numerical-methods

This is an application of functional analysis to the existence and smoothness of the Navier–Stokes equations using elementary weak solutions in Sobolev spaces. We solve the problem in mathematics. The problems are not in physics, so we do not use any physics or assumptions-falsified mathematics. We use mathematics only. We can solve the problem by using an exactly and completely FALSIFIED resolut…

For decades, numerical weather prediction (NWP) models have leaned on physics-based equations to simulate the atmosphere. These models pack a […]

Say that I have a representation $V_\ell$ of $\rm SO(3)$ , with angular momentum number $\ell$ . In Dirac notation, states in $V_\ell$ can be represented in the form $$ |v⟩ = \sum_{m=-\ell}^\ell c_m |\ell,m⟩, $$ where, if the representation is real-valued, we require $c_{-m} = (-1)^m c_m^*$ . I would like to randomly sample from this space , i.e., the vector $|v⟩$ and/or its coefficients $c_m$ , …

A GPU-friendly algorithm for tight certified singular-value endpoint bounds using scaled Gram matrices, SYRKs, Frobenius reductions, and a small scalar moment problem.

A hardware-aware hybrid polar decomposition for ML: one Dynamic Weighted Halley (rational) step to handle the hard early regime, then two Polar Express (polynomial) cleanup steps once the spectrum is easy. The result is exactly two rectangular GEMMs, no eigendecomposition or power iteration, and robust convergence from condition numbers up to 1000.

Geomagnetic field forecasting is critical for mitigating space weather hazards, yet single-station prediction remains a challenge due to the complex, non-linear coupling of vector components. In this work, we propose a Graph Neural Network (GNN) architecture enhanced with Temporal Convolutional Networks (TCN) to forecast the H, D, and Z components. By modeling the observatory’s sensors as nodes i…

If you ever need more precision than what 15 decimal digits of the double format can offer, there is a neat trick: glue two doubles together and treat them as one number. This gives you ~31 decimal digits for roughly 9x the cost of a plain double in a real kernel (4-12x per isolated operation). With no heap allocation and no dependencies, this puts it almost exactly halfway between a double and a…

I was going through a worksheet and got presented with the following problem: Given $$ \int_0^\infty e^{-tx}\sin(x)\,\mathrm dx = \frac{1}{1+t^2}, $$ find the resulting equation after taking the ...

Journal of Mathematics and Statistics, Published online: 4 August 2026; doi:10.3844/jmssp.2026.61.76 Four absolute measures and one relative measure of interindividual variability have been defined, together with their point and interval estimation (bootstrap and asymptotic), and an R script has been...

I was digging into the Runge-Kutta method for solving differential equations and a line from [1] piqued my curiosity. These calculations, which are not reproduced in Kutta’s paper (they are however in Huen (1900)), are very tedious. The calculations are a set of eight constraints that the parameters of a fourth-order Runge-Kutta method must satisfy. […] The post Solving the RK4 design equations f…

Abstract. Elliptic partial differential equations are ubiquitous in graphics and engineering, but remain challenging to solve on complex or evolving geometries. Traditional discretization schemes (e.g., FEM/FDM) provide stable, globally coupled solutions but require heavy meshing or extreme refinement to accurately resolve geometric detail. In contrast, grid-free Monte Carlo methods (e.g., Walk o…

Accurate evaluation of transmissibility is central to reliable numerical simulation of gas flow in porous media, because of its pressure-dependent properties such as density and viscosity. In commercial reservoir simulators, inter-block transmissibility is calculated using upstream averaging and in the well-containing grid-block, it is evaluated at the grid-block pressure. This procedure does not…