The accelerating decarbonization of energy systems has transformed electrical power systems into complex infrastructures exposed to threats whose interactions generate systemic vulnerabilities that conventional resilience approaches fail to capture. Although resilience assessment has expanded across multiple dimensions, existing studies largely examine them in isolation or adjacent pairs, leaving cross-dimensional couplings insufficiently explored. This study demonstrates i) that single-dimension assessments fail to capture the degradation produced by simultaneous cross-dimensional failures, ii) the nonlinear amplification emerging when physical, operational, and digital-cyber dimensions are jointly compromised, and iii) the intensification imposed by climatic and economic-regulatory stressors. To this end, we leverage a hybrid quantitative methodology. A PRISMA 2020 review with backward and forward snowballing identifies methodological gaps and unresolved dependencies across five resilience dimensions: physical, operational, digital-cyber, climatic-external, and economic-regulatory. Following this analysis, a Multidimensional Resilience Index (MDRI) is developed to capture endogenous couplings and exogenous amplification effects and is validated under escalating cyber-physical attack scenarios inspired by the December 2025 attack on Polish energy infrastructure. Results show that degradation under cascading and simultaneous failures is nearly eight times greater than under isolated stress, while exogenous conditions amplify degradation by an additional factor approaching six, with 72% of this amplification driven by exogenous stressors. Combined, these mechanisms produce a 46-fold increase in resilience loss compared to a single-vector reference.

We study the second largest spectral gap from $1$ of the normalized Laplacian of a graph, a quantity that appears in the literature in connection with random walks, expander graphs, and Ramanujan graphs. We relate it to the classical Cheeger and dual Cheeger constants, and we introduce a new Cheeger-type constant admitting a probabilistic interpretation in terms of two-step random walks. For this constant, we establish sharp inequalities analogous to the classical Cheeger inequalities.

Building upon the algebraic framework of trace monoids introduced by Giscard and Rochet, we establish a new determinant formula for the Ihara zeta functions of certain digraphs. We present our results through two main theorems. First, at the algebraic level, we show a general determinant formula expressed by a Cayley determinant over the hike monoid ring, which provides a unifying perspective on several known determinant expressions. Second, by evaluating the lengths of hikes through a natural ring homomorphism, we prove that the reciprocal of the Ihara zeta function can be explicitly expressed as a determinant whose dimension is equal to the number of simple cycles of the line digraph without backtrack.

In this paper, we introduce and study the notion of weakly Q-ideals in commutative rings.

In 2011, Csikvári [Electron. J. Combin. {\bf 18} (2011), $\#$P182] proved that among all graphs with a prescribed number of edges, the largest matching root is attained by a threshold graph, and conjectured that the extremal graph should be `as star-like as possible.' In this paper, we give a complete and affirmative answer to this problem and extend it to the setting of uniform hypergraphs. We prove that for every $k$-graph $\mathcal{H}$ with $m$ edges, its largest matching root satisfies $$λ(\mathcal{H})\le m^{1/k},$$ with equality if and only if $\mathcal{H}$ is intersecting. For $k=2$, after deleting all isolated vertices, the resulting graph must be the star $K_{1,m}$ or a triangle, thereby confirming Csikvári's conjecture. Moreover, if the matching number $ν(\mathcal{H})\ge 2$, then \[ λ(\mathcal{H})\le \left(\frac{m+\sqrt{m^2-4(ν(\mathcal{H})-1)}}{2}\right)^{1/k}, \] with equality if and only if $ν(\mathcal{H})=2$ and $\mathcal{H}$ has exactly one $2$-matching.

