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A subset $S$ of real numbers is called bi-Sidon if it is a Sidon set with respect to both addition and multiplication, i.e., if all pairwise sums and all pairwise products of elements of $S$ are distinct. Imre Ruzsa asked the following question: What is the maximum number $f(N)$ such that every set $S$ of $N$ real numbers contains a bi-Sidon subset of size at least $f(N)$? He proved that $f(N)\geq cN^{\frac13}$, for a constant $c>0$. In this note, we improve this bound to $N^{\frac13+\frac7{78}+o(1)}$., Comment: 9 pages, updated literature and exposition

Over the last decades, theoretical photochemistry has produced multiple techniques to simulate the nonadiabatic dynamics of molecules. Surprisingly, much less effort has been devoted to adequately describing the first step of a photochemical or photophysical process: photoexcitation. Here, we propose a formalism to include the effect of a laser pulse in trajectory-based nonadiabatic dynamics at the level of the initial conditions, with no additional cost. The promoted density approach (PDA) decouples the excitation from the nonadiabatic dynamics by defining a new set of initial conditions, which include an excitation time. PDA with surface hopping leads to nonadiabatic dynamics simulations in excellent agreement with quantum dynamics using an explicit laser pulse and highlights the strong impact of a laser pulse on the resulting photodynamics and the limits of the (sudden) vertical excitation. Combining PDA with trajectory-based nonadiabatic methods is possible for any arbitrary-sized molecules using a code provided in this work., Comment: SI included in the pdf

This paper proposes a control technique for autonomous RC car racing. The presented method does not require any map-building phase beforehand since it operates only local path planning on the actual LiDAR point cloud. Racing control algorithms must have the capability to be optimized to the actual track layout for minimization of lap time. In the examined one, it is guaranteed with the improvement of the Stanley controller with additive control components to stabilize the movement in both low and high-speed ranges, and with the integration of an adaptive lookahead point to induce sharp and dynamic cornering for traveled distance reduction. The developed method is tested on a 1/10-sized RC car, and the tuning procedure from a base solution to the optimal setting in a real F1Tenth race is presented. Furthermore, the proposed method is evaluated with a comparison to a more simple reactive method, and in parallel to a more complex optimization-based technique that involves offline map building the global optimal trajectory calculation. The performance of the proposed method compared to the latter, referring to the lap time, is that the proposed one has only 8% lower average speed. This demonstrates that with appropriate tuning, a local planning-based method can be comparable with a more complex optimization-based one. Thus, the performance gap is lower than 10% from the state-of-the-art method. Moreover, the proposed technique has significantly higher similarity to real scenarios, therefore the results can be interesting in the context of automotive industry.

We investigate the effect of focused-ion-beam (FIB) irradiation on spin waves with sub-micron wavelengths in Yttrium-Iron-Garnet (YIG) films. Time-resolved scanning transmission X-ray (TR-STXM) microscopy was used to image the spin waves in irradiated regions and deduce corresponding changes in the magnetic parameters of the film. We find that the changes of Ga$^+$ irradiation can be understood by assuming a few percent change in the effective magnetization $M_\mathrm{eff}$ of the film due to a trade-off between changes in anisotropy and effective film thickness. Our results demonstrate that FIB irradiation can be used to locally alter the dispersion relation and the effective refractive index $n_\textrm{eff}$ of the film, even for submicron wavelengths. To achieve the same change in $n_\textrm{eff}$ for shorter wavelengths, a higher dose is required, but no significant deterioration of spin wave propagation length in the irradiated regions was observed, even at the highest applied doses., Comment: 5 pages, 6 figures

Quantum computing has tremendous potential to overcome some of the fundamental limitations present in classical information processing. Yet, today's technological limitations in the quality and scaling prevent exploiting its full potential. Quantum computing based on superconducting quantum processing units (QPUs) is among the most promising approaches towards practical quantum advantage. In this article the basic technological approach of IQM Quantum Computers is described covering both the QPU and the rest of the full-stack quantum computer. In particular, the focus is on a 20-qubit quantum computer featuring the Garnet QPU and its architecture, which we will scale up to 150 qubits. We also present QPU and system-level benchmarks, including a median 2-qubit gate fidelity of 99.5% and genuinely entangling all 20 qubits in a Greenberger-Horne-Zeilinger (GHZ) state.

Nodal loop semimetals are topological materials with drumhead surface states characterized by reduced kinetic energy. If the Fermi energy of such a system is near these nondispersive states interaction among charge carriers substantially impacts their electronic structure. The emergence of magnetism in these surface states is one of the possible consequences. Ca3P2 an already synthesized material possesses a remarkably large nodal loop which is situated exactly at the Fermi level of the bulk system. In the present work, we investigate how surface magnetism is impacted by surface reconstruction and lattice termination in finite slabs in this material. We show that a slight deviation from the stoichiometric occupation of Ca sites results in the stabilization of magnetic patterns.

