In the nearby universe, quasars are well known to exhibit variability in their brightness over time, offering a powerful tool to probe the physics of accretion onto the SMBH and directly measure the mass of the SMBH. However, detecting variability in early quasars remains challenging. Here, we report the detection of multi-wavelength infrared and X-ray variability in a quasar observed just 850 million years after the Big Bang. The infrared variability spans five filters, tracing rest-frame ultraviolet and optical emission from the accretion disk, while the X-ray variability probes the corona. The variable spectrum reveals that the accretion disk has a geometrically thin, optically thick structure. This provides observational constraints on the accretion disk structure at early times, when quasars are accreting at high Eddington ratios and reside in extreme environments. Our findings demonstrate the feasibility of characterizing accretion physics using variability in the early universe, laying the groundwork for studies exploiting upcoming facilities such as the Rubin Observatory and Roman Space Telescope. These facilities will discover large samples of variable high-redshift quasars, enabling population-level variability studies of accretion physics and black hole masses, filling key missing ingredients in understanding early SMBH growth.

A Keplerian binary immersed in a bath of lighter particles hardens by ejecting them through gravitational slingshots. This process drives, for example, the evolution of supermassive black hole binaries following galaxy mergers, and has long been described with just two parameters: the hardening rate and the eccentricity growth rate. Here we show that the secular dynamics is substantially richer. Combining symmetry arguments with extensive three-body scattering experiments, we demonstrate that the medium exerts a net force on the binary's center of mass (CoM), induces apsidal precession, and rotates the orbital plane when the CoM velocity has an out-of-plane component. Remarkably, these deterministic effects persist even in a perfectly uniform and isotropic medium, as the binary's own asymmetry provides the propulsion. The interplay of self-acceleration, precession, and dynamical friction drives the CoM along an outward spiral. For supermassive black hole binaries, this displacement dominates over Brownian motion and approaches the radius of influence, suggesting they may be significantly offset from their host galaxies' centers. The displacement also enlarges the stellar loss cone, with direct implications for the final-parsec problem. We further show that the previously reported circularization of small-mass-ratio binaries is a numerical artifact of truncating long-lived encounters: all binaries undergo eccentricity growth. Our results enrich the standard picture of binary hardening and have implications in a variety of astrophysical contexts, including gravitational-wave source populations.

We introduce a preparation-dual notion of contextuality, formulated as an obstruction to stochastic extension. In parallel with the sheaf-theoretic formulation of measurement contextuality, preparation contextuality arises when locally specified preparation statistics cannot be extended to a single global response matrix compatible with all source contexts. Whereas measurement contextuality concerns the incompatibility of restriction maps (marginalisation), the preparation setting requires stochastic extension of partial conditioning data, which is inherently non-unique. We identify minimal structural and preparation compatibility conditions on admissible extension matrices and show that they enforce a rigid product form. This leads to a notion of preparation contextuality in which the absence of any admissible global response representation witnesses contextuality, while preparation compatibility identifies the cases in which this obstruction is nontrivial. The framework is formulated explicitly in matrix form and illustrated by a quantum-mechanical example exhibiting preparation contextuality.

The rapid advancements in foundation models and sixth-generation (6G) wireless communication systems necessitate the development of efficient, scalable, and privacy-preserving machine learning approaches. For foundation models in 6G, split learning (SL) and aggregation learning (AL) have emerged as promising paradigms that address key challenges in distributed artificial intelligence (AI), such as communication efficiency, resource allocation, and data privacy. SL enables multiple entities to collaboratively train deep learning models by partitioning neural networks, while AL focuses on aggregating intermediate results or model updates from multiple participants, improving robustness, optimizing resource utilization, and mitigating data leakage risks. Specifically, SL is ideal for scenarios requiring strict data isolation (e.g., vertical collaborations), whereas AL suits homogeneous horizontal data settings; they can be combined to balance privacy and communication efficiency. This survey provides a comprehensive analysis of SL and AL in 6G communication systems, exploring their architectures, technical methodologies, and integration with AI-native 6G communication technologies. We examine different SL configurations, aggregation techniques, and their roles in optimizing distributed foundation models. Furthermore, we discuss their applications in emerging wireless networks, including semantic communication, reconfigurable intelligent surfaces (RIS), space-air-ground integrated networks (SAGINs), and quantum communication. By analyzing the impact of SL and AL, this survey provides insights into their role in shaping distributed AI-driven communication systems in the 6G era, focusing on efficiency, privacy preservation, and scalability.

