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Functions in higher dimensions
*the physical space, dimension $3$, ...numbers, the graph of a function of one variable lies in the $xy$-plane, a space of dimension $2$.

113 KB (19,680 words) - 00:08, 23 February 2019
Products
...he product of a space with the segment ${\bf I}$ means “thickening” of the space. For example, the product ...oduct'' $X \times Y$ of $X$ and $Y$ is a topological space defined on the product set $X \times Y$ with the following basis:

44 KB (7,951 words) - 02:21, 30 November 2015
Multilinear algebra
...to handle directions appears, separately, at every point of the Euclidean space. The set of all possible directions at point $A\in {\bf R}^n$ form a vector space of the same dimension. It is $V_A$, a copy of ${\bf R}^n$, attached to each

49 KB (8,852 words) - 00:30, 29 May 2015
Functions
...r 1, we visualized a sequence of position of a falling ball by “separating space and time”. We gave the former a real line and the latter a line of intege Accommodating finer and finer representations of space or time will require to continue to divide the intervals in half until it s

151 KB (25,679 words) - 17:09, 20 February 2019
Relative topology
This idea applies to all topological spaces. ...--\index{topology}--> via ''neighborhoods'', a subset $A$ of a topological space $X$ with basis $\gamma$ will acquire its own collection $\gamma _A$ as the

34 KB (6,089 words) - 03:50, 25 November 2015
Homotopy
...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.

46 KB (7,846 words) - 02:47, 30 November 2015
Metric tensor
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''.

42 KB (7,131 words) - 17:31, 30 November 2015
Forms
...need for considering directions becomes clearer when the dimension of the space is $2$ or higher. We use ''vectors''. ...The set of all possible directions at point $A\in {\bf R}^n$ form a vector space of the same dimension. It is $V_A$, a copy of ${\bf R}^n$, attached to each

13 KB (2,459 words) - 03:27, 25 June 2015
Calculus of sequences
They can be used for studying the space and locations, as follows. ...more popular approach is the following. The idea is to ''separate time and space'', give a separate real line, an axis, to each moment of time, and then bri

113 KB (18,425 words) - 13:42, 8 February 2019
Computing homology
...t $\partial$ is a [[homomorphism]] (or a [[linear operator]] in the vector space case): is a linear operator between two copies of ${\bf R}^3$ with [[basis of vector space|bases]] $\{a, b, c \}$ and $\{A, B, C \}$ respectively. The values of $\par

26 KB (4,370 words) - 21:55, 10 January 2014
Vector fields
As we progress in time and space, new numbers are placed in the next row of our spreadsheet. There is a ''se We continue with the rest in the same manner. As we progress in time and space, numbers and vectors are supplied and placed in each of the four sets of co

91 KB (16,253 words) - 04:52, 9 January 2019
Quotients
These realizations, however, were placed within a specific Euclidean space ${\bf R}^N$. We will see that this is unnecessary. ...ays start with $X$ assumed to be a topological space, the quotient has been, so far, ''just a set''. We can't simply assume

26 KB (4,538 words) - 23:15, 26 November 2015
Calculus on cubical complexes
...d'', i.e., the infinite cubical complex acquired by dividing the Euclidean space into small, simple pieces (cubes). We denote it by ${\mathbb R}^n$. ...associativity, commutativity, distributivity, etc. Thus, we have a vector space:

36 KB (6,218 words) - 16:26, 30 November 2015
Configuration spaces
Its space of parameters is the torus: ...space'' as the set of all positions reachable by a robot's end-effector in space;

6 KB (921 words) - 17:14, 27 August 2015
Set-valued maps
...umber of degrees of freedom of the robot, to the $3$-dimensional operating space. ...is called a ''control system''. For example, in cruise control, $M$ is the space of all possible values of the car's speed and $U$ is the engine's possible

24 KB (4,382 words) - 15:52, 30 November 2015
Discrete forms
...d'', i.e., the infinite cubical complex acquired by dividing the Euclidean space into small, simple pieces. We denote it by ${\mathbb R}^n$. ...associativity, commutativity, distributivity, etc. Thus, we have a vector space:

35 KB (6,055 words) - 13:23, 24 August 2015
Continuous functions
'''Definition.''' Suppose we have a topological space $X$, then we can define the ''identity function''.

42 KB (7,138 words) - 19:08, 28 November 2015
Metric complexes
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''.

35 KB (5,871 words) - 22:43, 7 April 2016
Applications of integral calculus
...shape up is the following. Suppose we have a quantity $Q$ “contained” in a space region $R$: area, volume, mass (below), etc. Then, A certain quantity, $f$, is “spread” around locations in space; for now, it is an interval within the $x$-axis. This quantity may be: leng

103 KB (18,460 words) - 01:01, 13 February 2019
Functions of several variables
\text{dimension}&\text{ambient space}&\text{“hyperplane”}&\\ A hyperplane is something very “thin” relative the whole space but not as thin as, say, a curve.

