Moreover, the algebra generated by the three matrices is isomorphic to the 3-dimensional Euclidean real Clifford Algebra.

Moreover, the axioms assume that the vacuum is " cyclic ", i. e., that the set of all vectors which can be obtained by evaluating at the vacuum state elements of the polynomial algebra generated by the smeared field operators is a dense subset of the whole Hilbert space.

Moreover a supergroup has a super Lie algebra which plays a role similar to that of a Lie algebra for Lie groups in that they determine most of the representation theory and which is the starting point for classification.

108: 171 .</ ref > Moreover, Jaynes elaborated an interpretation of probability theory as generalized Aristotelian logic, a view very convenient for linking fundamental physics with digital computers, because these are designed to implement the operations of classical logic and, equivalently, of Boolean algebra.

Moreover, an element of a C *- algebra is positive if and only if there is some in such that.

Moreover if we define scalar multiplication in the obvious manner, the split-complex numbers actually form a commutative and associative algebra over the reals of dimension two.

Moreover, an alternative axiomatization of monadic Boolean algebras consists of the ( reinterpreted ) axioms for an interior algebra, plus ∀(∀ x )' = (∀ x )' ( Halmos 1962: 22 ).

Moreover, if U is a commutative unipotent group, the exponential map induces an isomorphism from the Lie algebra of U to U itself.

Moreover, MV-algebras characterize infinite-valued Łukasiewicz logic in a manner analogous to the way that Boolean algebras characterize classical bivalent logic ( see Lindenbaum-Tarski algebra ).

When viewed as a factor-algebra of g, this semisimple Lie algebra is also called the Levi factor of g. Moreover, Malcev ( 1942 ) showed that any two Levi subalgebras are conjugate by an ( inner ) automorphism of the form

Moreover, the places where B ramifies determines B up to isomorphism as an algebra.

Moreover, the algebra generated by the e < sub > n </ sub > is a C *- algebra in which the e < sub > n </ sub > are self-adjoint, and such that tr ( xe < sub > n </ sub >) = λ tr ( x ) when x is in the algebra generated by e < sub > 1 </ sub > up to e < sub > n − 1 </ sub >.

Moreover, plans are underway to build a-long sub-alpine tunnel connecting the older Lötschberg and Gotthard tunnels built in the 19th century.

Moreover, it would allow entirely new operating systems to be " built up " on a common core, aiding OS research.

Moreover, tunnels and key bridges are built with tank traps.

Moreover, many Japanese businesses built outside Japan have had ceremonies performed by a Shinto priest, with occasionally an annual visitation by the priest to re-purify.

Moreover, the road was blocked by a trench, and, towards the forest, an earthen wall had been built along the roadside, permitting the Germanic tribesmen to attack the Romans from cover.

Moreover, some revolvers have been built with spring-loaded extractor lips that enter the rim cut of “ rimless cartridges ” and extract those cases normally.

Moreover, Mong Kok station was planned to be built a bit further north of its present location, which would have taken the place of Prince Edward Station ; and Sham Shui Po was planned to be built a bit further south of its present location.

Moreover, a train extension has been built from the nearby station of Ottignies, which allows passengers to travel to or from Brussels in under an hour.

Moreover, at the end of World War II, SNCF inherited more than 200 units of German decapods built in France, mostly BR 44.

Moreover, King Sinmun built the Gomun Temple ( the Temple of Appreciated Blessing ) and dedicated it to his father, he built a waterway for the sea dragon to come to and from the sea and land, and he built a pavilion, Eegun, overlooking the islet so that future kings could pay their respects to the great King Munmu.

Moreover, the Pentax placed controls in locations that would become standard on 35 mm SLRs from all manufacturers, such as the right-handed rapid wind lever, the bottom right mounted rewind release, and film speed reminder around the film rewind crank, a location that remained standard even when the dial went from being merely a reminder to the photographer to actually controlling the light-meter built into later SLRs.

Moreover a number of forts built by the Ikkeri Nayakas have failed to stand the test of time and perished.

Moreover, the Mumbai Suburban Railway, though extensive, is not built to rapid transit specifications.

Moreover, when the Malaspina ( one of the leading Lunigianese dynasties during the Middle Ages ) played an import part in both the local politics of Lunigiana and the politics of northern Italy, they built a great number of castles, which were used as residences and fortifications by which several branches of the dynasty defended the territory.

Moreover, Elbegdorj assisted Mongolia's first Wrestling Palace which was half built for a long time.

Moreover, authorities evicted protestors from occupying the elementary school and working the fields where the base will be built by placing barbed wire at the boundary of the future base.

