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• A graph, non-tree representation of the topology of a gray scale image by Saveliev
  ...f these sets, and the [[topology]] of the image, by means of a graph. This graph contains the well-known [[inclusion tree]]s, but it is not a tree in genera ...e also a clear hierarchy based on [[inclusion]]. This hierarchy provides a graph representation of the topology of the image. It is called the [[inclusion t
  41 KB (6,854 words) - 15:05, 28 October 2011
• Topology of graphs
  ...structure that needs to be analyzed topologically is ''graphs''<!--\index{graph}-->: [[Image:graph examples.png|center]]
  25 KB (4,214 words) - 16:08, 28 November 2015
• Algebra on graphs
  ...structure that needs to be analyzed topologically is ''graphs''<!--\index{graph}-->: [[Image:graph examples.png| center]]
  36 KB (6,177 words) - 02:47, 21 February 2016
• Maps of graphs
  ...mations with their discrete counterparts, as functions of graphs<!--\index{graph}-->: So, we have learned that a function<!--\index{graph function}--> $f:G\to J$ between graphs should be a combination of two, the
  29 KB (5,042 words) - 17:57, 28 November 2015
• The topology of a binary image
  ...d holes is constructed. The algorithm for the construction of the topology graph is incremental – every time a pixel is added, the topology of the resulti ==A graph representation of the topology of a binary image==
  16 KB (2,639 words) - 15:00, 27 October 2012
• Binary images - implementation
  ...of the objects. Geometrically, these boundaries are circular sequences of directed edges of pixels in the image. In other words, these are 0- and 1-cycles. Th .... In particular, the 0- and 1-Betti numbers are updated at every step. The graph is implemented as a linked list. Each element of the list has pointers to t
  26 KB (4,036 words) - 21:05, 28 December 2009
• A graph representation of the topology of color images by Saveliev
  ''A graph representation of the topology of color images'' by [[Peter Saveliev]] ...nsion of this lower/upper level set approach by introducing the [[topology graph]].
  6 KB (1,011 words) - 15:33, 28 October 2011
• Higher order votes
  but now ''all'' of the edges are also present -- in the complete graph over $A$. The set of possible ratings is still the integers. The ratings ar ...There are $n$ alternatives/candidates and they are the nodes of a directed graph $K$:
  41 KB (6,942 words) - 05:04, 22 June 2016
• Elections
  ...al, a ''valid ballot'' (an element of $Z$) is a complete, acyclic directed graph on the set of $n$ vertices/candidates. Let's drop both of these restriction *a ''ballot'' is a directed graph on the set of $n$ vertices.
  33 KB (5,872 words) - 13:13, 17 August 2015
• Functions in higher dimensions
  ...ough in the early calculus (Chapters 7 -13) we deal with only numbers, the graph of a function of one variable lies in the $xy$-plane, a space of dimension ...ordinates of $P$. If we ignore the third variable ($z=0$), we can plot the graph of the resulting function of two variables:
  113 KB (19,680 words) - 00:08, 23 February 2019
• Pseudocode for binary images
  .... Then this correspondence takes the form of a table, T, so that for every directed edge E, T(E) is the cycle that passes through E. This table is updated at e ...he face of the pixel. Adding an edge means assigning cycles to both of the directed edges: forward E and backward -E. After all the edges have been added, ther
  5 KB (905 words) - 05:31, 6 February 2011
• The algebra of cubical cells
  <!--150-->[[image:Graph example 4 boundaries.png| center]] ...added complexity. We will rebuild the algebra of chains and boundaries for directed edges and, furthermore, “oriented” cells. It appears that all we need t
  46 KB (7,844 words) - 12:50, 30 March 2016
• The algebra of oriented cells
  ...added complexity. We will rebuild the algebra of chains and boundaries for directed edges and, furthermore, “oriented” cells. ...as if cell $x$ is visited twice. These are the interpretations of the new, directed chains:
  36 KB (6,395 words) - 14:09, 1 December 2015
• Vector fields
  [[image:gravity graph.png| center]] This force is directed down, just as the movement of the ball. The work done on the ball by this f
  91 KB (16,253 words) - 04:52, 9 January 2019
• Data made Euclidean
  A graph<!--\index{graphs}--> is pure data. It consists of two sets: <!--s-->[[Image:realizations of a graph.png|center]]
  31 KB (5,219 words) - 15:07, 2 April 2016
• Topology Illustrated -- Index
  *[[directed edge|directed edge]] *[[graph|graph]]
  16 KB (1,773 words) - 00:41, 17 February 2016
• PageRank: testing
  ...of edges of a [[directed graph]] and $N$ the total number of nodes in the graph then the [[PageRank]] of the $i$-th node is: [[image:graph PR.png|right|200px]]
  3 KB (546 words) - 16:08, 23 October 2011
• Graph
  ...ected graph'' is a collection of points, ''nodes'', connected by arrows, ''directed edges''. An ''undirected graph'' can be thought of as a one-dimensional [[simplicial complex]].
