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• 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

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