numerical-methods

We introduce Prob-GParareal, a probabilistic extension of the GParareal algorithm designed to provide uncertainty quantification for the Parallel-in-Time (PinT) solution of (ordinary and partial) differential equations (ODEs, PDEs). The method employs Gaussian processes (GPs) to model the Parareal correction function, in line with GParareal, further enabling the propagation of numerical uncertain…

UMBC mathematician Matthew Kvalheim has received an Air Force Office of Scientific Research Young Investigator Program Award to develop better ways of simplifying hybrid systems, like a drone swarm, humanoid robot, or aircraft. Kvalheim aims to create mathematical tools that make hybrid dynamics—which govern almost all real-world systems—easier to study and control, giving researchers a clearer “…

Scientific Reports, Published online: 11 September 2026; doi:10.1038/s41598-026-70795-w Numerical analysis of complete period-adding structures in a novel chaotic system for message encoding

Solve analytically the following coupled PDEs: \begin{cases} \partial_t F(x,t) = -2\alpha_0 x \left[ \partial_x G(x,t)-\partial_x^2 G(x,t) \right], \\[6pt] \partial_t G(x,t) = -2\alpha_0 x \left[ \partial_x F(x,t)+\partial_x^2 F(x,t) \right]. \tag{1} \end{cases} with a domain $x\in[x_0,\infty), \qquad x_0\equiv c+1, \qquad \alpha_0=\text{constant}, \qquad c>\frac{1}{2}. $ The initial conditions a…

A Millennium Prize Problem and a Fight Over Credit Two announcements landed within hours of each other on

arXiv:2603.22188v2 Announce Type: replace-cross Abstract: Simulation methods have become important tools for quantifying partisan and racial bias in redistricting plans. We generalize the Sequential Monte Carlo (SMC) algorithm of McCartan and Imai (2023), one of the commonly used approaches. First, our generalized SMC (gSMC) algorithm can split off regions of arbitrary size, rather than a single …

I am new to fitting histograms and as a result i have an issue understanding what to consider as error propagation when calculating a quantity. The following diagram shows entries for 4 different x-ray sources resulting in for different data files generating 4 histograms which are fitted separately. Lets say i want to calculate the quantity u= a*channel for the Rb source (check the following diag…

For decades, researchers wondered if the Navier-Stokes equations, which describe fluid flow, break down. OpenAI put thousands of agents on the question.

A breakthrough on the Navier-Stokes equations could be worth a $1 million prize, but a mathematician has serious questions about how OpenAI got there so fast.

NYU professor Tristan Buckmaster announced three proofs on Tuesday, including preliminary progress toward the Navier-Stokes existence and smoothness problem, one of mathematics’ seven Millennium Prize problems. The work, conducted with Anthropic mathematician Levent Alpöge using a mix of AI models including Codex, was overshadowed by controversy after OpenAI published a full proof of the same pro…

We’re sharing a solution to the Navier–Stokes existence and smoothness problem, one of the Millennium Prize Problems. This proof, produced by an internal OpenAI system, shows that the dynamics of the Navier-Stokes equations for fluid motion can develop a singularity in finite time. We’re sharing both a writeup of the proof and a formalization in […]

One of the trickiest problems in mathematics has now fallen to AI after just days of work. The groundbreaking result was announced amid rumour after similar, but less complete work, also created with AI, was announced just hours before

Three key findings produced in collaboration between humans and AI bring us closer to a solution to the infamous Navier-Stokes puzzle

Mathematicians at OpenAI showed that the Navier-Stokes equations, which describe how fluids flow, can sometimes “blow up.” But the massive result is not without controversy. The post AI Has Solved One of Math’s $1 Million Millennium Prize Problems first appeared on Quanta Magazine

We develop an exact and parameterized theory of iterated complex roots under explicit branch conventions. The principal negative-root recursion z n+1 = F r (z n) := Root pr r (−z n), r ≥ 2, separates completely into radial and angular dynamics. The radius obeys the exact law |z n | = |z 0 | 1/r n, while the principal argument is governed by a two-branch contraction. This yields a closed formula f…

Fred James (1939 – 2026) — Fred James, CERN’s renowned expert on data analysis and statistics, pioneer of numerical methods, and co-author of the popular book Statistical Methods in Experimental Physics, sadly passed away on 31 July.  Fred was born in Detroit, USA, on 28 May 1939, and graduated in 1960 from Princeton University with a degree in physics. […]

We proved that linear partial differential operator with constant coefficients has locally integrable fundamental solutions, elementarily.