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Power system operators must ensure that dispatch decisions remain feasible in case of grid outages or contingencies to prevent cascading failures and ensure reliable operation. However, checking the feasibility of all $N - k$ contingencies -- every possible simultaneous failure of $k$ grid components -- is computationally intractable for even small $k$, requiring system operators to resort to heuristic screening methods. Because of the increase in uncertainty and changes in system behaviors, heuristic lists might not include all relevant contingencies, generating false negatives in which unsafe scenarios are misclassified as safe. In this work, we propose to use input-convex neural networks (ICNNs) for contingency screening. We show that ICNN reliability can be determined by solving a convex optimization problem, and by scaling model weights using this problem as a differentiable optimization layer during training, we can learn an ICNN classifier that is both data-driven and has provably guaranteed reliability. Namely, our method can ensure a zero false negative rate. We empirically validate this methodology in a case study on the IEEE 39-bus test network, observing that it yields substantial (10-20x) speedups while having excellent classification accuracy., Comment: 11 pages, 4 figures

Machine learning can significantly improve performance for decision-making under uncertainty in a wide range of domains. However, ensuring robustness guarantees requires well-calibrated uncertainty estimates, which can be difficult to achieve in high-capacity prediction models such as deep neural networks. Moreover, in high-dimensional settings, there may be many valid uncertainty estimates, each with their own performance profile - i.e., not all uncertainty is equally valuable for downstream decision-making. To address this problem, this paper develops an end-to-end framework to learn the uncertainty estimates for conditional robust optimization, with robustness and calibration guarantees provided by conformal prediction. In addition, we propose to represent arbitrary convex uncertainty sets with partially input-convex neural networks, which are learned as part of our framework. Our approach consistently improves upon two-stage estimate-then-optimize baselines on concrete applications in energy storage arbitrage and portfolio optimization.

We present a chemical kinetics model of the solid-phase chemical evolution of a comet, beginning with a long period of cold-storage in the Oort Cloud, followed by five orbits that bring the comet close to the Sun. The chemical model is based on an earlier treatment that considered only the cold-storage phase, and which was based on the interstellar ice chemical kinetics model MAGICKAL. The comet is treated as 25 chemically distinct layers. Updates to the previous model includes: (i) Time- and depth-dependent temperature profiles according to heliocentric distance; (ii) a rigorous treatment of back-diffusion for species capable of diffusing through the bulk-ice layers; (iii) adoption of recent improvements in the kinetic treatment of nondiffusive chemical reaction rates. Starting from an initially simple ice composition, interstellar UV photons drive a rapid chemistry in the upper micron of material, but diminished by absorption of the UV by the dust component. Galactic cosmic rays (GCRs) drive a much slower chemistry in the deeper ices over the long cold-storage period down to 10 m. The first solar approach drives off the upper layers of ice material via thermal desorption and/or dissociation, bringing closer to the surface the deeper material that previously underwent long-term processing by GCRs. Subsequent orbits are more uniform in their chemical behavior. Loss of molecular material leads to concentration of the dust in the upper layers. Substantial quantities of complex organic molecules are formed in the upper 10 m during the cold storage phase, with some of this material released during solar approach; however, their abundances with respect to water appear too low to account for the observed gas-phase values for comet Hale-Bopp, indicating that the majority of complex molecular material observed, at least in comet Hale-Bopp, is an inheritance of primordial material., Comment: 22 pages, 14 figures, 6 tables

We study carbon-aware spatiotemporal workload management, which seeks to address the growing environmental impact of data centers. We formalize this as an online problem called spatiotemporal online allocation with deadline constraints ($\mathsf{SOAD}$), in which an online player completes a workload (e.g., a batch compute job) by moving and scheduling the workload across a network subject to a deadline $T$. At each time step, a service cost function is revealed, representing, e.g., the carbon intensity of servicing a workload at each location, and the player must irrevocably decide the current allocation. Furthermore, whenever the player moves the allocation, it incurs a movement cost defined by a metric space $(X,d)$ that captures, e.g., the overhead of migrating a compute job. $\mathsf{SOAD}$ formalizes the open problem of combining general metrics and deadline constraints in the online algorithms literature, unifying problems such as metrical task systems and online search. We propose a competitive algorithm for $\mathsf{SOAD}$ along with a matching lower bound that proves it is optimal. Our main algorithm, ${\rm C{\scriptsize ARBON}C{\scriptsize LIPPER}}$, is a learning-augmented algorithm that takes advantage of predictions (e.g., carbon intensity forecasts) and achieves an optimal consistency-robustness trade-off. We evaluate our proposed algorithms for carbon-aware spatiotemporal workload management on a simulated global data center network, showing that ${\rm C{\scriptsize ARBON}C{\scriptsize LIPPER}}$ significantly improves performance compared to baseline methods and delivers meaningful carbon reductions., Comment: 50 pages, 21 figures

