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• Vector and complex variables
  '''Example (rotation with stretch-shrink).''' Let's consider a more complex function: ==How complex numbers emerge==
  113 KB (18,750 words) - 02:33, 10 December 2018
• Triangulations
  *''simplicial complexes''<!--\index{simplicial complex}-->: cells are homeomorphic to points, segments, triangles, tetrahedra, ... *''cell complexes''<!--\index{cell complex}-->: cells are homeomorphic to points, closed segments, disks, balls, ...,
  30 KB (5,172 words) - 21:52, 26 November 2015
• Systems of ODEs
  More complex is the situation when the rate of change of the location depends on the loc $$\text{rabbits' gain }=\alpha\cdot x \cdot \Delta t,$$
  63 KB (10,958 words) - 14:27, 24 November 2018
• Partial differential equations
  $$\frac{\partial u}{\partial t}=\alpha\left(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}\ri A more complex model is the following. We still imagine that the string is made of springs
  53 KB (9,682 words) - 23:19, 18 November 2018
• PDEs
  ...ace'' is the cubical complex ${\mathbb R}^n$. The ''time'' is the standard complex ${\mathbb R}$. For now, we ignore the geometry of time and space. ...t of material'' $U=U(\alpha,t)$ is simply a number assigned to each room $\alpha$ which makes it an $n$-form. It also depends on time which makes it a $0$-f
  44 KB (7,469 words) - 18:12, 30 November 2015
• Cubical complex
  We defined a cubical complexes<!--\index{cubical complex}--> as a collection of cubical cells $K\subset {\mathbb R}^N$ for which the <!--200-->[[Image:cubical complex example 2.png|center]]
  29 KB (4,800 words) - 13:41, 1 December 2015
• Metric tensor
  [[image:cubical complex distorted.png|center]] [[image:cubical complex bent.png|center]]
  42 KB (7,131 words) - 17:31, 30 November 2015
• Applications of differential calculus
  The particles are flying away from the center. For more complex patterns, the vertical and horizontal will have to be interdependent. For e Thus, for each $x=c$, we indicate the angle $\alpha$, with $g(c)=\tan \alpha$, of the intersection of the graph of the unknown function $y=y(x)$ and the
  59 KB (10,063 words) - 04:59, 21 February 2019
• Simplicial complexes
  <!--s-->[[Image:example graph and simplicial complex.png|center]] This data set is called a ''simplicial complex''<!--\index{simplicial complex}--> (or sometimes even a “multi-graph”). Its elements are called $0$-,
  30 KB (5,021 words) - 13:42, 1 December 2015
• Metric complexes
  <!--75-->[[image:cubical complex distorted.png| center]] <!--75-->[[image:cubical complex bent.png| center]]
  35 KB (5,871 words) - 22:43, 7 April 2016
• Data made Euclidean
  <!--s-->[[Image:example graph and simplicial complex.png|center]] This data set is called a ''simplicial complex''<!--\index{simplicial complex}--> (or sometimes even a “multi-graph”). Its elements are called $0$-,
  31 KB (5,219 words) - 15:07, 2 April 2016
• Cubical complexes
  '''Definition.''' A cubical complex<!--\index{cubical complex}--> is a collection of cubical cells $K\subset {\mathbb R}^N$ for which the <!--200-->[[Image:cubical complex example 2.png| center]]
  20 KB (3,319 words) - 14:18, 18 February 2016
• More metric complexes
  Let's recall the mechanical interpretation of a realization $|K|$ of a metric complex $K$ of dimension $n=1$: ...rods using an extra set of rods (and hinges) that form a new, Hodge-dual, complex $K^{\star}$.
  21 KB (3,445 words) - 13:53, 19 February 2016
• Geometric complexes
  ...recall the mechanical interpretation of a realization $|K|$ of a geometric complex $K$ of ambient dimension $n=1$: ...rods using an extra set of rods (and hinges) that form a new, Hodge-dual, complex $K^{\star}$.
  20 KB (3,354 words) - 17:37, 30 November 2015
• Geometry
  [[image:cubical complex distorted.png|center]] [[image:cubical complex bent.png|center]]
  41 KB (6,928 words) - 17:31, 26 October 2015
• Cohomology
  $$V^* := \{ \alpha :V \to R, \alpha \text{ linear }\}.$$ ...' The dual $V^*$ of module $V$ is also a module, with the operations for $\alpha, \beta \in V^*,\ r \in R$ given by:
  45 KB (6,860 words) - 16:46, 30 November 2015
• Multilinear algebra
  '''Definition.''' For each vertex $A$ in a cell complex $K$, the (dimension $1$) ''tangent space'' at $A$ of $K$ is the set of $1$- Next, a subcomplex $L\subset K$ inherits its tangents from the ambient complex.
  49 KB (8,852 words) - 00:30, 29 May 2015
• Duality: forms as cochains
  $$V^* := \{\alpha :V \to R,\ \alpha \text{ linear}\}.$$ ....''' The dual $V^*$ of module $V$ is also a module, with the operations ($\alpha, \beta \in V^*,\ r \in R$) given by
  29 KB (4,540 words) - 13:42, 14 March 2016
• Diffusion equation
  ...standard cubical complex ${\mathbb R}^n$ and the ''time'' is the standard complex ${\mathbb R}$. For now, we ignore the geometry of time and space. ...t of material'' $U=U(\alpha,t)$ is simply a number assigned to each room $\alpha$ which makes it an $n$-form. It also depends on time which makes it a $0$-f
  35 KB (5,917 words) - 12:51, 30 June 2016
• Maps of polyhedra
  Previously, we proved that if complex $K^1$ is obtained from complex $K$ via a sequence of elementary collapses, then Suppose the circle is given by the simplest cell complex with just two cells $A,a$. Let's list ''all'' maps that can be represented
  51 KB (9,162 words) - 15:33, 1 December 2015

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