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• Functions in higher dimensions
  ...seen on the map, are known. The grade of the road is also known. How fast is the car climbing? The first variable is time, $t$. We also have two ''spatial'' variables: the horizontal location
  113 KB (19,680 words) - 00:08, 23 February 2019
• A graph, non-tree representation of the topology of a gray scale image by Saveliev
  ...as a combination of pixels as well as edges and vertices. The second tool is [[cycle]]s: both the connected components and the holes are captured by cir ...s of the gray level function of the image. The rationale for this approach is that the connected components of these sets are arguably building blocks of
  41 KB (6,854 words) - 15:05, 28 October 2011
• Applications of integral calculus
  ...half so that the area of the whole circle is then twice this number. This is too limiting... Let's start over. ...le with vertical bars based on these segments. Then the area of the circle is approximated by the sum of the areas of the bars: we add a column of the wi
  103 KB (18,460 words) - 01:01, 13 February 2019
• The gradient
  ...all functions of several variables. The two have one cell in common; that is numerical functions. This time we will see how everything is interconnected. We show with the red arrows for different types of function
  74 KB (13,039 words) - 14:05, 24 November 2018
• Continuous functions
  <center>if $x$ is close to $a$ then $f(x)$ is close to $f(a)$:</center> <center>for any $\epsilon > 0$ there is a $\delta > 0$ such that $|x - a| < \delta \Rightarrow | f(x) - f(a) | < \e
  42 KB (7,138 words) - 19:08, 28 November 2015
• Maps of polyhedra
  Previously, we proved that if complex $K^1$ is obtained from complex $K$ via a sequence of elementary collapses, then ...proof was ''straightforward''. However, the result, as important as it is, is a very limited instance of the invariance of homology. We explore next what
  51 KB (9,162 words) - 15:33, 1 December 2015
• Homotopy
  The answer we have been giving is: they all have one hole. However, there is a profound reason ''why'' they must all have one hole. These space are home The reasoning, still not fully justified, is transparent:
  46 KB (7,846 words) - 02:47, 30 November 2015
• Products
  The idea of the product may be traced to the image of a stack, which is a simple arrangement of multiple copies of $X$: We can think of it as if a copy of the $y$-axis is attached to every point on the $x$-axis. Or, we can think in terms of ''pro
  44 KB (7,951 words) - 02:21, 30 November 2015
• Social choice
  We discovered that there is no such solution when the homology of the forest is non-trivial, such as one with a lake in the middle. This is the general setup. There are $m$ voters, or agents, making their selections
  47 KB (8,030 words) - 18:48, 30 November 2015
• Quotients
  A new way of building new things from old is ''gluing'': ...flexivity Axiom, $A \sim A$, can be understood as: every spot of the sheet is glued to itself. The Symmetry Axiom, $A \sim B \Rightarrow B \sim A$, becom
  26 KB (4,538 words) - 23:15, 26 November 2015
• Topological spaces
  ...s to equip each of the sets involved with an additional structure called ''topology''. ...X$, a collection $\tau$ of subsets of $X$ is called a ''topology<!--\index{topology}--> on'' $X$ if it satisfies the following conditions:
  27 KB (4,693 words) - 02:35, 20 June 2019
• Euler and Lefschetz numbers
  We already know, and will prove below, that the meaning of the formula is topological. ...Euler characteristic}--> $\chi (K)$ of an $n$-dimensional cell complex $K$ is the alternating sum of the number of cells in $K$ for each dimension:
  41 KB (7,169 words) - 14:00, 1 December 2015
• PDEs
  Instead of being carried around, the heat is ''exchanged'' -- with adjacent locations. The process is also known as ''diffusion.''
  44 KB (7,469 words) - 18:12, 30 November 2015
• Diffusion
  ...e average of the temperature of the four adjacent rooms. This simple model is implemented with an Excel simulation with the following short formula: Normally, only a proportion $k$ of this amount is shared.
  39 KB (6,850 words) - 15:29, 17 July 2015
• Homotopy and homotopy equivalence
  The answer we have been giving is: they all have one hole. However, there is a profound reason ''why'' they must all have one hole. These space are home The reasoning, still not fully justified, is transparent:
  45 KB (7,738 words) - 15:18, 24 October 2015
• Cell complex
  ...<!--\index{cells}-->, and do it in a gradual and orderly manner. The point is to be able to build and compute homology<!--\index{homology}-->, and do it The main difference is in the manner these cells are found. In the case of cubical complexes<!--\i
  40 KB (6,459 words) - 23:27, 29 November 2015
• Triangulations
  ...nion of a collection of subsets of a Euclidean space, while a cell complex is built via the quotient construction<!--\index{quotient}-->, which always re ...lex $K$ has $n+1$ faces<!--\index{face}-->, $\sigma < \tau$, each of which is an $(n-1)$-simplex, illustrated on the left:
  30 KB (5,172 words) - 21:52, 26 November 2015
• Simplicial complexes
  A graph<!--\index{graph}--> is pure data. It consists of two sets: ...ustrate the data by a subset of the Euclidean space, as follows. Each node is plotted as a distinct point, but otherwise arbitrarily, and these points ar
  30 KB (5,021 words) - 13:42, 1 December 2015
• Diffusion equation
  Instead of being carried around, the heat is ''exchanged'' -- with adjacent locations. It's a circle. The process is also known as ''diffusion.''
  35 KB (5,917 words) - 12:51, 30 June 2016
• Data made Euclidean
  A graph<!--\index{graphs}--> is pure data. It consists of two sets: ...ustrate the data by a subset of the Euclidean space, as follows. Each node is plotted as a distinct point, but otherwise arbitrarily, and these points ar
  31 KB (5,219 words) - 15:07, 2 April 2016

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