Optimal control problems with symmetries often admit a non stationary turnpike property called trim turnpike, which characterizes the convergence of optimal solutions to certain symmetry induced trajectories called trim primitives. In this paper we establish an exponential trim turnpike property for a class of optimal control problems with structural properties related to Abelian Lie group symmetries. The key ingredient of our approach is the introduction of an appropriate reduced optimal control problem. We show that extremals of the original problem can be characterized through a reduced Hamiltonian boundary value problem that coincides with the optimality system of the reduced problem. Under a hyperbolicity assumption on the equilibrium of the corresponding reduced Hamiltonian system we prove that optimal trajectories remain exponentially close, up to boundary layers near the endpoints, to a trim primitive defined by the static reduced problem. The theoretical results are illustrated on three representative examples: linear and nonlinear problems with quadratic cost and the Kepler orbital transfer problem.

The Pareto distribution plays a crucial role in various disciplines, necessitating robust goodness-of-fit tests for its validation. This article introduces a novel tests based on Stein's characterization and the Laplace transform, offering a fresh perspective on model assessment. We establish the asymptotic properties of the proposed test and evaluate its empirical performance against existing methods in terms of size and power. Our findings demonstrate that the new test often outperforms or performs comparably to established tests. In addition, real data applications illustrate its practical utility.

Searches for neutrino electromagnetic interactions at reactor sites require an unusual combination of ultra-low thresholds and a stable low-background environment. It is shown here that cryogenic undoped cesium iodide (CsI) naturally satisfies these conditions in a way prior detectors have not. Although suppression of nuclear recoil ionization efficiency at low energies limits the use of this scintillator for coherent elastic neutrino-nucleus scattering, that same property renders the detector effectively blind to those nuclear recoils from MeV-scale reactor antineutrinos. This leaves the low-energy regime free to expose neutrino-electron ($\barν_{e} -e^{-}$) scattering as the dominant observable channel and converts cryogenic CsI into a targeted probe of electromagnetic couplings. This work presents a conceptual design based on pure CsI crystals immersed in an active xenon-doped liquid argon veto evaluated under realistic intrinsic and environmental backgrounds. Under present detector capabilities, order-of-magnitude improvements over current reactor limits on the neutrino magnetic moment and millicharge are achievable. Cryogenic pure CsI therefore offers a distinctive and scalable route to leading studies of $\barν_{e} -e^{-}$ physics.

The magnetic excitation spectrum of itinerant magnets exhibits rich and complex spectral features that often complicate interpretation of the underlying physics. For perturbations in the long wavelength limit, one obtains a well defined pole at zero frequency in the spectral function, the Goldstone magnon. However, for optical modes and finite wavevectors, the magnon spectrum may become damped, exhibit branching, or be completely washed out. In the present work, we show how the physical mechanism of all such features can be understood from careful analysis of the eigenmodes of the many-body spectral function. We perform first principles computations of elemental itinerant ferromagnets using a novel implementation of the linear response time-dependent density functional theory (LR-TDDFT) framework and classify the collective nature of individual spectral features based on the self-enhancement function, the product of the noninteracting Kohn-Sham susceptibility and the exchange-correlation kernel. In particular, we distinguish between coherent and incoherent collective excitations, depending on whether the real part of the self-enhancement function crosses unity at the spectral peak of the magnon, which may or may not be subject to Landau damping as quantified by the imaginary part. Classifying the computed magnon spectra accordingly, we observe coexistence of coherent magnon branches in bcc-Fe, as well as decoherence of the primary magnon branch in fcc-Ni for wave vectors near the BZ boundary where incoherent valley magnons instead carry substantial spectral weight. The analysis also naturally leads to a definition of the many-body Stoner spectrum and allows us to quantify the binding energy of the Stoner pair excitations.

