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• Tangent bundles
  ...need for considering directions becomes clearer when the dimension of the space is $2$ or higher. We use ''vectors''. First, as we just saw, the work of th ...e set of all possible directions at point $A\in V={\bf R}^2$ form a vector space of the same dimension. It is $V_A$, a copy of $V$, attached to each point $
  16 KB (2,753 words) - 13:55, 16 March 2016
• The algebra of oriented cells
  ...free finitely-generated abelian group</TD> <TD>a finite-dimensional vector space over ${\bf R}$</TD> </TR> ...rt (${\bf Z}_2$ for the Klein bottle)</TD> <TD>a finite-dimensional vector space</TD> </TR>
  36 KB (6,395 words) - 14:09, 1 December 2015
• Higher order votes
  ...ncept of the ''orthogonal complement'' of a subset $P$ of an inner product space $V$: '''Proposition.''' Suppose $P$ is a subset of an inner product space $V$. Then its orthogonal complement is a summand:
  41 KB (6,942 words) - 05:04, 22 June 2016
• A graph, non-tree representation of the topology of a gray scale image by Saveliev
  The union of any collection of pixels is a subset of the [[Euclidean space|Euclidean plane]]. Therefore it acquires its topology from the plane [12] ( .... R., Harvey, R., and Cawley, G. C., “The segmentation of images via scale-space trees”, British Machine Vision Conference, 33-43 (1998).
  41 KB (6,854 words) - 15:05, 28 October 2011
• Topological spaces
  The pair $(X,\tau)$ is called a ''topological space''<!--\index{topological space}-->. The elements of $\tau$ are called ''open sets''<!--\index{open sets}-- “Open” disks on the plane, and balls in the Euclidean space, are also open.
  27 KB (4,693 words) - 02:35, 20 June 2019
• Vector spaces: introduction
  '''Definition:''' A ''vector space'' is a set $V$, where two operations are defined: Main idea: '''A vector space is "closed" under these operations'''.
  14 KB (2,238 words) - 17:38, 5 September 2011
• Homotopy and homotopy equivalence
  ...ver, there is a profound reason ''why'' they must all have one hole. These space are homeomorphic! Informally, we say that one space can be “deformed into” the other.
  45 KB (7,738 words) - 15:18, 24 October 2015
• Inner product spaces: part 1
  Note there is no measuring in a vector space. But, in that case, there are ''no distances, no limits, no calculus''... '''Plan:''' Take a vector space and equip it with extra structure, so that we ''can'' measure.
  14 KB (2,404 words) - 15:04, 13 October 2011
• Data made Euclidean
  ...o see it we often have to illustrate the data by a subset of the Euclidean space, as follows. Each node is plotted as a distinct point, but otherwise arbitr ==Simplices in the Euclidean space==
  31 KB (5,219 words) - 15:07, 2 April 2016
• Cubical complex
  For objects located in a Euclidean space, we would like to devise a data structure that we can use to first represen Suppose the Euclidean space ${\bf R}^N$ is given and so is its cubical grid ${\bf Z}^N$. Suppose also t
  29 KB (4,800 words) - 13:41, 1 December 2015
• Calculus of chain maps
  ...b R}$, ${\mathbb R}_x$ and ${\mathbb R}_y$, possibly representing time and space respectively. We are to study functions, that will possibly represent motion in space. They have to somehow respect the cell structure of ${\mathbb R}$. Let's re
  41 KB (7,344 words) - 12:52, 25 July 2016
• Precalculus
  One can already see how harder is to visualize things in the $3$-dimensional space, which further justifies the need for the algebraic treatment of geometry t
  100 KB (16,148 words) - 20:04, 18 January 2017
• Classes of functions
  One can already see how harder is to visualize things in the $3$-dimensional space, which further justifies the need for the algebraic treatment of geometry t ...on we take up the second geometric task, ''directions'', in the Euclidean space equipped with the Cartesian coordinate system.
