Active nematics exhibit spontaneous flows through a well-known linear instability of the uniformly-aligned quiescent state. Here we show that even a linearly stable uniform state can experience a nonlinear instability, resulting in a discontinuous transition to spontaneous flows. In this case, quiescent and flowing states may coexist. Through a weakly nonlinear analysis and a numerical study, we trace the bifurcation diagram of striped patterns and show that the underlying pitchfork bifurcation switches from supercritical (continuous) to subcritical (discontinuous) by varying the flow-alignment parameter. We predict that the discontinuous spontaneous flow transition occurs for a wide range of parameters, including systems of contractile flow-aligning rods. Our predictions are relevant to active nematic turbulence and can potentially be tested in experiments on either cell layers or active cytoskeletal suspensions.

We provide a simple proof that the unit map from the sphere spectrum to the connective image-of-$J$ spectrum $\mathrm{j}$ is surjective on homotopy groups. This is achieved using a novel $t$-structure on the category of $E$-synthetic spectra and a specific construction of $\mathbf{F}_p$- and BP-synthetic lifts of $\mathrm{j}$. These synthetic lifts then easily produce modified Adams and Adams--Novikov spectral sequences for $\mathrm{j}$ which we use the prove the above detection statement, all without ever calculating $\mathbf{F}_p$- or BP-homology nor the associated Ext groups.

Moiré materials have emerged as a rich platform for exploring strong correlation effects in low dimensions, with twisted bilayer graphene (TBG) as a paradigmatic example. To distill the essential ingredients driving moiré-induced phases, a simplified one-dimensional analog -- a two-leg ladder with spatially modulated interleg hopping and a uniform magnetic flux -- was recently introduced. This model, which we refer to as the moiré ladder, features a nearly flat lowest-energy band in a suitable parameter regime, capturing the band-flattening mechanism of TBG. We investigate the ground-state phase diagram of the moiré ladder using a combination of bosonization and density matrix renormalization group (DMRG) techniques, and systematically disentangle the respective roles of the flux and the hopping modulation. At half filling, previous numerical work identified a metal-insulator transition at finite interaction strength and an unexpected ferromagnetic ground state. Revisiting this, we show that the metal-insulator transition can be understood perturbatively within bosonization, governed by the number of Fermi points. In contrast, the ferromagnetic correlations are nonperturbative and require both flux and spatial modulation -- neither alone is sufficient. We extend our analysis to other fillings: one-quarter, three-quarters, slightly above half filling (half filling plus two electrons), and slightly below half filling (half filling minus two electrons). At moderate interactions, we observe ferromagnetism below half filling and antiferromagnetism above; at stronger interactions, ferromagnetism dominates across all studied fillings. Crucially, the analysis demonstrates that periodic interleg hopping alone does not engender new correlated phases; the magnetic flux is essential for the observed unconventional behavior.

We consider evolutionary Stokes system, coupled with the so-called dynamic slip boundary condition, in the simple geometry of a $d$-dimensional half-space. Using the standard technique of the Fourier transform in tangential directions, we obtain an explicit formula for the resolvent. We then deduce estimates for both the weak (i.e. $W^{1,p}$) and strong (hence $W^{2,p}$) solutions, which are optimal in terms of the data belonging to appropriate negative Sobolev or fractional Besov space. In the latter case $L^p$-integrability of the pressure gradient is included. We allow for solutions with non-zero divergence, thus preparing the way for extensions to general domains. As a by-product, we show that the system generates an analytic semigroup in $L^p(Ω)\times L^p(\partial Ω)$. Our approach remains elementary in the sense that only the classical Mikhlin multiplier theorem will be used. The methods of $\mathcal{H}^{\infty}$-calculus are implicitly present; but we stay away from the concept of $R$-boundedness and related heavy functional analytic machinery.