Cleavable metallic antiferromagnets may be of use for low-dissipation spintronic devices; however, few are currently known. Here we present orthorhombic MnRhBi3 as one such compound and present a thorough study of its physical properties. Exfoliation is demonstrated experimentally, and the cleavage energy and electronic structure are examined by density functional theory calculations. It is concluded that MnRhBi3 is a van der Waals layered material that cleaves easily between neighboring Bi layers, and that the Bi atoms have lone pairs extending into the van der Waals gaps. A series of four phase transitions are observed below room temperature, and neutron diffraction shows that at least two of the transitions involve the formation of antiferromagnetic order. Anomalous thermal expansion points to a crystallographic phase transition and/or strong magnetoelastic coupling. This work reveals a complex phase evolution in MnRhBi3 and establishes this cleavable antiferromagnetic metal as an interesting material for studying the interplay of structure, magnetism, and transport in the bulk and ultrathin limits as well as the role of lone pair electrons in interface chemistry and proximity effects in van der Waals heterostructures.

Magnetic skyrmion crystals are traditionally associated with non-centrosymmetric crystal structures; however, it has been demonstrated that skyrmion crystals can be stabilized by competing interactions in centrosymmetric crystals. To understand and optimize the physical responses associated with topologically-nontrivial skyrmion textures, it is important to quantify their magnetic interactions by comparing theoretical predictions with spectroscopic data. Here, we present neutron diffraction and spectroscopy data on the centrosymmetric skyrmion material GdRu$_2$Si$_2$, and show that the key spectroscopic features can be explained by the magnetic interactions calculated using density-functional theory calculations. We further show that the recently-proposed 2-$\mathbf{q}$ "topological spin stripe" structure yields better agreement with our data than a 1-$\mathbf{q}$ helical structure, and identify how the magnetic structure evolves with temperature., Comment: 6 pages, 3 figures

Micro-credentials are increasingly being adopted across a variety of industries and countries, including the kindergarten through grade 12 education system in the United States, as a way to identify workers with specific skills, knowledge, and competencies. This report presents the results of a study conducted by RAND researchers who investigated the implementation and impact of teacher micro-credentials intended to improve high school science, technology, engineering, and math teaching and learning in Louisiana. The creation, design, and implementation of micro-credentials was undertaken through a collaboration between the Louisiana Department of Education, Louisiana State University, and BloomBoard. RAND researchers designed and executed a randomized controlled trial of the effectiveness of micro-credentials and an implementation study. The study indicated that teachers' engagement with micro-credentials was relatively low and that micro-credentials did not lead to an increase in student achievement. That said, participating teachers reported that the micro-credentials were well aligned with their courses and supported their teaching practice. Study implications include that micro-credential providers and stakeholders should clearly communicate the value proposition and time requirements of micro-credentials to teachers, as well as provide ample peer and asynchronous supports to teachers for undertaking them.

A codimension-two spiral spin-liquid is a correlated paramagnetic state with one-dimensional ground state degeneracy hosted within a three-dimensional lattice. Here, via neutron scattering experiments and numerical simulations, we establish the existence of a codimension-two spiral spin-liquid in the effective honeycomb-lattice compound Cs$_3$Fe$_2$Cl$_9$ and demonstrate the selective visibility of the spiral surface through phase tuning. In the long-range ordered regime, competing spiral and spin density wave orders emerge as a function of applied magnetic field, among which a possible order-by-disorder transition is identified., Comment: 21 pages, 23 figures

We study the design of risk-sensitive online algorithms, in which risk measures are used in the competitive analysis of randomized online algorithms. We introduce the CVaR$_\delta$-competitive ratio ($\delta$-CR) using the conditional value-at-risk of an algorithm's cost, which measures the expectation of the $(1-\delta)$-fraction of worst outcomes against the offline optimal cost, and use this measure to study three online optimization problems: continuous-time ski rental, discrete-time ski rental, and one-max search. The structure of the optimal $\delta$-CR and algorithm varies significantly between problems: we prove that the optimal $\delta$-CR for continuous-time ski rental is $2-2^{-\Theta(\frac{1}{1-\delta})}$, obtained by an algorithm described by a delay differential equation. In contrast, in discrete-time ski rental with buying cost $B$, there is an abrupt phase transition at $\delta = 1 - \Theta(\frac{1}{\log B})$, after which the classic deterministic strategy is optimal. Similarly, one-max search exhibits a phase transition at $\delta = \frac{1}{2}$, after which the classic deterministic strategy is optimal; we also obtain an algorithm that is asymptotically optimal as $\delta \downarrow 0$ that arises as the solution to a delay differential equation., Comment: Accepted for presentation at the Conference on Learning Theory (COLT) 2024. Updated with an additional reference and minor edits

