🔢 Mathematical Tricks Applied in Physics
By connecting mathematical theory with real-world physics, this post highlights how abstract techniques simplify complex physical problems. The discussion on Bloch-Floquet theory and hydrogen atom eigenstates offers a concrete example of these applications.
This article explores mathematical tricks, like diagonalizing commuting matrices, used by physicists to solve wave equations in various fields such as light, acoustics, and quantum mechanics. It also touches on applications in systems with specific symmetries like SU(3) in particle physics.
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🔺 A Triangle with Zero Angle Sum
A fascinating exploration of non-Euclidean geometry, demonstrating how geometry behaves differently on curved surfaces like spheres and hyperbolic planes. The example of a hyperbolic triangle with infinite perimeter challenges conventional geometric thinking.
This article delves into spherical and hyperbolic geometry, explaining how the area of a triangle is defined by the excess of its angles in spherical space, and by the defect in hyperbolic space. It also describes a hyperbolic triangle with infinite perimeter and finite area.
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⚡ Schubfach: Cutting-edge Double-to-String Conversion
Schubfach is a major leap forward in numerical computation, providing a highly efficient and minimalistic solution to floating-point number conversion. Its impressive performance is crucial for developers dealing with numerical systems.
This article presents the Schubfach algorithm for converting floating-point numbers to the shortest, correctly rounded decimal strings. It emphasizes its non-iterative approach, based on the pigeonhole principle, and high efficiency, along with a comparison to other algorithms.
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⚙️ Predicting the Resting States of Rigid Bodies
This breakthrough in geometric design and simulation allows for predicting and controlling the resting behavior of objects, offering new possibilities in gaming and product design without the need for expensive simulations.
This paper introduces a method for analyzing resting configurations of rigid bodies without simulation, using stationary points and probability calculations. The method aids in automatic orientation, stability feedback, and inverse design, such as for dice with specific resting behaviors.
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⏱️ Stopwatch Under the Hood (2016)
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