Thompson proved that every finite group admitting a fixed-point-free automorphism of prime order is nilpotent, and Kegel showed that the same conclusion holds for finite groups admitting a splitting automorphism of prime order. Motivated by these results, Sozutov asked whether a \(p'\)-group admitting a splitting automorphism of prime order is locally nilpotent if \[ \langle g, g^φ, \dots, g^{φ^{p-1}} \rangle \] is nilpotent for every \(g \in G\), \cite[Problem 10.59]{kourovka21}. We prove that if \(G\) is a periodic residually finite group admitting a splitting automorphism of prime order \(p\) then \(G\) is nilpotent of class bounded in terms of \(p\). This gives an affirmative answer, for residually finite groups, to the problem of Sozutov. We also prove that a possible counterexample to Sozutov's problem cannot be a Tarski monster.

We address the classical problem of constructing confidence intervals (CIs) for the mean of a distribution, given \(N\) i.i.d. samples, such that the CI contains the true mean with probability at least \(1 - δ\), where \(δ\in (0,1)\). We characterize three distinct learning regimes based on the minimum achievable limiting width of any CI as the sample size \(N_δ \to \infty\) and \(δ\to 0\). In the first regime, where \(N_δ\) grows slower than \(\log(1/δ)\), the limiting width of any CI equals the width of the distribution's support, precluding meaningful inference. In the second regime, where \(N_δ\) scales as \(\log(1/δ)\), we precisely characterize the minimum limiting width, which depends on the scaling constant. In the third regime, where \(N_δ\) grows faster than \(\log(1/δ)\), complete learning is achievable, and the limiting width of the CI collapses to zero, converging to the true mean. We demonstrate that CIs derived from concentration inequalities based on Kullback--Leibler (KL) divergences achieve asymptotically optimal performance, attaining the minimum limiting width in both sufficient and complete learning regimes for distributions in two families: single-parameter exponential and bounded support. Additionally, these results extend to one-sided CIs, with the width notion adjusted appropriately. Finally, we generalize our findings to settings with random per-sample costs, motivated by practical applications such as stochastic simulators and cloud service selection. Instead of a fixed sample size, we consider a cost budget \(C_δ\), identifying analogous learning regimes and characterizing the optimal CI construction policy.

The scalar potential of any particle-physics model must be bounded from below (BFB). We consider the extension of the Standard electroweak Model with three $SU(2)$ doublets of scalars and a symmetry under which each of those doublets changes sign. In the absence of necessary and sufficient conditions for boundedness from below (BnessFB) for this specific model, we argue that one may use increasingly stringent sets of necessary conditions. We introduce a Mathematica code, StableWein, that implements this idea. The user is allowed to choose the level of accuracy that they want in the determination of BnessFB; more precision means the use of more necessary conditions, and usually entails a longer running time for the code. Our investigation suggests that our procedure and code can be extremely precise in the determination of the potentials that are BFB. In addition, we introduce a machine-learning code that identifies, with more than 99% accuracy, which potentials are BFB.

Increasing penetration of electric vehicles, heat pumps, and rooftop photovoltaics is creating thermal and voltage stress in low-voltage distribution grids. This work links the German Federal Government energy transition pathway (2025-2045) with state estimation performance requirements, evaluated on two SimBench reference networks across three equipment quality levels (good, medium, poor) and three VDE Forum Netztechnik/Netzbetrieb (VDE FNN) measurement constellations that differ in the availability of transformer and feeder-level instrumentation. Within this work's analysis, congestion is caused exclusively by transformer overloading and voltage-band violations. No individual line exceeds its thermal rating (maximum: 89.5%). Equipment quality governs congestion onset for a given deployment trajectory: under good equipment, congestion remains absent through 2045, under medium equipment it emerges from 2035 (3/6 scenarios), under poor equipment from 2025 (6/6). Without transformer instrumentation, median voltage estimation errors reach 6-42% regardless of smart meter penetration. Adding a single transformer measurement reduces errors by an order of magnitude, achieving median errors of 0.5-1.7%. In urban networks, transformer-level instrumentation meets the VDE FNN voltage accuracy target (99th percentile voltage error below 2%) in all configurations. In rural networks under poor equipment, the target is approached but not met. These findings motivate prioritizing transformer instrumentation as an effective first step for grid observability and supplementing the current consumption-driven metering rollout with risk-based deployment criteria linked to local congestion exposure.

