Symmetries rigidly delimit the landscape of quantum matter. Recently uncovered spatially modulated symmetries, whose actions vary with position, enable excitations with restricted mobility, while Lieb-Schultz-Mattis (LSM) type anomalies impose sharp constraints on which lattice phases are realizable. In one dimensional a spin chain, gauging procedures have linked modulated symmetry to LSM type anomaly, but a general understanding beyond 1D remains incomplete. We show that spatially modulated symmetries and their associated dipole algebras naturally emerge from gauging ordinary symmetries in the presence of generalized LSM type anomalies. We construct explicit lattice models in two and three spatial dimensions and develop complementary field theoretic descriptions in arbitrary spatial dimensions that connect LSM anomaly inflow to higher-group symmetry structures governing the modulated symmetries. Our results provide a unified, nonperturbative framework that ties together LSM constraints and spatially modulated symmetries across dimensions.

We present an improved zodiacal light model, optimized for optical wavelengths, using archival Hubble Space Telescope (HST) imaging from the SKYSURF program. The Kelsall et al. 1998 model used infrared imaging from the Diffuse Infrared Background Experiment (DIRBE) on board the Cosmic Background Explorer to create a 3D structure of the interplanetary dust cloud. However, this model cannot accurately represent zodiacal light emission outside of DIRBE's nominal wavelength bandpasses, the bluest of which is 1.25 micron. We present a revision to this model (called ZodiSURF) that incorporates analytical forms of both the scattering phase function and albedo as a function of wavelength, which are empirically determined across optical wavelengths (0.3-1.6 micron) from over 5,000 HST sky surface brightness (sky-SB) measurements. This refined model results in significantly improved predictions of zodiacal light emission at these wavelengths and for Sun angles greater than 80 deg. Fits to HST data show an uncertainty in the model of ~4.5%. Remarkably, the HST sky-SB measurements show an excess of residual diffuse light (HST Sky - ZodiSURF - Diffuse Galactic Light) of 0.013 +/- 0.006 MJy/sr. We suggest that a very dim spherical dust cloud may need to be included in the zodiacal light model, which we present here as a toy model.

We experimentally study a scallop-like swimmer with reciprocally flapping wings in a nearly frictionless, cohesive granular medium consisting of hydrogel spheres. Significant locomotion is found when the swimmer's flapping frequency matches the inverse relaxation time of the material. Remarkably, the swimmer moves in the opposite direction compared to its motion in a cohesion-free granular material of hard plastic spheres. At higher or lower frequencies, we observe no motion of the swimmer, apart from a short initial transient phase. X-ray radiograms reveal that the wing motions create low-density zones, which in turn give rise to a hysteresis in drag and propulsion forces. This time-dependent effect, combined with the swimmer's inertia, accounts for locomotion at intermediate frequencies.

The IceCube Neutrino Observatory has detected several hundred high-energy neutrinos from cosmic sources. Despite numerous studies searching for their origin, it is still not known which sources emit them. A few likely individual associations exist with active galactic nuclei (AGNs), mostly comprising blazars (AGNs with jets pointed toward Earth). Nonetheless, on a population level, blazar-neutrino correlation strengths are rather weak. This could mean that blazars as a population do not emit high-energy neutrinos or that the detection power of the tests is insufficient due to the strong atmospheric neutrino background. By assuming an increase in high-energy neutrino emission during major blazar flares, in our previous studies, we leveraged the neutrino arrival time to boost the detection power. Here, we utilize the same principle, while substantially increasing the number of blazars in the sample. We searched for the spatiotemporal correlation of 356 IceCube high-energy neutrinos with major optical flares of 3225 radio- and 3814 $γ$-ray-selected blazars. We found that despite the increase in data size, the number of likely spatiotemporal associations remained low and the overall correlation strengths weak. Two individual associations were shown to drive our strongest correlation, namely, the only $>$2$σ$ post-trial spatiotemporal correlation, occurring with the BL Lac objects of the radio-selected blazar sample. We estimated that $\lesssim$8\% of the detected cosmic neutrinos were emitted by blazars during major optical flares. As a complementary analysis, we compared the synchrotron peak frequency, redshift, Doppler factor, X-ray brightness, and optical variability of spatially neutrino-associated blazars to those of the general blazar population. We found that spatially neutrino-associated blazars have a Doppler factor and X-ray brightness that are higher than average.

