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A very general hypersurface of dimension $n$ and degree $d$ in complex projective space is rational if $d \leq 2$, but is expected to be irrational for all $n, d \geq 3$. Hypersurfaces in weighted projective space with degree small relative to the weights are likewise rational. In this paper, we introduce rationality constructions for weighted hypersurfaces of higher degree that provide many new rational examples over any field. We answer in the affirmative a question of T. Okada about the existence of very general terminal Fano rational weighted hypersurfaces in all dimensions $n \geq 6$., Comment: 13 pages, 1 figure. Comments welcome!

The conventional BIM authoring process typically requires designers to master complex and tedious modeling commands in order to materialize their design intentions within BIM authoring tools. This additional cognitive burden complicates the design process and hinders the adoption of BIM and model-based design in the AEC (Architecture, Engineering, and Construction) industry. To facilitate the expression of design intentions more intuitively, we propose Text2BIM, an LLM-based multi-agent framework that can generate 3D building models from natural language instructions. This framework orchestrates multiple LLM agents to collaborate and reason, transforming textual user input into imperative code that invokes the BIM authoring tool's APIs, thereby generating editable BIM models with internal layouts, external envelopes, and semantic information directly in the software. Furthermore, a rule-based model checker is introduced into the agentic workflow, utilizing predefined domain knowledge to guide the LLM agents in resolving issues within the generated models and iteratively improving model quality. Extensive experiments were conducted to compare and analyze the performance of three different LLMs under the proposed framework. The evaluation results demonstrate that our approach can effectively generate high-quality, structurally rational building models that are aligned with the abstract concepts specified by user input. Finally, an interactive software prototype was developed to integrate the framework into the BIM authoring software Vectorworks, showcasing the potential of modeling by chatting.

Selective exposure, individuals' inclination to seek out information that supports their beliefs while avoiding information that contradicts them, plays an important role in the emergence of polarization. In the political domain, selective exposure is usually measured on a left-right ideology scale, ignoring finer details. Here, we combine survey and Twitter data collected during the 2022 Brazilian Presidential Election and investigate selective exposure patterns between the survey respondents and political influencers. We analyze the followship network between survey respondents and political influencers and find a multilevel community structure that reveals a hierarchical organization more complex than a simple split between left and right. Moreover, depending on the level we consider, we find different associations between network indices of exposure patterns and 189 individual attributes of the survey respondents. For example, at finer levels, the number of influencer communities a survey respondent follows is associated with several factors, such as demographics, news consumption frequency, and incivility perception. In comparison, only their political ideology is a significant factor at coarser levels. Our work demonstrates that measuring selective exposure at a single level, such as left and right, misses important information necessary to capture this phenomenon correctly.

Cable-driven exosuits have the potential to support individuals with motor disabilities across the continuum of care. When supporting a limb with a cable, force sensors are often used to measure tension. However, force sensors add cost, complexity, and distal components. This paper presents a design and control approach to remove the force sensor from an upper limb cable-driven exosuit. A mechanical design for the exosuit was developed to maximize passive transparency. Then, a data-driven friction identification was conducted on a mannequin test bench to design a model-based tension controller. Seventeen healthy participants raised and lowered their right arms to evaluate tension tracking, movement quality, and muscular effort. Questionnaires on discomfort, physical exertion, and fatigue were collected. The proposed strategy allowed tracking the desired assistive torque with an RMSE of 0.71 Nm (18%) at 50% gravity support. During the raising phase, the EMG signals of the anterior deltoid, trapezius, and pectoralis major were reduced on average compared to the no-suit condition by 30%, 38%, and 38%, respectively. The posterior deltoid activity was increased by 32% during lowering. Position tracking was not significantly altered, whereas movement smoothness significantly decreased. This work demonstrates the feasibility and effectiveness of removing the force sensor from a cable-driven exosuit. A significant increase in discomfort in the lower neck and right shoulder indicated that the ergonomics of the suit could be improved. Overall this work paves the way towards simpler and more affordable exosuits., Comment: 19 pages, 10 figures, and 3 tables in the main manuscript. 12 pages, 4 figures, and 13 tables in the supplementary materials