The ability of thin materials to shape-shift is a common occurrence that leads to dynamic pattern formation and function in natural and man-made structures. However, harnessing this concept to design inorganic structures at the nanoscale rationally has remained far from reach due to a lack of fundamental understanding of the essential physical components. Here, we show that the interaction between organic ligands and the nanocrystal surface is responsible for the full range of chiral shapes seen in colloidal nanoplatelets. The adsorption of ligands results in incompatible curvatures on the top and bottom surfaces of NPL, causing them to deform into helicoïds, helical ribbons, or tubes depending on the lateral dimensions and crystallographic orientation of the NPL. We demonstrate that nanoplatelets belong to the broad class of geometrically frustrated assemblies and exhibit one of their hallmark features: a transition between helicoïds and helical ribbons at a critical width. The effective curvature $\barκ$ is the single aggregate parameter that encodes the details of the ligand/surface interaction, determining the nanoplatelets' geometry for a given width and crystallographic orientation. The conceptual framework described here will aid the rational design of dynamic, chiral nanostructures with high fundamental and practical relevance.

Collective variables (CVs) are low-dimensional projections of high-dimensional system states. They are used to gain insights into complex emergent dynamical behaviors of processes on networks. The relation between CVs and network measures is not well understood and its derivation typically requires detailed knowledge of both the dynamical system and the network topology. In this work, we present a data-driven method for algorithmically learning and understanding CVs for binary-state spreading processes on networks of arbitrary topology. We demonstrate our method using four example networks: the stochastic block model, a ring-shaped graph, a random regular graph, and a scale-free network generated by the Albert-Barabási model. Our results deliver evidence for the existence of low-dimensional CVs even in cases that are not yet understood theoretically.

Given a set of overlapping local views (patches) of a dataset, we consider the problem of finding a rigid alignment of the views that minimizes a $2$-norm based alignment error. In general, the views are noisy and a perfect alignment may not exist. In this work, we characterize the non-degeneracy of an alignment in the noisy setting based on the kernel and positivity of a certain matrix. This leads to a polynomial time algorithm for testing the non-degeneracy of a given alignment. Subsequently, we focus on Riemannian gradient descent for minimizing the alignment error, providing a sufficient condition on an alignment for the algorithm to converge (locally) linearly to it. \revadd{Additionally, we provide an exact recovery and noise stability analysis of the algorithm}. In the case of noiseless views, a perfect alignment exists, resulting in a realization of the points that respects the geometry of the views. Under a mild condition on the views, we show that a non-degenerate perfect alignment \revadd{characterizes the infinitesimally rigidity of a realization, and thus the local rigidity of a generic realization}. By specializing the non-degeneracy conditions to the noiseless case, we derive necessary and sufficient conditions on the overlapping structure of the views for \revadd{a perfect alignment to be non-degenerate and equivalently, for the resulting realization to be infinitesimally rigid}. Similar results are also derived regarding the uniqueness of a perfect alignment and global rigidity.

Multiway data analysis aims to uncover patterns in data structured as multi-indexed arrays, with multiway covariance playing a crucial role in many applications. However, the high dimensionality of multiway covariance presents significant computational challenges. To overcome these challenges, factorized covariance models have been proposed that rely on a separability assumption: the multiway covariance can be accurately expressed as a sum of Kronecker products of mode-wise covariances. This paper addresses the representability, certification, and approximation of such separable models, leaving statistical estimation or finite-sample properties aside. We reduce the question of whether a given covariance can be decomposed into a separable multiway form to an equivalent question about the separability of quantum states. Leveraging results from quantum information theory, we show that generic multiway covariances are typically \emph{not} separable and that determining the best separable approximation is NP-hard. These findings suggest that factorized covariance models can be overly restrictive and difficult to fit without additional structural assumptions. Nevertheless, our numerical experiments indicate that standard iterative algorithms, namely Frank-Wolfe and gradient descent, often converge close to the best separable approximation. As NP-hardness concerns worst-case computational complexity, Kronecker-separable approximations to multiway covariance could still be tractable to apply for analyzing many real-world datasets.

