An elastic space curve is a critical point of the bending energy subject to appropriate constraints. An analytic representation, equivalent to the spherical pendulum equation, leads to an 11-parameter description of the space of 3D elastic curve segments. We give a numerically stable method for recovering the 11 parameters from a given elastic curve segment. Using this, we give a fast and stable method to approximate an arbitrary space curve segment by a 3D elastica. Applications include interactive design with exact elastic curves and CAD surface rationalization for robotic hot-blade cutting.

WepresentatwolevelSchwarzmethodasanalternativetoAlgebraicMultigridmethod(AMG) used as the last level (coarse) solver of the p-multigrid pMG preconditioner for pressure Poission equation resulting from Spectral/Finite element descretization of incompressible Navier-Stokes eqaution. Proposed Schwarz method consits of a local problem in the original pMG coarse space and a global coarse problem. Main contribution of the paper is a novel, structured and a non-nested coarse space for the global coarse problem. Structured nature of the proposed global coarse space enable communication-free interpolation between the original p-multgrid coarse space and the global coarse problem. We demonstrate the effectiveness of the proposed method compared to the state of the art AMG solver BoomerAMG by a series of experiments performed using Nek5000/RS, a suite of highly scalable incompressible Navier-Stokes solvers, on Summit/Frontier supercomputers at Oak Ridge Leadership Computing Facility.

Nonlinear geotechnical analysis often involves complex geometries, staged construction, local failure, and mesh-dependent stress and plastic strain responses. This study develops a polygonal cell-based smoothed finite element method (CS-FEM) for nonlinear geotechnical analysis and implements it in ABAQUS through the user element subroutine. The proposed method combines Wachspress interpolation with cell-based strain smoothing, in which the smoothed strain--displacement matrix is evaluated by boundary integration over polygonal smoothing subcells. This formulation avoids direct calculation of shape-function derivatives inside polygonal elements and enables standard polygonal meshes and hybrid quadtree meshes with hanging nodes to be handled in a unified framework. Nonlinear geomaterial behavior is incorporated through incremental elasto-plastic constitutive updates, including the Mohr--Coulomb model and the Duncan--Chang model. Several benchmark and engineering examples, including a perforated plate, strip footing, core rockfill dam, tunnel excavation, and slope stability problems, are presented for verification. The results show that the proposed method accurately predicts displacement, stress, plastic strain, bearing capacity, and factor of safety, while providing improved mesh flexibility and computational efficiency for nonlinear geotechnical analysis.

The extended complex plane is a fundamental object in complex analysis, hyperbolic geometry, and mathematical physics. Its geometry is governed by Möbius transformations, with the cross ratio serving as a central invariant. We present a formalization of these concepts in the Lean4 theorem prover. The extended complex plane is represented using Mathlib's Option type over $\mathbb{C}$, where the additional element represents the point at infinity. On this foundation, we define Möbius transformations, their action on the extended complex plane, and the cross ratio. We formalize several basic properties of Möbius transformations, including their group structure, and identify them with a projective general linear group. We also prove the uniqueness of a Möbius transformation mapping any three distinct points to any other three distinct points, and the invariance of the cross ratio. All proofs are machine-checked in Lean 4. The complete development comprises approximately 6,000 lines of Lean code, including about 40 definitions and 150 lemmas and theorems. This work provides a verified foundation for future formalizations of conformal geometry, hyperbolic models, modular forms, and applications in mathematical physics.

In this article, we present a robust $Q$-learning algorithm for discrete-time mean-field control problems under Wasserstein uncertainty in the common noise law. The algorithm combines a quantization-and-projection scheme with a Wasserstein dual reformulation on the common-noise space. We establish its convergence together with finite-time iteration bounds for both synchronous and asynchronous learning schemes. Numerical experiments on systemic risk and epidemic models compare the asynchronous implementation with an idealized Bellman iteration, illustrate the robustness-performance tradeoff under common-noise misspecification, and report the observed convergence behavior of the asynchronous $Q$-learning algorithm.

In this article, we introduce a novel data-dependent Shepard interpolation method inspired by the adaptive strategies proposed in [2]. In this case, as Shepard interpolation does not produce oscillations, our approach has the core objective of reducing the smearing near jump discontinuities in the data in one and two dimensions. While the original work in [2] focuses in on Radial Basis Function (RBF) interpolation, we extend these ideas to the Shepard framework by incorporating a data-dependent adaptation mechanism. Specifically, we modify the classical Shepard interpolation by adaptively adjusting the influence weights based on local smoothness indicators that modify the shape parameter. These indicators, similar to those used in [2], are designed to detect discontinuities: for grid-based data, we use squared undivided second-order differences, and for scattered data, we employ squared least-squares approximations of the Laplacian scaled by the square of the mean local separation of stencil points. The resulting data-dependent weighting scheme forces the kernels close to a discontinuity to behave like a local delta function, effectively reducing the smearing of the discontinuities introduced by the classical Shepard approach. We establish the theoretical foundation of the method, including the properties of the new interpolation and we theoretically prove that the reduction of the smearing of discontinuities is possible. Numerical experiments in one and two dimensions confirm that the proposed data-dependent Shepard interpolation significantly reduces the smearing of jump discontinuities while maintaining high accuracy in smooth regions.