Logical qubits can be protected against environmental noise by encoding them into a highly entangled state of many physical qubits and actively intervening in the dynamics with stabilizer measurements. In this work, we numerically optimize the rate of these interventions: the number of stabilizer measurement rounds for a logical qubit encoded in a surface code patch and idling for a given time. We model the environmental noise on the circuit level, including gate errors, readout errors, amplitude and phase damping. We find, qualitatively, that the optimal number of stabilizer measurement rounds is getting smaller for better qubits and getting larger for better gates or larger code sizes. We discuss the implications of our results to some of the leading architectures, superconducting qubits, and neutral atoms.

We present a simple variant of the Gaussian mechanism for answering differentially private queries when the sensitivity space has a certain common structure. Our motivating problem is the fundamental task of answering $d$ counting queries under the add/remove neighboring relation. The standard Gaussian mechanism solves this task by adding noise distributed as a Gaussian with variance scaled by $d$ independently to each count. We show that adding a random variable distributed as a Gaussian with variance scaled by $(\sqrt{d} + 1)/4$ to all counts allows us to reduce the variance of the independent Gaussian noise samples to scale only with $(d + \sqrt{d})/4$. The total noise added to each counting query follows a Gaussian distribution with standard deviation scaled by $(\sqrt{d} + 1)/2$ rather than $\sqrt{d}$. The central idea of our mechanism is simple and the technique is flexible. We show that applying our technique to another problem gives similar improvements over the standard Gaussian mechanism.

Sparse autoencoders (SAEs) are an unsupervised method for learning a sparse decomposition of a neural network's latent representations into seemingly interpretable features. Despite recent excitement about their potential, research applications outside of industry are limited by the high cost of training a comprehensive suite of SAEs. In this work, we introduce Gemma Scope, an open suite of JumpReLU SAEs trained on all layers and sub-layers of Gemma 2 2B and 9B and select layers of Gemma 2 27B base models. We primarily train SAEs on the Gemma 2 pre-trained models, but additionally release SAEs trained on instruction-tuned Gemma 2 9B for comparison. We evaluate the quality of each SAE on standard metrics and release these results. We hope that by releasing these SAE weights, we can help make more ambitious safety and interpretability research easier for the community. Weights and a tutorial can be found at https://huggingface.co/google/gemma-scope and an interactive demo can be found at https://www.neuronpedia.org/gemma-scope, Comment: 12 main text pages, and 14 pages of acknowledgements, references and appendices

Assume that $R_1,R_2,\dots,R_t$ are disjoint parallel lines in the plane. A $t$-interval (or $t$-track interval) is a set that can be written as the union of $t$ closed intervals, each on a different line. It is known that pairwise intersecting $2$-intervals can be pierced by two points, one from each line. However, it is not true that every set of pairwise intersecting $3$-intervals can be pierced by three points, one from each line. For $k\ge 3$, Kaiser and Rabinovich asked whether $k$-wise intersecting $t$-intervals can be pierced by $t$ points, one from each line. Our main result provides an asymptotic answer: in any set $S_1,\dots,S_n$ of $k$-wise intersecting $t$-intervals, at least $\frac{k-1}{k+1}n$ can be pierced by $t$ points, one from each line. We prove this in a more general form, replacing intervals by subtrees of a tree. This leads to questions and results on covering vertices of edge-colored complete graphs by vertices of monochromatic cliques having distinct colors, where the colorings are chordal, or more generally induced $C_4$-free graphs. For instance, we show that if the edges of a complete graph $K_n$ are colored with red or blue so that both color classes are induced $C_4$-free, then at least ${4n\over 5}$ vertices can be covered by a red and a blue clique, and this is best possible. We conclude by pointing to new Ramsey-type problems emerging from these restricted colorings., Comment: 15 pages, 3 figures

Poverty map inference is a critical area of research, with growing interest in both traditional and modern techniques, ranging from regression models to convolutional neural networks applied to tabular data, images, and networks. Despite extensive focus on the validation of training phases, the scrutiny of final predictions remains limited. Here, we compare the Relative Wealth Index (RWI) inferred by Chi et al. (2022) with the International Wealth Index (IWI) inferred by Lee and Braithwaite (2022) and Esp\'in-Noboa et al. (2023) across six Sub-Saharan African countries. Our analysis focuses on identifying trends and discrepancies in wealth predictions over time. Our results show that the predictions by Chi et al. and Esp\'in-Noboa et al. align with general GDP trends, with differences expected due to the distinct time-frames of the training sets. However, predictions by Lee and Braithwaite diverge significantly, indicating potential issues with the validity of the model. These discrepancies highlight the need for policymakers and stakeholders in Africa to rigorously audit models that predict wealth, especially those used for decision-making on the ground. These and other techniques require continuous verification and refinement to enhance their reliability and ensure that poverty alleviation strategies are well-founded., Comment: 14 pages (main) + 11 pages (appendix). Accepted at the 9th Workshop on Data Science for Social Good, SoGood 2024, held in conjunction with ECML PKDD 2024, at Vilnius, Lithuania

We prove a general zero density theorem on the Selberg class of functions. The result verifies the Density Hypothesis in the strip when the real part of the variable is at least 0.9 under the assumption that the degree of the function does not exceed 2. In the other cases we obtain a density theorem weaker than the Density Hypothesis.

