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Let F? =(5z+5x4)i? +(6y+7z+7sin(y4))j? +(5x+7y+6ez4)k? . (a) Find curl F? . curl

Let F? =(5z+5x4)i? +(6y+7z+7sin(y4))j? +(5x+7y+6ez4)k? . (a) Find curl F? . curl F? = 0 (b) What does your answer to part (a) tell you about ?CF? ?dr?  where C is the circle (x?10…

Let F_n be the free group on n elements. Give (a) an injective homomorphisms fro

Let F_n be the free group on n elements. Give (a) an injective homomorphisms from F_2 to F_3; (b) a surjective homomorphism from F_3 to F_2; (c) an injective homomorphism from F_3…

Let G (V, E) be an undirected graph with n nodes. Recall that a subset of the no

Let G (V, E) be an undirected graph with n nodes. Recall that a subset of the nodes is called an independent set if no two of them are joined by an edge. Finding large independent…

Let G (x,y,z) = be a vector field in R3. Let C be a circular helix cu

Let G(x,y,z) = <-y,1-x,z> be a vector field in R3. Let C be a circular helix curve parametrized by the equation r(t) = <cos(t), sin(t), t>, where t belongs to [0,2p]. …

Let G = ( V , E ) be a graph with n nodes in which each pair of nodes is joined

Let G = (V, E) be a graph with n nodes in which each pair of nodes is joined by an edge. There is a positive weight wij on each edge (i, j); and we will assume these weights satis…

Let G = ( V , E ) be a graph with n nodes in which each pair of nodes is joined

Let G = (V, E) be a graph with n nodes in which each pair of nodes is joined by an edge. There is a positive weight wij on each edge (i, j); and we will assume these weights satis…

Let G = ( V , E ) be an (undirected) graph with costs ce 0 on the edges e E . As

Let G = (V,E) be an (undirected) graph with costs ce 0 on the edges e E. Assume you are given a minimum-cost spanning tree T in G. Now assume that a new edge is added to G, connec…

Let G = (N, A) be a network with n nodes and m arcs, and integral capacities (u_

Let G = (N, A) be a network with n nodes and m arcs, and integral capacities (u_e: e A). For two nodes s, t N, denote by P_s, t the set of directed paths from s to t in G. Conside…

Let G = (Q - {0}, ), and let if be the subgroup H= {a/b | a and b are odd intege

Let G = (Q - {0}, ), and let if be the subgroup H= {a/b | a and b are odd integers}. Use the Fundamental Theorem to show that G/H = (z, +).

Let G = (T,*,e) be a group and U T. (a) Show that G\' = (U, *, e) is a subgroup

Let G = (T,*,e) be a group and U T. (a) Show that G' = (U, *, e) is a subgroup of G (if and only if the following fact holds: for all x, y T, if x U and y U then x* y-1 U.) (b) As…

Let G = (V, E) be a complete directed acyclic graph that has an edge between eve

Let G = (V, E) be a complete directed acyclic graph that has an edge between every pair of vertices and whose vertices are labeled 1, 2, ..., n, where n = |V|. To determined the d…

Let G = (V, E) be a connected, undirected graph. (In this course we follow the u

Let G = (V, E) be a connected, undirected graph. (In this course we follow the usual convention that, unless indicated otherwise, an undirected graph does not have loops or multip…

Let G = (V, E) be a directed graph in which each vertex u V is labeled with a un

Let G = (V, E) be a directed graph in which each vertex u V is labeled with a unique integer L(u) from the set {1, 2, ..., |V |}. For each vertex u V , let R(u) be the set of vert…

Let G = (V, E) be a directed graph in which each vertex v has an integer label `

Let G = (V, E) be a directed graph in which each vertex v has an integer label `(v) that denotes the influence of v. Assume for simplicity that all the influences are distinct. Fo…

Let G = (V, E) be a directed graph in which each vertex v has an integer label `

Let G = (V, E) be a directed graph in which each vertex v has an integer label `(v) that denotes the influence of v. Assume for simplicity that all the influences are distinct. Fo…

Let G = (V, E) be a directed graph with nodes v1, v2, . . . vn. We say that G is

Let G = (V, E) be a directed graph with nodes v1, v2, . . . vn. We say that G is an ordered graph if it has the following properties. (i) Each edge goes from a node with a lower i…