We present the findings of a study based on a new inelastic electron-scattering experiment on the ${}^{12}\mathrm{C}$ nucleus focusing on the kinematic region of $Q^2=0.8\,\mathrm{GeV}^2/{c}^2$. The measured cross section is sensitive to the transverse response function and provides a stringent test of theoretical models, as well as of the theoretical assumptions made in Monte-Carlo event-generator codes developed for the interpretation of neutrino-nucleus experiments, such as DUNE and HyperK. We find that modern generators such as GENIE and GiBUU reproduce our new experimental data within 10$\%$.

This paper presents a new model of textures, obtained as realizations of a new class of fractional Brownian fields. These fields, called weighted tensorized fractional Brownian fields, are obtained by a relaxation of the tensor-product structure that appears in the definition of fractional Brownian sheets. Statistical properties such as self-similarity, stationarity of rectangular increments and regularity properties are obtained. An operator scaling extension is defined and we provide simulations of the fields using their spectral representation.

We construct exceptional Fano varieties with the smallest known minimal log discrepancies in all dimensions. These varieties are well-formed hypersurfaces in weighted projective space. Their minimal log discrepancies decay doubly exponentially with dimension, and achieve the optimal value in dimension 2., Comment: 28 pages

A detailed analysis of ptychography for 3D phase reconstructions of thick specimens is performed. We introduce multi-focus ptychography, which incorporates a 4D-STEM defocus series to enhance the quality of 3D reconstructions along the beam direction through a higher overdetermination ratio. This method is compared with established multi-slice ptychography techniques, such as conventional ptychography, regularized ptychography, and multi-mode ptychography. Additionally, we contrast multi-focus ptychography with an alternative method that uses virtual optical sectioning through a reconstructed scattering matrix ($\mathcal{S}$-matrix), which offers more precise 3D structure information compared to conventional ptychography. Our findings from multiple 3D reconstructions based on simulated and experimental data demonstrate that multi-focus ptychography surpasses other techniques, particularly in accurately reconstructing the surfaces and interface regions of thick specimens.

We describe a compact ultra-high vacuum (UHV) scanning tunneling microscope (STM) system that does not need any external supply of cooling liquids. It achieves temperatures down to 1.5 K and a z-noise down to 300 fmRMS for the frequency range of 0.1 Hz - 5 kHz (feedback loop off). It employs a pulse tube cryocooler (PTC) and a Joule-Thomson (JT) stage inducing only small temperature oscillations at the STM with amplitude below 1 mK. The challenge to combine an effective vibrational decoupling from the PTC with sufficient thermal conduction is tackled by a multipartite approach. We realize a minimal stiffness of the UHV bellows that connect the PTC and the STM chamber. Fine Copper wires mechanically decouple the PTC stages from cooling plates that carry the thermal shields, the JT stage and the STM. Soft springs decouple the STM from the JT stage. Finally, the STM body has an optimized conical shape and is made of the light and stiff material Shapal Hi MSoft such that a strong reduction of low frequency vibrations results for the tunnel junction. The voltage noise in the tunnel junction is 0.12 mV and an RF antenna close to the tunnel junction provides radio frequency excitations up to 40 GHz with amplitudes up to 10 mV., Comment: 12 pages, 8 figures

We find sharp upper bounds on the order of the automorphism group of a hypersurface in complex projective space in every dimension and degree. In each case, we prove that the hypersurface realizing the upper bound is unique up to isomorphism and provide explicit generators for the automorphism group. Similar results were recently proven independently by S. Yang, X. Yu, and Z. Zhu using different methods., Comment: 33 pages, 2 tables