We consider time-continuous Markovian discrete-state dynamics on random networks of interacting agents and study the large population limit. The dynamics are projected onto low-dimensional collective variables given by the shares of each discrete state in the system, or in certain subsystems, and general conditions for the convergence of the collective variable dynamics to a mean-field ordinary differential equation are proved. We discuss the convergence to this mean-field limit for a continuous-time noisy version of the so-called ``voter model'' on Erdős-Rényi random graphs, on the stochastic block model, and on random regular graphs. Moreover, a heterogeneous population of agents is studied.

We construct Adams operations on the cohomology theory Tmf of topological modular forms; the first such stable operations on this cohomology theory. These Adams operations are then calculated on the Tmf-cohomology of spheres using a combination of descent spectral sequences and Anderson duality. Applications of these operations are then given, including constructions of connective height 2 analogues of Adams summands and image of J spectra.

For an $S^1$-framed modular operad $P$, we introduce its "Feynman compactification" denoted by $FP$ which is a modular operad. Let $\{\mathbb{M}^{\sf fr}(g,n)\}_{(g,n)}$ be the $S^1$-framed modular operad defined using moduli spaces of smooth curves with framings along punctures. We prove that the homology operad of $F\mathbb{M}^{\sf fr}$ is isomorphic to $H_*(\overline{M})$, the homology operad of the Deligne-Mumford operad. Using this isomorphism, we obtain an explicit formula of the fundamental class of $[\overline{M}_{g,n}/S_n]$ in terms of Sen-Zwiebach's string vertices. As an immediate application, under mild assumptions, we prove that Costello's categorical enumerative invariants of the ground field match with the Gromov-Witten invariants of a point.

We present an algorithm to compute the Hodge ideals of $\mathbb{Q}$-divisors associated to any reduced effective divisor $D$. The computation of the Hodge ideals is based on an algorithm to compute parts of the $V$-filtration of Malgrange and Kashiwara on $ι_{+}\mathscr{O}_X(*D)$ and the characterization of the Hodge ideals in terms of this $V$-filtration. In particular, this gives a new algorithm to compute the multiplier ideals and the jumping numbers of any effective divisor.

A bosonic Laplacian, which is a generalization of Laplacian, is constructed as a second order conformally invariant differential operator acting on functions taking values in irreducible representations of the special orthogonal group, hence of the spin group. In this paper, we firstly introduce some properties for homogeneous polynomial null solutions to bosonic Laplacians, which give us some important results, such as an orthogonal decomposition of the space of polynomials in terms of homogeneous polynomial null solutions to bosonic Laplacians, etc. This work helps us to introduce Bergman spaces related to bosonic Laplacians, named as bosonic Bergman spaces, in higher spin spaces. Reproducing kernels for bosonic Bergman spaces in the unit ball and a description of bosonic Bergman projection are given as well. At the end, we investigate bosonic Hardy spaces, which are considered as generalizations of harmonic Hardy spaces. Analogs of some well known results for harmonic Hardy spaces are provided here. For instance, connections to certain complex Borel measure spaces, growth estimates for functions in the bosonic Hardy spaces, etc.

For any 4-variate quartic form $f\geq 0$ (i.e. $f$ nonnegative, homogeneous polynomial of degree $4$ with real coefficients) there exist quadratic forms $q$ and $q'$ so that $qq'f$ is a sum of squares (s.o.s.) of quartics, by reducing to the case of $f=au^2+2bu+c$ with $a$, $b$, $c$ $3$-variate forms of degrees 2, 3, 4, respectively, and invoking on its discriminant $Δ=ac-b^2$ a theorem by Hilbert (1893) asserting that for any ternary sextic $h\geq 0$ there exists a quadric $q''$ so that $q''h$ is s.o.s. of quartics. Towards deciding whether just one $q$ always suffices to make $qf$ a s.o.s, we give explicit examples of non-s.o.s. $f=au^2+2bu+c\geq 0$ with non-s.o.s. $Δ$. However, in all these examples $af$ are s.o.s. That is, the straightforward s.o.s. decomposition via Hilbert (1893) need not be the best possible. While it remains open whether one $q$ always suffices (and we conjecture that $q=a$ suffices), we describe how the existence of such $q$ is related to particular types of s.o.s. decompositions for $Δ$.

