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• More on vector spaces
  ...des [[Euclidean space]]es, another important class of examples of [[vector space]]s is... ...-wise, input-wise: for each $x$, $f(x) \in {\bf R}$. ${\bf R}$ is a vector space! Use it.)
  14 KB (2,471 words) - 21:48, 5 September 2011
• Applications of ODEs
  ...o any mutual location of the rabbit and the hound as well to pursuits in a space of any dimension. ...cise.''' Implement a simulation of planetary motion in the $3$-dimensional space. Demonstrate that the motion is planar.
  50 KB (8,692 words) - 14:29, 24 November 2018
• Multilinear forms
  Given a vector space $V$, how does one ''compute'' the (algebraic) lengths, areas, volumes, etc ...of such $k$-forms over $V$ is denoted by $\Lambda ^k(V)$. It is a [[vector space]].
  18 KB (3,325 words) - 13:32, 26 August 2013
• Compact spaces
  ...nd $f(b)$. It follows from this theorem that the image of a path-connected space<!--\index{path-connectedness}--> (under a continuous map<!--\index{continuo ...will rely on the following familiar concept. A point $x$ in a topological space $X$ is called an accumulation point<!--\index{accumulation point}--> of sub
  19 KB (3,207 words) - 13:06, 29 November 2015
• Vector and complex variables
  Of course, any Euclidean space ${\bf R}^n$ can be -- in a similar manner -- rotated (around various axes), ...ppose we have addition and scalar multiplication carried out in the domain space of $A$:
  113 KB (18,750 words) - 02:33, 10 December 2018
• Cell complex
  ...y have seen two ways to construct topological spaces<!--\index{topological space}--> from cells. Cubical sets<!--\index{cubical sets}--> are unions of cubes ...exes are built from data and its cells can then be realized in a Euclidean space.
  40 KB (6,459 words) - 23:27, 29 November 2015
• Forms in Euclidean spaces
  ...space of continuous $k$-forms is denoted by $\Omega^k({\bf R}^n)$ and the space of discrete forms is $T^k({\mathbb R}^n)$. The above argument applies to show that in $3$-space the direction variables are independent from the location variables $x$, $y
  44 KB (7,778 words) - 23:32, 24 April 2015
• Systems of ODEs
  ...are placed in the first row of the spreadsheet. As we progress in time and space, new numbers are placed in the next row of our spreadsheet: ...i.e., the collections of $(t,x(t),y(t))$ lie in the $3$-dimensional $txy$-space. That is why, we, instead, plot their ''images'', i.e., the collections of
  63 KB (10,958 words) - 14:27, 24 November 2018
• Geometry
  In linear algebra, we learn how an inner product adds geometry to a vector space. We choose a more general setting. A module equipped with an inner product is called an ''inner product space''.
  41 KB (6,928 words) - 17:31, 26 October 2015
• The gradient
  ...the $y$-axis representing the dimensions of the input space and the output space. The first column consists of all parametric curves and the first row of al ...st dimension $3$).''' If $G$ is exact on a partition of a box in the $xyz$-space with component functions $p$, $q$, and $r$, then
  74 KB (13,039 words) - 14:05, 24 November 2018
• ODEs
  ...e continuous? The time is $K={\mathbb R}$, which seems discrete, while the space is $R={\bf R}$, which seems continuous. Let's take an alternative point of *the space is algebraic.
  47 KB (8,415 words) - 15:46, 1 December 2015
• Applications of differential calculus
  As we progress in time and space, new numbers are placed in the next row of our spreadsheet. This is how the We continue with the rest in the same manner. As we progress in time and space, a number is supplied and are placed in each of the columns of our spreadsh
  59 KB (10,063 words) - 04:59, 21 February 2019
• Algebra of differential forms
  For now, $1$-forms in the $3$-space appear to be functions of $x$, $y$, $z$, $dx$, $dy$, and $dz$ that are line First, we are given the "ambient space" which will be assumed to be Euclidean, ${\bf R}^n$.
