Automated market makers (AMMs) quote prices from pool state rather than from a limit order book. AMM pools often stay close to a reference price because arbitrageurs correct profitable mispricing. A large part of decentralized finance therefore relies on a simple economic premise: once the AMM price drifts away from the reference price, arbitrage incentives push it back. This paper studies when that premise is strong enough to guarantee block-scale stability. We model the gap between the reference price and the AMM price as a stochastic tracking error, treat arbitrage as the corrective input, and place blockchain execution inside the loop through fees, discrete blocks, transaction ordering, delays, and transaction failure. The detailed execution layer is reduced to the total successful correction confirmed in each block. Under a block-level correction condition, we prove geometric ergodicity of the tracking error and obtain explicit one-step bounds that connect tracking quality to liquidity and execution quality. We also show in a constant-product example how fees, fixed execution costs, and local liquidity map into the no-trade band and the optimal corrective trade. Finally, we build empirical proxies for the theorem quantities from realized block data and use them to organize reduced and mechanism-focused simulations whose comparative statics are consistent with the theory. The contribution is to turn a basic economic intuition behind decentralized finance into a quantitative stability statement together with a tractable calibration interface.

We introduce and study the multi-agent stochastic shortest path (MSSP) problem, in which $k$ agents strive to reach a target state, aiming to minimize the expected time to reach the target by any agent. We analyze the computational and strategy-complexity of the problem in both autonomous and coordinated settings, and we design efficient strategy-synthesis algorithms. The algorithms are experimentally evaluated on instances of increasing size against natural baselines.

The widespread adoption of mixed-precision quantization in large language models (LLMs) has created demand for hardware that can efficiently perform multiply-accumulate (MAC) operations across mixed datatypes and switch datatypes at runtime. Existing FPGA-based MAC solutions fall short due to limitations in fixed-datatype design, inefficient spatial or temporal resource sharing, and poor support for mixed-precision execution. These limitations collectively lead to under-utilization of DSP resources, limiting achievable parallelism and throughput. In this work, we present XtraMAC, a novel MAC architecture that unifies integer, floating-point, and mixed-precision operations within a single, datatype-adaptive microarchitecture. XtraMAC decomposes all supported MAC formats into a shared integer mantissa product with lightweight sign and exponent handling, enabling dynamic operand packing and efficient DSP resource sharing with constant latency and initiation interval of one across all datatypes. Evaluated on an AMD Xilinx U55c FPGA, XtraMAC achieves 1.4-2.0x higher compute density, reduces per-operation LUT, FF, and DSP consumption by 27-51%, and delivers up to 1.9x greater energy efficiency and 1.2x speedup on representative mixed-precision LLM workloads. The implementation of XtraMAC is open-sourced at https://github.com/Xtra-Computing/XtraMAC.

Optimizing Reconfigurable Intelligent Surfaces (RIS) is a high-dimensional combinatorial challenge. Current quantum algorithms often simplify this problem by ignoring physical constraints like mutual coupling, which significantly degrades real-world performance. Rather than targeting a fully realistic RIS description, we embed progressively more physics-informed models of mutual coupling into Quadratic Unconstrained Binary Optimization (QUBO) formulations. We evaluate four Ising interaction models ($J_{ij}$) for the Quantum Approximate Optimization Algorithm (QAOA), ranging from idealized phase-only to fully dense physical models. Analyzing a $5 \times 5$ grid, our results expose a critical trade-off between spatial pointing accuracy and quantum hardware feasibility. While complete global coupling maximizes beamforming precision, dense Hamiltonians introduce prohibitive routing overhead and complicate convergence on near-term processors. Ultimately, we demonstrate that while physics-informed quantum optimization is mathematically viable, sparse, distance-penalized models remain a necessary compromise for execution on current noisy intermediate-scale quantum (NISQ) devices.