Let G = (V, E) be a directed graph with weighted edges; edge weights could be po

Let G = (V, E) be a directed graph with weighted edges; edge weights could be positive, negative, or zero. How could we delete an arbitrary vertex v from this graph, without chang…

Let G = (V, E) be a directed graph. A black hole is a vertex v ? V such that and

Let G = (V, E) be a directed graph. A black hole is a vertex v ? V such that and v has no outgoing edge. (a) Prove or disprove: in a directed graph, there is at most one black hol…

Let G = (V, E) be a graph with |V| = n, and let k be an integer, where 1 lesstha

Let G = (V, E) be a graph with |V| = n, and let k be an integer, where 1 lessthanorequalto k lessthanorequalto n. Prove the following theorem: "Suppose the vertices in V can be or…

Let G = (V, E) be a weighted undirected, connected graph. Let (u, v) be an edge

Let G = (V, E) be a weighted undirected, connected graph. Let (u, v) be an edge of G. We want to find a Minimum Spanning Tree of G subject to the constraint that it must contain (…

Let G = (V, E) be an undirected connected graph and let c : E R + be a function

Let G = (V, E) be an undirected connected graph and let c : E R + be a function specifying the costs of the edges, i.e., every edge has a positive cost. Assume that no two edges h…

Let G = (V, E) be an undirected graph and let w(e) > 0 for edge weights for e G

Let G = (V, E) be an undirected graph and let w(e) > 0 for edge weights for e G E. Recall that a subset M E is called a matching if the edges in M do not share a node. The Maxi…

Let G = (V, E) be an undirected graph where |V| = n and |E| = m. Suppose for two

Let G = (V, E) be an undirected graph where |V| = n and |E| = m. Suppose for two vertices u, v elementof G, we want to know if u and v are reachable from one another, i.e., if the…

Let G = (V, E) be an undirected graph with a cost C 0 on each edge. You are give

Let G = (V, E) be an undirected graph with a cost C 0 on each edge. You are given a minimum spanning tree T in G. Now we are moving to a dynamic scenario: a new edge is added to G…

Let G = (V, E) be an undirected, connected graph with n vertices and m edges. Al

Let G = (V, E) be an undirected, connected graph with n vertices and m edges. All vertices .ire initially un-marked. Consider the following algorithm: Algorithm traverse(G, u) Inp…

Let G = (V, E) be an undirected, connected graph with n vertices and m edges. Al

Let G = (V, E) be an undirected, connected graph with n vertices and m edges. All vertices .ire initially un-marked. Consider the following algorithm: Algorithm traverse(G, u) Inp…

Let G = (V, E) be an undirected, connected graph. A vertex u E Vis bridge vertex

Let G = (V, E) be an undirected, connected graph. A vertex u E Vis bridge vertex if removal of v (and edges incident on v) makes the graph disconnected. Suppose that v EV be a bri…

Let G = (V,E) be a unit-capacity graph with n vertices and m edges. Let T denote

Let G = (V,E) be a unit-capacity graph with n vertices and m edges. Let T denote all the spanning trees in G. If we run Karger's algorithm, we will get a random spanning tree in T…

Let G = (V,E) be an edge-weighted connected graph, and T = (V,E\') be its minimu

Let G = (V,E) be an edge-weighted connected graph, and T = (V,E') be its minimum, and e be any edge in T. a) Consider the graph T' = (V, E - {e}). How many connected component(s) …

Let G = (V,E) be an undirected graph that is connected and has acost associated

Let G = (V,E) be an undirected graph that is connected and has acost associated with every one of its edges. Let T denote a minimum-cost spanning treeof G (as computed by, for exa…

Let G = (V,E) be an undirected graph. For two vertices u and v in G, the distanc

Let G = (V,E) be an undirected graph. For two vertices u and v in G, the distance from u to v, written d(u,v), is the length of a path with the fewest number of edges from u to v.…

Let G = (VE) be a graph and M a matching on G. Suppose G has no M-augmenting pat

Let G = (VE) be a graph and M a matching on G. Suppose G has no M-augmenting paths. Following the steps below to show that M is maximum. Let M' be a maximum matching, and let H be…

Let G = R*=

Let G = <[I, A, B, C, D, K}, Matrix Multiplication> R*= <x is a real number, where x cannot equal 0}, x> R* is all positive real numbers Define f: G --> R* by f(X) …

