Explore >> Select a destination
|
You are here 
|
francisbach.com | ||
| | | | |
djalil.chafai.net
|
|
| | | | | This tiny post is adapted from the introduction of a recent work with David García-Zelada and Paul Jungon the macroscopics and edge of a planar jellium seen as a Coulomb gas. Potential theory.The Coulomb kernel $g$ in $\mathbb{R}^d$, $d\geq1$, is given for all $x\in\mathbb{R}^d$ by \[ g(x)=\begin{cases}\displaystyle\log\frac{1}{|x|}&\text{if $d=2$}\\[1em]\displaystyle\frac{1}{(d-2)|x|^{d-2}}&\text{if $d\neq2$}\end{cases}.\] The Coulomb... | |
| | | | |
djalil.chafai.net
|
|
| | | | | This post is mainly devoted to a probabilistic proof of a famous theorem due to Schoenberg on radial positive definite functions. Let us begin with a general notion: we say that \( {K:\mathbb{R}^d\times\mathbb{R}^d\rightarrow\mathbb{R}} \) is a positive definite kernel when \[ \forall n\geq1, \forall x_1,\ldots,x_n\in\mathbb{R}^d, \forall c\in\mathbb{C}^n, \quad\sum_{i=1}^n\sum_{j=1}^nc_iK(x_i,x_j)\bar{c}_j\geq0. \] When \( {K} \) is symmetric, i.e. \( {K(x,y)=K(y,x)} \) for... | |
| | | | |
blog.ml.cmu.edu
|
|
| | | | | The latest news and publications regarding machine learning, artificial intelligence or related, brought to you by the Machine Learning Blog, a spinoff of the Machine Learning Department at Carnegie Mellon University. | |
| | | | |
stephenmalina.com
|
|
| | | Selected Exercises # 5.A # 12. Define $ T \in \mathcal L(\mathcal P_4(\mathbf{R})) $ by $$ (Tp)(x) = xp'(x) $$ for all $ x \in \mathbf{R} $. Find all eigenvalues and eigenvectors of $ T $. Observe that, if $ p = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + a_4 x^4 $, then $$ x p'(x) = a_1 x + 2 a_2 x^2 + 3 a_3 x^3 + 4 a_4 x^4. | ||