This chapter examines the challenges of the revised opt out system and the secondary use of health data in England. The analysis of this data could be very valuable for science and medical treatment as well as for the discovery of new drugs. For this reason, the UK government established the care.data program in 2013. The aim of the project was to build a central nationwide database for research and policy planning. However, the processing of personal data was planned without proper public engagement. Research has suggested that IT companies, such as in the Google DeepMind deal case, had access to other kinds of sensitive data and failed to comply with data protection law. Since May 2018, the government has launched the national data opt out system with the hope of regaining public trust. Nevertheless, there are no evidence of significant changes in the ND opt out, compared to the previous opt out system. Neither in the use of secondary data, nor in the choices that patients can make. The only notorious difference seems to be in the way that these options are communicated and framed to the patients. Most importantly, according to the new ND opt out, the type 1 opt out option, which is the only choice that truly stops data from being shared outside direct care, will be removed in 2020. According to the Behavioral Law and Economics literature (Nudge Theory), default rules, such as the revised opt out system in England, are very powerful, because people tend to stick to the default choices made readily available to them. The crucial question analyzed in this chapter is whether it is desirable for the UK government to stop promoting the type 1 opt outs, and whether this could be seen as a kind of hard paternalism.

While dynamical tides only become relevant during the last couple of orbits for circular inspirals, orbital eccentricity can increase their impact during earlier phases of the inspiral by exciting tidal oscillations at each close encounter. We investigate the effect of dynamical tides on the orbital evolution of eccentric neutron star binaries using post-Newtonian numerical simulations and constructing an analytic stochastic model. Our study reveals a strong dependence of dynamical tides on the pericenter distance, with the energy transferred to dynamical tides over that dissipated in gravitational waves (GWs) exceeding $\sim1\%$ at separations $r_\mathrm{p}\lesssim50$ km for large eccentricities. We demonstrate that the effect of dynamical tides on orbital evolution can manifest as a phase shift in the GW signal. We show that the signal-to-noise ratio of the GW phase shift can reach the detectability threshold of 8 with a single aLIGO detector at design densitivity for eccentric neutron star binaries at a distance of $40$ Mpc. This requires a pericenter distance of $r_\mathrm{p0}\lesssim68$ km ($r_\mathrm{p0}\lesssim76$ km) at binary formation with eccentricity close to 1 for a reasonable tidal deformability and f-mode frequency of 500 and $1.73$ kHz (700 and $1.61$ kHz), respectively. The observation of the phase shift will enable measuring the f-mode frequency of neutron stars independently from their tidal deformability, providing significant insights into neutron star seismology and the properties of the equation of state. We also explore the potential of distinguishing between equal-radius and twin-star binaries, which could provide an opportunity to reveal strong first-order phase transitions in the nuclear equation of state., Comment: 23 pages, 14 figures, submitted to Phys. Rev. D

Sparse autoencoders (SAEs) are a promising unsupervised approach for identifying causally relevant and interpretable linear features in a language model's (LM) activations. To be useful for downstream tasks, SAEs need to decompose LM activations faithfully; yet to be interpretable the decomposition must be sparse -- two objectives that are in tension. In this paper, we introduce JumpReLU SAEs, which achieve state-of-the-art reconstruction fidelity at a given sparsity level on Gemma 2 9B activations, compared to other recent advances such as Gated and TopK SAEs. We also show that this improvement does not come at the cost of interpretability through manual and automated interpretability studies. JumpReLU SAEs are a simple modification of vanilla (ReLU) SAEs -- where we replace the ReLU with a discontinuous JumpReLU activation function -- and are similarly efficient to train and run. By utilising straight-through-estimators (STEs) in a principled manner, we show how it is possible to train JumpReLU SAEs effectively despite the discontinuous JumpReLU function introduced in the SAE's forward pass. Similarly, we use STEs to directly train L0 to be sparse, instead of training on proxies such as L1, avoiding problems like shrinkage., Comment: v2: new appendix H comparing kernel functions & bug-fixes to pseudo-code in Appendix J v3: further bug-fix to pseudo-code in Appendix J

In two previous papers the second author proved some Carlson type density theorems for zeroes in the critical strip for Beurling zeta functions satisfying Axiom A of Knopfmacher. In the first of these invoking two additonal conditions were needed, while in the second an explicit, fully general result was obtained. Subsequently, Frederik Broucke and Gregory Debruyne obtained, via a different method, a general Carlson type density theorem with an even better exponent, and recently Frederik Broucke improved this further, getting $N(\sigma,T) \le T^{a(1-\sigma)}$ with any $a>\dfrac{4}{1-\theta}$. Broucke employed a new mean value estimate of the Beurling zeta function, while he did not use the method of Hal\'asz and Montgomery. Here we elaborate a new approach of the first author, using the classical zero detecting sums coupled with a kernel function technique and Hal\'asz' method, but otherwise arguing in an elementary way avoiding e.g. mean value estimates for Dirichlet polynomials. We will make essential use of the additional assumptions that the Beurling system of integers consists of natural numbers, and that the system satisfies the Ramanujan condition, too. This way we give a new variant of the Carlson type density estimate with similar strength as Tur\'an's 1954 result for the Riemann $\zeta$ function, coming close even to the Density Hypothesis for $\sigma$ close to 1.

