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 a specific task to solve, and a class of functions, learning means using a set of observations to find which solves the task in some optimal sense.

Given the fluid flow conditions and the desired shaft output speed, the specific speed can be calculated and an appropriate turbine design selected.

Given a general statement such as all ravens are black, a form of the same statement that refers to a specific observable instance of the general class would typically be considered to constitute evidence for that general statement.

Given a context pertaining to a specific OSI layer, PDU is sometimes used as a synonym for its representation at that layer.

Given that noise-equivalent power can be expressed as a function of the responsivity ( in units of or ) and the noise spectral density ( in units of or ) as, it's common to see the specific detectivity expressed as.

Given a day mandated for seasonal rejoicing ( Shemini Atzeret ) with no specific obligations as to the focus of the rejoicing ( such as sukkah, lulav or etrog ), it is not surprising that the celebration of the conclusion of the annual reading was absorbed into the celebration of Shemini Atzeret.

Given a rebbe's physical awareness of God, and the Rebbe's transcendent perception of Godliness, many hasidim take special care to observe the specific and sometimes minute practices of their Rebbe.

) Given the diffusion of different Piers / Ploughman texts, it is usually not possible to be certain about what someone means to refer to when they mention " Piers Plowman " unless they provide specific identifying details — and most writers do not.

Given the finite supply of natural resources at any specific cost and location, agriculture that is inefficient or damaging to needed resources may eventually exhaust the available resources or the ability to afford and acquire them.

The specific problems are: Given two, three or four points on a straight line, find another point on it such that its distances from the given points satisfy the condition that the square on one or the rectangle contained by two has a given ratio either ( 1 ) to the square on the remaining one or the rectangle contained by the remaining two or ( 2 ) to the rectangle contained by the remaining one and another given straight line.

Given that the tune was developed in an oral tradition, it is impossible to say for certain which of these influences may have played a specific role in the creation of this tune, but it is certain that numerous folk influences from different cultures such as these were prominent in the musical culture of the camp meeting, and that such influences were freely combined in the music-making that took place in the revival movement.

Given at the discretion of the Council of the Academy on the recommendation of a nominating committee, the prize is awarded to a person for his or her total literary achievement rather than for a specific work.

Given the specific gravity of a substance, its actual density can be calculated by rearranging the above formula:

Many of these procedures are under investigation in the U. S. Given that PRK is not reversible, a patient considering PRK is recommended to contact an eye-care practitioner for assistance in making an informed decision concerning the potential benefits and liabilities that may be specific to him or her.

Given the total mass and the scalars r and v at a single point of the orbit, one can compute the specific orbital energy, allowing an object orbiting a larger object to be classified as having not enough energy to remain in orbit, hence being " suborbital " ( a ballistic missile, for example ), having enough energy to be " orbital ", but without the possibility to complete a full orbit anyway because it eventually collides with the other body, or having enough energy to come from and / or go to infinity ( as a meteor, for example ).

Given the specific case of N being the product of distinct odd prime numbers p and q, the structure of the squaring map:

Given that field experiments necessarily take place in a specific geographic and political setting, the extent to which findings can be extrapolated to formulate a general theory regarding economic behavior is a concern.

" Given the Rules ' permissive backdrop and their inclusion of a specific rule on expert testimony that does not mention ' general acceptance ,' the assertion that the Rules somehow assimilated Frye is unconvincing.

Given the cost of an item, one can compute the selling price required to achieve a specific gross margin.

Given a finite symmetric group S < sub > n </ sub > and specific Young tableau λ corresponding to a numbered partition of n, define two permutation subgroups and of S < sub > n </ sub > as follows:

Given the large number of rDNA genes ( several hundreds ) available for transcription, the first mechanism involves adjustments in the number of genes being transcribed at a specific time.

: Given any family of nonempty sets, their Cartesian product is a nonempty set.

: 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 a set of integers, does some nonempty subset of them sum to 0?

Given any vector space V over a field F, the dual space V * is defined as the set of all linear maps ( linear functionals ).

