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Formally, a boolean circuit is a directed acyclic graph. Nodes with indegree zero are input nodes, labeled <math>x_1, x_2, \ldots , ...>v</math> shakes hand. The handshaking lemma states that in any undirected graph, the number of vertices whose degrees are odd is even. It is sufficient to ...

Formally, a boolean circuit is a directed acyclic graph. Nodes with indegree zero are input nodes, labeled <math>x_1, x_2, \ldots , ...>v</math> shakes hand. The handshaking lemma states that in any undirected graph, the number of vertices whose degrees are odd is even. It is sufficient to ...

'''Probability Theory''' <br> & '''Mathematical Statistics'''</font> This is the webpage for the ''Probability Theory and Mathematical Statistics'' (概率论与数理统计) class of Spring 2024. Students who ...

== Problem 5 (Probability meets graph theory) == Let <math>G = (V, E)</math> be a <strong>fixed</strong> undirected graph without isolating vertex. ...

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

...h> and <math>W</math> denotes the maximum edge weight. When the underlying graph is super dense, namely, the total number of insertions <math>m</math> is <m :In this work, we provide two algorithms for this problem when the graph is sparse. The first one is a simple deterministic algorithm with <math>\ti ...

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

[[File:Hyperbola E.svg|thumb|The area shown in blue (under the graph of the equation y=1/x) stretching from 1 to e is exactly 1.]] [[Category:Number theory]] ...

* [http://theory.stanford.edu/~yuhch123/ Huacheng Yu] (Harvard) |align="center"|[http://theory.stanford.edu/~yuhch123/ Huacheng Yu] (俞华程)<br> ...

...been used in proofs of many important results in computational complexity theory, such as [http://en.wikipedia.org/wiki/SL_(complexity) SL]=[http://en.wikip Consider an undirected (multi-)graph <math>G(V,E)</math>, where the parallel edges between two vertices are allo ...

Consider a graph <math>G(V,E)</math> which is randomly generated as: Such graph is denoted as '''<math>G(n,p)</math>'''. This is called the '''Erdős–Rényi ...

...article. The number following the name of the group is the [[order (group theory)|order]] of the group. | [[File:Complete graph K5.svg|105px]] ...

...s usually stated as a theorem for the existence of matching in a bipartite graph. In a graph <math>G(V,E)</math>, a '''matching''' <math>M\subseteq E</math> is an indep ...

...s usually stated as a theorem for the existence of matching in a bipartite graph. In a graph <math>G(V,E)</math>, a '''matching''' <math>M\subseteq E</math> is an indep ...

...s usually stated as a theorem for the existence of matching in a bipartite graph. In a graph <math>G(V,E)</math>, a '''matching''' <math>M\subseteq E</math> is an indep ...

...s usually stated as a theorem for the existence of matching in a bipartite graph. In a graph <math>G(V,E)</math>, a '''matching''' <math>M\subseteq E</math> is an indep ...

...s usually stated as a theorem for the existence of matching in a bipartite graph. In a graph <math>G(V,E)</math>, a '''matching''' <math>M\subseteq E</math> is an indep ...

...s usually stated as a theorem for the existence of matching in a bipartite graph. In a graph <math>G(V,E)</math>, a '''matching''' <math>M\subseteq E</math> is an indep ...

...s usually stated as a theorem for the existence of matching in a bipartite graph. In a graph <math>G(V,E)</math>, a '''matching''' <math>M\subseteq E</math> is an indep ...

...been used in proofs of many important results in computational complexity theory, such as [http://en.wikipedia.org/wiki/SL_(complexity) SL]=[http://en.wikip Consider an undirected (multi-)graph <math>G(V,E)</math>, where the parallel edges between two vertices are allo ...

== Graph Expansion == ...been used in proofs of many important results in computational complexity theory, such as [http://en.wikipedia.org/wiki/SL_(complexity) SL]=[http://en.wikip ...