Understanding the magnetotransport behaviors in topological systems remains alluring, as a lot of intrinsic information could be extracted, e.g., the band structures, Berry phase, Fermi surface, carrier density, and so on. Motivated by the recent magnetotransport developments in nodal-line semimetal, EuGa4, in this paper, we will study the magnetotransport properties of the system, focusing on the quantum oscillations and nonsaturating magnetoresistivity (MR). Firstly, we analyze the chemical potential and magnetoconductivity oscillations with the magnetic field and reveal that there exist two distinct oscillation frequencies, which are caused by the characteristic torus Fermi surface and can be regarded as an important experimental signature of nodal-line semimetals. Then we calculate the MR and find that although the MR is nonsaturating with the magnetic field in the low-energy region, the MR ratio is much smaller than that reported in the experiment.

Let $X$ be a projective algebraic $d$-variety endowed with isolated determinantal singularities, and let $ω$ be a $1$-form on $X$ exhibiting a finite number of singularities (in the stratified sense). Under some technical conditions, we use two generalizations of Poincaré-Hopf index with the goal of proving a Poincaré-Hopf Type Theorem for $X$.

This work conceives a Ring-Bayes channel learning framework that unifies Bayesian learning with near-field channel estimation in millimeter-wave (mmWave) hybrid MIMO systems. As the number of antennas scales up, users increasingly fall within the near-field region, rendering the conventional planar-wave assumption invalid. Moreover, the widely studied uniform linear arrays (ULAs) at the base station are impractical for large-scale deployment, whereas uniform circular arrays (UCAs) achieve superior beamforming gain and spatial directivity with the same antenna aperture. To exploit these advantages, we design a near-field concentric-ring codebook that captures channel features jointly in angular and distance domains. Leveraging this structure, the proposed Ring-Bayes framework enables highly accurate recovery of UCA near-field channels. Extensive simulations confirm that our approach delivers substantial improvements over existing methods, establishing Ring-Bayes as a powerful and scalable solution for next-generation mmWave communications.

We consider standing waves of the nonlinear Schrödinger equation $i\partial_t u = -Δ_αu + |u|^{p-1}u$ in the defocusing case in dimensions $N=2$ and $N=3$. Here, $-Δ_α$ denotes the Laplacian with a point interaction. This operator is bounded from below by a negative constant; consequently, unlike in the free case, the associated energy functional admits non-trivial minimizers. We establish existence and uniqueness of standing waves, and prove further qualitative properties, including radial symmetry, positivity, and stability. Moreover, we build an appropriate functional space for the zero-mass case and establish sharp decay estimates in this case.

Recently, there has been growing interest in unconventional computing as an approach for solving NP-hard problems, by developing dedicated hardware to find solutions more efficiently than conventional CPUs. In many of these approaches, however, certain problem geometries must be transformed into forms that are more amenable to the available hardware topology through techniques such as embedding, sparsification, and quadratisation, leading to a deterioration in solution quality. A probabilistic computing architecture based on high speed photonic quantum random number generators was recently proposed which utilises virtual hardware connections (Aboushelbaya et al., 2025), circumventing the necessity for such procedures. Here, we discuss the applicability of virtually connected hardware for running heuristic solving methods to solve a selection of problems, which due to their geometry, would suffer from topological hardware restrictions. We also employ greedy graph colouring algorithms for hardware parallelisation, allowing favourable scaling for desirable solution qualities. To emphasise the difficulty in solving these problems on physically connected hardware, we demonstrate the increase in problem size that would occur with quadratisation or sparsification. Using simulations to emulate hardware, we predict that a photonic probabilistic computer would outperform the time to solution recently reported for digital annealing units, on the ground state approximation of Erdos-Renyi graph spin-glasses, by orders of magnitude.

