The dark matter flux in a direct detection experiment depends on its local speed distribution. This distribution has been inferred from simulations of Milky Way-like galaxies, but such models serve only as proxies, given that no simulation directly captures the detailed evolution of our own Galaxy. This motivates alternative approaches that obtain this distribution directly from observations. In this work, we utilize 98 Milky Way analogues from the TNG50 simulation to develop and validate a procedure for inferring the dark matter speed distribution using the kinematics of nearby stars. We find that the dark matter that originated from old mergers, plus that from recent nonluminous accretions, is well described by a Maxwell-Boltzmann speed distribution centered at the local standard-of-rest velocity. Meanwhile, recently accreted dark matter from massive mergers has speeds that can be traced from the associated stellar debris of these events. The stellar populations systematically underestimate the velocity dispersion of their dark matter counterparts, but a simple kinematic boost brings the two into good alignment. Using the TNG50 host galaxies, we demonstrate that combining these two contributions provides an accurate reconstruction of the local dark matter speeds. As an application of the procedure to our own Galaxy, we utilize stellar kinematic data from Gaia to quantify how the dark matter remnants from the Milky Way's last major merger impact its speed distribution in the solar neighborhood.

Little Red Dots, discovered by the James Webb Space Telescope, are hypothesized to be active galactic nuclei containing a supermassive black hole, possibly surrounded by a dense stellar cluster, large amounts of gas, and likely by a population of stellar-mass black holes. We develop a simple nuclear star cluster model to evolve the rapid mass growth of black hole seeds into the supermassive regime. The combined processes of tidal disruption events, black hole captures, and gas accretion are accounted for self-consistently in our model. Given the observed number density of Little Red Dots, and under reasonable assumptions, we predict at least a few tens of tidal disruption events and at least a few black hole captures at z=4-6, with a tidal disruption event rate an order of magnitude larger than the black hole capture rate. We also estimate the uncertainties in these estimates. Finally, we comment on the low x-ray luminosity of Little Red Dots.

Model Predictive Control (MPC) is well understood in the deterministic setting, yet rigorous stability and performance guarantees for stochastic MPC remain limited to the consideration of terminal constraints and penalties. In contrast, this work analyzes stochastic economic MPC with an expected cost criterion and establishes closed-loop guarantees without terminal conditions. Relying on stochastic dissipativity and turnpike properties, we construct closed-loop Lyapunov functions that ensure $P$-practical asymptotic stability of a particular optimal stationary process under different notions of stochastic convergence, such as in distribution or in the $p$-th mean. In addition, we derive tight near-optimal bounds for both averaged and non-averaged performance, thereby extending classical deterministic results to the stochastic domain. Finally, we show that the abstract stochastic MPC scheme requiring distributional knowledge shares the same closed-loop properties as a practically implementable algorithm based only on sampled state information, ensuring applicability of our findings. Our findings are illustrated by a numerical example.

Altermagnets, which encompass a broad landscape of materials, are compensated collinear magnetic phases in which the antiparallel magnetic moments are related by a crystalline rotation. Here, we argue that collinear altermagnetic-like states can also be realized in lattices with an odd number of sublattices, provided that the electronic interactions promote non-uniform magnetic moments. We demonstrate this idea for a kagome metal whose band filling places the Fermi level close to the van Hove singularity. Combining phenomenological and microscopic modeling, we show that the intertwined charge density-wave and loop-current instabilities of this model lead to a wide parameter range in which orbital ferromagnetic, antiferromagnetic, and altermagnetic phases emerge inside the charge-ordered state. In the presence of spin-orbit coupling, their electronic structures display the usual spin-split fingerprints associated with the three types of collinear magnetic order. We discuss the possible realization of orbital altermagnetic phases in the $A$V$_3$Sb$_5$ family of kagome metals.