Using the framework of semi-classical Landau-Lifshitz dynamics (LLD), we conduct a systematic investigation of the temperature-dependent spin dynamics in the S = 1/2 Heisenberg square-lattice antiferromagnet (SqAF). By performing inelastic neutron scattering measurements on Zn2VO(PO4)2 (ZVPO) and corresponding finite-temperature spin dynamics simulations based on LLD, we present a comprehensive analysis that bridges quantum and classical spin dynamics over a broad temperature range. First, a remarkable agreement between experimental data and LLD simulations is found in the paramagnetic phase of ZVPO, demonstrating the capability of LLD in accurately determining the spin Hamiltonian of S = 1/2 systems and capturing the quantum-to-classical crossover of their spin dynamics. Second, by analyzing the discrepancies between the experimental data and the LLD simulations at lower temperatures, we determine the experimental temperature dependence of the quantum effects in the excitation spectrum of the S = 1/2 SqAF: the quantum renormalization factor for the magnon energies and the quantum continuum above the one-magnon bands. Notably, the emergence of each quantum effect is found to correlate with the formation of three-dimensional long-range order. This work demonstrates the utility of LLD in gaining experimental insights into the temperature-induced modifications of quantum spin dynamics and their convergence towards classical expectations at higher temperatures. This motivates further applications to more challenging quantum antiferromagnets dominated by stronger quantum fluctuations., Comment: 8 pages, 4 figures, Supplemental Material can be found on the journal's website

For sequences of quantum ergodic eigenfunctions, we define the quantum flux norm associated to a codimension $1$ submanifold $\Sigma$ of a non-degenerate energy surface. We prove restrictions of eigenfunctions to $\Sigma$, realized using the quantum flux norm, are quantum ergodic. We compare this result to known results from \cite{CTZ} in the case of Euclidean domains and hyperfurfaces. As a further application, we consider complexified analytic eigenfunctions and prove a second microlocal analogue of \cite{CTZ} in that context.

The S = 1/2 triangular lattice antiferromagnet (TLAF) is a paradigmatic example of frustrated quantum magnetism. An ongoing challenge involves understanding the influence of exchange anisotropy on the collective behavior within such systems. Using inelastic neutron scattering (INS) and advanced calculation techniques, we have studied the low and high-temperature spin dynamics of Ba2La2CoTe2O12 (BLCTO): a Co2+-based Jeff = 1/2 TLAF that exhibits 120{\deg} order below TN = 3.26 K. We determined the spin Hamiltonian by fitting the energy-resolved paramagnetic excitations measured at T > TN, revealing exceptionally strong easy-plane XXZ anisotropy. Below TN, the excitation spectrum exhibits a high energy continuum having a larger spectral weight than the single-magnon modes, suggesting a scenario characterized by a spinon confinement length that markedly exceeds the lattice spacing. We conjecture that this phenomenon arises from the proximity to a quantum melting point, even under strong easy-plane XXZ anisotropy. Finally, we highlight characteristic flat features in the excitation spectrum, which are connected to higher-order van Hove singularities in the magnon dispersion directly induced by easy-plane XXZ anisotropy. Our results provide a rare experimental insight into the nature of highly anisotropic S = 1/2 TLAFs between the Heisenberg and XY limits., Comment: 5 figures, Supplementary Information can be found on the journal's website

We introduce and study a family of online metric problems with long-term constraints. In these problems, an online player makes decisions $\mathbf{x}_t$ in a metric space $(X,d)$ to simultaneously minimize their hitting cost $f_t(\mathbf{x}_t)$ and switching cost as determined by the metric. Over the time horizon $T$, the player must satisfy a long-term demand constraint $\sum_{t} c(\mathbf{x}_t) \geq 1$, where $c(\mathbf{x}_t)$ denotes the fraction of demand satisfied at time $t$. Such problems can find a wide array of applications to online resource allocation in sustainable energy/computing systems. We devise optimal competitive and learning-augmented algorithms for the case of bounded hitting cost gradients and weighted $\ell_1$ metrics, and further show that our proposed algorithms perform well in numerical experiments., Comment: Accepted to ICML 2024. 31 pages, 12 figures

