numerical-methods

A battle between “slimes” and “zoglins” could be the best way to calculate pi—at least for fans of this megahit game

Friends, I will come back with a detailed latex post and commented code later tonight. But very quickly for friends trying to play with the program, I have changed the order of variables and indexing and it is different in new output variable cnOptim that comes out of Newton optimization. Please let me explain here. In the expansion of nth power of Gaussian, if there are K correlated variables, t…

A couple months ago I wrote about how to compute the sine and cosine of a complex number using only real functions of real variables using the equations You can do something analogous for all the elementary functions, though some of the equations are quite a bit more complicated than the ones above. See the […] The post Building complex functions out of real parts first appeared on John D. Cook .

© Copyright 2026 Chen, Ho Yiing (ORCID: 0009-0006-6816-9891), Independent researcher, charenix.com. All rights reserved by the author. This work is licensed under the Creative Commons Attribution 4.0. Restoring exactly one hidden-state coordinate (d=758, last four blocks of a passthrough-merged qwen2.5 hybrid) flips a Chinese narrative response into a bilingual codeblock-with-emoji response on a …

This paper establishes the structural prerequisites in AQM needed for a later conditional derivation of the fine-structure constant. It does not take the low energy fine-structure constant as the final numerical target of the present pa per. The task is to organize finite spectral bare stiffness, the UV spectral-shell endpoint, the IR boundary spectral gap, active central projection weights, and …

Scientific Reports, Published online: 22 May 2026; doi:10.1038/s41598-026-53831-7 Parameter-uniform numerical method for a coupled system of singularly perturbed turning point problems with Robin boundary conditions

This paper introduces the pole ratio metric and presents a sphere-based view of symmetric positive-definite matrix rotations on the Riemannian manifold of symmetric positive-definite matrices equipped with the affine-invariant Riemannian metric. The pole ratio quantifies whether data from different users lie on this Riemannian manifold in a way that enables effective transfer learning. The sphere…

American Journal of Engineering and Applied Sciences, Published online: 21 May 2026; doi:10.3844/ajeassp.2026.88.116 Numerical modeling of transient solidification under convective boundary conditions presents a significant challenge because accurately tracking the solid–liquid interface is essential for obtaining...

This paper extends our previously developed numerically stable solvers for cubic and quartic equations from the real to the complex domain. The core novelty is the systematic use of principal branches for square roots, cube roots, and fourth roots, combined with explicit branch cut analysis (negative real axis), which eliminates ambiguity and ensures backward stability for arbitrary complex coeff…

In many engineering and applied mathematics textbooks, linear ordinary differential equations (ODEs) are often solved quite casually using the Fourier transform. For example $$\ddot{x}(t) + 2x(t) = ...

In this article, we show that the minimal vectors of the extremal even unimodular lattices in R^{32} define T-avoiding universally optimal spherical codes for suitable sets T. Moreover, these codes are minimal T-avoiding spherical designs and maximal Tavoiding codes for appropriate choices of T.

Friends, I could not do the correlated case by the recursive method I was suggesting but I gained some insights. I think we can use some more numerical method to find the weights on correlated \(Z_{x_k}\) in the expansions. For example if we have ten underlying predictor variables and there are seven hermite polynomials, the total number of parameters to be found is only seventy which can easily …

I have a time dependent function as follows: $$f(x + \delta (t)) = \cos(2\pi \cdot (x - \delta(t)) + 0.05\sin(60\pi(x - \delta(t)))$$ where $\delta(t) = 0.05t$ and $ f(x,t=0) = f(x)$. Using the Taylor ...

I built a free Korean astrology (Saju) calculator and wanted to share what makes the math interesting. What's Saju? Saju is the Korean tradition of Chinese 4-pillar astrology (BaZi). Your birth year, month, day, and hour each get encoded as a Heavenly Stem + Earthly Branch pair, giving you 8 characters that describe your "energetic signature." The interesting parts 1. Solar terms via Meeus algori…

I'd like to store a list of functions that are derivatives of a given function. This is a working example of what I want f = Function[{x,y},Exp[y x]] then assoc= <"Function"->f, ...

This technical report presents a comparative study of three numerical methods for solving ordinary differential equations: Euler, RK2, and RK4. All methods were implemented from scratch in Python at step size h=0.1 over the interval [0,5]. RK4 achieves a maximum error of approximately 1.0 x 10^-6 compared to 0.0192 for Euler — roughly 19,000 times more accurate at the same step size. Source code …

The mining-induced failure zone is one of the main water flow channels. Due to the complexity of fracture distribution and groundwater flow in the fractures, calculating the water conductivity of fractures in the mining failure zone is a hot and difficult research topic at present. In order to simultaneously simulate the permeability of the complete rock mass of the floor and the hydraulic conduc…

Friends, I was thinking of other possible improvements on the basic Hermite regression algorithm so it works better even away from the mean of the density. Suppose we use the following notation \[Y(Z_y)\,=\,\sum\limits_{n=0}^{N}\,ah_n\,H_n(Z_y)\,=\,\,\,\sum\limits_{n=0}^{N}\,Y_n(Z_y)\,\]  and \[X(Z_x)\,=\,\sum\limits_{n=0}^{N}\,bh_n\,H_n(Z_x)\,=\,\,\,\sum\limits_{n=0}^{N}\,X_n(Z_x)\,\] basically …