Finding the first generation of (Population III or Pop III) stars is one of the most ambitious and exciting challenges of astrophysics. JWST opened concrete prospects for their detection during the Epoch of Reionization (EoR), where increasing evidence suggests that residual Pop III formation may persist, even within pristine pockets of high-mass halos, due to inhomogeneous enrichment. However, the identification of Pop III stars within globally enriched environments will be challenging. We investigate the detectability of a subdominant Pop III component in/around massive ($M_\star \gtrsim 10^9 ~\mathrm{M_\odot}$) galaxies at $z \approx 6.5 - 9$ from the dustyGadget cosmological simulation suite, and the confusion arising from second-generation (Pop II) stars in their surroundings. We find that young ($\lesssim 1$ Myr), massive ($M_\mathrm{III} \sim 6 \times 10^5 ~\mathrm{M_\odot}$) Pop III clusters forming within these galaxy environments are responsible for strong HeII1640 line emission ($L_\mathrm{HeII1640} \gtrsim 10^{41} ~\mathrm{erg \, s^{-1}}$), which would be detectable with $\approx 10 (50)$ h of medium-resolution observations with NIRSpec/IFU at $z \approx 6 (10)$. These bright luminosities cannot be produced by standard Pop II populations alone. On the other hand, the dominant Pop II component within massive ``hybrid'' Pop III hosts powers strong metal line emission ($L_\mathrm{[OIII]5007} \gtrsim 10^{42} ~\mathrm{erg \, s^{-1}}$), indicating that the detection of metal lines alone cannot exclude the presence of Pop IIIs in high-$z$ galaxy environments. We further discuss candidate selection strategies based on Ly$α$, H$α$ and H$β$ emission, and how spatially resolved observations may enable the detection of isolated, pristine pockets in the outskirts of massive halos.

Property-based testing is a mainstay of functional programming, boasting a rich literature, an enthusiastic user community, and an abundance of tools~ -- so many, indeed, that new users may have difficulty choosing. Moreover, any given framework may support a variety of strategies for generating test inputs; even experienced users may wonder which are better in any given situation. Sadly, the PBT literature, though long on creativity, is short on rigorous comparisons to help answer such questions. We present ETNA, a platform for empirical evaluation and comparison of PBT techniques. ETNA incorporates a number of popular PBT frameworks and testing workloads from the literature, and its extensible architecture makes adding new ones easy, while handling the technical drudgery of performance measurement. To illustrate its benefits, we use ETNA to carry out several experiments with popular PBT approaches in Rocq, Haskell, OCaml, Racket, and Rust, allowing users to more clearly understand best practices and tradeoffs.

The Nordic countries have adopted ambitious climate targets that require far-reaching power-sector transformations, making energy system modelling an important input to long-term planning. However, model-based evidence is produced using different model structures, assumptions, scopes, and scenario designs. This paper examines what can be learned from comparing such independently developed scenarios by assessing Nordic power-sector climate-neutrality pathways across eight structurally diverse energy system models. The comparison covers Denmark, Finland, Norway, and Sweden for 2030, 2040, and 2050, and focuses on electricity demand, generation capacity, CCS deployment, and power-sector CO2 emissions. Inputs are not harmonised; instead, outputs are compared using a common reporting basis reflecting how modelling evidence is encountered in applied policy contexts. The results show broad agreement on the direction of transition. Wind power, mainly onshore but complemented by offshore wind in some countries, is the clearest cross-model finding and forms the backbone of the Nordic power system by 2050. At the same time, installed capacities, CCS deployment, nuclear outcomes, and emissions levels vary substantially. These differences are interpreted considering renewable-resource potentials, technology availability, policy constraints, sectoral and geographical scope, emissions-accounting boundaries, and different implementations of climate-neutrality targets. The study shows that non-harmonised model comparisons can support policy analysis by identifying where models point in the same direction, such as wind expansion, and where outcomes depend more strongly on model and scenario assumptions, such as CCS, nuclear, and net-negative emissions. For policy use, the findings underline the need to report model scope, technology representation, policy constraints, and emissions-accounting boundaries..