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

Given also a measure on set, then, sometimes also denoted or, has as its vectors equivalence classes of measurable functions whose absolute value's-th power has finite integral, that is, functions for which one has

Given a trigonometric series f ( x ) with S as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that had S ' as its set of zeros, where S ' is the set of limit points of S. If p ( 1 ) is the set of limit points of S, then he could construct a trigonometric series whose zeros are p ( 1 ).

Given a topological space X, let G < sub > 0 </ sub > be the set X.

Given an equilateral triangle, the counterclockwise rotation by 120 ° around the center of the triangle " acts " on the set of vertices of the triangle by mapping every vertex to another one.

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 a binary operation ★ on a set S, an element x is said to be idempotent ( with respect to ★) if

Given a set

The knapsack problem or rucksack problem is a problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible.

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 a simple mathematical or functional description of an input or output to a system, the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications.

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 set of training examples of the form, a learning algorithm seeks a function, where is the input space and

Given a point x in a topological space, let N < sub > x </ sub > denote the set of all neighbourhoods containing x.

Given the similarities in the two characters ' names, professions, written works and generally dark subject matter, it is likely that Lovecraft's Alhazred provided the main inspiration for al-Hazir.

Given Chrétien's known tendency to create new stories and characters, being the first to mention the hero Lancelot and his love affair with Queen Guinevere for example, the name might also be entirely invented.

Given the book's fluid and changeable approach to plot and characters, a definitive, critically agreed-upon plot synopsis remains elusive ( see Critical response and themes: Difficulties of plot summary below ).

Given that much of the first half of the novel focuses on the initial encounter, blossoming friendship and eventual estrangement of these central characters, this issue continues to pique the curiosity of readers.

Given that each letter has uppercase and lowercase forms, and the number of various punctuation marks and ligatures ( e. g. the sequence ' fi ' combined in one character, commonly used in writing ) the Gutenberg Bible needed a set of 290 master characters.

Given the nature of Arthur's activities, he was always in the spotlight of the local police and crossed paths with several regular and occasional characters:

Given any finite flat commutative group scheme G over S, its Cartier dual is the group of characters, defined as the functor that takes any S-scheme T to the abelian group of group scheme homomorphisms from the base change G < sub > T </ sub > to G < sub > m, T </ sub > and any map of S-schemes to the canonical map of character groups.

Given that the details of characters ' physical appearances are not often given in His Dark Materials ( for example, it is a long time before it is mentioned that the protagonist, Lyra, is blonde ), this is an unusual attention to physical detail.

Given that he's more of a military man than a scientific one, his interests tend to conflict with the other main characters.

Given the characters ' blindness, some of these names seem sharply ironic (" the boy with the squint " or " the girl with the dark glasses ").

Given Hollywood's reluctance to hire Asian actors for substantial roles during that period ( with only a few reluctant exceptions, such as Anna May Wong, Sessue Hayakawa, and Philip Ahn ), he portrayed a variety of Asian characters in several movies before being offered the leading role in the 1929 film, The Mysterious Dr. Fu Manchu as the first onscreen portrayal of the title character.

Given the ordinary irreducible and irreducible Brauer characters of a particular finite group, the irreducible ordinary characters may be decomposed as non-negative integer combinations of the irreducible Brauer characters.

Given this world of " kill or be killed ", the main goals of characters in HoL are survival and escape.

Given the anticipated audience for a film, the screenwriter, director, or movie studio may wish to increase character time or invent new characters.

* Given the fact that the characters are animated, they do not have the physical limitations of the original puppets.

Given the large-scale interconnecting relationships in the series, this is a serious issue affecting most if not all of the characters.

Given that this population exhibits mixed characters, it has also been proposed that, rather than being a separate subspecies, it is actually a result of interbreeding between these two forms.

Given that he or she will be playing in pairs of characters, the other characters will be computer-controlled, or a second player can join in at any time.

Given such contradictions, these characters ' actual names and thus identities remain unclear.