97 KB (17,654 words) - 13:59, 24 November 2018
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 (under a continuous map 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 from cells. 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 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''. 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'' and each of them determines such a new function. That's why we further ...tter of ''calculus'', the calculus of differential forms:

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
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
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
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'' 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. 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'' 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. The way to construct it is by treating the list of vertices and simpli

34 KB (5,929 words) - 03:31, 29 November 2015
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
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
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
Algebra of discrete differential forms
==Discrete differential forms as a vector space== ...to define algebraic operations with them that make the set into a [[vector space]].

6 KB (1,000 words) - 18:30, 22 August 2015
Linear algebra
The result is the "parametric solution" of the system. It is plane in the 3-space. ''n'' = dimension of the space and

27 KB (4,667 words) - 01:07, 19 February 2011
Linear algebra of Euclidean space
...result is the "parametric solution" of the system. It is plane in the $3$-space. <center>$n=$ dimension of the space and 

26 KB (3,993 words) - 19:48, 26 August 2011
Elementary ODEs
...exerted on an object, we are compelled to specify the ''geometry'' of the space, in contrast to the previous examples. Broadly, an ODE is a dependence of directions on locations in ''space'' provided by ${\bf R}$ while its solutions exist over ''time'' ${\mathbb R

16 KB (2,913 words) - 22:40, 15 July 2016
Constructions
These realizations, however, were placed within a specific Euclidean space ${\bf R}^N$. We will see that this is unnecessary. '''Exercise.''' What if, this time, the target space $Y$ has an equivalence relation too? Analyze the possibility of a map $[f]:

13 KB (2,270 words) - 22:14, 18 February 2016
Homology as an equivalence relation
<center>''subsets of a Euclidean space'', $X\subset {\bf R}^N$.</center> ...mans, can see the whole thing by being ''outside'', in the $2$-dimensional space.

21 KB (3,530 words) - 19:54, 23 June 2015
Homology classes
<center>''subsets of a Euclidean space'', $X\subset {\bf R}^N$.</center> ...mans, can see the whole thing by being ''outside'', in the $2$-dimensional space.

21 KB (3,581 words) - 15:51, 28 November 2015
Clustering
...d(C,A), no matter how you define the distance d(,) between points in this space. The conclusion: if A and B are in the same cluster, then so is C. So adjac ...same “physical” object), which means higher [[dimension]] of the Euclidean space, which means higher computational costs. Not a good sign.

3 KB (477 words) - 19:14, 28 August 2010
Modern Thermodynamics by John Denker
...etc.) take on well-defined values. We further assume that this macrostate-space has dimensionality M, and that M is not very large." ...f each other, so it may be impossible to use them to parametrize the state space.

8 KB (1,251 words) - 03:54, 29 March 2011
Equilibria of dynamical systems
...sible price vectors is an $n$-[[simplex]] $S_n$ in the $(n+1)$-dimensional space. ...luded as they influence and are being influenced by the prices. The "state space" is then $S=S_n\times [0,R]$, where $S_n$ is our price simplex and $R$ is t

7 KB (1,251 words) - 15:00, 4 April 2014
Introductory algebraic topology: review exercises
*[[Is a closed subset of a compact space always compact? ]] 1. Suppose A is a subset of a topological space X and τ is the topology of X. Define a collection of subsets of A as $τ_A

9 KB (1,553 words) - 20:10, 23 October 2012
Discrete calculus article
...of a unit cube <math>[0,1]^n</math> [[Embedding|embedded]] in [[Euclidean space]] <math>\mathbf{R}^d</math> (for some <math>n,d\in\mathbf{N}\cup\{0\}</math A '''[[chain complex]]''' <math>(A_*, d_*)</math> is a sequence of [[vector space]]s ..., ''A''0, ''A''1, ''A''2, ''A''

27 KB (4,329 words) - 16:02, 1 September 2019
Parametric complexes
...(point clouds).''' A ''point cloud'' is a finite set $S$ in some Euclidean space of dimension $d$. Given a threshold $r$, we deem any two points that lie wi .... These homomorphisms record how the homology changes as this “parametric” space grows at each step. For example, a component appears, grows, and then merge

45 KB (7,255 words) - 03:59, 29 November 2015
Homology groups of graphs
We are given an abelian group $L$ and a subgroup $M$ (or a vector space $L$ and a subspace $M$). How do we “remove” $M$ from $L$? The simple an ...group (or the affine subspace of the vector space) produced when $M$ is “shifted” by $v$:

28 KB (4,685 words) - 17:25, 28 November 2015
Cohomology
from the reals to the vector space of all linear functions on the reals. In fact, it's an isomorphism! '''Definition.''' Let the ''dual'' of $V$ be defined by

45 KB (6,860 words) - 16:46, 30 November 2015
Matrices: part 1
Recall, that matrices -- of given, fixed dimensions -- form a [[vector space]] with respect to addition and scalar multiplication. Now, this new operation, matrix multiplication, may be outside this vector space. Just look at the dimensions of the product (unless it's $n \times n$)

8 KB (1,275 words) - 14:18, 9 September 2011
Compactness
...' For any topological space $X$ (or a subset $X$ of some other topological space), a collection of open sets $\alpha$ '''Definition.''' A topological space $X$ is called ''compact'' if every open cover contains a finite subcover.