Among the emerging excess of massive, bright galaxies at Cosmic Dawn $z \gtrsim 9$ seen by the James Webb Space Telescope, several exhibit spectral features associated with active galactic nuclei (AGN). These AGN candidates suggest that supermassive black holes (SMBHs) grow rapidly in the early Universe. In a series of numerical experiments, we investigate how SMBHs grow within and influence the most massive galaxies at Cosmic Dawn using cosmological hydrodynamic zoom-in simulations run with the adaptive mesh refinement code \textsc{ramses}. Our suite of simulations explore how super-Eddington accretion, seed mass, and the strength of feedback influence SMBH-galaxy co-evolution in the most massive galaxies ($M_\star \gtrsim 10^8 M_\odot$) of the early Universe ($z \sim 15 - 9$). The environment which our numerical experiments reside in is an overdensity that collapses into a $\sim 10^{11} M_\odot$ halo by $z \sim 9$. Within this type of environment we find that SMBH growth is sensitive to stellar feedback which generates a turbulent-multiphase interstellar medium (ISM) that stochastically starves the SMBH. In the absence of AGN feedback, we find that the SMBH is starved $\sim 50\%$ of the time after the onset of star formation in the galaxy. SMBH growth can become self-regulated by AGN feedback if the SMBH becomes massive enough, either by accretion or seeding, for its feedback to dominate the surrounding nuclear region. We find no evidence of galaxy-scale, AGN-driven quenching in the star formation rate (SFR) across all simulations in our suite.

Classical simulation of quantum circuits plays a crucial role in validating quantum hardware and delineating the boundaries of quantum advantage. Among the most effective simulation techniques are those based on the stabilizer extent, which quantifies the overhead of representing non-Clifford operations as linear combinations of Clifford unitaries. However, finding optimal decompositions rapidly becomes intractable as it constitutes a superexponentially large optimization problem. In this work, we exploit symmetries in the computation of the stabilizer extent, proving that for real, diagonal, and real-diagonal unitaries, the optimization can be restricted to the corresponding subgroups of the Clifford group without loss of optimality. This ``strong symmetry reduction'' drastically reduces computational cost, enabling optimal decompositions of unitaries on up to seven qubits using a standard laptop -- far beyond previous two-qubit limits. Additionally, we employ a ``weak symmetry reduction'' method that leverages additional invariances to shrink the search space further. Applying these results, we demonstrate exponential runtime improvements in classical simulations of quantum Fourier transform circuits and measurement-based quantum computations on the Union Jack lattice, as well as new insights into the nonstabilizer properties of multicontrolled phase gates and unitaries generating hypergraph states. Our findings establish symmetry exploitation as a powerful route to scale classical simulation techniques and deepen the resource-theoretic understanding of quantum advantage.

A system $\boldsymbolλ_θ$ is developed that combines modal logic and simply-typed lambda calculus, and that generalizes the system studied by Montague and Gallin. Whereas Montague and Gallin worked with Church's simple theory of types, the system $\boldsymbolλ_θ$ is developed in the typed base theory most commonly used today, namely the simply-typed lambda calculus. Further, the system $\boldsymbolλ_θ$ is controlled by a parameter $θ$ which allows more options for state types and state variables than is present in Montague and Gallin. A main goal of the paper is to establish some basic metatheory of $\boldsymbolλ_θ$: (i) an Andrews-like characterization of its models in terms of combinatory logic is given, and this combinatory logic involves a $\mathsf{BCKW}$-like basis rather than an $\mathsf{SKI}$-like basis and (ii) semantic conservation and expressibility results relating $\boldsymbolλ_θ$ to the maximal system $\boldsymbolλ_ω$ are proven. Similar results are proven for the relation between $\boldsymbolλ_ω$ and$\boldsymbolλ$, the corresponding ordinary simply-typed lambda calculus. This answers a question of Zimmermann in the semantics of the simply typed setting. In a companion paper this is extended to Church's simple theory of types. We further develop a partial correspondence between a pure combinatory logic centered on the $\mathsf{BCKW}$-like basis and the weak deductive system for $\boldsymbolλ_ω$ wherein $β$-reduction is not allowed under a lambda abstract, and we use this to show partial deductive conservation between the maximal system $\boldsymbolλ_ω$ and the intermediary systems $\boldsymbolλ_θ$.

