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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}\}.$$ ....''' The dual $V^*$ of module $V$ is also a module, with the operations ($\alpha, \beta \in V^*,\ r \in R$) given by
  29 KB (4,540 words) - 13:42, 14 March 2016
• Diffusion equation
  ...standard cubical complex ${\mathbb R}^n$ and the ''time'' is the standard complex ${\mathbb R}$. For now, we ignore the geometry of time and space. ...t of material'' $U=U(\alpha,t)$ is simply a number assigned to each room $\alpha$ which makes it an $n$-form. It also depends on time which makes it a $0$-f
  35 KB (5,917 words) - 12:51, 30 June 2016
• Maps of polyhedra
  Previously, we proved that if complex $K^1$ is obtained from complex $K$ via a sequence of elementary collapses, then Suppose the circle is given by the simplest cell complex with just two cells $A,a$. Let's list ''all'' maps that can be represented
  51 KB (9,162 words) - 15:33, 1 December 2015
• Fundamental group
  Here, if one thinks of the circle as a subset of the complex plane, the projection is given by $\pi (x)=e^{2\pi ix}$. In particular, the standard $n$-times wrapping loop $\alpha _n$ of the circle is lifted to $\gamma_n$ given by $\gamma_n(s)=ns$:
  10 KB (1,673 words) - 18:23, 2 December 2012
• ODEs
  Given a complex $K$, this is the most elementary ODE with respect to a $0$-form $f$: ...model ''motion''. Our domain is then the standard $1$-dimensional cubical complex $K={\mathbb R}$ and we are to study differential forms over ring $R={\bf R}
  47 KB (8,415 words) - 15:46, 1 December 2015
• Applications of ODEs
  *Case #3: complex conjugate roots. $$x = C e^{\alpha t} \cos(\beta t) + K e^{\alpha t} \sin(\beta t),$$
  50 KB (8,692 words) - 14:29, 24 November 2018
• Forms in Euclidean spaces
  The idea is as follows. Suppose cell complex $K$ is realized in ${\bf R}^n$. Then the tangent space $T_A(K)$ at vertex $ Let's review. The complex ${\mathbb R}^n$ comes with a standard orientation of all edges -- along the
  44 KB (7,778 words) - 23:32, 24 April 2015
• Computing homology
  Recall that given a [[cell complex]] $K$, a $k$-[[the algebra of chains|chain]] is a "formal" [[linear combina ...hains of different dimensions in order to capture the topology of the cell complex. This relation is given by the ''[[boundary operator]]''.
  26 KB (4,370 words) - 21:55, 10 January 2014
• The algebra of cells
  <!--150-->[[image:Cubical complex in 3d.png|center]] Now, what about boundaries of more complex objects?
  34 KB (5,644 words) - 13:35, 1 December 2015
• Diffusion
  ...one cell to the next. So, $F=F(p,t)$ is a $(n,1)$-form, but over the dual complex. ...cs (the generalized Hodge star) and the simulations for progressively more complex situations.
  39 KB (6,850 words) - 15:29, 17 July 2015
• The algebra of cubical cells
  <!--150-->[[image:Cubical complex in 3d.png| center]] Now, what about boundaries of more complex objects?
  46 KB (7,844 words) - 12:50, 30 March 2016
• The algebra of oriented cells
  ...nation of directions and our evaluation of the topology of a given cubical complex should remain the same. ...a cubical complex $K$ is a “formal linear combination of $k$-cells” in the complex:
  36 KB (6,395 words) - 14:09, 1 December 2015
• Discrete forms
  ...se here to concentrate on the ''cubical grid'', i.e., the infinite cubical complex acquired by dividing the Euclidean space into small, simple pieces. We deno ...they “look” identical. Frequently, one just assigns numbers to cells in a complex as we did above.
  35 KB (6,055 words) - 13:23, 24 August 2015
• Calculus on cubical complexes
  ...for now to concentrate on the ''cubical grid'', i.e., the infinite cubical complex acquired by dividing the Euclidean space into small, simple pieces (cubes). ...they “look” identical. Frequently, one just assigns numbers to cells in a complex as we did above.
  36 KB (6,218 words) - 16:26, 30 November 2015
• Differential calculus
  ...but let's review the tools at our disposal that allow us to deal with more complex functions. Examining the graph reveals that the maximum value lies somewher '''Example (quadratic polynomials).''' Things become much more complex if we need to analyze a quadratic function,
  84 KB (14,321 words) - 00:49, 7 December 2018
• Hodge duality of cubical forms
  [[image:dual complex dim 1.png|center]] ...x]] $K$ then the set of all of the duals of the cells of $K$ is the ''dual complex'' $K^*$.
