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 such estimates of first-order derivatives, the gradient magnitude is then computed as:

Given that John the Baptist was executed before the defeat of Herod by Aretas IV of Nabatea in the AD 36, and based on the scholarly estimates for the approximate date of the marriage of Herod Antipas and Herodias AD 28-35, Josephus establishes a key connection between the time frame of the ministry of Jesus and his execution.

Given reasonable estimates of the time needed to compute the hash, this would take about 1 second to find.

Given that investments are subject to revisions of future expectations the Nerbrand Z utilises uncertainty of consensus estimates to assess how much earnings forecasts can be revised in standard deviation terms before P / E rations return to normalised levels.

Given that this mass is confined inside a 44 million km diameter sphere, this yields a density ten times higher than previous estimates.

Given that it took 46 years of construction, scholarly estimates for the Temple visit in the Gospel of John are around 27-29 AD.

Given the fact that these gestation lengths are only estimates of an average, it is helpful to consider gestation time as a range of dates rather than a single " due date ".

Given the many problems with the record, the dating of Lóegaire's floruit is imprecise, estimates placing it in the second half of the fifth century, circa 450 to perhaps the late 480s.

Given that observational < ref name = tak02 > and modelling of anthropogenic emissions of CO < sub > 2 </ sub > estimates suggest that the ocean is currently responsible for the uptake of approximately 2 Gt C y < sup >− 1 </ sup >, and that these estimates are poor for the shelf regions, the continental shelf pump may play an important role in the ocean's carbon cycle.

* Optical character recognition ( OCR ): Given an image representing printed text, determine the corresponding text.

Given any embedding of a Tychonoff space X in a compact Hausdorff space K the closure of the image of X in K is a compactification of X.

Given a bounded sequence, there exists a closed ball that contains the image of ( is a subset of the scalar field ).

Given that Brigid's cross is a prevalent swastika-like image in Ireland, there may be further links here between Verbeia, imported Gaulish cults, and the swastika image.

" Given that declaration, along with seven years of bombings and the wording of their communiques throughout that time that strove to present an image of a powerful organization spread secretly throughout all sectors of society, the authorities took significant action.

Given this outlook, whatever the level of affection Whistler may have felt for his own mother, one finds an even more divergent use of the image in the Victorian era and later, especially in the United States, as an icon for motherhood, affection for parents, and " family values " in general.

Given a ring R and a subset S, one wants to construct some ring R * and ring homomorphism from R to R *, such that the image of S consists of units ( invertible elements ) in R *.

Given an algebra, a homomorphism h thus defines two algebras homomorphic to, the image h () and The two are isomorphic, a result known as the homomorphic image theorem.

Given this closure property for CSAs, they form a monoid under tensor product, compatible with Brauer equivalence, and the Brauer classes are all invertible: the inverse class to that of an algebra A is the one containing the opposite algebra A < sup > op </ sup > ( the opposite ring with the same action by K since the image of K → A is in the center of A ).

Given only one share, a second share can be crafted to reveal any possible image ; therefore, individual shares reveal no information about the original image.

Given the image of a curve one can define several different parameterizations of the curve.

Given any injective immersion f: N → M the image of N in M can be uniquely given the structure of an immersed submanifold so that f: N → f ( N ) is a diffeomorphism.

Given any embedding f: N → M of a manifold N in M the image f ( N ) naturally has the structure of an embedded submanifold.

Given a large database of images ( in effect, a book of mug shots ) and a photograph, the problem was to select from the database a small set of records such that one of the image records matched the photograph.

# Given the purpose this image is being used for in the article, it is unlikely that any such image will be located in the future.

Given the existence of other raw image formats, Adobe's creation of DNG as a competing format implies that DNG is unusual and satisfies objectives that other raw image formats do not.

# Given the purpose this image is being used for in the article, it is unlikely that any such image will be located in the future ( copyright for departmental logos and the Royal Arms are protected by Letters Patent and will never expire under the terms of the Copyrights Designs and Patents Act 1988.

