Search results

...om walk on an undirected graph is determined by the expansion ratio of the graph. We now consider the random walks in a more general setting, and study the ...the transition matrix <math>P</math> instead of the adjacency matrix of a graph. With the same argument as the spectrum of graphs, we can show that <math>\ ...

...The function ''f'' is a bijection if every horizontal line intersects the graph of ''f'' in '''exactly''' one point. ...very horizontal line (in the codomain) intersects exactly one point of the graph. ...

.../Expander Graphs and Mixing|Expander Graphs and Mixing]]: expander graphs, graph spectrum, spectral gap, Cheeger's inequality, rapid mixing of expander walk = The Probability Theory Toolkit = ...

* a directed graph <math>G(V,E)</math>; A fundamental fact in flow theory is that cuts always upper bound flows. ...

* a directed graph <math>G(V,E)</math>; A fundamental fact in flow theory is that cuts always upper bound flows. ...

.../Expander Graphs and Mixing|Expander Graphs and Mixing]]: expander graphs, graph spectrum, spectral gap, Cheeger's inequality, rapid mixing of expander walk = The Probability Theory Toolkit = ...

= Graph Coloring = A '''coloring''' of a graph <math>G(V,E)</math> is a mapping <math>\sigma:V\rightarrow[q]</math> for so ...

.../Expander Graphs and Mixing|Expander Graphs and Mixing]]: expander graphs, graph spectrum, spectral gap, Cheeger's inequality, rapid mixing of expander walk = The Probability Theory Toolkit = ...

...nt is adjacent to the events which are dependent with it in the dependency graph. |Definition (dependency graph)| ...

Formally, a boolean circuit is a directed acyclic graph. Nodes with indegree zero are input nodes, labeled <math>x_1, x_2, \ldots , ...ns that no matter how we color the edges of <math>K_6</math> (the complete graph on six vertices), there must be a '''monochromatic''' <math>K_3</math> (a t ...

== Principles in probability theory == * '''Basics of probability theory''': probability space, events, the union bound, independence, conditional p ...

== Principles in probability theory == * '''Basics of probability theory''': probability space, events, the union bound, independence, conditional p ...

=== Ramsey's theorem for graph === {{Theorem|Ramsey's Theorem (graph, multicolor)| ...

We introduce a general theory of counting permutations with restricted positions. In the derangement prob ...permutation <math>\pi</math> of <math>\{1,\ldots,n\}</math>, define the '''graph''' <math>G_\pi(V,E)</math> as ...

We introduce a general theory of counting permutations with restricted positions. In the derangement prob ...permutation <math>\pi</math> of <math>\{1,\ldots,n\}</math>, define the '''graph''' <math>G_\pi(V,E)</math> as ...

We introduce a general theory of counting permutations with restricted positions. In the derangement prob ...permutation <math>\pi</math> of <math>\{1,\ldots,n\}</math>, define the '''graph''' <math>G_\pi(V,E)</math> as ...

We introduce a general theory of counting permutations with restricted positions. In the derangement prob ...permutation <math>\pi</math> of <math>\{1,\ldots,n\}</math>, define the '''graph''' <math>G_\pi(V,E)</math> as ...

We introduce a general theory of counting permutations with restricted positions. In the derangement prob ...permutation <math>\pi</math> of <math>\{1,\ldots,n\}</math>, define the '''graph''' <math>G_\pi(V,E)</math> as ...

...s]] (like [[light]]) work. It is also called "quantum physics" or "quantum theory". ...e to explain the light that comes from glowing [[hydrogen]]. The quantum [[theory]] of the [[atom]] also had to explain why the [[electron]] stays in its [[o ...

We introduce a general theory of counting permutations with restricted positions. In the derangement prob ...permutation <math>\pi</math> of <math>\{1,\ldots,n\}</math>, define the '''graph''' <math>G_\pi(V,E)</math> as ...

We introduce a general theory of counting permutations with restricted positions. In the derangement prob ...permutation <math>\pi</math> of <math>\{1,\ldots,n\}</math>, define the '''graph''' <math>G_\pi(V,E)</math> as ...

We introduce a general theory of counting permutations with restricted positions. In the derangement prob ...permutation <math>\pi</math> of <math>\{1,\ldots,n\}</math>, define the '''graph''' <math>G_\pi(V,E)</math> as ...

;edge exposure in a random graph ...osure of a random graph, as we "exposing" the edges one by one in a random graph. ...

...ynchronized''', such that for each '''round''', every link (an edge of the graph) can forward at most one packet. ..., such that the number of edges is significantly smaller than the complete graph, yet the distance between any pair of vertices is small, so that the packet ...

Coupling is a powerful proof technique in probability theory. It allows us to compare two unrelated variables (or processes) by forcing ...-dimensional [http://en.wikipedia.org/wiki/Hypercube '''hypercube'''] is a graph on vertex set <math>\{0,1\}^n</math>. For any <math>x,y\in\{0,1\}^n</math>, ...

In probability theory, the '''total variation distance''' measures the difference between two pro Coupling is a powerful proof technique in probability theory. It allows us to compare two unrelated variables (or processes) by forcing ...

We introduce a general theory of counting permutations with restricted positions. In the derangement prob ...permutation <math>\pi</math> of <math>\{1,\ldots,n\}</math>, define the '''graph''' <math>G_\pi(V,E)</math> as ...

...icted number (with the distance measured in standard deviations), then the theory being tested may not be right. See [[prediction interval]]. ...l curve''' because the numbers spread out to make the shape of a bell on a graph. ...

We introduce a general theory of counting permutations with restricted positions. In the derangement prob ...permutation <math>\pi</math> of <math>\{1,\ldots,n\}</math>, define the '''graph''' <math>G_\pi(V,E)</math> as ...

...ie, Leipzig: Hirzel English translation The Physical Principles of Quantum Theory. Chicago: University of Chicago Press, 1930.</ref> ...f a reed) and there was an [[wikt:emit|emitted]] wave that could be [[wikt:graph|graphed]] as a [[wikt:sine|sine]] wave. Much of what had earlier been figur ...

* Each instance is a (nonnegatively)weighted directed graph <math>G(V,E,w)</math> along with two distinct vertices <math>s,t\in V</math ...duling is a class of problems. We consider a central problem in scheduling theory: the '''minimum makespan scheduling'''. ...