The evolved cosmological matter density field is fully determined by the initial matter density field at fixed cosmological parameters. However, the two-dimensional cosmological projected matter density field, relevant for weak-lensing and photometric galaxy studies, is fully determined by the initial projected matter density field only at the linear order. At non-linear order, the entire volume of initial matter contributes. We study a model for the evolved projected density field that is deterministic in the initial projected density fields and probabilistic in the effects of the remaining modes in the initial conditions. We write down predictions for the mean evolved projected field model using Lagrangian perturbation theory. We run a suite of small $N$-body simulations with fixed projected initial conditions and measure the statistical properties of the ensemble of evolved projected fields. Measurements and theory are in good agreement and show that the information on the initial projected fields is exponentially suppressed on non-linear scales. We implement this approach in a likelihood code and use Hamiltonian Monte-Carlo sampling to show that initial fields can be reconstructed even in the presence of non-trival mask features.

The approach taken by Gheorghiu, Gu and Pym in their paper on giving a base-extension semantics for Intuitionistic Multiplicative Linear Logic is an interesting adaptation of the work of Sandqvist for IPL to the substructural setting. What is particularly interesting is how naturally the move to the substructural setting provided a semantics for the multiplicative fragment of intuitionistic linear logic. Whilst ultimately the Gheorghiu, Gu and Pym used their foundations to provide a semantics for bunched implication logic, it begs the question, what of the rest of intuitionistic linear logic? In this paper, I present just such a semantics. This is particularly of interest as this logic has as a connective the bang, a modal connective. Capturing the inferentialist content of formulae marked with this connective is particularly challenging and a discussion is dedicated to this at the end of the paper.

Galaxy mergers play an important role in galaxy evolution. Therefore, accurate merger identifications are paramount for achieving a complete understanding of how galaxies evolve. As we enter the era of large, deep, high-resolution imaging surveys, we can observe mergers extending to even lower masses and higher redshifts. Despite low-mass galaxies being more common, many previous merger identification methods were calibrated for high-mass galaxies, which are easier to identify. To prepare for upcoming surveys, we train a convolutional neural network (CNN) using mock $\textit{HST}$ CANDELS images at $z\sim1$ created from the IllustrisTNG50 cosmological simulation. We successfully identify galaxy mergers between a wide range of galaxies ($10^8M_\odot < M_\star < 10^{12.5}M_\odot$, and $q\geq1:10$), achieving overall accuracy, purity, and completeness of $\sim65\%$. We show, for the first time, that a CNN trained on this diverse set of galaxies is capable of identifying major mergers, especially at early stages (74% accuracy), similar to that of networks trained at lower redshifts and/or higher masses (with accuracies between $66-80$%). We discuss the inherent limits of galaxy merger identification due to orientation angle, finding 98% of mergers are correctly identified from at least one angle, and 61% from the majority of angles. We additionally explore the confounding variables, such as star formation, to consider when applying to real data.This network enables the exploration of the impact of previously overlooked mergers of high mass ratio and low stellar masses on galaxy evolution in CANDELS, and can be expanded to surveys from $\textit{JWST}$, Rubin, $\textit{Roman}$, and $\textit{Euclid}$.