Tests of gravity-induced entanglement have been proposed as a route to probing the quantum nature of gravity, but existing schemes rely on free-fall interferometry of massive spatial superpositions, imposing severe experimental constraints. We show that systems exhibiting effectively inertial dynamics in the short-time regime reproduce the same gravitational phase accumulation responsible for entanglement generation. Deviations from the free-fall phase enter at order $(t/T)^2$, where $t$ is the interferometer timescale and $T$ is the characteristic period of the constrained motion. We analyse a representative mechanically constrained implementation using carbon nanotube pendula and show that the resulting correction to the entangling phase remains below $10^{-6}$ in experimentally relevant regimes, leading to a negligible modification of the interference visibility used to certify entanglement. These results demonstrate that gravity-induced entanglement protocols extend beyond free-fall implementations to a broader class of constrained dynamical systems, significantly relaxing the requirements for experimental realisation of the Bose-Marletto-Vedral protocol.

Oftentimes, the Shapley value becomes infeasible for games with many players. However, establishing symmetry allows for polynomial-time computation. To examine this reduction, we identify the spectrum of homogeneous group games by using an induced representation from a Young subgroup. We then prove that such games are supported solely by irreducible representations, via the Littlewood-Richardson rule, where the depth of interactions is strictly bounded by the size of the minority group. Therefore, the algebraic structure of the game filters out the complexities of the general kernel $W$. We then show that this filtration constrains any symmetric linear value to a specific subspace. This recovers the Shapley value uniquely for $m=2$ under standard axioms. Finally, we explore applications to the UN Security Council and complementary goods markets to illustrate the practical power of this approach.

We develop a unified mathematical formulation for post-quantum authenticated telemetry and actuation in FADEC-coupled dual-spool turbofan cyber-physical systems. The formulation integrates lattice-based key establishment under LWE/SIS-style assumptions, PUF-derived attestation entropy, authenticated encryption, radar-altimeter integrity, avionics-bus timing, and Kalman residual monitoring in a stochastic hybrid model. Within this model, plant evolution, communication latency, leakage, adversarial channel quality, and cryptographic state evolve under a common filtration. We show that channel uncertainty tightens admissible key-renewal periods, that ciphertext expansion enters bus-level schedulability constraints, and that sensing and actuator limits shape integrity thresholds and allowable control delay. We further relate PUF smooth min-entropy to distinguishing advantage and connect innovation statistics to conservative alarm design. Overall, the results characterize how post-quantum security, real-time schedulability, and closed-loop stability interact in safety-critical aerospace control architectures within a defensive analytical treatment that does not provide operational guidance for interference with real platforms.

We propose a laser isotope separation mechanism in which selectivity arises from nuclear spin rather than isotope shifts, enabling the use of broadband ultrafast lasers. A Ramsey pulse sequence is applied to paramagnetic molecular isotopologues possessing two electronic states coupled by a dipole transition. For even-even isotopologues (nuclear spin $I = 0$), each electronic state is a single level and the time-reversed sequence returns all population to the ground state exactly. For even-odd isotopologues ($I > 0$), the hyperfine interaction splits each state into multiple levels with coupling amplitudes set by Wigner $6j$ symbols; incommensurate phase evolution during the dark interval prevents the echo from closing, trapping a fraction $P_m$ of the population in the excited manifold. In the impulsive limit ($Ω\gg A_{\rm HF}$), $P_m$ depends only on the angular momentum quantum numbers $(J_g, J_m, I)$ and is independent of laser intensity or bandwidth. Density matrix simulations confirm $P_m = 0$ for $I = 0$ and $P_m \approx 0.23$-$0.47$ for $I > 0$ across representative systems including ${}^{235}$U, ${}^{87}$Sr, and ${}^{57}$Fe. Under realistic collisional conditions, single-pass enrichment exceeding 90% from natural feed is achievable without cascading.