In this work, we propose an operator-splitting variational quantum algorithm, termed Dirac-sVQA, for simulating the nonlinear Dirac equation (NLDE). The main difficulty arises from the state-dependent nonlinear interaction, its time-discrete update depends explicitly on the intermediate spinor state and, in general, cannot be implemented as a fixed state-independent unitary circuit. To address this difficulty, we decompose the NLDE evolution into a structured linear Dirac substep and a nonlinear variational correction. The linear substep is implemented by a spinor-Fourier Dirac propagator on a joint position-spin register, preserving the spin-momentum coupling and mass-induced spin evolution of the Dirac operator. The nonlinear correction is reformulated as a measurement-based variational update through a small set of overlap, self-channel, and cross-channel observables. We provide the corresponding quantum circuits and derive measurement-aware resource and complexity estimates. Numerical experiments in several nonlinear regimes show that Dirac-sVQA accurately captures both the total density and the componentwise spinor dynamics, agrees well with classical Fourier pseudospectral splitting solutions, and exhibits stable error behavior over time. These results provide numerical evidence for the feasibility of operator-splitting variational quantum simulation for nonlinear relativistic wave equations.

We investigate the spectral energy dynamics of turbulent flows with variable viscosity using direct numerical simulation of homogeneous isotropic turbulence of generalised Newtonian fluids described by the Carreau constitutive model, covering both shear-thinning and shear-thickening regimes. The spectral evolution equations for the variable viscosity Navier-Stokes system show that the viscous term becomes nonlinear and gives rise to a convolution product in spectral space, formally analogous to that of the convective term. Unlike the constant viscosity case, where it acts as a purely local dissipation mechanism, the variable viscosity term carries both conservative (transfer) and non-conservative (dissipation) contributions entangled in the convolution product. We present novel computations of the viscous \mtm coupling $\hat{V}(\k, \kP)$, which does not satisfy a detailed conservation property analogous to that of the convective term. The viscous coupling maps reveal two distinct spectral regions: a sign-definite non-conservative region near $\kP \approx \bm{0}$, and a transfer-like dipole near $\kP \approx \k$ in shear-thickening fluids. The dipole satisfies the approximate antisymmetry $\hat{V}(\k, \kP) \approx -\hat{V}(\kP, \k)$, which is the defining signature of a conservative energy transfer. This demonstrates that energy transfer across scales, a role traditionally attributed exclusively to the convective nonlinearity, can arise from any nonlinear term in the momentum equation. The viscous energy transfer participates in the forward cascade alongside the convective transfer, eventually taking over the latter in the dissipation range. Its presence is connected to the emergence of power-law spectral decay replacing the classical exponential cutoff in shear-thickening fluids.

In this paper we discuss an ongoing multi-year project that aims to make open text feedback more accessible and useful to UX practitioners by automating classification and providing real time access to comments, themes, and analysis. By significantly lowering the time and knowledge cost of implementing automated solutions, we aim to effectively democratize our data analysis processes, allowing and encouraging non-technical stakeholders to access and leverage data on their own. We share both the organizational and technical constraints we have encountered over the course of this project, and the solutions we have prototyped as a result of those constraints.

On four-dimensional (pseudo-)Riemannian manifolds $\mathcal{M}$ the curvature tensor (viewed as an endomorphism on 2-forms) admits a chiral $6 \times 6$ matrix representation which decomposes into four $3 \times 3$ blocks. $\mathcal{M}$ is Einstein if and only if the off-diagonal blocks vanish. If the manifold is a warped product $\mathcal{M} = F \times_f B$, then there exists an alternative matrix representation relative to the decomposition of the 2-forms into spaces induced by the exterior algebra on both the base and the fiber. These two representations are not independent and a similarity transformation can be found between them. We construct these matrices and associated transformations for $1+3$, $2+2$ and $3+1$ warped products, giving classifications for the Einstein limits from this algebraic perspective. Using this, one can easily Petrov classify the Einstein warped products for each case considered: $3+1$ are generically type-\RNum{1}, $2+2$ are type-D while $1+3$ are constrained to be type-O. In the closed Riemannian case, there are a number of topological restrictions on these manifolds that we discuss: in the half-conformally flat limit, each of these Einstein warped products must be flat.