Active galactic nuclei (AGN) are some of the brightest and most variable objects in the Universe. Those with relativistic jets observed at small viewing angles are blazars. Due to Doppler boosting, blazars exhibit extreme stochastic variability. While the origin of this variability is thought to be changes in the accretion flow and jet dynamics, much about blazar variability remains unknown. In this paper we use several blazar-dominated AGN samples to form a catalog of 7918 blazars and candidates -- the largest to date. We also collected source types, redshifts, peak frequencies of the spectral energy distribution, radio variability Doppler factors, and X-ray flux densities for as many sources as possible. We used all-sky surveys (CRTS, ATLAS, and ZTF, abbreviated as ``CAZ'') to extract their optical multiband flux density on a nightly basis between 2007 and 2023, and we constructed as long and as high cadence light curves as possible for as many sources as attainable. We quantified the variability of the light curves and applied the Bayesian blocks algorithm to determine their flaring periods. The CAZ catalog and light curves as well as the corresponding Bayesian blocks and flaring periods are all provided in the accompanying electronic tables, with the goal of enabling analyses involving jetted AGN variability with unprecedented sample sizes. Overall, we find (1) optical flares generally have a faster rise than decay; (2) optical brightness and variability are strongly dependent on the synchrotron peak frequency; (3) flat spectrum radio quasars and BL Lac objects have comparable optical variability and flare characteristics at the same synchrotron peak frequency; and (4) optical flare times tend to decrease while amplitudes increase with an increasing radio variability Doppler factor.

Ferrons are a type of quasiparticle corresponding to elementary excitations of the ferroelectric order. Analogously to how magnons modulate and transport magnetization, ferrons modulate and transport electric polarization. Here, we introduce multiferrons as elementary excitations with both electric and magnetic character. Multiferrons lead to a tilt and elliptical precession of the polarization and at the same time create a magnetization through the mechanism of dynamical multiferroicity. Using first-principles calculations for LiNbO$_3$, we show that the electric polarization of multiferrons is perpendicular to the equilibrium ferroelectric polarization, whereas the magnetization is parallel to it. Our calculations further demonstrate that multiferrons carry net electric and magnetic quadrupole and octupole moments, which we term multipolons. These multipolons could couple to internal multipolar degrees of freedom, for example in altermagnets, or to external probes such as neutrons, leading to potentially experimentally observable phenomena following coherent or thermal excitation of multiferrons.

We introduce a new analytical framework for modelling degree sequences in individual communities of real-world networks, e.g., citations to papers in different fields. Our work is inspired by a recent modification of the Price's model, which assumes that citations are gained partly accidentally, and to some extent preferentially. Our work addresses the need to represent the heterogeneity of various scientific domains, as standard homogeneous models fail to capture the distinct growth ratios and citing cultures of different fields. Extending the model to networks with a community structure allows us to devise the analytical formulae for, amongst others, citation counts in each cluster and their inequality as described by the Gini index. We also show that a citation count distribution in each community tends to a Pareto type II distribution. Thanks to the derived model parameter estimators, the new model can be fitted to real citation and similar networks.

Late-time cosmic acceleration has motivated the exploration of various extensions of general relativity, among which $f(Q,\mathcal{T})$ gravity, based on the non-metricity scalar $Q$ and the trace of the energy--momentum tensor $\mathcal{T}$, has gained increasing attention. In this study, we explore the thermodynamic aspects of $f(Q,\mathcal{T})$ gravity by establishing the first law and generalized second law of thermodynamics at the apparent horizon of a flat FLRW universe. By applying the Gibbs relation, we determined the rate of change of the total entropy and assessed the conditions under which the generalized second law remains valid for various choices of $f(Q,\mathcal{T})$. Our analysis focuses on linear, power-law, quadratic trace, exponential, and cross-coupling models, inspired by frameworks such as $f(R,\mathcal{T})$, $f(T)$, and modifications motivated by string theory. Our analysis showed that linear and mildly nonlinear models are generally thermodynamically consistent, whereas strongly nonlinear or interaction-type models require fine-tuned parameters for the generalized second law to hold. The present analysis underscores that thermodynamic considerations serve as effective criteria for assessing the viability of modified gravity models and their relevance to cosmological dynamics.

