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I recently read the paper “Practical Guide to Automated TEM Image Analysis for Increased Accuracy and Precision in the Measurement of Particle Size and Morphology.” In Figure 8, the spherical nanoparticles appear as faintly overlapping circles,

which immediately brought to mind the Hough Circle detection algorithm. By voting in the circle parameter space, Hough Circle can pick out weak or partially obscured arcs and thus avoid lumping overlapping particles into a single object.

The following note explains how to deploy a high-performance VPN using WireGuard on VPP with DPDK. This lets lab members connect to computation servers from anywhere on the campus network with minimal overhead.

The following note outlines how to set up a VPN so that lab members can connect to the computation servers from anywhere on the campus network.

1) Requirements

• Goal: Allow remote VPN clients to access lab servers on a different subnet.
• VPN software: SoftEther VPN Server (L2-bridge mode) + SoftEther VPN Client.
• Network device: Layer-3 switch that supports SVIs and inter-VLAN routing.

Periodic boundary conditions (PBC) are ubiquitous in molecular simulations. However, when visualizing results, one often prefers to see whole molecules rather than fragments cut by the simulation box. If the molecule percolates through the box, minimizing artificial cuts becomes essential. Below are two methods for reconstructing intact structures under PBC:

Problem: given a series molecules $i=1,\dots,N$, connected with junction atoms $\mathbf{x}_i^k$, for $k=1,2,…m$ if there are $m_i$ atoms from molecule $i$ that are bonded with other molecules. The connection bonds are defined as $\mathbf{b}_i := \lbrace\mathbf{x}_j^{k_1}-\mathbf{x}_i^{k_2}|\forall k_1, k_2\, \mathrm{and}\, \forall j\in \mathrm{neighbor\, of}\, i\rbrace$, assume that the center-of-mass of the molecules are fixed, but are rotatable, minimize $\sum_i |\mathbf{b}_i|^2$ with a series of rotation matrices $R_i$:

$$ R_i = \min_{R_i}(L:= \sum_i \sum_{j \in \mathrm{neighbor\, of}\, i} \sum_{k_1,k_2} |\mathrm{pbc}(R_j \mathbf{x}_{j}^{k_1} – R_i \mathbf{x}_{i}^{k_2})|^2) $$

with constraint

$$ c=\sum_i |R_i^TR_i – I|^2=0 $$

the constraint ensures $R^T R =I$ for each rotation matrix thus $R_i$ are rotation matrices. The periodic boundary condition is introduced for most simulation systems.

Minimization procedure includes calculation of jacobian of loss and constraint functions:

$$ \frac{\partial L}{\partial R_i} = -2 \sum_{j \in \mathrm{neighbor\, of}\, i} \sum_{k_1, k_2} \mathrm{pbc}(R_{j} \mathbf{x}_{j}^{k_1} – R_i \mathbf{x}_i^{k_2})^T \mathbf{x}_i^{k_2} $$

and

$$\frac{\partial c}{\partial R_i} = 4 (R_i R_i^T – I) R_i$$