This paper introduces a new combinatorial framework for modeling the growth of binary trees through a discrete evolution process that incorporates a growing rule and an extinction rule. Building upon the theory of increasingly labeled structures and the analysis of polynomial iterates, we extend previous models of increasing trees with label repetitions by allowing growth branches to terminate. This mechanism enables a direct connection between dynamic evolutionary processes and classical unlabeled binary trees. We provide a combinatorial outlook for this model, linking our new approach to essential but traditionally complex parameters such as tree height, the maximum number of leaves at the deepest level (for a given tree size), and the overall tree profile. Our approach reveals structural links with Mandelbrot polynomials and coding theory. Furthermore, we leverage these structural insights to develop an efficient, iterative uniform random sampler for binary trees with a prescribed profile, achieving optimal complexity in both time and space and in random bit consumption.

Single-snapshot FDA-MIMO-GPR requires clutter models that account for dispersive-medium uncertainty, yet the statistical link between complex-medium characterization and clutter covariance analysis has remained unclear. This paper develops a propagation-side statistical framework that maps random perturbations of the relaxation spectrum to complex permittivity, complex wavenumber, steering-vector perturbation, medium-induced clutter covariance, and total clutter covariance. Within this framework, the effects of medium uncertainty on effective rank, effective clutter-subspace dimension, and target--clutter separability are characterized through a KL-based modal decomposition and a subspace-projection analysis. Numerical validation uses five literature-informed dielectric families to define physically traceable prior scenarios, a controlled random-field model to exercise the main propagation chain, and gprMax-based full-wave FDTD snapshots for an independent solver-level consistency check. Monte Carlo closure shows stage-wise numerical consistency, identifies steering linearization as the dominant approximation-sensitive step, and supports a weak perturbation regime with a bounded extension into a moderate regime. In a representative whitening-and-detection benchmark, the structured covariance model raises AUC from 0.593 for a diagonal baseline to 0.753, while prior-mismatch experiments indicate gradual rather than abrupt degradation. These results provide an explicit and interpretable interface for embedding complex-medium uncertainty into FDA-MIMO-GPR clutter analysis within a first-order, propagation-dominated setting.

Forbidden $β$ transitions are important for understanding the reactor antineutrino anomaly. Starting from chiral two- plus three-nucleon forces, we have derived the valence-space effective Hamiltonian and effective operators of first-forbidden transitions using the many-body perturbation theory. 20 dominant first-forbidden transitions have been investigated, which provide important contributors to the reactor antineutrino spectrum anomaly. Calculated $\log ft$ values are in reasonable agreement with experimental data. Obtained shape factors exhibit significant deviations from the values approximated with forbidden transitions treated as allowed transitions. The ``5 MeV bump'' observed in the experimental ${}^{235}$U-fission antineutrino spectrum was discussed with self-consistent shape factors obtained in the present \textit{ab initio} calculations. Unlike phenomenological models that require empirical quenching factors to reproduce $β$-decay data, the present \textit{ab initio} calculations do not need to introduce quenching factors for calculations of the first-forbidden transitions.

A Moser-type construction for the kinetic Liouville equation is proposed, which is based on a characteristic-adapted interpolation. For Hamiltonian accelerations, the construction is reduced to a family of weighted elliptic problems in the velocity variable. The corresponding kinetic compatibility condition is derived.

We establish that adaptive collision-finding queries are strictly more powerful than non-adaptive ones by proving that the complexity class PWPP (Polynomial Weak Pigeonhole Principle) is not closed under adaptive Turing reductions in the black-box setting. Previously, PWPP was known to be closed under non-adaptive Turing reductions (Jeřábek 2016). We demonstrate this black-box separation by introducing the NESTED-COLLISION problem, a natural collision-finding problem defined on a pair of shrinking functions. We show that while this problem is solvable via two adaptive calls to a PWPP oracle, it cannot be solved via an efficient black-box non-adaptive reduction to the canonical PWPP-complete problem COLLISION.