Recently, Danielson, Satishchandran, and Wald (DSW) have proposed a novel source of decoherence for uniformly accelerated charges and masses in spatial superposition in spacetimes containing a bifurcating Killing horizon. Such an effect can be traced back to the emission and absorption of soft photons and gravitons, which effectively act as "which-path'' information probes. This results in the decoherence of any such superposition in a finite proper time. With this in mind, we study the DSW mechanism using a gapless finite-time detector prepared in a spatial superposition of uniformly accelerated paths in Minkowski spacetime that interacts with a massive scalar field. The calculation is nonperturbative. Such a model will enable us to analyze the decoherence process in a more controlled manner, highlighting the main factors that give rise to this interesting mechanism.

In the ordered phase of a $\mathbb{Z}_2$-symmetric collective spin system, two natural bases -- localised pointer states $\{|P\rangle,|R\rangle\}$ and energy eigenstates $\{|E_0\rangle,|E_1\rangle\}$ -- yield Lindblad dephasing rates that differ by a factor approaching $2$ as $N\to\infty$ and reaching $2.42$ near the quantum-critical crossover. The discrepancy has a single algebraic origin: parity forces $\langle E_i|\hat{J}_z|E_i\rangle=0$ exactly, eliminating the cross-term that doubles the localised-state rate. Two distinct protection factors are identified: $η_{\rm MF}=(Nm_*)^2/(2G_{01})\approx2.42$, where $m_*$ is the order parameter and $G_{01}=\frac{1}{2}(\langle E_0|\hat{J}_z^2|E_0\rangle+\langle E_1|\hat{J}_z^2|E_1\rangle)$ (advantage over the classical mean-field estimate), and $η_{\rm exact}=(G_{01}+J_{01}^2)/G_{01}\approx1.86$, where $J_{01}=\langle E_0|\hat{J}_z|E_1\rangle$ (exact physical ratio of pointer-state to eigenstate decay rate). In the thermodynamic limit the secular approximation fails, the doublet degenerates, and both rates converge. The three-regime structure is demonstrated in the Lipkin-Meshkov-Glick model via exact diagonalisation, and the algebraic origin of the discrepancy is established via the $\mathbb{Z}_2$ parity of the Lindblad jump operator.

A number field $K$ is called \emph{monogenic} if its ring of integers $\mathbb{Z}_K$ can be expressed as a simple ring extension $\mathbb{Z}[α]$ for some $α\in \mathbb{Z}_K$. A monic irreducible polynomial $f(x)\in\mathbb{Z}[x]$ is said to be monogenic if one of its roots generates both the number field and its ring of integers. In this article, we establish the necessary and sufficient conditions for $[\mathbb{Z}_{K_i}:\mathbb{Z}[α_i]]=1$, where $K_i=\mathbb{Q}(α_i)$ and $α_i$ is a root of the composed polynomial $f_i(x^k+b)$ for $i=1,2$. Here, $f_1(x)=x^n+c\sum_{j=1}^{n}(ax)^{n-j}\in\mathbb{Z}[x]$ and $f_2(x)=x^n+c\sum_{j=1}^{n}a^{j-1}x^{n-j}\in\mathbb{Z}[x]$ are irreducible polynomials of degree $n\ge 3$. In addition, we derive asymptotic estimates for the number of monogenic polynomials in these families under natural assumptions. As an application of our main results, we construct a class of polynomials with non-square-free discriminants. We also analyze the behavior of solutions to certain related differential equations.

We study the problem of deciding universal termination of linear and affine loops over the reals in the bit-model of real computation. We show that both problems are as close to decidable as one can expect them to be: there exist sound partial algorithms that halt on all problem instances whose answer is robust under all sufficiently small perturbations. We further show that in each case the set of non-robust problem instances has Lebesgue measure zero.

The growing relevance of multi-agent systems has drawn increasing focus on communication-efficient filters for collaborative perception to alleviate the system's communication burden. While the event-triggered (ET) mechanism can improve communication efficiency in collaborative state estimation, an inevitable trade-off exists between estimation accuracy and communication cost in ET filters. This paper proposes a fast and accurate ET diffusion-based filter for real-time multi-agent collaborative target tracking, aiming to reduce the system's data transmission without compromise in tracking performance. The proposed filter achieves improved tracking accuracy, reduced data transmission, and accelerated convergence using an error-minimized ET cubature information filter (CIF) for local estimation, and a correlation-aware diffusion strategy for global fusion. The experimental results confirm the scalability of the proposed EDC-CIF algorithm and demonstrate its efficacy in simultaneously reducing estimation error and computation time while significantly enhancing communication efficiency.

