Given the first n digits of Ω and a k ≤ n, the algorithm enumerates the domain of F until enough elements of the domain have been found so that the probability they represent is within 2 < sup >-( k + 1 )</ sup > of Ω.

Given a complex-valued function ƒ of a single complex variable, the derivative of ƒ at a point z < sub > 0 </ sub > in its domain is defined by the limit

Given a function f: X → Y, the set X is the domain of f ; the set Y is the codomain of f. In the expression f ( x ), x is the argument and f ( x ) is the value.

Given a partial isometry V, the deficiency indices of V are defined as the dimension of the orthogonal complements of the domain and range:

Given on the periodic domain

Given any set of points in the desired domain of your functions, take a multivariate Gaussian whose covariance matrix parameter is the Gram matrix of your N points with some desired kernel, and sample from that Gaussian.

Given an integral domain, let be an element of, the polynomial ring with coefficients in.

Given a fundamental domain for the group action ( for the full, orientation-reversing symmetry group, a ( 2, 3, 7 ) triangle ), the reflection domains ( images of this domain under the group ) give a tiling of the quartic such that the automorphism group of the tiling equals the automorphism group of the surface – reflections in the lines of the tiling correspond to the reflections in the group ( reflections in the lines of a given fundamental triangle give a set of 3 generating reflections ).

Given an action of a group G on a topological space X by homeomorphisms, a fundamental domain ( also called fundamental region ) for this action is a set D of representatives for the orbits.

Given any morphism between objects X and Y, if there is an inclusion map into the domain, then one can form the restriction fi of f. In many instances, one can also construct a canonical inclusion into the codomain R → Y known as the range of f.

Given an element a and a non-zero element b in a Euclidean domain R equipped with a Euclidean function d, there exist q and r in R such that and either or.

Given the space X = Spec ( R ) with the Zariski topology, the structure sheaf O < sub > X </ sub > is defined on the D < sub > f </ sub > by setting Γ ( D < sub > f </ sub >, O < sub > X </ sub >) = R < sub > f </ sub >, the localization of R at the multiplicative system

: Given any set A and any set B, there is a set C such that, given any set D, D is a member of C if and only if D is equal to A or D is equal to B.

: Given any set A, there is a set B such that, for any element c, c is a member of B if and only if there is a set D such that c is a member of D and D is a member of A.

Given its lack of a D ' Oyly Carte or Savoy connection, Thespis has a tenuous claim to be a " Savoy Opera.

Given a triangle ABC, let the lines AO, BO and CO be drawn from the vertices to a common point O to meet opposite sides at D, E and F respectively.

D., Given Chiefly Through Letters and Journals.

On September 18, the EPA released a report in which Whitman said, " Given the scope of the tragedy from last week, I am glad to reassure the people of New York and Washington, D. C. that their air is safe to breathe and their water is safe to drink.

Given under my hand this, the 1st day of March A. D. 1862.

Given constants C, D and V, there are only finitely many ( up to diffeomorphism ) compact n-dimensional Riemannian manifolds with sectional curvature | K | ≤ C, diameter ≤ D and volume ≥ V.

Given constants C, D and V, there are only finitely many homotopy types of compact n-dimensional Riemannian manifolds with sectional curvature K ≥ C, diameter ≤ D and volume ≥ V.

Given the team's poor results in the last few years, which she attributes to Coach D ' Amato's " old-school methods ", she attempts to take a more hands-on approach to the team, including bringing in an innovative Offensive Coordinator Nick Crozier.

An eyewitness account of the last day of Major André can be found in the book The American Revolution: From the Commencement to the Disbanding of the American Army Given in the Form of a Daily Journal, with the Exact Dates of all the Important Events ; Also, a Biographical Sketch of the Most Prominent Generals by James Thacher, M. D., a surgeon in the American Revolutionary Army:

id =" CITEREFYoung1856 "> Young, Brigham &# 32 ;( September 21, 1856d ), " The People of God Disciplined by Trials — Atonement by the Shedding of Blood — Our Heavenly Father — A Privilege Given to all the Married Sisters in Utah ", in &# 32 ; Watt, G. D .,&# 32 ; Journal of Discourses by Brigham Young, President of the Church of Jesus Christ of Latter-day Saints, His Two Counsellors, and the Twelve Apostles, vol.

Given a circle whose center is point O, choose three points V, C, and D on the circle.

Given a circle whose center is point O, choose three points V, C, and D on the circle.

Idealists are skeptics about the physical world, maintaining either: 1 ) that nothing exists outside the mind, or 2 ) that we would have no access to a mind-independent reality even if it may exist ; the latter case often takes the form of a denial of the idea that we can have unconceptualised experiences ( see Myth of the Given ).

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

Given a function f ∈ I < sub > x </ sub > ( a smooth function vanishing at x ) we can form the linear functional df < sub > x </ sub > as above.

Given the existence of a Godlike object in one world, proven above, we may conclude that there is a Godlike object in every possible world, as required.

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

On poverty, Hoover said that " Given the chance to go forward with the policies of the last eight years, we shall soon with the help of God, be in sight of the day when poverty will be banished from this nation ", and promised, " We in America today are nearer to the final triumph over poverty than ever before in the history of any land ," but within months, the Stock Market Crash of 1929 occurred, and the world's economy spiraled downward into the Great Depression.

Given the state at some initial time ( t = 0 ), we can solve it to obtain the state at any subsequent time.

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 function ƒ defined over the reals x, and its derivative ƒ < nowiki > '</ nowiki >, we begin with a first guess x < sub > 0 </ sub > for a root of the function f. Provided the function is reasonably well-behaved a better approximation x < sub > 1 </ sub > is

Given that both A and not-A are seen to be “ true ,” Kant concludes that it ’ s not that “ God doesn ’ t exist ” but that there is something wrong with how we are asking questions about God and how we have been using our rational faculties to talk about universals ever since Plato got us started on this track!

Given how little we can know for sure, our focus should be on this earth and life ; beauty, justice, love.

Given any vector space V over K we can construct the tensor algebra T ( V ) of V. The tensor algebra is characterized by the fact:

Given metric spaces ( X, d < sub > 1 </ sub >) and ( Y, d < sub > 2 </ sub >), a function f: X → Y is called uniformly continuous if for every real number ε > 0 there exists δ > 0 such that for every x, y ∈ X with d < sub > 1 </ sub >( x, y ) < δ, we have that d < sub > 2 </ sub >( f ( x ), f ( y )) < ε.

Given the symmetry of circularly polarized light, we could have in fact selected any other two orthogonal components and found the same phase relationship between them.

Given an evaluation e of variables by elements of M < sub > w </ sub >, we

Given that any proposition containing conjunction, disjunction, and negation can be equivalently rephrased using conjunction and negation alone ( the conjunctive normal form ), we can now handle any compound proposition.

Given any energy eigenstate, we can act on it with the lowering operator, a, to produce another eigenstate with-less energy.

Given two ultrafilters and on, we define their sum by

Given a Boolean ring R, for x and y in R we can define

Given a prime, we define the height of, written to be the supremum of the set

Given such possibilities, we can expect TC to be used to suppress everything from pornography to writings that criticize political leaders.

Given a testing procedure E applied to each prepared system, we obtain a sequence of values

Given objects and in an additive category, we can represent morphisms as-by-matrices