We study a semilinear equation involving the fractional Laplacian on the hyperbolic space $\mathbb{H}^n$. Unlike in conformally compact Einstein manifolds, the fractional Laplacian on $\mathbb{H}^n$ does not enjoy conformal covariance. By employing Helgason-Fourier analysis, we explicitly derive the Green's function of the fractional Laplacian on $\mathbb{H}^n$ as well as its asymptotic behaviors. We then apply a direct method of moving planes to the integral form of the equation, and show that nonnegative weak solutions are symmetric. In addition, we extend several maximum principles to hyperbolic space.

Software transactional memory (STM) allows programmers to easily implement concurrent data structures. STMs simplify atomicity. Recent STMs can achieve good performance for some workloads but they have some limitations. In particular, STMs typically cannot support long-running reads which access a large number of addresses that are frequently updated. Multiversioning is a common approach used to support this type of workload. However, multiversioning is often expensive and can reduce the performance of transactions where versioning is not necessary. In this work we present Multiverse, a new STM that combines the best of both unversioned TM and multiversioning. Multiverse features versioned and unversioned transactions which can execute concurrently. A main goal of Multiverse is to ensure that unversioned transactions achieve performance comparable to the state of the art unversioned STM while still supporting fast versioned transactions needed to enable long running reads. We implement Multiverse and compare it against several STMs. Our experiments demonstrate that Multiverse achieves comparable or better performance for common case workloads where there are no long running reads. For workloads with long running reads and frequent updates Multiverse significantly outperforms existing STMS. In several cases for these workloads the throughput of Multiverse is several orders of magnitude faster than other STMs.

Radiation-hydrodynamics (RHD) determines the bulk evolution and observable emission in a wide variety of high-energy astrophysical phenomena. Due to their complexity, RHD problems must usually be studied through numerical simulation. We have extended the publicly available RICH code, which previously solved the equations of RHD in the limit of grey flux-limited diffusion (FLD), to operate with a multigroup FLD solver. RICH is a semi-Lagrangian code that solves the equations of RHD on an unstructured moving mesh, and is the first multigroup RHD moving mesh code, making it uniquely applicable to problems with extreme dynamic range and dynamically important radiation forces. We validate our multigroup module against multiple analytic benchmarks, including a novel test of the RHD Doppler term. The computational efficiency of the code is aided by a novel scheme to accelerate convergence in optically thick cells by limiting the absorption coefficients. Finally, we apply multigroup \textsc{rich} in a pilot three dimensional study of a stellar tidal disruption event (TDE), using a $10^4 M_\odot$ intermediate-mass black hole. Our simulations self-consistently produce a bright early-time X-ray flash prior to peak optical/UV light, in qualitative agreement with post-processing of (grey) RICH simulations of supermassive black hole TDEs, as well as X-ray observations of the TDE AT 2022dsb.

We explore the evolution of sub-Neptune (radii between $\sim$1.5 and 4 R$_\oplus$) exoplanet interior structures using our upgraded evolution code, \texttt{APPLE}, which self-consistently couples the thermal and compositional evolution of the whole structure. We incorporate stably stratified regions with convective mixing and, for the first time, ab initio results on the phase separation of silicate-hydrogen mixtures to model silicate rain in sub-Neptune envelopes. We demonstrate that inefficient mantle cooling can retain sufficient heat to Gyr ages: inefficient heat transport from mantle to envelope alone keeps radii $\sim$10\% larger than predicted by adiabatic models at late times. Silicate rain can contribute an additional $\sim$5\% to the radius, depending on envelope mass and initial metal abundance. The silicate-hydrogen immiscibility region may lie in the middle or even upper envelope, far above the envelope-mantle boundary layer, and bifurcates the envelope into two an upper, hydrogen-rich region and a lower, metal-rich region above the mantle. If silicate rain occurs, atmospheres should appear depleted of silicates while radii remain inflated at late ages. To demonstrate the effects of inefficient mantle cooling, we present interior evolution models for GJ 1214 b, K2-18 b, TOI-270 d, and TOI-1801 b, showing that hot, liquid silicate mantles with thin envelopes reproduce their radii and mean densities, providing an alternative to water-world interpretations. These results imply that bulk compositions inferred from mean density must account for the mantle thermal state and the envelope mixing/phase-separation history; such thermal ``memories'' may constrain formation entropies and temperatures when metallicities are more precisely measured.