We present a study on cubic Euler sums of degree four, five and six, where three different types of denominators $1/k^n$, $1/((2k-1)^n)$ and $1/(k(2k-1))$ will be considered We demonstrate that for all three orders the complete variety of corresponding nonlinear Euler sums belonging to the eight different families can be explicitly calculated in terms of zeta values and polylogarithmic values $Li_4(1/2)$, $Li_5(1/2)$, $Li_6(1/2)$, $Li_6(-1/2)$ and $Li_6(-1/8)$.

We derive novel guaranteed lower bounds for eigenvalues of the Euler-Bernoulli beam with variable bending stiffness. While the standard finite element Rayleigh-Ritz method automatically yields upper bounds, we obtain lower bounds by employing interpolation error estimates with the explicitly known value of the associated constant. This approach is especially efficient and easy to apply for piecewise constant bending stiffness. For general variable material parameters, we obtain guaranteed lower bounds through an auxiliary beam-bending problem. The first eigenvalue is of primary interest in applications because it represents the critical load that causes buckling of the beam. Our method is, however, suitable also for the higher buckling modes. In addition, it can be applied to the physically more relevant nonlinear Gao beam model with piecewise constant bending stiffness, which has the same first eigenvalue as the classical Euler--Bernoulli beam. The presented numerical experiments illustrate the performance of the proposed eigenvalue bounds, demonstrating their convergence rates.

Caching and reusing intermediate features across consecutive frames is a common technique to reduce redundant computation and transmission for edge-cloud video analytics in mobile edge computation. Existing methods manage the cache in a fixed or globally shifted coordinate system, treating it as an indivisible whole. Under the non-uniform motion patterns of mobile scenes, this whole-scene granularity invalidates large portions of the cache even when most content has merely shifted spatially, wasting computation and bandwidth. The root cause is a granularity mismatch: the cache is managed per scene, yet motion varies per region. In this paper, we present FluxShard, a motion-aware edge-cloud video analytics system that uses codec-level block motion vectors (MVs) to manage feature cache reuse and recomputation at the granularity of individual motion regions. By re-indexing cached features along per-block MVs, FluxShard separates spatial displacement from content changes, recovering reusable content that whole-scene methods would otherwise discard. To ensure correct reuse under heterogeneous motion, the Receptive Field Alignment Principle (RFAP) identifies, from the input-level MV field alone, the positions that must be recomputed due to inconsistent spatial composition within receptive fields. To maintain cache coherence across frames, MV-guided cache remapping warps the entire feature cache to the current coordinate system each frame, sustaining a high reuse ratio over time. A profiling-driven dispatcher routes the remaining sparse workload between edge and cloud for lower latency. Evaluation across multiple vision tasks, dynamic video benchmarks, and network conditions shows that FluxShard reduces latency by 32.6-83.8% and energy by 14.9-64.0% over all baselines under the prescribed accuracy budget.

This paper presents a topology optimization method for designing two-fluid heat exchangers under turbulent conditions using a Darcy flow-based low-fidelity (LF) model. The LF model is calibrated against a high-fidelity (HF) model based on the Reynolds-averaged Navier-Stokes (RANS) equations to increase the accuracy of predictions for fluid flow and heat transfer characteristics. Since the discrepancies between the LF and HF models can be significant, particularly for pressure drops, a multifidelity topology optimization framework is adopted to leverage the strengths of both models. Using the calibrated LF model, we perform topology optimization for various inlet velocities in the boundary conditions and trade-off parameters in the objective function to obtain diverse optimized designs. The optimized designs are then evaluated using the HF model to assess their performance with higher accuracy. The results demonstrate that the optimized designs significantly improve overall heat transfer coefficients while maintaining manageable pressure drops, achieving up to a 22% higher performance evaluation criterion (PEC) compared to a reference design enhanced by conventional twisted tape insertion. The improvements are attributed to the optimized configurations that promote enhanced fluid mixing and increased surface area for heat exchange, yet maintain streamlined flow paths to minimize pressure losses. Overall, the proposed topology optimization method using the Darcy flow-based LF model proves effective in designing high-performance double pipe heat exchangers, showcasing the potential of the multifidelity approach in overcoming the challenges of optimizing heat exchangers under turbulent flow conditions.