Here, we experimentally demonstrate the unique properties of Sb$_2$S$_3$ metasurfaces on monolithic and hybridised Sb$_2$S$_3$-Si platforms. Their advantages are particularly pronounced in the telecommunication spectral range, where both phases remain nearly lossless together with a large and stable refractive index contrast compared to other counterparts. The monolithic metasurface enables transmission modulation depths of up to 92\% and resonance shifts as large as $\sim$150~nm at the telecommunication wavelengths through laser-induced phase transitions from amorphous to $\sim$ 50\% polycrystalline states. Furthermore, we demonstrate that integrating a silicon overlayer introduces high-Q hybrid resonances with enhanced near-field confinement, enabling comparable modulation, experimentally, with nearly half the laser switching power required as compared to the monolithic structure. Beyond linear optical switching, the broad resonance tunability of the Sb$_2$S$_3$ metasurface, together with the strong third-order nonlinearity of these materials, provides a promising platform for tunable nonlinear light generation. By exciting the Sb$_2$S$_3$ metasurfaces with a fixed broadband pump, we experimentally demonstrate tunable third-harmonic generation emission over a $\sim$40~nm spectral range through phase-change transition. Such tunability provides a versatile route for integrating Sb$_2$S$_3$ with intrinsically highly nonlinear materials to enable the combination of large optical tunability and efficient nonlinear light generation. We have proven this concept via our hybrid Sb$_2$S$_3$-Si platform, which shows a threefold enhancement in tunable THG emission arising from the combined nonlinear responses of Sb$_2$S$_3$ and silicon, and benefiting from enhanced electromagnetic field confinement enabled by the metasurface's high-Q resonances.

We study the asymptotic behavior of random dimer coverings of the one-periodic Aztec diamond in random environment. We investigate quenched limit theorems for the height function and we extend annealed limit theorems that were recently studied in [arXiv:2507.08560]. We consider more general choices of random edge weights (independence is not assumed) and we distinguish two cases where the random edge weights satisfy the Central Limit Theorem (CLT) under different scalings. For both cases, we prove convergence to the Gaussian Free Field for the quenched fluctuations. For the annealed version, it had been shown in [arXiv:2507.08560], that Gaussian Free Field fluctuations can be dominated by the much larger fluctuations of the random environment. To access quenched fluctuations we analyze the Schur process with random parameters in a way that allows to prove the annealed CLT for the height function for non i.i.d. weights. We consider specific examples where we determine the asymptotic fluctuations.

We present optical + near-infrared (NIR) + mid-infrared (MIR) observations of the normal Type Ia supernovae (SN Ia) 2022aaiq and 2024gy in the nebular phase, continuously spanning 0.35-28 microns. Medium-resolution JWST spectroscopy reveals novel narrow ($v_{\mathrm{FWHM}}<1500$ km s$^{-1}$) [Ni II] 1.94 and 6.64 micron cores in both events. The MIR [Ni II] 6.64 micron line exhibits a distinct narrow core atop a broader base, indicating a central enhancement of stable Ni. This structure points to high central densities consistent with a near-Chandrasekhar-mass ($M_{\text{Ch}}$) progenitor or a high-metallicity sub-$M_{\text{Ch}}$ progenitor. From detailed line-profile inversions of SN 2024gy, we derive emissivity profiles for stable iron-group elements (IGEs), radioactive material, and intermediate-mass elements (IMEs), revealing spatially distinct ejecta zones. The [Ni III] 7.35 micron line shows a shallow-to-steep slope transition - a "broken-slope" morphology - that matches predictions for delayed detonation explosions with separated deflagration and detonation ashes. We also reanalyze and compare to archival JWST spectra of SN 2021aefx and the subluminous SN 2022xkq. From the stable Ni luminosities, we infer that SN 2024gy produced ~5-10 times more stable Ni mass than SN 2022xkq, favoring a near-$M_{\text{Ch}}$ scenario for SN 2024gy and sub-$M_{\text{Ch}}$ scenario for SN 2022xkq. These results demonstrate that resolved line profiles, now accessible with JWST, provide powerful diagnostics of explosion geometry, central density, and progenitor mass in SN Ia.

Pure spinor formalism and RNS formalism are related by a chain of equivalences constructed by introducing and integrating-out BRST quartets. This is known as B-RNS-GSS formalism. The first step is to add BRST quartets to the RNS model and do a similarity transformation on the space of fields. We show that this step can be understood as a strictification procedure which lifts a strong homotopy action of the supersymmetry algebra to a quasiisomorphic strict action. This observation allows us to clarify the details of the B-RNS-GSS derivation. We obtain a closed formula for the similarity transformation as a path-ordered exponential, and derive the supersymmetric currents.

