The affine Yangian associated with $\widehat{\mathfrak{sl}}(n)$ has several presentations: the current presentation, the minimalistic presentation and so on. The evaluation map for the affine Yangian was given by using the minimalistic presentation. One of the issues about the evaluation map is that the images of the evaluation maps are unkown except on finitely many generators. In this article, we write down the images of the higher generators of the current presentation via the evaluation map for the affine Yangian of type $A$ explicitly.

Given a graph $G$ and a real $\varepsilon>0$, an edge-coloring of $G$ is called $\varepsilon$-balanced if each color appears on at least an $\varepsilon$-fraction of the edges in $G$. A classical result of Erdős and Szemerédi asserts that if a $2$-edge-coloring of a complete graph $K_n$ is not $\varepsilon$-balanced for some $0<\varepsilon\leq1/2$, then there exists a large monochromatic clique. This theorem has been used extensively in Ramsey-type arguments, as it allows one to focus on reasonably balanced colorings. However, in its original formulation the dependence between $n$ and $\varepsilon$ was left implicit, occasionally leading to inaccurate applications. In this short note, we revisit the Erdős--Szemerédi theorem and specify all parameter dependencies.

We generalize the theory of base norm spaces to the complex case, and further to the noncommutative setting relevant to `quantum convexity'. In particular, we establish the duality between complex Archimedean order unit spaces and complex base norm spaces, as well as the corresponding duality between their noncommutative counterparts. Additional topics include an exploration of natural connections with various notions of quantum convexity and regularity of noncommutative convex sets, and an analysis of how these concepts interact with complexification. We also define, as in the classical case, a class that contains and generates the noncommutative base norm spaces, but is defined by fewer axioms. We show how this may be applied to provide new and interesting examples of noncommutative base norm spaces.

For the LISA mission, Tilt-To-Length (TTL) coupling is expected to be one of the dominant instrumental noise contributions after laser frequency noise is suppressed based, on assumptions on the size of the coupling and angular jitter levels. This work uses for the first time a closed-loop, non-linear, and time-varying dynamics implementation to simulate detailed angular jitters for the spacecraft and optical benches. In turn, this gives an improved expectation of the TTL contribution to the interferometric output. It is shown that the TTL coupling impact is limited given current estimates on the size of coupling coefficients. A time-domain Least Squares estimator is used to infer the TTL parameters from the simulated measurements. The bias and correlations limit the estimator in the case of regular datasets with amplified TTL coefficients to a relative error of $10\%$, but the subtraction of the TTL signal still works well. For lower readout noises, the estimation error diverges, which can be mitigated using a regularization term. Alternatively, using sinusoidal maneuvers improves the inference to a high accuracy of $0.1\%$ for TTL coefficients around the expected level, removing all correlations in the inferred parameters. This validates the maneuver design by Wegener et al. (2025) in this closed-loop setting.

Ill-founded (or non-wellfounded) proof systems have emerged as a natural framework for inductive and coinductive reasoning. In such systems, soundness relies on global correctness criteria, such as the progressivity condition. Ensuring that these criteria are preserved under infinitary cut elimination remains a central technical challenge in ill-founded proof theory. In this paper, we present two cut elimination arguments for ill-founded $μ\mathsf{MALL}$ - a fragment of linear logic extended with fixed-points - based on the reducibility candidates technique of Tait and Girard. In both arguments, preservation of progressivity follows directly from the defining properties of the reducibility candidates. In particular, the second argument is derived from the topological notion of internally closed set developed in previous work by Afshari and Leigh.

A new Real Fluid Quasi-Conservative (RFQC) finite volume method is developed to address the numerical simulation of real fluids involving shock waves in transcritical and phase-change flows. To eliminate the spurious pressure oscillations inherent in fully conservative schemes, we extend the classic quasi-conservative method, originally designed for two-phase flows, to real fluids governed by arbitrary equations of state (EoS). The RFQC method locally linearizes the real fluid EoS at each grid point and time step, constructing and evolving the frozen Grüneisen coefficient $Γ$ and the linearization remainder $E_0$ via two advection equations. At the end of each time step, the evolved $Γ$ and $E_0$ are utilized to reconstruct the oscillation-free pressure field, followed by a thermodynamic re-projection applied to the conserved variables. Theoretical analysis demonstrates that, in smooth regions, the energy conservation error introduced by the RFQC method is a second-order small term dominated by the time-step. In discontinuous regions, this error is determined by the entropy increase rate, thereby maintaining consistency with the inherent truncation error of shock-capturing methods. A series of numerical tests verifies that the method can robustly simulate complex flow processes with only minor energy conservation errors, including transcritical flows, phase transitions, and shock-interface interactions. The RFQC method is proven to be both accurate and robust in capturing shock waves and phase transitions.

