We study the Wasserstein gradient flow of semi-discrete energies in the space of probability measures, that is functionals depending on two measures-one being an absolutely continuous density and the other an atomic measure. These energies appear naturally in the field of urban planning. This is done via the celebrated JKO scheme, for which we prove convergence to a limiting system composed of a parabolic PDE with singular advection coupled with an ODE, also presenting singular dynamics. This is first done under more general assumptions using classical tools, and in a second moment convergence is proven to hold in $L^2_tH^1_x$ for the cases of linear and Porous-Medium type diffusions. We then pass to the study of some qualitative properties of this system, such as the convergence of the atoms towards the baricenters of their corresponding Laguerre cells. We finish this work with extensive numerical simulations that aid in formulating conjectures for the qualitative behavior of this system; in the case of linear diffusion, for instance, we observe a dynamic crystallization phenomenon.

The KID_READOUT board, developed for the CONCERTO millimeter-wave astronomy instrument, implements FPGA-based digital frequency multiplexing to read out large arrays of Microwave Kinetic Inductance Detectors (MKIDs). The complexity of the implemented multirate DSP chain, which combines tones synthesis, interpolation, digital frequency translation, polyphase filter-bank (PFB) channelization, and digital down-conversion (DDC), makes analytical performance optimization difficult. To address this, we developed a cycle-and bitaccurate, Python-based digital twin of the FPGA readout firmware DSP chain. Using this model, we identified the origin of previously measured and unexplained spurs in the readout channels, tracing them to periodicity mismatches between the excitation and analysis paths and to insufficient suppression of negative-frequency components by the DDC filters. Based on these insights, we implemented a mitigation strategy that aligns the periodicities and improves the DDC filter characteristics, effectively eliminating the spurs with a minor increase in FPGA resource usage.

We establish unweighted Hardy-type inequalities on step-two Carnot groups with one-dimensional vertical layer, with explicit lower bounds for the optimal Hardy constant. The approach is based on a quantitative integration-by-parts mechanism that replaces the non-horizontal Euler vector field by a suitably constructed horizontal vector field with controlled norm. As applications, we obtain fully explicit bounds in the Heisenberg group for both the Kor{à}nyi gauge and the Carnot--Carath{é}odory distance, and we extend the results to non-isotropic step-two structures through a generalized Kor{à}nyi-type homogeneous norm.

In stellarator design, the coil winding surfaces $Σ\subset\mathbb R^3$ support current distributions $j$ that shape the magnetic field. This work provides a theoretical framework explaining the emergence of centre and saddle point regions, a key feature in coil optimisation. For coil winding surfaces with a toroidal shape, we prove a dichotomy principle: the current distribution has both centre and saddle point regions or is no-where vanishing. For coil winding surfaces that consist of piecewise cylinders, we show that if $j$ is oppositely oriented on the two boundary circles, centre and saddle points appear, and all but finitely many field lines of $j$ are periodic. When $j$ admits a harmonic potential, all field lines are closed poloidal orbits. These results offer insights into current patterns on winding surfaces, with implications for coil design strategies and their simplification.

Van der Waals (vdW) ferroelectrics are emerging nonlinear photonic materials that combine large second-order susceptibility \c{hi}(2) with heterostructure compatibility, offering an attractive route toward miniaturized spontaneous parametric down-conversion (SPDC) sources. However, vdW SPDC sources operating under continuous irradiation in air remain limited in low brightness and poor operational stability, as oxygen and moisture exposure, together with pump-induced heating, lead to material degradation and permanent damage. Here we demonstrate an air-stable, bright SPDC source based on ferroelectric NbOI2 enabled by graphene encapsulation. Graphene provides robust environmental protection and can effectively supress pump induced degradation by enhancing heat dissipation. We report a record photon-pair generation absolute rate of 258 Hz and a normalized brightness of 19,900 Hz/(mW.mm). Leveraging this stabilized platform, we further generate polarization entangled photon pairs with 94% fidelity with respect to the maximally entangled Bell states from graphene-encapsulated 90° twisted bilayer NbOI2. Our results establish a practical and air-stable vdW ferroelectric SPDC platform that overcomes key limitations of existing vdW quantum light sources and provides a viable pathway toward scalable, integrated entangled photon sources for on chip quantum photonics.