The solution of potential-driven steady-state flow in large networks is required in various engineering applications, such as transport of natural gas or water through pipeline networks. The resultant system of nonlinear equations depends on the network topology, and its solution grows more challenging as the network size increases. We present an algorithm that utilizes a given partition of a network into tractable sizes to compute a global solution for the full nonlinear system through local solution of smaller subsystems induced by the partitions. When the partitions are induced by interconnects or transfer points corresponding to networks owned by different operators, the method ensures data is shared solely at the interconnects, leaving network operators free to solve the network flow system corresponding to their own domain in any manner of their choosing. The proposed method is shown to be connected to the Schur complement and the method's viability demonstrated on some challenging test cases.

The formulation of two-dimensional quantum gravity at finite cutoff remains an open problem. We revisit this question in JT gravity from two perspectives: the closed-channel bulk path integral and the path integral over boundary curves. First, we study the radial evolution of a closed universe and derive the trumpet wavefunction as a transition amplitude between a geodesic boundary and a finite Dirichlet boundary. Our analysis recovers the Hartle--Hawking wavefunction without imposing asymptotic boundary conditions, allowing the trumpet to be glued to a cap wavefunction to reconstruct the smooth disk. Second, we derive an exact Riccati equation for the extrinsic curvature of a finite-cutoff boundary curve in the Euclidean Poincaré disk. A WKB expansion of this equation yields all perturbative corrections in the cutoff parameter and captures nonperturbative effects. From this, we compute the quadratic boundary action and the one-loop partition function at finite cutoff, finding agreement with both the bulk approach and the expected one-loop effective action for the $T\bar{T}$ deformation of the Schwarzian theory. Extracting lessons from JT gravity, we then argue that similar relationships hold for general dilaton gravities with arbitrary potentials $V(ϕ)$ and propose an exact expression for their finite cutoff partition functions. We finally investigate several signatures of UV completeness in these settings, introducing a canonical quantization approach within the finite cutoff framework.

Some recent work pointed out the usefulness of taking a large-deviation perspective when trying to extract anything resembling a macroscopic order parameter from a computer simulation. In this paper we note that the end-to-end distance of polymers is such an order parameter. The presence of long-ranged excluded volume interactions leads to significant qualitative differences between the conformations of two- and three-dimensional polymers, some of which are difficult to quantify in computer simulations of realistic (off-lattice) polymer models. But we show here that phenomena such as the greatly enlarged non-Hooke's-law elasticity present in 2D are straightforward to extract from simulation using a large-deviation framework - even though simulating that nonlinearity is tantamount to simulating a 4th order susceptibility. The large-deviation perspective includes both a set of thermodynamic-like tools suitable for studying finite-size systems and a realization that an accurate description of the system's average behavior needs to be consistent with how improbably large fluctuations would behave in that system. The latter is key because strong correlations are absent in this asymptotic large fluctuation regime, so the regime's far-reaching effects can be analytically incorporated into the analysis of simulation data. That, in turn, allows us to direct the efforts of simulations away from difficult-to-sample rare-event domains. We illustrate this point with two- and three-dimensional Monte Carlo simulations (and exact results) on two models of a single isolated polymer chain: a chain of linked hard spheres, which has long-ranged excluded volume effects, and a discretized worm-like chain, which does not.

We study the cohomology of divisors on a Burniat surface $X$ with $K_X^2=6$. We provide an algorithm for computing the cohomology groups of arbitrary divisors on $X$. As an application, we prove that there are no Ulrich line bundles\,(with respect to an arbitrary polarization), and that there exists an Ulrich vector bundle of rank 2 with respect to $3K_X$. The existence of Ulrich vector bundle of rank 2 was previously established by Casnati, but our construction yields one that cannot be obtained by his method.