Due to the pseudo-scalar nature of the axion-like particle (ALP), the CP- conserving production of two Higgs bosons via the ALP necessarily involves an additional Z or $\gamma$ boson. We examine the existing constraints from di-Higgs searches at Run 2 of the LHC and find that, despite the presence of extra objects in the final state, these searches are sensitive to a combination of ALP couplings to gluons and three-bosons in the TeV scale range. Additionally, we propose a specialized search strategy incorporating an energetic leptonic Z boson. This refined ALP-induced production process would allow for the identification of the h h $\rightarrow$ 4 b-jet final state and could potentially probe the TeV scale using data from Run 2 of the LHC. This production process can also occur through a coupling between the top quark and the ALP. We translate the current constraints on di-Higgs production into new limits on the ALP-top coupling., Comment: 19 pages, 9 figures

Near IR spectroscopic reverberation of Active Galactic Nuclei (AGN) potentially allows the infrared (IR) broad line region (BLR) to be reverberated alongside the disc and dust continua, while the spectra can also reveal details of dust astro-chemistry. Here, we describe results of a short pilot study (17 near-IR spectra over a 183 d period) for Mrk 509. The spectra give a luminosity-weighted dust radius of $\langle R_{\mathrm{d,lum}} \rangle = 186 \pm 4$ light-days for blackbody (large grain dust), consistent with previous (photometric) reverberation campaigns, whereas carbon and silicate dust give much larger radii. We develop a method of calibrating spectral data in objects where the narrow lines are extended beyond the slit width. We demonstrate this by showing our resultant photometric band light curves are consistent with previous results, with a hot dust lag at >40 d in the K band, clearly different from the accretion disc response at <20 d in the z band. We place this limit of 40 d by demonstrating clearly that the modest variability that we do detect in the H and K band does not reverberate on time-scales of less than 40 d. We also extract the Pa$\beta$ line light curve, and find a lag which is consistent with the optical BLR H$\beta$ line of $\sim$70-90 d. This is important as direct imaging of the near-IR BLR is now possible in a few objects, so we need to understand its relation to the better studied optical BLR., Comment: 16 pages, 12 figures

Tendon-driven continuum robots have been gaining popularity in medical applications due to their ability to curve around complex anatomical structures, potentially reducing the invasiveness of surgery. However, accurate modeling is required to plan and control the movements of these flexible robots. Physics-based models have limitations due to unmodeled effects, leading to mismatches between model prediction and actual robot shape. Recently proposed learning-based methods have been shown to overcome some of these limitations but do not account for hysteresis, a significant source of error for these robots. To overcome these challenges, we propose a novel deep decoder neural network that predicts the complete shape of tendon-driven robots using point clouds as the shape representation, conditioned on prior configurations to account for hysteresis. We evaluate our method on a physical tendon-driven robot and show that our network model accurately predicts the robot's shape, significantly outperforming a state-of-the-art physics-based model and a learning-based model that does not account for hysteresis., Comment: 8 pages, 9 figures, Submitted to IEEE Robotics and Automation Letters

In recent years, theoretical results and simulation evidence have shown Bayesian additive regression trees to be a highly-effective method for nonparametric regression. Motivated by cost-effectiveness analyses in health economics, where interest lies in jointly modelling the costs of healthcare treatments and the associated health-related quality of life experienced by a patient, we propose a multivariate extension of BART applicable in regression and classification analyses with several correlated outcome variables. Our framework overcomes some key limitations of existing multivariate BART models by allowing each individual response to be associated with different ensembles of trees, while still handling dependencies between the outcomes. In the case of continuous outcomes, our model is essentially a nonparametric version of seemingly unrelated regression. Likewise, our proposal for binary outcomes is a nonparametric generalisation of the multivariate probit model. We give suggestions for easily interpretable prior distributions, which allow specification of both informative and uninformative priors. We provide detailed discussions of MCMC sampling methods to conduct posterior inference. Our methods are implemented in the R package `suBART'. We showcase their performance through extensive simulations and an application to an empirical case study from health economics. By also accommodating propensity scores in a manner befitting a causal analysis, we find substantial evidence for a novel trauma care intervention's cost-effectiveness.

