Privacy Policy

Privacy Policy

This Privacy Policy explains how this website collects, uses, and protects your information. By using this site, you agree to the terms described below.

Information Collected

This website may collect the following types of information:

  • Log data: browser type, IP address, referring pages, and the date/time of visits.
  • Cookies: small files stored on your device to improve your browsing experience.
  • Voluntary information: data you submit through contact forms or comments.

Use of Cookies

This website uses cookies to:

  • remember user preferences
  • analyse site traffic
  • deliver personalised advertisements

You can disable cookies in your browser settings, though some site functions may not operate correctly.

Google AdSense and Third-Party Advertising

This site uses Google AdSense, a third-party advertising service. Google uses cookies, including the DoubleClick cookie, to:

  • serve ads based on your visit to this site and other websites
  • display personalised or interest-based advertisements

Users may opt out of personalised advertising through Google’s Ads Settings:

https://www.google.com/settings/ads

Third-party vendors may also use cookies to deliver ads. This site has no control over those cookies.

How Your Information Is Used

The information collected may be used to:

  • improve the performance and content of the website
  • customise advertising and user experience
  • respond to messages or inquiries
  • analyse traffic and user behaviour

Data Protection and Security

This website does not sell, trade, or rent personal data to third parties. Reasonable technical measures are used to protect the data collected; however, no online system is entirely secure.

GDPR (EU/UK) Compliance

If you are located in the EU or UK, you have the right to:

  • access your personal data
  • request correction or deletion
  • object to processing
  • withdraw consent

To make a GDPR-related request, use the contact information below.

CCPA Compliance (California Residents)

California residents have the right to:

  • know what personal information is collected
  • request deletion of personal data
  • opt out of the sale of personal information

This site does not sell personal information.

External Links

This site may contain links to external websites. This Privacy Policy does not apply to those websites, and this site is not responsible for their content or practices.

Changes to This Privacy Policy

This Privacy Policy may be updated occasionally. Any changes will be posted on this page with a revised “Last Updated” date.

Contact

If you have any questions about this Privacy Policy or wish to make a data request, please contact:

Email: tiago.hands@tutanota.com

Last Updated: 20.11.2025

Popular posts from this blog

The Method of Differences — A Clean Proof of the Sum of Cubes

The Method of Differences — A Clean Proof of the Sum of Cubes The method of differences is a remarkably elegant tool for evaluating finite sums. When each term of a series can be written in the form f(r+1) − f(r) , the sum “collapses” — all interior terms cancel, leaving only a boundary expression. This behaviour is called a telescoping sum . 1) Telescoping Sums Assume the general term u r can be written as: u r = f(r+1) − f(r). Then the finite sum from r = 1 to r = n becomes: Σ u r = Σ ( f(r+1) − f(r) ). To see what happens, write out a few terms: u₁ = f(2) − f(1) u₂ = f(3) − f(2) u₃ = f(4) − f(3) ⋮ uₙ = f(n+1) − f(n) When these are added, everything cancels except the first and last pieces: Σ u r = f(n+1) − f(1). This is the essence of the method: interior structure disappears, leaving just the difference between the final and initial states. 2) A Classic Application — The Sum of Cubes We will use this technique to prove the well-known ...
Read more

2×2 Orthogonal Matrix Mastery — A Generalised Construction

Image
2×2 Orthogonal Matrix Mastery — A Generalised Construction Orthogonal matrices in two dimensions reveal one of the cleanest structures in linear algebra. A 2×2 matrix is orthogonal when its columns (and rows) satisfy two conditions: They are perpendicular (their dot product is zero); They have unit length (their magnitude is one). This article presents a clear generalisation: any pair of perpendicular vectors with equal magnitude can be normalised to form an orthogonal matrix. 1. Begin with two perpendicular vectors Let the first vector be: (a, b) A perpendicular vector can be chosen as: (−b, a) Their dot products confirm orthogonality: (a, b) · (−b, a) = −ab + ab = 0 (a, −b) · (b, a) = ab − ab = 0 2. Compute their shared magnitude Both vectors have the same length: |(a, b)| = √(a² + b²) We can therefore normalise each one by dividing by √(a² + b²). 3. Form the matrix using the normalised vectors Place the two normalised vectors...
Read more

The Maclaurin Series — A Clean Derivation

Image
The Maclaurin Series — A Clean Derivation Many smooth functions can be written as an infinite polynomial. When this expansion is centred at x = 0 , we obtain the Maclaurin series . This article derives the Maclaurin formula directly from repeated differentiation, showing precisely why the coefficients involve derivatives and factorials. 1) Begin with a General Power Series Suppose a function f(x) can be expressed as f(x) = a₀ + a₁x + a₂x² + a₃x³ + … + aᵣxʳ + … The constants aᵣ are real coefficients whose values we wish to determine. 2) Evaluate at x = 0 f(0) = a₀ so a₀ = f(0). 3) Differentiate Once f′(x) = a₁ + 2a₂x + 3a₃x² + … + r·aᵣxʳ⁻¹ + … Setting x = 0 eliminates all higher powers: f′(0) = a₁. Thus, a₁ = f′(0). 4) Differentiate Again f″(x) = 2·1·a₂ + 3·2·a₃x + … + r(r−1)aᵣxʳ⁻² + … Evaluate at x = 0 : f″(0) = 2! · a₂ Hence a₂ = f″(0) / 2!. 5) The General Pattern Differentiate repeatedly. After r differentiations, a...
Read more