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• Functions in higher dimensions
  ...seen on the map, are known. The grade of the road is also known. How fast is the car climbing? The first variable is time, $t$. We also have two ''spatial'' variables: the horizontal location
  113 KB (19,680 words) - 00:08, 23 February 2019
• A graph, non-tree representation of the topology of a gray scale image by Saveliev
  ...as a combination of pixels as well as edges and vertices. The second tool is [[cycle]]s: both the connected components and the holes are captured by cir ...s of the gray level function of the image. The rationale for this approach is that the connected components of these sets are arguably building blocks of
  41 KB (6,854 words) - 15:05, 28 October 2011
• Applications of integral calculus
  ...half so that the area of the whole circle is then twice this number. This is too limiting... Let's start over. ...le with vertical bars based on these segments. Then the area of the circle is approximated by the sum of the areas of the bars: we add a column of the wi
  103 KB (18,460 words) - 01:01, 13 February 2019
• The gradient
  ...all functions of several variables. The two have one cell in common; that is numerical functions. This time we will see how everything is interconnected. We show with the red arrows for different types of function
  74 KB (13,039 words) - 14:05, 24 November 2018
• Continuous functions
  <center>if $x$ is close to $a$ then $f(x)$ is close to $f(a)$:</center> <center>for any $\epsilon > 0$ there is a $\delta > 0$ such that $|x - a| < \delta \Rightarrow | f(x) - f(a) | < \e
  42 KB (7,138 words) - 19:08, 28 November 2015
• Maps of polyhedra
  Previously, we proved that if complex $K^1$ is obtained from complex $K$ via a sequence of elementary collapses, then ...proof was ''straightforward''. However, the result, as important as it is, is a very limited instance of the invariance of homology. We explore next what
  51 KB (9,162 words) - 15:33, 1 December 2015
• Homotopy
  The answer we have been giving is: they all have one hole. However, there is a profound reason ''why'' they must all have one hole. These space are home The reasoning, still not fully justified, is transparent:
  46 KB (7,846 words) - 02:47, 30 November 2015
• Products
  The idea of the product may be traced to the image of a stack, which is a simple arrangement of multiple copies of $X$: We can think of it as if a copy of the $y$-axis is attached to every point on the $x$-axis. Or, we can think in terms of ''pro
  44 KB (7,951 words) - 02:21, 30 November 2015
• Social choice
  We discovered that there is no such solution when the homology of the forest is non-trivial, such as one with a lake in the middle. This is the general setup. There are $m$ voters, or agents, making their selections
  47 KB (8,030 words) - 18:48, 30 November 2015
• Quotients
  A new way of building new things from old is ''gluing'': ...flexivity Axiom, $A \sim A$, can be understood as: every spot of the sheet is glued to itself. The Symmetry Axiom, $A \sim B \Rightarrow B \sim A$, becom
  26 KB (4,538 words) - 23:15, 26 November 2015
• Topological spaces
  ...s to equip each of the sets involved with an additional structure called ''topology''. ...X$, a collection $\tau$ of subsets of $X$ is called a ''topology<!--\index{topology}--> on'' $X$ if it satisfies the following conditions:
  27 KB (4,693 words) - 02:35, 20 June 2019
• Euler and Lefschetz numbers
  We already know, and will prove below, that the meaning of the formula is topological. ...Euler characteristic}--> $\chi (K)$ of an $n$-dimensional cell complex $K$ is the alternating sum of the number of cells in $K$ for each dimension:
  41 KB (7,169 words) - 14:00, 1 December 2015
• PDEs
  Instead of being carried around, the heat is ''exchanged'' -- with adjacent locations. The process is also known as ''diffusion.''
  44 KB (7,469 words) - 18:12, 30 November 2015
• Diffusion
  ...e average of the temperature of the four adjacent rooms. This simple model is implemented with an Excel simulation with the following short formula: Normally, only a proportion $k$ of this amount is shared.
  39 KB (6,850 words) - 15:29, 17 July 2015
• Homotopy and homotopy equivalence
  The answer we have been giving is: they all have one hole. However, there is a profound reason ''why'' they must all have one hole. These space are home The reasoning, still not fully justified, is transparent:
  45 KB (7,738 words) - 15:18, 24 October 2015
• Cell complex
  ...<!--\index{cells}-->, and do it in a gradual and orderly manner. The point is to be able to build and compute homology<!--\index{homology}-->, and do it The main difference is in the manner these cells are found. In the case of cubical complexes<!--\i
  40 KB (6,459 words) - 23:27, 29 November 2015
• Triangulations
  ...nion of a collection of subsets of a Euclidean space, while a cell complex is built via the quotient construction<!--\index{quotient}-->, which always re ...lex $K$ has $n+1$ faces<!--\index{face}-->, $\sigma < \tau$, each of which is an $(n-1)$-simplex, illustrated on the left:
  30 KB (5,172 words) - 21:52, 26 November 2015
• Simplicial complexes
  A graph<!--\index{graph}--> is pure data. It consists of two sets: ...ustrate the data by a subset of the Euclidean space, as follows. Each node is plotted as a distinct point, but otherwise arbitrarily, and these points ar
  30 KB (5,021 words) - 13:42, 1 December 2015
• Diffusion equation
  Instead of being carried around, the heat is ''exchanged'' -- with adjacent locations. It's a circle. The process is also known as ''diffusion.''
