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Let $\Delta_{k}(x)$ be the error term in the classical asymptotic formula for the sum $\sum_{n\leq x}d_{k}(n)$, where $d_{k}(n)$ is the number of ways $n$ can be written as a product of $k$ factors. We study the analytic properties of the Dirichlet series $\sum_{n=1}^{\infty}\Delta_{k}(n)n^{-s}$ and use Perron's formula to estimate the sums $\sum_{n\leq x}\Delta_{3}(n)$ and $\sum_{n\leq x}\Delta_{4}(n)$ for large $x>0$.

While autoregressive machine-learning-based emulators have been trained to produce stable and accurate rollouts in the climate of the present-day and recent past, none so far have been trained to emulate the sensitivity of climate to substantial changes in CO$_2$ or other greenhouse gases. As an initial step we couple the Ai2 Climate Emulator version 2 to a slab ocean model (hereafter ACE2-SOM) and train it on output from a collection of equilibrium-climate physics-based reference simulations with varying levels of CO$_2$. We test it in equilibrium and non-equilibrium climate scenarios with CO$_2$ concentrations seen and unseen in training. ACE2-SOM performs well in equilibrium-climate inference with both in-sample and out-of-sample CO$_2$ concentrations, accurately reproducing the emergent time-mean spatial patterns of surface temperature and precipitation change with CO$_2$ doubling, tripling, or quadrupling. In addition, the vertical profile of atmospheric warming and change in extreme precipitation rates up to the 99.9999th percentile closely agree with the reference model. Non-equilibrium-climate inference is more challenging. With CO$_2$ increasing gradually at a rate of 2% year$^{-1}$, ACE2-SOM can accurately emulate the global annual mean trends of surface and lower-to-middle atmosphere fields but produces unphysical jumps in stratospheric fields. With an abrupt quadrupling of CO$_2$, ML-controlled fields transition unrealistically quickly to the 4xCO$_2$ regime. In doing so they violate global energy conservation and exhibit unphysical sensitivities of and surface and top of atmosphere radiative fluxes to instantaneous changes in CO$_2$. Future emulator development needed to address these issues should improve its generalizability to diverse climate change scenarios., Comment: 31 pages, 13 figures

Existing machine learning models of weather variability are not formulated to enable assessment of their response to varying external boundary conditions such as sea surface temperature and greenhouse gases. Here we present ACE2 (Ai2 Climate Emulator version 2) and its application to reproducing atmospheric variability over the past 80 years on timescales from days to decades. ACE2 is a 450M-parameter autoregressive machine learning emulator, operating with 6-hour temporal resolution, 1{\deg} horizontal resolution and eight vertical layers. It exactly conserves global dry air mass and moisture and can be stepped forward stably for arbitrarily many steps with a throughput of about 1500 simulated years per wall clock day. ACE2 generates emergent phenomena such as tropical cyclones, the Madden Julian Oscillation, and sudden stratospheric warmings. Furthermore, it accurately reproduces the atmospheric response to El Ni\~no variability and global trends of temperature over the past 80 years. However, its sensitivities to separately changing sea surface temperature and carbon dioxide are not entirely realistic., Comment: 31 pages, 23 figures

Foundational verification considers the functional correctness of programming languages with formalized semantics and uses proof assistants (e.g., Coq, Isabelle) to certify proofs. The need for verifying complex programs compels it to involve expressive Separation Logics (SLs) that exceed the scopes of well-studied automated proof theories, e.g., symbolic heap. Consequently, automation of SL in foundational verification relies heavily on ad-hoc heuristics that lack a systematic meta-theory and face scalability issues. To mitigate the gap, we propose a theory to specify SL predicates using abstract algebras including functors, homomorphisms, and modules over rings. Based on this theory, we develop a generic SL automation algorithm to reason about any data structures that can be characterized by these algebras. In addition, we also present algorithms for automatically instantiating the algebraic models to real data structures. The instantiation reuses the algebraic models of component structures and preserves their data abstractions. Case studies on formalized imperative semantics show our algorithm can instantiate the algebraic models automatically for a variety of complex data structures. Experimental results indicate the automatically instantiated reasoners from our generic theory show similar results to the state-of-the-art systems made of specifically crafted reasoning rules. The presented theories, proofs, and the verification framework are formalized in Isabelle/HOL., Comment: Accepted by POPL'25

