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A binomial model for sports matches is developed making use of the maximum possible score $n$ in a game. In contrast to previous approaches the scores of the two teams are negatively correlated, abstracting from a scenario whereby teams cancel each other out. When $n$ is known, analytical results are possible via a Gaussian approximation. Model calibration is obtained via generalized linear modelling, enabling elementary econometric and strategic analysis to be performed. Inter alia this includes quantifying the Last Stone First End effect, analogous to the home-field advantage found in conventional sports. When $n$ is unknown the model behaviour is richer and leads to heavy-tailed non-Gaussian behaviour. We present an approximate analysis of this case based on the Variance Gamma distribution.

The celebrated It\^o-Michler theorem asserts that a prime $p$ does not divide the degree of any irreducible character of a finite group $G$ if and only if $G$ has a normal and abelian Sylow $p$-subgroup. The principal block case of the recently-proven Brauer's height zero conjecture isolates the abelian part in the It\^o-Michler theorem. In this paper, we show that the normal part can also be isolated in a similar way. This is a consequence of work on a strong form of the so-called Brauer's height zero conjecture for two primes of Malle and Navarro. Using our techniques, we also provide an alternate proof of this conjecture., Comment: Revised following Gunter Malle's suggestions

GenEO (`Generalised Eigenvalue problems on the Overlap') is a method from the family of spectral coarse spaces that can efficiently rely on local eigensolves in order to build a robust parallel domain decomposition preconditioner for elliptic PDEs. When used as a preconditioner in a conjugate gradient, this method is extremely efficient in the positive-definite case, yielding an iteration count completely independent of the number of subdomains and heterogeneity. In a previous work this theory was extended to the cased of convection--diffusion--reaction problems, which may be non-self-adjoint and indefinite, and whose discretisations are solved with preconditioned GMRES. The GenEO coarse space was then defined here using a generalised eigenvalue problem based on a self-adjoint and positive definite subproblem. The resulting method, called $\Delta$-GenEO becomes robust with respect to the variation of the coefficient of the diffusion term in the operator and depends only very mildly on variations of the other coefficients. However, the iteration number estimates get worse as the non-self-adjointness and indefiniteness of the operator increases, which is often the case for the high frequency Helmholtz problems. In this work, we will improve on this aspect by introducing a new version, called $H_k$-GenEO, which uses a generalised eigenvalue problem based directly on the indefinite operator which will lead to a robust method with respect to the increase in the wave-number. We provide theoretical estimates showing the dependence of the size of the coarse space on the wave-number., Comment: arXiv admin note: text overlap with arXiv:2403.18378

Dark matter makes up approximately 85% of total matter in our universe, yet it has never been directly observed in any laboratory on Earth. The origin of dark matter is one of the most important questions in contemporary physics, and a convincing detection of dark matter would be a Nobel-Prize-level breakthrough in fundamental science. The ABRACADABRA experiment was specifically designed to search for dark matter. Although it has not yet made a discovery, ABRACADABRA has produced several dark matter search results widely endorsed by the physics community. The experiment generates ultra-long time-series data at a rate of 10 million samples per second, where the dark matter signal would manifest itself as a sinusoidal oscillation mode within the ultra-long time series. In this paper, we present the TIDMAD -- a comprehensive data release from the ABRACADABRA experiment including three key components: an ultra-long time series dataset divided into training, validation, and science subsets; a carefully-designed denoising score for direct model benchmarking; and a complete analysis framework which produces a community-standard dark matter search result suitable for publication as a physics paper. This data release enables core AI algorithms to extract the signal and produce real physics results thereby advancing fundamental science. The data downloading and associated analysis scripts are available at https://github.com/jessicafry/TIDMAD

We characterize when a finite group G possesses a Sylow 3-subgroup P with abelianization of order 9 in terms of the number of height zero characters lying in the principal 3-block of G, settling a conjecture put forward by Navarro, Sambale, and Tiep in 2018. Along the way, we show that a recent result by Laradji on the number of character of height zero in a block that lie above a given character of some normal subgroup holds, without any hypothesis on the group for blocks of maximal defect., Comment: slight title change and other minor changes, following helpful comments thanks to Gunter Malle and Bejamin Sambale