Long-range interactions are the source of many equilibrium and out-of-equilibrium quantum many-body phenomena. Analog simulators based on ionic, atomic, superconducting, and molecular systems provide a natural platform to obtain these interactions using vibration- and photon-mediated processes. Recent experimental advances, such as their integration in multi-mode cavities and waveguides, or the use of Raman-assisted transitions, enable dynamical control over both the strength and the spatial range of these interactions, thereby rendering them programmable. Here, we develop a hybrid classical-quantum toolbox that exploits this tunability to enhance many-body state preparation in analog simulators beyond fixed-connectivity architectures. Our approach is based on classical pre-compilation in homogeneous small systems, whose optimized parameters are extrapolated iteratively to larger system sizes, and then refined on the quantum hardware using noise-aware hybrid re-optimization and error-mitigation techniques. We benchmark this strategy across several fermionic, spin-1/2, and spin-1 models, demonstrating orders-of-magnitude improvements in fidelity and energy estimation for system sizes ranging from 100 to 1000 particles. Finally, we show that the combination of such high-fidelity programmable state preparation techniques with tunable-range out-of-equilibrium dynamics enables controlled studies of many-body thermalization in regimes accessible to current experimental platforms. Our results establish programmable long-range interactions as a powerful resource for next-generation analog quantum simulators.

We present the standard double slit experiment with non-relativistic particles in the Heisenberg Picture of quantum mechanics. Our motivation is threefold. First and foremost, and contrary to some claims in the literature, we show that there is no need to talk about non-locality when explaining the interference fringes. Secondly, we emphasise the fact that even in the non-relativistic regime, and in order to preserve locality, we should define the position and momentum observables of a particle as functions of both space and time (and not just time). Thirdly, our presentation compares the projective measurements in the Heisenberg picture with the "Church of the Larger Hilbert Space", the latter of which is seldom discussed in the Heisenberg picture of quantum mechanics.

Advancements in 3D modeling,digital display technologies,and the growing availability of digital cultural heritage data have significantly improved the accuracy of heritage depictions and expanded opportunities for analysis.However,while many studies focus on presenting specific cultural heritage figurines,an often overlooked aspect is the visualization of the Terracotta Warriors as a unified entity.This involves concisely representing the distribution of features and their relationships,providing a clear and insightful presentation that engages practitioners, academics,and wider audiences.To tackle the challenges mentioned above,this research seeks to explore the application of AI methods in processing cultural heritage data.It aims to optimize and augment the dataset,analyze the distribution and relationships of various attributes, and interpret the analysis results through visualization techniques.The Terracotta Warriors,among China's most significant cultural heritages and renowned for their abundance,exquisite workmanship,and magnitude,are chosen as a case study.The contribution of this paper is primarily twofold.Firstly,we constructed a dataset of Terracotta Warriors from Pit No.1,detailing the attributes significant for identifying different Terracotta Warriors.Secondly,we employ various AI methods,such as generative adversarial network and random forest,to process and analyze these attributes,followed by visualizing the analysis results for an intuitive presentation.This study introduces a novel scheme for presenting information on a collection of cultural relics,offering a practical case for analyzing and visualizing the Terracotta Warriors'attributes as a whole entity,rather than showcasing individual relics'information in isolation.

We present a novel framework for line-of-sight (LoS) delay-Doppler (DD) estimation in dense scattering propagation environments. We present two time-frequency (TF) domain pilot sequences inspired by the Zadoff-Chu sequence that exhibit desirable autocorrelation properties. Further, we present a twisted convolution-based approach for LoS DD estimation directly from the TF-domain received signal, avoiding an additional TF to DD transformation, which is commonly found in literature. Numerical results from simulations demonstrate that the proposed framework significantly outperforms traditional single-carrier Zadoff-Chu sequences in both delay and Doppler estimation over a wide range of Rician fading factor and SNR values.

