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• Maps of polyhedra
  ==Maps v. cell maps== ...exes, a map $f:|K|\to |L|$ between their realizations doesn't have to be a cell map. As a result we are unable to define its chain map $f_{\Delta}:C(K)\to
  51 KB (9,162 words) - 15:33, 1 December 2015
• Applications of differential calculus
  ...=0$. One is to use the ''linearization'' of $f$ as a substitute, one such approximation at a time. ...of the equation $f(x)=0$. We replace $f$ in this equation with its linear approximation $L$ at this initial point:
  59 KB (10,063 words) - 04:59, 21 February 2019
• ODEs
  ...e concerned with geometry in the rest of this section. We suppose $K$ is a cell complex. '''Definition.''' Given a cell complex $K$ and a function
  47 KB (8,415 words) - 15:46, 1 December 2015
• Combinatorial cell maps
  Below, we will see how the theory of simplicial maps is extended to general cell complexes. ...esenting the spaces as cell complexes, we examine to what cell in $L$ each cell in $K$ is taken by $f$:
  31 KB (5,330 words) - 22:14, 14 March 2016
• Functions of several variables
  ...f this table. It is now time to move to the right. We retreat to the first cell because the new material does not depend on the material of Chapter 17 -- o ...ical functions. Hence the need for parametric curves. Similarly, the first cell of the second column has surfaces but not all of them because some of them
  97 KB (17,654 words) - 13:59, 24 November 2018
• Triangulations
  ==Simplicial vs cell complexes== *''cell complexes''<!--\index{cell complex}-->: cells are homeomorphic to points, closed segments, disks, ball
  30 KB (5,172 words) - 21:52, 26 November 2015
• Topology Illustrated -- Index
  *[[best rating approximation|best rating approximation]] *[[boundary of a cubical cell|boundary of a cubical cell]]
  16 KB (1,773 words) - 00:41, 17 February 2016
• Tangent bundles
  ...cal complexes. Below, we make a step toward discrete calculus over general cell complexes. Next, we apply this idea to cell complexes.
  16 KB (2,753 words) - 13:55, 16 March 2016
• Euler and Lefschetz numbers
  ...stic''<!--\index{Euler characteristic}--> $\chi (K)$ of an $n$-dimensional cell complex $K$ is the alternating sum of the number of cells in $K$ for each d '''Proposition.''' If $C_k(K)$ is the $k$-chain group of cell complex $K$, then
  41 KB (7,169 words) - 14:00, 1 December 2015
• Forms in Euclidean spaces
  ==Euclidean cell complexes== ...and well as its cubical counterpart ${\mathbb R}^n$. We now consider other cell representations of ${\bf R}^n$. In addition, calculus would be incomplete u
  44 KB (7,778 words) - 23:32, 24 April 2015
• The gradient
  ...and the first row of all functions of several variables. The two have one cell in common; that is numerical functions. ...cation vs. velocity). That's why we have arrows that come back to the same cell. Once again, every continuous parametric curve is somebody's derivative. In
  74 KB (13,039 words) - 14:05, 24 November 2018
• Several variables
  ...spectively. In the next column, the initial velocity is entered in the top cell, $200$ and $0$ respectively. Below, the velocity is computed as a Riemann s In the next column, the initial location is entered in the top cell, $0$ and $200$ respectively. Below, the location is computed as a Riemann s
  76 KB (13,017 words) - 20:26, 23 February 2019
• Discrete differential forms
  ...d by taking more sample values (we look at the dials more often), then the approximation improves and, mathematically, the Riemann sums [[convergence|converge]] to Question: What about [[convergence]] of the approximation of the velocity? Suppose $f_n \rightarrow f$, does it mean that $f'_n \righ
  10 KB (1,471 words) - 12:50, 12 August 2015
• Convergence of the discrete to the continuous
  <!--100-->[[image:integral approximation.png| center]] <!--100-->[[image:derivative approximation.png| center]]
  21 KB (3,664 words) - 02:02, 18 July 2018
• Discrete Calculus: Applied Analysis on Graphs for Computational Science by Grady and Polimeni
  ...ignificant. Indeed, if you know the [[boundary]] of each k-[[cell]] in a [[cell complex]] in terms of $(k-1$-cells, you also know the exterior derivative o ...]] and the [[exterior derivative]] are ''defined by the structure of the [[cell complex|complex]] itself''. In other words, the structure of the derivative
  11 KB (1,663 words) - 16:03, 26 November 2012
• Discrete functions
  ...aking more sample values (i.e., we look at the dials more often). Then the approximation improves too and, mathematically, the Riemann sums above converge to the Ri To summarize, no specific approximation satisfies the laws of calculus (or the laws of physics).
  15 KB (2,532 words) - 12:21, 11 July 2016
• Set-valued maps
  ...e simplicial complexes and $Q_a$ is an acyclic subcomplex of $L$ for every cell $a$ in $K$. (1) Suppose $g_0:K^{(0)}\to L^{(0)}$ is a map of vertices that ...map (extension) $g=\{g_i\}:C(K)\to C(L)$ that $g(a) \subset Q_a$ for every cell $a$ in $K$. (2) If two chain maps $g$ and $g'$ satisfy this condition, $g(a
  24 KB (4,382 words) - 15:52, 30 November 2015
• CellAnalyst
  ...alyze his [[microscopy]] images and manage the output data. [[Particle and cell analysis]] is the most immediate application.</p> ...ed to automate the analysis of digital images, especially ones coming from cell biology. It works as follows:</p>
  6 KB (966 words) - 18:49, 19 February 2011
• Discrete forms and cochains
  ...aking more sample values (i.e., we look at the dials more often), then the approximation improves too and, mathematically, the Riemann sums above converge to the Ri *Any given approximation violates the laws of calculus (and the laws of physics).
  34 KB (5,619 words) - 16:00, 30 November 2015
• Topology II -- Spring 2014 -- final exam
  ...maps and the homology maps of cell maps. Give an example of two different cell maps with the same homology map. ...e the Simplicial Approximation Theorem with definitions. Find a simplicial approximation of the rotation of the triangulated circle through $\sqrt{2}\,\pi$.
  1 KB (154 words) - 23:07, 5 May 2014

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