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

Given the military strength of the Great Powers, these interventions were nearly always decisive and quickly ended the civil wars.

Given these connotations, dandyism can be seen as a political protestation against the rise of levelling egalitarian principles, often including nostalgic adherence to feudal or pre-industrial values, such as the ideals of " the perfect gentleman " or " the autonomous aristocrat ", though paradoxically, the dandy required an audience, as Susann Schmid observed in examining the " successfully marketed lives " of Oscar Wilde and Lord Byron, who exemplify the dandy's roles in the public sphere, both as writers and as personae providing sources of gossip and scandal.

Given these origins, there have been various suggestions over the years to rename the town ( for example, to " Invernevis ").

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, a new term Anthropocene, is specifically proposed and used informally for the latest part of this epoch since approximately synchronous lithospheric evidence, or more recently atmospheric evidence, of human impacts have been found on the Earth and its ecosystems ; these impacts may be considered of global significance for future evolution of living species.

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.

We have seen a few cavities of the appropriate size and shape for ivory-bills, but these can be old, or exceptionally large Pileated Woodpecker cavities, or mammal-enlarged Pileated Woodpecker cavities .… Given the results, it is unlikely a population of any meaningful size of Ivory-billed Woodpeckers exists in south Florida.

Given that the Qur ' an is vague regarding the punishment of homosexual sodomy, Islamic jurists turned to the collections of the hadith and seerah ( accounts of Muhammad's life ) to support their argument for Hudud punishment ; these are perfectly clear but particularly harsh.

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

" Given the elimination of these pernicious influences, Marx allowed that " normal conditions of spontaneous development " of the rural commune could exist.

Given the properties of a concept, these are necessary and sufficient conditions to classify the concept.

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 a memex, a scholar could create her own knowledge tools as connections within reams of information, share these tools, and use complexes of tools to create yet more sophisticated knowledge that could in turn be deployed toward this work.

Given these issues, Microsoft started to work in parallel on a version of Windows which was more future-oriented and more portable.

Given an arbitrary group G, there is a related profinite group G < sup >^</ sup >, the profinite completion of G. It is defined as the inverse limit of the groups G / N, where N runs through the normal subgroups in G of finite index ( these normal subgroups are partially ordered by inclusion, which translates into an inverse system of natural homomorphisms between the quotients ).

Given the difficulty in specifying exact distributions of sample statistics, many methods have been developed for approximating these.

Given that the validity of conclusions drawn from a statistical analysis depend on the validity of any assumptions made, it is clearly important that these assumptions should be reviewed at some stage.

" Given enough time, these formations can meet and fuse to create columns of calcium carbonate.

Given India's historic position that the original intent of the line was to separate the two nations by the highest mountains in the world, in these locations India extended its forward posts northward to the ridges, regarding this move as compliant with the original border proposal, although the Simla Convention did not explicitly state this intention.

Given Jerome's conservative methods, and that manuscript evidence from outside Egypt at this early date is very rare ; these Vulgate readings have considerable critical interest.

Given these facts, algorithm defined by the following pseudocode:

Given a model of some atomic positions, these positions and their respective Debye-Waller factors ( or B-factors, accounting for the thermal motion of the atom ) can be refined to fit the observed diffraction data, ideally yielding a better set of phases.

Given these assumptions, the constraint that time travel must not lead to inconsistent outcomes could be seen merely as a tautology, a self-evident truth that cannot possibly be false, because if you make the assumption that it is false this would lead to a logical paradox.

Given the radically different motivating assumptions, it may come as a surprise that ESSes and Nash equilibria often coincide.

Given the truth of the assumptions, a valid deduction guarantees the truth of the conclusion.

Given that the four trials under consideration can indeed be consistently considered and experimentally realized, the assumptions concerning hidden variables which lead to the indicated mathematical contradiction are therefore collectively unsuitable to represent all experimental results ; namely the assumption of local hidden variables which occur equally in distinct trials.

Given that a dichotomous criterion violates these assumptions of linear regression, conducting linear regression with a dichotomous criterion may lead to errors in inference and at the very least, interpretation of the outcome will not be straightforward.

Given the assumptions of the model ( for example, that the profits are reinvested and that capital accumulation does not substitute for skilled labor in production ), the process becomes self-sustaining and leads to modernization and economic development.

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 assumptions of biological positivism, the only reasonable rationale for punishing offenders is to incapacitate them for as long as possible so that they no longer posed a threat to the peace and security of society.

Given the assumptions of the Hotelling model, consumers will choose either firm as long as the combined price and transportation cost of the product is less than the competitive firm.

While tests have been done to support this with fixed pieces of flat metal, the result is inconclusive with respect to the curved armour of the period. Given the following actions of the archers, it seems likely Froissart was correct.

Given the above expression, evidently the result of her ( local ) measurement is that the three-particle state would collapse to one of the following four states ( with equal probability of obtaining each ):

Given the apparent magnitude (> 20 ) of all but the biggest trans-Neptunian objects, the physical studies are limited to the following:

Informally it solves the following problem: Given an integer N, find its prime factors.

A Clifford algebra Cℓ ( V, Q ) is a unital associative algebra over K together with a linear map satisfying for all defined by the following universal property: Given any associative algebra A over K and any linear map such that

Given a ring R and a proper ideal I of R ( that is I ≠ R ), I is a maximal ideal of R if any of the following equivalent conditions hold:

Given the following XML document:

Given the slow speed and the coarse resolution this was not a feasible technique for printing large images, but could usefully print a small logo onto a letterhead and then the following letter, all in a single unattended print run without changing the print element.

Given a general triangle ABC, the following conditions would need to be fulfilled for the case to be ambiguous:

Given all of the above premises are true, the angle B may be acute or obtuse ; meaning, one of the following is true:

Given that the UK Government opted not to pursue a tidal scheme in the Severn Estuary following the completion of the feasibility study, the chances of a third crossing anytime in the near future seem doubtful.

Given an SVD of M, as described above, the following two relations hold:

Given a natural number x, h outputs the index of the partial computable function that performs the following computation:

Given the fact that the following Popes John, from John XV ( 985 – 996 ) until John XIX ( 1024 – 1034 ), seemed to have neglected the existence of John XIV " bis ", Petrus Hispanus meant to correct this error by electing for himself the name John XXI.

Given a set S with three subsets, J, K, and L, the following holds:

Given this definition, the following are all equivalent:

Given a non-empty totally ordered set R with the following four properties:

Given the following:

De Tactionibus embraced the following general problem: Given three things ( points, straight lines, or circles ) in position, describe a circle passing through the given points and touching the given straight lines or circles.

Given Vanier's physical state, Pearson was hesitant to advise the Queen to act along those lines, but his worry was short lived, as the following day, after hearing the Holy Sacrifice of the Mass and receiving Holy Communion in the chapel, the Governor General died.

Given the preceding information, the knowledge gap hypothesis can be expressed using the following set of related propositions:

Given the angular velocity vector and velocity of the object, the resulting force can be calculated using the following formula:

Given a bounded lattice with largest and smallest elements 1 and 0, and a binary operation, these together form a Heyting algebra if and only if the following hold: