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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
• Diffusion with various geometry
  ...defined as a [[linear operator]] between the primal and the dual [[cochain complex]]es: ...the ''discrete (geometric) Hodge star'' of $\phi$ is a cochain on the dual complex and it is defined by its values on the dual cells: for a $m$-[[chain]] $a$
  13 KB (2,121 words) - 16:33, 7 June 2013
• Axioms of chain complexes
  '''2. [[Refinement]]''': Refinement/subdivision doesn't change the chain complex. That is, if $(X,A)$ is a pair and $\gamma$ is an open cover of $X$, then t ...t space $P$ is acyclic. That is, the boundary operator $\partial$ of chain complex $C(P)$ of $P$ satisfies
  4 KB (592 words) - 14:13, 4 August 2013
• Combinatorial cell maps
  ...all possible ways complex $K=\{A,a,\alpha\}$ can be mapped to another cell complex. '''Exercise.''' Choose a different cell complex to represent $Y$ above in such a way that the projection is then a cell map
  31 KB (5,330 words) - 22:14, 14 March 2016
• Boundary operator
  #redirect[[Chain complex]] [[Image:1dim complex.jpg|center|150px]]
  5 KB (837 words) - 16:24, 1 June 2014
• Cell maps
  ...all possible ways complex $K=\{A,a,\alpha\}$ can be mapped to another cell complex. '''Exercise.''' Choose a different cell complex to represent $Y$ above in such a way that the projection is then a cell map
  42 KB (7,005 words) - 03:10, 30 November 2015
• Integration in dimension n
  Note: we can compute the volume of a complex figure G by putting it in a box and setting the density equal to zero in th What about more complex, curved domains? What if the domain of integration $G$ is represented as a
  33 KB (5,415 words) - 05:58, 20 August 2011
• Cubical calculus
  ...for now to concentrate on the ''cubical grid'', i.e., the infinite cubical complex acquired by dividing the Euclidean space into cubes, ${\mathbb R}^n$. ...they “look” identical. Frequently, one just assigns numbers to cells in a complex as we did above. The difference is that these numbers aren't the coefficien
  25 KB (4,238 words) - 02:30, 6 April 2016
• Precalculus
  '''Theorem (Slope).''' The angle $\alpha$ between the $x$-axis and the line from $O$ to $P=(x,y)\ne O$ is given by i $$\tan \alpha =\frac{y}{x}.$$
  100 KB (16,148 words) - 20:04, 18 January 2017
• The gradient
  ...ctor attached to that point. This is just a clever way to visualize such a complex -- in comparison to the ones we have seen so far -- function. It's a ''loca '''Example.''' Three-dimensional vector fields are more complex. The one below is similar to the first example above:
  74 KB (13,039 words) - 14:05, 24 November 2018
• Dual cells and dual forms
  ...$(2-k)$-cell (dual) $\alpha^*$ with $\alpha^*$ centered at the center of $\alpha$. $$\alpha^{**}=\alpha$$
  2 KB (266 words) - 18:11, 27 August 2015
• Cochain complexes and cohomology
  Generally, if $K$ is a [[cubical complex]], $\dim C^k(K) = $ number of $k$-cells in $K$. Consider the [[cochain complex]]:
  17 KB (2,592 words) - 14:38, 14 April 2013
• Geometric Hodge duality
  ==Dual complex== [[image:metric tensor vs dual complex.png|center]]
  5 KB (867 words) - 13:24, 19 May 2013
• Convergence of the discrete to the continuous
  Suppose we have a sequence of copies of the standard cubical complex ${\mathbb R}$, ...\subset {\bf R}$ and suppose $f$ is integrable on $[A,B]$. Then, for every complex $K$ representing $[A,B]$ we define $g$ as the $1$-form acquired by integrat
  21 KB (3,664 words) - 02:02, 18 July 2018
• Applications of Computational Topology by Christopher Johnson
  ...s to bypass the computationally intensive process of building a simplicial complex. Using persistent homology allows us to locate important features in the im
  2 KB (282 words) - 16:46, 20 February 2011
• What shape of sword is best for cutting?
  ...hrough but, because the sword is rigid, the rest is also slowed down. This complex motion-control problem is routinely solved by a skillful swordsman. However ...help of the rotation matrix (shown at the top) for an angle of rotation $\alpha$:
  14 KB (2,504 words) - 14:59, 17 September 2019
• Modelling with discrete vector fields and forms
  ...[u,v]$, let $f(\alpha u+(1-\alpha)v)=\alpha f(u)+(1-\alpha)f(v)$ for all $\alpha \in [0,1]$; Dynamical systems are known for exhibiting a complex behavior even with a simple choice of $F$:
  9 KB (1,561 words) - 16:06, 27 August 2015
• Hodge decomposition
  *''[[co-exact]]'' if $\omega=\delta\alpha$ for some form $\alpha$; $$\phi = d\alpha +\delta \beta + \gamma \,$$
  1 KB (241 words) - 20:52, 13 March 2013
• Products
  More complex outcomes result from attaching to every point of $X$ a copy of $Y$: $$\alpha := \{V_b:b\in Y\}.$$
  44 KB (7,951 words) - 02:21, 30 November 2015
• Cup product
  where $\sigma$ is a $(p+q)$-[[simplex]] in the [[simplicial complex]] $X$ and $\sigma _{0,1, ..., p}$ is the $p$-th "front face" and $\sigma_{p For a [[cubical complex]] in the $n$-dimensional space, cochains are defined on the [[cube]]s:
  3 KB (460 words) - 20:57, 13 March 2013
• Calculus of discrete differential forms
  Our definition, and the theorem, is applicable to any [[cubical complex]] $R \subset {\bf R}$ because, by definition, it is a set of cells such tha We need to find the value of the $2$-form $d \varphi$ on the $2$-cells $\alpha$ and $\beta$. We subtract vertically and horizontally, as before, and then
  9 KB (1,503 words) - 18:30, 22 August 2015
• Advanced Topology: midterm
  ..., \gamma \in \Omega ^1({\bf R}^3)$ are linearly independent. Assuming $dd(\alpha)=dd(\beta)=dd(\gamma)=0$, prove that $dd(\psi ^1)=0$. ...wedge \psi ^2$, where the latter is equal to $1$ on a single square, say $\alpha$, parallel to the $xy$-plane and equal to $0$ elsewhere.