The exteriors of stellar and galactic dynamos are usually modeled as a current-free potential field. A more realistic description might be that of a force-free magnetic field. Here, we suggest that, in the absence of outflows, neither of those reflect the actual behavior when the magnetic field spreads diffusively into a more poorly conducting turbulent exterior outside dynamo. In particular, we explain why the usual ordering of the dipole magnetic field being the most slowly decaying one is altered, and that the quadrupole can develop a toroidal component that decays even more slowly with radial distance. This is a robust feature that persists even for spatially nonuniform magnetic diffusivities. It is best seen for spherical dynamo volumes and becomes more complicated for oblate ones. In either case, however, those fields are confined within a magnetosphere beyond which the field drops exponentially. We demonstrate that the Faraday displacement current, which plays a role in a vacuum, can safely be neglected in all cases. The superposition of magnetic fields from galaxies in the outskirts of the voids between galaxy clusters can therefore not explain the void magnetization of the intergalactic medium, reinforcing the conventional expectation that those fields are of primordial origin. For quadrupolar configurations, the synchrotron emission from the magnetosphere is found to be constant along concentric rings. The dipolar and quadrupolar configurations display large-scale radial trends that are potentially distinguishable with existing radio telescopes.

Two different dynamical system formulations are presented for the generic $f(R,ϕ,X)$ family of gravity theories. As illustrative examples, the first and the second formulation is applied to study the phase space of a toy model of the Non-Minimal Derivative Coupling (NMDC) without a potential, and the mixed $R^2$-Higgs inflation model, respectively. The first dynamical system formulation applied to the toy NMDC model, although able to identify several invariant submanifolds, fails to fully investigate the fixed point structure, as all the fixed points turn out to be non-hyperbolic. We, however, discover an interesting feature that the qualitative dynamics are independent of the coupling strength between the Ricci scalar and the scalar field derivative. The second dynamical system formulation applied to the mixed $R^2$-Higgs inflation model performs much better, being able to correctly reduce to the individual phase spaces of the $R^2$ and Higgs inflation separately in special cases, as well as correctly delivering the expected invariant submanifolds and fixed points. For the mixed $R^2$-Higgs case, illustrative phase portraits are provided for a somewhat better understanding of the dynamics.

We revisit Bourgain's 1981 counterexample to the sequential completeness of the `pointwise plus domination' convergence on $\ell_1$ from the perspective of vector lattices. In this setting, we show that for sequences the associated notion of Bourgain--uo convergence coincides with ordinary order convergence. Motivated by Bourgain's construction, we introduce a strengthened, subsequence-invariant notion of Cauchy sequence: a sequence $(x_n)$ in a vector lattice $E$ is called Buo-Cauchy if for every strictly increasing sequence $(n_k)$ the differences $x_{n_{k+1}}-x_{n_k}$ converge to $0$ in order in $E$. We first show that sequential Buo-completeness forces $σ$-order completeness. Thus every non-$σ$-order complete vector lattice fails sequential \Buo-completeness. In particular, free Banach lattices $\mathrm{FBL}(E)$ are not sequentially Buo-complete whenever $\dim E>1$. On the positive side, we prove that the classical sequence lattices $c_0$ and $\ell_\infty$ are sequentially Buo-complete: every Buo-Cauchy sequence converges in order, and hence in the Buo sense. Finally, we obtain a sharp metric characterisation for bounded Lipschitz function lattices: the vector lattice $\mathrm{Lip}_b(X)$ of bounded Lipschitz functions on a metric space $(X,d)$ is sequentially Buo-complete if and only if $X$ is uniformly discrete.