  143 KB (24,052 words) - 13:11, 23 February 2019
• Pattern recognition in computer vision
  ...nsight into the nature of the problem. Once all the data is in a Euclidean space (no matter how large) all [[statistics|statistical]], [[topological data an ...k at the [[distance]] between points – images - in this 10,000-dimensional space. It can be defined in a number of ways, but as long as it is symmetric we h
  9 KB (1,526 words) - 17:54, 1 July 2011
• Cubical calculus
  ...c. These “expressions” are called ''differential forms''<!--\index{acyclic space}--> and each of them determines such a new function. That's why we further ...tter of ''calculus'', the calculus of differential forms<!--\index{acyclic space}-->:
  25 KB (4,238 words) - 02:30, 6 April 2016
• Maps of polyhedra
  Without refining the target space, repeating this approximation doesn't produce a sequence $g_n$ convergent t ...ether the set is included in one of the elements of the cover. In a metric space, it's simpler:
  51 KB (9,162 words) - 15:33, 1 December 2015
• Partial differential equations
  ...because our temperature distribution function $w$ is then a $0$-form in a space of any dimension. Recall that a partition of a ''box'' $B$ in the $txy$-space comes from partitions of its three edges as described in Chapter 20:
  53 KB (9,682 words) - 23:19, 18 November 2018
• Euler and Lefschetz numbers
  *(2) If $Y$ is a subspace of vector space $X$, then '''Fixed Point Problem.''' If $X$ is a topological space and $f:X \to X$ is a self-map, does $f$ have a fixed point: $x\in X$ such t
  41 KB (7,169 words) - 14:00, 1 December 2015
• The algebra of oriented cubes
  ...case of higher dimensions will require using the product structure of the space. ==The boundary of a cube in the $N$-dimensional space==
  32 KB (5,480 words) - 02:23, 26 March 2016
• Topology
  ...ns and coming (or not coming) back will produce information about loops in space. These loops, or $1$-cycles, are used to detect tunnels in the Universe. ...a new item: space, or, more accurately: a 3-dimensional space. How such a space creates a 3-''cycle'' may be hard or impossible to visualize. Nonetheless,
  20 KB (3,407 words) - 21:46, 30 November 2015
• Chain complexes
  If $R$ is a field, an $R$-module is a vector space. ...uld like to be able to study functions defined on subsets of the Euclidean space.
  33 KB (5,293 words) - 03:06, 31 March 2016
• Calculus exercises: part III
  ...et tall. Represent the motion as a parametric curve in the $3$-dimensional space. *Find an equation of the spiral in space converging to the origin as below (view from above):
  46 KB (8,035 words) - 13:50, 15 March 2018
• The algebra of cells
  We know that we can decompose the $n$-dimensional Euclidean space into $n$-dimensional blocks, the $n$-cells. For example, this is how an obj Thus, our approach to decomposition of space, in any dimension, boils down to the following:
  34 KB (5,644 words) - 13:35, 1 December 2015
• How to compute homology
  ...rticle we summarize the procedure for computing the [[homology as a vector space|homology]] of a [[cell complex]], by hand. ...[[chain group]] $C_k(K)$ is given as a vector space with [[basis of vector space|basis]] consisting of the cells of the complex:
  6 KB (1,049 words) - 09:21, 3 September 2011
• Stability Analysis
  ...roblem from which they were obtained, when the values of both the time and space steps are allowed to all tend to zero. Due to known results from Numerical ...OGETHER WITH OUR BOUNDARY CONDITIONS, to find expressions for the time and space components of our separated solution; using this, and a clever trick from a
  12 KB (2,051 words) - 03:51, 11 August 2012
• Diffusion
  Note: When the domain isn't the whole space, the pipes at the border of the region have to be “removed”. Here we us ...n be written simply as $Qdt$, where $Q$ is a dual $1$-form with respect to space.
  39 KB (6,850 words) - 15:29, 17 July 2015
• Wedge product of continuous forms
  or in $3$-space Now, for the $2$-dimensional space we've got all we need. All $2$-forms are given by:
  14 KB (2,417 words) - 18:16, 22 August 2015
• Simplicial complexes and maps
  ...see it, we often have to illustrate the data by a subset of the Euclidean space, as follows. Each node is plotted as a distinct point, but otherwise arbitr ...uclidean space; for example, a plane (and a square) in the $3$-dimensional space is oriented by a choice of one of the two unit normal vectors.
  27 KB (4,625 words) - 12:52, 30 March 2016
• Algebra on graphs
  ...a+sb,\ \forall r,s \in {\bf R}$. Then $\{a,b \}$ is a basis of this vector space and the idea becomes:
  36 KB (6,177 words) - 02:47, 21 February 2016
• Calculus I, the discrete version
  ...e standard domain, the discrete representation of ${\bf R}$. Second, the ''space'' is given by ${\bf R}$, at the simplest. ...that the only possible type of motion in this force-less and distance-less space-time is uniform; i.e., it is a repeated addition:
  64 KB (11,521 words) - 19:48, 22 June 2017
• Chain maps
  ...have proven the identity for all basis elements, simplices, of the vector space, $C_k(K)$, then the two linear operator coincide. $\blacksquare$
  47 KB (8,115 words) - 16:19, 20 July 2016
• Lefschetz theory for coincidences
  ...ility of control systems dictates the need for a higher dimensional domain space $N$. For example, the projection of the [[torus]] on the [[circle]] is such ...y differential equation: $M$ is the space, $F$ is the time, and $N$ is the space-time; and
  19 KB (3,563 words) - 15:20, 9 December 2012
• Introduction to discrete calculus
  Second, the ''space'' is given by any ring $R$, in general. For all the derivatives to make sen ...that the only possible type of motion in this force-less and distance-less space-time is uniform; i.e., it is a repeated addition:
  40 KB (6,983 words) - 19:24, 23 July 2016
• How to compute Betti numbers
  One can acquire the [[Betti numbers]] from the [[homology as a vector space|homology groups]] (and [[cohomology]]) by taking their dimensions/ranks. Ho ...[[chain group]] $C_k(K)$ is given as a vector space with [[basis of vector space|basis]] consisting of the cells of the complex:
  5 KB (890 words) - 14:47, 24 August 2014
• Cochain complexes and cohomology
  both are [[vector space]]s, very familiar objects. ...hat the set $P = \{1, x, x^2, \ldots\}$ is [[linearly independent]] in the space of functions $C({\bf R}) = \Omega^0({\bf R})$.