Moreover, to its summit at Ladmanlow, it would climb a thousand feet from Cromford, making it one of the highest lines ever built in Britain.

Moreover, the to be +-ing verb form could be used to express a passive meaning without any additional markers: " The house is building " could mean " The house is being built.

Moreover the intention was to double the line and rather than widen the Glenfield Tunnel a deviation was built from Desford to meet the main line south of Leicester London Road station.

Moreover, the turnstiles for the three newer stands are built into a perimeter wall rather than the stadium itself, and there are very large open air spaces before the stands themselves.

Moreover, pack ice built up behind the iceberg in the Ross Sea creating a nearly frozen barrier that blocked two cargo ships enroute to supply McMurdo Station, according to the National Science Foundation.

Moreover, early Christian Church Fathers used canon to rank the authoritative texts of the New Testament, preserving them, given the expense of vellum and papyrus and mechanical book reproduction, thus, being comprehended in a canon ensured a book ’ s preservation as the best way to retain information about a civilization.

Moreover, it was traditionally considered in the hardware manufacturer's interest to guarantee that their clients can use their hardware in an optimum way.

Moreover, owing to the middle-class aspirations of much of the peasantry, Leon Trotsky proposed that the proletariat should lead the revolution, as the only way for it to be truly socialist and democratic ; although Lenin initially disagreed with Trotsky ’ s formulation, he adopted it before the Russian Revolution in October 1917.

Moreover, more recently than other standards above, MPEG has started following international standards ; each of the standards holds multiple MPEG technologies for a way of application.

Moreover, he studied the way people lived and aimed to inform leaders how they should rule and even how they themselves should live.

Moreover, he argued that the death and resurrection of Jesus was for the salvation of man, stating: " to achieve each resurrection of ours, the savior paid with his single life, and he pre-enacted and presented his one and only one by way of sacrament and by way of model.

Moreover, any element u ∈ F ( A ) defines a natural transformation in this way.

Moreover, some communitarians " experience this less as a case of being used for others ' ends and more as a way of contributing to the purposes of a community I regard as my own ".

Moreover, this construction is formulaic in the sense that it works in essentially the same way for any rng.

Moreover, Gödel's second incompleteness theorem shows that the consistency of sufficiently strong effective theories of arithmetic can be tested in a particular way.

Moreover, Paris and Helen had stopped in Mysia on their way to Troy, and had asked Telephus to fight off the Achaeans should they come.

Moreover, Möbius, as one of the judges, criticized the way Grassmann introduced abstract notions without giving the reader any intuition as to why those notions were of value.

For example, when searching for a proper divisor of a random number n, it is better to enumerate the candidate divisors in increasing order, from 2 to n-1, than the other way around — because the probability that n is divisible by c is 1 / c. Moreover, the probability of a candidate being valid is often affected by the previous failed trials.

Moreover, the government is obliged to empower members of agencies with political rights to run for elections and be elected ; or, appoint them into the public sector as a way to hold the government representative and ensure voices from all constituencies are included in policy-making process.

Moreover, his commitment to the Principle of the correlation of parts caused him to doubt that any mechanism could ever gradually modify any part of an animal in isolation from all the other parts ( in the way Lamarck proposed ), without rendering the animal unable to survive.

Moreover, Overy asserted that the repressive capacity of the German state as a way of dealing with domestic unhappiness was somewhat downplayed by Mason.

Moreover, D & C algorithms can be designed for important algorithms ( e. g., sorting, FFTs, and matrix multiplication ) to be optimal cache-oblivious algorithms – they use the cache in a provably optimal way, in an asymptotic sense, regardless of the cache size.

Moreover, there is no way to determine the energy densities in particles and magnetic fields from observation: the same synchrotron emissivity may be a result of a few electrons and a strong field, or a weak field and many electrons, or something in between.

Moreover, scholars view his unorthodox methodology ( for example, using comparative mythology to derive scenarios in celestial mechanics ) as an unacceptable way to arrive at conclusions.

Moreover, students often seek out clinical programs because doctrinal courses offer little in the way of practical training.

Moreover, the only way the stars could be so distant and still appear the sizes they do in the sky would be if even average stars were gigantic — at least as big as the orbit of the Earth, and of course vastly larger than the sun.

Moreover, the eight Batavian auxiliary units of Vitellius ' army were on their way home and could be easily persuaded to join the rebellion for an independent Batavia.

Moreover, the technique can be further generalized in a straightforward way to also include an entropy constraint for vector data.