* Given any topological space X, the continuous real-or complex-valued functions on X form a real or complex unitary associative algebra ; here the functions are added and multiplied pointwise.

Given a finite dimensional real quadratic space with quadratic form, the geometric algebra for this quadratic space is the Clifford algebra Cℓ ( V, Q ).

Given a vector space V over the field R of real numbers, a function is called sublinear if

Given a function f of a real variable x and an interval of the real line, the definite integral

Given the tendency for real negotiations in Japan to be conducted privately ( in the nemawashi, or root binding, process of consensus building ), the shingikai often represented a fairly advanced stage in policy formulation in which relatively minor differences could be thrashed out and the resulting decisions couched in language acceptable to all.

Given m real values t < sub > i </ sub >, called knots, with

Given metric spaces ( X, d < sub > 1 </ sub >) and ( Y, d < sub > 2 </ sub >), a function f: X → Y is called uniformly continuous if for every real number ε > 0 there exists δ > 0 such that for every x, y ∈ X with d < sub > 1 </ sub >( x, y ) < δ, we have that d < sub > 2 </ sub >( f ( x ), f ( y )) < ε.

Given two metric spaces ( X, d < sub > X </ sub >) and ( Y, d < sub > Y </ sub >), where d < sub > X </ sub > denotes the metric on the set X and d < sub > Y </ sub > is the metric on set Y ( for example, Y might be the set of real numbers R with the metric d < sub > Y </ sub >( x, y )

: Given: a function f: A R from some set A to the real numbers

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.

Given two C < sup > k </ sup >- vector fields V, W defined on S and a real valued C < sup > k </ sup >- function f defined on S, the two operations scalar multiplication and vector addition

Given real life prisons are same-sex communities, this fetish does lend itself to male on male or female on female activities and settings.

Given that the Liber historiae Francorum admits four years for Chlothar's reign, it remains possible that Chlothar began to exert some real authority when his mentors deemed him an adult, 669.

Given an ordered sequence of real numbers: the first difference is defined as

Given that his predecessors were monarchists who tried without success to restore the French monarchy, Grévy is seen as the first real republican President of France.

Given the variety of approaches, Economic Geography has taken to many different subject matters, including: the location of industries, economies of agglomeration ( also known as " linkages "), transportation, international trade, economic development, real estate, gentrification, ethnic economies, gendered economies, core-periphery theory, the economics of urban form, the relationship between the environment and the economy ( tying into a long history of geographers studying culture-environment interaction ), and globalization.

Given the nature of this interrogation and the motive behind it, it is not known for certain what elements of Number Six's life so portrayed are real and which are fiction.

Given two normed vector spaces V and W ( over the same base field, either the real numbers R or the complex numbers C ), a linear map A: V → W is continuous if and only if there exists a real number c such that

Given a real dynamical system ( R, M, Φ ), I ( x ) is an open interval in the real numbers, that is.

* Given a positive real number ε, an ε-isometry or almost isometry ( also called a Hausdorff approximation ) is a map between metric spaces such that

Given an m × n matrix with real entries ( or entries from any other field ) and rank r, then there exists at least one non-zero r × r minor, while all larger minors are zero.

Given key responsibilities for the country's diplomacy in the Napoleonic Wars, he was responsible for planning the attack on Copenhagen in September 1807, much of which he undertook at his country estate, South Hill Park at Easthampstead in Berkshire.

Given the choice of a harbour or a new church, in 1875 the islanders asked the estate to provide a place of worship ; a Methodist chapel was built.

Given the town's proximity to Monaco, real estate in Beausoleil is usually prohibitively expensive for people who work in the town.

Given the ensuing decline in world commodity prices, countries heavily dependent on the export of raw materials such as oil were among those most severely hit.

Given the very conservative engineering of the early IBM personal computers and their higher than average prices, this was not a terribly difficult task at first, bar only the great technical challenge of crafting a BIOS that duplicated the function of the IBM BIOS exactly but did not infringe on copyrights.

Given the guaranteed low prices during contract days and the continued high cost of logging in Southeast Alaska today, one analysis concludes that, since 1980, the forest service has lost over a billion dollars in Tongass timber sales.

Given that the problem of high textbook prices is linked to the " broken " economics of the market, requiring publishers to disclose textbook prices to faculty is a solution pursued by a number of legislatures.

Given that the program was cancelled in 1986, the NEP was active for five years which are amongst the most expensive for oil prices on record and the NEP prevented Alberta's economy from fully realising those prices. Fluctuations: Oil Prices & Alberta Per Capita Federal Contributions 1975-1981

Given the fairly unsophisticated stock research of the era, the company appeared to be growing without bound, and its share prices rose.