Despite its amazing quantitative successes and contributions to revolutionary technologies, physics currently faces many unsolved mysteries ranging from the meaning of quantum mechanics to the nature of the dark energy that will determine the future of the Universe. It is clearly prohibitive for the general reader, and even the best informed physicists, to follow the vast number of technical papers published in the thousands of specialized journals. For this reason, we have asked the leading experts across many of the most important areas of physics to summarise their global assessment of some of the most important issues. In lieu of an extremely long abstract summarising the contents, we invite the reader to look at the section headings and their authors, and then to indulge in a feast of stimulating topics spanning the current frontiers of fundamental physics from The Future of Physics by William D Phillips and What characterises topological effects in physics? by Gerard t Hooft through the contributions of the widest imaginable range of world leaders in their respective areas. This paper is presented as a preface to exciting developments by senior and young scientists in the years that lie ahead, and a complement to the less authoritative popular accounts by journalists., Comment: 54 pages, 13 figures

The purpose of this work is to improve the estimates for the $\Delta$-GenEO method from the paper "Overlapping Schwarz methods with GenEO coarse spaces for indefinite and nonself-adjoint problems" by N. Bootland, V. Dolean, I. G Graham, C. Ma, R. Scheichl (https://doi.org/10.1093/imanum/drac036) when applied to the indefinite Helmholtz equation. We derive k-dependent estimates of quantities of interest ensuring the robustness of the method.

In this report, we introduce the Gemini 1.5 family of models, representing the next generation of highly compute-efficient multimodal models capable of recalling and reasoning over fine-grained information from millions of tokens of context, including multiple long documents and hours of video and audio. The family includes two new models: (1) an updated Gemini 1.5 Pro, which exceeds the February version on the great majority of capabilities and benchmarks; (2) Gemini 1.5 Flash, a more lightweight variant designed for efficiency with minimal regression in quality. Gemini 1.5 models achieve near-perfect recall on long-context retrieval tasks across modalities, improve the state-of-the-art in long-document QA, long-video QA and long-context ASR, and match or surpass Gemini 1.0 Ultra's state-of-the-art performance across a broad set of benchmarks. Studying the limits of Gemini 1.5's long-context ability, we find continued improvement in next-token prediction and near-perfect retrieval (>99%) up to at least 10M tokens, a generational leap over existing models such as Claude 3.0 (200k) and GPT-4 Turbo (128k). Finally, we highlight real-world use cases, such as Gemini 1.5 collaborating with professionals on completing their tasks achieving 26 to 75% time savings across 10 different job categories, as well as surprising new capabilities of large language models at the frontier; when given a grammar manual for Kalamang, a language with fewer than 200 speakers worldwide, the model learns to translate English to Kalamang at a similar level to a person who learned from the same content.

Accurate blind docking has the potential to lead to new biological breakthroughs, but for this promise to be realized, docking methods must generalize well across the proteome. Existing benchmarks, however, fail to rigorously assess generalizability. Therefore, we develop DockGen, a new benchmark based on the ligand-binding domains of proteins, and we show that existing machine learning-based docking models have very weak generalization abilities. We carefully analyze the scaling laws of ML-based docking and show that, by scaling data and model size, as well as integrating synthetic data strategies, we are able to significantly increase the generalization capacity and set new state-of-the-art performance across benchmarks. Further, we propose Confidence Bootstrapping, a new training paradigm that solely relies on the interaction between diffusion and confidence models and exploits the multi-resolution generation process of diffusion models. We demonstrate that Confidence Bootstrapping significantly improves the ability of ML-based docking methods to dock to unseen protein classes, edging closer to accurate and generalizable blind docking methods.