The difference-in-differences (DiD) design is a quasi-experimental method for estimating treatment effects. In staggered DiD with multiple treatment groups and periods, estimation based on the two-way fixed effects model yields negative weights when averaging heterogeneous group-period treatment effects into an overall effect. To address this issue, we first define group-period average treatment effects on the treated (ATT), and then define groupwise, periodwise, dynamic, and overall ATTs nonparametrically, so that the estimands are model-free. We propose doubly robust estimators for these types of ATTs in the form of augmented inverse variance weighting (AIVW). The proposed framework allows time-varying covariates that partially explain the time trends in outcomes. Even if part of the working models is misspecified, the proposed estimators still consistently estimate the parameter of interest. The asymptotic variance can be explicitly computed from influence functions. Under a homoskedastic working model, the AIVW estimator is simplified to an augmented inverse probability weighting (AIPW) estimator. We demonstrate the desirable properties of the proposed estimators through simulation and an application that compares the effects of a parallel admission mechanism with immediate admission on the China National College Entrance Examination.

For over two decades, the gravity sector of the Standard-Model Extension (SME) has served as a phenomenological framework for testing spacetime symmetry breaking in the presence of gravity. During this time, various theoretical features have been examined in greater detail and some refinements have been made. In particular, differences between spontaneous and explicit breaking of diffeomorphisms, local translations, and local Lorentz transformations in Riemann-Cartan geometry, as well as their corresponding consistency issues with geometric and mathematical identities, have been probed more deeply. This has led to a modified version of the SME being developed that is suitable for investigating explicit breaking in gravity theories, which can be used as well to search for new geometries that go beyond Riemann-Cartan. A selective overview of some of these features is presented here.

In this paper, we consider the problem of minimizing a smooth function on a Riemannian manifold and present a Riemannian gradient method with momentum. The proposed algorithm represents a substantial and nontrivial extension of a recently introduced method for unconstrained optimization. We prove that the algorithm, supported by a safeguarding rule, produces an $ε$-stationary point with a worst-case complexity bound of $\mathcal{O}(ε^{-2})$. Extensive computational experiments on benchmark problems are carried out, comparing the proposed method with state-of-the-art solvers available in the Manopt package. The results demonstrate competitive and often superior performance. Overall, the numerical evidence confirms the effectiveness and robustness of the proposed approach, which provides a meaningful extension of the recently introduced momentum-based method to Riemannian optimization.

We present a novel gradient regularization to completely eliminate the magnetic divergence error in meshless magnetohydrodynamics (MHD), which offers a high spatial resolution and conservative advantage, due to its Lagrangian nature. Comparing with the counterpart of constrained-gradient (CG) technique, we reform $\nabla \cdot \mathbf{B}=0$ by an implicit projection method to modify the magnetic-field gradients. The accuracy of modified-gradient (MG) method is verified and it achieves exact divergence-free results with round-off precision, by using tests of shock tube, 2D and 3D vortex, magneto-rotational instability, and especially, advection experiment, compared with CG method and the GIZMO code. It leads to noticeable improvement in pattern, amplitude and numerical dissipation of divergence error of magnetic field.

We study the empirical spectral distribution of the normalized Laplacian of linear preferential attachment graphs in the Barab{á}si-Albert regime with fixed out-degree. For the resulting sequence of random multigraphs, we prove that the empirical spectral distribution converges weakly in probability to a deterministic probability measure supported on the interval [0, 2]. The limit is characterized via the local weak limit of preferential attachment graphs (the P{ó}lya-point graph): the limiting Stieltjes transform is given by the expected diagonal Green function at the root of the normalized Laplacian operator on this infinite random graph. The proof combines a resolvent approach with a uniform Neumann-series expansion for the normalized Laplacian, a random-walk representation in terms of return probabilities on decorated neighborhoods, a truncation and Doob martingale-Azuma-Hoeffding concentration argument along the PA filtration, and an analytic continuation argument based on normal families.