# Given the purpose this image is being used for in the article, it is unlikely that any such image will be located in the future ( copyright for departmental logos and the Royal Arms are protected by Letters Patent and will never expire under the terms of the Copyrights Designs and Patents Act 1988.

* Given the weight of the train and the gradients on the line, both brake-vans should have been at the rear of the train

Given the extraordinary competition and technological craftsmanship required for top search engine placement, the implication of the term " search engine optimization " has deteriorated over the last decade.

Given the current size of the Web, even large search engines cover only a portion of the publicly available part.

Given the choice, she chose Heracles over Zeus who went off in search of other pursuits.

Given the shortcomings of the beneficial acclimation hypothesis, researchers are continuing to search for a theory that will be supported by empirical data.

Given source material, Rabin – Karp can rapidly search through a paper for instances of sentences from the source material, ignoring details such as case and punctuation.

Given police permission to search Sobhraj's apartment ( a full month after the suspect had left the country ), Knippenberg found much evidence, such as victims ' documents and poison-laced medicines.

Given the ability to geocode street addresses and other features, one can create a lot of interesting spatial analysis, location-based service, political campaigning apps and localised search services.

According to Peter C. Newman ’ s analysis financing could not have come from investors, “ who would have had to be kept in ignorance of the project ’ s significance .” The government had the mine immediately “ drained and cemented … and employed prospectors to search for additional uranium deposits .” The miners hired to reopen Eldorado were screened by the RCMP and sworn to secrecy Given this notion it seems secrecy was conducted the same way uranium contracts had been allocated ; efficiency and development of an atomic weapon took precedent over political concerns like communist attitude amongst the workforce or homage to Britain.

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

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 a field F, the assertion “ F is algebraically closed ” is equivalent to other assertions:

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

" Given that Thomas Aquinas died six years before Albertus Magnus ' death, this legend as stated is unlikely.

Given Ealdred's known support of Godwin's family, John of Worcester is probably correct.

* Given any Banach space X, the continuous linear operators A: X → X form a unitary associative algebra ( using composition of operators as multiplication ); this is a Banach algebra.

* Given an R-module M, the endomorphism ring of M, denoted End < sub > R </ sub >( M ) is an R-algebra by defining ( r · φ )( x ) = r · φ ( x ).

Given the other ZF axioms, the axiom of regularity is equivalent to the axiom of induction.

After Christians in Ephesus first wrote to their counterparts recommending Apollos to them, he went to Achaia where Paul names him as an apostle ( 1 Cor 4: 6, 9-13 ) Given that Paul only saw himself as an apostle ' untimely born ' ( 1 Cor 15: 8 ) it is certain that Apollos became an apostle in the regular way ( as a witness to the risen Lord and commissioned by Jesus-1 Cor 15: 5-9 ; 1 Cor 9: 1 ).< ref > So the Alexandrian recension ; the text in < sup > 38 </ sup > and Codex Bezae indicate that Apollos went to Corinth.

Given its " blue-water " bias, the navy is even less inclined to become involved in counterdrug operations than the army or air force.

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 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.

While tests have been done to support this with fixed pieces of flat metal, the result is inconclusive with respect to the curved armour of the period. Given the following actions of the archers, it seems likely Froissart was correct.

Given a ket of norm 1, the orthogonal projection onto the subspace spanned by is

Given this influence, it is unfortunate that many of the details of his life remain shrouded in mystery, perhaps forever ; even the only known picture of him, shown above, is heavily retouched, with a fake tie painted in by hand.

Given points P < sub > 0 </ sub > and P < sub > 1 </ sub >, a linear Bézier curve is simply a straight line between those two points.

Given the then GDP ($ 7. 095 bln ) of the country, military spending is roughly estimated to be about $ 300 million.

Given some encrypted data (" ciphertext "), the goal of the cryptanalyst is to gain as much information as possible about the original, unencrypted data (" plaintext ").

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 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 such a sweeping definition, it is apparent that cognition is involved in everything a human being might possibly do ; that every psychological phenomenon is a cognitive phenomenon.