Estimating natural effects is a core task in causal mediation analysis. Existing triply robust (TR) frameworks (Tchetgen Tchetgen & Shpitser 2012) and their extensions have been developed to estimate the natural effects. In this work, we introduce a new quadruply robust (QR) framework that enlarges the model class for unbiased identification. We study two modeling strategies. The first is a nonparametric modeling approach, under which we propose a general QR estimator that supports the use of machine learning methods for nuisance estimation. We also study high-dimensional settings, where the dimensions of covariates and mediators may both be large. In these settings, we adopt a parametric modeling strategy and develop a model quadruply robust (MQR) estimator to limit the impact of model misspecification. Simulation studies and a real data application demonstrate the finite-sample performance of the proposed methods.

We report the detection of the nucleus of interstellar object 3I/ATLAS, using a nucleus extraction technique on Hubble Space Telescope (HST) observations taken between December 2025 and January 2026. The product of the V-band geometric albedo, $p_V$, with the physical cross-section of the nucleus is $0.22 \pm 0.07$ km$^{2}$, which corresponds to an effective radius of $1.3 \pm 0.2$ km if assuming a comet-like albedo $p_{V} = 0.04$. This size is in agreement with an independent estimate based on the reported nongravitational acceleration and activity of the interstellar object. If the measured photometric variations are solely due to the rotation of an aspherical nucleus, the axis ratio must be $2:1$ or greater, and the rotation period $\gtrsim\!1$ hr. Leveraging the range of covered phase angles, we identified a significant opposition surge of $\sim\!0.2$ mag with a width of $3^{\circ} \pm 1^{\circ}$, which may include concurrent contributions from orbital plane crossing and tail projection, and determined a linear phase slope of $0.026 \pm 0.006$ mag degree$^{-1}$ for the coma dust. Compared to the preperihelion brightening trend, 3I faded more rapidly on the outbound leg, following an activity index of $4.5 \pm 0.3$, not unusual in the context of solar system comets. This activity asymmetry is further corroborated by a postperihelion coma surface brightness profile that is significantly shallower than its preperihelion counterpart. From discovery statistics, we infer that multiple interstellar objects resembling 3I probably went undetected prior to the discovery of 1I/`Oumuamua, unless the overall population possesses a steep size distribution.

The formation of primordial black holes (PBHs) generally requires large density perturbations, which is widely supported by researchers. This paper studies the local coupling properties of the Starobinsky and Kachru-Kallosh-Linde-Trivedi (KKLT) potentials by introducing a linear Lorentzian-type coupling, which locally breaks the slow-roll conditions. We find that both positive and negative couplings can generate a considerable abundance of PBH. Additionally, we study the scalar-induced gravitational waves (SIGWs) generated by this model.

Let $\mathcal G\simeq H\rtimesΓ$ be the semidirect product of a finite group $H$ and $Γ\simeq\mathbb Z_p$. Let $ F/\mathbb Q_p$ be a finite extension with ring of integers $\mathcal O_F$. Then the total ring of quotients $\mathcal Q^F(\mathcal G)$ of the completed group ring $\mathcal O_F[[\mathcal G]]$ is semisimple artinian. We determine its Wedderburn decomposition in full generality in terms of the Wedderburn decomposition of the group ring $ F[H]$. Such a description was previously available only for those simple components for which a certain associated field extension is totally ramified.

Detecting the stochastic gravitational wave background (SGWB) from our Universe under the inflationary era is one of the primary scientific objectives of DECi-hertz Interferometer Gravitational wave Observatory (DECIGO), a space-borne gravitational wave detector sensitive in the 0.1Hz frequency band. This frequency band is dominated by the gravitational waves from inspiraling compact object binaries. Subtracting these signals is necessary to search for the primordial SGWB. In this paper, we assess the feasibility of the subtraction of such binary signals by employing the population model inferred from the latest gravitational wave event catalogue of the LIGO-Virgo-KAGRA Collaboration. We find that the projection scheme, which was originally proposed by Cutler & Harms (2005), is necessary to reduce the binary signals to the level where DECIGO can detect the primordial background.