We determine the automorphism group of an open log K3 surface with irreducible boundary.

Scalable oversight protocols aim to enable humans to accurately supervise superhuman AI. In this paper we study debate, where two AI's compete to convince a judge; consultancy, where a single AI tries to convince a judge that asks questions; and compare to a baseline of direct question-answering, where the judge just answers outright without the AI. We use large language models (LLMs) as both AI agents and as stand-ins for human judges, taking the judge models to be weaker than agent models. We benchmark on a diverse range of asymmetries between judges and agents, extending previous work on a single extractive QA task with information asymmetry, to also include mathematics, coding, logic and multimodal reasoning asymmetries. We find that debate outperforms consultancy across all tasks when the consultant is randomly assigned to argue for the correct/incorrect answer. Comparing debate to direct question answering, the results depend on the type of task: in extractive QA tasks with information asymmetry debate outperforms direct question answering, but in other tasks without information asymmetry the results are mixed. Previous work assigned debaters/consultants an answer to argue for. When we allow them to instead choose which answer to argue for, we find judges are less frequently convinced by the wrong answer in debate than in consultancy. Further, we find that stronger debater models increase judge accuracy, though more modestly than in previous studies., Comment: 15 pages (53 including appendices). V2: minor correction to Figure 3; add Figure A.9 comparing open vs assigned consultancy; add a reference

Members of a society can be characterized by a large number of features, such as gender, age, ethnicity, religion, social status, and shared activities. One of the main tie-forming factors between individuals in human societies is homophily, the tendency of being attracted to similar others. Homophily has been mainly studied with focus on one of the features and little is known about the roles of similarities of different origins in the formation of communities. To close this gap, we analyze three datasets from Information and Communications Technology (ICT) services, namely, two online social networks and a network deduced from mobile phone calls, in all of which metadata about individual features are available. We identify communities within egocentric networks and surprisingly find that the larger the community is, the more overlap is found between features of its members and the ego. We interpret this finding in terms of the effort needed to manage the communities; the larger diversity requires more effort such that to maintain a large diverse group may exceed the capacity of the members. As the ego reaches out to her alters on an ICT service, we observe that the first alter in each community tends to have a higher feature overlap with the ego than the rest. Moreover the feature overlap of the ego with all her alters displays a non-monotonic behaviors as a function of the ego's degree. We propose a simple mechanism of how people add links in their egocentric networks of alters that reproduces all the empirical observations and shows the reason behind non-monotonic tendency of the egocentric feature overlap as a function of the ego's degree., Comment: 8 pages, 7 figures, 1 table

In this work, we introduce an open-source integrated CAD-CFD tool, Anvil, which combines FreeCAD for CAD modeling and OpenFOAM for CFD analysis, along with an AI-based optimization method (Bayesian optimization) and other sampling algorithms. Anvil serves as a scientific machine learning tool for shape optimization in three modes: data generation, CFD evaluation, and shape optimization. In data generation mode, it automatically runs CFD evaluations and generates data for training a surrogate model. In optimization mode, it searches for the optimal design under given requirements and optimization metrics. In CFD mode, a single CAD file can be evaluated with a single OpenFOAM run. To use Anvil, experimenters provide a JSON configuration file and a parametric CAD seed design. Anvil can be used to study solid-fluid dynamics for any subsonic flow conditions and has been demonstrated in various simulation and optimization use cases. The open-source code for the tool, installation process, artifacts (such as CAD seed designs and example STL models), experimentation results, and detailed documentation can be found at \url{https://github.com/symbench/Anvil}.

In this work, we derived formulae concerning the electric and magnetic field characteristics of a focused radially polarized Gaussian vector beam. Such a beam is consistent with Maxwell's equations contrary to plane waves having uniform field distribution. Hence a realistic picture is provided of the focused field distributions having importance before designing applications such as particle acceleration. For focusing a perfectly reflecting large numerical aperture on-axis parabolic mirror was supposed to have practical importance. The computation technique was based on the Stratton-Chu vector diffraction method. We pointed out that this offers a unique opportunity in the long wavelength regime, where the Richards-Wolf theory becomes unreliable. In the terahertz frequency range longitudinal electric field component with an amplitude of $\sim$160 $\text{MV}/\text{cm}$ was predicted, which is ideal for particle acceleration applications. Based on the field characteristics experienced as a function of the focusing angle, the possibility of using a paraboloid ring for particle acceleration was suggested. Its advantage is reflected not only in the strong available longitudinal field but also in ensuring the unobstructed transfer of particles as a practical point of view. The axial and radial distributions of the longitudinal electric field component for different incident beam divergences were analyzed in detail. It was found that the shift of the physical focus relative to the geometrical focus along the longitudinal direction shows a linear dependence on the divergence. The effect of the divergence angle on the field enhancement factor was also studied.