We propose a boomerang mechanism for the explanation of the excess radio background detected by ARCADE 2. In an early stage of the Universe, at a temperature $T$ in the range $\sim 0.1\,{\rm keV}$--$1\,{\rm MeV}$, a fraction of relic neutrinos is resonantly converted into dark neutrinos by mixing induced by a pre-existing lepton asymmetry. Dark neutrinos decay much later into a dark-standard photon state and a dark fermion, with a lifetime longer than the age of the Universe, as required by a solution to the excess radio background. This scenario circumvents the upper bound on the neutrino magnetic moment but still implies a testable lower bound.

A second-order block coordinate descent method is proposed for the unconstrained minimization of an objective function with a Lipschitz continuous Hessian. At each iteration, a block of variables is selected by means of a greedy (Gauss-Southwell) rule which considers the amount of first-order stationarity violation, then an approximate minimizer of a cubic model is computed for the block update. In the proposed scheme, blocks are not required to have a predetermined structure and their size may change during the iterations. For non-convex objective functions, global convergence to stationary points is proved and a worst-case iteration complexity analysis is provided. In particular, given a tolerance $ε$, we show that at most ${\cal O(ε^{-3/2})}$ iterations are needed to drive the stationarity violation with respect to a selected block of variables below $ε$, while at most ${\cal O(ε^{-2})}$ iterations are needed to drive the stationarity violation with respect to all variables below $ε$. Numerical results are finally given, comparing the proposed approach with other second-order methods and block selection rules.

Managing stateful resources safely and expressively is a longstanding challenge in programming languages, especially in the presence of aliasing. For example, scope-based constructs like Java's synchronized blocks offer ease of reasoning, but they restrict expressiveness and parallelism. Conversely, imperative, flow-sensitive approaches enable fine-grained control, but they require sophisticated typestate analyses and often burden programmers with explicit state tracking. In this work, we present a novel approach that unifies the ease of scoped reasoning with the expressiveness of imperative typestate management. Our design extends traditional flow-insensitive capability mechanisms to a flow-sensitive setting. In particular, we decouple capability lifetimes from lexical scopes, allowing functions to receive, revoke, or return capabilities in a flow-sensitive manner, building on existing mechanisms for the safety and ergonomics of scoped capability programming. We implement our approach as an extension to the Scala 3 compiler, leveraging path-dependent types and implicit resolution to enable concise, statically safe, and expressive typestate programming. Our prototype generically supports a wide range of patterns, including file operations, advanced locking protocols, DOM construction, and session types, showing that expressive and safe typestate management can be achieved with minimal extensions to an existing language with capability support.

We introduce coarse scrambling, a novel randomization for digital sequences that permutes blocks of digits in a mixed-radix representation. This construction is designed to preserve the powerful $(0,\mathbb{e},d)$-sequence property of the underlying points. For sufficiently smooth integrands, we prove that this method achieves the canonical $O(n^{-3+ε})$ variance decay rate, matching that of standard Owen's scrambling. Crucially, we show that its maximal gain coefficient grows only logarithmically with dimension, $O(\log d)$, thus providing theoretical robustness against the curse of dimensionality affecting scrambled Sobol' sequences. Numerical experiments validate these findings and illustrate a practical trade-off: while Owen's scrambling is superior for integrands sensitive to low-dimensional projections, coarse scrambling is competitive for functions with low effective truncation dimension.

Reconstructing latent state-space geometry from time series provides a powerful route to studying nonlinear dynamics across complex systems. Delay-coordinate embedding provides the theoretical basis but assumes long, noise-free recordings, which many domains violate. In many real-world domains, recordings are short, noisy, and coarsely sampled; in neuroimaging, for example, fMRI additionally contains autocorrelated background structure that can obscure oscillatory components and destabilize embeddings. We propose bootstrap Monte Carlo singular spectrum analysis (BMC-SSA), which combines Monte Carlo SSA with bootstrap stability to retain oscillatory modes that are statistically supported and reproducible across resampled data. This produces reconstructions that emphasize reliable oscillatory structure, enhancing determinism and stabilizing subsequent embeddings. Our results show that BMC-SSA improves the reliability of functional measures and uncovers differences in state-space dynamics in fMRI, offering a general framework for robust embedding of noisy, finite signals.

