra corresponding to a fixed value of c. Stability requires that the energy spectrum of the Hamiltonian be nonnegative.

* The primary algebra ( Chapter 6 of LoF ), whose models include the two-element Boolean algebra ( hereinafter abbreviated 2 ), Boolean logic, and the classical propositional calculus ;

Boundary algebra is Meguire's ( 2011 ) term for the union of the primary algebra ( hereinafter abbreviated pa ) and the primary arithmetic.

LoF argues that the pa ( primary algebra ) reveals striking connections among logic, Boolean algebra, and arithmetic, and the philosophy of language and mind.

Some of these authors have modified the primary algebra in a variety of interesting ways.

Modulo this detail, 2 is a model of the primary algebra.

Spencer-Brown referred to the mathematical system of Laws of Form as the " primary algebra " and the " calculus of indications "; others have termed it " boundary algebra.

Some of these authors have modified and extended the primary algebra, with interesting consequences.

Matrices are of primary importance in linear algebra.

Elementary mathematics encompasses topics from algebra, analysis, arithmetic, calculus, geometry and number theory that are frequently taught at the primary or secondary school level.

Too many people think that the primary purpose of a higher education is to help you make a living ; ;

This is the primary function of the imagination operating in the absence of the original experiential stimulus by which the images were first appropriated.

Neither is primary experience understood according to the attitude of modern empiricism in which nothing is thought to be received other than signals of sensory qualities producing their responses in the appropriate sense organs.

The principal defender of this view of primary experience as `` causal efficacy '' is Alfred North Whitehead.

Speaking as a non-Jew I believe that its primary contribution is in the realm of future policy.

The natural and primary aesthetic attitude is to enjoy contemporary art, to despise and dislike the art of the recent past, and wholly to ignore everything else ''.

Here we may observe that at least one modern philosophy of history is built on the assumption that ideas are the primary objectives of the historian's research.

But competent observers believe he is making progress, particularly toward what Sen. Jackson lists as the primary need -- `` a clearer understanding of where our vital national interests lie and what we must do to promote them ''.

Its primary advantage is that it is a moral choice ; ;

A primary function is the operation of a Government Bid Center, which receives bids daily from the Federal Government's principal purchasing agencies.

If the Department of State is to take primary responsibility for foreign policy in Washington, it follows that the ambassador is expected to take charge overseas.

For this reason, the more uncertain skywave service was denominated `` secondary '' in our rules, as compared to the steadier, more reliable groundwave `` primary service '', and, for both skywave service and skywave interference, signal strength is expressed in terms of percentage of time a particular signal-intensity level is exceeded -- 50 percent of the time for skywave service, 10 percent of the time for skywave interference.

As mentioned, the primary allocation objective to be followed in the allocation of stations on clear channels is the provision of widespread service, free from destructive interference.

The result is a color picture of the specimen where the primary colors correspond to the three different ultraviolet wavelengths.

A primary ingredient in these fields is imagination, and Tri-State Pipeline Corporation deserves a very good mark.

The theorem which we prove is more general than what we have described, since it works with the primary decomposition of the minimal polynomial, whether or not the primes which enter are all of first degree.

If Af are the projections associated with the primary decomposition of T, then each Af is a polynomial in T, and accordingly if a linear operator U commutes with T then U commutes with each of the Af, i.e., each subspace Af is invariant under U.

In the primary decomposition theorem, it is not necessary that the vector space V be finite dimensional, nor is it necessary for parts ( A ) and ( B ) that P be the minimal polynomial for T.

Though there is obviously great need for continued experimentation with various types of short-term intervention to further efforts in developing an operational definition of prevention at the secondary -- or perhaps, in some instances, primary -- level, the place of short-term intervention has already been documented by a number of investigators in a wide variety of settings.

Total inorganic carbon is not believed to limit primary production in the oceans, so its increasing availability in the ocean does not directly affect production ( the situation on land is different, since enhanced atmospheric levels of CO < sub > 2 </ sub > essentially " fertilize " land plant growth ).

According to this viewpoint, citizens are sovereign, morally autonomous beings with duties to pay taxes, obey the law, engage in business transactions, and defend the nation if it comes under attack, but are essentially passive politically, and their primary focus is on economic betterment.

