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Sympathetic cooling of molecular ions through the Coulomb interaction with laser-cooled atomic ions is an efficient tool to prepare translationally cold molecules. Even at relatively high collisional energies of about 1eV (T$\sim$10000K), the nearest approach in the ion-ion collisions never gets closer than $\sim$1nm such that naively perturbations of the internal molecular state are not expected. The Coulomb field may, however, induce rotational transitions changing the purity of initially quantum state prepared molecules. Here, we use estimates of rotational state changes in collisions of diatomic ions with atomic ions (arXiv:1905.02130) and determine the overall rotational excitation accumulated over the sympathetic cooling. We also estimate the cooling time, considering both a single atomic ion and a Coulomb crystal of atomic ions., Comment: arXiv admin note: text overlap with arXiv:1905.02130

We introduce \texttt{cymyc}, a high-performance Python library for numerical investigation of the geometry of a large class of string compactification manifolds and their associated moduli spaces. We develop a well-defined geometric ansatz to numerically model tensor fields of arbitrary degree on a large class of Calabi-Yau manifolds. \texttt{cymyc} includes a machine learning component which incorporates this ansatz to model tensor fields of interest on these spaces by finding an approximate solution to the system of partial differential equations they should satisfy., Comment: 35 pages, 12 figures

Calabi--Yau compactifications of the $E_8\times E_8$ heterotic string provide a promising route to recovering the four-dimensional particle physics described by the Standard Model. While the topology of the Calabi--Yau space determines the overall matter content in the low-energy effective field theory, further details of the compactification geometry are needed to calculate the normalized physical couplings and masses of elementary particles. In this work, we present numerical computations of physical Yukawa couplings in a number of heterotic models in the standard embedding and demonstrate the existence of natural hierarchies, a coveted feature in string model building., Comment: 7 pages, 3 figures

Recent advances in large language models (LLMs) have increased the demand for comprehensive benchmarks to evaluate their capabilities as human-like agents. Existing benchmarks, while useful, often focus on specific application scenarios, emphasizing task completion but failing to dissect the underlying skills that drive these outcomes. This lack of granularity makes it difficult to deeply discern where failures stem from. Additionally, setting up these environments requires considerable effort, and issues of unreliability and reproducibility sometimes arise, especially in interactive tasks. To address these limitations, we introduce the Massive Multitask Agent Understanding (MMAU) benchmark, featuring comprehensive offline tasks that eliminate the need for complex environment setups. It evaluates models across five domains, including Tool-use, Directed Acyclic Graph (DAG) QA, Data Science and Machine Learning coding, Contest-level programming and Mathematics, and covers five essential capabilities: Understanding, Reasoning, Planning, Problem-solving, and Self-correction. With a total of 20 meticulously designed tasks encompassing over 3K distinct prompts, MMAU provides a comprehensive framework for evaluating the strengths and limitations of LLM agents. By testing 18 representative models on MMAU, we provide deep and insightful analyses. Ultimately, MMAU not only sheds light on the capabilities and limitations of LLM agents but also enhances the interpretability of their performance. Datasets and evaluation scripts of MMAU are released at https://github.com/apple/axlearn/tree/main/docs/research/mmau.

We apply reinforcement learning (RL) to generate fine regular star triangulations of reflexive polytopes, that give rise to smooth Calabi-Yau (CY) hypersurfaces. We demonstrate that, by simple modifications to the data encoding and reward function, one can search for CYs that satisfy a set of desirable string compactification conditions. For instance, we show that our RL algorithm can generate triangulations together with holomorphic vector bundles that satisfy anomaly cancellation and poly-stability conditions in heterotic compactification. Furthermore, we show that our algorithm can be used to search for reflexive subpolytopes together with compatible triangulations that define fibration structures of the CYs.

Superpositions of black holes can be described geometrically using a combined canonical formulation for space-time and quantum states. A previously introduced black-hole model that includes quantum fluctuations of metric components is shown here to give full access to the corresponding space-time geometry of weak-field gravity in terms of suitable line elements with quantum corrections. These results can be interpreted as providing covariant formulations of the gravitational force implied by a distribution of black holes in superposition. They can also be understood as a distribution of quantum matter constituents in superposition for a single black hole. A detailed analysis in the weak-field limit reveals quantum corrections to Newton's potential in generic semiclassical states, as well as new bounds on quantum fluctuations, implied by the covariance condition, rather than the usual uncertainty principle. These results provide additional control on quantum effects in Newton's potential that can be used in a broad range of predictions to be compared with observations., Comment: 33 pages

Many natural language processing systems operate over tokenizations of text to address the open-vocabulary problem. In this paper, we give and analyze an algorithm for the efficient construction of deterministic finite automata designed to operate directly on tokenizations produced by the popular byte pair encoding technique. This makes it possible to apply many existing techniques and algorithms to the tokenized case, such as pattern matching, equivalence checking of tokenization dictionaries, and composing tokenized languages in various ways.