While the Einstein Telescope and Cosmic Explorer proposals for next-generation, ground-based detectors promise vastly improved sensitivities to gravitational-wave signals, only joint observations are expected to enable the full scientific potential of these facilities, making timing and coordination between the efforts crucial to avoid missed opportunities. This study investigates the impact of long-term delays on the scientific capabilities of next-generation detector networks. We use the Fisher information formalism to simulate the performance of a set of detector networks for large, fiducial populations of binary black holes, binary neutron stars, and primordial black-hole binaries. Bootstrapping the simulated populations, we map the expected observation times required to reach a number of observations fulfilling scientific targets for key sensitivity and localization metrics across various network configurations. We also investigate the sensitivity to stochastic backgrounds. We find that purely sensitivity-driven metrics such as the signal-to-noise ratio are not strongly affected by delays between facilities. This is contrasted by the localization metrics, which are very sensitive to the number of detectors in the network and, by extension, to delayed observation campaigns for a detector. Effectively, delays in one detector behave like network-wide interruptions for the localization metrics for networks consisting of two next-generation facilities. We examine the impact of a supporting, current-generation detector such as LIGO India operating concurrently with next-generation facilities and find such an addition will greatly mitigate the negative effects of delays for localization metrics, with important consequences on multi-messenger science and stochastic searches.

One or two negative mass singularities are found to occur in static inhomogeneous spatially closed solutions to the Einstein equations. The singularities produce a positive Komar mass, and this decreases the size of the cosmological constant relative to normal matter. The energy density of a perfect fluid vanishes at the singularities and is finite elsewhere. Numerical examples of these static solutions are provided, and their stability properties are found to be similar to the Einstein static universe. In an expanding universe, the effect of the singularities is to push the acceleration towards more positive values. Given the sentiment that naked singularities are to be avoided, we review just how benign the negative mass singularity is.

We introduce RGFlow, a deep neural network-based real-space renormalization group (RG) framework tailored for continuum scalar field theories. Leveraging generative capabilities of flow-based neural networks, RGFlow autonomously learns real-space RG transformations from data without prior knowledge of the underlying model. In contrast to conventional approaches, RGFlow is bijective (information-preserving) and is optimized based on the principle of minimal mutual information. We demonstrate the method on two examples. The first one is a one-dimensional Gaussian model, where RGFlow is shown to learn the classical decimation rule. The second is the two-dimensional phi^4 theory, where the network successfully identifies a Wilson-Fisher-like critical point and provides an estimate of the correlation-length critical exponent.

A calculation of the QED vacuum polarization potential in the Coulomb field of a pointlike nucleus was presented in an earlier publication by the author and his collaborators. Corrections up to order $α^2 (Zα)^7$ were evaluated, where $Z$ is the nuclear charge number and $Zα$ is treated as an independent variable. These corrections correspond to two-loop Feynman graphs with proper propagators of fermions in the external field. The calculation employed a reduction to free QED, leading to free QED Feynman graphs with up to eight independent loops. The method of calculation is described here in detail.

In recent years, significant research efforts have focused on improving blockchain throughput and confirmation speeds without compromising security. While decreasing the time it takes for a transaction to be included in the blockchain ledger enhances user experience, a fundamental delay still remains between when a transaction is issued by a user and when its inclusion is confirmed in the blockchain ledger. This delay limits user experience gains through the confirmation uncertainty it brings for users. This inherent delay in conventional blockchain protocols has led to the emergence of preconfirmation protocols -- protocols that provide users with early guarantees of eventual transaction confirmation. This article presents a Systematization of Knowledge (SoK) on preconfirmations. We present the core terms and definitions needed to understand preconfirmations, outline a general framework for preconfirmation protocols, and explore the economics and risks of preconfirmations. Finally, we survey and apply our framework to several implementations of real-world preconfirmation protocols, bridging the gap between theory and practice.