We investigate how galactic disk structures connect to the detailed properties of their host dark-matter halos using the TNG50 simulation. From the hydrodynamic and matched dark-matter-only runs, we measure a comprehensive list of halo properties describing density structure, angular momentum, shape, assembly history, and environment. Using the morphological decomposition developed in Paper I, we quantify the sizes, scale heights, and mass fractions of the disk components for galaxies at $0 \le z \le 4$. Random Forest (RF) regression shows that halo properties alone predict disk size and thickness with high accuracy, while Symbolic Regression (SR) provides compact empirical relations with slightly lower accuracy. Disk height is consistently easier to predict than disk size, and lower-mass halos yield higher accuracy than massive halos. Predictions based on halo properties measured in the hydro simulations outperform those based on halos matched in the dark-matter-only simulation, reflecting the imprint of baryonic restructuring on the inner halo. SHAP analysis reveals the most informative halo parameters include concentration, Einasto shape, global and inner spin, and recent mass accretion, though their importance varies across disk properties. We show correlations between disk size and the density-profile shape arise primarily from disk-induced modification of the inner halo, rather than a primordial connection. Finally, we point out that disks become more extended with respect to their host halos at higher redshift in low-mass halos, while massive high-redshift halos show the opposite trend. We provide SR-based prescriptions that accurately map halo properties to disk structures, offering practical tools for galaxy-halo modeling.

We investigate the conditions under which the perturbative treatment of the backreaction of spin-2 particles on the dynamics of an axion-SU(2) gauge field system breaks down during cosmic inflation. This condition is based on the ratio of the energy density of spin-2 particles from the SU(2) gauge field to that of the background field. The perturbative treatment breaks down when this ratio exceeds unity. We show that this occurs within a parameter space nearly identical to the strong backreaction regime identified in previous studies. However, in some cases, the ratio exceeds unity even before the system enters the strong backreaction regime. Our results suggest that attempts to study the strong backreaction regime using perturbation theory are necessarily limited. Reliable calculations require non-perturbative treatments, such as three-dimensional lattice simulations.

Despite direct searches at the LHC excluding tripletlike Higgs bosons up to several hundred GeV over much of the type-II seesaw model parameter space, parts of it -- most notably those featuring ``cascade decays'' of the charged Higgs bosons into their neutral partners and off-shell $W$ bosons -- still remain unconstrained. Meanwhile, measurements of the diphoton signal strength of the Standard Model (SM) Higgs boson -- potentially modified by loop contributions from tripletlike Higgs states -- are in good agreement with the SM expectation, with combined experimental uncertainties currently at approximately 8%. Given the trend in previous measurements, it is expected that future precision Higgs measurements at the HL-LHC and a future lepton collider such as the Circular Electron Positron Collider, Future Circular Collider, or Muon Collider will be consistent the standard diphoton signal strength, albeit with significantly reduced uncertainties, down to about 0.7%. Presuming this and considering all relevant constraints, we explore whether such increasingly precise diphoton measurements can indirectly probe the parameter space that currently evades direct searches. We find that subpercent-level determinations of the diphoton rate will decisively probe a substantial fraction of this otherwise elusive region.

We investigate flat morphisms of schemes of positive characteristic whose relative Frobenius is an isomorphism, which we call pristine. We show that these give rise to a natural Grothendieck topology that is fine tuned for the localization of Cartier modules.

A hybrid dual-readout calorimeter concept, comprising both electromagnetic and hadronic sections, has recently been proposed to meet the performance requirements of experiments at future e$^{+}$e$^{-}$ colliders. The front compartment consists of a homogeneous electromagnetic calorimeter made of high-density crystals, each coupled to a pair of Silicon Photomultipliers (SiPMs) providing the simultaneous readout of scintillation and Cherenkov light. To efficiently detect Cherenkov photons in the presence of dominant scintillation signals, an optical filter is placed in front of one of the two SiPMs to suppress photons in the wavelength region corresponding to that of scintillation emission. In this study, PWO, BGO, and BSO crystals with different dimensions were tested to measure their scintillation light yield and decay time, as well as their transmission and emission spectra. A set of $\sim 100~\rm μm$-thick optical filters was also characterized by measuring their transmittance curves. The experimental results were used to model and estimate the expected filter performance in attenuating scintillation light for the various crystals. The performance of each filter was experimentally validated by measuring the crystal light output with and without the filter using a $^{22}$Na radioactive source and a LYSO:Ce crystal, confirming the accuracy of the calculations. The results show that interference filters are unsuitable for this application because their transmittance strongly depends on the photon incidence angle. Conversely, two absorptive long-pass filters with cutoff wavelengths around 590 nm were found to block more than 99% of the scintillation light from PWO crystals, satisfying the calorimeter specifications.

