Given these changes, the classical terms " system ", " expansion " and " peripheral " no longer have the same connotations.

If V is finite-dimensional, then V * has the same dimension as V. Given a basis

Given the limited budget for schools, it is claimed that a voucher system would weaken public schools while at the same time not necessarily providing enough money for people to attend private schools.

Given a groupoid G, the vertex groups or isotropy groups or object groups in G are the subsets of the form G ( x, x ), where x is any object of G. It follows easily from the axioms above that these are indeed groups, as every pair of elements is composable and inverses are in the same vertex group.

Given these two assumptions, the coordinates of the same event ( a point in space and time ) described in two inertial reference frames are related by a Galilean transformation.

Given the same set of verifiable facts, some societies or individuals will have a fundamental disagreement about what one ought to do based on societal or individual norms, and one cannot adjudicate these using some independent standard of evaluation.

Given the proposition that ' a persons life prospects should not be decided by factors outside of their control or for which a person cannot claim personal credit ' ( i. e. social status, inherited wealth, race and other accidents of birth ) a meritocracy proposes a system where people are rewarded based on their efforts, and if everyone can start on equal footing with the same opportunity to advance, then the results are just.

Given printing practices at the time ( which included type-setting from dictation ), no two editions turned out to be identical, and it is relatively rare to find even two copies that are exactly the same.

* Coreference resolution: Given a sentence or larger chunk of text, determine which words (" mentions ") refer to the same objects (" entities ").

Given a general statement such as all ravens are black, a form of the same statement that refers to a specific observable instance of the general class would typically be considered to constitute evidence for that general statement.

Given two affine spaces and, over the same field, a function is an affine map if and only if for every family of weighted points in such that

Given the same set of resources, IS-IS can support more routers in an area than OSPF.

Given a diagram F: J → C ( thought of as an object in C < sup > J </ sup >), a natural transformation ψ: Δ ( N ) → F ( which is just a morphism in the category C < sup > J </ sup >) is the same thing as a cone from N to F. The components of ψ are the morphisms ψ < sub > X </ sub >: N → F ( X ).

Given any two similar polygons, corresponding sides taken in the same sequence are proportional and corresponding angles taken in the same sequence are equal in measure.

Given that multiple people often consume food from the same source, it is common for more than a single person to be affected simultaneously.

Given a gain of about 100 and a laser efficiency of about 1 %, HiPER produces about the same amount of fusion energy as electrical energy was needed to create it.

Given the symmetry of circularly polarized light, we could have in fact selected any other two orthogonal components and found the same phase relationship between them.

Given that radio communication channels are allocated by agencies such as the Federal Communication Commission giving a prescribed ( maximum ) bandwidth, the advantage of QPSK over BPSK becomes evident: QPSK transmits twice the data rate in a given bandwidth compared to BPSK-at the same BER.

Given the choice between playing a game at an arcade three or four times ( perhaps 15 minutes of play for a typical arcade game ), and renting, at about the same price, exactly the same game — for a video game console — the console became the preferred choice.

: Given any set A and any set B, if A is a nonempty set ( that is, if there exists a member C of A ), then if A and B have precisely the same members, then they are equal.

Pope Saint Adeodatus I or Deodatus I ( which is Given by God in Latin, also called Deusdedit, which is God Has Given ; both are now considered variants of the same name ) ( died 8 November 618 ) was Pope from 13 November 615 to his death.

Given a long enough barrel, there would eventually be a point at which friction between the bullet and the barrel, and air resistance, would equal the force of the gas pressure behind it, and from that point, the velocity of the bullet would decrease.

Given certain assumptions about how to treat the gravitational force, he argued that the bullet sinks with acceleration, where g is the acceleration due to gravity ( assumed to be uniform over the scale of the thought experiment ) and is the factor mentioned above.

Given the problem of the aerodynamic design of the nose cone section of any vehicle or body meant to travel through a compressible fluid medium ( such as a rocket or aircraft, missile or bullet ), an important problem is the determination of the nose cone geometrical shape for optimum performance.

Given the problem of the aerodynamic design of the nose cone section of any vehicle or body meant to travel through a compressible fluid medium ( such as a rocket or aircraft, missile or bullet ), an important problem is the determination of the nose cone geometrical shape for optimum performance.