Deforming fundamental domains of wallpaper groups provides a systematic way to generate non-convex blocks which admit topological interlocking assemblies (TIAs). We use this approach to construct TIAs that fully occupy the space between two parallel planes and incorporate multiple block types. In addition to wallpaper groups, semiregular tessellations are employed in the construction of such TIAs. These construction methods open up an extensive design space for TIAs, expanding the possibilities of feasible interlocking systems and creating new opportunities for architectural and material design. Several resulting block families can be interpreted as geometric realizations of generalized Truchet tiles or decorated lozenge tilings and, with suitable colouring rules, we establish a one-to-one correspondence between these tilings and specific TIAs. This framework enables a systematic investigation of symmetric and asymmetric assemblies derived from diverse block types.

Schatten class properties of commutators $[b,T]$ of pointwise multipliers $b$ and singular integral operators $T$ have been characterised in a variety of settings. An abstract framework, covering many of these results as special cases, was proposed by the author [arXiv:2411.02613]. However, recent results about commutators of the concrete Bessel-Riesz transforms by Fan-Li-Sukochev-Zanin [arXiv:2411.14928] are beyond this abstract setting. In this work, we present an extension of the framework of [arXiv:2411.02613], introducing two measures $μ$ and $ν$ that are $A_\infty$-equivalent to each other. The commutators act on a given space $L^2(μ)$, but the characterising function space norms of the multiplier $b$ are taken with respect to another measure $ν$. In this way, assumptions like Ahlfors regularity and Poincaré inequality on the original measure $μ$ may be relaxed, as long as there is an $A_\infty$-equivalent measure $ν$ that satisfies these assumptions. In the Bessel example, the original $μ$ fails to be Ahlfors regular, but $ν$ is simply the Lebesgue measure. Within this framework, the Schatten norm characterisations of commutators of the Bessel-Riesz transforms at the critical-index by Fan-Li-Sukochev-Zanin [op cit.] are recovered by a completely different argument, replacing non-commutative techniques by real-variable harmonic analysis and hardly using any specifics of the Bessel setting. As a by-product, we also obtain a simpler characterisation in the non-critical case, replacing an ad-hoc Besov space of Fan-Lacey-Li-Xiong [J. Funct. Anal. 2026] by a classical Besov space.

AI assistants can increasingly generate and evolve test cases. The challenge is no longer merely to produce them, but also to help engineers understand why a generated artefact exists and what supports it. Existing work has focused on classifying testing techniques, linking requirements to tests and structuring system assurance arguments, but it does not explicitly represent the argumentation behind individual test design decisions. We propose a conceptual taxonomy and a structured template for AI-assisted test generation that characterizes a test case by its test goal, claim, reason, and evidence. The taxonomy is intended for both constructive use during test design and retrospective use during review, to assess the quality of the attached argument rather than the plausibility or objective value of the generated test cases.

We investigate the integrability of Fréchet tangent distributions on Fréchet manifolds. We introduce the local well-posedness Condition W for split tangent subbundles, which reduces the local integrability problem to solving initial value problems with parameters whose solutions define curves tangent to the distribution. By applying a variational approach to establish the existence and uniqueness of these solutions, we prove a Frobenius theorem stating that involutivity and Condition W are sufficient for integrability. This yields the existence of a unique maximal foliation of the manifold. Furthermore, we provide a dual formulation of the theorem using differential forms, which characterizes the algebraic conditions for integrability via the exterior derivative of the subbundle's local annihilator.

