Given an asymmetric subunit on a triangular face of a regular isosahedron, with three subunits per face 60 such subunits can be placed in an equivalent manner.

Given that a natural language such as English contains, at any given time, a finite number of words, any comprehensive list of definitions must either be circular or rely upon primitive notions.

Given a simple, binary component feed, analytical methods such as the McCabe-Thiele method or the Fenske equation can be used.

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

Given good circumstances one might be able to discern the result of some human activity such as the changing of the Netherlands ' coast or the partial drying out of the Aral Sea, but even that would not be easy.

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 Greek fleet's unpreparedness resulting from the premature outbreak of the war, such an early Ottoman attack might well have been able to achieve a crucial victory.

Given a set S with a partial order ≤, an infinite descending chain is a chain V that is a subset of S upon which ≤ defines a total order such that V has no least element, that is, an element m such that for all elements n in V it holds that m ≤ n.

Given the toleration of Hinduism, such imperial leeway was not shocking.

Given a complete set of axioms ( see below for one such set ), modus ponens is sufficient to prove all other argument forms in propositional logic, and so we may think of them as derivative.

Given the need for such fine regulation of Ca < sup > 2 +</ sup > signaling, it is perhaps unsurprising that dysregulated mitochondrial Ca < sup > 2 +</ sup > has been implicated in several neurodegenerative diseases, while the catalogue of tumor suppressors includes a few that are enriched at the MAM.

Given its continental origin as a fragment of Zealandia, unlike many of the islands of the Pacific such as the Hawaiian chain, New Caledonia is not of geographically recent volcanic provenance.

* Named entity recognition ( NER ): Given a stream of text, determine which items in the text map to proper names, such as people or places, and what the type of each such name is ( e. g. person, location, organization ).

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 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 Poseidon's connection with horses as well as the sea, and the landlocked situation of the likely Indo-European homeland, Nobuo Komita has proposed that Poseidon was originally an aristocratic Indo-European horse-god who was then assimilated to Near Eastern aquatic deities when the basis of the Greek livelihood shifted from the land to the sea, or a god of fresh waters who was assigned a secondary role as god of the sea, where he overwhelmed the original Aegean sea deities such as Proteus and Nereus.

Given a preorder on S one may define an equivalence relation ~ on S such that a ~ b if and only if a b and b a.

Given X such that

Given that the authors were the primary developers of the Unix operating system, it is natural that QED had a strong influence on the classic UNIX text editors ed, sed and their descendants such as ex and sam.

Given that in such a case pits the freedom of an innocent child against the freedom of the evil-doer, it is not clear why God would not intervene for the sake of the child.

Given two subspaces with, this leads to a definition of angles called canonical or principal angles between subspaces.

Given the definition of above, we might fix ( or ' bind ') the first argument, producing a function of type.

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 the definition of morpheme as " the smallest meaningful unit " Nanosyntax aims to account for idioms where it is often an entire syntactic tree which contributes " the smallest meaningful unit.

Given a hierarchical netlist, the list of instance names in a path from the root definition to a primitive instance specifies the single unique path to that primitive.

Bernhard Riemann in his memoir " On the Number of Primes Less Than a Given Magnitude " published in 1859 extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation and established a relation between its zeros and the distribution of prime numbers.

Given the rapid orbital decay of objects below approximately, the commonly accepted definition for LEO is between and above the Earth's surface.

Given the philosophical definition that knowledge differs from belief ( knowledge is often defined as a justified, true belief ), Dorff's works explicitly analyze epistemological questions.

Given that the lapse rate is not a conservative quantity when the tropopause is considered for stratosphere-troposphere exchanges studies, there exists an alternative definition named dynamic tropopause.

Given this definition of matter, we can distinguish between what Aristotle calls " proximate matter " and what he calls " non-proximate matter ".

Given this definition, the following are all equivalent:

Given this definition, there are many publics ; each of them comes into being when an issue arises and ceases to exist when the issue is resolved.

Given that the logit is not intuitive, researchers are likely to focus on a predictor's effect on the exponential function of the regression coefficient-the odds ratio ( see definition ).

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 an orientable Haken manifold M, by definition it contains an orientable, incompressible surface S. Take the regular neighborhood of S and delete its interior from M. In effect, we've cut M along the surface S. ( This is analogous, in one less dimension, to cutting a surface along a circle or arc.

Given a homogeneous prime ideal P of, let X be a subset of P < sup > n </ sup >( k ) consisting of all roots of polynomials in P .< ref > The definition makes sense since if and only if for any nonzero λ in k .</ ref > Here we show X admits a structure of variety by showing locally it is an affine variety.

Given this definition,

Given that modern definition, a verse that condemns fornication ( such as which is often cited by conservative denominations as biblical opposition to pre-marital sex ) would appear to be clear.

Given the width and definition of the window function w ( t ), we initially require the height of the window function to be scaled so that

" Given that involuntarily celibate individuals by definition are compelled by circumstance, rather than choice, towards the state of not having a sex life, Galloway's proclamations may be viewed psychoanalytically as exhibiting some or another form of self-justifying behavior, but may be significant nevertheless.

Given the philosophical definition that knowledge differs from belief ( knowledge is often defined as a justified, true belief ), Gillman's works explicitly analyze epistemological questions.

Given the definition of E-democracy this act is seen as something that encourages citizens to get their own opinions in and possibly help write new legislation.

Given a method for establishing the satisfaction ( or not ) of every atomic sentence of the form A (... x < sub > k </ sub >...), the usual rules for truth-functional connectives and quantifiers yield a definition for the satisfaction condition of all sentences of the object language.

Given this definition, respondents were asked how many promotions they had received over the past 10 years.