18.06 Lecture Notes - Lecture 2: Zend, Lambada, Allodial Title

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amethystmuskrat145

22 May 2016

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The incidence matrix of a graph tells how the n nodes are connected by the m edges. Normally there are more edges than nodes, m > n. Every entry of an incidence matrix is 0, 1, or, -1. The incidence matrix takes differences in voltage across the edges of a graph. The voltages are x , x ,, x. A graph is complete when every pair of nodes is connected by an edge. A tree is a graph that has no closed loops. The rows of b match the nonzero rows of u. The maximum number of edges is n(n - 1) The minimum number of edges is m = n - 1. Rows are dependent when edges form a loop. V is in the row space if and only if it is perpendicular to the null-space. In this case, the null-space is the vector (1, 1, 1, 1) The small loops give a basis for the left null-space.

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33. Base10 is the set of numbers created from the digits {0,1,2,3,4,5,6,7,8,9}. Whereas base2, which is the set of numbers created with {0,1}. We could similarly define base; as the ordered set of characters created from {X,Y,Z}. What is the following basez number, Yxz, in base ? Random graph For this section you will need to look at the classes Graph and GraphNode present in the appendix. A Graph consists of a number of nodes, each of which has connections to other nodes in the graph. You can think of nodes as a representation of a person in a social network (e.g. Facebook). The ArrayList connectedTo variable in GraphNodes is the list of friends that particular person has. For example, in the following graph, the node O is friends with node 1 and 2. On the other hand, node 2 and node 1 are not friends, hence the lack of link. 0-0 In the constructor of the Graph, the process of connecting nodes is randomized. Moreover, we will not allow a node to be connected to itself and we are going to assume that the arguments passed to the Graph constructor respect the following conditions: n > 2, and n > nbConnections. Pay close attention to the character order

ey table is seen below xeecions tor Chapter 1 12, Fori, b â¬ Î.el * b-aa (0 (D 14 ForVEQ r the operation defined in Example 1 on the wr John Sue Henry. P iDFor 15ine if is ass ociative by determining if the final six engu defined ang property (ii)of Theorem 1 . So no element i if the operntion debined in Example 1 of Emle if each element has an inverse Section 1.3 Groups iw comutatityn ild Chapter 3 where the Funda the exercises that follow etermining why they are not Write out the Cayley table for the gru(o)nud lensify thee element the 10 elements of the group Z 18 its operation uses ts in the first coordinnte and +2 in the secotu Ldessity the Zn and write out the Cayley table. Reeall th of each element does Theorem 1.14 tell us aont the entries in the Cayley table ol s xrewp that is set with exactly one element. A {n), will always form group since there 19. What nly one way to define an operation on the set. Be sure to shhow that the properi combination of the usual notice what set the rule is be of a group all hold. 21. We can create even create groupe with games! Consider four cups placed in a ssare patter table. If we have a penny in one of the cups there are four ways we can move it to t cup: Horizontally, Vertically, Diagonally, or Stay nehere it is. We will p, S. To define nn operation, consider two movements in a row, v.e.,m ove the penny as r tells us to, then after that move the penny as y instruets. the set of integers the set of real numbers n the set of positive integers. e set of positive integers. n the set of positive integers. on the set of integers e set of negative integers. we ple, HV D since if we first move it horizontally and then vertically nltogether e moved it diagonally. Create the Cayley table for this group, identify the identity and the inverse of cach element 22. Prove that the set A 0 under matrix multiplication. Is it abelian? 23. Verify that 3(R) is a group using the addition of funetions as defined in Example 13 e set of nonzero rational n he set of nonzero real num n on the set R (-1 Section 1.4 Basic Algebra in Groups 24. Prove the Cancellation Law (Theorem 1.20). Do not use any theorems that occur after it in the text ssociative, commutative, has roof or counterexample fo 25. Assume G is a group and a e G. Consider a fixed integer k and use PMI (Theorem 0.3) to prove that for all nEN, a aYou will need to consider cases here since k can be positive, negative, or o 26. Prove Theorem 1.23 (ii) for the case of n > 0 and m

18.06 Lecture Notes - Lecture 2: Zend, Lambada, Allodial Title

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