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 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 a field ordering ≤ as in Def 1, the elements such that x ≥ 0 forms a positive cone of F. Conversely, given a positive cone P of F as in Def 2, one can associate a total ordering ≤< sub > P </ sub > by setting x ≤ y to mean y − x ∈ P. This total ordering ≤< sub > P </ sub > satisfies the properties of Def 1.

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 the reputation of oat grain to have invigorating properties and the obvious connection between plant seeds and human " seed ", it is not surprising that the meaning of the phrase shifted towards more or less explicitly referring to the destructive sexual liaisons of an unmarried young male, possibly resulting in unwanted children born out of wedlock.

Given a commutative diagram, a proof by diagram chasing involves the formal use of the properties of the diagram, such as injective or surjective maps, or exact sequences.

Given an isolated physical system, the allowed states of this system ( i. e. the states the system could occupy without violating the laws of physics ) are part of a Hilbert space H. Some properties of such a space are

Given a database or a set of rules, the computer tries to find a solution matching all the desired properties.

Given that X, Y, and Z are sets of attributes in a relation R, one can derive several properties of functional dependencies.

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

Given that they were dependent on the dogs, which gave them an advantage over other cattle breeders, it is understandable that the dogs were not distributed beyond the Hall's properties.

Euler generalised the problem considerably, and his ideas were taken up years later by Bernhard Riemann in his seminal 1859 paper On the Number of Primes Less Than a Given Magnitude, in which he defined his zeta function and proved its basic properties.

Given the above rules, mirrors, including half-silvered mirrors, have the following properties:

Given these properties of DNA damage and mutation, it can be seen that DNA damages are a special problem in non-dividing or slowly dividing cells, where unrepaired damages will tend to accumulate over time.

Given a vector space V over a subfield F of the complex numbers, a norm on V is a function with the following properties:

Given the selective properties of gillnet fishing, alternative methods of harvest are currently being studied.

Given a finite set of n properties, a thing is good if it is perceived to have all of the properties, fair if it has more than < sup > n </ sup >/< sub > 2 </ sub > of them, average if < sup > n </ sup >/< sub > 2 </ sub > of them, and bad if it has fewer than < sup > n </ sup >/< sub > 2 </ sub >.

Given these properties, it is not surprising that a strong correlation between Wnt signaling and the onset of cancer exists.

Given an undirected graph G = ( V, E ), a set of random variables X = ( X < sub > v </ sub >)< sub > v ∈ V </ sub > indexed by V form a Markov random field with respect to G if they satisfy the following equivalent Markov properties:

Given the difficulty of constructing explicit small families of solutions, much less presenting something like a " general " solution to the Einstein field equation, or even a " general " solution to the vacuum field equation, a very reasonable approach is to try to find qualitative properties which hold for all solutions, or at least for all vacuum solutions.

Given its rich nutrient properties, tomatoes, cucumbers and melons are grown with noteworthy results.

) Given the fact the novel's text is described as having been translated from " a language that has not yet achieved existence ", the substance referred to may not in fact be mercury, although its properties would seem to suggest that it is.

Given a Lorentzian manifold, we can find infinitely many frame fields, even if we require additional properties such as inertial motion.

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

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 the prominence of the concept of Pax Romana, historians have coined variants of the term to describe systems of relative peace that have been established, attempted or argued to have existed.

Given these objections to creole as a concept, articles such as Against Creole Exceptionalism and Deconstructing Creole have arisen which question the idea that creoles are exceptional in any meaningful way.

" Given the situation in Spain for several years regarding terrorist attacks, these ties can be considered objectively as a threat to democracy ," the court said, also adding that these parties " contradicted the concept of a ' democratic society ' and presented a major danger to Spain's democracy ".

Given the introduction of ligand concept, he was also able to further derive the idea of bite angle and other aspects of chelation.

Given that there are many historical definitions of the keystone species concept, and without a consensus on its exact definition, a list of examples best illustrates the concept of keystone species.

Given a vector bundle V over M, the corresponding field concept is called a section of the bundle: for m varying over M, a choice of vector

Given the utility of the design in combat in China, the Imperial Japanese Army was therefore content to retain the Type 94, although the design, and indeed the concept of the tankette, came to be regarded as obsolescent in Western armies.

Given Bonpensiero's middle adulthood, even the concept of facing the minimum thirty years in jail would have been an effective life sentence.

Given the extremes of life in the Urban Monads, law enforcement and the concept of justice employ a zero tolerance policy.

Given his changing definition of incommensurability Pérez Ransanz has identified three phases in Kuhn's work, or at least in how it deals with this concept.

Given an unramified finite extension L / K of local fields, there is a concept of Frobenius endomorphism which induces the Frobenius endomorphism in the corresponding extension of residue fields.

Given the large revenue generated by merchandise featuring Cusick's Dalek design, some feel that he should have been paid a royalty ( as was script writer Terry Nation, who created the concept of the Daleks but not their design or appearance ).

Given the tolerance, respect and understanding that Buddha's teaching observe, and more specifically the respect for other members, the concept of prostitution is highly ill-advised in order to follow the Fourth Noble Truths.

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 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.