Soft matters whose constituents are deformable are ubiquitous in nature especially in biological systems-including cells and their organelles-as well as in foams and emulsions. The capacity for deformation in these soft materials gives rise to a range of intriguing phenomena, such as glassy behavior without any size dispersity, cluster crystal formation, and re-entrant melting. Deformability also plays a crucial role in facilitating essential biological processes, such as the flow of blood through veins and arteries. In this work, we investigate assemblies of two-dimensional (2D) polymeric, non-overlapping rings, which mimic deformable particulates in 2D using extensive molecular dynamics simulations. The rings are confined in a rectangular channel with hard walls perpendicular to the flow direction, mimicking natural flow conditions. We analyze the flow properties of these deformable particle assemblies at two different stiffness values. To further asses the impact of deformability, we examine the same monodisperse system at higher densities for the stiffer rings, where deformation is necessary and a fluid layer emerges at the channel edges. Finally, we explore a mixture of rings with two distinct stiffnesses and observe effective segregation of soft and hard particles at small channel widths.

\noindent\textbf{Background} Prenatal glucocorticoid exposure alters cardiac development, but whether persistent cardiac effects in childhood follow a dose-response relationship remains unknown. We recently showed that ECG foundation models detect robust cardiac differences between steroid-exposed and control children, while traditional heart rate variability metrics lose significance after covariate adjustment. Here, we investigate the dose-response dimension using complementary analytical approaches. \noindent\textbf{Methods} We studied 49 children (ages 8--15) whose mothers received betamethasone during pregnancy for multiple sclerosis: 12 low-dose ({$<$}5\,g cumulative), 13 high-dose ({$\geq$}5\,g), and 24 controls. Five-minute ECG recordings during the Trier Social Stress Test yielded 251 observations. We computed 12 multiscale complexity features and tested 11 ECG foundation model (FM) dimensions using linear mixed models, Kruskal--Wallis tests with Dunn's post-hoc comparisons, Spearman correlations, and Jonckheere--Terpstra trend tests. \noindent\textbf{Findings} The binary exposed-versus-controls comparison showed no significant complexity effects ($p>0.39$). However, dose-based analysis revealed that high-dose children exhibited significantly faster entropy rate ($h$) decay rates than low-dose children ($p=0.031$); neither sample entropy nor approximate entropy decay rates reached significance ($p=0.18$ and $p=0.12$, respectively). Effects localized to the mental arithmetic stress segment (Kruskal--Wallis $p=0.005$; Dunn's $p=0.004$). A cross-condition robustness analysis confirmed that $h$ decay rate is invariant to input signal choice and normalization ($r>0.98$), while sample and approximate entropy are not. In contrast, the 11 FM dimensions showed weak dose-response evidence: only 1 of 22 covariate-adjusted contrasts survived FDR correction, with paradoxically stronger low-dose effects. \noindent\textbf{Interpretation} The entropy rate decay rate -- uniquely robust across input conditions -- reveals a dose-dependent effect on cardiac autonomic dynamics under cognitive stress, while FM dimensions detect a dose-independent morphological ``exposure fingerprint.'' These exploratory findings suggest a two-component model of prenatal glucocorticoid cardiac programming -- ~morphological (dose-independent) and dynamical (dose-dependent)~ -- providing more complete characterization than either approach alone. Given the small sample size, these results should be considered hypothesis-generating and require replication in larger cohorts.

It is mandatory to know how to operationally define and translate a reference frame into mathematics, in order that a physical interpretation of theory calculations in terms of observational data is possible. The situation is particularly challenging for gauge systems such as General Relativity where spacetime coordinates are subject to spacetime diffeomorphisms considered as gauge transformations turning coordinates into non-observables. This motivates the idea of operationally defined (material) reference frames which specify coordinates in terms of matter or geometry reference fields leading to the concept of relational observables, relational reference frames and gauge reduction. Upon quantisation, all fields become operator valued distributions. Now new conceptual and technical questions arise such as: Should one reduce before or after quantisation and how are the reference fields quantised respectively in either route? Is a reference frame itself subject to quantisation and how are different quantum reference frames related? How does the gauge reduction fit into this, i.e. how can it be that a certain reference field is considered a non-observable in one reference frame and an observable in another which upon quantisation even displays fluctuations? How precisely are gauge dependent fields interpreted in terms of the relational observables in a given reference frame? What is the relative dynamics, e.g. how exactly are physical Hamiltonians of two relational reference frames related? The present conceptual work addresses these and related questions in a non-perturbative field theory context of sufficient generality to cover General Relativity coupled to standard matter. A central role is played by the concept of the relational reference frame transformation (RRFT) for which a general formula is derived and its properties are explored.

