A global citizen 🇲🇽 🇪🇸 🇬🇧 | PhD in Applied Physics & Optics | Research Fellow at @QUBelfast | @STEMAmbassadors UK
Belfast, Northern Ireland, UK
Joined May 2018
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Y. Jauregui-Sánchez retweeted
A sawtooth, a square and a triangle wave, each summed from its first ten nonzero Fourier terms, f(t) = ∑ bₙ sin(nt). Each term is a circle of radius |bₙ| turning n times per period.
The sawtooth has bₙ = 2(−1)ⁿ⁺¹/(nπ), the square 4/(nπ) for odd n, the triangle ±8/(n²π²) for odd n. At ten terms, the square peaks at 1.18 and the sawtooth at 1.09; the triangle stays within 0.021 of exact.
A jump forces coefficients like 1/n, a corner only 1/n².
Y. Jauregui-Sánchez retweeted
A Riemann sum using ninety-nine equal partitions approximates the area under the sine curve from zero to pi, producing 1.99983 against the exact integral value of two and illustrating how finer partitions drive the discrete sum toward the continuous result.
Y. Jauregui-Sánchez retweeted
As a vector of fixed length rotates steadily around the origin, its height traces a sine wave and its horizontal position traces a cosine wave of the same period, showing how steady circular motion produces the two basic oscillations of trigonometry.
Y. Jauregui-Sánchez retweeted
Different observers can disagree about space and time without disagreeing about the laws of physics. In Minkowski spacetime, observers moving relative to one another use different space and time axes, connected by the Lorentz transformation.
What remains unchanged is the spacetime interval and the structure of the light cone. Space and time coordinates depend on the observer, but the speed of light and the causal structure of spacetime do not.
Y. Jauregui-Sánchez retweeted
Ten elementary functions y = f(x), each on a window fitted to its shape, from the cusp of √|x| at the origin to x/(1 + x²).
sin x / x tends to 1 at x = 0 and its integral over the whole line is π. x/(1 + x²) peaks at x = 1 with height ½. e⁻⁰·³ˣ sin 2x loses 61 % of its amplitude every period π, since e⁻⁰·³π ≈ 0.39.
The logistic curve and tanh are one shape, rescaled: 1/(1 + e⁻ˣ) = (1 + tanh(x/2))/2.
Y. Jauregui-Sánchez retweeted
I asked Opus 5.5 to explain camera focus by building an interactive lens lab
Here's what it came up with after 1 hour 26 minutes in one shot, $25.66 API cost
lens.lab.sael.net
Move the focus ring and you can see the glass elements shift the sharp plane through the scene
Y. Jauregui-Sánchez retweeted
A twist is a rotation whose angle grows along its own axis: Tₓ(s) turns the point (x, y, z) about the x axis through the angle sx.
The face x = 0 stays put, each slice x = c turns rigidly through sc, and edges along the axis bend into helices. With s = cos θ and side 2, the far face swings through ±2 radians, about ±115°.
The map is not linear, yet its Jacobian has determinant 1 everywhere, so the twist keeps volume.
Y. Jauregui-Sánchez retweeted
W(q, p) is allowed to be negative. That single liberty turns a classical phase-space density into a complete encoding of a quantum state.
Wigner defined it in 1932 so that the ordinary integrals
W(q, p) = (1/π) ∫ ψ*(q + y) ψ(q − y) exp(2 i p y) dy
recover the correct position and momentum probabilities as marginals, while expectation values of Weyl-ordered operators become ordinary averages against W. Only Gaussian wave packets stay non-negative everywhere; every other pure state must oscillate through negative regions, the visible imprint of quantum interference.
The same idea, discretized by parity, paints each point of the Bloch sphere with a characteristic pattern of positive and negative values.
Y. Jauregui-Sánchez retweeted
Visualization of Euler’s formula:
e^{iθ} = cos θ + i sin θ
A beautiful connection between exponential functions, trigonometry, and the complex plane.
Y. Jauregui-Sánchez retweeted
VISUALIZATION OF YOUNG’S DOUBLE SLIT EXPERIMENT :
Light passes through two slits and creates bright and dark bands as the waves interfere with each other.
Y. Jauregui-Sánchez retweeted
A coherent wave passes through two finite slits and the downstream field is rebuilt by a Huygens sum.
The filaments trace the probability current j = Im(ψ*∇ψ). The interference bands on detector grow from individual impacts sampled from |ψ|².
Y. Jauregui-Sánchez retweeted
Researchers have discovered a double neutron star system with the lowest known total mass and the second-shortest — and decreasing due to gravitational waves — known orbital period, making it a strong candidate for studying relativistic effects.
🔗 go.aps.org/4xWM6m7
ALT A still from a simulation of two colliding neutron stars shows two yellow circles surrounded by clouds of red over a black background. The red surroundings are merging into each other.
Y. Jauregui-Sánchez retweeted
The first rule of learning is to admit you don’t know. The second rule is to never stop asking why.
Y. Jauregui-Sánchez retweeted
How can an Electromagnetic wave pass an object as though the object were never there?
This animation demonstrates Transformation-Optics Cloaking.
It is a technique that uses deliberately engineered, spatially varying anisotropic material to redirect electromagnetic energy around a hidden region. It then guides the separated wavefronts back together so they merge together as though the obstacle had never been encountered.
Simulation Method: Time-domain P1 Finite-Element simulation with an anisotropic transformation-optics material tensor and energy flux tracers.
Y. Jauregui-Sánchez retweeted
A Luneburg lens bends an Electromagnetic wave without a curved boundary. Instead its dielectric constant changes continuously from the centre to the edge. This forces different parts of the wavelength to travel at different speeds.
Here, we solve the full 2D TM EM wave equation on an unstructured triangular mesh using the Finite Element Method (FEM).
Y. Jauregui-Sánchez retweeted
A Quantum state can represented in Phase Space using the Wigner function.
The two bright lobes behave almost classically, but the interference between them produces regions where the Wigner function becomes negative. This is something an ordinary probability distribution can never do.
Y. Jauregui-Sánchez retweeted
A Metric is the rule for measuring distance on a curved surface. A Geodesic is a path that counts as straight once measuring rule has been chosen.
When the metric changes from point to point, straightness can no longer be judged by eye.
This idea belongs mainly to Differential Geometry and Riemannian Geometry, but its also central to General Relativity where Spacetime carries a metric and freely falling particles follow Spacetime geodesics.
Y. Jauregui-Sánchez retweeted
Micro-sized water droplets + sunlight = a breathtaking "Iridescent Contrail"! ✈️🌈✨
Celebrate #WorldPhotographyDay by entering the @OPNmagazine 21st Annual Photo Contest!
📸 Submit your photo by 15 September: bit.ly/4wmEqIk
#PhotographyContest #Optics