4 KB (635 words) - 12:57, 12 August 2015
Nerve of cover
.... But how do you find this representation if all you have is a topological space, i.e., a collection of open sets. ...ogic to create simplicial complexes from any open cover of any topological space:

8 KB (1,389 words) - 13:35, 12 August 2015
Discrete Calculus: Applied Analysis on Graphs for Computational Science by Grady and Polimeni
...ishes a separate, equivalent calculus that operates purely in the discrete space without any reference to an underlying continuous process." ...rivative operator depends on the [[topology|topological structure]] of the space -- in a sense, the graph ''is'' the operator."

11 KB (1,663 words) - 16:03, 26 November 2012
Surfaces and their homology
...phism|homeomorphic]] to the plane which is the $2$-dimensional [[Euclidean space]] ${\bf R}^2$, we can say that surfaces are "locally Euclidean". '''Definition.''' A [[topological space]] $S$ is called a ''surface'' (without boundary) if $S$ is [[separation axi

17 KB (2,696 words) - 00:47, 12 January 2014
Linear functions in Euclidean space
...rightarrow T$, where $S, T$ are [[Linear_algebra_of_Euclidean_space|linear space]]s, is a ''linear function'' if: <center>$2 + 1 = 3 =$ [[dimension of vector space|dimension]] ${\bf R}^3$.</center>

23 KB (3,893 words) - 04:43, 15 February 2013
Discrete functions
...latter as the continuous space. It is the idea of infinite divisibility of space that makes the choice of ${\bf R}$ so plausible. There are many quantities besides space that are infinitely divisible: time, heat, mass, money. However, they can b

15 KB (2,532 words) - 12:21, 11 July 2016
Robustness of topology of digital images and point clouds by Saveliev
...s usually in the form of a real valued function defined on the topological space. [[Persistence]] is a measure of robustness of the homology classes of the The [[topological space]]s subject to such analysis are cell complexes. A ''[[cell complex]]'' is

27 KB (4,547 words) - 04:08, 6 November 2012
Calculus as a part of topology
==Three pillars of calculus -- three structures of the Euclidean space== ...where this is all happening. The locus is the ''Euclidean space''. Such a space has three different types of structures present at the same time.

13 KB (2,233 words) - 14:41, 20 February 2015
Dual space
from the reals to the vector space of all linear functions. In fact, it's an [[isomorphism]]! More generally, suppose $V$ is a [[vector space]]. Let

9 KB (1,390 words) - 16:14, 16 June 2014
Fixed points and selections of set valued maps on spaces with convexity by Saveliev
...ompact]] subset of a [[locally convex]] [[Hausdorff]] [[topological vector space]], and let $F:X \rightarrow Y$ be an [[upper semicontinuous]] [[multifuncti ...$X$ be a nonempty convex compact subset of a Hausdorff topological vector space, and let $F:X\rightarrow X$ be a multifunction with nonempty convex images

3 KB (469 words) - 16:12, 26 March 2011
Axioms of calculus
This is the analogue of the [[tensor space]]. ...equivalence relation, the space becomes a (non-degenerate) [[inner product space]]. We define an equivalence:

6 KB (1,124 words) - 14:17, 4 August 2013
Cell decomposition of images
...ut into pieces and these pieces can be [[glued]] together and the original space reappears intact. ...ng our collection of cells with cells of dimensions lower than that of the space itself.

11 KB (1,829 words) - 19:26, 10 February 2015
Tangent cone
In the continuous case, the tangent space $T_x(A)$ is a [[vector space]]. The reason is that it should be able to accommodate parametric curves in We need to amend our definition of the tangent space in order to have the tangent bundle which is a [[surface]] as the one we se

5 KB (882 words) - 02:14, 26 March 2013
Fundamental group
Let $(X,x_{0})$ be a ''pointed [[topological space]], ''i.e., a topological space with a chosen point $x_{0}$, called the ''basepoint''. The basepoints in the same path-component of the space will give [[isomorphic]] groups.

10 KB (1,673 words) - 18:23, 2 December 2012
Fantasy math
...tart with [[inner product]] spaces -- by adding this structure to [[vector space]]s. ...s. Derivatives and integrals of vector functions. [[Curvature]]. Motion in space: [[velocity]] and [[acceleration]].

8 KB (1,196 words) - 13:02, 24 August 2015
Wave equation
...are dealing with the second derivative of the $0$-form $u$ with respect to space: Compare it to the second derivative of a $1$-form $U$ with respect to space:

10 KB (1,775 words) - 02:40, 9 April 2016
Introduction to discrete calculus, continued
...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

22 KB (3,661 words) - 13:12, 18 July 2016
Computational science training: 2010 projects
...ted 1000 points, called the [[point cloud]], in a 100-dimensional [[vector space]]. ...''topological'' approach to the problem. For a point cloud in a euclidean space, suppose we are given a threshold r so that any two points within r from ea

6 KB (926 words) - 17:02, 7 February 2011
Calculus Illustrated -- preface
...quantities infinitely divisible either by nature or by assumption: time, space, mass, temperature, money, some commodities, etc. Calculus produces definit

17 KB (1,758 words) - 13:57, 25 August 2019
Chains vs cochains
*they form [[vector space]]s of the same dimension -- they are [[dual spaces]]! The vector space of all $k$-chains is $C_k.$ Meanwhile, the space of all $k$-cochains is $C^k.$

4 KB (635 words) - 18:28, 22 August 2015
Tensor product
...ations between the elements of $V,W,V \times W$ are lost in the new vector space. The ''tensor product'' is defined as the [[quotient]] vector space

2 KB (409 words) - 14:14, 1 August 2013
Integration of differential forms of degree 2
That's a $k$-form on $M$, but it's still a form on the whole space and satisfies all the conditions. This is a [[vector space]], $C_k(R)$, with $k$-cells serving as a [[basis]].