  35 KB (5,917 words) - 12:51, 30 June 2016
• Data made Euclidean
  A graph<!--\index{graphs}--> is pure data. It consists of two sets: ...ustrate the data by a subset of the Euclidean space, as follows. Each node is plotted as a distinct point, but otherwise arbitrarily, and these points ar
  31 KB (5,219 words) - 15:07, 2 April 2016
• Topology of graphs
  Here we take our first step toward ''algebraic topology''. *verifying that it is path-connected<!--\index{path-connectedness}--> requires testing ''infinite
  25 KB (4,214 words) - 16:08, 28 November 2015
• Compact spaces
  ...nnectedness}--> (under a continuous map<!--\index{continuous function}-->) is a path-connected. The second main topological theorem from Calculus 1 is:
  19 KB (3,207 words) - 13:06, 29 November 2015
• Cubical complex
  Suppose the Euclidean space ${\bf R}^N$ is given and so is its cubical grid ${\bf Z}^N$. Suppose also that we have its decomposition $ ...of cubical cells $K\subset {\mathbb R}^N$ for which the boundary operator is well defined. This requires us to include all “faces” of the cells alre
  29 KB (4,800 words) - 13:41, 1 December 2015
• Computing homology
  ..., typically, integer coefficients. Then the set of all $k$-chains $C_k(K)$ is an [[abelian group]] with respect to chain addition generated by the $k$-ce ...tyle\sum_i s_i \sigma_i \colon s_i \in {\bf Z}, \sigma_i {\rm \hspace{3pt} is \hspace{3pt} a \hspace{3pt}} k{\rm -cell \hspace{3pt} in \hspace{3pt}} K \r
  26 KB (4,370 words) - 21:55, 10 January 2014
• Simplicial maps and chain maps
  A graph map<!--\index{graph map}--> $f:K\to L$ is a function between graphs $K,L$ that satisfies, for each edge $e$, either: *1. (cloning) $f(e)$ is an edge $g$ and $f$ takes the end-points of $e$ to the end-points of $g$; o
  34 KB (5,897 words) - 16:05, 26 October 2015
• Robustness of topology of digital images and point clouds by Saveliev
  ''Robustness of topology of digital images and point clouds'' by [[Peter Saveliev]] ...ements of the filtration without double count. The second step of analysis is to discard the features that lie outside the user's choice of the acceptabl
  27 KB (4,547 words) - 04:08, 6 November 2012
• Modeling with discrete exterior calculus
  ...erential forms. Our main conclusion is that the isotropy of heat on a grid is heavily dependant upon the geometry of the grid. For example, the square gr ...T}{\partial t}=-k\nabla^{2} T$ as the [[Heat transfer |heat equation]]. It is continuous and its solution relies upon both initial and boundary condition
  31 KB (5,254 words) - 17:57, 21 July 2012
• Simplicial maps
  A graph map<!--\index{graph map}--> $f:K\to L$ is a function between graphs $K,L$ that satisfies, for each edge $e$, either: *1. (cloning) $f(e)$ is an edge $g$ and $f$ takes the end-points of $e$ to the end-points of $g$; o
  34 KB (5,929 words) - 03:31, 29 November 2015
• Selection votes
  ...seen as mutually exclusive but ''all'' of them may come true. What happens is determined by the probabilities assigned to the primary events. These conve ...epresents the lottery when either is equally likely to appear while “hail” is impossible. The probabilities of the three events give the vector $(\tfrac{
  24 KB (3,989 words) - 01:56, 16 May 2016
• Applied Topology and Geometry
  #REDIRECT[[Topology Illustrated]] ''Applied Topology and Geometry'' by [[Peter Saveliev]]
  16 KB (2,088 words) - 16:37, 29 November 2014
• Test
  ''Applied Topology and Geometry'' by [[Peter Saveliev]] ...appreciate your comments. If you are a beginner, you might want to start [[Topology Illustrated|here]] instead.