The maximum agreement forest (MAF) problem in phylogenetics takes as input a set t >= 2 of binary phylogenetic trees T on the same set of taxa X. It asks for a partition of X into the smallest number of blocks such that the subtrees induced by these blocks are disjoint and have common topology across all the trees in T. We produce a modified version of the well-known chain reduction rule in order to prove that after exhaustive application of reduction rules each tree has O( t * r * k ) leaves, where k is the natural parameter (the number of blocks) and r=min{max{k,3},t+1}}. We prove this bound for both the unrooted and rooted version of the problem, and demonstrate that the bound r, the length to which common chains are truncated, is tight. Our results constitute the first kernels for MAF in the t>2 regime.

We study the Heisenberg spin-1/2 model on a semi-infinite chain - or, equivalently, a trotterized unitary SU(2) symmetric six-vertex quantum circuit - with a boundary defect where the interaction between the two spins nearest the edge differs from that in the bulk. For sufficiently strong boundary interaction we explicitly construct a conserved operator quasi-localized near the boundary using a matrix-product ansatz. This quasi-local edge mode leads to non-decaying boundary correlation functions, corresponding to a nonzero boundary Drude weight. The correlation length of the edge mode diverges at a finite critical value of the boundary interaction, signaling a transition to ergodic boundary dynamics for subcritical interactions.

Using density functional calculations and multi-orbital, multi-atom cluster exact diagonalization that includes local exchange and Coulomb interactions, we explored the local low-energy electronic states of trilayer La$_4$Ni$_3$O$_{10}$ via a minimal Ni$_3$O$_{14}$ cluster. We find that, at ambient pressure, starting with all three Ni being nominally 2+ valence, one of the two extra holes is localized in the central NiO$_2$ layer forming a Zhang-Rice singlet (ZRS) with $d_{x^2-y^2}$ orbital. The other hole mainly occupies the antibonding combination of the two interplane apical O $p_z$ orbitals and thereby hybridizes with an out-of-plane three-spin-polaron (3SP) formed by the $d_{z^2}$ orbitals of three NiO$_2$ layers. At high pressure, however, the two extra holes are concentrated on one of two outer layers and the inner layer separately forming the ZRS with $d_{x^2-y^2}$ orbitals. We highlight the similarities between the bilayer La$_3$Ni$_2$O$_7$ and trilayer La$_4$Ni$_3$O$_{10}$ via speculated possible charge and spin configurations as well as the in-plane 3SP on two neighboring clusters suggested by our isolated cluster results.We thereby propose that the hole transfer from apical to in-plane oxygen orbitals of outer layer generates in-plane 3SP-like quasiparticles that act as mobile carriers coupled by interlayer superexchange; while the interplane 3SP-like states may provide the pairing glue. Since the low-pressure phase lacks freely propagating in-plane quasiparticles, this scenario naturally favors SC in the high-pressure phase.

We analyze branch decoherence in inflationary quantum cosmology by computing reduced density matrices and branch-overlap factors for long-wavelength perturbations. The Hartle-Hawking no-boundary state is real in the semiclassical regime and contains both expanding and contracting WKB components, whereas the tunneling state is selected as an outgoing complex WKB branch; expanding-contracting decoherence is therefore central for the former and mainly diagnostic for the latter. Using the influence-functional formalism, we derive the noise kernel for a light spectator environment and evaluate decoherence under horizon-based and EFT-motivated coarse grainings. We then compute the single-mode branch overlap directly from the Bunch-Davies mode functions, obtaining $|\mathcal{D}_k(z)|=[z^2/(z^2+1)]^{1/4}$ in the massless limit and $|\mathcal{D}_k(z)|\sim z^ν$ on superhorizon scales for massive fields, where $z=-kη$ is the dimensionless wavenumber with $η$ the conformal time. In the massless case, the accumulated geometric branch functional is evaluated in closed form, with a leading cutoff-sensitive phase-space term and a universal subleading contribution. The calculation provides an explicit quantitative bridge between quantum-cosmological boundary conditions, inflationary squeezing, and the emergence of effectively classical cosmological histories.

