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 manifold with a submanifold, one sometimes says can be knotted in if there exists an embedding of in which is not isotopic to.

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 point it is related to a line in projective twistor space where we can see the incidence relation as giving the linear embedding of

Given the definition of morpheme as " the smallest meaningful unit " Nanosyntax aims to account for idioms where it is often an entire syntactic tree which contributes " the smallest meaningful unit.

Given a sample of wood, the variation of the tree ring growths provides not only a match by year, it can also match location because the climate across a continent is not consistent.

Given a connected, undirected graph, a spanning tree of that graph is a subgraph that is a tree and connects all the vertices together.

Given the rules of any two-person game with a finite number of positions, one can always trivially construct a minimax algorithm that would exhaustively traverse the game tree.

Given that humanity cannot exist except within a covenantal relationship with God, and all covenants use symbols to give us " the attestation of his grace ", he gives the tree, " not because it could confer on man that life with which he had been previously endued, but in order that it might be a symbol and memorial of the life which he had received from God.

Given before her thirteenth birthday to a man who makes her a slave, she sheds her tears on the tree she planted, which bears the best fruit in the world.

The original problem was stated in the form that has become known as the Euclidean Steiner tree problem or geometric Steiner tree problem: Given N points in the plane, the goal is to connect them by lines of minimum total length in such a way that any two points may be interconnected by line segments either directly or via other points and line segments.

The realization problem for Euclidean minimum spanning trees is stated as follows: Given a tree T = ( V, E ), find a location D ( u ) for each vertex u ∈ V so that T is a minimum spanning tree of D ( u ): u ∈ V, or determine that no such locations exist.

Given a square of 2 < sup > 2k </ sup > cells, 2 < sup > k </ sup > on a side, at the kth level of the tree, the hash table stores the 2 < sup > k-1 </ sup >- by-2 < sup > k-1 </ sup > square of cells in the center, 2 < sup > k-2 </ sup > generations in the future.

Given the fact that the tree was actually taller than Calhoun ( itself a five and six story building in different places ), the tree posed a real danger to the college structure and Calhoun students.

Given the environmental impact of the brown tree snake, studies have attempted to provide a capturing methodology to alleviate the detrimental effects of the tree snake.

Given a list of n points, the following algorithm uses a median-finding sort to construct a balanced k-d tree containing those points.

Given that this is a plane wave, each vector represents the magnitude and direction of the electric field for an entire plane that is perpendicular to the axis.

Given a normalized light vector l ( pointing from the light source toward the surface ) and a normalized plane normal vector n, one can work out the normalized reflected and refracted rays:

Given a set of points in the Euclidean plane, selecting any one of them to be called 0 and another to be called 1, together with an arbitrary choice of orientation allows us to consider the points as a set of complex numbers.

* Given n points in the plane, find the two with the smallest distance from each other.

Given a sphere of unit radius, place its center at the origin of the complex plane, oriented so that the equator on the sphere coincides with the unit circle in the plane, and the north pole is " above " the plane.

Given a point in the plane, draw a straight line connecting it with the north pole on the sphere.

# Given a subdivision of the plane into vertical slabs, determine which slab contains a given point.

The problem in more mathematical terms is: Given a needle of length dropped on a plane ruled with parallel lines t units apart, what is the probability that the needle will cross a line?

Given a fixed oriented line L in the Euclidean plane R < sup > 2 </ sup >, a meander of order n is a non-self-intersecting closed curve in R < sup > 2 </ sup > which transversally intersects the line at 2n points for some positive integer n. Two meanders are said to be equivalent if they are homeomorphic in the plane.

Given three points in a plane as shown in the figure, the point is a convex combination of the three points, while is not.

Given any four points and in the compactified complex plane, the cross-ratio is defined by

Given a line in a plane, there exists at least one point in the plane that is not on the line.

Given a plane in space, there exists at least one point in space that is not in the plane.

Given any analytic function in the upper half plane, the function where is real will also be analytic in the upper half of the plane.

: Given five points in the plane in general position, prove that four of them form a convex quadrilateral.

Given a spread of, the André / Bruck-Bose construction < sup > 1 </ sup > produces a translation plane of order q < sup > 2 </ sup > as follows: Embed as a hyperplane of.

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 two manifolds M and N, a bijective map f from M to N is called a diffeomorphism if both

Given a subset X of a manifold M and a subset Y of a manifold N, a function f: X → Y is said to be smooth if for all p in X there is a neighborhood of p and a smooth function g: U → N such that the restrictions agree ( note that g is an extension of f ).

Given a group G, a factor group G / N is abelian if and only if ≤ N.

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.

Given that Gable and Cukor had worked together before, in Manhattan Melodrama and Gable had no objection to working with him then, and given Selznick's desperation to get Gable for Rhett Butler, if Gable had any objections to Cukor, certainly they would have been expressed before he signed his contract for the film.

Given a vector space V over the field R of real numbers, a function is called sublinear if

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

Given that the cost of replacing an executive can run over 100 % of his or her annual salary, any investment of time and energy in re-recruitment will likely pay for itself many times over if it helps a business retain just a handful of key players that would have otherwise left.

Given the proposition that ' a persons life prospects should not be decided by factors outside of their control or for which a person cannot claim personal credit ' ( i. e. social status, inherited wealth, race and other accidents of birth ) a meritocracy proposes a system where people are rewarded based on their efforts, and if everyone can start on equal footing with the same opportunity to advance, then the results are just.

: Given a determination as to the governing jurisdiction, a court is " bound " to follow a precedent of that jurisdiction only if it is directly in point.

Given a preorder on S one may define an equivalence relation ~ on S such that a ~ b if and only if a b and b a.

: Theorem: Given a set of FDs, if and only if.

Given Cerion's extensive geographic diversity, Gould later lamented that if Christopher Columbus had only cataloged a single Cerion it would have ended the scholarly debate about which island Columbus had first set foot on in America.

Given a functor F: J → C ( thought of as an object in C < sup > J </ sup >), the limit of F, if it exists, is nothing but a terminal morphism from Δ to F. Dually, the colimit of F is an initial morphism from F to Δ.

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.

noted, " Given that the bandwidth for conducting crawls is neither infinite nor free, it is becoming essential to crawl the Web in not only a scalable, but efficient way, if some reasonable measure of quality or freshness is to be maintained.

Given these assumptions, the constraint that time travel must not lead to inconsistent outcomes could be seen merely as a tautology, a self-evident truth that cannot possibly be false, because if you make the assumption that it is false this would lead to a logical paradox.

Given two affine spaces and, over the same field, a function is an affine map if and only if for every family of weighted points in such that

Given a ring R and a proper ideal I of R ( that is I ≠ R ), I is a maximal ideal of R if any of the following equivalent conditions hold:

# Given any elements a and b of G, if a ~ b, then b ~ a ( symmetry );