We introduce optimal transport (OT) as a physics-based intermediate event representation for weakly supervised anomaly detection. With only $0.5\%$ injection of resonant signals in the LHC Olympics benchmark datasets, the OT-augmented feature set achieves nearly twice the significance improvement of standard high-level observables provided in the benchmark, while end-to-end deep learning on low-level four-momenta struggles in the low-signal regime. The gains persist across signal types and classifiers, underscoring the value of structured representations in machine learning for anomaly detection.

We study constrained selection sets of random closed sets defined on a non-atomic probability space. Given a random interval $Y=[y_L,y_U]$ and scalar constraints on the expectation or the median of admissible selections, we characterize the restricted selection set and establish sharp bounds on the attainable ranges of means, medians, and event probabilities. In particular, we give conditions under which every value in the Aumann expectation range is realized as the mean of a measurable selection, and we obtain explicit formulas for the extremal expectations under median and higher-moment restrictions via rearrangement and convex-duality arguments. We further show that the selection set of any random compact convex set in $\R^d$ can be approximated in $L^1$ by selection sets of disjoint unions of random cubes, each of which decomposes coordinate-wise into one-dimensional interval selection problems. This gives us an approximation-based reduction of constrained selection problems for random compact convex sets in $\R^d$.

We present a universal quantum computing architecture which combines the measurement-driven aspect of MBQC with the circuit model's algorithm dependent generation of qubit entanglement. Our architecture, which we call QGATE, is tailored for discrete-variable photonic quantum computers with deterministic photon sources capable of generating 1D entangled photonic states. QGATE achieves universal quantum computing on a logical data qubit register via the implementation of Clifford operations, QGATE ancilla, and arbitrary angle single-qubit measurements. We realise unitary evolutions defined by multi-qubit Pauli strings via the generation of entanglement between a sub-set of logical qubits and a mutual QGATE ancilla qubit. Measurement of the QGATE ancilla in the appropriate basis then implements a given term of the desired unitary operation. This enables QGATE to both directly perform Hamiltonian evolutions in terms of a series of multi-qubit Pauli operators, in terms of projectors for an arbitrary sparse Hamiltonian, or realise multi-controlled gates enabling direct translation of circuit models to QGATE. We consider examples inspired by quantum chemistry and computational fluid dynamics. We propose an example photonic implementation of QGATE and calculate thresholds of $10.36\pm0.02\%$ or $25.98\pm0.28\%$ on the photonic loss for logical qubits constructed from foliated rotated surface codes, dependent on the deployment of intra-layer or inter-layer fusion respectively.

Let $A$ denote an affine algebra over an algebraically closed field $k$, with $\dim A=d\geq 3$. In the light of availability of cancellation theorems for stably free modules $P$ with $rank(P)=d-1$ (corank one), we try to implement the methods of complete intersections theory in corank zero, to the corank one case. Our conclusion is that cancellation theorems need to clean up some of the lack of minor generalities, for such an approach to work. However, we hypothesize and derive some of the consequences to complete intersections, of such hypotheses.