The creation of precise and high-resolution crop point clouds in agricultural fields has become a key challenge for high-throughput phenotyping applications. This work implements a novel calibration method to calibrate the laser scanning system of an agricultural field robot consisting of two industrial-grade laser scanners used for high-precise 3D crop point cloud creation. The calibration method optimizes the transformation between the scanner origins and the robot pose by minimizing 3D point omnivariances within the point cloud. Moreover, we present a novel factor graph-based pose estimation method that fuses total station prism measurements with IMU and GNSS heading information for high-precise pose determination during calibration. The root-mean-square error of the distances to a georeferenced ground truth point cloud results in 0.8 cm after parameter optimization. Furthermore, our results show the importance of a reference point cloud in the calibration method needed to estimate the vertical translation of the calibration. Challenges arise due to non-static parameters while the robot moves, indicated by systematic deviations to a ground truth terrestrial laser scan.

Diffusion models are the main driver of progress in image and video synthesis, but suffer from slow inference speed. Distillation methods, like the recently introduced adversarial diffusion distillation (ADD) aim to shift the model from many-shot to single-step inference, albeit at the cost of expensive and difficult optimization due to its reliance on a fixed pretrained DINOv2 discriminator. We introduce Latent Adversarial Diffusion Distillation (LADD), a novel distillation approach overcoming the limitations of ADD. In contrast to pixel-based ADD, LADD utilizes generative features from pretrained latent diffusion models. This approach simplifies training and enhances performance, enabling high-resolution multi-aspect ratio image synthesis. We apply LADD to Stable Diffusion 3 (8B) to obtain SD3-Turbo, a fast model that matches the performance of state-of-the-art text-to-image generators using only four unguided sampling steps. Moreover, we systematically investigate its scaling behavior and demonstrate LADD's effectiveness in various applications such as image editing and inpainting.

Contrastive learning is a paradigm for learning representations from unlabelled data that has been highly successful for image and text data. Several recent works have examined contrastive losses to claim that contrastive models effectively learn spectral embeddings, while few works show relations between (wide) contrastive models and kernel principal component analysis (PCA). However, it is not known if trained contrastive models indeed correspond to kernel methods or PCA. In this work, we analyze the training dynamics of two-layer contrastive models, with non-linear activation, and answer when these models are close to PCA or kernel methods. It is well known in the supervised setting that neural networks are equivalent to neural tangent kernel (NTK) machines, and that the NTK of infinitely wide networks remains constant during training. We provide the first convergence results of NTK for contrastive losses, and present a nuanced picture: NTK of wide networks remains almost constant for cosine similarity based contrastive losses, but not for losses based on dot product similarity. We further study the training dynamics of contrastive models with orthogonality constraints on output layer, which is implicitly assumed in works relating contrastive learning to spectral embedding. Our deviation bounds suggest that representations learned by contrastive models are close to the principal components of a certain matrix computed from random features. We empirically show that our theoretical results possibly hold beyond two-layer networks.

Skyrmion lattices (SkL) in centrosymmetric materials typically have a magnetic period on the nanometer-scale, so that the coupling between magnetic superstructures and the underlying crystal lattice cannot be neglected. Here, we reveal the commensurate locking of a SkL to the atomic lattice in Gd$_3$Ru$_4$Al$_{12}$ via high-resolution resonant elastic x-ray scattering (REXS). Weak easy-plane magnetic anisotropy, demonstrated here by a combination of ferromagnetic resonance and REXS, penalizes placing a skyrmion core on a site of the atomic lattice. Under these conditions, a commensurate SkL, locked to the crystal lattice, is stable at finite temperatures -- but gives way to a competing incommensurate ground state upon cooling. We discuss the role of Umklapp-terms in the Hamiltonian for the formation of this lattice-locked state, its magnetic space group, the role of slight discommensurations, or (line) defects in the magnetic texture, and contrast our findings with the case of SkLs in noncentrosymmetric material platforms., Comment: 17 pages, 4 figures, additional SI included (19 pages, 11 figures, 1 table)