We present a class and home problem in graduate transport phenomena and electrochemical engineering that clarifies a common misconception: electroneutrality implies the electric field is constant. Starting with one-dimensional Poisson--Nernst--Planck equations for a silver electroplating cell, students obtain concentration and potential profiles. A companion home problem with added background electrolyte introduces a new dimensionless ratio and admits a closed-form solution. Students conclude that electroneutrality is necessary but not sufficient for a constant electric field.

Agentic AI scientists equipped with domain-specific tools are rapidly entering scientific workflows across disciplines, with especially strong uptake in the life sciences where they can be used for literature synthesis, sequence analysis, and experimental planning support. While these systems accelerate biological research, they also introduce risks for dual-use applications that are not captured by current model-centric safety evaluations. We present evidence that current agentic AI scientists, including Biomni and K-Dense, are willing to assist with dual-use tasks that are blocked by base model safeguards. We also found that in a paired evaluation framework for biology and chemistry prompts involving Weapons of Mass Destruction proxies (WMDP), agentic scaffolding of Biomni increased the benchmark performance relative to the underlying standalone model, producing measurable capability uplift. We believe it is necessary to include additional safeguards in existing models and build future tools from the ground up with agentic vulnerabilities in mind. To systematically categorize broader risks, we introduce BioVeil MATRIX, a defensive taxonomy that maps AI-enabled biosecurity risks using 10 tactical categories (TA01--TA10) and 22 different techniques. We propose to use this taxonomy as a baseline for future AI scientist development and generate specialized benchmarks and protocols for red-teaming these vulnerabilities before public deployment. BioVeil MATRIX can be found at: https://bioveilmatrix.com/

We study hierarchical component selection under severe information constraints. Component quality is not directly observable, each selector observes only the outcome of the chosen pathway, and no explicit evaluation channel crosses module boundaries. We analyse a proportional-redistribution mechanism in which each selector maintains a weight vector over its children and updates that vector from observed outcomes. The sign of a parent's weight change can be read locally as an implicit binary evaluation signal by the selected child, yielding a decentralised evaluation mechanism with no explicit reporting channel. We give a full formal treatment. Proportional redistribution preserves market integrity algebraically. The sign of the weight change propagates without loss through the active path. The single-selector dynamics admit a unique interior equilibrium; for $N{=}2$ the equilibrium is exact and closed-form, while for general $N$ an equi-ratio condition yields an explicit affine equilibrium. Hierarchical composition is informationally clean, with each node's active-round dynamics identical to a standalone instance observed on a thinned clock. All structural results, the equilibrium formula, and the composition theorem are fully proved. Illustrative cases on synthetic hierarchies with up to 32,768 leaves and on three natural-hierarchy datasets confirm the mechanism's operation under constructed and applied conditions.

One of the key choices for numerical models of geophysical fluids is how parametrisations of physical processes interact with the numerical methods that handle the resolved flow, known in the atmospheric community as the dynamical core. As both the dynamical core and parametrisations of physics processes continue to evolve and improve, the issue of physics-dynamics coupling - how these two different parts of the model interact - becomes ever more important. In this paper we use two variations of the moist shallow water equations to develop a simplified framework that can be used to investigate some of the questions associated with physics-dynamics coupling. The shallow water equations act as a simplified dynamical core that is computationally cheap but still retains pertinent features of the atmosphere, and the introduction of moisture means the addition to the model of a physical parametrisation. This study uses 'split-physics' moist thermal shallow water equations which couple moisture to the shallow water equations via a parametrisation, and also develops a new 'integrated-physics' formulation of the moist thermal shallow water equations which re-formulates the model so that all the moist processes are captured in the dynamics. This integrated-physics model thus requires no physics-dynamics coupling and acts as a ground-truth to compare coupling strategies in the split-physics formulation against. We use both models to examine the effect of varying the approach to how physics is coupled to the dynamics in the semi-implicit quasi-Newton timestepping scheme. The results demonstrate the usefulness of the integrated-physics moist shallow water equations and provide insights into how best to deal with physics in a model's timestep.