Square-lattice systems offer a direct route for realizing 2D quantum magnetism with frustration induced by competing interactions. In this work, the square-lattice materials YbBi2IO4 and YbBi2ClO4 were investigated using a combination of magnetization and specific heat measurements on polycrystalline samples. Specific heat measurements provide evidence for long-range magnetic order below TN = 0.21 K (0.25 K) for YbBi2IO4 (YbBi2ClO4). On the other hand, a rather broad maximum is found in the temperature-dependent magnetic susceptibility, located at Tmax = 0.33 K (0.38 K) in YbBi2IO4 (YbBi2ClO4), consistent with the quasi-2D magnetism expected for the large separation between the magnetic layers. Estimation of the magnetic entropy supports the expected Kramers' doublet ground state for Yb3+ and the observed paramagnetic behavior is consistent with a well-isolated doublet. Roughly two-thirds of the entropy is consumed above TN, due to a combination of the quasi-2D behavior and magnetic frustration. The impact of frustration is examined from the viewpoint of a simplified J1-J2 square lattice model, which is frustrated for antiferromagnetic interactions. Specifically, a high-temperature series expansion analysis of the temperature-dependent specific heat and magnetization data yields J2/J1 = 0.30 (0.23) for YbBi2IO4 (YbBi2ClO4). This simplified analysis suggests strong frustration that should promote significant quantum fluctuations in these compounds, and thus motivates future work on the static and dynamic magnetic properties of these materials., Comment: 8 pages, 5 figures

The asymmetric distribution of chiral objects with opposite chirality is of great fundamental interests ranging from molecular biology to particle physics. In quantum materials, chiral states can build on inversion-symmetry-breaking lattice structures or emerge from spontaneous magnetic ordering induced by competing interactions. Although the handedness of a chiral state can be changed through external fields, a spontaneous chirality flipping has yet to be discovered. In this letter, we present experimental evidence of chirality flipping via changing temperature in a topological magnet EuAl$_4$, which features orthogonal spin and charge density waves (SDW/CDW). Using circular dichroism of Bragg peaks in the resonant magnetic x-ray scattering, we find that the chirality of the helical SDW flips through a first order phase transition with modified SDW wavelength. Intriguingly, we observe that the CDW couples strongly with the SDW and displays a rare commensurate-to-incommensurate transition at the chirality flipping temperature. Combining with first principles calculations and angle resolved photoemission spectroscopy, we establish the Fermi surface origin of the helical SDW with intertwined spin, charge, and lattice degrees of freedom in EuAl$_4$. Our results reveal an unprecedented spontaneous chirality flipping and lays the groundwork for a new functional manipulation of chirality through momentum dependent spin-charge-lattice interactions., Comment: Supplementary materials are available from the corresponding author upon request

We introduce and study online conversion with switching costs, a family of online problems that capture emerging problems at the intersection of energy and sustainability. In this problem, an online player attempts to purchase (alternatively, sell) fractional shares of an asset during a fixed time horizon with length $T$. At each time step, a cost function (alternatively, price function) is revealed, and the player must irrevocably decide an amount of asset to convert. The player also incurs a switching cost whenever their decision changes in consecutive time steps, i.e., when they increase or decrease their purchasing amount. We introduce competitive (robust) threshold-based algorithms for both the minimization and maximization variants of this problem, and show they are optimal among deterministic online algorithms. We then propose learning-augmented algorithms that take advantage of untrusted black-box advice (such as predictions from a machine learning model) to achieve significantly better average-case performance without sacrificing worst-case competitive guarantees. Finally, we empirically evaluate our proposed algorithms using a carbon-aware EV charging case study, showing that our algorithms substantially improve on baseline methods for this problem., Comment: Accepted to SIGMETRICS / Performance '24. 47 pages, 9 figures

The burgeoning field of algorithms with predictions studies the problem of using possibly imperfect machine learning predictions to improve online algorithm performance. While nearly all existing algorithms in this framework make no assumptions on prediction quality, a number of methods providing uncertainty quantification (UQ) on machine learning models have been developed in recent years, which could enable additional information about prediction quality at decision time. In this work, we investigate the problem of optimally utilizing uncertainty-quantified predictions in the design of online algorithms. In particular, we study two classic online problems, ski rental and online search, where the decision-maker is provided predictions augmented with UQ describing the likelihood of the ground truth falling within a particular range of values. We demonstrate that non-trivial modifications to algorithm design are needed to fully leverage the UQ predictions. Moreover, we consider how to utilize more general forms of UQ, proposing an online learning framework that learns to exploit UQ to make decisions in multi-instance settings.