15 KB (2,341 words) - 20:53, 13 March 2013
Multivectors
Given a [[vector space]] (or a [[module]]) $V$ over a [[field]] (or [[ring]]) $R$, we think of ''m It's a [[vector space]] with respect to the usual operations of addition and scalar multiplicatio

9 KB (1,564 words) - 17:19, 25 August 2013
Closed and exact forms
This is a ''sequence'' of [[vector space]]s and [[linear operator]]s. '''Example:''' Let $G$ be a [[vector space]] and $H$ a [[linear subspace]] of $G$. Then,

9 KB (1,423 words) - 20:53, 13 March 2013
Parametric curves
*'''$f(t)$ is thought of as the position in space at time $t$'''. Just the space happens to be $n$-dimensional...

34 KB (5,665 words) - 15:12, 13 November 2012
Tangent bundles and differential forms
To ensure that for each $a \in M$, there is a [[tangent space]], we assume that $M$ is [[smooth manifolds|smooth]]. ==The vector space of forms on a manifold==

6 KB (1,177 words) - 15:53, 5 November 2012
Topological data analysis
...1000 points, called a [[point cloud]], in the 100-dimensional [[Euclidean space]]. For a point cloud in a [[euclidean space]], suppose we are given a threshold $r$ so that any two points within $r$ f

4 KB (549 words) - 12:54, 12 August 2015
Vietoris-Rips complex
We realize that the "object" is just dots suspended in space! It is called a ''point cloud''. ...1000 points, called a [[point cloud]], in the 100-dimensional [[Euclidean space]].

4 KB (630 words) - 20:15, 27 August 2015
Modelling with discrete vector fields and forms
But which grid? There are two: $t$ and $x$, time and space. Which one should be discrete? Or both? Can we make both time ''and'' space discrete?

9 KB (1,561 words) - 16:06, 27 August 2015
Homology as a vector space
'''Theorem.''' $C_k(K)$ is a [[vector space]] with a basis consisting of the $k$-cells of $K$. ...the [[boundary operator]] is a [[linear operator]] defined on this vector space. Then the notation for the chain complex of $K$ is:

5 KB (868 words) - 18:41, 27 August 2015
Discrete parametric curves
Another measurement in the Euclidean space is ''angles''. *$f(t)$ is thought of as the position in space at time $t$.

32 KB (5,426 words) - 21:57, 5 August 2016
Products of complexes
...he product of a space with the segment ${\bf I}$ means “thickening” of the space. As an example, the product If we are able to decompose a topological space into the product of two others, $Z=X\times Y$, we expect $X$ and $Y$ to be

16 KB (2,892 words) - 22:39, 18 February 2016
Students research projects
...ted 1000 points, called the [[point cloud]], in a 100-dimensional [[vector space]]. It is impossible to visualize this data as any representation that one c ...''topological'' approach to the problem. For a point cloud in a euclidean space, suppose we are given a threshold r so that any two points within r from ea

11 KB (1,674 words) - 23:20, 25 October 2011
Euler characteristic of graphs
...nificance of this number becomes clear if we consider a simple path $C$ in space. It can be a realization of a various graphs $G$ but suppose $G$ is a seque (In the language of [[linear algebra]], these are two bases of this vector space.)

11 KB (1,876 words) - 19:23, 10 February 2015
Linear algebra: test 2
#Given basis $\{1,x,x^{2}\}$ of the space $\mathbf{P}_{2}$ of degree $\leq $2 polynomials, find the change of basis m #Suppose $V$ is the space of differentiable at $0$ functions of two variables. Suppose $A:V\longright

4 KB (583 words) - 01:13, 12 October 2011
Matrices as functions
Note: these are [[vector space]]s, ${\bf R}^2$. We just combined $u$ and $v$ in one vector $(u,v)$. These are, in fact, the two operations of a vector space.

13 KB (2,187 words) - 22:17, 9 September 2011
Modelling motion on manifolds
Let's recall how we describe motion in the Euclidean space. It is given by an [[ordinary differential equation]] (ODE): ...rs at $a$ they produce is a [[vector space]], $T_aM$, called the [[tangent space]].

2 KB (377 words) - 17:13, 27 August 2015
Elections
...ngful (i.e., transitive, non-circular) ranking we make the topology of the space of outcomes topologically non-trivial, non-acyclic. As we already demonstra ...ve of this cover will contain an edge between these two vertices. Then the space of choices becomes acyclic!