In this work, we provide a complete description of the scattering matrix elements and electron energy spectrum in one dimensional PT-symmetric hybrid finite systems, using the characteristic determinant approach. We present an analytical formulation of the problem and obtain a closed-form expression for the energy spectrum of the system, consisting of a region of real potential (passive region) surrounded by regions of gain and loss on the left and right, respectively. It has been shown that under certain conditions and a specific ratio between the real and imaginary parts of the complex potentials, it is possible to find analytical expressions for the spectral singularities at which the scattering matrix elements of the hybrid structure tend to infinity at a specific real energy. Within the framework of the same approach, we present a compact analytical expression for the quantization condition that determines the energy spectrum of a model corresponding to the placement of a rigid lattice within a finite-sized box.

The Big-Line-Big-Clique Conjecture of Kara, Por and Wood asserts that, for every fixed $k$ and $\ell$, every sufficiently large finite planar point set contains either $k$ collinear points or $\ell$ pairwise visible points. We prove a quantitative form in two structured regimes and isolate the precise ambient obstruction to the full conjecture. The main result is a deterministic cubic-container theorem. If $A \subset \mathbb{R}^2$ has $n$ points, no $k$ collinear points, and all but $s$ points of $A$ lie on a real cubic, then the cubic-supported part of $A$ has a visible clique cover of size $O_k(s+1)$; in particular $V(A)$ contains a clique of size $Ω_k(n/(s+1))$, unless the cubic is the excluded three-line case containing only $O_k(1)$ points. Combining this with the Green-Tao structure theorem, we obtain that every $n$-point set with no $k$ collinear points and at most $Kn$ ordinary lines contains a visible clique of size $Ω_{k,K}(n)$; more strongly, all but $O_K(1)$ points can be partitioned into $O_{k,K}(1)$ mutually visible sets. We also combine the cubic-container theorem with the Elekes-Szabo theorem on triple lines and cubic curves to prove the Big-Line-Big-Clique conclusion for point sets contained in any fixed irreducible algebraic curve. Finally, we prove a dense-orchard core lemma showing that the absence of a visible $K_\ell$ forces a positive-density subset in which every point lies on linearly many 3-rich lines, and we give a sharp one-blocker example showing why ambient blockers cannot be ignored.

We study the decision version of tensor spectral norm from the viewpoint of real algebraic complexity. For a rationally specified tensor, the tensor spectral threshold problem asks whether its spectral norm exceeds a prescribed rational threshold. Since the feasible domain is compact, attainment itself is trivial; the meaningful question is the threshold decision problem. We prove that this problem is $\exists\mathbb{R}$-hard by giving an explicit polynomial-time reduction from bounded quartic equality feasibility. The reduction first transforms bounded quartic feasibility into homogeneous quadratic sphere feasibility by homogenization, box encoding, and quadratic lifting. It then maps the resulting homogeneous quadratic system to a quartic form whose maximum over the unit sphere separates feasible from infeasible instances. Finally, the quartic form is represented as a symmetric order-four tensor, yielding the desired tensor spectral threshold instance. The result shows that the computational obstruction in tensor spectral norm is not merely non-convex optimization or combinatorial hardness, but real algebraic feasibility itself.

Accurate estimation of the human circadian phase plays an important role in personalized health monitoring, but most existing wearable-based approaches operate retrospectively and require full circadian cycle recordings, leading to high estimation latency and substantial data and computational burden for real-time deployment on edge devices. In this study, we investigated whether circadian phase can be estimated in real time using only short historical windows of wearable data. We propose a low latency framework that estimates instantaneous circadian phase from past observations, with a cosinor-fitted core body temperature rhythm serving as the reference. Data from a free-living field study involving 14 participants were used to systematically evaluate the effects of sensor modality selection, historical window length, and model class under participant-based cross-validation. The results showed that estimation accuracy improves with increasing window length but saturates at approximately 8 hours of history. Tree-based models reached a performance plateau beyond 480 minutes, whereas sequence-based models continued to benefit from longer temporal contexts. When relying solely on light exposure and physical activity, the proposed approach achieved a mean circular mean absolute error (CMAE) of 1.19 h. These findings provide practical guidance for efficient and deployable real-time circadian phase monitoring using wearables.