Let $(r_k)$ be an increasing sequence and $(w_k)$ a positive sequence. We study the following question: is it true that for every sequence $(a_k)$ satisfying $\sum_{k=0}^\infty |a_k|^2 w_k^2 < \infty$ there exists a function $f\in C(\mathbb{T})$ such that $\hat{f}(2^k) = a_k$ and $\hat{f}(n) = 0$ for $n\notin \cup_k [2^k-r_k,2^k+r_k]$? We show that this is possible if and only if $\sup_{k\in\mathbb{N}}\sum_{n=[\log_2 r_k]}^k w_k^{-2} < \infty$.

We show that first-order formulae are concise in acylindrically hyperbolic groups and certain extensions thereof. We study further classes of groups, including Burnside groups, icc groups, groups with the `Big Powers' condition, torus knot groups and more, and prove conciseness for wide classes of formulae. We also explore properties of definable sets in these groups, such as their finiteness, depending on the type of formula considered.

This paper discusses the computational resolution and presents numerical results for solving affine combinations of Heaviside composite optimization problems (abbreviated as A-HSCOPs) by a progressive integer programming (abbreviated as PIP) method. The characteristics of these problems are that the Heaviside functions, which appear in the objective and define the constraints, are discontinuous, and their mixed-signed combinations result in the overall objective lacking the matching semicontinuity needed for the optimization and in the feasible set being not necessarily closed. Added to these challenging properties is the nondifferentiability of the inner functions in the composition. In this paper, we propose resolutions to all these challenges by first an approximation to remedy the lack of semicontinuity in the objective and closedness in the constraints, followed by a progressive integer programming approach with successive decomposition to handle the intrinsically discrete nature of the Heaviside function. Convergence to the local optimizers of the given Heaviside optimization problem is established. The effectiveness of the overall solution strategy is supported by extensive computational experiments on the score-based and tree-based multiclass classification problems with precision constraints.

We introduce the geometric mean and the parallel sum of completely positive (CP) maps on von Neumann algebras, based on the Pusz--Woronowicz theory of positive sesquilinear forms. We provide a concrete characterization via a block matrix positivity condition and establish their fundamental properties, including the AM--GM--HM inequality with respect to the CP order. In finite-dimensional settings, our construction is compatible with the Choi--Jamiolkowski correspondence, under which the geometric mean of CP maps corresponds to the Kubo--Ando geometric mean of their Choi matrices. This yields a natural operator-theoretic framework for interpolating quantum channels. As an application, we obtain index-type inequalities for conditional expectations in subfactor theory. Finally, we establish a Lebesgue-type decomposition of CP maps via a parallel sum construction, thereby providing a unified framework that simultaneously generalizes Ando's decomposition of bounded positive operators and Kosaki's decomposition of normal positive functionals on von Neumann algebras.

In this paper, we conduct a detailed investigation on the effect of independent component (IC)-based noise rejection methods in neural network classifier-based decoding of electroencephalography (EEG) data in different task datasets. We apply a pipeline matrix of two popular different independent component (IC) decomposition methods (Infomax and Adaptive Mixture Independent Component Analysis (AMICA)) with three different component rejection strategies (none, ICLabel, and multiple artifact rejection algorithm [MARA]) on three different EEG datasets (motor imagery, long-term memory formation, and visual memory). We cross-validate processed data from each pipeline with three architectures commonly used for EEG classification (two convolutional neural networks and one long short-term memory-based model. We compare decoding performances on within-participant and within-dataset levels.Our results show that the benefit from using IC-based noise rejection for decoding analyses is at best minor, as component-rejected data did not show consistently better performance than data without rejections; especially given the significant computational resources required for independent component analysis (ICA) computations.

