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jeremykun.wordpress.com | ||
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www.jeremykun.com
Define the Ramsey number $ R(k,m)$ to be the minimum number $ n$ of vertices required of the complete graph $ K_n$ so that for any two-coloring (red, blue) of the edges of $ K_n$ one of two things will happen: There is a red $ k$-clique; that is, a complete subgraph of $ k$ vertices for which all edges are red. There is a blue $ m$-clique. It is known that these numbers are always finite, but it is very difficult to compute them exactly. |
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yufeizhao.wordpress.com
This post is adapted from my new expository survey Extremal regular graphs: independent sets and graph homomorphisms. The earliest result in extremal graph theory is usually credited to Mantel, who proved, in 1907, that a graph on $latex {n}$ vertices with no triangles contains at most $latex {n^2/4}$ edges, where the maximum is achieved for... |
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nhigham.com
The spectral radius $latex \rho(A)$ of a square matrix $latex A\in\mathbb{C}^{n\times n}$ is the largest absolute value of any eigenvalue of $LATEX A$: $latex \notag \rho(A) = \max\{\, |\lambda|: \lambda~ \mbox{is an eigenvalue of}~ A\,\}. $ For Hermitian matrices (or more generally normal matrices, those satisfying $LATEX AA^* = A^*A$) the spectral radius is just... |
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matt.might.net
[AI summary] This text explains how a single perceptron can learn basic Boolean functions like AND, OR, and NOT, but fails to learn the non-linearly separable XOR function. This limitation led to the development of modern artificial neural networks (ANNs). The transition from single perceptrons to ANNs involves three key changes: 1) Adding multiple layers of perceptrons to create Multilayer Perceptron (MLP) networks, enabling modeling of complex non-linear relationships. 2) Introducing non-linear activation functions like sigmoid, tanh, and ReLU to allow networks to learn non-linear functions. 3) Implementing backpropagation and gradient descent algorithms for efficient training of multilayer networks. These changes allow ANNs to overcome the limitations of ... |
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