...r function''' is a [[Function (mathematics)|function]] whose [[:wikt:graph|graph]] is a [[Line|straight line]] in 2-dimensions (see images).<ref>{{cite book ...a linear function is a function ''f''(''x''):'''R'''→'''R''' such that the graph of ''f'' is a line. This means the [[function (mathematics)|domain]] or inp ...

...been used in proofs of many important results in computational complexity theory, such as [http://en.wikipedia.org/wiki/SL_(complexity) SL]=[http://en.wikip Consider an undirected (multi-)graph <math>G(V,E)</math>, where the parallel edges between two vertices are allo ...

In probability theory, the word "condition" is a verb. "Conditioning on the event ..." means that = Min-Cut in a Graph = ...

...l type of equation is called the function. This is often used in making [[graph]]s. ...mber or numbers into and get a certain number out. When using functions, [[graph]]s can be powerful tools in helping us to study the solutions to equations. ...

...ing a leaf (along with the edge adjacent to it) from a tree, the resulting graph is still a tree. :let <math>T</math> be empty graph, and <math>v_{n-1}=n</math>; ...

'''Probability Theory''' <br> & '''Mathematical Statistics'''</font> This is the webpage for the ''Probability Theory and Mathematical Statistics'' (概率论与数理统计) class of Spring 2023. Students who ...

Consider a graph <math>G(V,E)</math> which is randomly generated as: Such graph is denoted as '''<math>G(n,p)</math>'''. This is called the '''Erdős–Rényi ...

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

In probability theory, the word "condition" is a verb. "Conditioning on the event ..." means that = Min-Cut in a Graph = ...

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

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

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

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

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

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

Consider a graph <math>G(V,E)</math> which is randomly generated as: Such graph is denoted as '''<math>G(n,p)</math>'''. This is called the '''Erdős–Rényi ...

<li>[<strong>Chernoff bound meets graph theory</strong>] ...pproaching 1 (as <math>n</math> tends to infinity), the Erdős–Rényi random graph <math>\textbf{G}(n,1/2)</math> has the property that the maximum degree is ...

Formally, a boolean circuit is a directed acyclic graph. Nodes with indegree zero are input nodes, labeled <math>x_1, x_2, \ldots , ...>v</math> shakes hand. The handshaking lemma states that in any undirected graph, the number of vertices whose degrees are odd is even. It is sufficient to ...

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

==== Transition graph ==== ...eighted directed graph <math>G(V,E,w)</math> is said to be a '''transition graph''' of a finite Markov chain with transition matrix <math>P</math> if: ...

== Transition graph == ...eighted directed graph <math>G(V,E,w)</math> is said to be a '''transition graph''' of a finite Markov chain with transition matrix <math>P</math> if: ...

== Transition graph == ...eighted directed graph <math>G(V,E,w)</math> is said to be a '''transition graph''' of a finite Markov chain with transition matrix <math>P</math> if: ...

== Transition graph == ...eighted directed graph <math>G(V,E,w)</math> is said to be a '''transition graph''' of a finite Markov chain with transition matrix <math>P</math> if: ...

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 ...

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 ...

...The function ''f'' is a surjection if every horizontal line intersects the graph of ''f'' in at least one point. ...,''y'')):&#8477;&sup2;→&#8477; defined by ''z''=''y'' is a surjection. Its graph is a plane in 3-dimensional space. The pre-image of ''z''_o is t ...

...re the challenges that arise at the interface of machine learning and game theory: selfish agents may interact with machine learning algorithms strategically ...on of "fairness" in real-world applications and how to model "fairness" in theory. Then I will present several recent progress in designing algorithms that m ...

*[[set theory]] *[[Banach space]] theory ...

...he problem and an output is called a '''solution''' to that instance. The theory of complexity deals almost exclusively with [http://en.wikipedia.org/wiki/D ...itself is a certificate. And for the later one, a Hamiltonian cycle in the graph is a certificate (given a cycle, it is easy to verify whether it is Hamilto ...