Mathelirium
@mathelirium • 34,407 subscribers
I work in applied maths, com physics, & scientific visualization. My hobby is turning mathematical models into unique, original simulated visuals
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Black Holes Don’t Simply Suck Things In. They Sort Motion by Geometry. A particle falling toward a black hole is not automatically doomed. Its path depends on how it approaches, the distance, the angle, and the momentum it carries. Some paths cross the horizon. Some skim the edge and whirl around before falling or escaping. Others get sharply deflected and fly back out into space.
Mathelirium274,920 次观看 • 1 个月前
A Hamiltonian system is a way of describing motion where position and momentum evolve together as one coupled system. The plot shows an energy landscape in phase space, with the motion of the system traced directly on top of it and projected onto the underlying phase portrait. #HamiltonianSystems #PhaseSpace #PhysicsSimulation #DynamicalSystems #MathematicalPhysics
Mathelirium35,263 次观看 • 6 天前
A neural network can begin as a flat sheet and learn the shape of hidden data A self-organizing map turns learning into geometry. Each data point pulls one winning neuron toward it, but nearby neurons move too, and so the whole lattice bends without losing its neighborhood structure. The strange part is that the network is not given the roll shape. It discovers the shape through competition and local cooperation. Paper: Self-Organized Formation of Topologically Correct Feature Maps Authors: Teuvo Kohonen Year: 1982
Mathelirium129,024 次观看 • 27 天前
In 1996, James Sethian showed something almost unfair...you can find shortest routes through a messy world by letting a wave expand once...no trial paths, no search beams, just one growing front. Here’s how: we solve for an arrival-time field T(x,y) so that T literally means how long the wave needs to reach this point. The rule is ||∇ T|| = 1/F, where the medium is fast (F large) the front sprints, where it’s slow it trudges, and obstacles are speed ≈ 0, so the front wraps around them because that’s the only way forward. Then comes the satisfying part: once T exists, a path doesn’t need to search at all...drop a bead anywhere and let it follow ẋ ∝ -∇ T, it slides downhill on the time landscape and traces a globally fastest route back to the source. This “wave = optimal control” viewpoint is exactly what Tsitsiklis (1995) made precise from the Hamilton-Jacobi side...compute the value/arrival-time function and the optimal trajectories fall out from it. #FastMarching #EikonalEquation
Mathelirium766,121 次观看 • 5 个月前
A Lens That Takes Derivatives US Patent Basis: US8610839B2 - Optical Processing System for Computing Derivatives. In a 4f Optical Processor, the first lens takes the incoming field and forms its Fourier spectrum. At that middle plane, a tiny optical mask multiplies the spectrum by iξ. This is the derivative operator written in Fourier language u(x) -> U(ξ) -> iξU(ξ) -> ∂u/∂x Then the second lens brings the field back to the real space. What comes out is no longer just a focused beam. It is the spatial derivative of the input field, computed by light as it propagates. So, this is the serious promise of optical computing. A physical optical train can perform operations that usually live inside numerical code: differentiation, filtering, convolution, edge detection, correlation, and many other linear transforms.
Mathelirium137,413 次观看 • 1 个月前
White Hole - A Black Hole With Time Reversed A White Hole is the time-reversed shadow of a Black Hole. It appears in the maximally extended Schwarzschild solution of General Relativity. The exterior geometry is the same ds² = −(1 − 2M/r)dt² + (1 − 2M/r)⁻¹dr² + r²dΩ² The difference is causal. A Black Hole lets paths enter but not escape. A White Hole lets paths emerge but not enter. No White Hole has been observed. But inside the equations of GR, the question is real: What does a Black Hole look like when time runs the other way?
Mathelirium74,665 次观看 • 1 个月前
In 1906, the world-renowned Russian mathematician Andrey A. Markov asked a heretic question of that time: if randomness has memory, do averages still behave or does probability theory collapse? His answer was a new kind of dependence where the next step only remembers the present, P(X_{n+1}=j | X_n=i, X_{n-1}, …) = P_ij, and yet the law of large numbers survives. The path stays noisy, while long-run state frequencies converge to a fixed profile pi = pi P. Markov’s idea is now the workhorse behind estimates by random-walks. MCMC (Markov Chain Monte Carlo) builds a chain whose stationary distribution is the target (posteriors, partition functions, constrained geometries), then uses time-averages as estimates. The same Markov structure shows up in hidden Markov models for time series, in biophysics as channels switching between states, and in control/RL as Markov decision processes. #ProbabilityTheory #Markov #MCMC
Mathelirium288,220 次观看 • 5 个月前
If you’re an engineer, physicist, or CS person and you’ve been telling yourself that Pure Mathematics is optional, and you’re anywhere between an ambitious undergrad and a beginning PhD, gravity is usually where that story ends. Not because mathematicians are trying to show off, but because the basic object in the theory is Spacetime, and Spacetime is already a pure mathematical structure. It’s a Manifold with Topology built in. Sure you can get pretty far by memorising formulas, but at some point you realise you’re no longer reading the theory, you’re just reciting it. In 2015, during the 100th anniversary of general relativity and the International Year of Light, the Scientific Organizing Committee released a central set of 24 lectures by Frederic P. Schuller. The series is titled: A thorough introduction to the theory of general relativity. It builds the subject carefully from first principles, step by step, across 24 self-contained lectures. #GeneralRelativity #Spacetime #MathematicalPhysics #DifferentialGeometry #PhysicsEducation #Relativity
Mathelirium197,684 次观看 • 4 个月前