Each node board contained 256 kB of I-cache and D-cache, essentially primary cache.

In contrast to submachine guns and autocannons, machine guns ( like rifles ) tend to share the characteristic of a very high ratio of barrel length to caliber ( a long barrel for a small caliber ); indeed, a true machine gun is essentially a fully automatic rifle, and often the primary criterion for a machine gun as opposed to an automatic rifle is the presence of a quick-change barrel or other cooling system ( see below ).

The Titan with its 120mm gun and heavy armor is essentially a walking main battle tank and as such the primary combat unit of the GDI.

According to Hart, law is essentially a system of primary social rules that guide the conduct of law's subjects, and secondary rules that regulate how the primary rules may be changed, how disputes about them are to be adjudicated and, especially, how the primary rules are to be identified.

This is supported by textual examples of the use of the primary dao in the verbal sense " to lead " ( e. g., Analects 1. 5 ; 2. 8 ) and seriously undermines the unspoken assumption implied in the common translation of Dao as " way " that the concept is essentially a nominal one.

Since each cylinder bank is essentially a straight-6, this configuration has perfect primary and secondary balance no matter which V angle is used and therefore needs no balance shafts.

This section has been interpreted to mean that for the entire passage of the poem, Aeneas who symbolizes pietas ( reason ) in a moment becomes furor ( fury ), thus destroying what is essentially the primary theme of the poem itself.

The primary purpose of a mode scrambler is to create a uniform, overfilled launch condition that can be easily reproduced on multiple measurement systems, so that measurement systems have essentially the same launch conditions and can measure approximately the same bandwidth despite having different laser sources.

( At the time, South Carolina was a one-party state: the blacks had essentially been disfranchised since the turn of the century, and the Democratic Party's primary was the deciding election for offices, in which only whites voted.

Since the fundamental theorem of arithmetic applied to a non-zero integer n that is neither 1 nor − 1 also asserts uniqueness of the representation for p < sub > i </ sub > prime and e < sub > i </ sub > positive, a primary decomposition of ( n ) is essentially unique.

At the beginning of the Civil War, Indian Territory had been essentially reduced to the boundaries of the future State of Oklahoma, and the primary residents of the Territory were members of the Five Civilized Tribes plus a number of Plains tribes that had been relocated to the western part of Indian Territory on land leased from the Five Civilized Tribes.

In addition, genetic variability can also be achieved by producing new primary triticales, which essentially means the reconstitution of triticale, and the development of various hybrids involving triticale, such as triticale-rye hybrids.

Dringenberg points out the primary difference between their vision for Desire being that Gaiman's idea for the character was essentially sexless and unsatisfying.

Anything other than the primary findings of Miller are not to be expected in any case, as by the time of Hoazinoides, essentially all modern bird families are either known or believed to have been present and distinct.

* In the Indian subcontinent, the jagir ( a type of fief ) was often thus assigned to individual junior relatives of the ruling house of a princely state, but not as a customary right of birth, though in practice usually hereditarily held, and not only to them but also to commoners, normally as an essentially meritocratic grant of land and taxation rights ( guaranteeing a ' fitting ' income, in itself bringing social sway, in the primary way in a mainly agricultural society ), or even as part of a deal.

So-called " Maltese cross formations " on the blood film are essentially diagnostic of babesiosis, since they are not seen in malaria, the primary differential diagnosis.

Two years later, after Speaker John W. McCormack retired, Hicks beat out 11 other candidates to win the Democratic primary for his South Boston-based congressional district ; in heavily Democratic Boston, the winner of that primary was essentially assured victory in the November election.

Malta's primary utility to Great Britain lay in its excellent natural harbours and strategic location, and for many decades, Malta was essentially a " fortress colony ".

This was primarily intended for the Apple IIe Card ( the primary reason for the Color Classic's switchable 560x384 display, essentially quadruple the IIe's 280x192 High-Resolution graphics ), which was offered with education models of the LCs.

This Shadow is essentially a secondary personality that, if the wraith's primary personality ( or Psyche ) becomes vulnerable, asserts control of the wraith's body and senses for a time.