This paper presents a technique for stress and fracture analysis by using the scaled boundary finite element method (SBFEM) with quadtree mesh of high-order elements. The cells of the quadtree mesh are modelled as scaled boundary polygons that can have any number of edges, be of any high orders and represent the stress singularity around a crack tip accurately without asymptotic enrichment or other special techniques. Owing to these features, a simple and automatic meshing algorithm is devised. No special treatment is required for the hanging nodes and no displacement incompatibility occurs. Curved boundaries and cracks are modelled without excessive local refinement. Five numerical examples are presented to demonstrate the simplicity and applicability of the proposed technique.

This paper consists mostly of pictures visualizing ideas leading to Godel's rotating cosmological model. The pictures are constructed according to concrete metric tensor fields. Information about these are in the last chapters. The main aim is to visualise ideas.

This paper presents a hierarchical decision-making framework for autonomous systems operating under uncertainty, demonstrated through autonomous driving as a representative application. Surrounding agents are modeled using Hybrid Markov Decision Processes (HMDPs) that jointly capture maneuver-level and dynamic-level uncertainties, enabling the multi-modal environmental prediction. The ego agent is modeled using a separate HMDP and integrated into a Model Predictive Control (MPC) framework that unifies maneuver selection with dynamic feasibility within a single optimization. A set of joint chance constraints serves as the bridge between environmental prediction and optimization, incorporating multi-modal environment predictions into the MPC formulation and ensuring safety across all plausible interaction scenarios. The proposed framework provides theoretical guarantees on recursive feasibility and asymptotic stability, and its benefits in terms of safety and efficiency are validated through comprehensive evaluations in highway and urban environments, together with comparisons against a rule-based baseline.

A classical result by Otter shows that the complete graph has an exponential number of non-isomorphic spanning trees. This was recently extended by Lee to every almost regular graph of sufficiently large degree. In this paper, we consider graphs of large minimum degree. We show that every connected graph $G$ with $n$ vertices and minimum degree $d$ has at least $n^{Ω(d)}$ non-isomorphic spanning trees. This is tight up to the constant factor in the exponent. In fact, we prove the following anticoncentration result: if $\mathcal{T}$ is a uniformly random spanning tree of $G$, then for every tree $T$, the probability that $\mathcal{T}$ is isomorphic to $T$ is at most $n^{-Ω(d)}$. This proves a conjecture of Lee in a strong form.

A novel generative site-specific beamforming (GenSSBF) approach, termed fast beam-brainstorm (F-BBS), is proposed to address the practical bottlenecks of slow beam generation and fixed channel probing lengths in existing GenSSBF. To accelerate beam generation, F-BBS utilizes a two-stage distillation strategy that learns an average velocity field, instead of an instantaneous one, to guide the beam generative process. This strategy enables larger generation steps, realizing few-step or even one-step beam generation. Furthermore, to accommodate flexible channel probing lengths, a stochastic masking mechanism and a beam index-aware masked condition encoder are proposed, enabling a single trained model to operate with variable-length channel probing observations without retraining. Therefore, FBBS achieves the fast generation of high-fidelity communication beams from coarse and variable-length channel probing feedback, i.e., reference signal received power (RSRP), from user equipments. Simulation results on accurate ray-tracing datasets show that 1) F-BBS achieves comparable performance while reducing the beam generation cost by over 90% compared with diffusion-based GenSSBF solutions, 2) F-BBS realizes robust performance across variable channel probing length, and 3) FBBS offers a desirable trade-off between beamforming gain and beam probing overhead.