The main goal of this work is to provide sample-path estimates for the solution of slowly time-dependent SPDEs perturbed by a cylindrical fractional Brownian motion. Our strategy is similar to the approach by Berglund and Nader for space-time white noise. However, the setting of fractional Brownian motion does not allow us to use any martingale methods. Using instead optimal estimates for the probability that the supremum of a Gaussian process exceeds a certain level, we derive concentration estimates for the solution of the SPDE, provided that the Hurst index $H$ of the fractional Brownian motion satisfies $H>\frac14$. As a by-product, we also obtain concentration estimates for one-dimensional fractional SDEs valid for any $H\in(0,1)$., Comment: 13 pages

Gromov-Witten invariants arise in the topological A-model as counts of worldsheet instantons. On the A-side, these invariants can be computed for a Fano or semi-Fano toric variety using generating functions associated to the toric divisors. On the B-side, the same invariants can be computed from the periods of the mirror. We utilize scattering diagrams (aka wall structures) in the Gross-Siebert mirror symmetry program to extend the calculation of Gromov-Witten invariants to non-Fano toric varieties. Following the work of Carl-Pumperla-Siebert, we compute corrected mirror superpotentials $\vartheta_1(\mathbb{F}_m)$ and their periods for the Hirzebruch surfaces $\mathbb{F}_m$ with $m \ge 2$., Comment: 40 pages, 19 figures

Standard toric geometry methods used to construct Calabi-Yau varieties may be extended to complete intersections in non-Fano varieties encoded by star triangulating non-convex polytopes. Similarly, mirror symmetry is conjectured to hold in terms of a transpolar duality generalizing the original construction of Batyrev and Borisov. The associated mirror pairs naturally include certain flip-folded, multi-layered multihedral objects, inclusively named VEX multitopes, and a correspondingly generalized transpolar duality. These self-overlaying VEX multitopes, long since known in pre-symplectic geometry, are found to correspond to certain non-algebraic but smooth toric spaces with Chern classes that satisfy the standard Todd-Hirzebruch identities. The computation of diffeomorphism invariants, including characteristic submanifold intersection numbers, corroborates their recent inclusion in the connected web of Calabi-Yau spaces and associated string compactifications: They arise together with the standard (Fano/reflexive polytope) constructions, within deformation families of generalized complete intersections in products of projective spaces., Comment: 48 pages, 8 figures; v.2: minor edits (typos and clarifications)

Update formulas for the Hessian approximations in quasi-Newton methods such as BFGS can be derived as analytical solutions to certain nearest-matrix problems. In this article, we propose a similar idea for deriving new limited memory versions of quasi-Newton methods. Most limited memory quasi-Newton methods make use of Hessian approximations that can be written as a scaled identity matrix plus a symmetric matrix with limited rank. We derive a way of finding the nearest matrix of this type to an arbitrary symmetric matrix, in either the Frobenius norm, the induced $l^2$ norm, or a dissimilarity measure for positive definite matrices in terms of trace and determinant. In doing so, we lay down a framework for more general matrix optimization problems with unitarily invariant matrix norms and arbitrary constraints on the set of eigenvalues. We then propose a trust region method in which the Hessian approximation, after having been updated by a Broyden class formula and used to solve a trust-region problem, is replaced by one of its closest limited memory approximations. We propose to store the Hessian approximation in terms of its eigenvectors and eigenvalues in a way that completely defines its eigenvalue decomposition, as this simplifies both the solution of the trust region subproblem and the nearest limited memory matrix problem. Our method is compared to a reference trust region method with the usual limited memory BFGS updates, and is shown to require fewer iterations and the storage of fewer vectors for a variety of test problems., Comment: 42 pages, 5 figures