Optimal control for fully observed diffusion processes is well established and has led to numerous numerical implementations based on, for example, Bellman's principle, model free reinforcement learning, Pontryagin's maximum principle, and model predictive control. In contrast, much fewer algorithms are available for optimal control of partially observed processes. However, this scenario is central to the digital twin paradigm, where a physical twin is partially observed and control laws are derived based on a digital twin. In this paper, we contribute to this challenge by combining data assimilation in the form of the ensemble Kalman filter with the recently proposed McKean-Pontryagin approach to stochastic optimal control. We derive forward evolving mean-field evolution equations for states and co-states which simultaneously allow for an online assimilation of data as well as an online computation of control laws. The proposed methodology is therefore perfectly suited for real time applications of digital twins. We present numerical results for controlled Lorenz-63 and Lorenz-96 systems as well as an inverted pendulum.

Graph states are entangled states that are essential for quantum information processing. As experimental advances enable the realization of large-scale graph states, efficient fidelity estimation methods are crucial for assessing their robustness against noise. However, calculations of exact fidelity become intractable for large systems due to the exponential growth in the number of stabilizers. In this work, we show that the fidelity between any ideal graph state and its noisy counterpart under IID Pauli noise can be mapped to the partition function of a classical spin system, enabling efficient computation via statistical mechanical techniques. Using this approach, we analyze the fidelity for regular graph states under depolarizing noise and uncover the emergence of phase transitions in fidelity between the pure-state regime and the noise-dominated regime. Specifically, in 2D, phase transitions occur only when the degree satisfies $d\ge 6$, while in 3D they already appear at $d\ge 5$. However, for graph states with excessively high degree, such as fully connected graphs, the phase transition disappears. Robustness of graph states against noise is thus determined by their connectivity and spatial dimensionality. Graph states with lower degree and/or dimensionality, which exhibit a smooth crossover, demonstrate greater robustness, while highly connected or higher-dimensional graph states are more fragile. Extreme connectivity, as the fully connected graph state possesses, restores robustness. Furthermore, we show that the fidelity can be rewritten in the form of the partition function of a constraint-percolation problem. Within this picture, we discuss the qualitative difference between 2D regular graph states with $d=6$ and $d=5$ regarding the presence or absence of a phase transition, as well as the suppressed critical behavior of fully connected graph states.

Let $X(\mathbb{R}^n)$ be a Banach function space and $Ω\subseteq\mathbb{R}^n$ be a measurable set of positive measure. For a Fourier multiplier $a$ on $X(\mathbb{R}^n)$, consider the Wiener-Hopf type operator $W_Ω(a):=r_ΩF^{-1}aF e_Ω$, where $F^{\pm 1}$ are the Fourier transforms, $r_Ω$ is the operator of restriction from $\mathbb{R}^n$ to $Ω$ and $e_Ω$ is the operator of extension by zero from $Ω$ to $\mathbb{R}^n$. Let $X_2(Ω)$ be the closure of $L^2(Ω)\cap X(Ω)$ in $X(Ω)$. We show that if $X(Ω)$ satisfies the so-called weak doubling property, then \[ \|a\|_{L^\infty(\mathbb{R}^n)} \le \|W_Ω(a)\|_{\mathcal{B}(X_2(Ω),X(Ω))}. \] Further, we prove that if $X(Ω)$ satisfies the so-called separated doubling property, then the Kuratowski measure of noncompactness of $W_Ω(a)$ admits the following lower estimate: \[ \frac{1}{2}\|a\|_{L^\infty(\mathbb{R}^n)} \le \|W_Ω(a)\|_{\mathcal{B}(X_2(Ω),X(Ω)),κ}. \] These results are specified to the case of variable Lebesgue spaces $L^{p(\cdot)}(C,w)$ with Muckenhoupt type weights $w$ over open cones $C\subseteq\mathbb{R}^n$ with the vertex at the origin.

We use FIRE-2 cosmological zoom-in hydrodynamic simulations to investigate the co-evolution between Milky Way-size galaxies and their host dark matter halos. We find that the formation of these galaxies follows a two-phase pattern, with an early phase featured by hot dynamics, bulge-dominated structure and bursty star formation, and a later phase featured by cold dynamics, disk-dominated structure and steady star formation. The transition times of these galaxy properties are correlated with the time when the host halo transits from fast to slow accretion, indicating the two-phase assembly of halos as a potential mechanism that drives the two-phase formation of galaxies. The physical origin of dynamical hotness can be summarized into two modes of star formation: a scattered mode in which stars form at large radii within cold gas streams associated with fast assembly of halos, and a concentrated mode in which stars form at small radii through violent fragmentation from globally self-gravitated gas when halo assembly is about to slow down. Cold gaseous and stellar disks can form when the conditions of the two modes are removed by the stall of fast halo assembly and the reduction of gas by feedback processes. The two modes of star formation leave distinct imprints on the structural properties of high-redshift galaxies, providing implications to be tested by JWST and future observations.