We present updated observational constraints on the spatially flat $ϕ$CDM model, where dark energy is described by a minimally coupled scalar field $ϕ$ with an inverse power-law potential $V=V_0 ϕ^{-α}$. Using Planck 2018 CMB temperature, polarization (P18), and lensing power spectra (lensing), along with a compilation of non-CMB data including baryon acoustic oscillation, type Ia supernova, Hubble parameter, and growth rate measurements, we constrain $ϕ$CDM and $ϕ$CDM+$A_L$ models where $A_L$ is the CMB lensing consistency parameter. The scalar field parameter $α$, which governs dark energy dynamics, is more tightly constrained by non-CMB data than by CMB data alone. For the full dataset, we obtain $α= 0.055 \pm 0.041$ in the $ϕ$CDM model and $α= 0.095 \pm 0.056$ in the $ϕ$CDM+$A_L$ model, mildly favoring evolving dark energy over a cosmological constant by $1.3σ$ and $1.7σ$. The Hubble constant is $H_0=67.55_{-0.46}^{+0.53}$ km s$^{-1}$ Mpc$^{-1}$ in the $ϕ$CDM model, consistent with median statistics and some local determinations, but in tension with other local determinations. The constraints for matter density and clustering amplitude ($Ω_m = 0.3096 \pm 0.0055$, $σ_8 = 0.8013_{-0.0067}^{+0.0077}$) of the flat $ϕ$CDM model statistically agree with $Λ$CDM model values. Allowing $A_L$ to vary reduces tensions between CMB and non-CMB data, although we find $A_L = 1.105 \pm 0.037$, $2.8σ$ higher than unity, consistent with the excess smoothing seen in Planck data. Model comparison using AIC and DIC indicates that the $ϕ$CDM model provides a fit comparable to $Λ$CDM, with the $ϕ$CDM+$A_L$ slightly preferred. Overall, while the $Λ$CDM model remains an excellent fit, current data leave open the possibility of mildly evolving quintessence-like dynamical dark energy.

Since the 1970s, it has been known that any open connected manifold of dimension 2, 4 or 6 admits a complex analytic structure whenever its tangent bundle admits a complex linear structure. For half a century, this has been conjectured to hold true for manifolds of any dimension. In this paper, we extend the result to manifolds of dimension 8. To prove the result new $Γ_n^\mathbb{C}$-structures on $\mathbb{CP}^n$ are constructed. As a consequence we derive a theorem concerning the homology of Haefligers classifying space, $BΓ_n^\mathbb{C}$. The result then follows from obstruction theory.

Government trust, as a core concept in political economy and public policy research, serves as a fundamental cornerstone of democratic legitimacy and state capacity. This paper examines how environmental conditions, particularly sunlight efficiency, influence reported government trust through both affective and cognitive mechanisms. Leveraging World Values Survey Wave 7 data merged with NASA POWER high-frequency weather data, we propose and validate a novel ``salience and attribution'' mechanism: clearer skies may paradoxically reduce government trust by heightening environmental awareness and triggering negative attributions. We further identify potential mediating pathways, including subjective well-being, political interest, political discussion, and health perception, and demonstrate that environmental conditions introduce measurement error in survey-based trust indicators. Our findings provide theoretical contributions to environmental psychology, behavioral political economy, and survey methodology, and yield practical implications for governance, policy design, and survey

A real number is called left-computable if there exists a computable increasing sequence of rational numbers converging to it. In this article we investigate the Kolmogorov complexity and the binary expansions of a very specific subset of the left-computable numbers. We show in our main result that a real number is reordered computable if, and only if, it is left-computable and not strongly Kurtz random. In preparation of this, we characterize strong Kurtz randomness by a suitable notion of randomness tests. We also look at the binary expansions of reordered computable numbers and clarify whether they can be immune, hyperimmune, hyperhyperimmune, strongly hyperhyperimmune, or cohesive. Then, we investigate the effective Hausdorff and packing dimensions of reordered computable numbers. Finally, we have a short look at regular reals in the context of immunity properties, Kolmogorov complexity and (strong) Kurtz randomness.