We propose a novel algorithm, termed soft quasi-Newton (soft QN), for optimization in the presence of bounded noise. Traditional quasi-Newton algorithms are vulnerable to such perturbations. To develop a more robust quasi-Newton method, we replace the secant condition in the matrix optimization problem for the Hessian update with a penalty term in its objective and derive a closed-form update formula. A key feature of our approach is its ability to maintain positive definiteness of the Hessian inverse approximation. Furthermore, we establish the following properties of soft QN: it recovers the BFGS method under specific limits, it treats positive and negative curvature equally, and it is scale invariant. Collectively, these features enhance the efficacy of soft QN in noisy environments. For strongly convex objective functions and Hessian approximations obtained using soft QN, we develop an algorithm that exhibits linear convergence toward a neighborhood of the optimal solution, even if gradient and function evaluations are subject to bounded perturbations. Through numerical experiments, we demonstrate superior performance of soft QN compared to state-of-the-art methods in various scenarios., Comment: 28 pages, 4 figures

The Monitoring the Future (MTF) study monitors historical and developmental changes in substance use prevalence among key subgroups of the general U.S. adolescent and adult population. The current study first devised and evaluated a cohort-specific pooled analysis weighing procedure for the MTF panel study that weighted back to the initial 12th grade samples. Following this, the cohort-specific weights were updated to age-specific weights in order to provide increased flexibility for a wide range of model specifications and to avoid the need to re-calculate a first half-sample respondent's final weight at a particular wave after the full cohort had completed that wave. The cohort-specific pooled analysis weights appear to result in an overall improvement in the degree to which the sociodemographic distributions of the initial 12th grade samples are retained, as well as likely producing slightly improved substance use estimates due to accounting for historical variation in panel sample selection and attrition over time. The updated age-specific pooled analysis weights continued to provide the benefits associated with the cohort-specific weights, but also brought increased flexibility for modeling both cohort- and age-specific research questions, and allowed all respondents' weights at each specific age to remain fixed across time.

One of the challenges of heterotic compactification on a Calabi-Yau threefold is to determine the physical $(\mathbf{27})^3$ Yukawa couplings of the resulting four-dimensional $\mathcal{N}=1$ theory. In general, the calculation necessitates knowledge of the Ricci-flat metric. However, in the standard embedding, which references the tangent bundle, we can compute normalized Yukawa couplings from the Weil-Petersson metric on the moduli space of complex structure deformations of the Calabi-Yau manifold. In various examples (the Fermat quintic, the intersection of two cubics in $\mathbb{P}^5$, and the Tian-Yau manifold), we calculate the normalized Yukawa couplings for $(2,1)$-forms using the Weil-Petersson metric obtained from the Kodaira-Spencer map. In cases where $h^{1,1}=1$, this is compared to a complementary calculation based on performing period integrals. A third expression for the normalized Yukawa couplings is obtained from a machine learned approximate Ricci-flat metric making use of explicit harmonic representatives. The excellent agreement between the different approaches opens the door to precision string phenomenology., Comment: 33 pages, 11 figures, 2 tables, 3 lemmas, 1 theorem. v2: Minor edits

High spin Chern-number insulator (HSCI) emerges as a novel 2D topological phase of condensed matter that is beyond the classification scheme of topological quantum chemistry. In this work, we report the observation of a semimetallic Sb monolayer with the same band topology as HSCI. Our first-principles calculations further indicate a moderate lattice strain can make Sb monolayer an insulator while retaining the nontrivial topological order. A quantized spin Hall conductivity is found in the HSCI phase of Sb monolayer. The results suggest strained Sb monolayer as a promising platform for exploring intriguing physical properties of the exotic HSCI topological matter, Comment: 4 figures

These notes are based on a series of three lectures given (online) by the first named author at the workshop "Higher Structures and Operadic Calculus" at CRM Barcelona in June 2021. The aim is to give a concise introduction to rational homotopy theory through the lens of higher structures. The rational homotopy type of a simply connected space of finite type is modeled by a $C_\infty$-algebra structure on the rational cohomology groups, or alternatively an $L_\infty$-algebra structure on the rational homotopy groups. The first lecture is devoted to explaining these models and their relation to the classical models of Quillen and Sullivan. The second lecture discusses the relation between Koszul algebras, formality and coformality. The main result is that a space is formal if and only if the rational homotopy $L_\infty$-algebra is Koszul and, dually, a space is coformal if and only if the cohomology $C_\infty$-algebra is Koszul. For spaces that are both formal and coformal, this collapses to classical Koszul duality between Lie and commutative algebras. In the third lecture, we discuss certain higher structure in the rational homotopy theory of automorphisms of high dimensional manifolds, discovered by Berglund and Madsen. The higher structure in question is Kontsevich's Lie graph complex and variants of it., Comment: 56 pages. Lecture notes for a lecture series given online by the first author at CRM Barcelona in June 2021. Comments welcome!