Future quantum architectures are expected to be modular, with quantum processors connecting multiple quantum processing units (QPUs) via photonic interconnects. In topological quantum error correction, such as color codes, this creates seam boundaries where parity checks require remote CNOT operations using heralded Bell pairs. These non-local checks are slower and noisier than bulk local checks because entanglement generation is probabilistic, causing data qubits to accumulate idle noise while waiting for remote operations. A natural way to reduce this overhead is to skip some seam measurements; however, doing so makes seam syndrome information stale and can degrade decoding. The central scheduling problem is therefore to determine how frequently seam checks should be measured so as to balance remote-operation and waiting noise against syndrome staleness. To address this trade-off, we develop a scheduling module that integrates directly into standard syndrome-extraction circuits. We consider two policies: Skip-Seam-$τ$ (SS-$τ$), which measures all bulk checks every round while measuring seam checks once every $τ$ rounds and copying the most recent syndrome in skipped rounds, and Adaptive Skip-$τ$ (AST), which selects $τ$ as a function of code distance and entanglement generation rate (EGR). We evaluate these policies on triangular color codes under circuit-level noise in Stim, including idling errors induced by Bell-pair generation delays. Our simulations show that SS-tau and AST reduce remote-operation overhead and can lower the logical error rate (LER) relative to the Measure-All (MA) baseline. For physical error rate $p = 10^{-3}$, we identify an EGR regime in which both SS-$τ$ and AST exhibit behavior consistent with fault-tolerant scaling, with LER decreasing as code distance increases. Across these regimes, SS-$τ$ and AST outperform MA.

We establish uniform error bounds of the L1 discretization of the Caputo fractional derivative of the function from the weighted Sobolev space with weight belonging to the Mucknenhoupt class. We present how our framework works for several examples of weight, which belong to the Muckenhoupt class. As and application, we show the convergence of the L1 scheme for the Fractional ODE. Finally, we verify the theoretical results with numerical illustrations.

This paper proposes and analyzes Riemannian optimization algorithms on the manifold of unitary and symmetric matrices, denoted ${\cal {U}}_s$, which naturally models the scattering matrices of passive and reciprocal devices such as beyond-diagonal reconfigurable intelligent surfaces (BD-RISs). Despite its relevance, the geometry of ${\cal {U}}_s$ has remained largely unexplored, and existing BD-RIS optimization methods either ignore the symmetry constraint or rely on costly Takagi-based parameterizations. We first provide a rigorous geometric characterization of ${\cal {U}}_s$, deriving its tangent space, a simple retraction, and closed-form expressions for geodesics. Building on these results, we develop two Riemannian manifold optimization (MO) algorithms tailored to ${\cal {U}}_s$: a line-search (LS) based scheme and a phase-optimization (PO) update along geodesics. We then apply the proposed framework to BD-RIS-assisted multiple-input multiple-output (MIMO) links, addressing sum-gain maximization, rate maximization, and minimum mean-square error problems, where they outperform existing approaches. Furthermore, we show that when the number of BD-RIS elements exceeds the total number of antennas, the optimal scattering matrix is low-rank, which motivates and enables efficient low-rank variants of the proposed algorithms.

We present a modular and thermodynamically consistent modeling framework for simulating steady-state and transient behavior in fixed-bed reactors. Accurate simulation of dynamic reactor behavior is essential for enabling flexible operation in Power-to-X (P2X) applications, such as Power-to-Ammonia and Power-to-Methanol, where fluctuating renewable energy inputs demand robust and responsive process control. The proposed models integrate non-ideal thermodynamics through cubic equations of state and account for both advective and dispersive transport phenomena. We derive consistent mass and energy balances using internal energy as the energy state variable, and obtain temperature and pressure from thermodynamic constraints. Our simulation methodology provides the necessary model functions for steady-state and dynamic simulations, as well as parametric sensitivity analysis. It is applied to two fundamental fixed-bed reactor units, the fixed-bed reactor (FBR) and the direct-cooled reactor (DCR). In the context of ammonia synthesis, we simulate representative reactor variants, the adiabatic fixed-bed reactor (AFBR) and the isothermal direct-cooled reactor (IDCR). Simulations assess the impact of real and ideal thermodynamic models, transport assumptions, and steady-state approximations. Results show that real-fluid effects at elevated pressures significantly influence steady-state outlet temperatures and conversions for the IDCR, while common literature model assumptions generally provide accurate dynamic predictions. Altogether, the framework supports systematic reactor model development and analysis under variable operating conditions and model assumptions relevant to Power-to-X applications.