33 KB (5,872 words) - 13:13, 17 August 2015
Euler-Poincare formula
where $H_k(K)$ are the [[homology as a vector space|homology group]]s (vector spaces) of $K$ ($\dim$ replaced with rank in case Fact 1. If $M, L$ are [[vector space]]s and $A \colon M \rightarrow L$ is a [[linear operator]], then

5 KB (790 words) - 13:01, 28 August 2015
Inner product spaces: part 2
Given vector space $V$, $u,v \in V$, $u \neq 0$, ''projection'' of $v$ onto $u$ is a vector $p ...h: $v$ is $\sin$, $u$ is $x$. Now find the projection $p$ of $\sin$ on the space of linear functions, $y=mx$.

10 KB (1,688 words) - 17:59, 13 October 2011
Advanced Topology -- Spring 2013
...and discrete differential forms and their relation to the topology of the space. ...inear algebra: course|linear algebra]], in the sense of theory of [[vector space]]s. Frequently, this material is only seen in more advanced linear algebra

5 KB (725 words) - 14:49, 8 May 2013
Calculus Two by Flanigan and Kazdan
*1.1 The space $R^2$ 7 *1.2 [[euclidean space|The space $R^n$]] 12

3 KB (311 words) - 14:51, 23 November 2011
Category
*[[dual space]] functor $D:{\mathscr Vec} \to {\mathscr Vec}$, ...|dual]] $V^{*}$ is not natural (arrow are reversed) but between a [[vector space]] and its second $V^{**}$ dual is.

7 KB (1,007 words) - 22:17, 18 April 2014
Linear algebra in elementary calculus
...as a [[basis]]. The obvious choice of operations makes $\Phi^*$ a [[vector space]]. We want to show $\Phi^*/_{\sim}$ is a vector space too.

4 KB (604 words) - 15:52, 27 August 2015
A new look at continuity
...the rest. This way it doesn't matter how the domain fits into some bigger space. Now, once again, what if the domain is a ''subset'' $X$ of the Euclidean space?

17 KB (2,946 words) - 04:51, 25 November 2015
Vector integrals
We want to compute the flow of liquid through a region in space. For that we need to understand the direction of the flow with respect to t ...on (at least locally). Let's understand orientation of the square, $Q$, in space first. We can still look at it as corresponding to the direction of the [[p

15 KB (2,545 words) - 19:47, 20 August 2011
Isotropy in numerical PDEs
==Deeper reason: curved space?==

10 KB (1,593 words) - 13:20, 8 April 2013
Properties of the exterior derivative
...d \colon \Omega^k \rightarrow \Omega^{k+1}$ is a function between [[vector space]]s, so we already know that $d$ is a [[linear operator]]: Let's prove this property in $3$-space using the formulas above.

8 KB (1,539 words) - 18:17, 22 August 2015
Applied Calculus -- Spring 2015
**9.2 Surfaces in Space **9.2 Surfaces in Space

9 KB (1,141 words) - 16:08, 26 April 2015
Modeling with discrete exterior calculus
However, we are interested in a discrete form of this idea. When time and space are broken up into discrete regions, the differential equation becomes

31 KB (5,254 words) - 17:57, 21 July 2012
Elementary PDEs
Note: When the domain isn't the whole space, the pipes at the border of the region have to be removed. In the spreadshe

16 KB (2,843 words) - 21:41, 23 March 2016
Functions of several variables OLD
...] we need to understand better the [[topology]] of the underlying (domain) space, ${\bf R}^n$. Its topology is much more complex than that of ${\bf R}$.

34 KB (5,636 words) - 23:52, 7 October 2017
Geometry of Euclidean space
Next, what is the [[dimension of linear space|dimension]] of $S$? Consider:

21 KB (3,396 words) - 20:31, 10 August 2011
Image search
...computer represents every 100x100 image a point in the 10,000-dimensional space and then runs clustering. First, this may be impractical and, second,.. doe

17 KB (2,793 words) - 17:40, 14 September 2008
Singular chain complexes
...use continuous maps of these cells. A ''singular $k$-cell'' in topological space $X$ is a map '''Corollary (Additivity).''' If a space is the disjoint union of a family of topological spaces $\{X_{\alpha}\}$:

8 KB (1,367 words) - 13:49, 4 August 2013
The topology of data
...ted 1000 points, called the [[point cloud]], in a 100-dimensional [[vector space]]. ...''topological'' approach to the problem. For a point cloud in a euclidean space, suppose we are given a threshold r so that any two points within r from ea

9 KB (1,431 words) - 16:57, 20 February 2011
Zachary Ahlers
...angles become equilateral and the rhombuses become square, i.e. making the space euclidean. The simulation incorporates many physical laws into discrete time and space, the most fundamental of which is conservation of mass and conservation of

8 KB (1,315 words) - 15:20, 10 August 2012
Examples of differential forms
...is in fact a [[vector space]]) of all $k$-forms (in a particular Euclidean space) is denoted by $\Omega^k$. ...in ${\bf R}^2$ or ${\bf R}^3$ because there are no 4-th variable in either space.