We expose a surprising failure of generalization in auto-regressive large language models (LLMs). If a model is trained on a sentence of the form "A is B", it will not automatically generalize to the reverse direction "B is A". This is the Reversal Curse. For instance, if a model is trained on "Valentina Tereshkova was the first woman to travel to space", it will not automatically be able to answer the question, "Who was the first woman to travel to space?". Moreover, the likelihood of the correct answer ("Valentina Tershkova") will not be higher than for a random name. Thus, models do not generalize a prevalent pattern in their training set: if "A is B" occurs, "B is A" is more likely to occur. It is worth noting, however, that if "A is B" appears in-context, models can deduce the reverse relationship. We provide evidence for the Reversal Curse by finetuning GPT-3 and Llama-1 on fictitious statements such as "Uriah Hawthorne is the composer of Abyssal Melodies" and showing that they fail to correctly answer "Who composed Abyssal Melodies?". The Reversal Curse is robust across model sizes and model families and is not alleviated by data augmentation. We also evaluate ChatGPT (GPT-3.5 and GPT-4) on questions about real-world celebrities, such as "Who is Tom Cruise's mother? [A: Mary Lee Pfeiffer]" and the reverse "Who is Mary Lee Pfeiffer's son?". GPT-4 correctly answers questions like the former 79% of the time, compared to 33% for the latter. Code available at: https://github.com/lukasberglund/reversal_curse., Comment: 21 pages, 11 figures

In this paper, we formalize practical byte pair encoding tokenization as it is used in large language models and other NLP systems, in particular we formally define and investigate the semantics of the SentencePiece and HuggingFace tokenizers, in particular how they relate to each other, depending on how the tokenization rules are constructed. Beyond this we consider how tokenization can be performed in an incremental fashion, as well as doing it left-to-right using an amount of memory constant in the length of the string, enabling e.g. using a finite state string-to-string transducer., Comment: In Proceedings NCMA 2023, arXiv:2309.07333

We aim to better understand the emergence of `situational awareness' in large language models (LLMs). A model is situationally aware if it's aware that it's a model and can recognize whether it's currently in testing or deployment. Today's LLMs are tested for safety and alignment before they are deployed. An LLM could exploit situational awareness to achieve a high score on safety tests, while taking harmful actions after deployment. Situational awareness may emerge unexpectedly as a byproduct of model scaling. One way to better foresee this emergence is to run scaling experiments on abilities necessary for situational awareness. As such an ability, we propose `out-of-context reasoning' (in contrast to in-context learning). We study out-of-context reasoning experimentally. First, we finetune an LLM on a description of a test while providing no examples or demonstrations. At test time, we assess whether the model can pass the test. To our surprise, we find that LLMs succeed on this out-of-context reasoning task. Their success is sensitive to the training setup and only works when we apply data augmentation. For both GPT-3 and LLaMA-1, performance improves with model size. These findings offer a foundation for further empirical study, towards predicting and potentially controlling the emergence of situational awareness in LLMs. Code is available at: https://github.com/AsaCooperStickland/situational-awareness-evals.

We elaborate on the new understanding of the cosmological constant and the gauge hierarchy problems in the context of string theory in its metastring formulation, based on the concepts of modular spacetime and Born geometry. The interplay of phase space (and Born geometry), the Bekenstein bound, the mixing between ultraviolet (UV) and infrared (IR) physics and modular invariance in string theory is emphasized. This new viewpoint is fundamentally rooted in quantum contextuality and not in statistical observer bias (anthropic principle). We also discuss the extension of this point of view to the problem of masses of quarks and leptons and their respective mixing matrices., Comment: 31 pages, minor edits, reference added

Calabi-Yau manifolds can be obtained as hypersurfaces in toric varieties built from reflexive polytopes. We generate reflexive polytopes in various dimensions using a genetic algorithm. As a proof of principle, we demonstrate that our algorithm reproduces the full set of reflexive polytopes in two and three dimensions, and in four dimensions with a small number of vertices and points. Motivated by this result, we construct five-dimensional reflexive polytopes with the lowest number of vertices and points. By calculating the normal form of the polytopes, we establish that many of these are not in existing datasets and therefore give rise to new Calabi-Yau four-folds. In some instances, the Hodge numbers we compute are new as well., Comment: 24 pages, 2 figures. Accepted version