Scaling extreme ultraviolet (EUV) source power for next-generation lithography demands higher conversion efficiency (CE) at reduced per-pulse energies. We demonstrated a 40% CE enhancement by simultaneous dual-beam irradiation of a planar Sn target with a 2090-nm, 20-ns Ho:YAG laser. Single-beam irradiation at 40 mJ yielded an EUV CE of 2.6%; splitting the same total energy equally into two beams of 20 mJ each - at identical peak intensity - raised the EUV CE to 3.6%, which was the highest reported for 2-μm-driven laser-produced plasma sources. The EUV source size (60-70 μm) and energetic-ion spectra were nearly identical across both configurations, confirming comparable plasma conditions. Because the scheme requires only passive beam splitting and scales readily to three or more beams, it offers a practical route toward multi-kW-class, energy-efficient EUV sources for high-NA and hyper-NA lithography.

The ALICE experiment will replace its three innermost tracking layers with the Inner Tracking System 3 (ITS3) during LHC Long Shutdown 3. This upgrade introduces the first fully cylindrical, wafer-scale silicon vertex detector, utilising Monolithic Active Pixel Sensors (MAPS) fabricated in a 65nm CMOS process. By thinning sensors to 50$μ$m and bending them to radii as small as 19mm, the design achieves a self-supporting structure that eliminates traditional support material. Wafer-scale stitching enables 27cm-long seamless sensors with integrated power and signal distribution, removing the need for flexible printed circuits within the active volume. These innovations, combined with a move from water to air cooling, reduce the material budget to less than 0.09%X$_0$ per layer. The R&D program has been validated through full-scale prototypes (MOSS, MOST), which demonstrated stitching feasibility, high yield, and radiation hardness. Engineering models confirmed the feasibility of air-convection cooling, indicating effective thermal management and structural stability. This contribution summarises the key advances in stitched sensor development, mechanical integration, and the path toward the final qualification model.

A classical theorem of Boshernitzan states that if $f$ is a function which belongs to a Hardy field and which satisfies $|f(x)|\prec x^{\ell}$ for some $\ell\in \mathbb{N}$, then the sequence $(f(n))_{n\in \mathbb{N}}$ is uniformly distributed modulo 1 if and only if $\lim_{x\to\infty}\frac{|f(x)-p(x)|}{\log(x)} = \infty$ for all $p(x)\in \mathbb{Q}[x]$. We provide a new proof of this result using methods from summability theory and we extend Boshernitzan's criterion by obtaining necessary and sufficient conditions for $f$ to be uniformly distributed modulo 1 with respect to a broad class of weighted averages. As an application of our results, we show that for the function $f(x) = x^{3/2}$ and for any $(a,b)\subset [0,1]$, and all sufficiently large $N\in\mathbb{N}$, there is an $n\in [N-N^{\frac{1}{4}},N]$ such that $f(n)\mod 1\in (a,b)$.

We investigate the monogenity of irreducible factors of the Fibonacci polynomials $F_n(x)$ and the Lucas polynomials $L_n(x)$. Our main results show that for every odd positive integer $n$, all irreducible factors of $F_n(x)$ are monogenic, and for every even positive integer $n$, all irreducible factors of $L_n(x)$ are monogenic.