Recently a novel multi-antenna architecture termed ray antenna array (RAA) was proposed, where several simple uniform linear arrays (sULAs) are arranged in a ray-like structure to enhance communication and sensing performance. By eliminating the need for phase shifters, it also significantly reduces hardware costs. However, RAA is prone to signal blockage and has no elevation angle resolution capability in three-dimensional (3D) scenarios. To address such issues, in this paper we propose a novel spherical directly-connected antenna array (DCAA), which composes of multiple simple uniform planar arrays (sUPAs) placed over a spherical surface. All elements within each sUPA are directly connected. Compared to conventional arrays with hybrid analog/digital beamforming (HBF), DCAA significantly reduces hardware cost, improves energy focusing, and provides superior and uniform angular res olution for 3D space. These advantages make DCAA particularly suitable for integrated sensing and communication (ISAC) in low-altitude unmanned aerial vehicles (UAV) swarm scenarios, where targets may frequently move away from the boresight of traditional arrays, degrading both communication and sensing performance. Simulation results demonstrate that the proposed spherical DCAA achieves significantly better angular resolution and higher spectral efficiency than conventional array with HBF, highlighting its strong potential for UAV swarm ISAC systems.

Two-dimensional (2D) magnets offer a rich platform for exploring emergent spin phenomena due to their unique and diverse magnetic properties. Beyond intrinsic magnetism, external manipulation$\unicode{x2013}$such as defect engineering, molecular adsorption, or charge doping$\unicode{x2013}$offers powerful routes to control their magnetic behavior. In this work, we demonstrate that localized electron polarons provide an effective means to modulate magnetism in 2D magnets. Using first-principles calculations, we investigate polaron formation in monolayer MnPS$_3$ and compute the resulting changes in magnetic exchange interactions. Our results reveal that polarons can locally break magnetic symmetry and induce anisotropic exchange couplings, highlighting a novel mechanism for tuning magnetic textures. This insight opens promising pathways for designing atomic-scale control of magnetism, with potential impact on spintronic technologies.

This survey-style note reviews constructive versions of the Peter--Weyl theorem in the Bishop--Coquand--Spitters line. Its main purpose is to clarify which parts of the classical Peter--Weyl package admit constructive reformulations, which parts survive only in weaker or reorganized form, and which questions still appear to remain open. The term ``constructive'' is used here primarily in the Bishop-style sense, together with the related locale-theoretic and formal-topological developments that occur in the work of Coquand and Spitters. We review the constructive compact-group results of Coquand and Spitters, the later role of almost periodic functions and compact completions, and the interaction with constructive Gelfand representation and locale-theoretic compactness. The guiding theme is that the constructive theory exists, but it is often most naturally expressed not as a literal transcription of the classical theorem in terms of irreducible decompositions alone, but rather through finite-rank approximation, characters, and compactifications attached to functions or groups. For orientation and comparison, we also include an appendix giving a standard classical form of the Peter--Weyl theorem together with a pedagogical Haar-measure-based proof, followed by comments indicating where the classical argument relies on steps that are not automatically constructive. Possible later extensions to topological loops and quasigroups are included as a programmatic direction rather than as part of the currently established core.

Conventional hybrid beamforming architectures are often compared with one another and with the fully-digital architecture under the same \emph{radiated} antenna power. However, the physically relevant budget is the power injected by the RF-chain outputs into the passive analog RF network, which is then usually transferred to the antenna ports in a contractive (lossy) manner. This issue is especially pronounced for fully-connected splitter--phase-shifter--combiner networks, whose physical power transfer remains contractive even under ideal passive-component assumptions. Motivated by this injected-power viewpoint, this paper proposes a hybrid beamforming architecture based on a programmable unitary RF network. Under ideal passive-component assumptions, all injected RF-chain power reaches the antenna ports without loss. The analog RF network is realized as an \emph{interlaced mixer--phase} architecture consisting of fixed (non-programmable) mixing layers interleaved with programmable diagonal phase-shifting layers. We derive a closed-form digital beamformer and a low-complexity programming method for the analog beamformer, yielding a hybrid precoder that closely matches the fully-digital precoder. Narrowband simulations with continuous and quantized phases, benchmarked against the fully-digital architecture, the physically modeled fully-connected phase-shifter baselines, and an ideal-lossless Butler/DFT beam-selection baseline under equal total injected RF-chain power, show that the continuous-phase and 6-bit realizations of the proposed architecture are nearly indistinguishable from the fully-digital benchmark and achieve significant gains over the baseline hybrid beamforming architectures.