We investigate the relationship between the Eklof-Mekler-Shelah Construction Principle for a variety of algebras $\mathbf{V}$ and the question of superstability of the free objects in $\mathbf{V}$, denoted as $\mathcal{F}_\mathbf{V}$. We consider this question in the general setting of AEC-coverings of $\mathcal{F}_\mathbf{V}$, with applications to first-order logic and beyond. Our main result is that if a strong form of the Construction Principle is satisfied, then almost all AEC-covering of $\mathcal{F}_\mathbf{V}$ are unsuperstable. Concrete applications to $R$-modules and varieties of groups are also considered.

Physical quantum systems are generically coupled to an environment, resulting in open system dynamics. A typical approach to simulating this dynamics is to propagate the density matrix of the system via the Lindblad master equation. This approach is numerically challenging due to the size of the density matrix, which has led to the development of quantum jump methods, which unravel the density matrix into an ensemble of state vectors. These methods utilize a stochastic sampling of the quantum jump times, which becomes inefficent for weakly dissipative dynamics, in which jumps are rare events. Here, we propose the deterministic quantum jump (DQJ) method, which we show to outperform standard quantum jump methods in the weakly dissipative regime, by removing the error of stochastic sampling. We describe the methodology at the single-jump and two-jump level, reconstructing the density matrix at the corresponding level. We demonstrate the performance of the method for two examples, the dissipative transverse-field Ising model, and the dissipative Kerr oscillator. Given that quantum technologies such as quantum computing have weakly dissipative quantum dynamics as their central focus, we propose this method to be utilized in that context, for exploring and understanding quantum technology platforms.

In modern e-commerce and service operations, firms must jointly manage inventory replenishment and real-time order fulfillment to maximize profit under demand uncertainty. While each component has been studied extensively in isolation, their interaction remains underexplored. This paper investigates a fundamental operational question: which lever plays a more decisive role in overall system performance, replenishment or fulfillment? We model the system as a one-location online order fulfillment problem with lost sales and stochastic customer arrivals, each offering heterogeneous rewards. Replenishment follows either a base-stock or constant-order policy, while real-time fulfillment decisions are made using online algorithms. Our core performance metric is the expected average profit per replenishment cycle, evaluated across all combinations of these policies and algorithms. Our main theoretical result shows that when the replenishment cycle is long, the cumulative regret of online fulfillment remains of the same order as in a corresponding single-cycle problem, even under repeated replenishment, revealing a form of regret stability. This phenomenon also extends to a multi-location setting. We further develop a regret-based framework that quantitatively compares the value of improving replenishment versus improving fulfillment, and we characterize regimes in which optimizing replenishment yields a larger revenue impact than refining the online fulfillment algorithm (and vice versa). Motivated by examples where myopic algorithms underperform, we introduce a novel look-ahead online algorithm that anticipates future replenishment and demand. Numerical experiments verify that this algorithm outperforms myopic baselines. Overall, our results provide both theoretical and managerial insights into situations where inventory replenishment policies are more influential and vice versa.

An eigenvalue $λ$ of a signed graph $S$ of order $n$ is called a main eigenvalue if its eigenspace is not orthogonal to the all-ones vector $j$. Characterizing signed graphs with exactly $k$ $(1\le k\le n)$ distinct main eigenvalues is a problem in algebraic and graph theory that has been studied since 2020. Du et al. (2024, 2026) characterized a class of signed graphs with exactly two main eigenvalues by analyzing a type of multigraph whose base graph is a tree. In this paper, we extend this study to the case where the associated multigraph has a unicyclic base graph, and we conclude by proposing several open problems.

