More Like this

1. Don’t Follow Reinforcement Learning Blindly: Lower Sample Complexity of Learning Optimal Inventory Control Policies with Fixed Ordering Cost

   https://doi.org/10.1177/10591478251378851  

   Fan, Xiaoyu; Chen, Boxiao; Olsen, Tava; Qin, Hanzhang; Zhou, Zhengyuan (April 2026, Production and Operations Management)

   We study the sample complexity of offline learning for a class of structured Markov decision processes (MDPs) describing the inventory control system with fixed ordering cost/setup cost, a fundamental problem in supply chains. We find that a naive plug-in sampling-based approach applied to the inventory MDPs leads to strictly lower sample complexity bounds compared to the optimal bounds recently obtained for the general MDPs. More specifically, in the infinite-horizon discounted cost setting, we obtain an $\tilde{O}(min\{\frac{(\overline{S}-\underset{\_}{s}{)}^2}{(1-\gamma {)}^2{ϵ}^2},\frac{1}{(1-\gamma {)}^4{ϵ}^2}\})$ sample complexity bound, where ${(\overline{S}-\underset{\_}{s})}^2$ corresponds to the number of state-action pairs in a generic MDP with state space $\mathcal{S}$ and action space $\mathcal{A}$ . As such, $\tilde{O}\left(\frac{(\overline{S}-\underset{\_}{s}{)}^2}{(1-\gamma {)}^2{ϵ}^2}\right)$ improves on the optimal generic reinforcement learning (RL) bound $\tilde{\mathrm{\Theta }}\left(\frac{(\overline{S}-\underset{\_}{s}{)}^2}{(1-\gamma {)}^3{ϵ}^2}\right)$ (when directly applying $\tilde{\mathrm{\Theta }}\left(\frac{|\mathcal{S}||\mathcal{A}|}{(1-\gamma {)}^3{ϵ}^2}\right)$ here) by a factor of $(1-\gamma {)}^{-1}$ , and $\tilde{O}\left(\frac{1}{(1-\gamma {)}^4{ϵ}^2}\right)$ is able to completely remove the dependence on state and action cardinality. In the infinite-horizon average cost setting, we obtain an $\tilde{O}\left(\frac{(\overline{S}-\underset{\_}{s}{)}^2}{{ϵ}^2}\right)$ bound, improving on the generic optimal RL bound $\tilde{\mathrm{\Theta }}\left(\frac{(\overline{S}-\underset{\_}{s}{)}^2{t}_{mix}}{{ϵ}^2}\right)$ (when directly applying $\tilde{\mathrm{\Theta }}\left(\frac{|\mathcal{S}||\mathcal{A}|t_{mix}}{{ϵ}^2}\right)$ here) by a factor of $t_{mix}$ , and hence removing the mixing time dependence. By carefully leveraging the structural properties of the inventory dynamics in various settings, we are able to improve on those “best-possible” bounds developed in the RL literature. Our results demonstrate the drawbacks one could face by blindly following RL algorithms and the necessity of designing sample efficient algorithms that properly incorporate the special structures of the inventory systems. 

   more » « less
2. The Eisenstein ideal at prime-square level has constant rank

   https://doi.org/10.1073/pnas.2500729122  

   Lang, Jaclyn; Wake, Preston (July 2025, Proceedings of the National Academy of Sciences)

   LetNandpbe prime numbers with $p\ge 5$ such that $p\mid \mid (N+1)$ . In a previous paper, we showed that there is a cuspformfof weight 2 and level ${\mathrm{\Gamma }}_0\left(N^2\right)$ whoseℓ-th Fourier coefficient is congruent to $\ell +1$ modulo a prime abovepfor all primesℓ. In this paper, we prove that this formfis unique up to Galois conjugacy, and the extension of ${\mathbb{Z}}_p$ generated by the coefficients offis exactly ${\mathbb{Z}}_p[{\zeta }_p+{\zeta }_p^{-1}]$ . We also prove similar results when a higher power ofpdivides $N+1$ . 

   more » « less
3. Comments on entire functions of the derivative operator

   https://doi.org/10.1088/1361-6382/ae9110  

   Woodard, R P (August 2026, Classical and Quantum Gravity)

