* Given any Banach space X, the continuous linear operators A: X → X form a unitary associative algebra ( using composition of operators as multiplication ); this is a Banach algebra.

* Given a partition of A, G is a transformation group under composition, whose orbits are the cells of the partition ‡;

Given a field K, the corresponding general linear groupoid GL < sub >*</ sub >( K ) consists of all invertible matrices whose entries range over K. Matrix multiplication interprets composition.

Given high enough cooling rates and appropriate alloy composition, metallic bonding can occur even in glasses with an amorphous structure.

# Given any point x in X, and any sequence in X converging to x, the composition of f with this sequence converges to f ( x )

Given the representation of T as a multiplication operator, it is easy to characterize the Borel functional calculus: If h is a bounded real-valued Borel function on R, then h ( T ) is the operator of multiplication by the composition.

Given a ring R and an R-module M, a composition series for M is a series of submodules

Given that socks directly contact the feet, their composition can have an impact on foot odor.

Given two subsets A and B of N and a set of functions F from N to N which is closed under composition, A is called reducible to B under F if

Position Statement Summary: Given that the focus of writing instruction is expanding: the curriculum of composition is widening to include not one but two literacies: a literacy of print and a literacy of the screen and that work in one medium is used to enhance learning in the other, this statement articulates principles of good practice governing these new pedagogical activities for digital literacy.

Given a map, the mapping cylinder is a space, together with a cofibration and a surjective homotopy equivalence ( indeed, Y is a deformation retract of ), such that the composition equals f.

Given the widely recognized importance of proteins as cellular catalysts and recognition elements, the ability to precisely control the composition and connectivity of polypeptides is a valued tool in the chemical biology community and is an area of active research.

Given the timing of Led Zeppelin's first live performances of " Since I've Been Loving You " ( early 1970 ), it has been suggested that " Never " is the original source for this composition.

Additionally, there are composition operations: Given a sequence of sequences of objects, a sequence of objects, and an object Z: if

Given its unusual composition for an igneous rock, the magmatic nature of the carbonatite was not proposed for a long time and remained doubtful subsequently.

Given current values

Put together: Given an input signal, current state, and stack symbol, the automaton can follow a transition to another state, and optionally manipulate ( push or pop ) the stack.

Given current laser energies this was impractical, but the concept was studied throughout the 1960s and later.

Given the current value of the port of Toledo to Ohio, it can be reasonably suggested that both sides benefitted from the conflict.

Given current trends in the UMC — with overseas churches growing, especially in Africa, and U. S. churches collectively losing about 1, 000 members a week — it has been estimated that Africans will make up at least 30 % of the delegates at the 2012 General Conference, and it is also possible that 40 % of the delegates will be from outside the U. S. One Congolese bishop has estimated that typical Sunday attendance of the UMC is higher in his country than in the entire United States.

Given the current size of the Web, even large search engines cover only a portion of the publicly available part.

Given that the right to be politically active has granted them the opportunity to expand their knowledge and influence in current affairs, they are now considered one of the main components in the country's decision-making in both politics and economy.

Given the closeness shared by the countries bordering the Mediterranean basin, and very topical interest to the consolidation of the European Union's enlargement policy, he also intended to establish trade relations with the Republic of Malta aimed at encouraging mutual knowledge and awareness in a profitable way their local communities on European issues of common interest and current.

Given the many available models, it appears that the material removal mechanism in EDM is not yet well understood and that further investigation is necessary to clarify it, especially considering the lack of experimental scientific evidence to build and validate the current EDM models.

Given the current estimated mass of the black hole, the progenitor star must have lost over 30 solar masses of material.

Given current values

Given the current employment opportunities, it is not likely that the number of specialists in SA Indian languages will increase fast enough to document most of the surviving SA languages before they go out of use, as most of them unavoidably will.

Given the age of the current Justices, this allowed a Supreme Court size of up to 15 Justices.

Given the current human population and birth-rate, this equals five per day or 1800 per year.

Given the direction of the current, it is clear that the buoys would be dragged in the opposite direction to that claimed by Mr Jim Brooks.

Given a sufficiently high and well known amount of electric current and voltage, the timing of the bridgewire vaporization is both extremely short ( a few microseconds ) and extremely precise and predictable ( standard deviation of time to detonate as low as a few tens of nanoseconds ).

Given the current climate Palmair has been forced to cease its entire operations starting from Summer 2011.

Given criticisms of the current DSM classification, a new disorder combining the characteristics of HSDD and sexual arousal disorder has been proposed for the next version of the DSM ( the DSM-V, due to be published in 2013 ).

Given the current economic climate, this is likely to remain in this state for at least another decade.

Given the current recession, there have been signs the project may be delayed indefinitely.

Given our current estimate of the parameters θ < sup >( t )</ sup >, the conditional distribution of the Z < sub > i </ sub > is determined by Bayes theorem to be the proportional height of the normal density weighted by τ:

Given the minister's " current crusade on financial literacy ," Fullard believed a suitable charitable cause would be a scholarship for business education.

Given an unchangeable Z < sub > L </ sub >, one can maximize both the voltage and current ( and therefore, the power ) at the load by minimizing Z < sub > S </ sub >.

: Given n men and n women, where each person has ranked all members of the opposite sex with a unique number between 1 and n in order of preference, marry the men and women together such that there are no two people of opposite sex who would both rather have each other than their current partners.

Given the small population of the region ( compared with the Central Canadian provinces or the New England states ), the regional economy is a net exporter of natural resources, manufactured goods, and services.

Given the shared nature of the Canadian monarch, most members of the Canadian Royal Family are also members of the British Royal Family, and thus the House of Windsor.

Given Canada's small market and its position next to the dominant producer of feature films, the Canadian film industry receives substantial assistance from the government.

Given his military service, the selection of Arthur proved to be prudent, as he acted as the King's, and thus the Canadian Commander-in-Chief's, representative through the first years of World War I.

Given his years of promoting baseball in Canada while a member of the Blue Jays, Whitt remained active as an ambassador of Canadian baseball Indeed, Whitt's coaching career first came to prominence in his capacity as manager of the Canadian national baseball team.

Given the size of the CAW with the Canadian labour movement, the CAW's support for the Liberals has caused significant problems for the CLC leadership in continuing to follow the Congress ' policy of NDP support.

Given the sufferings that some indigenous children experienced in the Canadian Residential School system, the Holy Father expressed his sorrow at the anguish caused by the deplorable conduct of some members of the Church and he offered his sympathy and prayerful solidarity.

Given here are the ethnic origins of Canadian residents ( citizens, landed immigrants, and non-citizen temporary residents ) as recorded by them on their 2006 census form.

Given the structure of the Canadian infantry platoon, the MCpl is roughly equivalent to the British rank of corporal, second in command of an infantry section.

Given that these communities have considerable mobility within Canada, they retain traits common in many Canadian cities.