Multi-speaker automatic speech recognition (ASR) aims to transcribe conversational speech involving multiple speakers, requiring the model to capture not only what was said, but also who said it and sometimes when it was spoken. Recent Speech-LLM approaches have shown the potential of unified modeling for this task, but jointly learning speaker attribution, temporal structure, and lexical recognition remains difficult and data-intensive. At the current stage, leveraging reliable speaker diarization as an explicit structural prior provides a practical and efficient way to simplify this task. To effectively exploit such priors, we propose DM-ASR, a diarization-aware multi-speaker ASR framework that reformulates the task as a multi-turn dialogue generation process. Given an audio chunk and diarization results, DM-ASR decomposes transcription into a sequence of speaker- and time-conditioned queries, each corresponding to one speaker in one time segment. This formulation converts multi-speaker recognition into a series of structured sub-tasks, explicitly decoupling speaker-temporal structure from linguistic content and enabling effective integration of diarization cues with the reasoning capability of large language models. We further introduce an optional word-level timestamp prediction mechanism that interleaves word and timestamp tokens, yielding richer structured outputs and better transcription quality. Our analysis shows that diarization systems provide more reliable speaker identities and segment-level boundaries, while LLMs excel at modeling linguistic content and long-range dependencies, demonstrating their complementary strengths. Experiments on Mandarin and English benchmarks show that the proposed approach achieves strong performance with relatively small models and training data, while remaining competitive with or outperforming existing unified approaches.

For a vector bundle $V$ over a curve $X$, the Segre invariant $s_n (V)$ encodes the maximal degree attained by rank $n$ subbundles of $V$. The functions $s_n$ define stratifications on moduli of $V$ which are well studied. Let $G$ be a connected reductive algebraic group, and $E \to X$ a principal $G$-bundle. For each parabolic subgroup $P \subset G$ there is a Segre number $s_P (E)$, generalising $s_n (V)$. We show that $s_P$ is semicontinuous in families of $G$-bundles, and thus defines stratifications on moduli spaces of $G$-bundles over $X$. We study the invariance properties of $s_P$, relating the behaviour of $s_P$ and $s_{ϕ(P)}$ for a surjective homomorphism $ϕ\colon G \to H$ and allowing us to compare the Segre stratifications for $G$ and $H$. Finally, we analyse the stratification for the Borel subgroup $B$ of ${\rm GL}_3$, identifying patterns in the geometry and proving, in particular, a sharp Hirschowitz-type bound on $s_B (E)$ for certain topological types.

The multi-messenger exploration of dark matter and physics beyond the Standard Model has emerged as a central direction in modern astro-particle physics, particularly following the discovery of gravitational waves. In this work, we present a comprehensive review and forward-looking perspective on machine-learning-enhanced multi-messenger approaches, combining information from gravitational waves, cosmic rays, gamma rays, neutrinos, and collider experiments. We summarize the current state of the field, discuss recent methodological developments, and outline a coherent research program aimed at integrating heterogeneous datasets within a unified inference framework. Our collaboration proposes here a plan for forthcoming analyses aiming at extracting information on the properties and interactions of dark matter, and finally on its genesis, combining multi-messenger astronomy techniques and inputs from laboratory physics. The main objectives planned in this line of research comprise: i) the multi-messenger analysis of new physics in cosmology, including mainly, but not only, several different models of dark matter; ii) the phenomenology of new physics signatures in ground-based cosmic rays experiments, with cross-correlation to the corresponding physical, astrophysical and cosmological observations; iii) the development of machine learning methods for data analysis in ground-based cosmic rays experiments, in light of the new physics signatures. We note that several groups have explored the use of multi-messenger observations, including gravitational waves, to probe alternative dark matter candidates. The present work builds on these developments by focusing on the role of machine learning in integrating heterogeneous datasets. We foresee that such a cross-fertilizing approach will represent the right path to extract information about the main questions left in fundamental physics.