  359 bytes (47 words) - 21:38, 19 February 2010
• PageRank as a flow
  ...be the set of edges of the graph and $N$ the total number of nodes in the graph. This is the formula for the PageRank: This formula can be interpreted as a [[PDE]] on a graph:
  4 KB (580 words) - 14:58, 5 January 2013
• Grayscale images - implementation
  ...of the objects. Geometrically, these boundaries are circular sequences of directed edges of pixels in the image. In other words, these are 0- and 1-cycles. Th ...ding Pixels]]. There are no circular sequences of arrows. Each node of the graph contains a pair of links (pointers) to next nodes. If the cycle is merged w
  13 KB (2,095 words) - 16:48, 4 December 2009
• Linear operators: part 4
  '''Example:''' Suppose $G$ is a directed graph. *$\ker D = $ all cycles of the graph.
  13 KB (2,067 words) - 01:11, 12 September 2011
• Metric tensor
  ...ample, here we have deformed the $x$-axis and changed the concavity of the graph at $x=1$: Concavity is an important concept that captures the shape of the graph of a function in calculus as well as the acceleration in physics. The upwar
  42 KB (7,131 words) - 17:31, 30 November 2015
• Geometry
  ...ample, here we have deformed the $x$-axis and changed the concavity of the graph at $x=1$: Concavity is an important concept that captures the shape of the graph of a function in calculus as well as the acceleration in physics. The upwar
  41 KB (6,928 words) - 17:31, 26 October 2015
• Applications of ODEs
  *3. directed along the segment between them. ...an just the gravity. The gravity is a ''central force'', i.e., one that is directed along the line between the location and the origin. Then the recursive form
  50 KB (8,692 words) - 14:29, 24 November 2018
• PageRank as a solution of a matrix equation
  Suppose $p_1,...,p_N$ are the $N$ pages/nodes in the graph (of the World Wide Web). Suppose also that $R(p_i)$ is the PageRank of node where $A$ is the [[adjacency matrix]] of the graph and $K$ is the diagonal matrix with the [[outdegree]]s $L(p_j)$ in the diag
  5 KB (811 words) - 18:36, 28 November 2012
• Social choice
  The simple formula fails when there are loops in the graph of links: '''Example.''' Consider the propagation of PageRank through this simple graph:
  47 KB (8,030 words) - 18:48, 30 November 2015
• ODEs
  ...''' Provide an analog of this iterative procedure for an IVP with domain a graph $K$ and find out under what circumstances this procedure works or fails. '''Exercise.''' (a) Explain how an arbitrary directed graph gives rise to advection and find its ODE. (b) What if, in addition to carry
  47 KB (8,415 words) - 15:46, 1 December 2015
• Metric complexes
  Suppose $K$ is a graph and its vertex $A$ has three adjacent edges $AB,AC,AD$. Then, in the dual c '''Proposition.''' If the complex $K$ is a directed graph with unit edges, the matrix $D$ of the second derivative is given by its el
  35 KB (5,871 words) - 22:43, 7 April 2016
• Calculus Illustrated -- preface
  ...s, etc.). However, a large proportion of time and effort can and should be directed elsewhere: ...shortest step from pen and paper: one cell - one number - one point on the graph. For more advanced tasks such as modelling, spreadsheets were chosen over o
  17 KB (1,758 words) - 13:57, 25 August 2019
• Second derivative and the Laplacian
  ...ntial forms]] of order $0$ and $1$. So the formula applies to a directed [[graph]]. In that case, the matrix $L$ of $\Delta$ can be computed directly: for e
  4 KB (608 words) - 13:13, 28 August 2015
• Web search
  That's why the web can be treated as [[directed graph]]s. So web search can be analyzed [[mathematics|mathematically]]. For how n
  469 bytes (65 words) - 14:50, 24 August 2011
• More ODEs
  '''Exercise.''' (a) Explain how an arbitrary directed graph gives rise to advection and present its ODE. (b) What if, in addition to ca
  26 KB (4,649 words) - 12:43, 7 April 2016
• Conferences and workshops
  ..., geographical information systems, protein folding, statistical analysis, graph drawing, visualization, circuit design, etc. ...has published over 30 articles on knot theory and related subjects. He has directed over 40 undergraduate students on research in knot theory and co-authored p
  8 KB (1,122 words) - 02:52, 24 October 2011