5 KB (959 words) - 13:15, 12 August 2015
Examples of maps
...t-set topology. However, it's unnecessary if only subsets of the Euclidean space are involved. In that case we can use what we know about [[continuity of fu Suppose we have a topological space $X$, then we we can define the ''[[identity function]]'' $i_X: X {\rightarr

5 KB (918 words) - 16:54, 27 August 2015
From continuity to point-set topology
...imited to that of the [[euclidean space]], "nice" subsets of the euclidean space, such as [[cells]], and "nice" combinations of those. ...isn't general enough as we want to deal with any subsets of the Euclidean space. Therefore we need to adjust the definition a bit more.

7 KB (1,207 words) - 13:01, 12 August 2015
Geometry of data
...(PCA), which finds a "basis" for the dataset as a subspace of the ambient space that reveals its structure. ...clidean topology on the plane|non-Euclidean topologies]] for the Euclidean space, even with Euclidean topology on each of the coordinates.

984 bytes (149 words) - 22:27, 26 February 2011
Computational topology
#Robotics: capturing the connectivity of the [[configuration space]] of a robot in order to plan optimal trajectories. ...l structural biology: finding optimal trajectories within the conformation space of a protein to define its folding path.

5 KB (748 words) - 19:24, 31 January 2015
Constant homotopy between constant maps
...s are homotopic under these circumstances: $Y$ is [[convex]]. The ''target space is too simple''! ...fact that this question seems too challenging indicates that the ''domain space is too complex''!

3 KB (575 words) - 14:51, 11 March 2013
Dot product
How do we compute distances and angles in [[vector space]]s? '''Plan:''' Take a vector space and equip it with extra structure, so that we ''can'' measure.

2 KB (410 words) - 15:09, 9 June 2012
Orientation
Suppose we want to compute the flow of liquid through a region in space. For that we need to understand the direction of the flow with respect to t Let's understand orientation of the square, $Q$, in space first. We can still look at it as corresponding to the direction of the [[p

4 KB (753 words) - 03:35, 21 October 2012
Boundary operator
...t $\partial$ is a [[homomorphism]] (or a [[linear operator]] in the vector space case): The key fact needed for [[homology as a vector space|homology theory]] is:

5 KB (837 words) - 16:24, 1 June 2014
Quotient sets
...sponds to its point of intersection with the $x$-axis. Hence, the quotient space is the real line. Algebraically, A special kind of a quotient space is when a subset is collapsed to a single point. Given $X$ and a subset $A$

3 KB (464 words) - 19:36, 31 October 2012
More about manifolds
*the set of straight lines through the origin in 3-space; *the configuration space of $n$ rigid bodies connected by rods consecutively with the ends fixed.

9 KB (1,542 words) - 19:58, 21 January 2014
Duality: forms as cochains
from the reals to the vector space of all linear functions on the reals. In fact, it's an isomorphism! Recall that a cell complex $K$ is called acyclic if its chain complex is an ''exact sequence''<!--\index{exact sequence}

29 KB (4,540 words) - 13:42, 14 March 2016
Examples of homology of cubical complexes
...ctions of cells. This is possible for any complex if we choose the ambient space of high enough dimension.

3 KB (519 words) - 18:06, 27 August 2015
Homology in dimension 2
...t $\partial$ is a [[homomorphism]] (or a [[linear operator]] in the vector space case). In the [[vector space]] environment,

8 KB (1,318 words) - 18:42, 27 August 2015
Computational Homology by Kaczynski, Mischaikow, Mrozek
**13.2 [[Field]]s and [[vector space|Vector Space]]s

5 KB (616 words) - 14:03, 6 October 2016
Computational science training: 2011 projects
...ted 1000 points, called the [[point cloud]], in a 100-dimensional [[vector space]]. It is impossible to visualize this data as any representation that one c ...''topological'' approach to the problem. For a point cloud in a euclidean space, suppose we are given a threshold r so that any two points within r from ea

6 KB (794 words) - 00:56, 1 June 2012
Review exercises
'''Homework:''' Let $V$ be the space of infinite sequences $\{x_1,\ldots,x_n,\ldots\}$. Find an infinite dimensi

4 KB (677 words) - 17:31, 13 October 2011
Tree representation of images
...o, Richard Harvey, Gavin C. Cawley, ''The Segmentation of Images via Scale-Space Trees'', British Machine Vision Conference, 1998.

8 KB (1,263 words) - 18:45, 9 February 2011
Machine learning
...computer represents every 100x100 image a point in the 10,000-dimensional space and then runs [[clustering]] or another pattern recognition method. Will th

8 KB (1,463 words) - 22:30, 4 November 2011
Homology in Calculus
<center>''subsets of a Euclidean space''.</center>

6 KB (945 words) - 22:56, 9 February 2015
Calculus by Rogawski
12.5 Planes in Three-Space 13.5 Motion in Three-Space

6 KB (634 words) - 16:38, 1 March 2013
Parametrized complexes
...s usually in the form of a real valued function defined on the topological space. ''Persistence'' is a measure of robustness of the homology classes of the Second, a [[point cloud]] is a finite set $S$ in some Euclidean space of dimension $d$. Given a threshold $r$, we deem any two points that lie wi

8 KB (1,240 words) - 13:26, 28 August 2015
Calculus on chains
With the help of [[dual space|duality]] we have a bird's-eye view of (a large part of) calculus, as follo The dimension of each of these spaces matches that of the space above while the matrices of the operators are the [[transpose]]s of the one

4 KB (556 words) - 14:03, 30 March 2013
Persistent homology
Suppose we are given a [[point cloud]] $K$ in a [[euclidean space]] of dimension $d$. Suppose also that we are given a threshold $r$ so that ...$K$, collectively $H_{\ast}(K)$. Commonly, $H_{k}(K)$ is simply a [[vector space]] and its dimension is equal to the corresponding Betti number $B_{k}$.