Contact tracing is one of the most important control measures deployed during epidemics. Relying on the identification of contacts of known infected individuals, it necessitates a network perspective. Although pairwise models have been used extensively to study contact tracing, their analysis typically depends on a decoupling assumption-most commonly that contact tracing operates on a much faster timescale than disease transmission. Furthermore, contact tracing models often assume that all infected individuals become contact tracing-triggering, which is unrealistic given partial compliance to treatment. We relax both of these restrictive assumptions and provide a full analytical characterisation of the epidemic threshold in the pairwise mean-field model. Our analysis uses a fast-variables approach that captures the rapid early stabilisation of key network quantities. Inspired by mechanisms from social adoption dynamics, we introduce triplewise contact tracing in which an infected individual can be traced not only through direct contact with a single tracing-triggering neighbor (pairwise tracing), but also indirectly when connected to two tracing-triggering nodes simultaneously. For pure pairwise and pure triplewise contact tracing, we derive analytical expressions for critical contact tracing levels and demonstrate that when many infected individuals bypass treatment, the epidemic can become uncontrollable. When both contact tracing mechanisms operate together, we map out their combined contribution and relative impact on epidemic control. This unified framework yields rigorous and tractable threshold conditions for contact tracing dynamics on networks, extending the applicability of pairwise models beyond the fast-tracing regime and providing new insight into the interplay between disease progression, partial treatment compliance, and higher-order tracing processes.

In this paper, we present a novel semi-implicit numerical scheme for the stochastic Cahn--Hilliard equation driven by multiplicative noise. By reformulating the original equation into an equivalent stochastic scalar auxiliary variable (SSAV) system, our method enables an efficient and stable treatment of polynomial nonlinearities in a semi-implicit fashion. In order to accurately capture the impact of stochastic perturbations, we carefully incorporate Itô correction terms into the SSAV approximation. Leveraging the smoothing properties of the underlying semigroup and the $H^{-1}$-dissipative structure of the nonlinear term, we establish the optimal strong convergence order of one-half for the proposed scheme in the trace-class noise case. Moreover, we show that the modified SAV energy asymptotically preserves the energy evolution law. Finally, numerical experiments are provided to validate the theoretical results and to explore the influence of noise near the sharp-interface limit.

Mermin's dielectric function [N.D. Mermin, Phys. Rev. B 1, 2362 (1970)] is widely assumed to satisfy the f-sum rule because he constrains his ansatz with the continuity equation. However, we identify a moment-closure problem in Mermin's use of the continuity equation. Further, we show that the Mermin's model can be derived without invoking continuity. We describe how other approaches such as the ``completed Mermin'' model of Chuna and Murillo [Phys. Rev. E 111, 035206 (2025)] remedy this closure issue. We then inspect the f-sum rule for both the original and completed Mermin models and find for the Mermin ansatz that collision frequencies scaling as $ω$ must violate the f-sum rule, whereas constant, real, positive collision frequencies will satisfy it, with the caveat that, in practice, convergence with respect to the upper integration limit $ω_{\max}$ is sufficiently slow that finite-domain numerical evaluations exhibit apparent violations, regardless of wavenumber $q$. We also find that collision frequencies with constant imaginary components cause f-sum rule violations. We conclude that if Mermin's model is fit to data via optimizing its collision frequency, then the f-sum rule is not inherently satisfied; constraints, though broad, are needed in order to assume the f-sum rule is satisfied. Further, if the f-sum rule is theoretically satisfied, but violations still appear, then these deviations ought to be included in the error estimates.

Performance indicators are essential tools for assessing the convergence behavior of multi-objective optimization algorithms, particularly when the true Pareto front is unknown or difficult to approximate. Classical reference-based metrics such as hypervolume and inverted generational distance are widely used, but may suffer from scalability limitations and sensitivity to parameter choices in many-objective scenarios. Indicators derived from Karush--Kuhn--Tucker (KKT) optimality conditions provide an intrinsic alternative by quantifying stationarity without relying on external reference sets. This paper revisits an entropy-inspired KKT-based convergence indicator and proposes a robust adaptive reformulation based on quantile normalization. The proposed indicator preserves the stationarity-based interpretation of the original formulation while improving robustness to heterogeneous distributions of stationarity residuals, a recurring issue in many-objective optimization.