A curve in the plane is $x$-monotone if every vertical line intersects it at most once. A family of curves are called pseudo-segments if every pair of them have at most one point in common. We construct $2^{\Omega(n^{4/3})}$ families, each consisting of $n$ labelled $x$-monotone pseudo-segments such that their intersection graphs are different. On the other hand, we show that the number of such intersection graphs is at most $2^{O(n^{3/2-\varepsilon})}$, where $\varepsilon>0$ is a suitable constant. Our proof uses a new upper bound on the number of set systems of size $m$ on a ground set of size $n$, with VC-dimension at most $d$. Much better upper bounds are obtained if we only count bipartite intersection graphs, or, in general, intersection graphs with bounded chromatic number.

Let $P\to X$ be a ${\mathbb P}^1$-bundle over a variety $X$. The aim of this note is to understand all connected, algebraic groups $$ \operatorname{Aut}^\circ(P)\subset G\subset \operatorname{Bir}( X\times {\mathbb P}^1). $$ We get a quite complete answer if $\operatorname{Aut}^\circ(X)$ is a maximal, connected, algebraic subgroup of $\operatorname{Bir}(X)$, and $P$ is sufficiently unstable. This gives examples of connected, algebraic subgroups of $\operatorname{Bir}({\mathbb P}^4)$ that are not contained in any maximal one. Version 2: references updated.

We consider the problem of computing tight privacy guarantees for the composition of subsampled differentially private mechanisms. Recent algorithms can numerically compute the privacy parameters to arbitrary precision but must be carefully applied. Our main contribution is to address two common points of confusion. First, some privacy accountants assume that the privacy guarantees for the composition of a subsampled mechanism are determined by self-composing the worst-case datasets for the uncomposed mechanism. We show that this is not true in general. Second, Poisson subsampling is sometimes assumed to have similar privacy guarantees compared to sampling without replacement. We show that the privacy guarantees may in fact differ significantly between the two sampling schemes. In particular, we give an example of hyperparameters that result in $\varepsilon \approx 1$ for Poisson subsampling and $\varepsilon > 10$ for sampling without replacement. This occurs for some parameters that could realistically be chosen for DP-SGD.

A \emph{complete geometric graph} consists of a set $P$ of $n$ points in the plane, in general position, and all segments (edges) connecting them. It is a well known question of Bose, Hurtado, Rivera-Campo, and Wood, whether there exists a positive constant $c<1$, such that every complete geometric graph on $n$ points can be partitioned into at most $cn$ plane graphs (that is, noncrossing subgraphs). We answer this question in the affirmative in the special case where the underlying point set $P$ is \emph{dense}, which means that the ratio between the maximum and the minimum distances in $P$ is of the order of $\Theta(\sqrt{n})$., Comment: 10 pages, 5 figures

As current experiments already realize small quantum circuits on error corrected qubits, it is important to fully understand the effect of physical errors on the logical error channels of these fault-tolerant circuits. Here, we investigate a lattice-surgery-based CNOT operation between two surface code patches under phenomenological error models. (i) For two-qubit logical Pauli measurements -- the elementary building block of the CNOT -- we optimize the number of stabilizer measurement rounds, usually taken equal to $d$, the size (code distance) of each patch. We find that the optimal number can be greater or smaller than $d$, depending on the rate of physical and readout errors, and the separation between the code patches. (ii) We fully characterize the two-qubit logical error channel of the lattice-surgery-based CNOT. We find a symmetry of the CNOT protocol, that results in a symmetry of the logical error channel. We also find that correlations between X and Z errors on the logical level are suppressed under minimum weight decoding.

In the vibrant landscape of AI research, decentralised learning is gaining momentum. Decentralised learning allows individual nodes to keep data locally where they are generated and to share knowledge extracted from local data among themselves through an interactive process of collaborative refinement. This paradigm supports scenarios where data cannot leave local nodes due to privacy or sovereignty reasons or real-time constraints imposing proximity of models to locations where inference has to be carried out. The distributed nature of decentralised learning implies significant new research challenges with respect to centralised learning. Among them, in this paper, we focus on robustness issues. Specifically, we study the effect of nodes' disruption on the collective learning process. Assuming a given percentage of "central" nodes disappear from the network, we focus on different cases, characterised by (i) different distributions of data across nodes and (ii) different times when disruption occurs with respect to the start of the collaborative learning task. Through these configurations, we are able to show the non-trivial interplay between the properties of the network connecting nodes, the persistence of knowledge acquired collectively before disruption or lack thereof, and the effect of data availability pre- and post-disruption. Our results show that decentralised learning processes are remarkably robust to network disruption. As long as even minimum amounts of data remain available somewhere in the network, the learning process is able to recover from disruptions and achieve significant classification accuracy. This clearly varies depending on the remaining connectivity after disruption, but we show that even nodes that remain completely isolated can retain significant knowledge acquired before the disruption., Comment: Supported by the H2020 HumaneAI Net (952026), CHIST-ERA-19-XAI010 SAI, PNRR - M4C2 - Investimento 1.3, Partenariato Esteso PE00000013 FAIR, PNRR - M4C2 - Investimento 1.3, Partenariato Esteso PE00000001 RESTART