We study the coherence of two coupled spin qubits in the presence of a bath of nuclear spins simulated using generalized cluster correlation expansion (gCCE) method. In our model, two electron spin qubits coupled with isotropic exchange or magnetic dipolar interactions interact with an environment of random nuclear spins. We study the time-evolution of the two-qubit reduced density matrix (RDM) and resulting decay of the off diagonal elements, corresponding to decoherence, which allows us to calculate gate fidelity in the regime of pure dephasing. We contrast decoherence when the system undergoes free evolution and evolution with dynamical decoupling pulses applied. Moreover, we study the dependence of decoherence on external magnetic field and system parameters which mimic realistic spin qubits, emphasizing magnetic molecules. Lastly, we comment on the application and limitations of gCCE in simulating nuclear-spin induced two-qubit relaxation processes.

Recently, Malle and Navarro obtained a Galois strengthening of Brauer's height zero conjecture for principal $p$-blocks when $p=2$, considering a particular Galois automorphism of order~$2$. In this paper, for any prime $p$ we consider a certain elementary abelian $p$-subgroup of the absolute Galois group and propose a Galois version of Brauer's height zero conjecture for principal $p$-blocks. We prove it when $p=2$ and also for arbitrary $p$ when $G$ does not involve certain groups of Lie type of small rank as composition factors. Furthermore, we prove it for almost simple groups and for $p$-solvable groups., Comment: a few minor improvements over version 1

Two natural ways of modelling Formula 1 race outcomes are a probabilistic approach, based on the exponential distribution, and econometric modelling of the ranks. Both approaches lead to exactly soluble race-winning probabilities. Equating race-winning probabilities leads to a set of equivalent parametrisations. This time-rank duality is attractive theoretically and leads to quicker ways of dis-entangling driver and car level effects., Comment: 9 pages

This article presents a quantum computing approach to designing of similarity measures and kernels for classification of stochastic symbolic time series. In the area of machine learning, kernels are important components of various similarity-based classification, clustering, and regression algorithms. An effective strategy for devising problem-specific kernels is leveraging existing generative models of the example space. In this study we assume that a quantum generative model, known as quantum hidden Markov model (QHMM), describes the underlying distributions of the examples. The sequence structure and probability are determined by transitions within model's density operator space. Consequently, the QHMM defines a mapping from the example space into the broader quantum space of density operators. Sequence similarity is evaluated using divergence measures such as trace and Bures distances between quantum states. We conducted extensive simulations to explore the relationship between the distribution of kernel-estimated similarity and the dimensionality of the QHMMs Hilbert space. As anticipated, a higher dimension of the Hilbert space corresponds to greater sequence distances and a more distinct separation of the examples. To empirically evaluate the performance of the kernels, we defined classification tasks based on a simplified generative model of directional price movement in the stock market. We implemented two widely-used kernel-based algorithms - support vector machines and k-nearest neighbors - using both classical and quantum kernels. Across all classification task scenarios, the quantum kernels consistently demonstrated superior performance compared to their classical counterparts., Comment: 25 pages, 18 figures

A common approach to constructing a Synthetic Control unit is to fit on the outcome variable and covariates in pre-treatment time periods, but it has been shown by Ferman and Pinto (2019) that this approach does not provide asymptotic unbiasedness when the fit is imperfect and the number of controls is fixed. Many related panel methods have a similar limitation when the number of units is fixed. I introduce and evaluate a new method in which the Synthetic Control is constructed using a General Method of Moments approach where units not being included in the Synthetic Control are used as instruments. I show that a Synthetic Control Estimator of this form will be asymptotically unbiased as the number of pre-treatment time periods goes to infinity, even when pre-treatment fit is imperfect and the number of units is fixed. Furthermore, if both the number of pre-treatment and post-treatment time periods go to infinity, then averages of treatment effects can be consistently estimated. I conduct simulations and an empirical application to compare the performance of this method with existing approaches in the literature.