  11 KB (1,947 words) - 18:14, 22 August 2015
• Triangulations
  ...omplex may be built as the union of a collection of subsets of a Euclidean space, while a cell complex is built via the quotient construction<!--\index{quot Representation of a topological space<!--\index{topological space}--> as a realization of a simplicial complex is also called a ''triangulati
  30 KB (5,172 words) - 21:52, 26 November 2015
• Lefschetz numbers in control theory
  *$U$ is the space of inputs, *$M$ is the space of states, and
  17 KB (3,052 words) - 22:12, 15 July 2014
• PDEs
  ...tandard complex ${\mathbb R}$. For now, we ignore the geometry of time and space. ...time. Since $a$ is an $(n-1)$-cell, $k$ is an $(n-1)$-form with respect to space. It is also a $0$-form with respect to time.
  44 KB (7,469 words) - 18:12, 30 November 2015
• Integer-valued calculus
  Because there is a hole in the space, there is a vector field that is irrotational but not gradient. And vice ve ...the integral over a half of the interval? An infinite divisibility of the space implies an infinite divisibility of the ring of numbers...
  27 KB (3,824 words) - 19:07, 26 January 2019
• Several variables
  ...pter. Of course, the motion metaphor -- $x$ and $y$ are coordinates in the space -- will be superseded. In contrast to this approach, we look at the two qua ...tead of plotting all points $(t,x,y)$, which belong to the $3$-dimensional space, we just plot $(x,y)$ on the $xy$-plane -- for each $t$.
  76 KB (13,017 words) - 20:26, 23 February 2019
• The algebra of cubical cells
  We know that we can decompose the $N$-dimensional Euclidean space into blocks, the $N$-cells. For instance, this is how an object can be repr Thus, our approach to decomposition of space, in any dimension, boils down to the following:
  46 KB (7,844 words) - 12:50, 30 March 2016
• Diffusion equation
  ...tandard complex ${\mathbb R}$. For now, we ignore the geometry of time and space. ...time. Since $a$ is an $(n-1)$-cell, $k$ is an $(n-1)$-form with respect to space. It is also a $0$-form with respect to time.
  35 KB (5,917 words) - 12:51, 30 June 2016
• Topology of graphs
  ...pological questions about specific spaces. Given a subset of the Euclidean space: One can also think of a graph as just a collection of points in space, also called “vertices”, or “nodes” connected by paths, called “e
  25 KB (4,214 words) - 16:08, 28 November 2015
• Differential calculus of parametric curves
  *$(f(t),g(t))$ is thought of as the position in space at time $t$. The motion may also be in the $3$-dimensional space:
  130 KB (22,842 words) - 13:52, 24 November 2018
• Simplicial complexes
  ...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==
  30 KB (5,021 words) - 13:42, 1 December 2015
• Simplicial homology
  ...ata'', and yet we can still study the topology of subsets of the Euclidean space -- via realizations of graphs. We will follow this latter route with simpli *A representation of a topological space $X$ as a homeomorphic image of a realization of a simplicial complex $K$ is
  31 KB (5,170 words) - 13:44, 1 December 2015
• Calculus exercises: part I
  ...unction $f$ is given below. Sketch the graph of the derivative $f′$ in the space under the graph of $f$. Identify all important points and features on the g ...unction $f$ is given below. Sketch the graph of the derivative $f'$ in the space under the graph of $f$. Identify all important points on the graph.
  49 KB (8,436 words) - 17:14, 8 March 2018
• Social choice
  *the space of choices is a simplicial complex $W$; '''Theorem (Impossibility).''' Suppose the space of choices $W$ is path-connected and has torsion-free homology. Then the so
  47 KB (8,030 words) - 18:48, 30 November 2015
• Homology of maps
  <center>How does a continuous functions change the topology of the space?</center> If $R$ is a field, an $R$-module is a vector space.
  41 KB (6,926 words) - 02:14, 21 October 2015
• Cell maps
  <center>How does a continuous functions change the topology of the space?</center> If $R$ is a field, an $R$-module is a vector space.
  42 KB (7,005 words) - 03:10, 30 November 2015
• Differential equations
  ...are placed in the first row of the spreadsheet. As we progress in time and space, new numbers are placed in the next row of our spreadsheet: ...placed in the first row of the spreadsheet and, as we progress in time and space, new numbers are placed in the next row of our spreadsheet:
  64 KB (11,426 words) - 14:21, 24 November 2018
• Motion planning in robotics
  Now, the [[configuration space]] of a ''two-joint'' arm is the [[torus]]: ...e the lengths of the arms. Under the assumption $R_1>R_2$, the operational space is the annulus.
  5 KB (786 words) - 20:58, 27 August 2015
• 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

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