9 KB (1,504 words) - 04:14, 6 November 2012
Projective plane
The $n$-dimensional [[projective space]] is In particular, the $1$-dimensional projective space is the ''projective line''. What is it?

2 KB (333 words) - 01:28, 1 December 2012
Topology I -- Fall 2013
#[[Discretization of the Euclidean space]] #[[Homology as a vector space]]

7 KB (881 words) - 19:04, 10 December 2013
Vector calculus: midterm
#(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 ...r map $L(x_1,x_2) = (3x_1 + x_2, -3x_1 - x_2)$ find the basis of the image space.

4 KB (674 words) - 02:48, 22 August 2011
Voting and social choice
...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'' if it is a contin

14 KB (2,570 words) - 17:10, 26 June 2016
Modules
<center>''modules are [[vector space]]s over [[ring]]s''</center> If $R$ is a field, an $R$-module is a vector space.

2 KB (412 words) - 19:37, 18 April 2013
Cone
A (geometric) ''cone'' is a subset of a [[vector space]] that contains all of its (positive) multiples: In topology, this is how cone is defined. Given a topological space $X$, first form the product $[0,1] \times X$, then the ''cone of'' $X$ is

717 bytes (125 words) - 01:02, 1 December 2012
Discrete differential forms
The grid: the [[Euclidean space]] is divided into small, [[disjoint]] parts. Within each of the piece, the | '''space:'''

10 KB (1,471 words) - 12:50, 12 August 2015
Calculus 1: final exam
...tion $f$ is given below. Sketch the graph of the derivative $f′(x)$ in the space under the graph of $f$. Identify all important points and features on the g ...is given below. Sketch a possible graph of the function $f$ itself in the space under the graph of $f′$. Identify all important points and features on th

2 KB (349 words) - 20:21, 13 June 2011
Diffusion with various geometry
Now, given a primal $m$-[[cochain]] $\phi$ in the $n$-dimensional space, the ''discrete (geometric) Hodge star'' of $\phi$ is a cochain on the dual

13 KB (2,121 words) - 16:33, 7 June 2013
Objects and holes
...images]]. It is based the [[partial order]] of the [[RGB color model|color space]].

2 KB (407 words) - 18:32, 30 January 2011
Vector functions
'''Example.''' Surface in space. Define $f: {\bf R}^2 {\rightarrow} {\bf R}^3$ by ...t's parametrize the [[cylinder]]. The idea is to take the plane (the input space) and roll it into the cylinder (the image).

28 KB (4,769 words) - 19:42, 18 August 2011
Geometric Hodge duality
Now, given a primal $k$-[[cochain]] $\phi$ in the $n$-dimensional space, the ''discrete (geometric) Hodge star'' of $\phi$ (denoted by $\star \phi$

5 KB (867 words) - 13:24, 19 May 2013
Graph representation of images
...Richard Harvey, Gavin C. Cawley, The Segmentation of Images via Scale-Space Trees, British Machine Vision Conference, 1998.

4 KB (617 words) - 17:23, 14 August 2009
Hodge duality of differential forms
...symmetry of the [[de Rham complex]]. Indeed, for $n$ the dimension of the space, consider this (non-commutative) ''Hodge duality diagram'':

8 KB (1,072 words) - 17:59, 24 April 2013
Homology of balls and spheres
Let's compute the [[homology as a vector space|homology]] of the $n$-[[ball]] ${\bf B}^n$ and the $(n-1)$-[[sphere]] ${\bf

3 KB (412 words) - 18:43, 27 August 2015
Homology of images
The homology groups of this image are [[vector space]]s generated by these [[cycle]]s, as follows. Consider first the image to t

2 KB (282 words) - 12:25, 12 August 2015
Image-to-image search
...computer represents every 100x100 image a point in the 10,000-dimensional space and then runs clustering. First, this may be impractical and, second,.. doe

3 KB (472 words) - 15:43, 20 April 2012
Conferences and workshops
...sciences (e.g., fair division problems and voting) and biology (e.g., the space of phylogenetic trees). Many interesting problems in geometric combinatori

8 KB (1,122 words) - 02:52, 24 October 2011
Christopher Means
grid will have four neighbors in space, as well as two in time. To reconfigure the continuous equation into a

6 KB (1,025 words) - 23:41, 15 July 2012
Integration of differential forms of degree 0 and 1
...\in {\bf R}$, $c_i$ are $1$-cells. They form $C_1({\bf R}^2)$, a [[vector space]] (assuming ${\bf R}^2$ has the grid).