Diffusion models create data from noise by inverting the forward paths of data towards noise and have emerged as a powerful generative modeling technique for high-dimensional, perceptual data such as images and videos. Rectified flow is a recent generative model formulation that connects data and noise in a straight line. Despite its better theoretical properties and conceptual simplicity, it is not yet decisively established as standard practice. In this work, we improve existing noise sampling techniques for training rectified flow models by biasing them towards perceptually relevant scales. Through a large-scale study, we demonstrate the superior performance of this approach compared to established diffusion formulations for high-resolution text-to-image synthesis. Additionally, we present a novel transformer-based architecture for text-to-image generation that uses separate weights for the two modalities and enables a bidirectional flow of information between image and text tokens, improving text comprehension, typography, and human preference ratings. We demonstrate that this architecture follows predictable scaling trends and correlates lower validation loss to improved text-to-image synthesis as measured by various metrics and human evaluations. Our largest models outperform state-of-the-art models, and we will make our experimental data, code, and model weights publicly available.

Searches for anomalous neutral triple gauge boson couplings (NTGCs) provide important tests for the gauge structure of the standard model. In SMEFT ("standard model effective field theory") NTGCs appear only at the level of dimension-8 operators. While the phenomenology of these operators has been discussed extensively in the literature, renormalizable UV models that can generate these operators are scarce. In this work, we study a variety of extensions of the SM with heavy fermions and calculate their matching to d = 8 NTGC operators. We point out that the complete matching of UV models requires four different CP-conserving d = 8 operators and that the single CPC d = 8 operator, most commonly used by the experimental collaborations, does not describe all possible NTGC form factors. Despite stringent experimental constraints on NTGCs, limits on the scale of UV models are relatively weak, because their contributions are doubly suppressed (being d = 8 and 1-loop). We suggest a series of benchmark UV scenarios suitable for interpreting searches for NTGCs in the upcoming LHC runs, obtain their current limits and provide estimates for the expected sensitivity of the high-luminosity LHC., Comment: 27 pages, 8 figures

We report on time-resolved nonlinear terahertz spectroscopy of a strongly correlated ruthenate, CaRuO$_3$, as a function of temperature, frequency and terahertz field strength. Third-harmonic radiation for frequencies up to 2.1 THz is observed evidently at low temperatures below 80 K, where the low-frequency linear dynamical response deviates from the Drude model and a coherent heavy quasiparticle band emerges by strong correlations associated with the Hund's coupling.Phenomenologically, by taking an experimentally observed frequency-dependent scattering rate, the deviation of the field driven kinetics from the Drude behavior is reconciled in a time-dependent Boltzmann description, which allows an attribution of the observed third-harmonic generation to the terahertz field driven nonlinear kinetics of the heavy quasiparticles., Comment: 5 pages, 3 figures

Antiferromagnets hosting structural or magnetic order that breaks time reversal symmetry are of increasing interest for 'beyond von Neumann computing' applications because the topology of their band structure allows for intrinsic physical properties, exploitable in integrated memory and logic function. One such group are the non-collinear antiferromagnets. Essential for domain manipulation is the existence of small net moments found routinely when the material is synthesised in thin film form and attributed to symmetry-breaking caused by spin canting, either from the Dzyaloshinskii-Moriya interaction or from strain. Although the spin arrangement of these materials makes them highly sensitive to strain, there is little understanding about the influence of local strain fields caused by lattice defects on global properties, such as magnetisation and anomalous Hall effect. This premise is investigated by examining non-collinear films that are either highly lattice mismatched or closely matched to their substrate. In either case, edge dislocation networks are generated and for the former case these extend throughout the entire film thickness, creating large local strain fields. These strain fields allow for finite intrinsic magnetisation in seemly structurally relaxed films and influence the antiferromagnetic domain state and the intrinsic anomalous Hall effect., Comment: 15 pages, 6 figures