X-ray thermally emitting isolated neutron stars (XINSs) are a rare population that provides insights into neutron star cooling, magnetic-field evolution, and Galactic demographics. Using more than two decades of observations from the European Space Agency's XMM-Newton Observatory, we searched the 4XMM-DR9 and 4XMM-DR12 catalogues for absorbed XINS candidates down to a flux of $10^{-14}$ erg cm$^{-2}$ s$^{-1}$ in the 0.5--1 keV band. Candidates were selected based on soft X-ray spectra and the absence of catalogued optical, ultraviolet, or infrared counterparts. Follow-up observations with XMM-Newton and FAST were complemented by data from the SRG/eROSITA All-Sky Survey, Chandra, and optical surveys. Of ten sources analysed, five are compelling XINS candidates, one is the known XINS 4XMM J022141.5-735632, two are extragalactic contaminants, and two remain ambiguous because of limited photon statistics. The five candidates exhibit soft ($kT\sim80-100$ eV), moderately absorbed, and stable X-ray emission consistent with distant XINSs. They are located primarily in the Galactic plane, with possible associations at distances of $\sim$1.8 and $\sim$6 kpc. Population-synthesis simulations predict $20\pm5$ XINSs within the 4XMM-DR12 footprint, of which $6^{+2}_{-3}$ exceed our flux threshold, consistent with the observed sample if additional candidates are confirmed. The model further predicts that $\sim$70% of the population remains below the detection threshold. Deep optical and additional X-ray observations are required to establish the nature of the candidates. Future missions such as NewAthena will enable more detailed studies of these distant populations and improve constraints on the Galactic XINS population.

We examine three degenerate spacetime configurations with wormhole topology, obtained via branching coordinate transformations of the Rindler, Minkowski, and Schwarzschild vacuum metrics.These configurations are, respectively, a Rindler wormhole with a planar throat, and the Klinkhamer and Schwarzschild Klinkhamer wormholes with spherical throats. Within the framework of the Einstein Palatini Cartan formulation, we demonstrate that these degenerate configurations are matter free. In this regard, they differ fundamentally from their nondegenerate wormhole counterparts in the thin shell model, which require exotic matter. Nevertheless, the degenerate configurations considered here exhibit distinct physical manifestations, ranging from purely topological structures to genuine gravitational effects. Furthermore, in all three examples, we demonstrate the absence of a limiting transition from the non-degenerate spacetime to the matter free degenerate configuration. This suggests that degenerate geometries constitute an independent sector of the configuration space of general relativity.

Heavy transition metal compounds with strong spin-orbit coupling appeared as a platform for $d$-electron multipolar physics. We report the electronic, magnetic, and structural properties of antifluorite-type tungsten chloride A$_2$WCl$_6$ (A = Cs, Rb, and K), comprising a face-centered cubic lattice of W$^{4+}$ ions. The 5$d^2$ configuration of W$^{4+}$ ions in a cubic environment yields a spin-orbit-entangled $J$ = 2 state, which has been discussed to give rise to multipolar ordering such as charge quadrupolar or magnetic octupolar ordering. We found that K$_2$WCl$_6$ undergoes a cubic-to-tetragonal structural transition which lifts the degeneracy of the $J$ = 2 state, leading to a non-magnetic singlet ground state. By contrast, Rb$_2$WCl$_6$ and Cs$_2$WCl$_6$ show no signs of phase transition and remain non-magnetic down to the lowest temperature measured. At low temperatures, signatures of weak structural anomalies were revealed, which may point to the presence of local distortions of the WCl$_6$ octahedra. We argue that the subtle structural distortion arises from the local quadrupolar component of the $J$ = 2 state but the frustrated quadrupolar interaction, together with chemical disorder, inhibits the formation of long-range quadrupolar ordering.