We establish the well-posedness of stationary solutions for a class of SPDEs with locally monotone coefficients, and prove the Freidlin--Wentzell large deviation principle (LDP) for these stationary solutions. The LDP for the associated invariant measures then follows via the contraction principle, avoiding the need to construct the quasi-potential and verify the Dembo--Zeitouni uniform LDP over bounded sets. By working directly with stationary solutions, we bypass these technical difficulties, thereby providing a more general and flexible framework that is adapted to additive noise, multiplicative noise, and transport-type noise. As applications, our results cover a range of SPDEs, including the stochastic reaction-diffusion equations, stochastic 1D viscous Burgers equation, stochastic 2D Navier--Stokes equations, stochastic 2D magneto-hydrodynamic equations and stochastic 3D hyper-dissipative Navier--Stokes equations.

Studying the interstellar medium (ISM) in merging high-redshift galaxies is crucial for understanding early galaxy assembly, star formation, and black hole growth, predicted by hierarchical $Λ$CDM models. Deep imaging and spatially resolved spectroscopy with JWST enable unprecedented insight into these processes, even for galaxies in the Epoch of Reionization. We present NIRSpec and MIRI integral field spectroscopy and MIRI imaging of the merging galaxy Gz9p3 at z=9.3 of the UV and optical rest-frame showing a clumpy morphology in the continuum as well as line emission covering the entire galaxy over a range of 5 kpc from the central clump to the tail region. We analyze the integrated spectrum as well as different apertures in the galaxy allowing a spatially resolved characterization of the ionized ISM of this galaxy. We compare our measurements with archival NIRCam imaging and ALMA data. We measure a total star formation rate of 13.4 $\pm$ 1.8 Msun yr$^{-1}$, a metallicity of 12+log(O/H) = 7.84 $\pm$ 0.05 and $ξ_{ion}$= 25.4 $\pm$ 0.1 erg$^{-1}$ Hz and a burstiness parameter of 0.9 $\pm$ 0.1 for the integrated spectrum. We find large spatial differences in these parameters between the central clump and the tail region. The optical [OIII] emission peaks in the main galaxy, the far-infrared [OIII] emission peaks towards the tail, indicating different physical conditions in the ISM of the tail and main galaxy. This study presents the spatially resolved ISM analyses of a galaxy at z>9, revealing nebular line emission and strong spatial variations in star formation, metallicity, physical conditions, and ionizing efficiency. The results indicate a recent, metal-poor starburst in a tail alongside a more evolved, enriched central clump with evidence for extreme excitation. This demonstrates the power of spatially resolved JWST spectroscopy of galaxies in the Epoch of Reionization.

We explore a fuzzy modal logic that can formalise probabilistic reasoning about actions and knowledge. In particular, we deal with contexts involving statements about events expressed via modal formulas, e.g., "after doing $a$, the probability of $A$ knowing that $p$ holds increases / decreases / is equal to $0.25$", "according to $A$, $p$ is equally likely to happen after doing $a$ or $b$", etc. We define the semantics of the logic on Kripke frames equipped with probability measures. We analyse the complexity of deciding the satisfiability of formulas of our logic over finitely branching models, for the full language and its fragments of varying expressivity. In particular, we identify several fragments of our logic where satisfiability is decidable in polynomial time.