Large language models (LLMs) excel in tasks requiring processing and interpretation of input text. Abstract screening is a labour-intensive component of systematic review involving repetitive application of inclusion and exclusion criteria on a large volume of studies identified by a literature search. Here, LLMs (GPT-3.5 Turbo, GPT-4 Turbo, GPT-4o, Llama 3 70B, Gemini 1.5 Pro, and Claude Sonnet 3.5) were trialled on systematic reviews in a full issue of the Cochrane Library to evaluate their accuracy in zero-shot binary classification for abstract screening. Trials over a subset of 800 records identified optimal prompting strategies and demonstrated superior performance of LLMs to human researchers in terms of sensitivity (LLM-max = 1.000, human-max = 0.775), precision (LLM-max = 0.927, human-max = 0.911), and balanced accuracy (LLM-max = 0.904, human-max = 0.865). The best performing LLM-prompt combinations were trialled across every replicated search result (n = 119,691), and exhibited consistent sensitivity (range 0.756-1.000) but diminished precision (range 0.004-0.096). 66 LLM-human and LLM-LLM ensembles exhibited perfect sensitivity with a maximal precision of 0.458, with less observed performance drop in larger trials. Significant variation in performance was observed between reviews, highlighting the importance of domain-specific validation before deployment. LLMs may reduce the human labour cost of systematic review with maintained or improved accuracy and sensitivity. Systematic review is the foundation of evidence synthesis across academic disciplines, including evidence-based medicine, and LLMs may increase the efficiency and quality of this mode of research., Comment: RS and AJT are joint-first authors

Interface-type resistive switching (RS) devices with lower operation current and more reliable switching repeatability exhibits great potential in the applications for data storage devices and ultra-low-energy computing. However, the working mechanism of such interface-type RS devices are much less studied compared to that of the filament-type devices, which hinders the design and application of the novel interface-type devices. In this work, we fabricate a metal/TiOx/TiN/Si (001) thin film memristor by using a one-step pulsed laser deposition. In situ transmission electron microscopy (TEM) imaging and current-voltage (I-V) characteristic demonstrate that the device is switched between high resistive state (HRS) and low resistive state (LRS) in a bipolar fashion with sweeping the applied positive and negative voltages. In situ scanning transmission electron microscopy (STEM) experiments with electron energy loss spectroscopy (EELS) reveal that the charged defects (such as oxygen vacancies) can migrate along the intrinsic grain boundaries of TiOx insulating phase under electric field without forming obvious conductive filaments, resulting in the modulation of Schottky barriers at the metal/semiconductor interfaces. The fundamental insights gained from this study presents a novel perspective on RS processes and opens up new technological opportunities for fabricating ultra-low-energy nitride-based memristive devices.

This paper develops a model of in-kind redistribution where consumers participate in either a private market or a government-designed program, but not both. We characterize when a social planner, seeking to maximize weighted total surplus, can strictly improve upon the laissez-faire outcome. We show that the optimal mechanism consists of three components: a public option, nonlinear subsidies, and laissez-faire consumption. We quantify the resulting distortions and relate them to the correlation between consumer demand and welfare weights. Our findings reveal that while private market access constrains the social planner's ability to redistribute, it also strengthens the rationale for non-market allocations.