Recent work has found that sparse autoencoders (SAEs) are an effective technique for unsupervised discovery of interpretable features in language models' (LMs) activations, by finding sparse, linear reconstructions of LM activations. We introduce the Gated Sparse Autoencoder (Gated SAE), which achieves a Pareto improvement over training with prevailing methods. In SAEs, the L1 penalty used to encourage sparsity introduces many undesirable biases, such as shrinkage -- systematic underestimation of feature activations. The key insight of Gated SAEs is to separate the functionality of (a) determining which directions to use and (b) estimating the magnitudes of those directions: this enables us to apply the L1 penalty only to the former, limiting the scope of undesirable side effects. Through training SAEs on LMs of up to 7B parameters we find that, in typical hyper-parameter ranges, Gated SAEs solve shrinkage, are similarly interpretable, and require half as many firing features to achieve comparable reconstruction fidelity., Comment: 15 main text pages, 22 appendix pages

Precision measurements of space and time, like those made by the detectors of the Laser Interferometer Gravitational-wave Observatory (LIGO), are often confronted with fundamental limitations imposed by quantum mechanics. The Heisenberg uncertainty principle dictates that the position and momentum of an object cannot both be precisely measured, giving rise to an apparent limitation called the Standard Quantum Limit (SQL). Reducing quantum noise below the SQL in gravitational-wave detectors, where photons are used to continuously measure the positions of freely falling mirrors, has been an active area of research for decades. Here we show how the LIGO A+ upgrade reduced the detectors' quantum noise below the SQL by up to 3 dB while achieving a broadband sensitivity improvement, more than two decades after this possibility was first presented.

Perturbation theory in asymptotically free quantum field theories is asymptotic. The factorially growing perturbative coefficients carry information about non-perturbative corrections, which can be related to renormalons and instantons. Using the Wiener-Hopf technique we determine the full analytic solution for the free energy density in the two dimensional $O(N)$ sigma models. For $N>3$ there are no instantons, and we found that the perturbative series carries all the information about the non-perturbative corrections. However, in the $O(3)$ case, we identify several non-perturbative sectors that are not related to the asymptotics of the perturbative series. The number of sectors depends on the observables: for the ground-state energy density we identify three sectors, which we attribute to instantons. For the free energy density in the running perturbative coupling we found infinitely many sectors., Comment: 76 pages, 7 figures

We are given a bounded Borel-measurable real-valued function on a product of countably many Polish spaces, and a product probability measure. We are interested in points in the product space that can be used to approximate the expected value of this function. We define two notions. A point is called a weak $\epsilon$-approximation, where $\epsilon \geq 0$, if the Dirac measure on this point, except in finitely many coordinates where another measure can be taken, gives an expected value that is $\epsilon$-close to the original expected value. A point is called a strong $\epsilon$-approximation if the same holds under the restriction that in those finitely many coordinates the measure is equal to the original one. We prove that both the set of weak 0-approximation points and the set of strong $\epsilon$-approximation points, for any $\epsilon>0$, have measure 1 under the original measure. Finally, we provide two applications: (i) in Game Theory on the minmax guarantee levels of the players in games with infinitely many players, and (ii) in Decision Theory on the set of feasible expected payoffs in infinite duration problems.

In many control problems there is only limited information about the actions that will be available at future stages. We introduce a framework where the Controller chooses actions $a_{0}, a_{1}, \ldots$, one at a time. Her goal is to maximize the probability that the infinite sequence $(a_{0}, a_{1}, \ldots)$ is an element of a given subset $G$ of $\mathbb{N}^{\mathbb{N}}$. The set $G$, called the goal, is assumed to be a Borel tail set. The Controller's choices are restricted: having taken a sequence $h_{t} = (a_{0}, \ldots, a_{t-1})$ of actions prior to stage $t \in \mathbb{N}$, she must choose an action $a_{t}$ at stage $t$ from a non-empty, finite subset $A(h_{t})$ of $\mathbb{N}$. The set $A(h_{t})$ is chosen from a distribution $p_{t}$, independently over all $t \in \mathbb{N}$ and all $h_{t} \in \mathbb{N}^{t}$. We consider several information structures defined by how far ahead into the future the Controller knows what actions will be available. In the special case where all the action sets are singletons (and thus the Controller is a dummy), Kolmogorov's 0-1 law says that the probability for the goal to be reached is 0 or 1. We construct a number of counterexamples to show that in general the value of the control problem can be strictly between 0 and 1, and derive several sufficient conditions for the 0-1 ``law" to hold.

Data deluge characteristic for our times has led to information overload, posing a significant challenge to effectively finding our way through the digital landscape. Addressing this issue requires an in-depth understanding of how we navigate through the abundance of information. Previous research has discovered multiple patterns in how individuals navigate in the geographic, social, and information spaces, yet individual differences in strategies for navigation in the knowledge space has remained largely unexplored. To bridge the gap, we conducted an online experiment where participants played a navigation game on Wikipedia and completed questionnaires about their personal information. Utilizing a graph embedding trained on the English Wikipedia, our study identified distinctive strategies that participants adopt: when the target is a famous person, participants typically use the geographical and occupational information of the target to navigate, reminiscent of hub-driven and proximity-driven approaches, respectively. We discovered that many participants playing the same game exhibit a "wisdom of the crowd" effect: The set of strategies provide a good estimate for the information landscape around the target indicating that the individual differences complement each other., Comment: 25 pages, 8 figures

