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

Given that Conan was well established in genealogies as the founder of Brittany this certainly is connected to an older tradition than Geoffrey.

Given this reality, scholars are continually challenged to research and understand how online communities are comprised, how they function, and how they are connected to offline social life.

Given the San Gabriel Valley's burgeoning population of Asian Americans ( specifically Chinese Americans ), several business districts were developed to serve their needs creating a collection of Southern California Chinatowns loosely connected along the Valley Boulevard Corridor.

Given two oriented submanifolds of complementary dimensions in a simply connected manifold of dimension, one can apply an isotopy to one of the submanifolds so that all the points of intersection have the same sign.

Given a partial solution to the puzzle, they use dynamic programming within each row or column to determine whether the constraints of that row or column force any of its squares to be white or black, and whether any two squares in the same row or column can be connected by an implication relation.

Given a perfect normal subgroup of the fundamental group of a connected CW complex, attach two-cells along loops in whose images in the fundamental group generate the subgroup.

Given Aeschylus ' tendency to write connected trilogies, three plays attested in the catalogue of his work have been supposed to constitute the Achilleis: Myrmidons, Nereids and Phrygians ( alternately titled The Ransoming of Hector ).

Given any compact Lie group G one can take its identity component G < sub > 0 </ sub >, which is connected.

Given an open and connected neighborhood U of p, a function

Given a connected graph G =( V, E ) with V the set of vertices and E the set of edges, and with a root vertex r, the level structure is a partition of the vertices into subsets L < sub > i </ sub > called levels, consisting of the vertices at distance i from r. Equivalently, this set may be defined by setting L < sub > 0 </ sub > =

Given a connected and orientable manifold M of dimension 4k, the cup product gives rise to a quadratic form Q on the ' middle ' real cohomology group

Given a simply connected open subset D of C < sup > n </ sup >, there is an associated sheaf O < sub > D </ sub > of holomorphic functions on D. Throughout, U is any open subset of D. Then the set O < sub > D </ sub >( U ) of holomorphic functions from U to C has a natural ( componentwise ) C-algebra structure and one can collate sections that agree on intersections to create larger sections ; this is outlined in more detail at sheaf.

Given a simply connected and open subset D of R < sup > 2 </ sup > and two functions I and J which are continuous on D then an implicit first-order ordinary differential equation of the form

Given an exact differential equation defined on some simply connected and open subset D of R < sup > 2 </ sup > with potential function F then a differentiable function f with ( x, f ( x )) in D is a solution if and only if there exists real number c so that

Given an undirected graph G = ( V, E ), a set of random variables X = ( X < sub > v </ sub >)< sub > v ∈ V </ sub > indexed by V form a Markov random field with respect to G if they satisfy the following equivalent Markov properties:

Given a table of costs and revenues at each quantity, we can either compute equations or plot the data directly on a graph.

Given a graph G with n vertices, the closure cl ( G ) is uniquely constructed from G by repeatedly adding a new edge uv connecting a nonadjacent pair of vertices u and v with until no more pairs with this property can be found.

Given a graph, a vertex-deleted subgraph of is a subgraph formed by deleting exactly one vertex from.

Given a graph, an edge-deleted subgraph of is a subgraph formed by deleting exactly one edge from.

Given a graph Λ ( for example, a d-dimensional lattice ), per each lattice site j ∈ Λ there is a discrete variable σ < sub > j </ sub > such that σ < sub > j </ sub > ∈

* Given a partial function f from the natural numbers into the natural numbers, f is a partial recursive function if and only if the graph of f, that is, the set of all pairs such that f ( x ) is defined, is recursively enumerable.

: Given the graph on the right, is it possible to construct a path ( or a cycle, i. e. a path starting and ending on the same vertex ) which visits each edge exactly once?

Given three vertices v, u, and w, where ( v, u ) and ( u, w ) are edges in the graph, the lifting of vuw, or equivalent of ( v, u ), ( u, w ) is the operation that deletes the two edges ( v, u ) and ( u, w ) and adds the edge ( u, w ).

Given a random graph of n nodes and an average degree < math >< k ></ math >.

Given a flow network, and a flow on, we define the residual graph of with respect to as follows.

Given a graph property P, an invariant u, and a set of graphs H, we wish to find the minimum value of m such that every graph in H which has u larger than m possess property P. In the example above, H was the set of n-vertex graphs, P was the property of being cyclic, and u was the number of edges in the graph.

Given a graph G = ( V, E ), a matching M in G is a set of pairwise non-adjacent edges ; that is, no two edges share a common vertex.

: Given a weighted directed graph and an integer K, is there a Hamiltonian path ( or Hamiltonian cycle if the salesman must return home ) with total weight less than or equal to K?

Given a perfect graph G, Lovász forms a graph G * by replacing each vertex v by a clique of t < sub > v </ sub > vertices, where t < sub > v </ sub > is the number of distinct maximum independent sets in G that contain v. It is possible to correspond each of the distinct maximum independent sets in G with one of the maximum independent sets in G *, in such a way that the chosen maximum independent sets in G * are all disjoint and each vertex of G * appears in a single chosen set ; that is, G * has a coloring in which each color class is a maximum independent set.

Given a graph G, its line graph L ( G ) is a graph such that

Given a graph G and given a set L ( v ) of colors for each vertex v ( called a list ), a list coloring is a choice function that maps every vertex v to a color in the list L ( v ).

Given a symmetric matrix we visualize the matrix as the adjacency matrix of a graph.

Given a vector space V over a field K, the span of a set S ( not necessarily finite ) is defined to be the intersection W of all subspaces of V which contain S. W is referred to as the subspace spanned by S, or by the vectors in S. Conversely, S is called a spanning set of W.

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 graph where all its edges have distinct weights, is a given edge in the minimum weight spanning forest?

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 an embedding of a rooted tree in the plane, if one fixes a direction of children ( starting from root, then first child, second child, etc.

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.

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.