Practical distributed deployments inherently suffer from asynchronous signal arrivals, which exacerbate multi-user interference and degrade system performance, especially for coherent transmission. To natively mitigate the asynchronous reception effect, this paper proposes integrating fluid antenna systems (FASs) into distributed cell-free massive MIMO systems, exploiting their reconfigurable spatial positions to release additional spatial degrees of freedom (DoFs). We establish the FAS-enabled data transmission model with asynchronous reception, i.e., delay phases. We also derive the analytical downlink spectral efficiency (SE) performance of the proposed system under coherent and non-coherent transmissions, using low-complexity Maximum Ratio (MR) precoding to provide fundamental theoretical bounds. Specifically, we propose a novel nonmonotone accelerated projected gradient ascent algorithm to jointly optimize FAS positions and power control coefficients, maximizing the downlink sum SE. Numerical results demonstrate that while asynchronous reception severely degrades system performance for coherent transmission, the spatial DoFs unlocked by optimized FAS positions, along with efficient power control, can significantly counteract the effects of unknown delay phases and outperform traditional fixed-position antennas. For non-coherent transmission, which inherently bypasses asynchronous reception, the application of FAS leverages spatial reconfigurability to natively maximize signal strength and achieve more pronounced SE gains. Ultimately, our proposed FAS-enabled system, coupled with efficient power control, mitigates performance degradation due to asynchronous reception and outperforms traditional fixed-position antennas, paving the way for the practical deployment of FASs in robust, highly efficient 6G cell-free massive MIMO systems.

The oracle problem, broadly understood as the difficulty of reliably incorporating external information into blockchain-based systems, has been widely examined by scholars and practitioners. Recent comparative research has shown that several challenges of modern blockchain oracles, including attributability, accountability, integrity, and query design, mirror procedural and epistemic constraints already present in ancient oracular institutions such as the Delphic Oracle. Yet the translation of these insights into applied oracle design remains largely unexplored. This paper introduces the Dodona Protocol, a modular, chain-agnostic oracle service inspired by procedural patterns identified in ancient and modern oracle systems. Named after the Oracle of Zeus at Dodona, one of the oldest oracular sanctuaries in ancient Greece, the protocol operationalizes principles such as structured consultation, access control, attributable resolution, constrained query formats, reputational accountability, and tiered service availability. Its first module implements a query and dispute resolution mechanism in which a named expert resolver provides binding answers to structured questions submitted by petitioners. The oracle does not claim to reveal objective truth; rather, it produces outcomes that parties have agreed in advance to accept. The paper presents the design rationale, architecture, and comparative positioning of the Dodona Protocol. It frames the protocol as a living research experiment within the Design Science Research tradition, where the deployed system functions as the research artifact and operational data support structured analysis, iterative refinement, and peer-reviewed dissemination. In doing so, the paper seeks to bridge the gap between oracle theory and oracle practice.

We theoretically investigate the DC conductivity of two-dimensional tilted Dirac systems subject to short- and long-range impurity scattering. Using the Kubo formalism, we systematically study transport across the subcritical (Type I), critical, and overcritical (Type II) tilt regimes. In the subcritical phase, short-range impurities yield a frequency-independent conductivity that decreases monotonically with tilt. Conversely, long-range Coulomb scattering results in a strongly energy-dependent conductivity governed by a tilt-independent scattering rate. At the Lifshitz transition ($t = 1$), the transport signatures of these impurities diverge fundamentally: the van Hove singularity in the density of states induces a localized conductivity dip for short-range disorder, but a pronounced macroscopic peak for Coulomb impurities. In the overcritical regime, an ultraviolet momentum cutoff is required to regularize the open Fermi surface, leading to distinct behaviors for each impurity type. Notably, the conductivity perpendicular to the tilt direction ($σ_{xx}$) exhibits a cutoff-dependent, non-monotonic peak near $t = \sqrt{2}$ for short-range defects, while it decays monotonically with increasing tilt for long-range scattering. For both potentials, the conductivity along the tilt axis ($σ_{yy}$) increases without bound, revealing extreme transport anisotropy. For long-range impurities, the energy dependence of the conductivity becomes nearly quadratic and linear for Type I and II, respectively. Furthermore, vertex corrections vanish identically at the Lifshitz transition for both impurity types. Finally, we provide a unified geometric framework for these phenomena, establishing the tilt parameter as a powerful knob for engineering macroscopic transport in Dirac materials.