Let $\Omega$ be a piecewise-smooth, bounded convex domain in $\R^2$ and consider $L^2$-normalized Neumann eigenfunctions $\phi_{\lambda}$ with eigenvalue $\lambda^2$. Our main result is a small-scale {\em non-concentration} estimate: We prove that for {\em any} $x_0 \in \overline{\Omega},$ (including boundary and corner points) and any $\delta \in [0,1),$ $$ \| \phi_\lambda \|_{B(x_0,\lambda^{-\delta})\cap \Omega} = O(\lambda^{-\delta/2}).$$ The proof is a stationary vector field argument combined with a small scale induction argument., Comment: This is an expanded version of the first half of the preprint arXiv:2012.15237 [math.AP] by the same authors

Micro-credentials are increasingly being adopted across a variety of industries and countries, including the kindergarten through grade 12 education system in the United States, as a way to identify workers with specific skills, knowledge, and competencies. This report presents the results of a study conducted by RAND researchers who investigated the implementation and impact of teacher micro-credentials intended to improve high school science, technology, engineering, and math teaching and learning in Louisiana. The creation, design, and implementation of micro-credentials was undertaken through a collaboration between the Louisiana Department of Education, Louisiana State University, and BloomBoard. RAND researchers designed and executed a randomized controlled trial of the effectiveness of micro-credentials and an implementation study. The study indicated that teachers' engagement with micro-credentials was relatively low and that micro-credentials did not lead to an increase in student achievement. That said, participating teachers reported that the micro-credentials were well aligned with their courses and supported their teaching practice. Study implications include that micro-credential providers and stakeholders should clearly communicate the value proposition and time requirements of micro-credentials to teachers, as well as provide ample peer and asynchronous supports to teachers for undertaking them.

A body immersed in a supersaturated fluid like carbonated water can accumulate a dynamic field of bubbles upon its surface. If the body is mobile, the attached bubbles can lift it upward against gravity, but a fluid-air interface can clean the surface of these lifting agents and the body may plummet. The process then begins anew, and continues for as long as the concentration of gas in the fluid supports it. In this work, experiments using fixed and free immersed bodies reveal fundamental features of force development and gas escape. A continuum model which incorporates the dynamics of a surface buoyancy field is used to predict the ranges of body mass and size, and fluid properties, for which the system is most dynamic, and those for which body excursions are suppressed. Simulations are then used to probe systems which are dominated by a small number of large bubbles. Body rotations at the surface are critical for driving periodic vertical motions of large bodies, which in turn can produce body wobbling, rolling, and damped surface 'bouncing' dynamics., Comment: 36 pages, 10 figures. Movies are posted here: https://people.math.wisc.edu/~spagnolie/publications.html

Magnetic excitations in the spin chain candidate Sr$_2$V$_3$O$_9$ have been investigated by inelastic neutron scattering on a single crystal sample. A spinon continuum with a bandwidth of $\sim22$ meV is observed along the chain formed by alternating magnetic V$^{4+}$ and nonmagnetic V$^{5+}$ ions. Incipient magnetic Bragg peaks due to weak ferromagnetic interchain couplings emerge when approaching the magnetic transition at $T_N\sim 5.3$ K while the excitations remain gapless within the instrumental resolution. Comparisons to the Bethe ansatz, density matrix renormalization group (DMRG) calculations, and effective field theories confirm Sr$_2$V$_3$O$_9$ as a host of weakly coupled $S = 1/2$ chains dominated by antiferromagnetic intrachain interactions of $\sim7.1$(1) meV., Comment: 19 pages, 9 figures

Automatic differentiation, also known as backpropagation, AD, autodiff, or algorithmic differentiation, is a popular technique for computing derivatives of computer programs accurately and efficiently. Sometimes, however, the derivatives computed by AD could be interpreted as incorrect. These pitfalls occur systematically across tools and approaches. In this paper we broadly categorize problematic usages of AD and illustrate each category with examples such as chaos, time-averaged oscillations, discretizations, fixed-point loops, lookup tables, and linear solvers. We also review debugging techniques and their effectiveness in these situations. With this article we hope to help readers avoid unexpected behavior, detect problems more easily when they occur, and have more realistic expectations from AD tools.