Accurate electroencephalography (EEG) and magnetoencephalography (MEG) source localization and reconstruction are essential for understanding brain function, yet remain challenging because the underlying EEG/MEG inverse problem is inherently ill-posed. Spatial filtering (beamforming) approaches, such as linearly constrained minimum variance (LCMV) spatial filters, are widely used and well supported by existing analysis software. In this work, we extend this framework by deriving a novel family of unbiased multi-source neural activity indices that form the localization stage of a two-stage spatial-filtering-based localization-reconstruction framework for the EEG/MEG inverse problem. In contrast to existing formulations, the proposed indices do not require knowledge of the target source covariance matrix, making them directly applicable in practical experimental settings. Their compact algebraic forms enable straightforward and numerically efficient implementation. The framework is validated on simulated EEG data and its applicability is illustrated through an example involving experimental EEG data from an oddball paradigm. To facilitate adoption, we provide a full open-source implementation extending MNE-Python, accompanied by a practical tutorial.

DES-5Y supernovae, combined with DESI BAO, appear to favour Chevallier-Polarski-Linder $(w_0, w_a)$ dynamical dark energy over $Λ$CDM. arXiv:2408.07175 suggested that this is driven by a systematic in the DES pipeline, which particularly affects the low-redshift supernovae brought in from legacy surveys. It is difficult to investigate these data in isolation, however, as the complicated supernovae pipelines must properly account for selection effects. In this work, we discover that the Bayesian evidence previously found for flexknot dark energy (arXiv:2503.17342) is beaten by a magnitude offset between the low- and high-redshift supernovae. In addition, we find that the possible tension between DES-5Y and DESI is significantly reduced by such an offset. We also take the opportunity to trial Nested Bridge Sampling with Sequential Monte Carlo as an alternative method for calculating Bayes factors.

We introduce $\texttt{RandomMeas$.$jl}$, a modular and high-performance open-source software package written in Julia for implementing and analyzing randomized measurement protocols in quantum computing. Randomized measurements provide a powerful framework for extracting properties of quantum states and processes such as expectation values, entanglement, and fidelities using simple experimental procedures combined with classical post-processing, most prominently via the classical shadow formalism. $\texttt{RandomMeas$.$jl}$ covers the full randomized measurement workflow, from the generation of measurement settings for use on a quantum computer, the optional classical simulation of randomized measurements with tensor networks, to a suite of estimators for physical properties based on classical shadows. The package includes advanced features such as robust and shallow shadow techniques, batch estimators, and built-in statistical uncertainty estimation. Its unified, composable design enables the scalable application and further development of randomized measurements protocols across theoretical and experimental contexts.

Nanofluidic memristors promise brain-inspired information processing with ions, yet their microscopic origin remains debated. So far, ionic memory has been attributed to ion-specific interactions, dynamic wetting, chemical reactions or mechanical deformations, yet typically without direct experimental evidence. Here, by combining operando interferometric imaging with electrokinetic measurements, we directly visualize voltage-induced blistering of the confining walls of two-dimensional (2D) nanochannels, as key origin of memristive hysteresis. We identify two distinct classes of blisters: unidirectional, driven by electrostatic forces on surface charges, and bidirectional, arising from osmotic pressure due to concentration polarization. This mechanistic framework explains device evolution and device-to-device variability, and reframes stochastic blistering as a functional design element. Our results constitute a direct proof of electromechanical coupling as a robust pathway to ionic memory in 2D nanochannels and open routes toward high-performance ionic memristors and electrically actuated nanofluidic valves.

We prove that equality in a sharp lower bound for the first $p$-eigenvalue of the Hodge Laplacian on closed submanifolds in space forms can occur only on topological spheres, assuming positivity.