Complex concentrated alloys with intrinsic chemical heterogeneity are promising candidates for nuclear applications, where local chemical order can strongly influence defect evolution under irradiation. Grain boundaries also contribute to radiation damage mitigation by serving as defect sinks, yet this interaction can alter interfacial structure, typically leading to destabilization and grain growth. This study investigates how chemical ordering influences grain boundary migration and stability during successive radiation events in CrCoNi. Using atomistic simulations, bicrystals were equilibrated to induce segregation-enhanced chemical order, followed by prolonged irradiation at 1100 K. Our results show that grain boundaries in random CrCoNi begin to migrate after only a few collision cascades, whereas those in the ordered alloy remain immobile until the chemical order is sufficiently disrupted. Single-cascade simulations reveal key mechanistic differences, where cascades near chemically ordered interfaces produce smaller damage volumes and reduced atomic displacement due to enhanced Frenkel pair combination within the cascade core. This limits both the residual defect population and the energetic driving force for interfacial rearrangement. Subsequent simulations of irradiated interfaces show that interstitial absorption induces a structural transition that modifies the segregation morphology at and near the grain boundary, demonstrating a dynamic coupling between ordering stability and defect evolution. These findings offer new insights into the role of local chemical order on defect-interface interactions under extreme conditions and highlight pathways for designing radiation-tolerant materials for next-generation nuclear systems.

We use inelastic neutron scattering (INS) and angle-resolved photoemission spectroscopy (ARPES) to study the impact of Li doping on the semiconducting altermagnet $α$-MnTe. Introducing Li results in a spin reorientation from in-plane to out-of-plane direction and increases the density of itinerant carriers. While our ARPES measurements do not indicate any notable doping-induced changes in the electronic band structure or the magnitude of the altermagnetic band splitting, our INS measurements reveal an abrupt carrier-induced decrease in the spin wave lifetime near the zone boundary at high energies. This finding is consistent with a new magnon decay channel driven by doping-induced subtle changes in the band structure and enhanced interactions between Mn$^{2+}$ local moments and itinerant electrons. By extracting the local dynamic susceptibility from INS spectra and applying the total moment sum rule, we find that both undoped and Li-doped $α$-MnTe exhibit the full expected Mn$^{2+}$ local moment of $\approx5.9~μ_\mathrm{B}$ with $S=5/2$. These findings suggest that $α$-MnTe hosts robust local-moment altermagnetism which shows a breakdown at high energies upon addition of itinerant carriers, highlighting the importance of carrier-spin coupling in magneto-transport and spin dynamic properties of altermagnets even in the dilute-carrier limit.

We provide a complete and explicit characterization of the exposed extreme rays of the cone of sums of nonnegative circuit (SONC) polynomials. The criterion we derive is purely combinatorial and depends only on the existence of certain circuits within the ground set and on the nature of the corresponding extreme ray. Our constructive proofs also yield explicit exposing functionals, offering a basis for algorithmic detection of exposed rays in SONC-based optimization.

We present a new classical mechanism for nucleation of bubbles of true vacuum. The mechanism arises when dense boson stars form in the false vacuum. As the boson stars collapse due to attractive self-interactions, the field inside the star cores is enhanced beyond the potential barrier. Subsequently the stars explode as true vacuum bubbles, and induce a cosmological phase transition. The mechanism raises the possibility that a vacuum that is stable against quantum tunneling can be vulnerable to ``astrophysical'' processes.

A basic postulate of modern compositional approaches to generalised physical theories is the generalised Born rule, in which probabilities are postulated to be computable from the composition of states and effects. In this paper we consider whether this postulate, and the strength of the identification between scalars and probabilities, can be argued from basic principles. To this end, we first consider the most naive possible process- theoretic interpretation of textbook quantum theory, in which physical processes (unitaries) along with states and effects (kets and bras) and a probability function from states and effects satisfying just some basic compatibility axioms are identified. We then show that any process theory equipped with such structure is equivalent to an alternative process theory in which the generalised Born rule holds. We proceed to consider introduction of noise into any such theory, and observe that the result of doing so is a strengthening of the identification between scalars and probabilities; from bare monoid homomorphisms to semiring isomorphisms.

