Notes

Diffusion is a physical process that models random motion, first analyzed by Brown when studying pollen grains in water. In this section, we will first analyze a simplified 1-dimensional version, and then delve into diffusion models for images, the ones closest to (Ho et al. 2020). The Diffusion Process This note follows original Einstein’s presentation, here we have a simplified version. Let’s suppose we have a particle at $t = 0$ at some position $i$. We have a probability of jumping to the left of $p$ to right of $q$, the rest is staying at the same position. ...

Questo problema è stato trattato in modo un po’ più semplificato (nel caso in cui la carica era esattamente a metà in Campo elettrico#Dipolo elettrico). Questo problema è stato storico, utilizzato per analizzare l’atomo. Potenziale del dipolo elettrico 🟩– $$ V(P) = V_{r^{+}} + V_{r^{-}} = \frac{q}{4\pi\varepsilon_{0}}\left( \frac{1}{r^{+}} - \frac{1}{r^{-}} \right) $$$$ r^{+} - r^{-} = -a \cos \theta $$$$ \left( \frac{1}{r^{+}} - \frac{1}{r^{-}} \right) = \frac{a\cos \theta}{r^{2}} $$$$ V(P) = \frac{1}{4\pi\varepsilon_{0}}\frac{qa\cos \theta}{r^{2}} = \frac{1}{4\pi\varepsilon_{0}}\frac{P\cos \theta}{r^{2}} = \frac{1}{4\pi\varepsilon_{0}} \frac{\vec{P}\cdot \hat{r}}{r^{2}} $$ Direttamente proporzionale al momento di tipolo Inversamente proporzionale al quadrato del raggio. Campo elettrico nel dipolo $$ \vec{E} = \frac{1}{4\pi\varepsilon_{0}} \vec{P} \cdot \frac{\hat{r}}{r^{3}} $$$$ \vec{E} = -\vec{\nabla} V $$Componente parallela 🟩 $$ \vec{E} = - \vec{\nabla}V = -\frac{\delta V}{\delta x}\hat{i} -\frac{\delta V}{\delta y}\hat{j} -\frac{\delta V}{\delta z}\hat{k} $$ Sappiamo che $\vec{P} = P\hat{k}$ e $\vec{r} = x\hat{i} + y \hat{j} + z \hat{k}$ allora abbiamo che $\vec{P} \cdot \vec{r} = Pz$ Poi abbiamo che $z = r \cos \theta$ ...

The DP (Dirichlet Processes) is part of family of models called non-parametric models. Non parametric models concern learning models with potentially infinite number of parameters. One of the classical application is unsupervised techniques like clustering. Intuitively, clustering concerns in finding compact subsets of data, i.e. finding groups of points in the space that are particularly close by some measure. The Dirichlet Process See Beta and Dirichlet Distributions for the definition and intuition of these two distributions. One quite important thing that Dirichlet allows to do is the ability of assigning an ever growing number of clusters to data. This models are thus quite flexible to change and growth. ...

We want to know how to handle systems that have a large number of data. In previous lesson we have discovered how to quickly access and make Scalable systems with huge dimensions, see Cloud Storage. Object storage could store billions of files, we want to handle millions of petabyte files. Designing DFSs The Use Case Remember that the size of the files where heavily limited for Cloud Storage. The physical limitation was due to the limited size of a single hard disk, which was usually in the order of the Terabytes. Here, we would like to easily store petabytes of data in a single file, for example big datasets. Another feature that should be easily supported is highly concurrent access to the filesystem, last but not least being able to set up permissions in the system. ...

Scalare Scalare e gradiente 🟩 $$ \varphi(x, y, z) : \mathbb{R}^{3} \to \mathbb{R} $$$$\vec{\nabla}\varphi = ( \frac{\delta\varphi}{\delta x}, \frac{\delta\varphi}{\delta y}, \frac{\delta\varphi}{\delta z}) = \frac{\delta\varphi}{\delta x} \hat{i} + \frac{\delta\varphi}{\delta y} \hat{j} + \frac{\delta\varphi}{\delta z} \hat{k}$$ Se consideriamo il gradiente da solo è un campo vettoriale (dice la direzione della derivata multidimensionale). Gradiente in coordinate polari 🟨 Questo è un po’ più difficile da gestire, però è abbastanza facile una volta che si fanno certe osservazioni. Sappiamo che $dV = \vec{\nabla} V \cdot d\vec{s}$, TODO: finire la dimostrazione, è descritta bene a pagina 47 del mazzoldi. ...