We prove that the Ginzburg-Landau equation correctly predicts the dynamics of quasilinear pattern-forming reaction-diffusion-advection systems, close to the first instability. We present a simple theorem which is easily applicable for such systems and relies on key maximal regularity results. The theorem is applied to the Gray-Scott-Klausmeier vegetation-water interaction model and its application to general reaction-diffusion-advection systems is discussed.

We show that for $\gg K^2$ of the half-integral weight Hecke cusp forms in the Kohnen plus subspaces with weight bounded by a large parameter $K$, the number of "real" zeroes grows at the expected rate. A key technical step in the proof is to obtain sharp bounds for the mollified first and second moments of quadratic twists of modular $L$-functions.

For $χ_k$ a self$-$dual primitive Dirichlet character mod $k$ several reduced identities of Dirichlet $L-$functions $L_k(s):=L(s,χ_k)$, expressed as linear combinations of Hurwitz $ζ$ functions, are found for $s=2,3$ and some selected values of $k$. By using a merged approach between the Wilf$-$Zeilberger method and a Dougall$'$s $_5H_5$ technique, new proven accelerated series of hypergeometric$-$type are derived for specific Hurwitz $ζ$ function values. These fast series that are computed by means of the binary splitting algorithm, enter into the reduced identities found producing very efficient formulas to compute selected $L-$function values. The new algorithms include $L_k(2)$ for $k = -4$ Catalan's constant, $-7, -8, -15, -20, -24$ together with $L_k(3)$ for $k = 1$ Apery's constant, $5, 8$ and $12$. Formulas were tested and verified up to 100 million decimal places for each $L-$value.

In this article, we explore the use of contour deformation for the numerical evaluation of Feynman integrals after sector decomposition. In existing codes, the contour of integration is determined heuristically for each phase-space point by sampling the integrand. In this work, we introduce a method for choosing the contour deformation for an entire phase-space region using only an initial sampling or training step. We demonstrate that the resulting integrand has a lower variance than that obtained with heuristic methods and show that optimising a contour to reduce the estimated error of a Quasi-Monte Carlo sample is an ill-defined problem. The a priori knowledge of the integration path obtained in this work can be used to improve the speed of conventional integration methods or be leveraged for integration using neural networks, where, crucially, it removes the need to retrain the neural network for each phase-space point. The techniques described in this work can be adapted to other problems where a non-trivial integration path has to be chosen subject to a set of constraints.

Evidence for fluvial features and standing liquid water indicate that Mars was a warmer and wetter place in its past; however, climate models have historically been unable to produce conditions to yield a warm early Mars under the faint young sun. Some models invoke thick greenhouse atmospheres to produce continuously warm conditions, but others have argued that available geologic evidence is more consistent with short-duration and transient warming events on an otherwise cold Mars. One possibility of harmonizing these perspectives is that early Mars experienced climate limit cycles that caused the climate to oscillate between short periods of warmth and prolonged periods of glaciation, due to modulation of greenhouse warming by the carbonate-silicate cycle. This study suggests that episodic limit cycling during the Noachian and Hesperian periods provides a hypothetical explanation for the timing and formation of fluvial features on Mars. A schematic time-forward trajectory of the full history of Mars is calculated using an energy balance climate model, which includes an active carbonate-silicate cycle, instellation changes due to the sun's main sequence evolution, variations in the obliquity of Mars, and supplemental warming from additional greenhouse gases beyond carbon dioxide alone. These calculations demonstrate the viability of a climate history for Mars involving episodic limit cycling to enable the formation of the valley networks at 4.1-3.5 Ga and delta features at 3.3-3.0 Ga, interspersed with cold stable climates and ending in the late Amazonian in a carbon dioxide condensation regime. This schematic climate trajectory provides a plausible narrative that remains consistent with available geologic data, and further exploration of warming mechanisms for the climate of Mars should consider the possibility of episodic transient events driven by carbonate-silicate limit cycling.

This paper focuses on phase retrieval from phaseless total-field data in biharmonic scattering problems. We prove that a phased biharmonic wave can be uniquely determined by the modulus of the total biharmonic wave within a nonempty domain. As a direct corollary, the uniqueness for the inverse biharmonic scattering problem with phaseless total-field data is established. Moreover, using the Atkinson-type asymptotic expansion, we derive explicit asymptotic formulas for the problem of phase retrieval.

