Computational electromagnetics
Joined August 2020
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I'm gullible and naive when it comes to Twitter. I assume people are who they say they are, believe what they say they believe, and argue their viewpoint in good faith. Sadly I know this is probably the exception not the rule.
Michael Glidewell retweeted
A rectangular array of numbers can rotate a spacecraft, train a network, or solve a circuit without ever drawing the geometry.
Each entry aᵢⱼ lives in row i and column j. The array is m × n when it has m rows and n columns; equal counts make it square. Zeros below the diagonal produce an upper-triangular matrix, zeros above a lower-triangular one.
Graphics pipelines, Kalman filters, and neural layers exploit exactly these zero patterns so most multiplications vanish.
Michael Glidewell retweeted
Branches of Ordinary Differential Equations:
From the quiet drift of slope fields y′ = f(t,y) to the sudden structural shifts of pitchfork bifurcations x′ = rx − x³, differential equations turn rate into reality.
Whether solved exactly or approximated step by step through Euler’s method yₙ₊₁ = yₙ + hf(tₙ, yₙ), they reveal the unseen architecture governing dynamic systems.
Major trial examines sugar-free sweetened soft drinks share.google/crs9uI5UQq0TvdP…
If stupidity is the reason the GOP loses the Senate then so be it.
youtu.be/uT2BGYa6RSc
Complete stupidity on the part of the United States
youtu.be/c5-1CXn9rng?is=u0Rl…
Michael Glidewell retweeted
Gold is getting dumped as yields surge... The war is far from over.
In a global sovereign debt crisis, rising yields aren't bearish for gold... They're bullish.
Because these high debt levels make high rates unsustainable.
In other words, higher yields in the short term will lead to artificially lower yields in the long run.
We don't own enough gold for what's coming.
Michael Glidewell retweeted
US Treasuries are experiencing one of the worst bear markets in history:
The 30Y US Treasury price return index has dropped -60% since 2020, to ~107, matching its lowest level in 2000.
In other words, in just 6 years, the index has erased nearly 20 years of gains.
By comparison, US nominal GDP has grown +63% over the same period.
This comes as the 30Y Treasury yield has surged +478 basis points from its intraday low of 0.71% set in March 2020.
To put this into perspective, its previous largest drawdown over the past 40 years was -35% during the 2008 Financial Crisis.
We are witnessing a historic bond market decline.
Michael Glidewell retweeted
Two integrals stand as exact inverses of each other:
X(f) = ∫_{-∞}^{∞} x(t) e^{−j 2π f t} dt
x(t) = ∫_{-∞}^{∞} X(f) e^{j 2π f t} df
One extracts every frequency hidden inside a waveform that evolves with time; the other rebuilds the original signal from those frequencies alone. A restless sum of two unequal sines therefore collapses, in the frequency domain, to a pair of isolated lines.
Fourier wrote the first version of the transform in 1822 while studying heat flow; the same pair now decomposes nearly every wave we measure.
Michael Glidewell retweeted
The product rule, once integrated and rearranged, becomes a method for trading one integral for another.
∫ u dv = uv − ∫ v du
One factor is differentiated until it simplifies; the other is integrated and carried across the equality with a change of sign. What remains is frequently easier than the original product.
Leibniz recorded a geometric form of the relation in unpublished notes of the 1670s; Brook Taylor printed the formula in 1715. A logarithm written first and the constant 1 written second yields, after a single step, the antiderivative x log x − x.
Michael Glidewell retweeted
Add one term, μ₀ε₀ ∂E/∂t, to Ampère’s law and the four equations close.
A changing electric field now produces a magnetic field, the precise counterpart of Faraday’s statement that a changing magnetic field produces an electric field.
The two curl equations together support traveling waves whose speed is 1/√(μ₀ε₀), already known, in 1861, to be the speed of light. Maxwell first wrote the extra term while building a mechanical picture of the ether as a lattice of molecular vortices; by 1865 the scaffolding could be discarded and the field itself remained.
∇ · E = ρ/ε₀
∇ × E = −∂B/∂t
∇ · B = 0
∇ × B = μ₀J + μ₀ε₀ ∂E/∂t
Source: Hackaday
share.google/LPKpZeu2J67FUP9…
Michael Glidewell retweeted
Henri Lebesgue introduced the integral that now bears his name in his 1902 doctoral thesis, Intégrale, Longueur, Aire. His aim was not simply to “patch” Riemann integration, but to develop a more general theory capable of handling functions and limiting processes for which the Riemann framework was too restrictive.
Lebesgue explained the difference between the two approaches with a memorable money analogy, later recorded by mathematician Paul Montel and his co-authors.
Suppose you must pay a certain amount using coins and banknotes taken from your pocket. In the Riemann-style picture, you take them out one by one and add their values in the order in which they appear. In Lebesgue’s picture, you first sort together coins and notes having the same value, and then add the contributions of each group.
The analogy reflects the mathematical distinction. A Riemann sum partitions the domain into intervals and samples the function on those intervals. Lebesgue integration instead organizes points according to the values taken by the function and measures the sets on which those values occur.
Lebesgue integration therefore extends Riemann integration: on a bounded interval, every Riemann-integrable function is Lebesgue integrable, and the two integrals have the same value. But the Lebesgue theory also applies to many functions and convergence problems beyond the reach of the Riemann integral.
Michael Glidewell retweeted
Every local minimum is global.
When both the objective and the feasible set are convex, a point that cannot be improved in any neighborhood cannot be improved anywhere. The geometry itself forbids hidden valleys; first-order stationarity is already optimality.
The same fact, isolated in the study of convex bodies around 1900, is why linear programs, positive-semidefinite quadratic programs, and a wide family of modern learning problems can be solved to proven global optimality.
Michael Glidewell retweeted
Dirac wrote it in 1928 so an electron could obey both quantum mechanics and special relativity; the γ matrices mix the components precisely as Lorentz transformations demand, while the mass term couples the two chiralities.
A first-order equation for a four-component spinor that simultaneously carries energy, momentum, spin and the seed of antimatter.
Gauged, the same compact line becomes the starting point of quantum electrodynamics and, with the appropriate internal symmetries, of the weak and strong interactions as well.
Michael Glidewell retweeted
USA! USA! USA!
Despite rising yields, Japan’s net interest payments as a share of GDP are projected to remain among the lowest in the OECD in 2027.
Source: Japan Economy Watch richardkatz.substack.com/p/f…
Michael Glidewell retweeted
P(A|B) = P(B|A) · P(A) / P(B)
A single quotient inverts the direction of conditioning. What had been the probability of the evidence given a hypothesis becomes, after multiplication by the prior and division by the total probability of the evidence, the revised probability of the hypothesis given that evidence.
The same relation that decides whether a laboratory result should change a diagnosis also updates a scientific theory after an experiment or the chance that a message is spam. Thomas Bayes left the argument unpublished; it appeared in 1763, two years after his death.