We propose a unified derivative-free proximal Newton-type algorithm framework for solving composite optimization problems formulated as the sum of a black-box function and a known regularization term. We establish the iteration and oracle complexity bounds for the algorithm to attain an $ε$-optimal solution under both nonconvex and strongly convex settings. We also establish its local R-superlinear convergence based on the Dennis--Moré condition, and theoretically address an open problem by showing that the BFGS scheme is more compatible with finite-difference gradient estimators than with smoothing-based ones. Numerical experiments are further presented to demonstrate the efficiency of the proposed method.

As a sequel to [2] and Theorem C of [3], this paper shows that for $Sp(p,q)$, the spin norm strictly increases along any Vogan pencil once it goes beyond the unitarily small convex hull.

Accurate prediction of adiabatic $0$-$0$ excited-state energies is crucial for modeling molecular photophysical processes. Here, we benchmark computational strategies for evaluating excited-state energies and singlet-triplet gaps obtained using different geometry-optimization strategies, including time-dependent density functional theory (TDDFT), spin-unrestricted Kohn-Sham (UKS) DFT for triplet states (${\rm T}_1$), and state-specific orbital-optimized UKS (ssUKS) DFT for singlet excited states (${\rm S}_1$). Zero-point vibrational energy corrections are evaluated consistently at the optimized geometries and combined with ADC(2) excitation energies for comparison with experimental anion photoelectron spectroscopy data for a representative set of molecules. Among the protocols considered, adiabatic $0$-$0$ energies evaluated at TDDFT-optimized ${\rm S}_1$ and ${\rm T}_1$ geometries show the best agreement with experiment, with a mean absolute error below 0.1 eV. Replacing these geometries with UKS-optimized ${\rm T}_1$ and ssUKS-optimized ${\rm S}_1$ structures yields comparable accuracy. Vertical excitation energies are substantially more sensitive to the choice of geometry than the corresponding ${\rm S}_1$-${\rm T}_1$ gaps, which are comparatively more robust because of partial error cancellation. As a larger case study, we examine rubrene and find that UKS/ssUKS-based geometries remain useful for evaluating singlet-fission energetics. Overall, UKS/ssUKS-based workflows provide an efficient and accurate route to excited-state geometry optimization and to the evaluation of adiabatic $0$-$0$ energies for states with dominant single-determinant character.

This paper presents a homogenized topology optimization (TO) method for spatially optimizing cell-size distribution of triply-periodic minimal surface (TPMS) structures, with high accuracy in the optimized structural response after de-homogenization. To achieve this, we introduce a novel de-homogenization technique that directly minimizes the difference between the wavenumbers obtained from the target and actual size distributions. This minimization problem is efficiently solved as a typical Poisson's equation utilizing the discrete cosine transform. We first verify the proposed de-homogenization method through numerical examples, showcasing its capability in significantly reducing the known distortion of the de-homogenized TPMS structures from the conventional periodic modulation (PM) method. Then, we apply the proposed method to a stiffness maximization problem, to demonstrate its effectiveness in improving the structural response compared to the PM method. The proposed method successfully reduced the distortion of the de-homogenized structures compared to the PM method, leading to 0.8% difference in the strain energy compared to the homogenized model, as opposed to 63.6% difference in the PM method. The optimized structure from the proposed method shows a significant improvement in the strain energy by 50.1% compared to the uniform case in the FE analysis on the de-homogenized models, while the PM method results in a significant decrease of 45.8%. The experimental validation shows that the effective stiffness of the optimized structure from the proposed method is 54.2% higher than that of the uniform case, while the PM method results in a significant decrease by 77.3%. These results exhibit the proposed method effectively increases the accuracy of the de-homogenization, thereby maximizing the potential of the homogenized TO for the spatial cell-size optimization of TPMS structures.

