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Pairs of commuting integer matrices
No full textWe prove upper and lower bounds on the number of pairs of commuting n x n matrices with integer entries in [-T, T], as T -> . Our work uses Fourier analysis and leads to an analysis of exponential sums involving matrices over finite fields. These are bounded by combining a stratification result of Fouvry and Katz with a new result about the flatness of the commutator Lie bracket
The ionizing photon production efficiency of star-forming galaxies at z ∼ 4–10
No full textContext. Investigating the ionizing emission of star-forming galaxies and the escape fraction of ionizing photons is critical to understanding their contribution to reionization and their impact on the surrounding environment. The number of ionizing photons available to reionize the intergalactic medium (IGM) depends on not only the abundance of galaxies but also their efficiency in producing ionizing photons (ξion). This quantity is thus fundamental to quantify the role of faint versus bright sources in driving this process, as we must assess their relative contribution to the total ionizing emissivity.
Aims. Our goal is to estimate the ξion using Balmer lines (Hα or Hβ) in a sample of 761 galaxies at 4 ≤ z ≤ 10 selected from different JWST spectroscopic surveys. We aim to determine the redshift evolution of ξion and the relation of ξion with the physical properties of the galaxies.
Methods. We used the available HST and JWST photometry to perform a spectral energy distribution (SED) fitting in the sample to determine their physical properties and relate them with ξion. We used the BAGPIPES code for the SED fitting and assumed a delayed exponential model for the star formation history. We used the NIRSpec spectra from prism or grating configurations to estimate Balmer luminosities, and then constrained ξion values after dust correction.
Results. We find a mean value of 1025.22 Hz erg−1 for ξion in the sample with an observed scatter of 0.42 dex. We find an increase in the median values of ξion with redshift from 1025.09 Hz erg−1 at z ∼ 4.18 to 1025.28 Hz erg−1 at z ∼ 7.14, confirming the redshift evolution of ξion found in other studies. Regarding the relation between ξion and physical properties, we find a decrease in ξion with increasing stellar mass, indicating that low-mass galaxies are efficient producers of ionizing photons. We also find an increase in ξion with increasing specific star formation rate (sSFR) and increasing UV absolute magnitude. This indicates that faint galaxies and galaxies with high sSFR are also efficient producers. We also investigated the relation of ξion with the rest-frame equivalent width (EW) of [OIII]λ5007 and find that galaxies with the higher EW([OIII]λ5007) are more efficient producers of ionizing photons, with the best fit leading to the relation log(ξion) = 0.43 × log(EW[OIII])+23.99. Similarly, we find that galaxies with higher O32 = [OIII]λ5007/[OII]λλ3727,3729 and lower gas-phase metallicities (based on the R23 = ([OIII]λλ4959,5007+[OII]λλ3727,3729)/Hβ calibration) show higher ξion values
A tutorial for mesoscale computer simulations of lipid membranes: Tether pulling, tubulation and fluctuations
No full textLipid membranes and membrane deformations are a long-standing area of research in soft matter and biophysics. Computer simulations have complemented analytical and experimental approaches as one of the pillars in the field. However, setting up and using membrane simulations can come with barriers due to the multidisciplinary effort involved and the vast choice of existing simulations models. In this review, we introduce the non-expert reader to coarse-grained membrane simulations at the mesoscale. Firstly, we give a concise overview of the modelling approaches to study fluid membranes, together with guidance to more specialized references. Secondly, we provide a conceptual guide on how to develop mesoscale membrane simulations. Lastly, we construct a hands-on tutorial on how to apply mesoscale membrane simulations, by providing a pedagogical examination of membrane tether pulling, shape and mechanics of membrane tubes, and membrane fluctuations with three different membrane models, and discussing them in terms of their scope and how resource-intensive they are. To ease the reader's venture into the field, we provide a repository with ready-to-run tutorials
Complex K-theory of moduli spaces of Higgs bundles
No full textWe establish an isomorphism of complex K-theory of the moduli space M of “SL n ”-Higgs bundles of degree d and rank n (in the sense of Hausel–Thaddeus) and twisted complex K-theory of the orbifold M of PGL n -Higgs bundles of degree e, where (n,d)=(n,e)=1. Along the way, we prove the vanishing of torsion for H ∗ ( M ) and certain twisted complex K-theory groups of M . We also extend Arinkin’s autoduality of compactified Jacobian to a derived equivalence between SL n - and PGL n -Hitchin systems over the elliptic locus. In the appendix, we develop a formalism of G-sheaves of spectra, generalising equivariant homotopy theory to a relative setting
The neural basis of species-specific defensive behaviour in Peromyscus mice
No full textEvading imminent threat from predators is critical for animal survival. Effective defensive strategies can vary, even between closely related species. However, the neural basis of such species-specific behaviours remains poorly understood1,2,3,4. Here we find that two sister species of deer mice (genus Peromyscus)5 show different responses to the same looming stimulus: Peromyscus maniculatus, which occupies densely vegetated habitats, predominantly escapes, whereas the open field specialist, Peromyscus polionotus, briefly freezes. This difference arises from species-specific escape thresholds, is largely context-independent, and can be triggered by both visual and auditory threat stimuli. Using immunohistochemistry and electrophysiological recordings, we find that although visual threat activates the superior colliculus in both species, the role of the dorsal periaqueductal grey (dPAG) in driving behaviour differs. Whereas dPAG activity scales with running speed in P. maniculatus, neural activity in the dPAG of P. polionotus correlates poorly with movement, including during visually triggered escape. Moreover, optogenetic activation of dPAG neurons elicits acceleration in P. maniculatus but not in P. polionotus, and their chemogenetic inhibition during a looming stimulus delays escape onset in P. maniculatus to match that of P. polionotus. Together, we trace species-specific escape thresholds to a central circuit node, downstream of peripheral sensory neurons, localizing an ecologically relevant behavioural difference to a specific region of the mammalian brain
