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• Topology of graphs
  ...pological questions about specific spaces. Given a subset of the Euclidean space: One can also think of a graph as just a collection of points in space, also called “vertices”, or “nodes” connected by paths, called “e
  25 KB (4,214 words) - 16:08, 28 November 2015
• Differential calculus of parametric curves
  *$(f(t),g(t))$ is thought of as the position in space at time $t$. The motion may also be in the $3$-dimensional space:
  130 KB (22,842 words) - 13:52, 24 November 2018
• Simplicial complexes
  ...see it, we often have to illustrate the data by a subset of the Euclidean space, as follows. Each node is plotted as a distinct point, but otherwise arbitr ==Simplices in the Euclidean space==
  30 KB (5,021 words) - 13:42, 1 December 2015
• Simplicial homology
  ...ata'', and yet we can still study the topology of subsets of the Euclidean space -- via realizations of graphs. We will follow this latter route with simpli *A representation of a topological space $X$ as a homeomorphic image of a realization of a simplicial complex $K$ is
  31 KB (5,170 words) - 13:44, 1 December 2015
• Calculus exercises: part I
  ...unction $f$ is given below. Sketch the graph of the derivative $f′$ in the space under the graph of $f$. Identify all important points and features on the g ...unction $f$ is given below. Sketch the graph of the derivative $f'$ in the space under the graph of $f$. Identify all important points on the graph.
  49 KB (8,436 words) - 17:14, 8 March 2018
• Social choice
  *the space of choices is a simplicial complex $W$; '''Theorem (Impossibility).''' Suppose the space of choices $W$ is path-connected and has torsion-free homology. Then the so
  47 KB (8,030 words) - 18:48, 30 November 2015
• Homology of maps
  <center>How does a continuous functions change the topology of the space?</center> If $R$ is a field, an $R$-module is a vector space.
  41 KB (6,926 words) - 02:14, 21 October 2015
• Cell maps
  <center>How does a continuous functions change the topology of the space?</center> If $R$ is a field, an $R$-module is a vector space.
  42 KB (7,005 words) - 03:10, 30 November 2015
• Differential equations
  ...are placed in the first row of the spreadsheet. As we progress in time and space, new numbers are placed in the next row of our spreadsheet: ...placed in the first row of the spreadsheet and, as we progress in time and space, new numbers are placed in the next row of our spreadsheet:
  64 KB (11,426 words) - 14:21, 24 November 2018
• Motion planning in robotics
  Now, the [[configuration space]] of a ''two-joint'' arm is the [[torus]]: ...e the lengths of the arms. Under the assumption $R_1>R_2$, the operational space is the annulus.
  5 KB (786 words) - 20:58, 27 August 2015
• Tangent bundles
  ...need for considering directions becomes clearer when the dimension of the space is $2$ or higher. We use ''vectors''. First, as we just saw, the work of th ...e set of all possible directions at point $A\in V={\bf R}^2$ form a vector space of the same dimension. It is $V_A$, a copy of $V$, attached to each point $
  16 KB (2,753 words) - 13:55, 16 March 2016
• The algebra of oriented cells
  ...free finitely-generated abelian group</TD> <TD>a finite-dimensional vector space over ${\bf R}$</TD> </TR> ...rt (${\bf Z}_2$ for the Klein bottle)</TD> <TD>a finite-dimensional vector space</TD> </TR>
  36 KB (6,395 words) - 14:09, 1 December 2015
• Higher order votes
  ...ncept of the ''orthogonal complement'' of a subset $P$ of an inner product space $V$: '''Proposition.''' Suppose $P$ is a subset of an inner product space $V$. Then its orthogonal complement is a summand:
  41 KB (6,942 words) - 05:04, 22 June 2016
• A graph, non-tree representation of the topology of a gray scale image by Saveliev
  The union of any collection of pixels is a subset of the [[Euclidean space|Euclidean plane]]. Therefore it acquires its topology from the plane [12] ( .... R., Harvey, R., and Cawley, G. C., “The segmentation of images via scale-space trees”, British Machine Vision Conference, 33-43 (1998).
  41 KB (6,854 words) - 15:05, 28 October 2011
• Topological spaces
  The pair $(X,\tau)$ is called a ''topological space''<!--\index{topological space}-->. The elements of $\tau$ are called ''open sets''<!--\index{open sets}-- “Open” disks on the plane, and balls in the Euclidean space, are also open.
  27 KB (4,693 words) - 02:35, 20 June 2019
• Vector spaces: introduction
  '''Definition:''' A ''vector space'' is a set $V$, where two operations are defined: Main idea: '''A vector space is "closed" under these operations'''.
  14 KB (2,238 words) - 17:38, 5 September 2011
• Homotopy and homotopy equivalence
  ...ver, there is a profound reason ''why'' they must all have one hole. These space are homeomorphic! Informally, we say that one space can be “deformed into” the other.
  45 KB (7,738 words) - 15:18, 24 October 2015
• Inner product spaces: part 1
  Note there is no measuring in a vector space. But, in that case, there are ''no distances, no limits, no calculus''... '''Plan:''' Take a vector space and equip it with extra structure, so that we ''can'' measure.
  14 KB (2,404 words) - 15:04, 13 October 2011
• Data made Euclidean
  ...o see it we often have to illustrate the data by a subset of the Euclidean space, as follows. Each node is plotted as a distinct point, but otherwise arbitr ==Simplices in the Euclidean space==
  31 KB (5,219 words) - 15:07, 2 April 2016
• Cubical complex
  For objects located in a Euclidean space, we would like to devise a data structure that we can use to first represen Suppose the Euclidean space ${\bf R}^N$ is given and so is its cubical grid ${\bf Z}^N$. Suppose also t
  29 KB (4,800 words) - 13:41, 1 December 2015

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