This paper studies the 1D stochastic Allen--Cahn equation on a bounded domain driven by localized white noise. We prove that the associated Markov process admits a unique invariant measure and is exponential mixing. The main challenge lies in the interaction between localized nature of the noise and non-trivial global dynamics of the system. To overcome this, our approach relies on two ingredients from PDE control theory: stabilization for the linearized system and global steady-state controllability for the nonlinear equation. The stabilization result is derived using the weak observability and Fenchel--Rockafellar duality, while the global controllability relies on quasi-static deformations combined with global dynamics.

The question regarding the location of Banach spaces inside their biduals has been investigated and answered reasonably satisfactorily in the linear theory of Banach spaces. Thus, for instance, whereas it is known that a dual Banach space is complemented inside its bidual, the space of all null sequences is not! However, the latter space is a Lipschitz retract of its bidual. In his famous paper of 1964, Lindenstrauss asked if every Banach space is a Lipschitz retract of its bidual. In this short note, we show how to relate the Lindenstrauss problem (LP) to certain other important and well-known questions that remain open in the Lipschitz theory of Banach spaces and how these latter questions may be settled in the affirmative under the assumption of (LP) having a positive solution.

Recent reports suggest that chemical reaction rates can change when reactants are placed inside an optical cavity. These effects have been attributed to the hybridization of molecular vibrational modes with cavity modes into polaritons, but the underlying mechanism remains debated. Recently, attempts to reproduce the key experiments have sometimes failed, which poses also ambiguity and impedes the determination of the possible mechanism. Without a reliable theoretical framework, polaritonic chemistry -- which seeks to use optical resonators as catalysts to control reactions -- has reached a pivotal stage. Standardized protocols for reproducible cavity experiments are therefore urgently needed. Here, we identify pitfalls in approaches that monitor reaction progress with UV/Vis spectroscopy. Using the Transfer Matrix Method, we analyze a model pseudo-first-order reaction and assess how transient cavity thickness variations, cavity inhomogeneity, and fitting protocols influence the extracted rate constant. We find that changes in cavity thickness upon reactant introduction can strongly distort apparent kinetics when monitoring at a single wavelength, an artifact that can be mitigated by spectral smoothing. Additionally, we demonstrate that, unlike in many previous studies, the asymptotic extinction should be treated as a fitting parameter rather than fixed to the final experimental value. By identifying these pitfalls, our work lays the foundation for more robust analyses and reliable measurements in polaritonic chemistry.

We present a perturbative formulation of quantum gravity for asymptotically flat vacuum spacetimes based on the Null Surface Formulation (NSF), in which the expansion is ultraviolet-finite term by term up to the orders computed, without the need for renormalization. The outgoing Bondi shear operators are constructed explicitly up to fourth order, with interaction kernels determined recursively from on-shell gravitational data at null infinity. Ultraviolet finiteness at each order follows from the on-shell structure of the construction and the restriction of all integrations to the compact celestial sphere, eliminating off-shell propagators. The map between the in and out states admits a perturbative construction, and unitarity is verified explicitly up to fourth order. The outgoing operators satisfy the same commutation relations as the incoming ones, indicating that the transformation is canonical and consistent with the unitary implementation. Collinear configurations give rise to infrared singularities, as expected in massless quantum field theories, but do not affect the ultraviolet behavior established here. In coherent states, the expectation value of the shear reproduces the known finite classical graviton scattering at lowest nontrivial order. These results provide a perturbative framework for quantum gravity with improved ultraviolet behavior relative to the covariant approach.