We introduce a new invariant of $G$-varieties, the dual complex, which roughly measures how divisors in the complement of the free locus intersect. We show that the top homology group of this complex is an equivariant birational invariant of $G$-varieties. As an application, we demonstrate the non-linearizability of certain large abelian group actions on smooth hypersurfaces in projective space of any dimension and degree at least $3$., Comment: 8 pages, comments welcome!

If primordial black holes constitute the dark matter, stars forming in dark-matter dominated environments with low velocity dispersions, such as ultra-faint dwarf galaxies, may capture a black hole at birth. The capture probability is non-negligible for primordial black holes of masses around $10^{20}$g, and increases with stellar mass. Moreover, infected stars are turned into virtually invisible black holes on cosmologically short timescales. Hence, the number of observed massive main-sequence stars in ultra-faint dwarfs should be suppressed if the dark matter was made of asteroid-mass primordial black holes. This would impact the measured mass distribution of stars, making it top-light (i.e. depleted in the high-mass range). Using simulated data that mimic the present-day observational power of telescopes, we show that already existing measurements of the mass function of stars in local ultra-faint dwarfs could be used to constrain the fraction of dark matter composed of primordial black holes in the -- currently unconstrained -- mass range of $10^{19}-10^{21}$g., Comment: Updated to match the MNRAS version. The mean captured PBH number (figure 1) was corrected. Most figures as well as some parts of the text are slightly modified

The ratio of the transverse and longitudinal component of polarization transfer to protons in quasi-elastic $(\vec{e}, e^{\prime} \vec{p}\,)$ reaction, $P^{\prime}_x/P^{\prime}_z$, is sensitive to the proton's electromagnetic form factor ratio, $G_E/G_M$. To explore density-dependent in-medium modifications, a comparison of polarization transfer ratios involving protons from distinct nuclear shells, each with different local nuclear densities, has been proposed. In this study, we present such comparisons between four shells, $1s_{1/2}$, $1p_{3/2}$ in $^{12}\mathrm{C}$ and $1d_{3/2}$, $2s_{1/2}$ in $^{40}\mathrm{Ca}$. In an effort to account for other many-body effects that may differ between shells, we use state-of-the-art relativistic distorted-wave impulse-approximation (RDWIA) calculation and present the double ratios, $(P^{\prime}_x/P^{\prime}_z)_{\rm Data}/(P^{\prime}_x/P^{\prime}_z)_{\rm RDWIA}$ as well as the super ratios, $\left[(P^{\prime}_x/P^{\prime}_z)_{\rm A}/(P^{\prime}_x/P^{\prime}_z)_{\rm B}\right]_{\rm Data}/\left[(P^{\prime}_x/P^{\prime}_z)_{\rm A}/(P^{\prime}_x/P^{\prime}_z)_{\rm B}\right]_{\rm RDWIA}$, for chosen shells A and B, as a function of effective local nuclear densities. We find that double ratios for individual shells show a dependence on the probed effective nuclear densities. Studying the ratios, we observed a systematic variation between pairs of higher- and lower-density shells.

We consider a stochastic individual-based model of adaptive dynamics for an asexually reproducing population with mutation, with linear birth and death rates, as well as a density-dependent competition. To depict repeating changes of the environment, all of these parameters vary over time as piecewise constant and periodic functions, on an intermediate time-scale between those of stabilization of the resident population (fast) and exponential growth of mutants (slow). Studying the growth of emergent mutants and their invasion of the resident population in the limit of small mutation rates for a simultaneously diverging population size, we are able to determine their effective growth rates. We describe this growth as a mesoscopic scaling-limit of the orders of population sizes, where we observe an averaging effect of the invasion fitness. Moreover, we prove a limit result for the sequence of consecutive macroscopic resident traits that is similar to the so-called trait-substitution-sequence., Comment: 54 pages, 9 figures

