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We study the time-dependent Hartree--Fock equation in $\R^3$ with physical Coulomb interactions in both direct and exchange channels, with exchange coupling $\eta$, linearized about homogeneous equilibria $\gamma_f=g(-i\nabla)$ having compact momentum support. Nguyen and You showed that, without exchange, two purely im...
We study the generalized Flint Hills series $\mathcal{F}_{s,t}(x)=\sum_{n\ge1} n^{-s}|\sin(\pi nx)|^{-t}$ for $s>0$ and $t>1$. An explicit comparison with a series over continued-fraction denominators yields the Hausdorff dimension $\min\{1,2t/(s+t)\}$ of its divergence set. At each critical exponent $\tau=1+s/t>2$ we...
A virtual classroom of 20 student agents who interact through rule-based chats, quarrels and consultations with friends and, when stressed, may instead consult a counselor AI under one of six style prompts, specifying the agent dynamics completely and discussing the limits of an LLM as generator of state updates.
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