Row column addressed (RCA) transducers present a promising solution for ultrafast volumetric imaging with a reduced channel count and a large field of view. However, RCA-based 3D imaging is fundamentally limited by severe sidelobe artifacts and a low signal-to-noise ratio (SNR), primarily due to weak transmit focusing inherent in RCA based ultrafast imaging strategies. To overcome these challenges, we propose a cross fusion and correlation (CFAC) method that leverages the incoherence of sidelobe artifacts and noise across datasets acquired using orthogonal apertures and multiple steering angle sets. The performance of the proposed method was validated through simulations, in vitro imaging of a multi-purpose ultrasound phantom, and in vivo experiments, and benchmarked against four established techniques: orthogonal plane wave (OPW) imaging, XDoppler method, row-column-specific frame-multiply-and-sum beamforming (RC-FMAS), and coherent factor (CF) imaging. Simulation results demonstrated that CFAC reduced sidelobe levels by 42.0 dB, 38.9 dB, 28.3 dB, and 25.5 dB compared to OPW, XDoppler, RC-FMAS, and CF, respectively. In phantom experiments, CFAC improved the CNR by up to 17.5 dB. Furthermore, in vivo imaging of a rat kidney showed that CFAC enables visualization of a significantly more detailed microvascular network, achieving a CNR improvement of over 25 dB against all benchmarked methods. In conclusion, the proposed CFAC method effectively suppresses sidelobe artifacts and noise in RCA-based imaging under low-SNR conditions, enabling high-contrast 3D visualization while preserving the high frame rate capabilities of ultrafast ultrasound imaging.

Microservice is a popular software architecture that relies on decentralized teams and clear service ownership to support modularity and scalability. However, in practice, developers frequently contribute across multiple services, creating organizational coupling (OC) that gradually erodes architectural boundaries and increases coordination overhead. This study proposes a vision for behavior-driven architectural governance through gamification in microservice systems to influence developer behavior and reduce OC. Our approach introduces a gamified framework that continuously mines repository data to detect architectural boundary violations and increasing service dependencies, and translates those signals into gameful designs, including points, badges, leaderboards, and architecture improvement quests. We outline a conceptual framework that integrates repository mining, architectural metrics, and gamification mechanisms to encourage developers to maintain service boundaries and improve architectural maintainability. Furthermore, we present an evaluation roadmap to assess the impact of gamified OC governance and developer engagement. This work aims to open a new research direction at the intersection of software architecture governance, socio-technical analysis, and gamification, highlighting the potential of behavioral incentives to support sustainable microservice evolution.

We investigate barycenters of Gaussian process laws in adapted Wasserstein space. The adapted Wasserstein distance refines classical optimal transport by enforcing compatibility of transport plans with the temporal flow of information, and is therefore well suited for stochastic systems with filtration constraints, as common in stochastic control, mathematical finance and sequential decision problems. Within this framework, we consider weighted Fréchet means of Gaussian process laws and prove that the associated barycenter problem admits Gaussian solutions. We derive a characterization of adapted Wasserstein barycenters in terms of the means and covariance operators of the underlying processes, and we analyze their existence, uniqueness, and regularity properties under natural assumptions. The Gaussian setting reveals a tractable and structurally rich subclass of adapted transport problems, bridging adapted optimal transport and Bures--Wasserstein geometry. Our results identify adapted Wasserstein barycenters as natural representatives of collections of Gaussian models and suggest new applications in stochastic optimization, robust finance, and sequential statistics.

We prove an integral formula for the spectral flow of differentiable loops of unitaries of the form ${\rm Id}+$Schatten. Our formula is in terms of a regularised winding number, expressed in terms of exact differential forms, and we show how the formula extends to non-closed paths. Applying these ideas to the scattering operator of Schrödinger scattering systems yields explicit formulae for the number of bound states, possibly modified by the presence of resonances, of the system in terms of the potential. We finish by briefly considering the paths of unbounded operators obtained from unitary loops via the Cayley transform. These include cases of moving domain as well as paths with non-constant Hilbert space.

We explore a cosmological model based on Rényi entropic corrections to the Friedmann equations and constrain it using DESI, P-BAO, CC, and gravitational-wave observations. Unlike earlier works, we obtain a direct and stringent constraint on the Rényi parameter $λ$ from late-time cosmic acceleration, with the resulting value even satisfying recent Big Bang Nucleosynthesis and baryogenesis bounds. The model predicts values of $H_0$ and $Ω_{m0}$ that are consistent with current observational data. It provides a successful description of late-time acceleration with a quintessence-like behavior, smoothly approaching the $Λ$CDM limit without crossing the phantom divide. Furthermore, the statical comparisons along with the evaluation of energy conditions and stability analyses reinforce its viability as a robust alternative to the standard cosmological model.