Let G = R*=

Let G = <[I,A,B,C,D,K} Matrix Multiplication> R*= <x is a real number, where x cannot equal zero> R* is all positive real numbers Define f: G --> R* by f(X) (the de…

Let G = [a] be a cyclic group of order 16. Let H = [a^4]. a. List the elements o

Let G = [a] be a cyclic group of order 16.  Let H = [a^4]. a.   List the elements of H, then give the index of H in G. b.   Explain why H must be a normal subgroup in G without ex…

Let G = {1, a, b, c} be the Klein 4-group and following the example below, label

Let G = {1, a, b, c} be the Klein 4-group and following the example below, label the elements a,b,c,d as 1,2,3,4. The example below shows that the left multiplication by the group…

Let G and H be groups,and :GH a homomorphism. (a) Show that ker is a subset of G

Let G and H be groups,and :GH a homomorphism. (a) Show that ker is a subset of G. _______________________________________________________________ (b) Fill in the blanks with appro…

Let G be a DAG with vertices labeled 1, 2, . . . ,n. Define the adjacency matrix

Let G be a DAG with vertices labeled 1, 2, . . . ,n. Define the adjacency matrix and adjacency list representations of G as in Rosen, page 669. Let A be the adjacency matrix that …

Let G be a DAG with vertices labeled 1,2, Define the adjacency matrix and adjace

Let G be a DAG with vertices labeled 1,2, Define the adjacency matrix and adjacency list representations of G as in Rosen, page 669. Let A be the adjacency matrix that represents …

Let G be a connected graph with vertices a, b, c, ...,h, i. Thus | V | = 9. A li

Let G be a connected graph with vertices a, b, c, ...,h, i. Thus |V| = 9. A list of edges, sorted by weight, is shown below. Execute both Kruskal's and Prims algorithms without dr…

Let G be a connected graph. For two vertices u and v. define d(u, u) to be the l

Let G be a connected graph. For two vertices u and v. define d(u, u) to be the length of the shortest path in G from u to v: we say that d(u. v) is the distance from it to v. Note…

Let G be a directed acyclic graph with n vertices and eedges, such that the undi

Let G be a directed acyclic graph with n vertices and eedges, such that the undirected version of G is connected. The out-degree (resp., in-degree) of a vertex isthe number of dir…

Let G be a directed acyclic graph with n vertices and eedges, such that the undi

Let G be a directed acyclic graph with n vertices and eedges, such that the undirected version of G is connected. The out-degree (resp., in-degree) of a vertex isthe number of dir…

Let G be a directed graph where each edge e has a capacity c(e)>0 associated wit

Let G be a directed graph where each edge e has a capacity c(e)>0 associated with it. That is, c(e) describes how much edge e can "carry". Let P=e1; e2; : : : ; ek be a path in…

Let G be a directed graph with nodes numbered 1 through 7 and the following edge

Let G be a directed graph with nodes numbered 1 through 7 and the following edges: (1, 2),(1, 6),(3, 4),(4, 1),(4, 5),(5, 6),(6, 3),(6, 7),(7, 5) Run the depth-first search algori…

Let G be a directed graph. The directed subgraph induced by X V , has X as verti

Let G be a directed graph. The directed subgraph induced by X V , has X as vertice, and has all edges with both endpoint in X. This graph is denoted by G(X). We say that the graoh…

Let G be a directed graph. The directed subgraph induced by X V , has X as verti

Let G be a directed graph. The directed subgraph induced by X V , has X as vertice, and has all edges with both endpoint in X. This graph is denoted by G(X). We say that the graoh…

Let G be a directed graph. The directed subgraph induced by X V , has X as verti

Let G be a directed graph. The directed subgraph induced by X V , has X as vertice, and has all edges with both endpoint in X. This graph is denoted by G(X). We say that the graoh…

Let G be a finite abelian group. The exponent of G is defined exp(G) = max{ord(g

Let G be a finite abelian group. The exponent of G is defined exp(G) = max{ord(g) | g is an element of H}, where ord(g) is the order of an element g in G. For this problem, standa…

Let G be a finite cyclic group (written multiplicatively). The discrete log prob

Let G be a finite cyclic group (written multiplicatively). The discrete log problem in G requires finding x (mod |G|), given u & g as elements of G with g as a generator of G …