   Abstract Many attempts to introduce fundamental nonlocality into quantum (or classical) field theory are based on the assumption that exponentials of the d’Alembertian are positive-definite, so that these operators can be employed without engendering the Ostrogradskian instability associated with higher derivative Lagrangians.This assumption is false.Working in the simple context of a 1-dimensional, point particle $q\left(t\right)$ , I demonstrate that the equation $\exp \left[T^2\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}\right]q\left(t\right)=0$ has an infinite number of rapidly oscillating, exponentially rising and falling solutions. This infinite kernel is in one-to-one correspondence with the ability to specify ‘initial value data’arbitrarilyoveranyfinite interval $t_1<t<t_2$ . 

   more » « less
4. Bounds on eigenfunctions of quantum cat maps

   https://doi.org/10.1088/1402-4896/ad0331  

   Kim, Elena; Koirala, Robert (October 2023, Physica Scripta)

   Abstract We studyℓ^∞norms ofℓ²-normalized eigenfunctions of quantum cat maps. For maps with short quantum periods (constructed by Bonechi and de Biévre in F Bonechi and S De Bièvre (2000,Communications in Mathematical Physics,211, 659–686)) we show that there exists a sequence of eigenfunctionsuwith $\parallel u{\parallel }_{\infty }\gtrsim {\left(\log N\right)}^{-1/2}$ . For general eigenfunctions we show the upper bound $\parallel u{\parallel }_{\infty }\lesssim {\left(\log N\right)}^{-1/2}$ . Here the semiclassical parameter is $h={\left(2\pi N\right)}^{-1}$ . Our upper bound is analogous to the one proved by Bérard in P Bérard (1977,Mathematische Zeitschrift,155, 249-276) for compact Riemannian manifolds without conjugate points. 

   more » « less
5. Characterizing the fundamental bending vibration of a linear polyatomic molecule for symmetry violation searches

   https://doi.org/10.1088/1367-2630/ace471  

   Jadbabaie, Arian; Takahashi, Yuiki; Pilgram, Nickolas H; Conn, Chandler J; Zeng, Yi; Zhang, Chi; Hutzler, Nicholas R (July 2023, New Journal of Physics)

   Abstract Polyatomic molecules have been identified as sensitive probes of charge-parity violating and parity violating physics beyond the Standard Model (BSM). For example, many linear triatomic molecules are both laser-coolable and have parity doublets in the ground electronic $\tilde{X}{}^2{\mathrm{\Sigma }}^{+}\left(010\right)$ state arising from the bending vibration, both features that can greatly aid BSM searches. Understanding the $\tilde{X}{}^2{\mathrm{\Sigma }}^{+}\left(010\right)$ state is a crucial prerequisite to precision measurements with linear polyatomic molecules. Here, we characterize the fundamental bending vibration of ${}^{174}$ YbOH using high-resolution optical spectroscopy on the nominally forbidden $\tilde{X}{}^2{\mathrm{\Sigma }}^{+}\left(010\right)$ $\to \tilde{A}{}^2{\mathrm{\Pi }}_{1/2}\left(000\right)$ transition at 588 nm. We assign 39 transitions originating from the lowest rotational levels of the $\tilde{X}{}^2{\mathrm{\Sigma }}^{+}\left(010\right)$ state, and accurately model the state’s structure with an effective Hamiltonian using best-fit parameters. Additionally, we perform Stark and Zeeman spectroscopy on the $\tilde{X}{}^2{\mathrm{\Sigma }}^{+}\left(010\right)$ state and fit the molecule-frame dipole moment to $D_{\mathrm{m}\mathrm{o}\mathrm{l}}=2.16\left(1\right)$ Dand the effective electrong-factor to $g_S=2.07\left(2\right)$ . Further, we use an empirical model to explain observed anomalous line intensities in terms of interference from spin–orbit and vibronic perturbations in the excited $\tilde{A}{}^2{\mathrm{\Pi }}_{1/2}\left(000\right)$ state. Our work is an essential step toward searches for BSM physics in YbOH and other linear polyatomic molecules. 

   more » « less