Gait speed is a vital health indicator for older adults, as changes in gait speed can reflect physiological and functional decline. Ambient sensors offer a promising, privacy-preserving solution for continuous in-home monitoring of gait speed; although it is often limited by methods requiring a home floor plan, which is frequently unfeasible. This paper proposes a novel, floor plan-agnostic method to detect gait speed drifts using only sparse ambient sensors. Our approach identifies informative sensor-to-sensor transitions and analyses fluctuations in their duration. For each sequence a non-parametric statistical test detects changes between a recent period and an initial baseline; and daily test results are aggregated to provide a robust drift detection response. We evaluate our method on a simulated dataset across four different home layouts, showing performance comparable to, and in some cases exceeding, a state-of-the-art baseline that requires floor plan information. This work demonstrates a feasible approach for scalable, cost effective gait drift detection monitoring, providing a foundation for future validation in complex real-world environments.

In a previous manuscript we addressed the possibility of generating a reflection in a region of spacetime inside a null surface under electromagnetic gauge transformations. In this manuscript we will deal with the possibility of full inversions in a region of spacetime inside a null surface under electromagnetic gauge transformations. Since new tetrads whose construction depends on electromagnetic gauge were introduced it has been proved that the physical kinematic nature of spacetime can be altered as proved previously. In this case we will see how to create a field configuration that will reverse the flow of time. We will prove that we can turn a future directed timelike vector into a past directed timelike vector by physical means. Experiments devised in order to achieve time reversal or full inversion have been never discussed before.

Symmetry-plane correlations (SPCs) are key observables sensitive to the medium's transport properties and are driven by participant-plane correlations (PPCs) in the nuclear overlap region. This study explores the possibility of nuclear-geometry-driven SPCs in Oxygen--Oxygen (OO) and Neon--Neon (Ne--Ne) collisions at $\sqrt{s_{\rm NN}}=5.36$ TeV using nuclear geometry simulations based on Nuclear Lattice Effective Field Theory (NLEFT) and Projected Generator Coordinate Method (PGCM) configurations. We investigate $\langle \cos[4(ψ_2 - ψ_4)]\rangle_{\rm GE}$ and $\langle \cos[6(ψ_3 - ψ_6)]\rangle_{\rm GE}$ in OO and Ne--Ne collisions at $\sqrt{s_{\rm NN}}=5.36$ TeV using the A Multi-Phase Transport (AMPT) model. We find that Ne--Ne collisions exhibit larger $\langle \cos[4(ψ_2 - ψ_4)]\rangle_{\rm GE}$ values than OO collisions, whereas $\langle \cos[6(ψ_3 - ψ_6)]\rangle_{\rm GE}$ is larger in OO than in Ne--Ne collisions. This behavior indicates a strongly deformed shape of the $^{20}$Ne nucleus and a tetrahedral structure of the $^{16}$O nucleus. We also explore SPCs for events with tip-tip and body-body collision configurations, which further support these findings.

While decentralized prediction markets like Polymarket have gained significant traction, their market microstructure and high-frequency pricing efficiency remain underexplored. This paper conducts a systematic empirical analysis of algorithmic arbitrage within Polymarket's NBA game markets. By reconstructing continuous market states from over 75 million limit order book snapshots across 173 games, we evaluate the frequency, duration, and profitability of both single-market and combinatorial arbitrage opportunities. Our findings demonstrate profound microstructural efficiency. Single-market anomalies are exceedingly rare, yielding only 7 executable in-game episodes that persist for a median duration of just 3.6 seconds. Combinatorial inefficiencies are more frequent, producing 290 active episodes overwhelmingly concentrated in the final minutes of live play. While combinatorial execution yields a statistically meaningful median return of 101 basis points, we find that the theoretical "Middle" jackpot is never empirically realized. Furthermore, execution is severely bottlenecked by shallow order book depth, with 76.9\% of combinatorial opportunities constrained to an average executable size of just 14.8 shares. Ultimately, while executable mispricings exist, they are structurally bounded by liquidity, confining risk-free extraction strictly to the retail scale.