We prove optimal non-commutative analogues of the classical Law of Iterated Logarithm (LIL) for both martingales and sequences of independent (non-commutative) random variables. The classical martingale version was established by Stout [Sto70b] and the independent case by Hartman-Wintner [HW41]. Our approach relies on a key exponential inequality essentially due to Randrianantoanina [Ran24] that improves that from Junge and Zeng [JZ15]. It allows to derive an optimal non-commutative Stout-type LIL just as in [Zen15], from that martingale result we then deduce a non-commutative Hartman-Wintner type LIL for independent sequences of random variables.

Fighting fish is a combinatorial configuration introduced by Duchi, Guerrini, Rinaldi and Schaeffer as a new model of branching surfaces that generalizes directed convex polyominoes. We come up with an alternative construction of fighting fish, using a tree structure built on the so-called stem cells of fighting fish. From this perspective, we establish a bijection between ternary trees and fighting fish with a marked strip of cells, which specializes to a direct bijection between left ternary trees and fighting fish. Using these results, we obtain a combinatorial enumeration of the fighting fish of size $n$ by establishing an $(n+1)$-to-2 bijection with the ternary trees with $n$ nodes. We present some additional enumerative results including that fighting fish with a marked tail and horizontally symmetric fighting fish are equinumerous with ordered pairs of ternary trees having a total of a given number of nodes.

We show that the statistics of chaotic systems can be predicted by constructing an associated sequence of periodic differential operators and computing their densities of states. For such operators, the density of states is well understood and can be computed straightforwardly, often yielding explicit formulas. As a case study, we investigate a nonlinear recursion relation that maps naturally onto a family of periodic operators generated by a Fibonacci tiling rule. This correspondence enables us to derive an explicit formula for the limiting statistics of the chaotic system and to demonstrate that the clustering near to critical values is equivalent to the van Hove singularities in the operators' densities of states.

We present integral field spectroscopic observations of 76 strong gravitational lens candidates identified with a residual neural network in the DESI Legacy Imaging Surveys, obtained with the Multi Unit Spectroscopic Explorer (MUSE) on the ESO's Very Large Telescope. These observations are part of an ongoing effort to build a large, spectroscopically confirmed sample of strong lensing systems for studies on dark matter, galaxy structure, and cosmology. Our MUSE program targets both lens and source redshifts, with particular emphasis on southern hemisphere systems. MUSE's wide spectral coverage and integral field capability allow for efficient identification of multiple sources, lens environments, and weak spectral features. Redshifts for lenses and sources were obtained via manual identification of spectral features in extracted 1D spectra. Our dataset includes systems with complex configurations, such as multiple source planes and group or cluster-scale environments. We extracted and analyzed 223 spectra, successfully determining both the lens and the source redshifts for 55 gravitational lens systems. For an additional 15 targets, we measured the redshifts of the lenses but were unable to determine the redshifts of the background sources. Six targets were confirmed to not be gravitational lenses. The results presented here complement space-based imaging from our HST SNAPshot program and spectroscopic follow-up with DESI and Keck, and have lasting legacy value for identifying interesting high redshift sources and complex lensing configurations.

We study the dynamics of multidimensional contracted rotations and address a problem posed by Y. Bugeaud and J-P. Conze in \textit{Acta Arithmetica} in 1999. More precisely, we show that if $A$ is an invertible linear contraction of $\mathbb{R}^d$, then the map $f: [0,1)^d\to [0,1)^d$ defined by $f(x) = Ax +b\,\,(\textrm{mod}\,\mathbb{Z}^d)$ is asymptotically periodic for Lebesgue almost all $b\in\mathbb{R}^d$. We also include an example of a family of multidimensional contracted rotations $(d>1)$ not conjugate to the product of one-dimensional contracted rotations $(d=1)$, showing that our result cannot be reduced to or derived from the one-dimensional result of Bugeaud and Conze.

