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Mar 23 Fourier transform Z-transform vector space complex analysis

Mar 23 complex analysis

Ordinary differential equations (ODE) can be solved via roots of auxiliary equations or complement equations, but the rationale is not so clear in text books I read.

Nov 06 Laplace transform transformation Fourier transform vector space differential equaiton Green's function vector

The worm-like chain model is the most widely used model to describe polymers. Here are some notes about the derivation of the relationship between wlc model and the theory of elasticity.

Oct 30 wlc persistence length bending rigidity bending energy polymers elastic rod correlation

Here are some fundamentals about thermodynamic concepts from a statistical mechanics perspective.

Feb 27 fluctuation Legendre thermodynamics statistical mechanics entropy approximation saddle point temperature chemical potential free energy

The phase of an electron wave is shifted when the electron wave passes near the nucleus of atoms. So the electron wave is imaged on the detected after focused by lens giving a phase contrast in cryo-EM imaging.

Feb 06 CTF PSF envelope function TEM

Fokker-Planck-Kramers equation is a stochastic differential function describing the motion in phase space. It can degenerate to diffusion function (in coordinate space) or Fokker-Planck function (in velocity space).

The post describes the derivation of FPK equation from moment generating function.

Jul 11 Langevin equation Fokker-Planck-Kramers equation stochastic differential eqaution phase space moment generating function

Fokker-Planck-Kramers equation is a stochastic differential function describing the motion in phase space. It can degenerate to diffusion function (in coordinate space) or Fokker-Planck function (in velocity space).

The post describes the derivation of FPK equation from Kramers Moyal expansion by Taylor series.

Jun 27 Langevin equation Fokker-Planck-Kramers equation stochastic differential eqaution phase space fluctuation diffusion equation

Almost every graduate student knows how to use t-test to test whether their experimental data is significantly different from the reference. As a widely used test statistic, it's useful to know the derivation of the t-distribution's analytic form, which prevents us from abusing or misinterpreting the t-test.

Jun 12 probability density function gamma t-test maximum likelihood estimation test statistic likelihood ratio type I error type II error power delta function

Periodic motion in dynamic systems are of especial importance. Very often we are interested not so much in the motion trajectories as in the frequencies of the motion.

May 21 AFM transformation generating function mechanics Hamilton Jacobi harmonic oscillation perturbation