We realize a cold-atom system to quantitatively test relational constructions of time. A well-isolated atomic Bose-Einstein condensate evolves in a conservative trap that is partitioned by a thin optical barrier into an observed and unobserved sector, with negligible dissipation on the experimental timescale. Motivated by relational-time approaches discussed in the Wheeler-DeWitt framework, we ask whether the dynamics of the observed sector can be ordered using only internal degrees of freedom. To this end, we construct an entropic time from an experimentally defined coarse-grained entropy, and demonstrate that it can robustly order the events in the observed sector across repeated cycles of expansion and recollapse. We finally derive an effective Schroedinger equation parameterized by this internal time and show that it is able to reproduce the measured evolution. These results establish a controlled experimental setting in which relational-time constructions can be quantitatively tested.

This work presents an efficient approach for neighbourhood searching in 3D point clouds by combining spatial reordering leveraging Space-Filling Curves (SFC), specifically Morton and Hilbert curves, with a linear Octree implementation. We also propose specialised search algorithms for fixed-radius and kNN queries, based on our linear Octree structures. Additionally, we introduce the novel concept of kNN locality histogram, which can be easily computed to characterise locality in data accesses, and we found to be directly related to cache misses and search performance. Our experiments reveal that SFC reordering significantly improves access to spatial data, reducing the number of cache misses from 25% to 75% and runtime by up to 50%. Moreover, we compare our proposal with several widely used Octree and KDTree implementations. Our method achieves a significant reduction in search time, up to 10$\times$ faster than existing solutions. Additionally, we analysed the performance of our neighbourhood searches (parallelised using OpenMP), demonstrating high scalability with the number of cores and the problem size. Notably, we observed a speedup of up to $36\times$ when executing fixed-radius searches in a system with 40 cores. The results obtained indicate that our methods provide a robust and efficient solution for applications that require fast access to large-scale 3D point neighbour sets.

While hidden, singly, doubly, and fully heavy tetraquark states have been experimentally observed, triply heavy tetraquark states remain experimentally unconfirmed. We systematically investigate the spectroscopic and decay properties of four triply heavy-flavor tetraquark systems ($cc\bar{c}\bar{n}$, $cc\bar{c}\bar{s}$, $bb\bar{b}\bar{n}$, $bb\bar{b}\bar{s}$; $n=u,d$) based on the nonrelativistic quark model. Using an effective Hamiltonian, we employ the Gaussian expansion method to solve the four-body Schrödinger equation and incorporate the effect of color-spin configuration mixing. Results show both $cc\bar{c}\bar{q}$ and $bb\bar{b}\bar{q}$ systems have two $J^{P}=0^{+}$, three $J^{P}=1^{+}$, and one $J^{P}=2^{+}$ states, with ground-state masses of 5.2-5.5 GeV and 15.0-15.3 GeV, respectively. Root-mean-square radius analysis supports compact tetraquark configurations. All states are unstable, with rearrangement strong decays dominant and negligible radiative decays. Narrow resonances (e.g., $T_{c^{2}\bar{c}\bar{s}}(5360,0^{+})$, $T_{b^{2}\bar{b}\bar{n}}(15148,2^{+})$) arise from Feynman amplitude cancellation. We propose experimental searches in $J/ψD^{*}_{s}$/ $η_{c}D_{s}$ (5.3-5.4 GeV) and $ΥB^{*}$ (15.1-15.2 GeV) channels, providing key guidance for triply heavy tetraquark identification.

The notion of poor manifolds was introduced by Bandman and Zarhin, who asked for their classification. In this paper, we answer this question completely in dimensions at most 3. We also classify poor compact Kähler manifolds of arbitrary dimension under the additional assumption that Kodaira dimension is not $-\infty$. We classify all poor $K3$ surfaces. Finally, give a sufficient condition for a compact Kähle manifold to have Kodaira dimension greater than or equal to zero.

We present new results about fully leafed induced subtrees in Penrose P2 tilings, also known as kites and darts tilings. We first determine the graph structure of these subtrees and show that they are caterpillars up to an appendix of at most six tiles. In other words, if we remove their degree one vertices, then they are path graphs up to an additional connected path of at most two tiles. We then study bi-infinite fully leafed induced caterpillars in P2 tilings and their geometric properties. In particular, we refute the conjecture proposed by C. Porrier, A. Goupil and A. Blondin Massé that there is a unique bi-infinite fully leafed caterpillar in Penrose P2 tilings.