We study the plus-pure threshold (ppt) of hypersurfaces in mixed characteristic. We show that the ppt limits to the $F$-pure threshold (fpt) as we ramify the base DVR. Additionally, we show that analogs of some positive characteristic extremal singularities cannot attain the same `extremal' ppt values in the unramified setting. We also study equations which have controlled ramification when we adjoin their $p$-th roots as well as equations which admit $p$-th roots modulo $p^2$ (or modulo other values), bounding their ppts. In particular, given a complete unramified regular local ring of mixed characteristic $p>0$, $f^p + p^2 g$ does not define a perfectoid pure singularity for any $f$ and $g$. Finally, we compute bounds on the ppt of hypersurfaces related to elliptic curves. This gives examples where the ppt is neither the corresponding fpt in characteristic $p > 0$ nor the lct in characteristic zero. This also provides examples where $p$ times the ppt is not a jumping number, in stark contrast with the characteristic $p > 0$ picture.

The slice decomposition is a bijective method for enumerating planar maps (graphs embedded in the sphere) with control over face degrees. In this paper, we extend the slice decomposition to the richer setting of hypermaps, naturally interpreted as properly face-bicolored maps, where the degrees of faces of each color can be controlled separately. This setting is closely related with the two-matrix model and the Ising model on random maps, which have been intensively studied in theoretical physics, leading to several enumerative formulas for hypermaps that were still awaiting bijective proofs. Generally speaking, the slice decomposition consists in cutting along geodesics. A key feature of hypermaps is that the geodesics along which we cut are directed, following the canonical orientation of edges imposed by the coloring. This orientation requires us to introduce an adapted notion of slices, which admit a recursive decomposition that we describe. Using these slices as fundamental building blocks, we obtain new bijective decompositions of several families of hypermaps: disks (pointed or not) with a monochromatic boundary, cylinders with monochromatic boundaries (starting with trumpets or cornets having one geodesic boundary), and disks with a "Dobrushin" boundary condition. In each case, the decomposition ultimately expresses these objects as sequences of slices whose increments correspond to downward-skip free (Lukasiewicz-type) walks subject to natural constraints. Our approach yields bijective proofs of several explicit expressions for hypermap generating functions. In particular, we provide a combinatorial explanation of the algebraicity and of the existence of rational parametrizations for these generating functions when face degrees are bounded.

The purpose of this expository article is to give a down-to-hearth introduction to the notion of an arithmetic group and arithmetic manifold. To achieve this we have decided to bring two geometrical questions relating the growth of systole and kissing number in hyperbolic manifolds, as a motivational guide, whose answers so far are given by the use of arithmetic manifolds. As a consequence, we answer these questions in detail for dimension 2, we mention was is known for hyperbolic manifolds of higher dimension, and also for other locally symmetric spaces. We end the exposition with some open questions.

We study the complexity of the parameterised counting constraint satisfaction problem: given a set of constraints over a set of variables and a positive integer $k$, how many ways are there to assign $k$ variables to 1 (and the others to 0) such that all constraints are satisfied. Existing work has so far exclusively focused on restricted settings such as finding and counting homomorphisms between relational structures due to Grohe (JACM 2007) and Dalmau and Jonsson (TCS 2004), or the case of finite constraint languages due to Creignou and Vollmer (SAT 2012), and Bulatov and Marx (SICOMP 2014). In this work, we tackle a more general setting of Valued Parameterised Counting Constraint Satisfaction Problems (VCSPs) with infinite constraint languages. In this setting we are able to model significantly more general problems such as (weighted) parameterised factor problems on hypergraphs and counting weight-$k$ solutions of systems of linear equations, not captured by existing complexity classifications. We express parameterised VCSPs as parameterised Holant problems on uniform hypergraphs, and we establish complete and explicit complexity dichotomy theorems. For resolving the $\mathrm{P}$ vs. $\#\mathrm{P}$ question, we mainly rely on hypergraph gadgets, the existence of which we prove using properties of degree sequences necessary for realisability in uniform hypergraphs. For the $\mathrm{FPT}$ vs. $\#\mathrm{W}[1]$ question, we build upon the recently established combinatorial toolkit for parameterised holants on the special case of graphs by Aivasiliotis et al. (ICALP 2025) and also rely on an extension of the framework of the homomorphism basis due to Curticapean, Dell and Marx (STOC 17) to uniform hypergraphs. As a technical highlight, we also employ Curticapean's "CFI Filters'' (SODA 2024) to establish polynomial-time algorithms for isolating vectors in the homomorphism basis.