Image classification usually requires connectivity and access to the cloud which is often limited in many parts of the world, including hard to reach rural areas. TinyML aims to solve this problem by hosting AI assistants on constrained devices, eliminating connectivity issues by processing data within the device itself, without internet or cloud access. This pilot study explores the use of tinyML to provide healthcare support with low spec devices in low connectivity environments, focusing on diagnosis of skin diseases and the ethical use of AI assistants in a healthcare setting. To investigate this, 10,000 images of skin lesions were used to train a model for classifying visually detectable diseases (VDDs). The model weights were then offloaded to a Raspberry Pi with a webcam attached, to be used for the classification of skin lesions without internet access. It was found that the developed prototype achieved a test accuracy of 78% and a test loss of 1.08., Comment: 6 pages, 5 figures

Considering digital ink as plane curves provides a valuable framework for various applications, including signature verification, note-taking, and mathematical handwriting recognition. These plane curves can be obtained as parameterized pairs of approximating truncated series (x(s), y(s)) determined by sampled points. Earlier work has found that representing these truncated series (polynomials) in a Legendre or Legendre-Sobolev basis has a number of desirable properties. These include compact data representation, meaningful clustering of like symbols in the vector space of polynomial coefficients, linear separability of classes in this space, and highly efficient calculation of variation between curves. In this work, we take a first step at examining the use of Chebyshev-Sobolev series for symbol recognition. The early indication is that this representation may be superior to Legendre-Sobolev representation for some purposes., Comment: Accepted at MathUI 2024

A Maple implementation of partitioned matrices is described. A recursive block data structure is used, with all operations preserving the block abstraction. These include constructor functions, ring operations such as addition and product, and inversion. The package is demonstrated by calculating the PLU factorization of a block matrix.

We report on a direct search for elastic photon-photon scattering using x-ray and $\gamma$ photons from a laser-plasma based experiment. A gamma photon beam produced by a laser wakefield accelerator provided a broadband gamma spectrum extending to above $E_\gamma = 200$ MeV. These were collided with a dense x-ray field produced by the emission from a laser heated germanium foil at $E_x \approx 1.4$ keV, corresponding to an invariant mass of $\sqrt{s} = 1.22 \pm 0.22$ MeV. In these asymmetric collisions elastic scattering removes one x-ray and one high-energy $\gamma$ photon and outputs two lower energy $\gamma$ photons. No changes in the $\gamma$ photon spectrum were observed as a result of the collisions allowing us to place a 95% upper bound on the cross section of $1.5 \times 10^{15}\,\mu$b. Although far from the QED prediction, this represents the lowest upper limit obtained so far for $\sqrt{s} \lesssim 1$ MeV., Comment: 6 pages, 10 figures

Particle dynamics in Earth's outer radiation belt can be modelled using a diffusion framework, where large-scale electron movements are captured by a diffusion equation across a single adiabatic invariant, $L^{*}$ $``(L)"$. While ensemble models are promoted to represent physical uncertainty, as yet there is no validated method to analyse radiation belt ensembles. Comparisons are complicated by the domain dependent diffusion, since diffusion coefficient $D_{LL}$ is dependent on $L$. We derive two tools to analyse ensemble members: time to monotonicity $t_m$ and mass/energy moment quantities $\mathcal{N}, \mathcal{E}$. We find that the Jacobian ($1/L^2$) is necessary for radiation belt error metrics. Components of $\partial\mathcal{E}/\partial t$ are explicitly calculated to compare the effects of outer and inner boundary conditions, and loss, on the ongoing diffusion. Using $t_m$, $\mathcal{N}$ and $\mathcal{E}$, we find that: (a) different physically motivated choices of outer boundary condition and location result in different final states and different rates of evolution; (b) the gradients of the particle distribution affect evolution more significantly than $D_{LL}$; (c) the enhancement location, and the amount of initial background particles, are both significant factors determining system evolution; (d) loss from pitch-angle scattering is generally dominant; it mitigates but does not remove the influence of both initial conditions and outer boundary settings, which are due to the $L$-dependence of $D_{LL}$. We anticipate this study will promote renewed focus on the distribution gradients, on the location and nature of the outer boundary in radiation belt modelling, and provide a foundation for systematic ensemble modelling.