12 KB (1,906 words) - 17:44, 31 December 2012
De Rham cohomology
...s a region. The $k^{\rm th}$ ''de Rham cohomology'' of $R$ is the [[vector space]]

6 KB (938 words) - 20:55, 13 March 2013
Orientation of manifolds
...es v_2$, where $v_1$,$v_2 \in T_aM$. These form the basis of the [[tangent space]] at this point.

6 KB (983 words) - 16:30, 28 January 2013
Introduction to Topology: Pure and Applied by Adams and Franzosa
** 0.4. Euclidean Space

3 KB (311 words) - 13:36, 26 October 2012
Discrete Calculus. An Introduction
##[[Dual space]]s

4 KB (466 words) - 19:07, 8 July 2014
Peter Saveliev's Academic Portfolio
...ral scheme'' (with N. Chmutin), Raketno-Kosmicheskaya Tekhnika (Rocket and Space Technology), 6 (1989) 3, 62-65.

25 KB (3,536 words) - 14:28, 17 January 2017
Klein bottle
Let's compute its [[homology as a vector space|homology]]:

2 KB (280 words) - 04:14, 10 November 2012
3D image analysis
*[[Homology as a vector space]]

7 KB (1,021 words) - 16:58, 20 February 2011
Product topology
...and $Y$. Then the ''product'' $X \times Y$ of $X$ and $Y$ is a topological space defined on the [[product set]] $X \times Y$ with the following [[Neighborho

8 KB (1,339 words) - 16:53, 27 August 2015
Linear operators: part 1
We discuss the existence of linear operators, in the [[vector space]] setting $A \colon V \rightarrow U$.

13 KB (2,086 words) - 19:58, 27 January 2013
Acyclic models
Indeed, for any topological space $X$

8 KB (1,126 words) - 15:32, 14 July 2013
Real analysis: course
*[[Metric space]]s

1 KB (173 words) - 17:18, 16 June 2011
Linear operators: part 3
*$V$ is a vector space,

10 KB (1,612 words) - 14:25, 16 October 2013
Algebraic topology and digital image analysis
...lassical topology studies images drawn on a piece of paper or 3D solids in space instead of digital images stored on a computer, so what? If your digital me

3 KB (542 words) - 14:34, 29 August 2010
Suspension
In topology, this is how the ''suspension'' is defined. Given a topological space $X$, first form the product $[0,1] \times X$, then the ''cone of'' $X$ is t

501 bytes (74 words) - 01:09, 1 December 2012
Motion analysis with Matlab
...ct would incorporate a form of [[stereo vision]] to be able to distinguish space and distance with the possible addition of a form of laser-based distance t

2 KB (287 words) - 16:48, 20 February 2011
Ayasdi
...cified as the [[preimage]]s of sets in the given covering of the reference space R). "

4 KB (561 words) - 14:46, 16 October 2011
Terms and conditions
...ding, but not limited to, limits on retention time, file size, and storage space for User Content. Intelligent Perception shall have no liability or respons

16 KB (2,535 words) - 04:15, 9 March 2011
Hodge duality of cubical forms
For the value of this sign depends on the dimension $n$ of the space and the degree $k$ of the cell: $(-1)^{k(n-k)}$. For details, see [[Hodge d

7 KB (1,114 words) - 18:10, 27 August 2015
Betti numbers
To sort this out one needs to turn the set of cycles into a [[vector space]].

4 KB (603 words) - 18:04, 27 August 2015
Topological equivalence
...new situation. In linear algebra the equivalence relation between [[vector space]]s is provided by [[isomorphisms]], same in modern algebra. There are also

2 KB (276 words) - 23:24, 30 November 2012
Homology and algebra
The latter groups are in fact [[vector space]]s. These are respectively homology over ${\bf Z}$ (or ''integer homology''

8 KB (1,386 words) - 18:40, 27 August 2015
Calculus III -- Fall 2017
...- Calculus with Analytic Geometry III.''' Vectors, curves, and surfaces in space. Derivatives and integrals of functions of more than one variable. A study

10 KB (1,596 words) - 13:34, 27 November 2017
Topology II -- Spring 2014
#[[Homology as a vector space]]

3 KB (445 words) - 16:04, 20 May 2014
Homology in dimension 1
...t $\partial$ is a [[homomorphism]] (or a [[linear operator]] in the vector space case): is a linear operator between two copies of ${\bf R}^3$ with [[basis of vector space|bases]] $\{a, b, c \}$ and $\{A, B, C \}$ respectively. The values of $\par

6 KB (1,046 words) - 18:42, 27 August 2015
Peter Saveliev
Such a symmetry is supposed to be an involution of the $3$-space, $A^2=I$; therefore, its diagonalized matrix has only $\pm 1$ on the diagon

4 KB (652 words) - 14:02, 15 April 2024
Topology of proteins
...-dimensional [[simplicial complex]]es, as follows. Each atom is a point in space and a vertex in the complex. If the distance between two atoms is less that

2 KB (316 words) - 21:16, 2 October 2011
Homology of homotopic maps
...[[cell complex]]es are [[homeomorphic]] then their [[homology as a vector space|homology groups]] are [[isomorphic]]:

2 KB (290 words) - 20:53, 27 August 2015
Vector space of infinite sequences
1. Is this a [[vector space]] in some sense?

2 KB (319 words) - 02:56, 13 October 2011

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