We formulate and prove an Agreement Theorem for quantum mechanics (QM), describing when two agents, represented by separate laboratories, can or cannot maintain differing probability estimates of a shared quantum property of interest. Building on the classical framework (Aumann, 1976), we define the modality of ``common certainty" through a hierarchy of certainty operators acting on each agent's Hilbert space. In the commuting case -- when all measurements and event projectors commute -- common certainty leads to equality of the agents' conditional probabilities, recovering a QM analog of the classical theorem. By contrast, when non-commuting operators are allowed, the certainty recursion can stabilize with different probabilities. This yields common certainty of disagreement (CCD) as a distinctive QM phenomenon. We show that agreement will nevertheless re-emerge if measurement outcomes are recorded in a classical register. We also establish an impossibility result stating that QM forbids a scenario where one agent is certain that a property of interest occurs, and is also certain that the other agent is certain that the property does not occur. In this sense, QM admits non-classical disagreement, but disagreement is still bounded in a disciplined way. We argue that our analysis offers a rigorous approach to the longstanding issue of how to understand intersubjectivity across agents in QM.

We investigate the origin of unidentified, extended TeV source 1LHAASO J0500$+$4454, considering three possible origins: cosmic rays interacting with a molecular cloud (MC), particles accelerated in a currently undetected supernova remnant (SNR), and an energetic outflow powered by a pulsar. Upper limits on the CO and X-ray emission from the $γ$-ray emitting region disfavor the MC and SNR scenarios, respectively. If a nebula of inverse Compton scattering $e^{\pm}$ powers 1LHAASO J0500$+$4454, then SED modeling indicates that the current particle energy in the nebula is $\sim 4 \times 10^{48}$ erg. If the coincident magnetar SGR 0501$+$4516's rotational energy powered 1LHAASO J0500$+$4454, then a conservative energy budget calculation requires an initial magnetar spin period $P_{0} \lesssim 5$ ms and a spin-down timescale $τ_{\rm sd} \lesssim 30$ yr, which has implications for the origins of magnetars.

We implement a feedback protocol to suppress the AC Stark shift in a two-photon rubidium optical frequency reference, reducing its sensitivity to optical power variations by a factor of 1000. This method alleviates the tradeoff between short-term and long-term stability imposed by the AC Stark shift, enabling us to simultaneously achieve instabilities of $3\times10^{-14}$ at 1 s and $2\times10^{-14}$ at $10^4$ s. We also quantitatively describe, and experimentally explore, a stability limit imposed on clocks using this method by frequency noise on the local oscillator.

The qualitative behavior of the Rabinowitz unbounded continuum of subcritical Gelfand problems is well known on balls in any dimension. We don't know of any such sharp and detailed description otherwise, which is our motivation to look for a new approach to the problem. The underlying idea is to describe solutions of Gelfand problems via suitably defined constrained problems of free boundary-type arising in plasma physics and to replace the usual $L^\infty$ norm of the solution with the energy of the plasma. Toward this goal, we first solve a long standing open problem of independent interest about the uniqueness of solutions of Grad-Shafranov type equations. Thus, we exploit these unique solutions to detect a curve containing both minimal and non minimal solutions of the associated Gelfand problem. In other words we come up with a new global parametrization of the Rabinowitz continuum, the monotonicity of the energy along the branch providing a meaningful generalization of the classical pointwise monotonicity property of minimal solutions, suitable to describe non minimal solutions as well. On a ball in any dimension, we come up as expected with a bell-shaped profile of the full branch of solutions of the Gelfand problem.

We present the results obtained by applying, in real-time, a correction method to precisely synchronize a time base generated from a free-running atomic clock with the Coordinated Universal Time (UTC). The method uses the Global Navigation Satellite System (GNSS) signals to have regular time comparisons between the atomic clock generated time base and the GPS Time, perform linear fits of the measurements and extrapolate a correction to apply to the free-running signal. In this work, we apply for the first time this method in real-time. Two atomic clocks were tested, a low-cost Rubidium clock and a more expensive magnetic Caesium clock. We demonstrate that we can obtain a residual difference between the clock time base and the French official realization of UTC (UTC(OP)) in the range of $\pm 15$ ns with no apparent residual drift.