Given a consumer's utility function, prices, and a utility target,

Given its easy access to Dublin City and relatively modest house prices compared to older more established areas in closer proximity to Dublin City many thousands of homes were built in a matter of years.

The optimization attempts to answer the question: " Given our operating constraints, what is the best mix of products and / or services for us to produce and sell in the period, and at what prices, to generate the highest expected revenue?

Given the Rover's equipment, prices were reasonably competitive in the large family car segment and considerably lower than the price of such compact executive cars such as the BMW 3 Series and Audi A4.

Given the pub's low prices and proximity to University College London, it is very popular with UCL students.

Given SP's history of encouraging settlement and development along its lines, land prices were expected to appreciate considerably.

Given that the bulk of family chores fell to women, and that they were five times more likely than men to head a single-parent family, the shortcomings of Finland's child day-care system affected women more than it did men.

Given the date of his publication and the widespread, permanent distribution of his work, it appears that he should be regarded as the originator of the concept of space sailing by light pressure, although he did not develop the concept further.

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 the range of his office-holding both regionally and at court, he did not need to draw ruinously on his own resources to dispense patronage on a grand scale and he was active in the arbitration of local disputes ; even state matters were regularly referred for his personal adjudication.

Given this s, we can compute the coordinates of 2 ( 2P ), just as we did above: 4P =( 259851, 116255 ).

Dr. Tennison states: " Given the account of Elliot Paul, and given that Lead Belly witnessed boogie-woogie in 1899 in the Arklatex ; and given the North to South migration of the Thomas family ; and given the Texas & Pacific headquarters in Marshall in the early 1870s ; and given that Harrison County had the largest slave population in the state of Texas ; and given the fact that the best-documented and largest-scale turpentine camps in Texas did not occur until after 1900 in Southeast Texas, it is most probable that boogie-woogie spread from Northeast to Southeast Texas, rather than from Southeast to Northeast Texas, or by having developed diffusely with an even density over all of the Piney Woods of East Texas.

Given the prohibition on sacrifices for sin contained in III Nephi 9: 19-20 ( Book of Mormon ), Strang did not require sin offerings.

Given that Orion's assets did not seem to merit the price paid, and that control of the company would have remained in friendly hands even without the buyout of Redstone, Wall Street observers were puzzled by Kluge's expenditure.

Given that the USA maintained diplomatic relations with Taiwan Republic of China since the end of World War II and did not establish recognition of so-called ' Red ' China or People's Republic of China until the latter part of the twentieth century, the introduction of Taiwanese style racing to America corresponded to the close business and social contacts between these two nations.

# Given an artifact whose performance has been evaluated, it is important to determine why and how the artifact worked or did not work within its environment.

Given the high death rate, many servants did not live to the end of their terms.

Given the lack of budget, in many regions systematization did not constitute an effective plan, good or bad, for development.

Given his later prowess, it is rather curious to note that during his martial training he apparently did not win many bouts.

Given, at her own request, a little freedom, she sprang quickly into the fire, but miracolously the fire did not do harm her.

Given this dismal track record there is no evidence that had Taichang reign lasted any longer than it did, he could have turned around the fortunes of the beleaguered Ming Dynasty after the long steady decline of the latter years of Wanli's reign.

Given the shortage of workers and consequent price rises the Ordinance of Labourers 1349 and the Statute of Labourers 1351 attempted to suppress sources of wage inflation by banning workers organisation, creating offences for any able-bodied person that did not work, and fixing wages at pre-plague levels.

Given the differences in facilities and intangibles such as the distance of the new school from Austin and other law students, the Supreme Court ruled it did not satisfy " separate but equal " provisions, and that African Americans must also be admitted to the University of Texas Law School at Austin.

Given that Jack the Ripper actually did show up on the station last year, the possibility that he is telling the truth cannot be dismissed out of hand.

Given did make his 400th appearance for the club during the season, however, in a New Year's Day match.

Given that the crinoline did eventually have a maximum diameter of up to 180 centimetres ( six feet ), it is easy to imagine difficulties in getting through doors, in and out of carriages, and the general problems of moving in such a large structure.

Given this poor performance, the USAAF placed its most experienced and accurate bombardiers aboard each formation's lead bombers ( primary and deputy lead ) and had the remaining airplanes open their bomb bays and drop their bombs when the leader did.

Given WSRP4J's incubator status, the project did not produce formal releases.

Given the costs of the road improvements, the monument itself besides the splendid funeral, and that the corporation it seems did not match a penny to these expenses having control of the Surfboat-men ’ s money, it is a lamentable indictment to Thanet ’ s officials that although their still should have been money enough left over, had it been wisely invested, to have yielded an income for life to the widows and orphans, who in the end personally received nothing out of the fund.