Antonio Lupetti

Artificial intelligence systems rely on complex automated deduction algorithms that originate from the formalism of logic. There are various types of logics, such as propositional logic, predicate logic, and descriptive logic. In this post, I will briefly explain the characteristics of propositional logic and how it is used to determine the truth values of propositional formulas. Propositional logic studies inferential relationships among sentences, focusing on logical operators called propositional connectives \( \neg \), \( \wedge \), \( \lor \), \( \rightarrow \), \( \leftrightarrow \), \( \oplus \).

Antonio Lupetti

This week’s issue of The Loop delves deeply into the recurrent argument regarding the concerns over emerging technologies and their impact on society. It’s fascinating to note that this debate is not a new phenomenon. In the modern era, it has been a part of our societal discourse since the 1950s, as highlighted by Isaac Asimov in his writings:

Whenever man develops a new and powerful technology, he inevitably worries about the potential negative consequences.

Isaac Asimov, The Naked Sun (1956)

The introduction of personal computers in the 1980s sparked similar discussions, and today, we revisit these concerns with the emergence of artificial intelligence.

Antonio Lupetti

When dealing with a large amount of text, it is essential to have tools that can help computers recognize and evaluate the similarity between documents. One of the most effective methods in this field is cosine similarity. This transformative approach allows for the semantic interpretation of human language in a format that machines can easily understand.

The cosine similarity between two vectors is calculated using the following formula:

\[ \text{cosine similarity} \ (V_x, V_y) = \frac{\sum_{i=1}^{n} V_{x_i} \cdot V_{y_i}}{\sqrt{\sum_{i=1}^{n} (V_{x_i})^2} \times \sqrt{\sum_{i=1}^{n} (V_{y_i})^2}} \]

or in compact form:

\[\text{cosine similarity} \ (V_x, V_y) = \frac{V_x \cdot V_y}{||V_x|| \ ||V_y||}\]

In this post, I propose a practical example of how to assess the similarity between different documents by referencing some simple cases.

Antonio Lupetti

THE LOOP n. 2

How Machine Think — Artificial intelligence merges methods to replicate human thought, driven by computing power, vast data, and advanced algorithms. Its rapid advancement transforms sectors and reshapes society by enhancing problem-solving capabilities through symbolic processing, knowledge representation, and logical inference.

Artificial intelligence is a field that combines different methods and approaches to imitate human thought processes, which a computerized system can replicate. The rapid progress we have observed in recent years in this field is driven by the growing computing power, the abundance of extensive datasets, and the ongoing research and development of advanced algorithms.

Read more

Antonio Lupetti

During the weekend, I dedicated much time and effort to improving the website’s user interface. The revisions have produced a more cohesive and visually pleasing appearance by exclusively utilizing grayscale tones. Additionally, I have incorporated unique background images to enhance the website’s design. Furthermore, I am currently in the process of developing and enriching the existing content.

I have included a button directly below the homepage to streamline navigation.


I have also included a series of links I recently curated, which may be interesting.

Antonio Lupetti

Over the past week, I’ve made some significant improvements to the website. I’ve revamped the about page and completed the section dedicated to the weekly newsletter, The Loop. In addition, I’ve introduced new topics like Ruffini’s Rule and made some enhancements to the exercise pages.

Finally, I have improved the internal search engine, which will gradually start to provide more comprehensive suggestions for the most popular search queries made by users.


Moreover, I came across some interesting links that I’d like to share with you. These links are insightful and informative, and I believe you’ll find them interesting too.

Antonio Lupetti

I added a new topic on logarithm theory and created a flashcard on the logarithmic function, its properties, and its graph. I’ll also include a summary section on well-known functions later on. Here is an example.


The logarithmic function is the inverse of the exponential function. Therefore, its domain and range are inverted compared to the exponential function. A logarithmic function is defined as a function of the form:

\[ y = \log{_a}x \quad \text{with} \quad a \gt 0 \quad a \neq 1 \quad \forall x \in \mathbb{R}^+\]

  • Domain: \(x \in \mathbb{R}^+ \)
  • Range: \(\mathbb{R}\)

We have two cases:

  • for \(a \gt 1\) the function in increasing.
  • for \(0 \lt a \lt 1\) the function is decreasing.

Graphical representation
of the function \(f(x) = \log_a(x)\)
with \(a > 1\).

Graphical representation
of the function \(f(x) = \log_a(x)\)
with \(0 \lt a \lt 1\).


  • Derivative: \[\frac{{d}}{{dx}} \log_a(x) = \frac{1}{x \ln(a)}\]

  • Indefinite integral: \[\int \frac{\log(x)}{\log(a)} \ dx = \frac{x \cdot (\log(x)-1)}{\log(a)} + c\]

More in general, considering the function \(y= \log(x)\), the natural logarithm, we have:

  • \(\log(x) = \log_e(x)\)
  • \(\log(x) = \log(a) \cdot \log_a(x)\)

  • Domain: \(x \in \mathbb{R}^+ \)
  • Range: \(\mathbb{R}\)
  • Roots: \(x=1\)

  • Derivative: \[\frac{{d}}{{dx}} \log(x) = \frac{1}{x}\]

  • Integral representation: \[\int_{0}^{x} \log(t) \ dt\]

  • Indefinite integral: \[\int \log(x)dx = x \cdot( \log(x)-1) + c\]

  • Definite integral \[\int_{0}^{1} \log(x) \ dx = -1\]
Antonio Lupetti

Two new topics, radicals and powers, have been added to the site within the last week. Additionally, we are pleased to announce the publication of the first issue of The Loop newsletter.


In this issue:

  • Artificial Intelligence is sparking widespread interest and debate due to its ethical, economic, and social implications.
  • As businesses explore AI’s potential, IT professionals are eager to lead internal adoption efforts driven by perceived benefits and by the potential benefits of its institutional storytelling.
  • However, the importance of pragmatic approaches to its adoption, clear goals, and strategic planning cannot be overstated, as they provide a solid foundation for understanding AI’s limitations.
Antonio Lupetti

Over the recent days, efforts have been dedicated to revising various articles, particularly those about quadratic equations. The attached module has also incorporated ten new exercises with detailed step-by-step explanations.

Practice with Quadratic Equations

Below are some examples. If you are beginning your learning journey: start from here.

  • \[\text{A1. } \quad x^2 = 5x-6\] solution

  • \[\text{A2. } \quad \frac{1}{2}x^2+\frac{\sqrt{3}}{2}x+\frac{5}{8}\] solution

  • \[\text{A3. } \quad -7x + 3 = -2x^2 \] solution

Antonio Lupetti

Algebrica has a new logo and a fresh new interface that is even cleaner and more essential. It aims to eliminate all distractions for readers, making content consumption more straightforward and enjoyable. The topics are better organized and are undergoing a comprehensive review to ensure quality. Some pages may be temporarily unavailable but will become available progressively in the coming weeks.