Let $R = k[x_1,\ldots,x_n]$ be a polynomial ring in $n$ variables over a field $k$, and let $I$ be a monomial ideal of $R$. If $I$ is an almost complete intersection, then we provide an explicit formula for the Castelnuovo-Mumford regularity of $I$ in terms of the powers of the dominant variables appearing in the regular sequence contained in $G(I)$ of length $|G(I)|-1$, where $G(I)$ is the set of minimal monomial generators of $I$. Furthermore, if $I$ is a dominant ideal or an almost complete intersection ideal, then we show that $\operatorname{reg}(\overline{I}) \leq \operatorname{reg}(I),$ where $\overline{I}$ denotes the integral closure of $I$. This provides a positive answer to the Küronya-Pintye conjecture for these two classes of monomial ideals. In addition, we give some examples to clarify these results.

In our previous work \cite{Omar-Shah2}, we initiated the study of $\CSX$- and $\XT$-groups associated with a fixed variety $\X$. A group belongs to the former class if all of its maximal $\X$-subgroups are malnormal, and to the latter if any two $\X$-subgroups with nontrivial intersection generate an $\X$-subgroup. In general, $\CSX$ does not imply $\XT$, however as shown in \cite{Omar-Shah2}, some varieties do satisfy this implication. In this article, we provide additional examples of varieties for which $\CSX$ implies $\XT$.

Let $M^4$ be a closed, orientable $4$--manifold carrying a transversely oriented $C^2$ codimension--one foliation whose leaves are diffeomorphic to $\mathbb{R}^3$. We prove that $M^4$ is homeomorphic to the $4$--torus $\mathbb{T}^4$. We also show that, whenever the original smooth structure on $M$ admits a smooth defining $1$--form, the conclusion sharpens to a diffeomorphism $M\cong\mathbb{T}^4$.

Metastable phases (MPs) are energetically unfavorable states typically suppressed in equilibrium phase diagrams. Rather than remaining ''hidden'', we show that they exist in the complex plane of thermal fields, as regions delineated by Lee-Yang zeros (LYZs). We demonstrate this numerically in a toy model with a tunable density of states featuring three Gaussian peaks and in a more realistic periodically driven system. In both cases, as artificial parameters or drive amplitudes increase, the LYZs bounding the MP approach the real axis and split into separated branches, signaling the emergence and stabilization of the MP within the enlarged gap between two adjacent stable phases. In the driven system, the imaginary part of LYZs correlates with drive strength, linking Lee-Yang theory to terahertz matter manipulation. These findings provide a scheme to describe MPs in phase diagram analysis. By viewing periodic drives as complex thermal fields, it also offers a new perspective for understanding and engineering non-equilibrium collective states.

The smeared null energy condition (SNEC) imposes a semilocal bound on the negative energy accumulated along null geodesics. In eternal inflation, rare stochastic upward fluctuations of the inflaton locally increase the Hubble parameter, creating an apparent tension with the SNEC. Focusing on a canonical single-field model, we investigate whether this quantum-induced self-reproduction violates the SNEC. Using the Fokker-Planck equation, we demonstrate that the ensemble drift of the Hubble parameter is parametrically bounded by slow-roll parameters and semiclassical suppression. Furthermore, a complementary single-trajectory analysis reveals a strong timescale hierarchy, $N_{\rm SNEC} \gg N_{\rm BR}$. This indicates that even for rare upward stochastic excursions, gravitational backreaction invalidates the background spacetime assumption long before the SNEC bound can be mathematically approached. We conclude that while standard stochastic diffusion drives eternal inflation, it does not inherently lead to SNEC violations within the semiclassical slow-roll regime.

We study natural smooth structures on Hausdorff locally convex spaces from the point of view of diffeology and their application to nonlinear operations on distributions. We show that a Hausdorff locally convex space is convenient if and only if the canonical map from the space to its internal tangent space is an isomorphism, thereby giving a geometric characterization of Mackey completeness. We also examine the behavior of these diffeologies under completion, duality, and inductive limits. As an application, we construct the space of microlocally multipliable distributions as a diffeological colimit of wavefront-restricted distribution spaces and prove that the induced multiplication map into the space of distributions is smooth. The construction uses only the standard wavefront-set criterion for multiplication and illustrates how diffeological colimits encode domains of partially defined nonlinear operations. This yields a diffeological framework for Hörmander-admissible multiplication beyond the classical manifold setting.

