Notes

This is also known as Lagrange Optimization or undetermined multipliers. Some of these notes are based on Appendix E of (Bishop 2006), others were found when studying bits of rational mechanics. Also (Boyd & Vandenberghe 2004) chapter 5 should be a good resource on this topic. $$ \begin{array} \\ \min f_{0}(x) \\ \text{subject to } f_{i}(x) \leq 0 \\ h_{j}(x) = 0 \end{array} $$Lagrangian function $$ \mathcal{L}(x, \lambda, \nu) = f_{0}(x) + \sum \lambda_{i}f_{i}(x) + \sum\nu_{j}h_{j}(x) $$ We want to say something about this function, because it is able to simplify the optimization problem a lot, but first we want to study this mathematically. ...

Introduzione alla normalizzazione Perché si normalizza? Cercare di aumentare la qualità del nostro database, perché praticamente andiamo a risolvere delle anomalie possibili al nostro interno, e questo aiuta per la qualità. Solitamente queste anomalie sono interessanti per sistemi write intensive, in cui vogliamo mantenere i nostri dati in una forma buona. Però capita non raramente che vogliamo solamente leggere. In quei casi sistemi come Cloud Storage, Distributed file systems potrebbero risultare più effettivi. ...

What are Banach Spaces? A Banach space is a complete normed vector space, meaning that every Cauchy sequence in the space converges to a limit within the space. See Spazi vettoriali for the formal definition. Examples of Banach Spaces In this section, we list some examples of the most common Banach Spaces $\ell^p$ Spaces (Sequence Spaces) Defined as: $$ \ell^p = \left\{ (x_n)_{n\in \mathbb{N}} \mid \sum_{n=1}^{\infty} |x_n|^p < \infty \right\}, \quad 1 \leq p < \infty $$ The norm is given by: $$ \|x\|_p = \left( \sum_{n=1}^{\infty} |x_n|^p \right)^{1/p} $$ When $p = \infty$, we define: $$ \ell^\infty = \left\{ (x_n)_{n\in \mathbb{N}} \mid \sup_n |x_n| < \infty \right\} $$ with the norm $\|x\|_{\infty} = \sup_n |x_n|$. These spaces are Banach under their respective norms. $L^p$ Spaces (Function Spaces) ...

Fatou’s lemma is a fundamental result in measure theory that deals with the relationship between limits and integrals of sequences of non-negative measurable functions. See the wikipedia page for further info. Statement of Fatou’s Lemma Let $(f_n)$ be a sequence of non-negative measurable functions on a measure space $(X,\mu)$. Then: $$\int \liminf_{n \to \infty} f_n \,d\mu \leq \liminf_{n \to \infty} \int f_n \,d\mu$$In words, this means that the integral of the limit inferior of a sequence of functions is less than or equal to the limit inferior of their integrals. ...

Riguardare Successioni per avere primo attacco sui limiti 4.1 Limiti finiti al finito 4.1.1 Intorno sferico Dato l’insieme $\mathbb{R}$ si definisce l’intorno sferico aperto di $x \in \mathbb{R}$ di raggio $r \in \mathbb{R}$ l’insieme $I_r(x) = (x -r, x + r)$ questa nozione è molto importante per definire il limite. Lo useremo subito su un punto di accumulazione 4.1.2 Punto di accumulazione Un punto di accumulazione $x$ di un insieme $A \subseteq \mathbb{R}$ è un punto tale per cui mi posso avvicinare in modo indefinito in quel punto. Infatti deve $\forall r > 0 \in R, \exists x_ 1 \in A : x_1 \in I_r(x) \wedge x_1 \not= x$ ossia per cui $A \cap I_r(x) \not= \varnothing$. ...

Questo è un tentativo di aggiungere un argomento che non era presente quando abbiamo fatto il corso due anni fa. Inizio la scrittura il 2024-03-03. Questo non è stato trattano nel corso, ma è importante per molte cose. Quindi introduco questo appunto. Introduzione alle serie Le serie infinite sono dei mostri strani perché non si comportano spesso come dovrebbero. Definizione di convergenza $$ \lim_{ n \to \infty } f_{n} = c $$ con $c$ un numero reale. ...

Geometria introduttiva Tangente e pendenza Si può trovare la relazione fra la pendenza della retta e la tangente. Possiamo analizzare la retta dal punto di vista analitico, della formula e si può dimostrare che data una retta nella forma $y = mx + q$ $m$ è la pendenza della retta. Formula generale delle rette Dati qualunque due punti .$(x_1, y_1), (x_2, y_2)$ possiamo dire che la pendenza è esprimibile come ...

The maximum entropy principle is one of the most important guiding motives in artificial artificial intelligence. Its roots emerge from a long tradition of probabilistic inference that goes back to Laplace and Occam’s Razor, i.e. the principle of parsimony. Let’s start with a simple example taken from Andreas Kraus’s Lecture notes in the ETH course of Probabilistic Artificial Intelligence: Consider a criminal trial with three suspects, A, B, and C. The collected evidence shows that suspect C can not have committed the crime, however it does not yield any information about sus- pects A and B. Clearly, any distribution respecting the data must assign zero probability of having committed the crime to suspect C. However, any distribution interpolating between (1, 0, 0) and (0, 1, 0) respects the data. The principle of indifference suggests that the desired distribution is $(\frac{1}{2}, \frac{1}{2}, 0)$, and indeed, any alterna- tive distribution seems unreasonable. ...

In this note we will briefly present one problem common in operation research. The practical needs that formulated this problem are quite obvious: choosing the best location to build some important services for communities. The optimal minimax facility location refers to the placement of a facility (such as a warehouse, hospital, or service center) in such a way that the maximum distance or cost between the facility and any of the demand points (such as customers, patients, or users) is minimized. This approach is particularly useful when the goal is to ensure that no demand point is too far from the facility, thus providing a form of equity in service delivery. ...

Little bits of history It was invented in 1970 in Almaden (San Jose) by IBM (Don Chamberlin, Raymond Boyce worked on this) for the first relational database, called system R. Then for copyright issues it hasn’t been called SEQUEL, so they branded it as SQL. SQL is a declarative language With declaratives language there is a separation between what I call the intentionality and the actual process. In declarative languages we just say what we want the result to be, and don’t care what the actual implementation is like. This allows queries to be executed and optimized in different ways, even if the query on the surface is the same ...