Sometimes only some digits of a numerical product or some terms of a polynomial or series product are required. Frequently these constitute the most significant or least significant part of the value, for example when computing initial values or refinement steps in iterative approximation schemes. Other situations require the middle portion. In this paper we provide algorithms for the general problem of computing a given span of coefficients within a product, that is the terms within a range of degrees for univariate polynomials or range digits of an integer. This generalizes the "middle product" concept of Hanrot, Quercia and Zimmerman. We are primarily interested in problems of modest size where constant speed up factors can improve overall system performance, and therefore focus the discussion on classical and Karatsuba multiplication and how methods may be combined., Comment: To appear in Computer Algebra in Scientific Computing (CASC 2024)

We study certain linear algebra algorithms for recursive block matrices. This representation has useful practical and theoretical properties. We summarize some previous results for block matrix inversion and present some results on triangular decomposition of block matrices. The case of inverting matrices over a ring that is neither formally real nor formally complex was inspired by Gonzalez-Vega et al., Comment: To appear in "LALO 60: Matrices and Polynomials in Computer Algebra"

Data-driven deep learning models are transforming global weather forecasting. It is an open question if this success can extend to climate modeling, where the complexity of the data and long inference rollouts pose significant challenges. Here, we present the first conditional generative model that produces accurate and physically consistent global climate ensemble simulations by emulating a coarse version of the United States' primary operational global forecast model, FV3GFS. Our model integrates the dynamics-informed diffusion framework (DYffusion) with the Spherical Fourier Neural Operator (SFNO) architecture, enabling stable 100-year simulations at 6-hourly timesteps while maintaining low computational overhead compared to single-step deterministic baselines. The model achieves near gold-standard performance for climate model emulation, outperforming existing approaches and demonstrating promising ensemble skill. This work represents a significant advance towards efficient, data-driven climate simulations that can enhance our understanding of the climate system and inform adaptation strategies., Comment: NeurIPS 2024; Code is available at https://github.com/Rose-STL-Lab/spherical-dyffusion

Symbolic mathematical computing systems have served as a canary in the coal mine of software systems for more than sixty years. They have introduced or have been early adopters of programming language ideas such ideas as dynamic memory management, arbitrary precision arithmetic and dependent types. These systems have the feature of being highly complex while at the same time operating in a domain where results are well-defined and clearly verifiable. These software systems span multiple layers of abstraction with concerns ranging from instruction scheduling and cache pressure up to algorithmic complexity of constructions in algebraic geometry. All of the major symbolic mathematical computing systems include low-level code for arithmetic, memory management and other primitives, a compiler or interpreter for a bespoke programming language, a library of high level mathematical algorithms, and some form of user interface. Each of these parts invokes multiple deep issues. We present some lessons learned from this environment and free flowing opinions on topics including: * Portability of software across architectures and decades; * Infrastructure to embrace and infrastructure to avoid; * Choosing base abstractions upon which to build; * How to get the most out of a small code base; * How developments in compilers both to optimise and to validate code have always been and remain of critical importance, with plenty of remaining challenges; * The way in which individuals including in particular Alan Mycroft who has been able to span from hand-crafting Z80 machine code up to the most abstruse high level code analysis techniques are needed, and * Why it is important to teach full-stack thinking to the next generation.

In this article we report on an initial exploration to assess the viability of using the general large language models (LLMs), recently made public, to classify mathematical documents. Automated classification would be useful from the applied perspective of improving the navigation of the literature and the more open-ended goal of identifying relations among mathematical results. The Mathematical Subject Classification MSC 2020, from MathSciNet and zbMATH, is widely used and there is a significant corpus of ground truth material in the open literature. We have evaluated the classification of preprint articles from arXiv.org according to MSC 2020. The experiment used only the title and abstract alone -- not the entire paper. Since this was early in the use of chatbots and the development of their APIs, we report here on what was carried out by hand. Of course, the automation of the process will have to follow if it is to be generally useful. We found that in about 60% of our sample the LLM produced a primary classification matching that already reported on arXiv. In about half of those instances, there were additional primary classifications that were not detected. In about 40% of our sample, the LLM suggested a different classification than what was provided. A detailed examination of these cases, however, showed that the LLM-suggested classifications were in most cases better than those provided.