  16 KB (2,139 words) - 23:01, 9 February 2015
• Lefschetz numbers in control theory
  The goal is to develop some applications of the Lefschetz fixed point theory techniques ...one, $x\in M$. The ''equilibrium'' set $C=\{x\in M:f(x)=x\}$ of the system is the set of fixed points of $f.$
  17 KB (3,052 words) - 22:12, 15 July 2014
• Discrete Calculus: Applied Analysis on Graphs for Computational Science by Grady and Polimeni
  This is an informal review... <center>'''[[Calculus is topology]].'''</center>
  11 KB (1,663 words) - 16:03, 26 November 2012
• Products of complexes
  The idea of the product may be traced to the image of a stack, which is a simple arrangement of multiple copies of $X$: We can think of it as if a copy of the $y$-axis is attached to every point on the $x$-axis. Or, we can think in terms of ''pro
  16 KB (2,892 words) - 22:39, 18 February 2016
• Calculus as a part of topology
  Topology, Algebra, and Geometry are disciplines within Mathematics. In calculus we u ...d Geometry. However, take a look at where this is all happening. The locus is the ''Euclidean space''. Such a space has three different types of structur
  13 KB (2,233 words) - 14:41, 20 February 2015
• More about manifolds
  Suppose the circle is centered at $0$ on the $xy$-plane. In other words, each point is identified with the one symmetric with respect to the $x$-axis. Then
  9 KB (1,542 words) - 19:58, 21 January 2014
• Nerve of cover
  ...ebraic invariants. But how do you find this representation if all you have is a topological space, i.e., a collection of open sets. To solve this problem, let's start with a study of the topology of a [[simplicial complex]].
  8 KB (1,389 words) - 13:35, 12 August 2015
• Constructions
  ...ical sense. The Reflexivity Axiom, $A \sim A$, is: every spot of the sheet is glued to itself. The Symmetry Axiom, $A \sim B \Longrightarrow B \sim A$, b Thinking of a ''zipper'' is also appropriate:
  13 KB (2,270 words) - 22:14, 18 February 2016
• Algebra of differential forms
  For example, we see below that $\phi = x^2 dx + xy dy$ is a [[linear function]] of $dx$ and $dy$, non-linear for $x,y$. This is called ''[[additivity]]''.
  11 KB (1,947 words) - 18:14, 22 August 2015
• Equilibria of dynamical systems
  ...em]] is simply a map $F:S \to S$. Therefore the meaning of its equilibrium is simple: $F(a)=a$. It's a [[fixed point]]! ...ith an example outside the realm of physics, for a change. The application is to the existence of equilibria supply and demand in a simple market economy
  7 KB (1,251 words) - 15:00, 4 April 2014
• Cell decomposition of images
  ...ixel]]s. Normally, it is assumed that objects are black and the background is white. ...ent the object simply as a list of pairs of numbers and this is the way it is commonly done. We will follow this convention.
  11 KB (1,829 words) - 19:26, 10 February 2015
• Algebraic topology: course
  ...uctory, two semester course on algebraic topology and its applications. It is intended for advanced undergraduate and beginning graduate students. ...dated. The source of material is currently in a draft of a book called ''[[Topology Illustrated]]''.
  3 KB (448 words) - 13:32, 17 March 2014
• Euclidean space made discrete
  ...<!--\index{finite differences}-->. Usually only the time (i.e., ${\bf R}$) is discretized: ...wave propagation<!--\index{ wave propagation}--> etc. This time the space is discretized as well:
  11 KB (1,801 words) - 15:50, 25 July 2014
• Pattern recognition in computer vision
  ...and then match them with those of the other image or images. This approach is complex and limited to face identification. In order to develop a content i ...and pattern recognition methods can be used. Nice! The most common method is probably [[clustering]] – looking for groups of points unusually close to
  9 KB (1,526 words) - 17:54, 1 July 2011
• Diffusion with various geometry
  Recall that, naively, [[diffusion]] is modeled with [[Excel]] as follows: On a deeper level, the geometry is supplied by [[Hodge duality]] and affects the forms that we deal with.
  13 KB (2,121 words) - 16:33, 7 June 2013
• Filtering output data
  ...ick only one. This is to be done by the user based on his criteria of what is [[noise]]: too small, low [[contrast]], low [[roundness]] etc. ...[[Pixcavator]]. In fact, [[Pixcavator's output table]] gives you a sample topology. After that you are on your own. And you have several ruler/sliders to use.
  3 KB (598 words) - 15:02, 9 October 2010
• Classification of points with respect to a subset
  For the background see [[Introduction to point-set topology]]. *a point $x$ is called an ''interior point'' of $A$ if there is a [[Neighborhoods and topologies|neighborhood]] $W$ of $x$ that lies entire
  4 KB (703 words) - 01:55, 1 October 2013
• Computational science training: 2010 projects
  <li>''[[The topology of data]]'' by Joseph Snyder, and</li> ...se issues were mostly resolved. Unfortunately, in either case there wasn't enough time to test the programs with real-life data.
  6 KB (926 words) - 17:02, 7 February 2011
• Geometry in calculus
  <center>$\lim_{x\to a}f(x)=L$ if for any $\epsilon >0$ there is a $\delta >0$ such that $0<|x-a|<\delta$ implies that $|f(x)-L|<\epsilon$. ...m_{x\to a}f(x)=L$ if for any open [[neighborhood]] $\epsilon$ of $L$ there is an open neighborhood $\delta$ of $a$ such that $x\in\delta$ implies $f(x)\i
  9 KB (1,604 words) - 18:08, 27 August 2015
• How to compute homology
  The $k$-[[chain group]] $C_k(K)$ is given as a vector space with [[basis of vector space|basis]] consisting of Its dimension is then obvious.
  6 KB (1,049 words) - 09:21, 3 September 2011

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