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

* 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 a set S and two binary operators · and + on S, we say that the operation ·

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

Given a bounded lattice with largest and smallest elements 1 and 0, and a binary operation, these together form a Heyting algebra if and only if the following hold:

Given a set with three binary operations and, and two distinguished elements 0 and 1, then is a Heyting algebra for these operations ( and the relation defined by the condition that when ) if and only if the following conditions hold for any elements and of:

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

Given a binary input, B, the corresponding gray code, G, is given by " G

Given a labelled state transition system ( S, Λ, →), a simulation relation is a binary relation R over S ( i. e. R ⊆ S × S ) such that for every pair of elements p, q ∈ S, if ( p, q )∈ R then for all α ∈ Λ, and for all p ' ∈ S,

Given a labelled state transition system (, Λ, →), a bisimulation relation is a binary relation over ( i. e., ⊆ × ) such that both < sup >- 1 </ sup > and are simulations.

Given a labelled state transition system, a bisimulation relation is a binary relation over ( i. e., ⊆ × ) such that

Given a labelled state transition system (, Λ, →), define to be a function from binary relations over to binary relations over, as follows:

Given that there is only one binary operation, distributivity obviously cannot be defined in the standard way.

* Given two formulas X and Y, and a binary connective b ( such as the logical conjunction ∧), then ( X b Y ) is a formula.

Given two objects, A and B, each with n binary attributes, the Jaccard coefficient is a useful measure of the overlap that A and B share with their attributes.

Given a binary relation R between fixed strings in the alphabet, called rewrite rules, denoted by, an SRS extends the rewriting relation to all strings in which the left-and right-hand side of the rules appear as substrings, that is, where s, t, u, and v are strings.

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

Given that most of the Indian targets ( Halwara, Pathankot and Adampur ) were deep into enemy territory only a dozen or so commandos made it back alive and the stealth operation proved ineffective.

Given the inherent goodness of Ṛta and its absolute power over the operation of the universe, the presence of gross inequality and injustice in the world represented a serious religious, philosophical and ethical dilemma.

Given three vertices v, u, and w, where ( v, u ) and ( u, w ) are edges in the graph, the lifting of vuw, or equivalent of ( v, u ), ( u, w ) is the operation that deletes the two edges ( v, u ) and ( u, w ) and adds the edge ( u, w ).

Given a Boolean algebra B, we turn B into a Boolean ring by using the symmetric difference as addition and the meet operation as multiplication.

Given an abelian group and two commuting automorphisms φ and ψ of, define an operation on by

Given the order to “ do something about it ,” Carr took baseball's faltering minor league system and transformed it into a healthy 41-league operation by 1939.

Given four operators ( F, G, U, and V ) together with a product operation (∙) defined for any pair of these four operators, and given that the following four pairs of operators commute:

An identity involving four operators ( F, G, U, and V ) and a product operation (·) has been pointed out by L. J. Landau: Given that the following four pairs of operators commute:

Given the imminent liberation of Paris by the Allies, they need only delay the train for a few days — still, it is an extremely dangerous operation and it must be done in such a way that does not risk damaging the priceless cargo.

Given the ( then highly unusual ) driver-only operation, the branch line was equipped with a two wire emergency telephone system at window level, a feature normally found only in tunnels on the London Underground.

Given the costs and resources involved in the construction and operation of manufacturing systems, the engineering process must be made more efficient.

Given two abelian groups ( A, ∗) and ( B, ·), their direct sum A ⊕ B is the same as their direct product, i. e. its underlying set is the Cartesian product A × B with the group operation ○ given componentwise:

Given a fertile mind and boundless energy, he pursued other fields of interest: growing vegetables in Puerto Rico ; owning a beautician school, restaurants, lounges, a marina, marine dredging, data processing services, multi-level marketing of vitamins and cosmetics, cement block manufacturing, country-western music publishing and recording studios, telecommunications companies, a high-end boutique shop, general insurance agency, internet connection and data center, a walk-in medical clinic, wholesale gasoline and diesel fuel distribution operation, a travel agency, home security monitoring, mail order medical and diabetic supply company, a Big & Tall men's clothing store, an AM radio station, a mobile medical screening company, a book publishing company, a warehouse fulfillment and distribution company, an advertising media consulting firm, television stations, mobile home parks, a 225 room hotel in Nassau, Bahamas, a 500, 000 square foot commercial warehouse complex in Las Vegas, and a five-star 23 story resort hotel and residences on Ft. Lauderdale Beach.

Given the unique nature of model aviation insofar as it requires both airspace as well as frequency allocation for radio control, the AMA serves as a liaison with the Federal Aviation Administration concerning aeromodeling safety and operation of model aircraft as it relates to full-scale aviation, most recently with the upcoming likelihood of the start of non-military, commercial and public-use UAV deployment in the National Airspace System over the United States, and how the FAA will regulate such commercial and public use of UAVs, simultaneous with accommodating the needs of recreational and sport flying of radio control model aircraft, with the AMA as an advocate for the aeromodeler.

The cylindrification operation takes a relation as input of arity k and outputs a relation of arity k + 1 as follows: Given a relation R of arity K, its cylindrification denoted by c ( R ), is the following set

* 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 the complexity of the calculations involved and the convoluted structure that a convertible bond can have, an arbitrageur often relies on sophisticated quantitative models in order to identify bonds that are trading cheap versus their theoretical value.

Given this, the bidding is said to start at the one-level when contracting for a total of seven tricks, at the two-level for eight tricks and so on to the seven-level to contract to take all thirteen tricks.

Given the federal nature of the holiday, celebrating Canada Day can be a cause of friction in the province of Quebec, where the holiday is overshadowed by Quebec's National Holiday, on June 24.

Given the rugged terrain of Ngazidja and Nzwani, and the dedication of extensive tracts to agriculture on all three islands, population pressures on Comoros are becoming increasingly critical.

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 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.

* Frederick Douglass lecture on Haiti – Given at the World's Fair in Chicago, January 1893.

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 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 difficulty of finding exact solutions, Einstein's field equations are also solved frequently by numerical integration on a computer, or by considering small perturbations of exact solutions.

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 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 that this etymology does not appear in the usual scholarly works on etymology, this claim is usually dismissed as a false etymology.

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 its heavy dependence on imported energy, Japan has aimed to diversify its sources.

Along with a hand-picked escort led by Given Campbell, Davis and his wife were captured on May 10, 1865, at Irwinville in Irwin County, Georgia.

Given the repeated failures and frustrations of workers ' revolutions and movements, Marx also sought to understand capitalism, and spent a great deal of time in the reading room of the British Museum studying and reflecting on the works of political economists and on economic data.

Given that estimation is undertaken on the basis of a least squares analysis, estimates of the unknown parameters β < sub > j </ sub > are determined by minimising a sum of squares function

Given that spinning a Dyson Sphere would result in the atmosphere pooling around the equator, the Ringworld removes all the extraneous parts of the structure, leaving a spinning band landscaped on the sun-facing side, with the atmosphere and inhabitants kept in place through centrifugal force and 1000 mile high perimeter walls ( rim walls ).