Colloidal crystals permeated by mobile ions exhibit a coupling between electrostatic and elastic degrees of freedom that renormalizes the effective screening length and induces wave-vector-dependent elastic softening. Building on a recently proposed continuum model [\textit{Commun. Theor. Phys.} \textbf{77}, 055602 (2025)], we perform a rigorous Gaussian fluctuation analysis to elucidate the stability limits of the homogeneous phase. By integrating out the electrostatic fluctuations, we derive the effective elastic modulus $Γ(q)$ as a function of wave vector $q$. We show that the long-wavelength modulus $Γ(0)$ remains identically equal to the bare modulus $βK$, protected by perfect ionic screening. In contrast, the short-wavelength modulus $Γ(q\to\infty) = βK(1-ξ)$ softens as the electrostatic-elastic coupling $ξ\equiv 2βn_0 v_0^2 K$ increases, vanishing at a critical value $ξ=1$. For $ξ>1$, the fluctuation spectrum exhibits a negative eigenvalue for all wave vectors $q > q_c = κ_0/\sqrt{ξ-1}$, signaling an ultraviolet instability of the uniform phase. In a real colloidal crystal, this divergence is regulated by the discrete lattice cutoff $q_{\max}\simπ/a$, confining the physical instability to a finite band $q_c < q < q_{\max}$. The macroscopic limit $q\to 0$ remains unconditionally stable for all $ξ$. The transition at $ξ=1$ thus marks the onset of short-wavelength mechanical failure, while macroscopic elastic stiffness remains intact. Our analysis clarifies the proper physical interpretation of the minimal coupling model and provides a consistent picture of how non-DLVO interactions can drive local structural collapse in charged colloidal crystals.

Motivated by the importance ascribed to correlations in random matrices used to model phenomena in various scientific disciplines, we report how algebraic correlations between matrix elements affect the eigenvalue statistics and spectral density of random matrices. These correlations, introduced through a long-range correlated percolation model, decay as a power law $\propto r^{-2H}$, with exponent $H > 0$. As $H$ varies, both the eigenvalue distribution and excess kurtosis undergo qualitative changes. At the threshold $H_c = 3/4$, characterized by emergent Gaussian statistics, a sign change in excess kurtosis marks a transition from a fat-tailed generalized $t$-distribution to one that gradually approaches the standard semicircle law for $H \gg H_c$. Our analytical results, based on scaling analysis and supported by extensive numerical simulations, provide clear predictions and uncover novel spectral regimes in random matrix theory. Our results connect techniques from statistical physics, percolation theory, and random matrix analysis, offering a new perspective on universality in correlated ensembles.

Surface functionalization plays a decisive role in the performance of biosensors, as it governs the efficiency and stability of biomolecule immobilization at the sensor interface and, consequently, the overall performance of the biosensing platforms. In this work, we present a comparative study of three organosilane chemistries - APTES, APDMS, and CPTES - applied to a SiO2 terminated 1D photonic crystal able to sustain Bloch surface waves and designed to operate as optical biosensors in both label free and fluorescence enhanced modes. Each chemistry was evaluated through a standardized label-free protocol based on the interaction between immobilized SARS CoV 2 spike protein and its corresponding antibodies, enabling quantitative assessment of binding efficiency, nonspecific adsorption, and signal repeatability. CPTES exhibited the most favorable balance between specific signals, reduced variability, and low nonspecific adsorption. The three chemistries were subsequently tested in fluorescence mode for the detection of anti SARS CoV 2 IgG antibodies in human serum, demonstrating the suitability of BSW enhanced fluorescence for rapid serological analysis. Overall, the study identifies CPTES as the most robust and reproducible functionalization strategy among the three investigated for BSW biosensing and highlights the potential of the platform for fast, sensitive detection of clinically relevant antibodies.