  17 KB (2,592 words) - 14:38, 14 April 2013
• Quotients of vector spaces
  Given a [[vector space]] $L$ and a subspace $M$. How do we "remove" $M$ from $L$? Unfortunately, $L \setminus M$ isn't a vector space!
  6 KB (1,115 words) - 16:03, 27 August 2015
• Selection votes
  Then, instead of a single simplex, ''the space of outcomes is a simplicial complex''. The complex is meant to represent al Do we ever face a space of outcomes with a more complex topology, such as one with holes, voids, et
  24 KB (3,989 words) - 01:56, 16 May 2016
• Cubical complexes
  For objects located in a Euclidean space, we would like to devise a data structure that we can use to first represen Suppose the Euclidean space ${\bf R}^N$ is given and so is its cubical grid ${\bf Z}^N$. Suppose also t
  20 KB (3,319 words) - 14:18, 18 February 2016
• Discrete forms and cochains
  ...ain''<!--\index{cochains}--> on $K$ is any linear function from the vector space of $k$-chains to $R$: '''Proposition.''' The $k$-cochains on complex $K$ form a vector space denoted by $C^k=C^k(K)$.
  34 KB (5,619 words) - 16:00, 30 November 2015
• Simplicial maps and chain maps
  ...have proven the identity for all basis elements, simplices, of the vector space, $C_k(K)$, then the two linear operator coincide. ...plex}--> $K$ can be realized as a topological space<!--\index{ topological space}-->. The way to construct it is by treating the list of vertices and simpli
  34 KB (5,897 words) - 16:05, 26 October 2015
• Torus
  ==As a product space== ==As a quotient space and a cell complex==
  5 KB (708 words) - 22:46, 3 September 2011
• More ODEs
  ...We also take as a model a fluid flow. The “phase space” ${\bf R}^2$ is the space of all possible locations. Then the position of a given particle is a funct ...point. Thus, there is one vector at each point picked from a whole vector space:
  26 KB (4,649 words) - 12:43, 7 April 2016
• Calculus exercises: advanced
  ...the linear map $L(x_1,x_2)=<3x_1+x_2,x_1-x_2>$ find the basis of the null space (the kernel). *(a) Give the definition of a basis of a linear space. (b) Show that the vectors $(1,0,0), (1,1,0), (1,1,1)$ form a basis of ${\b
  14 KB (2,538 words) - 18:35, 14 October 2017
• Means
  ...he group may have no division. If these are points in a subset of a vector space, the set may be non-convex. ...--\index{algebraic mean}--> if it is a homomorphism. For $X$ a topological space, a mean is ''topological''<!--\index{topological mean}--> if it is a contin
  10 KB (1,914 words) - 03:05, 6 November 2018
• Simplicial maps
  ...e have proven the identity for all basis elements, simplices of the vector space, $C_k(K)$, then the two linear operator coincide. ...mplex}--> $K$ can be realized as a topological space<!--\index{topological space}-->. The way to construct it is by treating the list of vertices and simpli
  34 KB (5,929 words) - 03:31, 29 November 2015
• Freshman's introduction to discrete calculus
  ...e standard domain, the discrete representation of ${\bf R}$. Second, the ''space'' is given by ${\bf R}$, at the simplest. ...that the only possible type of motion in this force-less and distance-less space-time is uniform; i.e., it is a repeated addition:
  42 KB (7,443 words) - 14:18, 1 August 2016
• Combinatorial cell maps
  If $R$ is a field, an $R$-module is a vector space. ...complex of time. What is the other chain complex $C$, the chain complex of space? Since these two forms take their values in ring $R$, we can choose $C$ to
  31 KB (5,330 words) - 22:14, 14 March 2016
• Vector calculus: review
  #(a) Give the definition of a basis of a linear space. (b) Show that the vectors $(1,0,0), (1,1,0), (1,1,1)$ form a basis of ${\b #Suppose that a mass $M$ is fixed at the origin in space. When a particle of unit mass is placed at the point $(x,y)$ other than the
  7 KB (1,394 words) - 02:36, 22 August 2011
• Integration
  '''Example (space shift).''' If $y$ is the location and we change the place from which we sta
  69 KB (11,727 words) - 03:34, 30 January 2019
• Inner product
  ...gles between them. An inner product is how one adds geometry to a [[vector space]]. Given a vector space $V$, an ''inner product'' on $V$ is a function that associates a number to
  4 KB (749 words) - 20:12, 1 May 2013

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