We introduce pseudofinite W*-probability spaces. These are W*-probability spaces that are elementarily equivalent to Ocneanu ultraproducts of finite-dimensional von Neumann algebras equipped with arbitrary faithful normal states. We are particularly interested in the case where these finite-dimensional von Neumann algebras are full matrix algebras: the pseudofinite factors. We show that these are indeed factors. We see as a consequence that pseudofinite factors are never of type $\mathrm{III}_0$. Mimicking the construction of the Powers factors, we give explicit families of examples of matrix algebra ultraproducts that are $\mathrm{III}_λ$ factors for $λ\in (0,1]$. We show that these examples share their universal theories with the corresponding Powers factor and thus have uncomputable universal theories. Finally, we show that pseudofinite factors are full. This generalizes a theorem of Farah-Hart-Sherman which shows that pseudofinite tracial factors do not have property $Γ$. It has the consequence that hyperfinite factors of type $\mathrm{III}$ (the Powers factors) are never pseudofinite. Our proofs combine operator algebraic insights with routine continuous logic syntactic arguments: using Łos' theorem to prove that certain sentences which are true for all matrix algebras are inherited by their ultraproducts.

We compute the gravitational fluxes and waveform for eccentric compact binaries including matter effects through adiabatic tidal interactions within the post-Newtonian approximation. The computations are performed at the relative 2.5PN order. Using the dynamics derived in the companion paper, we first derive the radiated energy and angular momentum, from which we deduce the equations describing the secular evolution of the orbital elements. We numerically solve for the secular dynamics for various systems. We find that the eccentric corrections to tidal terms induce a dephasing that could potentially be detectable in some regions of the parameter space of gravitational wave sources. Finally, we compute the amplitude of the strain, decomposed in spin-weighted spherical harmonics. Besides the memory contributions that are left for future works, we provide the amplitude modes containing the instantaneous, tail and post-adiabatic corrections expanded to the twelfth order in eccentricity. All relevant results are provided in an ancillary file.

We quantize the regularity properties of classical graphs that determine spin models for singly-generated Yang-Baxter planar algebras, including the Kauffman polynomial, and construct explicit examples. A source of examples comes from deforming graphs using higher-idempotent splittings of quantum isomorphisms for which we prove that the relevant algebraic, combinatorial, and topological properties of the original graphs are preserved along with the quantum automorphism group. We also obtain exotic examples of highly regular quantum graphs using the quantum Fourier transform and a method of iterated convolution. Our examples include quantum versions of the strongly regular $9$-Paley, $16$-Clebsch and the Higman-Sims graphs, yielding new models for their regularity parameters. As applications, we construct a compact quantum group that is monoidally equivalent to $SO_q(5)$ at the square of the golden ratio, whose dual is infinite with property (T), and exhibit a highly-regular quantum graph with no classical analogue. Finally, we introduce quantum spin models, construct explicit examples and make contact with quantum Hadamard matrices.

We present a versatile framework to study strong existence and uniqueness for stochastic differential equations (SDEs) in Hilbert spaces with irregular drift. We consider an SDE in a separable Hilbert space $H$ \begin{equation*} dX_t= (A X_t + b(X_t))dt +(-A)^{-γ/2}dW_t,\quad X_0=x_0 \in H, \end{equation*} where $A$ is a self-adjoint negative definite operator with purely atomic spectrum, $W$ is a cylindrical Wiener process, $b$ is $α$-Hölder continuous function $H\to H$, and a nonnegative parameter $γ$ such that the stochastic convolution takes values in $H$. We show that this equation has a unique strong solution provided that $α> α^*(γ)$, with an explicit function $α^*$ that takes values in $(0,1)$ for all $γ\in[0,3)$. This substantially extends the seminal work of Da Prato and Flandoli (2010) as no structural assumption on $b$ is imposed. The range of admissible $α$ is also extended. To obtain this result, we do not use infinite-dimensional Kolmogorov equations but instead develop a new technique combining Lê's theory of stochastic sewing in Hilbert spaces, Gaussian analysis, and a method of Lasry and Lions for approximation in Hilbert spaces.

Global ideal magnetohydrodynamic models of the heliosphere typically predict a greatly exaggerated magnetic field pile-up in the inner heliosheath (IHS), the region between the termination shock and heliopause. However, Voyager 1 and 2 observations show only a gradual increase throughout this region. This mismatch is largely attributed to the simplified assumption of a unipolar solar magnetic field in many global models, which neglects the complex, folded structure of the heliospheric current sheet (HCS). The IHS, especially at low heliolatitudes, contains these compressed sector boundaries, widely considered prime locations for magnetic dissipation via reconnection. To align global model simulations with observations without incurring the prohibitive computational cost of resolving the kinetic-scale current sheet, this work introduces a phenomenological term into the magnetic field induction equation. This term captures the macroscopic effect of magnetic energy dissipation due to unresolved HCS dynamics. It is designed to mitigate the artificial magnetic pile-up, preserve the topological integrity of the magnetic field lines, and avoid explicit magnetic diffusion. This study demonstrates that incorporating a phenomenological dissipation term into global heliospheric models helps to resolve the longstanding discrepancy between simulated and observed magnetic field profiles in the IHS. The proposed mechanism reduces exaggerated magnetic energy (converts it into thermal energy), aligns model output with Voyager measurements of both magnetic field and proton density, and produces the outward shift in termination shock position and a reduction of the IHS thickness. We found that the characteristic time for magnetic field dissipation of about 6 years provides improved agreement with Voyager data in the IHS.

