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In the Frequency-Modulation mode of AFM, the cantilever keeps harmonically oscillating during measurement. In contrast to optic microscopy or scanning tunneling microscopy, in which the electromagnetic signals are detected directly by detectors, FM-AFM detects the frequency shift of the cantilever's harmonic oscillation when the distance between sample surface and cantilever's tip changes.

May 18 AFM transformation generating function mechanics Hamilton Jacobi harmonic oscillation perturbation

Partial differential equations provide information about the relationship between the variables instead of the specific form of the solution function. Determining the specific form of solution needs other constraints such as boundary conditions.

May 15 diffusion equation differential equation wave function

Ordinary differential equations (ODE) are important in physical modeling and deduction.

May 02 Langevin equation differential equation integrating factor

A vector is typically regarded as a geometric entity characterized by a magnitude and a direction. Here are some notes about vector calculus and its basic application in small-angle x-ray scattering (SAXS).

Apr 21 vector calculus SAXS spherical average curvature gradient coordinate curl divergence Green's theorem Stokes's theorem Gauss's theorem

Let's say the concentration of the component \(j\) in the solution is \(C_j(\vec{r}, t)\), which is a scalar function of coordinates \(\vec{r}\) and time \(t\). For a well-mixed solution, the average of concentration over coordinates and time should be a constant \(<C_j(\vec{r}, t)> = \bar{C}_j\). The local concentration \(C_j(\vec{r}, t)\), however, fluctuates with coordinates and time. The amount of local fluctuation is \(\delta C_j(\vec{r}, t) = C_j(\vec{r}, t)-\bar{C}_j\).

Mar 27 fluctuation diffusion equation Fourier transform differential equation correlation function

Probability and statistics are one of the most common terminologies in scientific data interpretations. They provide us ways to find truths and make conclusions from this stochastic and noisy world. These different probability distributions are heavily coupled. In order to apply these distribution properly, we should understand their origin and derivation, which are the main purpose of this post (for continuous distributions).

Mar 10 probability density function exponential gaussian gamma beta chi-squared log-normal bayesian

Probability and statistics are one of the most common terminologies in scientific data interpretations. They provide us ways to find truths and make conclusions from this stochastic and noisy world. These different probability distributions are heavily coupled. In order to apply these distribution properly, we should understand their origin and derivation, which are the main purpose of this post (for discrete distributions).

Feb 13 probability mass function poisson binomial geometric hypergeometric bernoulli

Probability mass/density distribution have defined generating functions which faciliate the derivation and the understanding of different probability distributions.

Feb 11 probability density function probability mass function generating function

In scientific computing, integration by substitution is a very common skill. For example, we measured a series of practical values of a variable (\(x'\)), of which the distribution (\(g(x')\)) was unknown. But we know the theoretic distribution of the variable (\(f(x)\)) and the correction relationship or mapping relationship between theoretic and practical values (\(x=h(x')\)).

Feb 04 Jacobian determinant integration by substitution cross product infinitesimal area

Permutation

Let's first consider a set of \(n\) objects, which are all different. The number of all possible arrangements (permutations) is

\[ n(n-1) \cdots 1 = n! \]

Generally, if we select \(k (<n)\) objects from \(n\), the number of permuations is

\[ n(n-1)\cdot\cdot\cdot(n-k+1) = \frac{n!}{(n-k)!} = P(n, k) \]

Jan 21 permutation combination particle statistics