We investigate the optimization target of Contrast-Consistent Search (CCS), which aims to recover the internal representations of truth of a large language model. We present a new loss function that we call the Midpoint-Displacement (MD) loss function. We demonstrate that for a certain hyper-parameter value this MD loss function leads to a prober with very similar weights to CCS. We further show that this hyper-parameter is not optimal and that with a better hyper-parameter the MD loss function attains a higher test accuracy than CCS., Comment: Socially Responsible Language Modelling Research (SoLaR) NeurIPS 2023

JWST NIRSpec IFU (2.9-5.3 mu) and MIRI MRS (4.9-28.5 mu) observations were performed on both the leading and trailing hemispheres of Ganymede with a spectral resolution of ~2700. Reflectance spectra show signatures of water ice, CO2 and H2O2. An absorption feature at 5.9 mu is revealed and is tentatively assigned to sulfuric acid hydrates. The CO2 4.26-mu band shows latitudinal and longitudinal variations in depth, shape and position over the two hemispheres, unveiling different CO2 physical states. In the ice-rich polar regions, which are the most exposed to Jupiter's plasma irradiation, the CO2 band is redshifted with respect to other terrains. In the leading northern polar cap, the CO2 band is dominated by a high wavelength component at ~4.27 mu, consistent with CO2 trapped in amorphous water ice. At equatorial latitudes (and especially on dark terrains) the observed band is broader and shifted towards the blue, suggesting CO2 adsorbed on non-icy materials. Amorphous ice is detected in the ice-rich polar regions, and is especially abundant on the leading northern polar cap. In both hemispheres the north polar cap ice appears to be more processed than the south polar cap. A longitudinal modification of the H2O ice molecular structure and/or nano/micrometre-scale texture, of diurnal or geographic origin, is observed in both hemispheres. Ice frost is observed on the morning limb of the trailing hemisphere, possibly formed during the night from the recondensation of water subliming from the warmer subsurface. Reflectance spectra of the dark terrains are compatible with the presence of Na-/Mg-sulfate salts, sulfuric acid hydrates, and possibly phyllosilicates mixed with fine-grained opaque minerals, having an highly porous texture. Mid-IR brightness temperatures indicate a rough surface and a very low thermal inertia of 20-40 J m-2 s-0.5 K-1, consistent with a porous surface., Comment: 35 pages, 34 figures

Many modern datasets, such as those in ecology and geology, are composed of samples with spatial structure and dependence. With such data violating the usual independent and identically distributed (IID) assumption in machine learning and classical statistics, it is unclear a priori how one should measure the performance and generalization of models. Several authors have empirically investigated cross-validation (CV) methods in this setting, reaching mixed conclusions. We provide a class of unbiased estimation methods for general quadratic errors, correlated Gaussian response, and arbitrary prediction function $g$, for a noise-elevated version of the error. Our approach generalizes the coupled bootstrap (CB) from the normal means problem to general normal data, allowing correlation both within and between the training and test sets. CB relies on creating bootstrap samples that are intelligently decoupled, in the sense of being statistically independent. Specifically, the key to CB lies in generating two independent "views" of our data and using them as stand-ins for the usual independent training and test samples. Beginning with Mallows' $C_p$, we generalize the estimator to develop our generalized $C_p$ estimators (GC). We show at under only a moment condition on $g$, this noise-elevated error estimate converges smoothly to the noiseless error estimate. We show that when Stein's unbiased risk estimator (SURE) applies, GC converges to SURE as in the normal means problem. Further, we use these same tools to analyze CV and provide some theoretical analysis to help understand when CV will provide good estimates of error. Simulations align with our theoretical results, demonstrating the effectiveness of GC and illustrating the behavior of CV methods. Lastly, we apply our estimator to a model selection task on geothermal data in Nevada., Comment: 28 pages, 13 figures

We show that the Jordan decomposition of characters of finite reductive groups can be chosen so that if the centralizer of the relevant semisimple element is connected, then the map is Galois-equivariant. Further, in this situation, we show that there is a unique Jordan decomposition satisfying conditions analogous to those of Digne--Michel's unique Jordan decomposition in the connected center case.