Predictive inference requires balancing statistical accuracy against informational complexity, yet the choice of complexity measure is usually imposed rather than derived. We treat econometric objects as predictive rules, mappings from information to reported predictive distributions, and impose three structural requirements on evaluation: locality, strict propriety, and coherence under aggregation (coarsening/refinement) of outcome categories. These axioms characterize (uniquely, up to affine transformations) the logarithmic score and induce Shannon mutual information (Kullback-Leibler divergence) as the corresponding measure of predictive complexity. The resulting entropy-regularized prediction problem admits Gibbs-form optimal rules, and we establish an essentially complete-class result for the admissible rules we study under joint risk-complexity dominance. Rational inattention emerges as the constrained dual, corresponding to frontier points with binding information capacity. The entropy penalty contributes additive curvature to the predictive criterion; in weakly identified settings, such as weak instruments in IV regression, where the unregularized objective is flat, this curvature stabilizes the predictive criterion. We derive a local quadratic (LAQ) expansion connecting entropy regularization to classical weak-identification diagnostics.

Over the last 15 years, the Atacama Large Millimeter/submillimeter Array (ALMA) has revolutionized astrophysics by providing unprecedented resolution and sensitivity in observing the cold universe, including the formation of stars, planets, and galaxies. With groundbreaking discoveries ranging from the first detailed images of protoplanetary disks to the kinematics of galaxies in the Epoch of Reionization, ALMA has showcased the vast discovery potential of the (sub-)mm wavelength regime. However, in another 15 years from now--in the 2040s--the science landscape will have changed dramatically as new major observational facilities will have started their operations or have come towards advanced maturity in their scientific outcome (e.g., JWST, Rubin Observatory, ELT, Euclid, Gaia, Plato, Ariel, Roman Space Telescope, SPHEREx, LiteBIRD, LISA, SKA and others). At the same time, ALMA's current Wideband Sensitivity Upgrade will have been in place for ~10 years, and ALMA itself will have been operational for 30 years. To fully exploit this era, the community needs a next-generation facility operating at (sub-)mm wavelengths with capabilities far beyond those possible within ALMA's current infrastructure. To this end, ALMA2040 is a community-driven initiative to define the key scientific questions of the 2040s and translate them into a technical vision for a next-generation transformational (sub-)millimeter facility. Our goal with this document is to summarize the current status of the effort, synthesize outcomes from the 2025 workshops, outline next steps toward a reference design concept, and invite broad participation from the global mm/sub-mm community to help shape this future facility. In the following we provide details on the process and scope. We invite everyone who wishes to join the effort and/or contribute to the dedicated White Papers planned for 2026.

Starting from limited measurements of a turbulent flow, data assimilation (DA) attempts to estimate all the spatio-temporal scales of motion. Success is dependent on whether the system is observable from the measurements, or how much of the initial turbulent field is encoded in the available measurements. Adjoint-variational DA minimises the discrepancy between the true and estimated measurements by optimising the initial velocity or vorticity field (the `state space'). Here we propose to instead optimise in a lower-dimensional latent space which is learned by implicit rank minimising autoencoders. Assimilating in latent space, rather than state space, redefines the observability of the measurements and identifies the physically meaningful perturbation directions which matter most for accurate prediction of the flow evolution. When observing coarse-grained measurements of two-dimensional Kolmogorov flow at moderate Reynolds numbers, the proposed latent-space DA approach estimates the full turbulent state with a relative error improvement of two orders of magnitude over the standard state-space DA approach. The small scales of the estimated turbulent field are predicted more faithfully with latent-space DA, greatly reducing erroneous small-scale velocities typically introduced by state-space DA. Furthermore, latent-space DA is demonstrated to be robust to noisy measurements at the range of Reynolds numbers considered. These findings demonstrate that the observability of the system from available data can be greatly improved when turbulent measurements are assimilated in the right space, or coordinates.