  2 KB (282 words) - 16:11, 14 March 2013
• Algebraic operations with forms and cohomology
  We will assume the following: $R$ is any [[cubical complex]], i.e., a set of cells such that if cell $s \in R$, then all of its [[boun on $\alpha$.
  8 KB (1,289 words) - 15:11, 9 October 2012
• Transformations
  '''Example (rotation with stretch-shrink).''' Let's consider a more complex function: $$V_{\perp}=||V||\sin \alpha,$$
  46 KB (7,625 words) - 13:08, 26 February 2018
• Integration of differential forms of degree 0 and 1
  ...screte manifold = a [[combinatorial manifold]] = a manifold-like [[cubical complex]]. ...$p(t)=x$, $q(s)=x$, then we can assume $p'(t) = \alpha q'(s)$, for some $\alpha(x)$, for every $x$ on $C$. Then
  12 KB (1,906 words) - 17:44, 31 December 2012
• Vector fields
  $$F_{||}=||F||\cos \alpha,$$ where $\alpha$ is the angle of $F$ with $D$.
  91 KB (16,253 words) - 04:52, 9 January 2019
• Approaches to image analysis
  *Alpha Shapes [5] runs only in Linux/Unix/Sun and only computes Betti numbers of 3 ...used to represent its tunnels. However, this representation fails in more complex settings such as porous material. The representation of tunnels in 3D will
  13 KB (2,018 words) - 13:55, 12 May 2011
• Acyclic models
  $Ch(\mathcal{Mod})$ be the category of [[chain complex]]es of $R$-modules. *$\mathcal{F}=S_k$ is the functor of the $k$-th term of the singular chain complex;
  8 KB (1,126 words) - 15:32, 14 July 2013
• Calculus on chains
  .... Combined with the [[boundary operator]] $\partial$ they form the [[chain complex]] $\{C_*,\partial\}$ of $K$: But this is simply the [[cochain complex]] $\{C^*,d\}$:
  4 KB (556 words) - 14:03, 30 March 2013
• The derivative
  What about the latter, “touching”, line? It is much more complex. ...with which the cannonball leaves the muzzle -- no matter what the angle, $\alpha$, is. That's where the ''initial'' horizontal and the vertical velocities c
  75 KB (13,000 words) - 15:12, 7 December 2018
• Calculus of chain maps
  Suppose we have two copies of the complex ${\mathbb R}$, ${\mathbb R}_x$ and ${\mathbb R}_y$, possibly representing t $$d(\alpha f+\beta g)=\alpha df+\beta dg,$$
  41 KB (7,344 words) - 12:52, 25 July 2016
• Tangent cone
  A $2$-manifold is a [[surface]] and as a cubical complex it will look like this, compared to a $1$-manifold: $$TC_x(A)={\bf R} \cdot TC_x(A)=\{\alpha v: v \in TC_x(A),\alpha \in {\bf R} \}.$$
  5 KB (882 words) - 02:14, 26 March 2013
• Inner product
  ...cations when some measurements intended to evaluate the difference between complex entities, such molecules, may be more important than others. $$\cos \alpha = \frac{< u, v >}{ \lVert u \rVert \cdot \lVert v \rVert }$$
  4 KB (749 words) - 20:12, 1 May 2013
• JPlex
  ...ata as a list of vertices. After defining an epsilon value (similar to the alpha-value in [[persistent homology]]), PLEX calculates the appropriate edges fo ...logy]] is computed for [[point cloud]]s or [[filtration]]s of [[simplicial complex]]es.
  2 KB (231 words) - 03:21, 30 March 2011
• Visual image search engines
  ...nd if you can't solve a simpler problem how can you expect to solve a more complex one? I tried this color image. These are the tags: <em>texture, red, food, FAQ: "will be launching their alpha version to a small group of users on February 28th, 2006".
  15 KB (2,424 words) - 19:20, 14 June 2011
• Image search
  ...nd if you can't solve a simpler problem how can you expect to solve a more complex one? I tried this color image. These are the tags: <em>texture, red, food, FAQ: "will be launching their alpha version to a small group of users on February 28th, 2006".
  17 KB (2,793 words) - 17:40, 14 September 2008