Wide neural networks trained with weight decay provably exhibit neural collapse
No full textDeep neural networks (DNNs) at convergence consistently represent the training data in the last layer via a geometric structure referred to as neural collapse. This empirical evidence has spurred a line of theoretical research aimed at proving the emergence of neural collapse, mostly focusing on the unconstrained features model. Here, the features of the penultimate layer are free variables, which makes the model data-agnostic and puts into question its ability to capture DNN training. Our work addresses the issue, moving away from unconstrained features and
studying DNNs that end with at least two linear layers. We first prove generic guarantees on neural collapse that assume (i) low training error and balancedness of linear layers (for within-class variability collapse), and (ii) bounded conditioning of the features before the linear part (for orthogonality of class-means, and their alignment with weight matrices). The balancedness refers to the fact that W⊤ℓ+1Wℓ+1 ≈ WℓW⊤ℓfor any pair of consecutive weight matrices of the linear part, and the bounded conditioning requires a well-behaved ratio between largest and smallest non-zero singular values of the features. We then show that such assumptions hold for gradient descent training with weight decay: (i) for networks with a wide first layer, we prove low training error and balancedness, and (ii) for solutions that are either nearly optimal or stable under large learning rates, we additionally prove the bounded conditioning. Taken together, our results are the first to show neural collapse in the end-to-end training of DNNs
Unequal segregation of mitochondria during asymmetric cell division contributes to cell fate divergence in sister cells in vivo
No full textThe unequal segregation of organelles has been proposed to be an intrinsic mechanism that contributes to cell fate divergence during asymmetric cell division; however, in vivo evidence is sparse. Using super-resolution microscopy, we analysed the segregation of organelles during the division of the neuroblast QL.p in C. elegans larvae. QL.p divides to generate a daughter that survives, QL.pa, and a daughter that dies, QL.pp. We found that mitochondria segregate unequally by density and morphology and that this is dependent on mitochondrial dynamics. Furthermore, we found that mitochondrial density in QL.pp correlates with the time it takes QL.pp to die. We propose that low mitochondrial density in QL.pp promotes the cell death fate and ensures that QL.pp dies in a highly reproducible and timely manner. Our results provide in vivo evidence that the unequal segregation of mitochondria can contribute to cell fate divergence during asymmetric cell division in a developing animal
PMLR
No full textLearning-based systems are increasingly deployed across various domains, yet the complexity of traditional neural networks poses significant challenges for formal verification. Unlike conventional neural networks, learned Logic Gate Networks (LGNs) replace multiplications with Boolean logic gates, yielding a sparse, netlist-like architecture that is inherently more amenable to symbolic verification, while still delivering promising performance. In this paper, we introduce a SAT encoding for verifying global robustness and fairness in LGNs. We evaluate our method on five benchmark datasets, including a newly constructed 5-class variant, and find that LGNs are both verification-friendly and maintain strong predictive performance
Computational design and fabrication of modular robots with untethered control
No full textNatural organisms utilize distributed actuation through their musculoskeletal
systems to adapt their gait for traversing diverse terrains or to morph their
bodies for varied tasks. A longstanding challenge in robotics is to emulate
this capability of natural organisms, which has motivated the development of
numerous soft robotic systems. However, such systems are generally optimized
for a single functionality, lack the ability to change form or function on
demand, or remain tethered to bulky control systems. To address these
limitations, we present a framework for designing and controlling robots that
utilize distributed actuation. We propose a novel building block that
integrates 3D-printed bones with liquid crystal elastomer (LCE) muscles as
lightweight actuators, enabling the modular assembly of musculoskeletal robots.
We developed LCE rods that contract in response to infrared radiation, thereby
providing localized, untethered control over the distributed skeletal network
and producing global deformations of the robot. To fully capitalize on the
extensive design space, we introduce two computational tools: one for
optimizing the robot's skeletal graph to achieve multiple target deformations,
and another for co-optimizing skeletal designs and control gaits to realize
desired locomotion. We validate our framework by constructing several robots
that demonstrate complex shape morphing, diverse control schemes, and
environmental adaptability. Our system integrates advances in modular material
building, untethered and distributed control, and computational design to
introduce a new generation of robots that brings us closer to the capabilities
of living organisms