The association of persistent radio sources (PRSs) with repeating fast radio bursts (FRBs) offers unique insights into their circum-burst environments. Building upon the physical link between PRS luminosity ($L_ν$) and FRB rotation measure (RM), we introduce a novel diagnostic framework utilizing the intrinsic scatter of the $L_ν- |{\rm RM}|$ relation as a physical probe of nebula dynamics. We demonstrate that this scatter encodes critical information regarding the temporal evolution of the nebula radius ($R \propto t^α$). By deriving a generic scaling $L_ν\propto R^ε|\mathrm{RM}|$ and analyzing the residuals of the five confirmed FRB-PRS systems, we constrain the nebula's evolutionary index to $α|ε| = 1.5 \pm 0.7$ (1$σ$ uncertainty). This measurement provides a powerful diagnostic tool for distinguishing among different astrophysical scenarios. Its value deviates from the expectations for supernova remnants (SNRs) in the Sedov-Taylor phase ($α|ε| \sim 0.2-0.4$), reverse shocks during the free-expansion phase of SNR/interstellar medium (ISM) interactions ($α|ε| \gtrsim 3.5$), and pulsar wind nebulae (PWNe) powered by a decaying wind ($α|ε| \sim 0-0.15$). Instead, it is more consistent with forward shocks in the free-expansion phase of both SNR/ISM and PWN/SNR systems ($α|ε| \sim 2.0-2.8$), and young PWNe driven by a nearly constant wind ($α|ε| \sim 1$). These findings suggest that active repeaters are powered by dynamically young, rapidly expanding nebulae. While currently limited by the small sample size, this framework establishes a robust methodology for future population studies to constrain the physical origin of PRSs.

Sedna is a coded multi-proposer consensus protocol in which a sender shards a transaction payload into rateless symbols and disseminates them across parallel proposer lanes, providing high throughput and ``until decode'' privacy. This paper studies a sharp incentive failure in such systems. A cartel of lane proposers can withhold the bundles addressed to its lanes, slowing the chain's symbol accumulation while privately pooling the missing symbols. Because finalized symbols become public, the cartel's multi-slot information lead is governed by a chain level delay event where the chain fails to accumulate the $κ$ bundles needed for decoding by the honest horizon $t^\star=\lceil κ/m\rceil$. We characterize the resulting delay probability with KL-type large deviation bounds and show a knife edge pathology when the slack $Δ=t^\star m-κ$ is zero such that withholding a single bundle suffices to push inclusion into the next slot with high probability. We propose \textsf{PIVOT-$K$}, a Sedna native pivotal bundle bounty that concentrates rewards on the $κ$ bundles that actually trigger decoding, and we derive explicit incentive compatibility conditions against partial and coalition deviations. We further show that an adaptive sender ``ratchet'' that excludes lanes whose tickets were not redeemed collapses multi-slot withholding into a first slot deficit when $t^\star\ge 2$, reducing the required bounty by orders of magnitude. We close by bounding irreducible within slot decode races and providing parameter guidance and numerical illustrations. Our results show that for realistic parameters Sedna can reduce MEV costs to 0.04\% of the transaction value.

The inclusive photoproduction of heavy mesons in ultraperipheral $pPb$ and $PbPb$ collisions at the LHC energies is investigated considering the color dipole $S$-matrix formalism and assuming distinct models for the unintegrated gluon distribution, based on different assumptions for the description of the QCD dynamics. In particular, predictions for the $B^0$-meson photoproduction are presented here for the first time. The study of the $D^0$-meson photoproduction is revisited by estimating the impact of the treatment of the heavy charm fragmentation on the predictions and extended for $pPb$ collisions. Moreover, the contribution associated with the $b \rightarrow D^0$ transition is estimated. Our results indicate that a future experimental analysis of the heavy meson photoproduction will provide important constrains on the description of the hadronic structure at high energies.