The structure of personal networks reflects how we organise and maintain social relationships. The distribution of tie strengths in personal networks is heterogeneous, with a few close, emotionally intense relationships and a larger number of weaker ties. Recent results indicate this feature is universal across communication channels. Within this general pattern, there is a substantial and persistent inter-individual variation that is also similarly distributed among channels. The reason for the observed universality is yet unclear -- one possibility is that people's traits determine their personal network features on any channel. To address this hypothesis, we need to compare an individual's personal networks across channels, which is a non-trivial task: while we are interested in measuring the differences in tie strength heterogeneity, personal network size is also expected to vary a lot across channels. Therefore, for any measure that compares personal networks, one needs to understand the sensitivity with respect to network size. Here, we study different measures of personal network similarity and show that a recently introduced alter-preferentiality parameter and the Gini coefficient are equally suitable measures for tie strength heterogeneity, as they are fairly insensitive to differences in network size. With these measures, we show that the earlier observed individual-level persistence of personal network structure cannot be attributed to network size stability alone, but that the tie strength heterogeneity is persistent too. We also demonstrate the effectiveness of the two measures on multichannel data, where tie strength heterogeneity in personal networks is seen to moderately correlate for the same users across two communication channels (calls and text messages)., Comment: 25 pages, 10 figures

Fully decentralised federated learning enables collaborative training of individual machine learning models on distributed devices on a communication network while keeping the training data localised. This approach enhances data privacy and eliminates both the single point of failure and the necessity for central coordination. Our research highlights that the effectiveness of decentralised federated learning is significantly influenced by the network topology of connected devices. We propose a strategy for uncoordinated initialisation of the artificial neural networks, which leverages the distribution of eigenvector centralities of the nodes of the underlying communication network, leading to a radically improved training efficiency. Additionally, our study explores the scaling behaviour and choice of environmental parameters under our proposed initialisation strategy. This work paves the way for more efficient and scalable artificial neural network training in a distributed and uncoordinated environment, offering a deeper understanding of the intertwining roles of network structure and learning dynamics.

Activation Patching is a method of directly computing causal attributions of behavior to model components. However, applying it exhaustively requires a sweep with cost scaling linearly in the number of model components, which can be prohibitively expensive for SoTA Large Language Models (LLMs). We investigate Attribution Patching (AtP), a fast gradient-based approximation to Activation Patching and find two classes of failure modes of AtP which lead to significant false negatives. We propose a variant of AtP called AtP*, with two changes to address these failure modes while retaining scalability. We present the first systematic study of AtP and alternative methods for faster activation patching and show that AtP significantly outperforms all other investigated methods, with AtP* providing further significant improvement. Finally, we provide a method to bound the probability of remaining false negatives of AtP* estimates.

We give a lower bound for the essential dimension of isogenies of complex abelian varieties. The bound is sharp in many cases. In particular, the multiplication-by-$m$ map is incompressible for every $m\geq 2$, confirming a conjecture of Brosnan., Comment: 12 pages. v3: Added a new author. Sections 3 and 5 are new. Main results are strengthened to cover the case of small primes. Comments are welcome!

The differential equations of chemical kinetics are systems of nonlinear (polynomial) differential equations, therefore their solutions cannot usually be found in symbolic form. Here we offer a method to solve classes of kinetic differential equations based on the Cauchy--Riemann equations. It turns out that the method can be used to symbolically solve some polynomial differential equations that are not necessarily kinetic, as well., Comment: 26 pages, 7 figures

This work is part of a prediction challenge that invited theoretical/computational chemists to predict the photochemistry of cyclobutanone in the gas phase, excited at 200 nm by a laser pulse, and the expected signal that will be recorded during a time-resolved megaelectronvolt ultrafast electron diffraction (MeV-UED). We present here our theoretical predictions based on a combination of trajectory surface hopping with XMS-CASPT2 (for the nonadiabatic molecular dynamics) and Born-Oppenheimer molecular dynamics (BOMD) with MP2 (for the athermal ground-state dynamics following internal conversion), coined (NA+BO)MD. The initial conditions were sampled from BOMD coupled to a quantum thermostat. Our simulations indicate that the main photoproducts after 2 ps of dynamics are CO + cyclopropane (50%), CO + propene (10%), and ethene and ketene (34%). The photoexcited cyclobutanone in its second excited electronic state S$_2$ can follow two pathways for its nonradiative decay: (i) a ring-opening in S$_2$ and a subsequent rapid decay to the ground electronic state, where the photoproducts are formed, or (ii) a transfer through a closed-ring conical intersection to S$_1$, where cyclobutanone ring opens and then funnels to the ground state. Lifetimes for the photoproduct and electronic populations were determined. We calculated a stationary MeV-UED signal [difference pair distribution function - $\Delta$PDF$(r)$] for each (interpolated) pathway as well as a time-resolved signal [$\Delta$PDF$(r,t)$ and $\Delta I/I(s,t)$] for the full swarm of (NA+BO)MD trajectories. Furthermore, our analysis provides time-independent basis functions that can be used to fit the time-dependent experimental UED signals and potentially recover the population of photoproducts. We also offer a detailed analysis of the limitations of our model and their potential impact on the predicted experimental signals.