We consider solving a probably infinite dimensional operator equation, where the operator is not modeled by physical laws but is specified indirectly via training pairs of the input-output relation of the operator. Neural operators have proven to be efficient to approximate infinite dimensional operators. In this paper we analyze Tikhonov regularization with neural operators as surrogates for solving ill-posed operator equations. The analysis is based on balancing approximation errors of neural operators, regularization parameters, and noise. Moreover, we extend the approximation properties of neural operators from sets of continuous functions to Sobolev and Lebesgue spaces, which is crucial for solving inverse problems and we discuss the problem of finding an appropriate network structure of neural operators (training). Finally, we present some numerical experiments.

Basis pursuit is the problem of finding a vector with smallest $\ell_1$-norm among the solutions of a given linear system of equations. It is a well-known convex relaxation of the sparse affine feasibility problem, where sparse solutions to underdetermined systems are sought. Since basis pursuit admits a linear programming reformulation, standard LP solvers are directly applicable. We instead address the basis pursuit directly in its $\ell_1$-minimization form, without LP reformulation, via a scheme that uses alternating projections in its subproblems. These subproblems are designed to be inconsistent in the sense that they relate to two non-intersecting sets. Recently in [R. Behling, Y. Bello-Cruz and L.-R. Santos, SIAM J. Optim., 31 (2021), pp. 2863-2892], inconsistency coming from infeasibility has been shown to accelerate convergence of alternating projections. We deliberately enforce this inconsistency by constructing subproblems whose feasible sets are disjoint by design. We prove that the resulting $\ell_1$-radii converge linearly to the optimal value, and that when the solution is unique, all generated sequences converge linearly to it at a rate governed by a natural error bound between the feasible set and the optimal $\ell_1$-ball. The proposed method is numerically competitive against state-of-the-art open-source solvers on synthetic and real-world instances.

We prove Howe duality for an exceptional theta correspondence. To that end we exploit a pair of see-saw identities and relate the $K$-types of corresponding representations.

We investigate whether viscous cold dark matter (vCDM) in a $Λ$-dominated FLRW universe can alleviate the Hubble tension while satisfying thermodynamic constraints, examining both flat and curved geometries. We model vCDM with bulk viscosity $ζ= ζ_0\,(Ω_{vc}/Ω_{vc0})^m$, where $m$ determines the viscosity evolution and $Ω_{vc}$ is the density parameter of vCDM. We explore two particular scenarios: constant viscosity ($m=0$), and variable viscosity ($m$ free). Using Bayesian inference, we constrain these models with the latest datasets: the Pantheon+ SN Ia sample (both with SH0ES calibration, PPS, and without it, PP), $H(z)$ measurements from CC and BAO as separate datasets, and a Gaussian prior on $H_0$ from 2022 SH0ES baseline, $H_0=73.04 \pm 1.04$ km/s/Mpc (R22 prior). We compare the models via information criteria such as AIC, BIC, DIC, and Bayesian evidence. Our results reveal that the Hubble tension persists, although it shows partial alleviation ($\sim 1σ$ tension) in all investigated scenarios when local measurements are included. For the flat $m=0$ case, the joint analysis yields $H_0 = 71.05^{+0.62}_{-0.60}$ km/s/Mpc. Curved model initially favors $Ω_{K0} > 0$ (at more than $2σ$), but this preference shifts toward flatness once the PPS+R22 prior are included. Notably, the current viscosity is constrained to $ζ_0 \sim 10^6$ Pa s in all scenarios, in agreement with the thermodynamic requirements. Although model selection via BIC and Bayesian evidence favors $Λ$CDM, AIC and DIC show mild support for viscous models in some datasets. Bulk viscous models moderately improve fits but neither resolve the Hubble tension nor outperform the $Λ$CDM model. To achieve more robust constraints, future analyses should incorporate CMB observations, which are expected to break parameter degeneracies involving $m$ and $\tildeζ_0$.