Form-finding of unilateral membrane structures is commonly addressed by solving equilibrium equations with Finite Element Methods (FEMs). This paper investigates Physics-Informed Neural Networks (PINNs) as an alternative, where the equilibrium equation is enforced by minimizing its residual at collocation points during neural-network training rather than by solving a mesh-based discretized system. This approach is well suited to form-finding problems based on Membrane Equilibrium Analysis (MEA), in which the unknown membrane surface is governed by a second-order elliptic Partial Differential Equation (PDE) with Dirichlet boundary conditions. Two PINN formulations are proposed and compared: a soft-Boundary Condition (soft-BC) approach, where the boundary conditions are imposed through a penalty term, and a hard-BC approach, where they are satisfied exactly by construction through distance and lift functions. The methods are assessed on three case studies with different geometrical complexity, including compression-only and tension-only stress states, and combined self-weight, concentrated vertical loads, and horizontal actions. Both formulations produce membrane surfaces in close agreement with solutions obtained using an FEM-based PDE solver. The hard-BC formulation gives smaller errors and a smoother residual distribution, especially near the boundary, showing that exact enforcement of the Dirichlet conditions improves overall accuracy. The soft-BC formulation still provides structurally meaningful solutions and remains attractive when simpler implementation is preferred and limited relaxation of the boundary data is acceptable. Overall, the results show that PINNs are a viable alternative for MEA-based form-finding.

We study cash-flow forecasting for derivatives used in liquidity management and clarify its relation to risk-neutral valuation and replication. While it is well known that expectations under different measures (e.g., $\mathbb{P}$ vs. $\mathbb{Q}$) can yield different undiscounted cash-flows, further inconsistencies arise when payment times are stochastic. We show that using discounting sensitivities (funding-curve hedge ratios) instead of "expected cash-flows" aligns forecasting with the self-financing replication strategy and avoids measure-mixing/aggregation issues. We then illustrate how a standard valuation model delivers pathwise funding requirements and propose a simple liquidity valuation adjustment to capture settlement lags and related timing frictions. The note provides implementation hints (American Monte Carlo with adjoint differentiation) and clarifies when "expected cash-flows" are informative and when sensitivities should be used instead.

With an increasing number of prospective geological CO$_2$ storage projects and potential pressure communication between different projects, risk sharing under uncertain geological conditions are relevant to many project operators. In this work, the project operators are modeled as agents in a stochastic cooperative game. The agents can have different risk attitudes, here defined as being willing to accept more or less uncertainty in the predicted storage of CO$_2$. This uncertainty stems from lack of knowledge of geological parameters as well as unknown future actions of competing agents, and the corresponding probability distributions need to be estimated by numerical simulation. The agents can choose to share commercial risk if collaboration is preferable to a baseline scenario of individual injection. If their operations affect each other by means of, e.g., pressure communication, there may be no unique natural definition of a baseline scenario. As a remedy, we suggest belief distributions that combine uncertainty in physical data with maximum entropy prior distributions over the sets of viable injection actions. For a realistic storage site, exemplified by the Utsira Formation in the North Sea, we present numerical results for both cases of pressure competition, and no hydraulic connections between different project operations. It is shown that risk averse agents benefit from collaboration when there is no pressure communication or other interference between agents. It is also demonstrated that pressure communication leads to large variability in the feasible injection rates, but the resulting belief distributions are nevertheless informative and useful for decision making about collaboration.

This paper proposes a simple and parsimonious discrete-time simulation model to describe the endogenous formation and periodic collapse of financial bubbles. While existing literature has extensively explored the statistical properties of locally explosive bubble dynamics, capturing the micro-level interplay of investor herd behavior and panic selling within a unified framework remains a challenge. Our model addresses this by introducing a cubic function of market momentum to determine the balance of trading directions. This mechanism drives both trend-following behavior during the bubble phase and sudden market crashes when the momentum exceeds a critical threshold. Furthermore, inspired by the self-exciting nature of the Hawkes process, the model endogenizes``market frenzy" by linking trading frequency directly to the accumulated momentum. Simulation results demonstrate that this minimal setup successfully replicates the complex, nonlinear dynamics of bubbles, including simultaneous surges in liquidity and price, followed by dramatic crashes.

Optical imaging technologies are central to discovery in the life and physical sciences, yet their impact depends on how readily they can be built, adapted, and sustained across laboratories. Digital fabrication, including desktop 3D printing, offers new ways to engineer imaging instruments by simplifying assembly, lowering replication barriers, and enabling modular integration and local refinement. Here we examine, using open microscopy as a transparent case, how digitally fabricated components support adaptable, research-grade optical systems while enabling faster innovation cycles and distributed refinement. We outline practical design guidelines and discuss emerging developments that may further advance accessible, high-performance imaging.