In-situ plasma processing of superconducting radio frequency (SRF) cavities is a performance recovery technique used to mitigate the field emission limiting phenomenon. It has been proved very effective at major particle accelerator facilities such as SNS, CEBAF, FRIB, FNAL and C-ADS. This technique is based on the ignition of a noble-gas/oxygen plasma inside the cavity over several hours to remove hydrocarbon-based contamination, responsible for the parasitic field emission degradation observed after several years of operation. Despite a large experimental R\&D effort from the community, plasma parameters and cleaning rates under various experimental conditions have never been directly evaluated. In this study, plasma parameters were measured using a Langmuir probe and cleaning rates thanks to a quartz crystal microbalance (QCM) coated with an amorphous carbon film to simulate a carbon-based contamination. In this article, the main results from a large parameter space are discussed along with guidelines for improving the plasma processing effectiveness in SRF cavities. The encountered technical challenges are also discussed, as the SRF cavity is by design not intended to be a plasma reactor.

We introduce localization operators on weighted Bergman and Fock spaces and show that, under a natural scaling of symbols and window functions, localization operators on the weighted Bergman space $A_{βr^2}^2$ converge, in the weak sense, to localization operators on the Fock space $F_β^2$ as $r\to\infty$. From this we derive several applications, including one about sharp norm estimates for certain Toeplitz operators on Fock spaces, one about windowed Berezin transforms for weighted Bergman spaces, and another about Szegö-type theorems for localization operators on weighted Bergman spaces.

Rosenbloom and Tsfasman, in their foundational work on the $m$-metric, introduced algebraic-geometric codes defined by multiple points on a smooth projective curve $X$. This construction involves a divisor $G$ and another divisor $D=\sum n p_i$, where $p_i$ are distinct rational points with $p_i \notin \text{supp}(G)$ and $n\in\mathbb{N}$. Although these codes are significant, their formal development for arbitrary genus remains incomplete in the literature, as most studies have concentrated on the genus $0$ case. We present a rigorous treatment of this class of codes. Starting with a smooth projective curve $X$, an invertible sheaf $L$, and an effective divisor $D=\sum n_i p_i$ where the $n_i$ are not necessarily equal, as well as tuples of uniformizers $t_D$ at the points of $D$ and trivializations $γ_D$ for the localizations $L_{p_i}$, the associated differential Goppa code is defined. This code arises from the theory of $n$-jets of invertible sheaves on curves, which enables the description of codewords using Hasse-Schmidt derivatives of sections of $L$. The variation of the code under changes in the data $(t_D, γ_D)$ is examined, and the group acting on these parameters is described. The behavior of the minimum Hamming distance under such variations is analyzed, with explicit examples provided for curves of genus $0$ and $1$. A duality theorem is established, involving principal parts of meromorphic differential forms. It is demonstrated that Goppa codes constitute a proper subclass of differential Goppa codes, and that every linear code admits a differential Goppa code structure on $\mathbb P^1$ using only two rational points.

We develop a systematic framework for formulating and solving the conditions that lead to separability in stationary, axisymmetric spacetimes in the presence of matter fields. Guided by Carter's metric form, we introduce a general stationary, axisymmetric metric ansatz that allows for a transparent separation of radial and angular variables. This construction yields a broad family of stationary rotating solutions admitting separability structures. To illustrate the applicability of the formalism, we explicitly construct several examples, including a rotating black hole with a global monopole supported by anisotropic matter, as well as a new class of rotating wormhole geometries.

