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 a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers.

* Voronoi diagram: Given a set of points, partition the space according to which points are closest to the given points.

Given the similarities with readings from people living on loess soils, the general direction of the local movement according to Price et al., is from the northeast to the southwest.

Given a pair of elastic moduli, all other elastic moduli can be calculated according to formulas in the table below at the end of page.

Given their perceived piety, The Times noted the irony when a major investigation by their reporters uncovered a link between Islamic Jihadis and child pornography ; a discovery that, according to the London paper, " is expected to improve understanding of the mindsets of both types of criminals and has been hailed as a potentially vital intelligence tool to undermine future terrorist plots .".

Given a set of independent identically distributed data points where according to some probability distribution parameterized by θ, where θ itself is a random variable described by a distribution, i. e. the marginal likelihood in general asks what the probability is, where θ has been marginalized out ( integrated out ):

Given that the C-series is, like the B-series, a set of events cast in the same sequence forever, it is also unchanging, according to McTaggart.

Given that the air velocity is reduced to zero without losses the pressure increase can be calculated according.

Given a target optimal product mix, an independent oil refiner can attempt to hedge itself against adverse price movements by buying oil futures and selling futures for its primary refined products according to the proportions of its optimal mix.

: SIA: Given the fact that you exist, you should ( other things equal ) favor hypotheses according to which many observers exist over hypotheses on which few observers exist.

Given that the virus did not evolve into two separate strains until around 1900, and given how many transfers of fish there were, according to this research, " At present it is therefore not possible to suggest a geographical origin of the ISA virus.

Given the conditions, the organizers declared that each team member only needed to complete a minimum of 1 lap to qualify for a placing ( normally a minimum of 2 laps are required ) and that teams retiring early ( ie before the full 24 hours had elapsed ) would be placed according to the number of laps.

Given a surface through which a metal wire passes, electrons move in both directions across the surface at an equal rate.

Given this basic definition, further criteria are needed to establish a dividing line between postlexical clitics and morphological affixes, since both are characterized by a lack of prosodic autonomy.

Given two manifolds M and N, a bijective map f from M to N is called a diffeomorphism if both

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 this kind of data, the estimated coefficient on Years of Education in the equation above reflects both the effect of education on wages and the effect of other variables on wages, if those other variables were correlated with education.

Given the post-war shortages, both parties agreed not to publish it.

Given both Einstein's equations and suitable equations for the properties of matter, such a solution consists of a specific semi-Riemannian manifold ( usually defined by giving the metric in specific coordinates ), and specific matter fields defined on that manifold.

Given the dual status of science as objective knowledge and as a human construct, good historiography of science draws on the historical methods of both intellectual history and social history.

Given sufficient travel time and engineering work, both unmanned and generational interstellar travel seem possible, though present considerable technological and economic challenges are unlikely to be met in the near future, particularly for manned probes.

* Given a binary operation, an idempotent element ( or simply an idempotent ) for the operation is a value for which the operation, when given that value for both of its operands, gives the value as the result.

Given the realities of the Cold War, German reunification and recovery of lost territories in the east were not realistic goals as both of Stalin's notes specified the retention of the existing " Potsdam "- decreed boundaries of Germany.

Given that both A and not-A are seen to be “ true ,” Kant concludes that it ’ s not that “ God doesn ’ t exist ” but that there is something wrong with how we are asking questions about God and how we have been using our rational faculties to talk about universals ever since Plato got us started on this track!

# Given any two distinct points, there is exactly one line incident with both of them.

# Given any two distinct lines, there is exactly one point incident with both of them.

Given any of the above definitions of " myth ", the myths of many religions, both ancient and modern, share common elements.

Given the versatility of both vegetables, they were usually boiled or steamed, and eaten by itself or as ingredients in soups and strews.

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

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.

* Shay Given – former goalkeeper for both the Republic of Ireland and Newcastle United F. C ..

" Given their penchant for blacksmithing, many scholars believe the legend of the Cyclopes ' single eye arose from an actual practice of blacksmiths wearing an eyepatch over one eye to prevent flying sparks from blinding them in both eyes.

Given the type of warfare that had been carried out in the conflict — wherein highly mobile groups of soldiers and rebels continued to move to and from on the desert terrain, retreating at a time and then suddenly attacking to regain control of small towns and villages in the Eastern rebel held parts of Libya — had led to the technical becoming a vehicle of choice for both sides.

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 the tense nature of racial tension in America during the time of the film's production, Poitier felt he was " under close observation by both " Tracy and Hepburn during their first dinner meetings prior to production.

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.

He wrote three other books on names: A Concise Dictionary of American Place-Names ( 1970 ), Names on the Globe ( 1975 ), and American Given Names ( 1979 ).

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.

: Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains exactly one element in common with each of the sets in X.

* Given an algebraic number, there is a unique monic polynomial ( with rational coefficients ) of least degree that has the number as a root.

Given any element x of X, there is a function f < sup > x </ sup >, or f ( x ,·), from Y to Z, given by f < sup > x </ sup >( y ) := f ( x, y ).

Given that many journeys are for relatively short distances, there is considerable scope to replace car use with walking or cycling, though in many settings this may require some infrastructure modification, particularly to attract the less experienced and confident.

Given a subset X of a manifold M and a subset Y of a manifold N, a function f: X → Y is said to be smooth if for all p in X there is a neighborhood of p and a smooth function g: U → N such that the restrictions agree ( note that g is an extension of f ).

Given its different forms, there are various ways of representing uncertainty and modelling economic agents ' responses to it.

* Given any set X, there is an equivalence relation over the set of all possible functions X → X.

* Given a transformation group G over A, there exists an equivalence relation ~ over A, whose equivalence classes are the orbits of G.

Given a left neutral element and for any given then A4 ’ says there exists an such that.

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

Given the universality of free fall, there is no observable distinction between inertial motion and motion under the influence of the gravitational force.

Given the existence of a Godlike object in one world, proven above, we may conclude that there is a Godlike object in every possible world, as required.

Given that non-violence has priority, all other principles yield to it whenever there is a conflict.

Given this assumption, there are four categories in which a firm's profit may be considered to be.

Given the overall size of trade between Mexico and the United States, there are remarkably few trade disputes, involving relatively small dollar amounts.

Given the above commonalities there appear to be only two string theories: the heterotic string theory ( which is also the type I string theory ) and the type II theory.

Given infinite space, there would, in fact, be an infinite number of Hubble volumes identical to ours in the universe.

Given the diversity of functions performed by neurons in different parts of the nervous system, there is, as expected, a wide variety in the shape, size, and electrochemical properties of neurons.

: Given any positive number ε, there is a sequence

Given a base for the topology, in order to prove convergence of a net it is necessary and sufficient to prove that there exists some point x, such that ( x < sub > α </ sub >) is eventually in all members of the base containing this putative limit.

Given the immense expanse of the entire Universe, it has been argued that there is a higher probability that there exists ( or has existed ) another Earth-like planet that has yielded life ( geogenesis ) than not.

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 ).