Scientific progress is traditionally narrated through the interplay of theoretical insights and experimental findings. Yet this view of science underplays a third and central pillar of progress: the methods that underlie both conceptual advances and empirical evidence. By analysing more than 3 million articles across science published between 1980 and 2019, we find that science has undergone a fundamental structural transition. The share of papers that primarily contribute new methods-methods papers-has doubled across science over the past four decades, rising universally across disciplines and citation impact levels. Rather than a gradual evolution, this transition marks a pivotal shift beginning in the early 1990s, aligning with the computational revolution and the emergence of data-intensive science. The surge in methodological research is not confined to the most cited, elite publications; it spans the full spectrum of scientific output. These findings reveal a systemic reorientation of the scientific ecosystem where reusable methods increasingly serve as the essential infrastructure of scientific advances, challenging the traditional dichotomy of theory and experimental research. As science becomes increasingly methods-driven, our results call for rethinking how research is evaluated, funded and organised-towards better incentivising method innovations. This is especially the case as expanding AI must be effectively integrated with scientific instruments to realise its full potential.

We give a reformulation of tangential morphisms (which is a generalization of Deligne's tangential base point) via log geometry.

Topology provides a powerful language for describing global invariants in physical systems, yet optical topology has been explored predominantly with fully coherent light. Recent studies have shown that incoherent light can host topological structures mediated by coherence singularities; however, a general framework for their construction and control has been lacking. Here, we introduce an incoherent Milnor polynomial, which establishes a theoretical framework for real-space incoherent topological structured light, in which topology and statistical coherence emerge as independent and jointly addressable degrees of freedom. This framework overcomes a fundamental limitation of coherent topological structured light, enabling arbitrary intensity engineering without altering the underlying topological configuration. Experimentally, we realize incoherent Hopf-linked and trefoil-knotted coherence singularities with programmable statistical coherence. We further demonstrate a robust optical information-encoding scheme inspired by Rubik's-cube-like rotations, where statistical coherence determines far-field intensity patterns associated with the cube's initial states, and topological structures govern controlled rotations acting as encryption keys. Our results advance incoherent topological structured light from a physical curiosity to a programmable photonic platform, opening new avenues for optical information encoding, statistical photonics, and coherence-engineered functionalities beyond coherent optical topology.

We investigate the joint distribution of the trivariate statistics (des, maj, inv) on classical permutations, Andre permutations of the first and second kinds, and Simsun permutations. By decomposing permutations according to the position of the smallest element, we obtain explicit recurrence relations for the generating functions of these statistics. In the classical permutation setting, our recurrence relation yields the generating function for the trivariate statistics (des, maj, inv) due to Gessel, which is typically proved using MacMahon's technique.

Sunrise III is a stratospheric balloon-borne solar observatory with a 1-m diameter telescope and three post-focus instruments, along with an image stabilisation system, all within a protective gondola. It samples the lower solar atmosphere, from the solar surface to the middle chromosphere, at a resolution approaching 50~km on the Sun. Sunrise III flew successfully for 6.5 days suspended from a zero-pressure stratospheric balloon from northern Sweden to north-western Canada in July 2024, gathering around 200 TB of data. The present issue of ApJL focuses on the first scientific results from the data collected during that flight. This paper introduces this Focus Issue, providing a very brief overview of the capabilities of the instrumentation, the flight and of the gathered data. Challenges for the measurements, data reduction and interpretation are also briefly touched upon. The paper ends with an overview of the first set of science results obtained from these data, as presented in the current Focus Issue.