Given these issues, Microsoft started to work in parallel on a version of Windows which was more future-oriented and more portable.

BONUS: " Given a pair of resistors, give their equivalent resistance if they were connected in parallel with each other.

#( Desargues ) Given seven distinct points A, A ', B, B ', C, C ', O, such that AA ', BB ', and CC ' are distinct lines through O and AB is parallel to A ' B ' and BC is parallel to B ' C ', then AC is parallel to A ' C '.

Given a covariant derivative ∇, the parallel transport along a curve γ is obtained by integrating the condition.

* Given a point and line there is a unique line which contains the point and is parallel to the line

Given a motor with mass attached via motor mounts to the ground, the motor vibrates as it operates and the soft motor mounts act as a parallel spring and damper, and.

Given four points in general linear position ( no three collinear ; in particular, no two coincident ), there are exactly three pairs of lines ( degenerate conics ) passing through them, which will in general be intersecting, unless the points form a trapezoid ( one pair is parallel ) or a parallelogram ( two pairs are parallel ).

Given three points, if they are non-collinear, there are three pairs of parallel lines passing through them – choose two to define one line, and the third for the parallel line to pass through, by the parallel postulate.

An attempt by Adrian Davies to launch parallel proceedings in Belgium without notifying the Belgian Court of English proceedings was blocked by a landmark anti-suit injunction granted by Miss Sarah Asplin QC on 20 May 2011, where she stated " Given that the present defendants had submitted to the hearing of that issue and allowed costs to be incurred, in my judgement such conduct is vexatious and oppressive ".

Let E be a rank k vector bundle over a smooth manifold M and let ∇ be a connection on E. Given a piecewise smooth loop γ: → M based at x in M, the connection defines a parallel transport map P < sub > γ </ sub >: E < sub > x </ sub > → E < sub > x </ sub >.

* Given a point and line there is a unique line which contains the point and is parallel to the line ( Playfair's Axiom ).

Given straight lines l and m, the following descriptions of line m equivalently define it as parallel to line l in Euclidean space:

Given the equations of two non-vertical parallel lines

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 that the eponymous Miss Lonelyhearts and his fellow protagonists are struggling for existential meaning and interpersonal relationships in the uncertain, threatening sociopolitical context of the Great Depression, Dick may have intended this citation as a parallel narrative to the similar alienation, existential and interpersonal crises of his own protagonists in a world on the brink of Nazi-Japanese nuclear war after the Axis won World War II in its alternative universe ( and, as an ironic contrast, our own ).

Given a rotation matrix R, a vector u parallel to the rotation axis must satisfy

Given two parallel walls whose facing surfaces have respective emissivities

Given the notorious unreliability of the Historia Augusta, the veracity of this list is debatable ; there is a scholarly consensus that the author deliberately inflated the number of pretenders in order to parallel the Thirty Tyrants of Athens.

Given the similarity of names, and the parallel identification of Ahasuerus with her husband Xerxes I, it is possible that Amestris is the Biblical Vashti.

Given a groupoid in the category-theoretic sense, let G be the disjoint union of all of the sets G ( x, y ) ( i. e. the sets of morphisms from x to y ).

Given a diagram F: J → C ( thought of as an object in C < sup > J </ sup >), a natural transformation ψ: Δ ( N ) → F ( which is just a morphism in the category C < sup > J </ sup >) is the same thing as a cone from N to F. The components of ψ are the morphisms ψ < sub > X </ sub >: N → F ( X ).

Given a list of operations and axioms in universal algebra, the corresponding algebras and homomorphisms are the objects and morphisms of a category.

Given objects A < sub > 1 </ sub >,..., A < sub > n </ sub > in C, their biproduct is an object A < sub > 1 </ sub > ⊕ ··· ⊕ A < sub > n </ sub > together with morphisms

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

Given a statement regarding the category C, by interchanging the source and target of each morphism as well as interchanging the order of composing two morphisms, a corresponding dual statement is obtained regarding the opposite category C < sup > op </ sup >.

#( TR 3 ) Given a map between two morphisms, there is a morphism between their mapping cones ( which exist by axiom ( TR 1 )), that makes everything commute.

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 a function f of type, currying it makes a function.

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 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 two groups G and H and a group homomorphism f: G → H, let K be a normal subgroup in G and φ the natural surjective homomorphism G → G / K ( where G / K is a quotient group ).

Given a trigonometric series f ( x ) with S as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that had S ' as its set of zeros, where S ' is the set of limit points of S. If p ( 1 ) is the set of limit points of S, then he could construct a trigonometric series whose zeros are p ( 1 ).

Given f ∈ G ( x * x < sup >- 1 </ sup >, y * y < sup >-1 </ sup >) and g ∈ G ( y * y < sup >-1 </ sup >, z * z < sup >-1 </ sup >), their composite is defined as g * f ∈ G ( x * x < sup >-1 </ sup >, z * z < sup >-1 </ sup >).

Given the laws of exponents, f ( x )

Given a function f of a real variable x and an interval of the real line, the definite integral

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 any point x in X, and any sequence in X converging to x, the composition of f with this sequence converges to f ( x )

Given f

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 a morphism f: B → A the associated natural transformation is denoted Hom ( f ,–).

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