The parametrisation method for invariant manifolds is a powerful technique for deriving reduced-order models in the context of nonlinear vibrating systems, allowing accurate computations of nonlinear normal modes. Thanks to arbitrary order asymptotic expansions, converged results are within reach and directly applicable to finite element structures. However, since it relies on a local theory and asymptotic expansions, the results are only valid up to a given amplitude, which defines the convergence radius of the approximation. The aim of this contribution is to investigate the validity limits of the approach and review the existing error estimates, with the concrete objective of proposing a practical approach to estimate the validity range during the computation, thus producing safe bounds within which the reduced-order model can be used. Three different criteria are assessed. The first one uses the error in the invariance equation as the distance to the fixed point increases. The second one is adapted from an upper bound criterion derived for normal form transforms and based on the potential singularities of the homological operator. The third one uses Cauchy and d'Alembert convergence rules for series. The criteria are tested on a number of different examples that are representative of the situations encountered when dealing with nonlinear vibrations. The Duffing equation serves as a first benchmark that allows considering conservative oscillations, forced systems at primary resonance, and superharmonic resonance. The investigations are then extended to a vibrating system with two degrees of freedom. Finally, the different criteria are assessed on a finite element beam structure, and guidelines are formulated to generalise their practical use and produce accurate and easy-to-use error bounds in the context of model order reduction for nonlinear vibrating structures.

Phase separation in passive systems leads to uncontrolled droplet growth, limiting structural control in soft materials and cells. We identify a generic mechanism to arrest coarsening based on chemical interconversion between molecular species with different diffusivities. Sharp-interface theory and simulations show that when the faster-diffusing species becomes enriched inside droplets, composition gradients emerge that oppose mass influx. This transport asymmetry stabilizes droplet sizes even without interaction asymmetries, offering a minimal route to regulate structure formation in active emulsions.

Sterile neutrinos are well-motivated extensions of the Standard Model, introduced to address fundamental questions such as the origin of neutrino masses and the nature of dark matter. Exploiting the precise data reconstruction achieved by the CUPID-0 experiment, we searched for spectral distortions in the double $β$-decay of $^{82}$Se compatible with the emission of a sterile neutrino. The analysis relies on the construction of a detailed background model down to 200 keV, enabling an accurate characterization of the main sources of contamination. Using a Zn$^{82}$Se exposure of 9.95 kg$\cdot$yr, we explored sterile neutrino mass hypotheses between 0.5 MeV and 1.5 MeV. No evidence for a signal was observed in any scenario; therefore, we derived 90% C.I. upper limits on the active-sterile mixing probability $\sin^2θ$, obtaining the most stringent bound, $\sin^2θ<8\times 10^{-3}$, for a sterile neutrino mass of 0.7 MeV.

Travelling wave solutions of reaction-diffusion equations are widely used to model the spatial spread of populations and other phenomena in biology and physics. In this article, we reinterpret the classical variational principle approach through an optimal control formulation, in order to obtain a lower bound on the invasion speed of travelling wave solutions in systems of nonlinear partial differential equations. We begin by analysing single-species models, where the evolution of the density is governed by a scalar equation with a density-dependent diffusion term and a nonlinear reaction term. We show that for any admissible test function, maximising with respect to the parameter of interest yields a bound on the travelling wave speed. We apply this framework to several examples, including the porous-Fisher equation, and examine when nonlinear selection mechanisms dominate over the classical linear marginal stability criterion. Extending this approach, we then consider multi-species systems of reaction-diffusion equations and, reframed as Pontryagin-type optimality systems, we derive analogous bounds on the travelling wave speed using a variational framework under weak coupling. Finally, we employ numerical simulations to confirm the accuracy of the predicted wave speeds across a range of illustrative examples.