We introduce a representation of a 2D steady vector field ${{\mathbf v}}$ by two scalar fields $a$, $b$, such that the isolines of $a$ correspond to stream lines of ${{\mathbf v}}$, and $b$ increases with constant speed under integration of ${{\mathbf v}}$. This way, we get a direct encoding of stream lines, i.e., a numerical integration of ${{\mathbf v}}$ can be replaced by a local isoline extraction of $a$. To guarantee a solution in every case, gradient-preserving cuts are introduced such that the scalar fields are allowed to be discontinuous in the values but continuous in the gradient. Along with a piecewise linear discretization and a proper placement of the cuts, the fields $a$ and $b$ can be computed. We show several evaluations on non-trivial vector fields.

Quantum error correction is a key challenge for the development of practical quantum computers, a direction in which significant experimental progress has been made in recent years. In solid-state qubits, one of the leading information loss mechanisms is dephasing, usually modelled by phase flip errors. Here, we introduce quasistatic phase damping, a more subtle error model which describes the effect of Larmor frequency fluctuations due to 1/f noise. We show how this model is different from a simple phase flip error model, in terms of multi-cycle error correction. Considering the surface code, we provide numerical evidence for an error threshold, in the presence of quasistatic phase damping and readout errors. We discuss the implications of our results for spin qubits and superconducting qubits., Comment: 12 pages main text, + 3 pages bibliography and appendix, 6 figures

FTLE (Finite Time Lyapunov Exponent) computation is one of the standard approaches to Lagrangian flow analysis. The main features of interest in FTLE fields are ridges that represent hyperbolic Lagrangian Coherent Structures. FTLE ridges tend to become sharp and crisp with increasing integration time, where the sharpness of the ridges is an indicator of the strength of separation. The additional consideration of uncertainty in flows leads to more blurred ridges in the FTLE fields. There are multiple causes for such blurred ridges: either the locations of the ridges are uncertain, or the strength of the ridges is uncertain, or there is low uncertainty but weak separation. Existing approaches for uncertain FTLE computation are unable to distinguish these different sources of uncertainty in the ridges. We introduce a new approach to define and visualize FTLE fields for flow ensembles. Before computing and comparing FTLE fields for the ensemble members, we compute optimal displacements of the domains to mutually align the ridges of the ensemble members as much as possible. We do so in a way that an explicit geometry extraction and alignment of the ridges is not necessary. The additional consideration of these displacements allows for a visual distinction between uncertainty in ridge location, ridge sharpness, and separation strength. We apply the approach to several synthetic and real ensemble data sets.

A plurisubharmonic weight is log canonical if it is at the critical point of turning non-integrable. Given a log canonical plurisubharmonic weight, we show that locally there always exists a log canonical `holomorphic' weight having the same non-integrable locus. The proof uses the theory of quasimonomial valuations developed by Jonsson-Musta\c{t}\v{a} and Xu, as well as the minimal model program over complex analytic spaces due to Fujino and Lyu-Murayama. As a consequence, we obtain that the non-integrable locus is seminormal.

The layered van der Waals material ZrTe$_5$ is known as a candidate topological insulator (TI), however its topological phase and the relation with other properties such as an apparent Dirac semimetallic state is still a subject of debate. We employ a semiclassical multicarrier transport (MCT) model to analyze the magnetotransport of ZrTe$_5$ nanodevices at hydrostatic pressures up to 2 GPa. The temperature dependence of the MCT results between 10 and 300 K is assessed in the context of thermal activation, and we obtain the positions of conduction and valence band edges in the vicinity of the chemical potential. We find evidence of the closing and subsequent re-opening of the band gap with increasing pressure, which is consistent with a phase transition from weak to strong TI. This matches expectations from ab initio band structure calculations, as well as previous observations that CVT-grown ZrTe$_5$ is in a weak TI phase in ambient conditions., Comment: Main Text: 10 pages, 5 figures; Supporting Information: 12 pages, 8 figures

We consider partitions of a point set into two parts, and the lengths of the minimum spanning trees of the original set and of the two parts. If $w(P)$ denotes the length of a minimum spanning tree of $P$, we show that every set $P$ of $n \geq 12$ points admits a bipartition $P= R \cup B$ for which the ratio $\frac{w(R)+w(B)}{w(P)}$ is strictly larger than $1$; and that $1$ is the largest number with this property. Furthermore, we provide a very fast algorithm that computes such a bipartition in $O(1)$ time and one that computes the corresponding ratio in $O(n \log{n})$ time. In certain settings, a ratio larger than $1$ can be expected and sometimes guaranteed. For example, if $P$ is a set of $n$ random points uniformly distributed in $[0,1]^2$ ($n \to \infty$), then for any $\eps>0$, the above ratio in a maximizing partition is at least $\sqrt2 -\eps$ with probability tending to $1$. As another example, if $P$ is a set of $n$ points with spread at most $\alpha \sqrt{n}$, for some constant $\alpha>0$, then the aforementioned ratio in a maximizing partition is $1 + \Omega(\alpha^{-2})$. All our results and techniques are extendable to higher dimensions., Comment: 2 figures, 11 pages