PPT-Graphs, Vectors, and Matrices

Daniel A Spielman Yale University AMS Josiah Willard Gibbs Lecture January 6 2016 From Applied to Pure Mathematics Algebraic and Spectral Graph Theory Sparsification a pproximating graphs by graphs with fewer edges. A graph is a set of points called vertices and lines connecting some pairs of vertices called edges Two vertices connected by an edge are said to be adjacent Figure 1 As we can see from this example vertices can be connected by m ore than one edge In particular they are useful for compactly representing and discussing the linear programming problem Maximize subject to i j This appendix reviews several properties of vectors and matrices that are especially relevant to this problem We shoul It is essential that you do some reading but the topics discussed in this chapter are adequately covered in so many texts on linear algebra that it would be arti64257cial and unnecessarily limiting to specify precise passages from precise texts The Ther e ar e many applications of linear algebra for example chemists might use ow eduction to get a lear er pictur e of what elements go into a complicated eaction In this lectur e we explor e the linear algebra associated with electrical networks G Sometimes, two graphs have exactly the same form, in the sense that there is a one-to-one correspondence between their vertex sets that preserves edges. In such a case, we say that the two graphs are . Lecture 18. N. Harvey. TexPoint. fonts used in EMF. . Read the . TexPoint. manual before you delete this box. .: . A. A. A. A. A. A. A. A. A. A. Topics. Semi-Definite Programs (SDP). Solving SDPs by the Ellipsoid Method. Matrices. Definition: A matrix is a rectangular array of numbers or symbolic elements. In many applications, the rows of a matrix will represent individuals cases (people, items, plants, animals,...) and columns will represent attributes or characteristics. Daniel A. Spielman. Yale University. AMS Josiah Willard Gibbs Lecture. January . 6. , 2016 . From Applied to Pure Mathematics. Algebraic and Spectral Graph Theory. . . Sparsification. :. a. pproximating graphs by graphs with fewer edges. Section . 10.3. Representing Graphs: . Adjacency Lists. Definition. : An . adjacency list . can be used to represent a graph with no multiple edges by specifying the vertices that are adjacent to each vertex of the graph.. All Lectures. David Woodruff. IBM Almaden. Massive data sets. Examples. Internet traffic logs. Financial data. etc.. Algorithms. Want nearly linear time or less . Usually at the cost of a randomized approximation. Niebles. . and Ranjay Krishna. Stanford Vision and Learning . Lab. 10/2/17. 1. Another, very in-depth linear algebra review from CS229 is available here:. http://cs229.stanford.edu/section/cs229-linalg.pdf. Fei-Fei. Li. Stanford Vision Lab. 23-Sep-14. 1. Another, very in-depth linear algebra review from CS229 is available here:. http://. cs229.stanford.edu/section/cs229-linalg.pdf. And a video discussion of linear algebra from EE263 is here . K-means. Input: set of data points, k. Randomly pick k points as means. For . i. in [0, . maxiters. ]:. Assign each point to nearest center. Re-estimate each center as mean of points assigned to it. Rotation of coordinates -the rotation matrixStokes Parameters and unpolarizedlight1916 -20041819 -1903Hans Mueller1900 -1965yyxyEEEElinear arbitrary anglepolarization right or left circularpolarizati