In this paper, we study how global index formulas arise in the theory of one-dimensional holomorphic foliation from the microlocal point of view. We give short proofs and generalizations to a few exisiting index formulas concerning Schwartz, GSV and logarithmic indices.

We investigate finite-temperature magnetic and topological phase transitions in two-dimensional honeycomb ferromagnets using an extended self-consistent renormalized spin-wave theory (SRSWT) that incorporates higher-order corrections from the Holstein--Primakoff expansion. Focusing on the combined effects of single-ion anisotropy, Zeeman field, next-nearest-neighbor (NNN) exchange, and Dzyaloshinskii--Moriya interaction, we analyze how these parameters influence the magnetization curves and magnon spectra. This work serves two main goals. First, we critically examine the limitations of SRSWT, showing that in the absence of external or interaction tuning, the theory tends to overestimate magnon self-energy corrections, often predicting first-order magnetic transitions with multivalued magnetization and metastable solution branches (i.e., self-consistent but thermodynamically unstable states). Second, we demonstrate that topological transitions -- signaled by magnon gap closings at the Dirac points -- can be tuned to occur below the magnetic transition temperature and within the thermodynamically stable regime. In particular, we identify two practical tuning strategies: applying an external Zeeman field of appropriate sign depending on the anisotropy strength, and introducing a small antiferromagnetic NNN exchange coupling. These findings not only clarify the predictive scope and limitations of SRSWT but also provide experimentally relevant guidance for realizing thermally driven topological transitions in two-dimensional honeycomb magnetic insulators.

Open quantum systems play a central role in contemporary nanoscale technologies, including molecular electronics, quantum heat engines, quantum computation and information processing. A major theoretical challenge is to construct dynamical models that are simultaneously consistent with classical thermodynamics and complete positivity. In this work, we develop a framework that addresses this issue by extending classical stochastic dynamics to the quantum domain. We begin by formulating a generalized Langevin equation in which both friction and noise act symmetrically on the two Hamiltonian equations. From this, we derive a generalized Klein-Kramers equation expressed in terms of Poisson brackets, and we show that it admits the Boltzmann distribution as its stationary solution while satisfying the first and second laws of thermodynamics along individual trajectories. Applying canonical quantization to this classical framework yields two distinct quantum master equations, depending on whether the friction operators are taken to be Hermitian or non-Hermitian. By analyzing the dynamics of a harmonic oscillator, we determine the conditions under which these equations reduce to a Lindblad-type generator. Our results demonstrate that complete positivity is ensured only when friction and noise are included in both Hamiltonian equations, thus fully justifying the classical construction. Moreover, we find that the friction coefficients must satisfy the same positivity condition in both the Hermitian and non-Hermitian formulations, revealing a form of universality that transcends the specific operator representation. The formalism offers a versatile tool for deriving quantum versions of the thermodynamic laws and is directly applicable to a wide class of nonequilibrium nanoscale systems.

We analyze the properties of weakly compact sets in Lipschitz free spaces. Prior research has established that, for a complete metric space $M$, weakly precompact sets in the Lipschitz free space $\mathcal F(M)$ are tight. In this paper, we prove that these sets actually exhibit a stronger property, which we call uniform regularity. However, this condition alone is not sufficient to characterize weakly compact sets, except in the case of scattered metric spaces. On the other hand, if $T$ is an $\mathbb R$-tree, we leverage Godard's isometry between $\mathcal F(T)$ and $L^1(λ_T)$ to obtain an intrinsic characterization of weakly compact sets in $\mathcal F(T)$. This approach allows us to identify conditions that may describe weak compactness across a wider range of spaces. In particular, we provide a characterization of norm-compactness in terms of sums of "large molecules'', while we show that sums of "small molecules'' contain an $\ell_1$-basis.

In the present research, a bistable delayed-feedback oscillator with two delayed-feedback loops is shown to replicate a network of bistable nodes with nonlocal coupling. It is demonstrated that certain aspects of the nonlocal interaction impact on wavefront propagation identified in networks of bistable elements are entirely reproduced in the dynamics of a single oscillator with two delays. In particular, adding the second delayed-feedback loop allows speeding up both deterministic and stochastic wavefront propagation, achieving stabilization of propagating fronts at lower noise intensity and preventing fronts from noise-induced destruction occurring in the presence of single delayed-feedback. All the revealed effects are studied in numerical simulations and confirmed in physical experiments, showing an excellent correspondence.