Given our formula φ, we group strings of quantifiers of one kind together in blocks:

Given that hearing impairments can vary by frequency and that audiograms are plotted with a logarithmic scale, the idea of a percentage of hearing loss is somewhat arbitrary, but where decibels of loss are converted via a recognized legal formula, it is possible to calculate a standardized " percentage of hearing loss " which is suitable for legal purposes only.

Given a k-digit automorphic number, an automorphic number with at most 2k-digits can be found by from the formula:

Given a vector a in Euclidean space R < sup > n </ sup >, the formula for the reflection in the hyperplane through the origin, orthogonal to a, is given by

Given the lengths of the sides of any cyclic quadrilateral, Brahmagupta gave an approximate and an exact formula for the figure's area,

Given the angular velocity vector and velocity of the object, the resulting force can be calculated using the following formula:

Given a simple polygon constructed on a grid of equal-distanced points ( i. e., points with integer coordinates ) such that all the polygon's vertices are grid points, Pick's theorem provides a simple formula for calculating the area A of this polygon in terms of the number i of lattice points in the interior located in the polygon and the number b of lattice points on the boundary placed on the polygon's perimeter:

Given all this information, the probability of the observer having spotted a girl given that the observed student is wearing trousers can be computed by substituting these values in the formula:

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

Given a modular form f ( z ) of weight k, the mth Hecke operator acts by the formula

Given a set, a set is defined to be if it is definable by a formula in which the symbol is interpreted as ; similar definitions for and apply.

Given any valid primary arithmetic expression, insert into one or more locations any number of Latin letters bearing optional numerical subscripts ; the result is a pa formula.

Given some pa formula containing N variables, this decision procedure requires simplifying 2 < sup > N </ sup > PA formulae.

Given an initial configuration, and a twist, the homogeneous transformation to a new location and orientation can be computed with the following formula:

Given a recursive formula for producing a series, one can define a transfinite series by transfinite recursion by defining the series at limit ordinals by

* Given a formula X the negation ¬ X is a formula.

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

Given that v < sup > 2 </ sup > ignores direction, it is logical to assume that the formula can be extended to the entire sample, replacing m with the entire sample's mass, equal to the molar mass times the number of moles n yielding

In his 1859 paper On the Number of Primes Less Than a Given Magnitude Riemann found an explicit formula for the normalized prime-counting function π < sub > 0 </ sub >( x ) which is related to the prime-counting function π ( x ) by

Given a real or complex matrix and a real or complex vector, respectively, the method computes an approximation of the solution of the linear systems of equations as in the following formula,

Given a system described by a formula it is sometimes possible to make a Taylor expansion of the expression and then identify negligible terms.

Given a propositional formula, its circumscription is the formula having only the models of that do not assign a variable to true unless necessary.

Given a first-order logic formula containing a predicate, circumscribing this predicate amounts to selecting only the models of in which is assigned to true on a minimal set of tuples of values.

Given a supervaluationist semantics, one can define the predicate ' supertrue ' as meaning " true on all precisifications ".

: Given any x and y, x = y if, given any predicate P, P ( x ) if and only if P ( y ).

Given a Blum complexity measure and a total computable function with two parameters, then there exists a total computable predicate ( a boolean valued computable function ) so that for every program for, there exists a program for so that for almost all

: 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.