The Hopfield network (HN) is a classical model of associative memory with stored patterns encoded as minima of an energy function shaped by a Hebbian learning rule. Dense Associative Memory (DAM) models introduce n-body interactions among neurons with n greater than 2 and, more recently, also exponential interaction functions, which significantly improve the network's storing capacity. While the emergence of phase transitions in HN and DAM were extensively studied, the investigation of exponential DAM is still in its early stages. Further, an equilibrium thermodynamical condition is typically assumed, while out-of-equilibrium dynamics are not considered. Here, we study the temporal dynamics of a stochastic exponential DAM (SEDAM) with a multiplicative salt-and-pepper noise and trained on the MNIST dataset. While taking the noise probability p as control parameter, the time-averaged overlap Q and the diffusion scaling H are taken as order parameters, being H related to the network's time correlation features. The MNIST-based SEDAM is also compared with a SEDAM trained on standard Rademacher patterns and with a stochastic HN (SHN). We found the emergence of a phase transition in both Q and H, with the critical noise level pc decreasing as the load K increases. For each load K, Q highlights a transition between a sub-critical and a super-critical regime, both with short-time correlated dynamics. Conversely, in the critical regime of the MNIST-based SEDAM the network displays long-time correlated dynamics with highly persistent temporal memory marked by the high value H around 1.25. Similar behaviors are observed for both models trained with Rademacher patterns, but with a slightly higher temporal memory index H around 1.5.

Let $G=(V,E)$ be an unweighted undirected graph with $n$ vertices and $m$ edges. Let $g$ be the girth of $G$, that is, the length of a shortest cycle in $G$. We present a randomized algorithm with a running time of $\tilde{O}\big(\ell \cdot n^{1 + \frac{1}{\ell - \varepsilon}}\big)$ that returns a cycle of length at most $ 2\ell \left\lceil \frac{g}{2} \right\rceil - 2 \left\lfloor \varepsilon \left\lceil \frac{g}{2} \right\rceil \right\rfloor, $ where $\ell \geq 2$ is an integer and $\varepsilon \in [0,1]$, for every graph with $g = polylog(n)$. Our algorithm generalizes an algorithm of Kadria \etal{} [SODA'22] that computes a cycle of length at most $4\left\lceil \frac{g}{2} \right\rceil - 2\left\lfloor \varepsilon \left\lceil \frac{g}{2} \right\rceil \right\rfloor $ in $\tilde{O}\big(n^{1 + \frac{1}{2 - \varepsilon}}\big)$ time. Kadria \etal{} presented also an algorithm that finds a cycle of length at most $ 2\ell \left\lceil \frac{g}{2} \right\rceil $ in $\tilde{O}\big(n^{1 + \frac{1}{\ell}}\big)$ time, where $\ell$ must be an integer. Our algorithm generalizes this algorithm, as well, by replacing the integer parameter $\ell$ in the running time exponent with a real-valued parameter $\ell - \varepsilon$, thereby offering greater flexibility in parameter selection and enabling a broader spectrum of combinations between running times and cycle lengths. We also show that for sparse graphs a better tradeoff is possible, by presenting an $\tilde{O}(\ell\cdot m^{1+1/(\ell-\varepsilon)})$ time randomized algorithm that returns a cycle of length at most $2\ell(\lfloor \frac{g-1}{2}\rfloor) - 2(\lfloor \varepsilon \lfloor \frac{g-1}{2}\rfloor \rfloor+1)$, where $\ell\geq 3$ is an integer and $\varepsilon\in [0,1)$, for every graph with $g=polylog(n)$. To obtain our algorithms we develop several techniques and introduce a formal definition of hybrid cycle detection algorithms. [...]