Given these limitations, Basic Programming cartridge never took off and is considered to be a novel but a failed product.

Given the advent of high-throughput protein analysis and lower limits of detection, however, there has been interest in developing one-step probing systems that would allow the process to occur faster and with less consumables.

Given a strictly increasing integer sequence / function ( n ≥ 1 ) we can produce a faster growing sequence ( where the superscript n denotes the n < sup > th </ sup > functional power ).

Given its much higher density, a CPU designer may be inclined to use MRAM to offer a much larger but somewhat slower cache, rather than a smaller but faster one.

Given a faster than light technology and weapons scaled similar to those of the Age of Sail, some of the battles described in the Honor Harrington series resemble real-life sea battles in our historical record.

Given the right overall conditions of the compressed fuel — high enough density and temperature — this heating process will result in a chain reaction, burning outward from the center where the shock wave started the reaction.

Given the right overall conditions of the compressed fuel – high enough density and temperature – this heating process can result in a chain reaction, burning outward from the center where the shock wave started the reaction.

Given the right overall conditions of the compressed fuel — high enough density and temperature — this heating process can result in a chain reaction, burning outward from the center where the shock wave started the reaction.

Given the powder burns on her shirt, he determines that she shot herself.

Given this kind of business climate, what competitive marketing problems will your company face in the next 10 yrs.??

Given the magnitude of the disaster, reconstruction and economic recovery will remain the primary focus of the Salvadoran Government for some time to come.

Given the distances between the stars, the great majority of stellar systems in colliding galaxies will be unaffected.

On poverty, Hoover said that " Given the chance to go forward with the policies of the last eight years, we shall soon with the help of God, be in sight of the day when poverty will be banished from this nation ", and promised, " We in America today are nearer to the final triumph over poverty than ever before in the history of any land ," but within months, the Stock Market Crash of 1929 occurred, and the world's economy spiraled downward into the Great Depression.

Given these conditions, Lovelock expects human civilization will be hard pressed to survive.

Given that bodhisattvas are known to incarnate at will as living people according to the sutras, the princess Miaoshan is generally viewed as an incarnation of Avalokitesvara.

Given two secondary stations, the time difference ( TD ) between the primary and first secondary identifies one curve, and the time difference between the primary and second secondary identifies another curve, the intersections of which will determine a geographic point in relation to the position of the three stations.

Given that the cost of replacing an executive can run over 100 % of his or her annual salary, any investment of time and energy in re-recruitment will likely pay for itself many times over if it helps a business retain just a handful of key players that would have otherwise left.

Given enough time and repeated exposure to the antibiotic, a population of antibiotic-resistant bacteria will emerge.

" Given, a certain population, with various needs and powers of production, in possession of certain lands and other sources of material: required, the mode of employing their labour which will maximize the utility of their produce.

Given the poorly understood nature of the endocannabinoid system, it will take many more generations of progress to even scratch the surface of the extraordinarily complex mechanisms that control neurological development / degeneration.

Given sufficient time, oxygen, and water, any iron mass will eventually convert entirely to rust and disintegrate.

Given any two cards from the deck, there will be one and only one other card that forms a set with them.

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 time, the simple mixture of sand, silt, and clay will evolve into a soil profile which consists of two or more layers called horizons that differ in one or more properties such as texture, structure, colour, porosity, consistency, and reaction.

Given this announcement the prisoner can deduce that the hanging will not occur on the last day of the week.

# Given a large enough beta-tester and co-developer base, almost every problem will be characterized quickly and the fix obvious to someone.

Given the CIA's purposeful destruction of most records, its failure to follow informed consent protocols with thousands of participants, the uncontrolled nature of the experiments, and the lack of follow-up data, the full impact of MKUltra experiments, including deaths, will never be known.

Given sufficient resources for growth, a biofilm will quickly grow to be macroscopic ( visible to the naked eye ).

Given the objectivity and lowered costs of satellite-based land cover analysis, it appears likely that remote sensing technology will be an integral part of assessing the extent and damage of deforestation in the basin.

Given a grammar in GNF and a derivable string in the grammar with length n, any top-down parser will halt at depth n.

Given expectations about returns on fixed investment, every level of the real interest rate ( i ) will generate a certain level of planned fixed investment and other interest-sensitive spending: lower interest rates encourage higher fixed investment and the like.