This paper explores practical aspects of using a high-level functional language for GPU-based arithmetic on ``midsize'' integers. By this we mean integers of up to about a quarter million bits, which is sufficient for most practical purposes. The goal is to understand whether it is possible to support efficient nested-parallel programs with a small, flexible code base. We report on GPU implementations for addition and multiplication of integers that fit in one CUDA block, thus leveraging temporal reuse from scratchpad memories. Our key contribution resides in the simplicity of the proposed solutions: We recognize that addition is a straightforward application of scan, which is known to allow efficient GPU implementation. For quadratic multiplication we employ a simple work-partitioning strategy that offers good temporal locality. For FFT multiplication, we efficiently map the computation in the domain of integral fields by finding ``good'' primes that enable almost-full utilization of machine words. In comparison, related work uses complex tiling strategies -- which feel too big a hammer for the job -- or uses the computational domain of reals, which may degrade the magnitude of the base in which the computation is carried. We evaluate the performance in comparison to the state-of-the-art CGBN library, authored by NvidiaLab, and report that our CUDA prototype outperforms CGBN for integer sizes higher than 32K bits, while offering comparable performance for smaller sizes. Moreover, we are, to our knowledge, the first to report that FFT multiplication outperforms the classical one on the larger sizes that still fit in a CUDA block. Finally, we examine Futhark's strengths and weaknesses for efficiently supporting such computations and find out that a compiler pass aimed at efficient sequentialization of excess parallelism would significantly improve performance.

Large Language Models (LLMs) are promising analytical tools. They can augment human epistemic, cognitive and reasoning abilities, and support 'sensemaking', making sense of a complex environment or subject by analysing large volumes of data with a sensitivity to context and nuance absent in earlier text processing systems. This paper presents a pilot experiment that explores how LLMs can support thematic analysis of controversial topics. We compare how human researchers and two LLMs GPT-4 and Llama 2 categorise excerpts from media coverage of the controversial Australian Robodebt scandal. Our findings highlight intriguing overlaps and variances in thematic categorisation between human and machine agents, and suggest where LLMs can be effective in supporting forms of discourse and thematic analysis. We argue LLMs should be used to augment, and not replace human interpretation, and we add further methodological insights and reflections to existing research on the application of automation to qualitative research methods. We also introduce a novel card-based design toolkit, for both researchers and practitioners to further interrogate LLMs as analytical tools., Comment: 18 pages, 6 figures

Downscaling, or super-resolution, provides decision-makers with detailed, high-resolution information about the potential risks and impacts of climate change, based on climate model output. Machine learning algorithms are proving themselves to be efficient and accurate approaches to downscaling. Here, we show how a generative, diffusion-based approach to downscaling gives accurate downscaled results. We focus on an idealised setting where we recover ERA5 at $0.25\degree$~resolution from coarse grained version at $2\degree$~resolution. The diffusion-based method provides superior accuracy compared to a standard U-Net, particularly at the fine scales, as highlighted by a spectral decomposition. Additionally, the generative approach provides users with a probability distribution which can be used for risk assessment. This research highlights the potential of diffusion-based downscaling techniques in providing reliable and detailed climate predictions.

Generative Large Language Models enable efficient analytics across knowledge domains, rivalling human experts in information comparisons. However, the applications of LLMs for information comparisons face scalability challenges due to the difficulties in maintaining information across large contexts and overcoming model token limitations. To address these challenges, we developed the novel Abstractive Summarization & Criteria-driven Comparison Endpoint (ASC$^2$End) system to automate information comparison at scale. Our system employs Semantic Text Similarity comparisons for generating evidence-supported analyses. We utilize proven data-handling strategies such as abstractive summarization and retrieval augmented generation to overcome token limitations and retain relevant information during model inference. Prompts were designed using zero-shot strategies to contextualize information for improved model reasoning. We evaluated abstractive summarization using ROUGE scoring and assessed the generated comparison quality using survey responses. Models evaluated on the ASC$^2$End system show desirable results providing insights on the expected performance of the system. ASC$^2$End is a novel system and tool that enables accurate, automated information comparison at scale across knowledge domains, overcoming limitations in context length and retrieval., Comment: 11 pages, 7 figures, 5 tables