  7 KB (1,114 words) - 18:10, 27 August 2015
• Elementary PDEs
  *the ''topology'' of the cell complex $L$ of the objects and springs, *the ''geometry'' given to that complex such as the lengths of the springs, and
  16 KB (2,843 words) - 21:41, 23 March 2016
• Manifolds
  ....''' Prove that for any ''countable'' ordinal $\alpha$, pasting together $\alpha$ copies of $[0,1)$ gives a space still homeomorphic to $[0,1)$. $$\alpha = \{U_y:y \in F\},\ \beta = \{V_y:y \in F\},$$
  51 KB (8,919 words) - 01:58, 30 November 2015
• Modeling with discrete exterior calculus
  ...the heat spreads though a grid or lattice of cells. These form a cellular complex composed of 0-cells, 1-cells, and 2-cells. In discretizing the heat equatio ...his combination of "rooms" and "walls" (and "columns") is called [[cubical complex]]. This approach is different from the numerical approach to the heat equat
  31 KB (5,254 words) - 17:57, 21 July 2012
• The algebra of oriented cubes
  ...nation of directions and our evaluation of the topology of a given cubical complex should remain the same. ...a cubical complex $K$ is a “formal linear combination of $k$-cells” in the complex:
  32 KB (5,480 words) - 02:23, 26 March 2016
• Advanced Topology: exercises
  ..., \gamma \in \Omega ^1({\bf R}^3)$ are linearly independent. Assuming $dd(\alpha)=dd(\beta)=dd(\gamma)=0$, prove that $dd(\psi ^1)=0$. ...example of a graph that cannot be represented by a one-dimensional cubical complex.
  9 KB (1,487 words) - 18:18, 9 May 2013
• Classes of functions
  '''Example.''' Constant functions are convenient building blocks for more complex functions. This is a familiar example of how we build from three constant f '''Example (quadratic polynomials).''' Things become much more complex if we need to analyze a quadratic function,
  143 KB (24,052 words) - 13:11, 23 February 2019
• Homology software
  ...sted below compute [[homology groups]] of [[cell complex]]es, [[simplicial complex]]es etc in a variety of applied scenarios, including [[persistence]]. ...tation, no support, etc. Proceed at your own risk. The commercial ones are Alpha Shapes by GeoMagic and Iris by [[Ayasdi]].
  4 KB (648 words) - 03:16, 30 March 2011
• More ODEs
  ...the gain of the prey population per unit of time is $\alpha x$ for some $\alpha\in {\bf R}^+$. The rate of predation upon the prey is assumed to be proport $$dx = \alpha x - \beta x y.$$
  26 KB (4,649 words) - 12:43, 7 April 2016
• Forms
  ==Directions in a cell complex== '''Definition.''' For each vertex $A$ in a cell complex $K$, the (dimension $1$) ''tangent space'' at $A$ of $K$ is the set of $1$-
  13 KB (2,459 words) - 03:27, 25 June 2015
• Elections
  [[image:complex of all ballots.png|center]] ...e are no empty intersections. Therefore, the ''space of all ballots'' $N_{\alpha}$ is a simplex!
  33 KB (5,872 words) - 13:13, 17 August 2015
• Algebra of discrete differential forms
  Suppose we have a [[cubical complex]]. ...tter means that there is a number associated with each cell present in the complex.
  6 KB (1,000 words) - 18:30, 22 August 2015
• Functions in higher dimensions
  We thus replace the study the complex geometry of ''locations'' in a multi-dimensional space with a study of dist $$A=<0,-32>,\ V_0=<100\cos \alpha,\ 100\sin \alpha>,\ P_0=(6,0),$$
  113 KB (19,680 words) - 00:08, 23 February 2019
• Selection votes
  ...over death, the continuity implies that, for a small enough probability $\alpha$, he would see a positive value in the following extreme lottery: *death: probability $\alpha >0$; and
  24 KB (3,989 words) - 01:56, 16 May 2016
• Introductory algebraic topology: review exercises
  2. Sketch the realization of the following cubical complex: 4.Prove that the cubical complex $K$ given below:
  9 KB (1,553 words) - 20:10, 23 October 2012
• Advanced Topology -- Spring 2013 -- final exam
  #Construct the dual cubical complex of the cubical complex of the figure 8 (the one with 7 edges). ...wedge \psi ^2$, where the latter is equal to $1$ on a single square, say $\alpha$, parallel to the $xy$-plane and equal to $0$ elsewhere.
  3 KB (532 words) - 15:09, 8 May 2013
• Computing persistent homology of filtrations
  ...ration but its computation does not require computing the homology of each complex of the filtration. Meanwhile, the above algorithm may have to compute the s Given a filtration, is there a complex with its homology equal to the homology of the filtration?
  8 KB (1,192 words) - 03:40, 30 October 2012
• Homology in dimension 2
  Recall that given a [[cell complex]] $K$, a $k$-[[the algebra of chains|chain]] is a "formal" [[linear combina In order to capture the topology of the cell complex we use the ''[[boundary operator]]''.
  8 KB (1,318 words) - 18:42, 27 August 2015

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