X-ray interferometry provides valuable information in terms of attenuation, small-angle scatter, and differential-phase contrast. This multi-modal contrast can aid in many clinical applications, such as lung diseases and breast cancer. However, standard interferometry has an analyzer grating that can increase the dose requirement to maintain the same image quality as a standard X-ray. We propose the use of super-resolution methods for X-ray grating interferometry without an analyzer, with detectors that fail to meet the Nyquist sampling rate needed for traditional image recovery algorithms. Detector phase steps are used to nominally recover the fringe sampling, followed by iterative recovery of the visibility and object parameters. This method enables Talbot-Lau interferometry without the X-ray absorbing analyzer. Removing the absorbing analyzer grating may improve dose efficiency and reduce system complexity. We demonstrate the use of super-resolution methods to iteratively reconstruct attenuation, differential-phase, and dark-field images using simulations of two-dimensional lung phantoms with lesions. A direct CdTe detector was simulated with pixel sizes of 55, 75, and 150 micron. The simulation results show that the proposed super-resolution iterative reconstruction method for Talbot-Lau Interferometry remains stable under the simulated noise conditions and can recover image parameters in cases where traditional algorithms cannot be used.

We develop a formulation of the strong deflection limit for the scattering of particles following timelike geodesics in asymptotically flat, static, and spherically symmetric spacetimes. For fixed specific energy, as the angular momentum approaches its critical value from above, the particle passes arbitrarily close to the associated unstable circular orbit, undergoes many windings around it, and the deflection angle diverges logarithmically. Using the geodesic deviation equation, we show covariantly that the coefficient of this logarithmic divergence is determined by the radial instability exponent of the critical trajectory, defined per unit azimuthal angle. We express this instability exponent in terms of local curvature data on the unstable circular orbit, thereby providing both kinematic and geometric interpretations of the strong deflection limit. In general relativity, its matter dependence enters only through a single local scalar combination constructed from the static-frame energy density and the principal radial and tangential pressures.

Motivated by simulations of ultrasound-enhanced drug delivery, this work presents the numerical analysis of a mathematical model that captures the influence of ultrasound waves on the diffusivity of the drug. The system under study consists of the Westervelt wave equation, accounting for the nonlinear propagation of ultrasound, coupled to a convection-diffusion equation modeling the drug concentration. In particular, drug delivery is affected by ultrasound through a pressure-dependent diffusion coefficient. The Westervelt equation is supplemented by linear absorbing boundary conditions as a means of reducing spurious reflections off the boundaries of computational domains. For spatial discretization of this multiphysics system, we employ a discontinuous Galerkin approach on simplicial meshes. Under suitable assumptions on the exact pressure and the mesh size, we first establish well-posedness, non-degeneracy, and optimal convergence rates in the energy norm for the semi-discrete pressure subproblem. The smallness of the semi-discrete pressure is then used to establish the well-posedness and convergence of the wave--convection-diffusion system under suitable regularity of the exact concentration. Finally, theoretical findings are illustrated through numerical experiments.

Automated visualization design navigates a tension between symbolic systems and generative models. Constraint solvers enforce structural and perceptual validity, but the rules they require are difficult to author and too rigid to capture situated design knowledge. Large language models require no formal rules and can reason about contextual nuance, but they prioritize popular conventions over empirically grounded best practices. We address this tension by proposing a cataloging scheme that structures visualization design knowledge as natural-language guidelines with semantically typed metadata. This allows experts to author knowledge that machines can query. An expert study ($N=18$) indicates that practitioners routinely adapt heuristics to situational factors such as audience and communicative intent. To capture this reasoning, guideline sections specify not only advice but also the contexts where it applies, exceptions that invalidate it, and the sources from which it derives. We demonstrate the scheme's expressiveness by cataloging 744 guidelines drawn from cognitive science, accessibility standards, data journalism, and research on rhetorical aspects of visual communication. We embed guideline sections in a vector space, opening the knowledge itself to structural analysis. This reveals conflicting advice across sources and transferable principles between domains. Rather than replacing constraint-based tools, our scheme provides what they lack: situated guidance that generative systems can retrieve to ground their reasoning, users can verify against cited sources, and experts can author as knowledge evolves.