Symbolic library -- or Koopman dictionary -- selection is a fundamental challenge in data-driven dynamical systems. Extended Dynamic Mode Decomposition (EDMD), Sparse Identification of Nonlinear Dynamics (SINDy), and Kolmogorov--Arnold Networks for Dynamics (KANDy) all require the practitioner to commit to a function library at training time; Deep-Koopman Operators avoid this commitment but produce uninterpretable latent observables. We propose Deep-Koopman-KANDy, a structured approach to post-hoc symbolic dictionary readout that combines Deep-Koopman modeling with Kolmogorov-Arnold Networks for Dynamics (KANDy). The encoder and decoder of a Deep-Koopman Operator are replaced with two-layer Kolmogorov--Arnold Networks (KANs), and a level-set construction together with a chain-rule gradient identity exposes the compositional structure of the learned observables in a basis chosen \emph{after} training. We evaluate the method on the Lorenz system, the Chirikov standard map, the Ikeda map, and the Arnold cat map. On Lorenz it recovers the target dictionary $\{x,y,z,xy,xz\}$ with perfect recall and Jaccard score $0.79\pm0.06$; on the standard map it recovers a low-order Fourier basis matching the analytical structure; on Ikeda -- which has no sparse polynomial representation -- a misspecified polynomial readout still recovers the correct foliation coordinate $g\approx x^2+y^2$ together with a nontrivial outer function; and on the Arnold cat map -- used as a negative control because finite-dimensional Koopman closure is provably impossible -- the method fails to find a sparse closure, as expected.

Optical spin-orbit coupling provides a promising, fabrication-free route for developing ultra-compact optical vortex generators. However, the conversion efficiency has been theoretically limited to 0.5. Here, we demonstrate enhanced vortex generation efficiency by employing a Bessel beam as the input and propagating it through van der Waals (vdW) crystals. The large birefringence of vdW crystals and the single transverse wave vector of a Bessel beam allow a near unity spin-orbit conversion efficiency and a topological charge transition of $\ell \rightarrow \ell + 2$. Through combined analytical and experimental investigations, we demonstrate a conversion efficiency of up to 0.82 in hexagonal boron nitride (hBN) crystals with a thickness of $27.4\,μ\mathrm{m}$. The higher efficiency of Bessel input beams over Gaussian beams is attributed to their distinct transverse wave vector distribution of constituent plane wave components. Furthermore, we demonstrate the dependence of conversion efficiency on the numerical aperture (NA) of the objective lens, which is in good alignment with theoretical predictions. These demonstrations provide a fabrication-free route to highly efficient optical vortex generation via microscale vdW materials platforms.

Earlier, there were defined two generalized (``motivic'') versions of the Poincaré series of a collection of plane valuations on the algebra ${\mathcal O}_{{\mathbb C}^2,0}$ of germs of holomorphic functions in two variables. One of them was defined as an integral with respect to the generalized Euler characteristic over the projectivization of the extended semigroup of the collection. One has a natural version of it for valuations on the algebra ${\mathcal E}_{{\mathbb K}^2,0}$ of germs of holomorphic functions in two variables whose Taylor coefficients are from a fixed subfield ${\mathbb K}$ of the field ${\mathbb C}$ of complex numbers. In this setting the usual Poincaré series were computed for one plane curve or divisorial valuation on ${\mathcal E}_{{\mathbb K}^2,0}$. We give equations for the corresponding generalized Poincaré series.

The integration of massive offshore wind into hybrid AC-HVDC grids demands robust DC voltage regulation, yet conventional fixed-gain droop controllers struggle under severe stochastic volatility. This paper bridges the gap between system-level economic dispatch and converter-level control by proposing a novel Stochastic Optimal Power Flow (SOPF)-based adaptive droop framework. Rather than relying on heuristic or reactive tuning, wind forecast uncertainty is modeled using a zone-wise Beta distribution that accurately captures the heteroscedastic nature of wind errors across low, mid, and high power regimes. By leveraging Polynomial Chaos Expansion (PCE) within a chance-constrained SOPF, the system's stochastic states are formulated analytically. Crucially, the optimal adaptive droop gain is extracted directly from the first-order PCE coefficients via a Jacobian-free sensitivity analysis, embedding statistical voltage-security guarantees directly into the local converter control. Validation on a 4-terminal AC-HVDC system demonstrates that scenario-adaptive gains significantly outperform standard fixed-coefficient approaches, effectively minimizing active-power tracking errors during extreme wind disturbances.