Notes

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. ...

URI Sono stata LA vera invenzione di Berners Lee accennati in Storia del web. Il problema è avere un modo per identificare una risorsa in modo univoco sull’internet. Introduzione La risorsa Una risorsa è qualunque struttura che sia oggetto di scambio tra applicazioni all’interno del World Wide Web. Ora una risorsa può essere qualunque cosa, non solamente solo un file! Quindi è agnostico rispetto a contenuto oppure metodo di memorizzazione del dato, appare anche in questo ambiente importante vedere quanto siano importanti standard che permettano una comunicazione ...

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. ...

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 ...

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 ...

3.1 Successioni $$ \begin{cases} f: \mathbb{N} \to \mathbb{R} \\ n \to f(n) \\ \{a\}_{n \in \mathbb{N}} \vee a_n \end{cases} $$$$ \left\{ a \right\} _{n \in \mathbb{N}} $$3.1.1 Immagine e successione L’immagine di una successione (l’insieme dei suoi elementi) non è una successione! la successione è anche ordinata. 3.1.2 Limitazioni della successione Come per gli insiemi si può definire se l’insieme è limitato superiormente, inferiormente o entrambi, a seconda di come lo definiamo in questo modo possiamo poi farci altri ragionamenti ...

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. ...

Introduzione all’algebra relazionale Confronto con relazioni matematiche Le relazioni come le intendiamo in database sono leggermente diverse rispetto a quelle presenti per le relazioni matematiche: Non conta l’ordine Ci sono gli attributi Per il resto se introduciamo questo sistema per tenere conto delle astrazioni, possiamo analizzarle matematicamente, e questo ci fornisce qualche sicurezza in più diciamo. Four types of operations Set operations: union, intersection, difference Filter queries: Projecting or selecting Renaming queries: renames Join: correlare tuple di relazioni diverse Definition of tuples Le relazioni sono esattamente quelle definite in matematica, però noi aggiungiamo anche gli attributi, in modo da poter considerare l’ordine delle colonne non importante. ...

How Bloom Filters Work A Bloom filter is a space-efficient probabilistic data structure used to test whether an element is possibly in a set or definitely not in a set. It allows for false positives but never false negatives. One example of application is the membership query in Wide Column Storage, HBase. They make document lookup faster by completely skipping some HFiles. Structure and Initialization A Bloom filter consists of: A bit array of size $m$, initialized to all 0s. $k$ independent hash functions, each mapping an input element to one of the $m$ positions in the bit array. Common operations Insertion of an Element To insert an element $x$: ...