The spontaneous breaking of $SO(10)$ via flipped $SU(5)$ to the Standard Model yields a novel scenario in which the superheavy topologically stable GUT monopole carrying a single unit ($2π/e$) of Dirac magnetic charge emerges from the merger of a confined but topologically distinct monopole-antimonopole pair that are pulled together by a string. The $SO(10)$ breaking via the subgroup $SU(4)_c\times SU(2)_L\times SU(2)_R$, following a similar reasoning, produces a topologically stable monopole that carries two units ($4π/e$) of Dirac charge. We explore the cosmological consequences of this scenario by assuming that the monopoles and strings experience a limited number of inflationary $e$-foldings, before re-entering the horizon and ultimately forming a network of quasi-stable strings bounded by monopole-antimonopole pairs. We identify regions of the parameter space that yield an observable number density of the GUT monopole from the collapse of the appropriate string segments. The gravitational waves emitted by these quasi-stable cosmic strings lie in the Hz to kHz range, which can be tested in a number of proposed and ongoing experiments.

Magnetoencephalography measures the magnetic fields generated by neural currents using quantum sensors such as superconducting quantum interference devices and atomic magnetometers. Here we combine the energy resolution limit of magnetic sensing with the metabolic power available to neural currents to derive a technology-independent bound on the information capacity of MEG. The bound factorizes into geometry, metabolism, and Planck's constant, and gives an estimated maximum information rate of 2.2~Mbit/s for representative human-brain parameters. Further, we show that the externally measurable magnetic field has a finite angular bandwidth, with high multipole components being geometrically attenuated and falling below the quantum-limited noise floor. This yields an information-limited spatial scale of order $1~{\rm cm}$ and renders the accessible measurement space effectively finite-dimensional. The energy resolution limit therefore defines an information-theoretic Nyquist scale for magnetoencephalography, beyond which denser spatial sampling provides redundant measurements rather than additional recoverable information. Since the energy resolution limit also makes the noise variance grow linearly with measurement bandwidth, temporal and spatial bandwidths compete, producing a fundamental spatio-temporal trade-off. These results show how quantum-limited measurements constrain the observable complexity and information content of noninvasive brain imaging, providing a quantitative link between fundamental physics and neuroscience.

We present a physics-embedded Bayesian neural network (PE-BNN) framework that integrates fission product yields (FPYs) with prior nuclear physics knowledge to predict energy-dependent FPY data with fine structure. By incorporating an energy-independent phenomenological shell factor as a single input feature, the PE-BNN captures both fine structures and global energy trends. The combination of this physics-informed input with hyperparameter optimization via the Watanabe-Akaike Information Criterion (WAIC) significantly enhances predictive performance. Our results demonstrate that the PE-BNN framework is well-suited for target observables with systematic features that can be embedded as model inputs, achieving close agreement with known shell effects and prompt neutron multiplicities.

We prove that the unconditional uniqueness of mild solutions to the Navier-Stokes equations fails in all the Besov spaces with negative regularity index, by constructing non-trivial stationary singular solutions via convex integration. We also establish uniqueness of stationary weak solutions in an endpoint critical space. Similar results are proved for the fractional Navier-Stokes equations with arbitrarily large power of the Laplacian in both Lebesgue and Besov spaces.

Building on the earlier work by Araki and Tanii, Aschieri {\it et al.}, and Buratti {\it et al.}, we demonstrate that every model for self-dual nonlinear electrodynamics in four dimensions has a $\mathsf{U}(1)$ duality-invariant extension to $4p>4$ dimensions and construct new self-dual nonlinear theories for a gauge $(2p-1)$-form. We present a family of models for self-dual nonlinear $(2p-1)$-form electrodynamics in which the trace of the energy-momentum tensor determines the flow with respect to a duality-invariant deformation parameter. Finally, we propose the interesting problem of computing the so-called induced action in the case that self-dual $(2p-1)$-form electrodynamics is coupled to dilaton and axion fields and the compact duality group $\mathsf{U}(1)$ is enhanced to the non-compact group $\mathsf{SL}(2, {\mathbb R})$.