Gas accretion process can fuel both star formation and black hole activity, playing a critical role in galaxy evolution. The counter-rotating structures are believed to originate from gas accretion, serving as an ideal laboratory for studying its impact on galaxy evolution. Based on the Mapping Nearby Galaxies at Apache Point Observatory (MaNGA) survey, we built a sample of 147 galaxies with counter-rotating stellar disks (CRDs). This is the largest CRD sample to date, accounting for $\sim$1.5% of the MaNGA survey. For a subset of 138 CRDs, global stellar mass ($M_\ast$) and star formation rate (SFR) were measured in reference. We constructed a control sample with similar $M_\ast$ and SFR but lacking counter-rotating structures. The CRDs relatively exhibit more bulge-dominated morphology, lower molecular gas mass fraction and reside in less dense environment, supporting the hypothesis that they primarily originate from gas accretion. We classified 96 out of 138 CRDs into four types based on their stellar and gas kinematics following the criteria from Bao et al. (2022). There are two additional CRD types: 8 CRDs show misalignment between both stellar disks and gas disk, indicating multiple gas accretion events with differing angular momentum directions; 34 CRDs lack ionized gas emission, showing the highest $M_\ast$ among all the CRD types, which may represent a final stage of CRD evolution. We compared the radial gradients of gas-phase metallicity and stellar population properties between CRD types, and found that the impact of gas accretion on galaxy evolution primarily depends on the abundance of pre-existing gas in progenitors.

We present a new short proof of Van der Waerden's Theorem about the existence of arbitrarily long monochromatic arithmetic progressions. The proof uses algebra in the compact space of ultrafilters $β\N$, but contrarily to the other existing proofs, neither minimal nor idempotent ultrafilters are involved.

The cross-domain oceanic connectivity ranging from underwater to the sky has become increasingly indispensable for a plethora of data-consuming maritime applications, such as maritime meteorological monitoring and offshore exploration. However, broadband implementations can be severely hindered by the isolation from terrestrial networks, limited satellite resources, and the fundamental inability of radio waves to bridge the water-air interface at high rates. To this end, this paper introduces an optical network bridging underwater, air and near space, which features a number of cooperative low-altitude platforms (LAPs), serving as compute-capable, sensing-aware, and mission-adaptive agents. The network architecture consists of three scenario-specific segments, i.e., water-air direct link, low-altitude mesh network, and the near-space access network. With coordinate sensing and intelligent control, the system tightly couples beam tracking and resource optimization, enabling resilient networking under high mobility and harsh maritime dynamics. Furthermore, we review enabling technologies spanning from water-air channel modeling, adaptive beam alignment under sea-surface perturbations, to swarm-intelligence networking for decentralized control, integrated pose-topology planning, and optical Integrated sensing and communication (ISAC) for near-space target detection and beam alignment. Finally, open issues are also highlighted, constituting a clear roadmap toward scalable, secure, and ultra-broadband oceanic optical networks.

This work presents a description of the scientific goals and objectives of OPTIMus (OPTical, Infrared, Millimeter survey of massive star-forming regions), a survey of massive star-forming regions in the optical, infrared, and millimeter wavelengths. The survey is aimed at constructing a comprehensive characterization of the multicomponent and structurally complex interstellar medium in the vicinity of young massive stars, combining both observational and theoretical aspects. Using multi-wavelength observational data, we will reconstruct the three-dimensional structure and determine the physical parameters of HII regions, photodissociation regions, and the surrounding molecular clouds. The paper describes the observational data obtained with the BTA 6-m and Zeiss-1000 telescopes of the Special Astrophysical Observatory of the Russian Academy of Sciences, the 2.5-m telescope of the Caucasian Mountain Observatory of the Sternberg Astronomical Institute of Moscow State University, and the 20-m telescope of the Onsala Space Observatory.

We consider the inverse problem of calculating a refracting surface that generates a prescribed irradiance distribution in the far field for a collimated incident beam. This problem can be formulated as a mass transportation problem (MTP) with a quadratic cost function. To solve this problem, we propose a version of the supporting quadric method (SQM), in which the calculation of the quadric parameters is reduced to the problem of minimizing a convex function. We obtain simple analytical expressions for the second derivatives of this function, making it possible to calculate the quadric parameters using second-order optimization methods. This allows us to refer to the proposed method as the second-order SQM. We demonstrate high efficiency of this approach by designing several optical surfaces that generate complex irradiance distributions. We also consider the application of the second-order SQM to nonimaging optics problems described by MTPs with a non-quadratic cost function.