The link between in vitro hERG ion channel inhibition and subsequent in vivo QT interval prolongation, a critical risk factor for the development of arrythmias such as Torsade de Pointes, is so well established that in vitro hERG activity alone is often sufficient to end the development of an otherwise promising drug candidate. It is therefore of tremendous interest to develop advanced methods for identifying hERG-active compounds in the early stages of drug development, as well as for proposing redesigned compounds with reduced hERG liability and preserved on-target potency. In this work, we present CardioGenAI, a machine learning-based framework for re-engineering both developmental and commercially available drugs for reduced hERG activity while preserving their pharmacological activity. The framework incorporates novel state-of-the-art discriminative models for predicting hERG channel activity, as well as activity against the voltage-gated NaV1.5 and CaV1.2 channels due to their potential implications in modulating the arrhythmogenic potential induced by hERG channel blockade. We applied the complete framework to pimozide, an FDA-approved antipsychotic agent that demonstrates high affinity to the hERG channel, and generated 100 refined candidates. Remarkably, among the candidates is fluspirilene, a compound which is of the same class of drugs (diphenylmethanes) as pimozide and therefore has similar pharmacological activity, yet exhibits over 700-fold weaker binding to hERG. We envision that this method can effectively be applied to developmental compounds exhibiting hERG liabilities to provide a means of rescuing drug development programs that have stalled due to hERG-related safety concerns. We have made all of our software open-source to facilitate integration of the CardioGenAI framework for molecular hypothesis generation into drug discovery workflows.

Extreme debris disks can show short term behaviour through the evolution and clearing of small grains produced in giant impacts, and potentially a longer period of variability caused by a planetesimal population formed from giant impact ejecta. In this paper, we present results of numerical simulations to explain how a planetesimal populated disk can supply an observed extreme debris disk with small grains. We simulated a sample of giant impacts from which we form a planetesimal population. We then use the $N$-body code {\sc Rebound} to evolve the planetesimals spatially and collisionally. We adopt a simplistic collision criteria in which we define destructive collisions to be between planetesimals with a mutual impact velocity that exceeds two times the catastrophic disruption threshold, $V^*$. We find that for some configurations, a planetesimal populated disk can produce a substantial amount of dust to sustain an observable disk. The semi-major axis at which the giant impact occurs changes the mass added to the observed disk substantially while the orientation of the impact has less of an effect. We determine how the collision rate at the collision point changes over time and show that changes in semi-major axis and orientation only change the initial collision rate of the disk. Collision rates across all disks evolve at a similar rate., Comment: 20 pages, 21 figures, 3 tables, accepted by MNRAS

WebAssembly (Wasm) is a low-level bytecode language and virtual machine, intended as a compilation target for a wide range of programming languages, which is seeing increasing adoption across diverse ecosystems. As a young technology, Wasm continues to evolve -- it reached version 2.0 last year and another major update is expected soon. For a new feature to be standardised in Wasm, four key artefacts must be presented: a formal (mathematical) specification of the feature, an accompanying prose pseudocode description, an implementation in the official reference interpreter, and a suite of unit tests. This rigorous process helps to avoid errors in the design and implementation of new Wasm features, and Wasm's distinctive formal specification in particular has facilitated machine-checked proofs of various correctness properties for the language. However, manually crafting all of these artefacts requires expert knowledge combined with repetitive and tedious labor, which is a burden on the language's standardization process and authoring of the specification. This paper presents Wasm SpecTec, a technology to express the formal specification of Wasm through a domain-specific language. This DSL allows all of Wasm's currently handwritten specification artefacts to be error-checked and generated automatically from a single source of truth, and is designed to be easy to write, read, compare, and review. We believe that Wasm SpecTec's automation and meta-level error checking will significantly ease the current burden of the language's specification authors. We demonstrate the current capabilities of Wasm SpecTec by showcasing its proficiency in generating various artefacts, and describe our work towards replacing the manually written official Wasm specification document with specifications generated by Wasm SpecTec., Comment: 5 pages, 7 figures