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• Maps of polyhedra
  ...ons doesn't have to be a cell map. As a result we are unable to define its chain map $f_{\Delta}:C(K)\to C(L)$ and unable to define or compute its homology ==The Simplicial Approximation Theorem==
  51 KB (9,162 words) - 15:33, 1 December 2015
• Cell maps
  Further, we compute the chain maps <center>topological spaces $\longrightarrow$ cell complexes $\longrightarrow$ chain complexes $\longrightarrow$ homology groups.</center>
  42 KB (7,005 words) - 03:10, 30 November 2015
• ODEs
  where $AX$ is any $1$-chain from $A$ to $X$ in $K$, i.e., any $1$-chain $AX=a\in K^{(1)}$ with $\partial a=X-A$. ...h is always the case when $K$ is acyclic). Indeed, if $a'$ is another such chain, we have
  47 KB (8,415 words) - 15:46, 1 December 2015
• Topology Illustrated -- Index
  *[[Arrow's Impossibility Theorem|Arrow's Impossibility Theorem]] *[[Arzela-Ascoli Theorem|Arzela-Ascoli Theorem]]
  16 KB (1,773 words) - 00:41, 17 February 2016
• Homology of maps
  Further, we compute the chain maps <center>topological spaces $\longrightarrow$ cell complexes $\longrightarrow$ chain complexes $\longrightarrow$ homology groups.</center>
  41 KB (6,926 words) - 02:14, 21 October 2015
• Combinatorial cell maps
  Further, we compute the chain maps ...-chains to $k$-chains. More precisely, $g$ is a collection of functions (a chain map):
  31 KB (5,330 words) - 22:14, 14 March 2016
• More ODEs
  '''Theorem (Uniqueness).''' The solution to the IVP above, if exists, is given iterati '''Theorem (Existence).''' If the values of the vector field $P$ are ''edges'' of $K$,
  26 KB (4,649 words) - 12:43, 7 April 2016
• The algebra of cubical cells
  <center> the boundary of a $k$-cell is a chain of $(k-1)$-cells.</center> <center> '' $0$ is a $k$-chain, for any $k$.'' </center>
  46 KB (7,844 words) - 12:50, 30 March 2016
• The algebra of cells
  <center>the boundary of a $k$-cell is a chain of $(k-1)$-cells.</center> ...', then its boundary $\partial\sigma = \sum_i \partial\sigma _i$ is also a chain.
  34 KB (5,644 words) - 13:35, 1 December 2015
• Euler and Lefschetz numbers
  '''Proposition.''' If $C_k(K)$ is the $k$-chain group of cell complex $K$, then '''Theorem (Inclusion-Exclusion Formula).''' For sets $A,B \subset X$, we have:
  41 KB (7,169 words) - 14:00, 1 December 2015
• The algebra of oriented cubes
  *“the chain $q=a+b+c+d$ goes once around the hole”. $\\$ *$p:=q+q=(a+b+c+d)+(a+b+c+d)=0$, even though it seems that “the chain $p$ goes twice around the hole”.
  32 KB (5,480 words) - 02:23, 26 March 2016
• Cohomology
  Therefore, by the Classification Theorem of Vector Spaces, we have the following: '''Theorem.'''
  45 KB (6,860 words) - 16:46, 30 November 2015
• Algebra on graphs
  A chain of edges These sets include $0$ understood as the chain with all coefficients equal to $0$.
  36 KB (6,177 words) - 02:47, 21 February 2016
• Chain complex
  ==Sequences of chain groups== ...of''<!--\index{chain groups}--> $K$ and we call $C_k$ the ''total'' $k$-''chain group''.
  14 KB (2,374 words) - 02:54, 16 October 2014
• Set-valued maps
  but only if the inverse exists! The theorem below gives us a sufficient condition when this is the case. ==The Vietoris Mapping Theorem==
  24 KB (4,382 words) - 15:52, 30 November 2015
• Tangent bundles
  ...e oriented edge in ${\mathbb R}$, or a combination of edges. It is a $1$-''chain''. Furthermore, the force $F$ defines $W$ as a linear function of $D$. It i Our conclusion doesn't change: $D$ is a $1$-chain and $F$ is a $1$-form. Even though this idea allows us to continue our stud
  16 KB (2,753 words) - 13:55, 16 March 2016
• Elementary ODEs
  where $AX$ is the $1$-chain from $A$ to $X$, i.e., $\partial AX=X-A$. ...ppose $b:=BC\in {\mathbb R}$. Then, by the Stokes Theorem<!--\index{Stokes Theorem}-->, we have
  16 KB (2,913 words) - 22:40, 15 July 2016
• Differential equations
  '''Theorem.''' An ODE the right-hand side of which is a function independent of $y$ an '''Theorem.''' An ODE the right-hand side of which is a function independent of $y$ an
  64 KB (11,426 words) - 14:21, 24 November 2018
• Homology groups of graphs
  A chain of edges These sets include $0$ as the chain with all coefficients equal to $0$.
  28 KB (4,685 words) - 17:25, 28 November 2015
• Calculus exercises: advanced
  ...$T$, provide the Riemann sum for the integral and an illustration for the theorem. *(a) State Stokes' Theorem for “simple regions”. (b) Use part (a) to compute the area of a circle.
  14 KB (2,538 words) - 18:35, 14 October 2017
• Higher order votes
  Once again, we apply the Uniqueness Theorem to conclude that there is, essentially, just one such decision function. '''Theorem.''' A tally $f$ for comparisons that satisfies Stability and Triviality is
  41 KB (6,942 words) - 05:04, 22 June 2016
• Elections
  ...the non-emptiness of this set is a necessary condition for existence of an extension of $g$ from $M(K)$ to $K$. We will demonstrate later that this is also a su '''Theorem.''' A rating tally satisfies Independence if and only if there is such a ho
  33 KB (5,872 words) - 13:13, 17 August 2015
• Cochain complexes and cohomology
  '''Theorem.''' $$\dim C({\bf R}) = \infty.$$ '''Theorem:''' $H^0(K) = {\bf R}^m$, where $m$ is the number of path components of $|K
  17 KB (2,592 words) - 14:38, 14 April 2013
• Continuous vs discrete differential forms
  ...ative produces two discrete functions that don't satisfy the [[Fundamental Theorem of Calculus]].''</center> But the original function and its derivative do satisfy the theorem! The problem is created by the sampling itself. In fact, we can now describ
  9 KB (1,483 words) - 13:54, 13 April 2013
• Discrete functions
  The integral provides may be the answer. The ''Fundamental Theorem of Calculus'' states: Then the Fundamental Theorem of Calculus fails and, with it, one of ''the laws of physics''!
  15 KB (2,532 words) - 12:21, 11 July 2016
• Real analysis: final 2
  #State the extension of the Mean Value Theorem to functions $f:\mathbf{R}^{n}\rightarrow\mathbf{R}.$ ...prove the Contraction Principle. Give examples of functions for which the theorem does or does not apply.
  4 KB (582 words) - 20:29, 13 June 2011