With the need to feed a growing world population, the efficiency of crop production is of paramount importance. To support breeding and field management, various characteristics of the plant phenotype need to be measured -- a time-consuming process when performed manually. We present a robotic platform equipped with multiple laser and camera sensors for high-throughput, high-resolution in-field plant scanning. We create digital twins of the plants through 3D reconstruction. This allows the estimation of phenotypic traits such as leaf area, leaf angle, and plant height. We validate our system on a real field, where we reconstruct accurate point clouds and meshes of sugar beet, soybean, and maize., Comment: Accepted for IEEE Robotics and Automation Magazine

Unsupervised and self-supervised representation learning has become popular in recent years for learning useful features from unlabelled data. Representation learning has been mostly developed in the neural network literature, and other models for representation learning are surprisingly unexplored. In this work, we introduce and analyze several kernel-based representation learning approaches: Firstly, we define two kernel Self-Supervised Learning (SSL) models using contrastive loss functions and secondly, a Kernel Autoencoder (AE) model based on the idea of embedding and reconstructing data. We argue that the classical representer theorems for supervised kernel machines are not always applicable for (self-supervised) representation learning, and present new representer theorems, which show that the representations learned by our kernel models can be expressed in terms of kernel matrices. We further derive generalisation error bounds for representation learning with kernel SSL and AE, and empirically evaluate the performance of these methods in both small data regimes as well as in comparison with neural network based models.

Self-Supervised Learning (SSL) is an important paradigm for learning representations from unlabelled data, and SSL with neural networks has been highly successful in practice. However current theoretical analysis of SSL is mostly restricted to generalisation error bounds. In contrast, learning dynamics often provide a precise characterisation of the behaviour of neural networks based models but, so far, are mainly known in supervised settings. In this paper, we study the learning dynamics of SSL models, specifically representations obtained by minimising contrastive and non-contrastive losses. We show that a naive extension of the dymanics of multivariate regression to SSL leads to learning trivial scalar representations that demonstrates dimension collapse in SSL. Consequently, we formulate SSL objectives with orthogonality constraints on the weights, and derive the exact (network width independent) learning dynamics of the SSL models trained using gradient descent on the Grassmannian manifold. We also argue that the infinite width approximation of SSL models significantly deviate from the neural tangent kernel approximations of supervised models. We numerically illustrate the validity of our theoretical findings, and discuss how the presented results provide a framework for further theoretical analysis of contrastive and non-contrastive SSL.

We study Fano varieties endowed with a faithful action of a symmetric group, as well as analogous results for Calabi--Yau varieties, and log terminal singularities. We show the existence of a constant $m(n)$, so that every symmetric group $S_k$ acting on an $n$-dimensional Fano variety satisfies $k \leq m(n)$. We prove that $m(n)> n+\sqrt{2n}$ for every $n$. On the other hand, we show that $\lim_{n\to \infty} m(n)/(n+1)^2 \leq 1$. However, this asymptotic upper bound is not expected to be sharp. We obtain sharp bounds for certain classes of varieties. For toric varieties, we show that $m(n)=n+2$ for $n\geq 4$. For Fano quasismooth weighted complete intersections, we prove the asymptotic equality $\lim_{n\to \infty} m(n)/(n+1)=1$. Among the Fano weighted complete intersections, we study the maximally symmetric ones and show that they are closely related to the Fano--Fermat varieties, i.e., Fano complete intersections in $\mathbb P^N$ cut out by Fermat hypersurfaces. Finally, we draw a connection between maximally symmetric Fano varieties and boundedness of Fano varieties. For instance, we show that the class of $S_8$-equivariant Fano $4$-folds forms a bounded family. In contrast, the $S_7$-equivariant Fano $4$-folds are unbounded., Comment: 32 pages, 3 tables