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• 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 <br />
  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.</p>
  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''<sub>
  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<!--\index{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''<!--\index{dual space}--> 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• Isotropy in numerical PDEs
  ==Deeper reason: curved space?==
  10 KB (1,593 words) - 13:20, 8 April 2013
• 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
• 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
• 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
• 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
• 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<!--\index{acyclic space}--> 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
• Homology in Calculus
  <center>''subsets of a Euclidean space''.</center>
  6 KB (945 words) - 22:56, 9 February 2015
• 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.</p>
  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
• 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

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