We introduce and study the online pause and resume problem. In this problem, a player attempts to find the $k$ lowest (alternatively, highest) prices in a sequence of fixed length $T$, which is revealed sequentially. At each time step, the player is presented with a price and decides whether to accept or reject it. The player incurs a switching cost whenever their decision changes in consecutive time steps, i.e., whenever they pause or resume purchasing. This online problem is motivated by the goal of carbon-aware load shifting, where a workload may be paused during periods of high carbon intensity and resumed during periods of low carbon intensity and incurs a cost when saving or restoring its state. It has strong connections to existing problems studied in the literature on online optimization, though it introduces unique technical challenges that prevent the direct application of existing algorithms. Extending prior work on threshold-based algorithms, we introduce double-threshold algorithms for both the minimization and maximization variants of this problem. We further show that the competitive ratios achieved by these algorithms are the best achievable by any deterministic online algorithm. Finally, we empirically validate our proposed algorithm through case studies on the application of carbon-aware load shifting using real carbon trace data and existing baseline algorithms., Comment: 34 pages, 12 figures

We introduce the control conditions for 0th order pseudodifferential operators $\mathbf{P}$ whose real parts satisfy the Morse--Smale dynamical condition. We obtain microlocal control estimates under the control conditions. As a result, we show that there are no singular profiles in the solution to the evolution equation $(i\partial_t-\mathbf{P})u=f$ when $\mathbf{P}$ has a damping term that satisfies the control condition and $f\in C^{\infty}$. This is motivated by the study of a microlocal model for the damped internal waves., Comment: 20 pages, 3 figures. To appear in Int. Math. Res. Not. IMRN

Bayesian Optimization (BO) links Gaussian Process (GP) surrogates with sequential design toward optimizing expensive-to-evaluate black-box functions. Example design heuristics, or so-called acquisition functions, like expected improvement (EI), balance exploration and exploitation to furnish global solutions under stringent evaluation budgets. However, they fall short when solving for robust optima, meaning a preference for solutions in a wider domain of attraction. Robust solutions are useful when inputs are imprecisely specified, or where a series of solutions is desired. A common mathematical programming technique in such settings involves an adversarial objective, biasing a local solver away from ``sharp'' troughs. Here we propose a surrogate modeling and active learning technique called robust expected improvement (REI) that ports adversarial methodology into the BO/GP framework. After describing the methods, we illustrate and draw comparisons to several competitors on benchmark synthetic exercises and real problems of varying complexity., Comment: 27 pages, 17 figures, 1 table

Magnetic materials with noncoplanar magnetic structures can show unusual physical properties driven by nontrivial topology. Topologically-active states are often multi-q structures, which are challenging to stabilize in models and to identify in materials. Here, we use inelastic neutron-scattering experiments to show that the insulating double perovskites Ba2YRuO6 and Ba2LuRuO6 host a noncoplanar 3-q structure on the face-centered cubic lattice. Quantitative analysis of our neutron-scattering data reveals that these 3-q states are stabilized by biquadratic interactions. Our study identifies double perovskites as a highly promising class of materials to realize topological magnetism, elucidates the stabilization mechanism of the 3-q state in these materials, and establishes neutron spectroscopy on powder samples as a valuable technique to distinguish multi-q from single-q states, facilitating the discovery of topologically-nontrivial magnetic materials., Comment: 5 pages, 4 figures

In this paper, we continue the study of eigenfunctions on triangles initiated by the first author in \cite{Chr-tri} and \cite{Chr-simp}. The Neumann data of Dirichlet eigenfunctions on triangles enjoys an equidistribution law, being equidistributed on each side. The proof of this result is remarkably simple, using only the radial vector field and a Rellich type integrations by parts. The equidistribution law, including on higher dimensional simplices, agrees with what Quantum Ergodic Restriction would predict. However, distribution of the Neumann data on subsets of a side is not well understood, and elementary methods do not appear to give enough information to draw conclusions. In the present note, we first show that an "obvious" conjecture fails even for the simplest right isosceles triangle using only Fourier series. We then use a result of Marklof-Rudnick \cite{Marklof-Rudnick} in which the authors show an interior {\it spatial} equidistribution law for a density-one subsequence of eigenfunctions to give an estimate on energy distribution of eigenfunctions on the interior. Finally we present some numerical computations suggesting the behaviour of eigenfunctions on almost isosceles triangles is quite complicated.