We propose a conservative adaptive rank method for the 1D1V Wigner-Poisson system. The method targets a central challenge in deterministic quantum kinetic simulations: reducing the cost of phase-space evolution while preserving the macroscopic invariants needed for physical fidelity. The scheme combines a sampling-based adaptive rank Wigner-Poisson update [7] with a conservative macroscopic correction. A conservative density-momentum solve provides local macroscopic updates, a Fermi-Dirac-type reconstruction transfers them to the kinetic solution, and a global quadratic moment correction enforces the discrete total energy constraint at the kinetic level. Unlike Maxwell-Boltzmann-type corrections commonly used in classical kinetic settings, the reconstruction uses a Fermi-Dirac-type form motivated by the model's quantum-statistical structure. The corrected state is incorporated into an ACA SVD representation, allowing the numerical rank to adapt to the phase-space complexity generated by the nonlocal Wigner operator and self-consistent Poisson field. Numerical experiments for the two-stream instability, strong Landau damping, and bump-on tail instability show that the method captures benchmark Wigner-Poisson dynamics for several values of the quantum parameter H, maintains bounded adaptive ranks, and preserves the specified global discrete invariants with conservation errors near machine precision. We also compare this formulation, which uses local density-momentum correction plus global total energy correction, with a related globally conservative formulation for mass, momentum, and energy [8]. The two approaches produce nearly identical phase-space and diagnostic results for the periodic benchmark test considered here, indicating that both correction strategies are compatible with adaptive rank compression for Wigner-Poisson dynamics in the tested 1D1V periodic setting.

Pinching Waveguide Antennas (PWAs) offer significant potential for simultaneous wireless information and power transfer (SWIPT) by enabling precise near-field energy focusing. However, existing optimization frameworks are largely point-based (targeting a single coordinate for maximum gain), and thus highly sensitive to positioning errors and mobility, as near-field signals fluctuate significantly even over small spatial displacements. In this paper, we propose a spatially robust design framework based on discrete antenna selection optimized for service area (SA) coverage. Unlike point-based approaches, our model guarantees quality of service within predefined SAs for both information decoding (ID) and energy harvesting (EH) receivers, thereby improving robustness to user displacements. We formulate the problem as a non-convex binary quadratic program aimed at maximizing harvested energy within the EH SA subject to robust rate constraints in the ID SA. To characterize fundamental performance limits, we develop a semidefinite relaxation (SDR) framework that provides an upper bound on the achievable rate-energy (R-E) region. For the lower bound, we employ a low-complexity swap-based local search algorithm enforcing binary hardware constraints. Numerical results demonstrate that the proposed coverage-oriented design yields a robust R-E tradeoff and maintains stable performance across service regions, highlighting the advantages of discrete antenna activation over point-based near-field optimization approaches.

This paper explores the use of generative models to synthesize high-quality, site-specific multiple-input multiple-output (MIMO) channel data, addressing the high cost of the extensive measurement campaigns required to acquire real-world data for AI-native wireless networks. Two location-conditioned generative paradigms are compared: a conditional denoising diffusion implicit model (cDDIM), and a conditional flow matching model (cFMM). Both these models generate MIMO channel matrices conditioned on user coordinates, to preserve the spatial structure of the deployment site. The approaches are evaluated across three dimensions: statistical fidelity (including beam consistency and effective rank), generation efficiency, and utility in downstream tasks such as channel-state information compression and beam alignment. Results across diverse propagation scenarios (28 GHz and 3.5 GHz, both line-of-sight and non-line-of-sight) demonstrate that both models accurately capture site-specific characteristics, even when trained on scarce ground-truth data. Notably, cFMM achieves a quality comparable to cDDIM with roughly an order of magnitude less inference time. Augmenting scarce site-specific datasets with these synthetic channels yields hefty performance gains in downstream physical layer tasks compared to using scarce data alone or stochastic channels.

The John ellipsoid of a symmetric polytope $P=\{\mathbf{x}\in\mathbb{R}^d:\|\mathbf{A}\mathbf{x}\|_\infty\le1\}$, $\mathbf{A}\in\mathbb{R}^{n\times d}$, is computed by a long line of leverage-score algorithms, from Cohen, Cousins, Lee and Yang (COLT 2019) to its successors [WY24, CLS+25], all reaching a $(1+\varepsilon)$-approximation in $Θ(\varepsilon^{-1}\log(n/d))$ iterations. We separate this complexity into three costs the modern line conflates (certification, identification, and accuracy) and locate the historical $\varepsilon^{-1}$ in the first alone. In the equivalent D-optimal-design form $\min_{\mathbf{p}\inΔ_n}-\log\det(\sum_i p_i\mathbf{a}_i\mathbf{a}_i^\top)$, the leverage-score oracle is exactly the first-order oracle and the $(1+\varepsilon)$-John guarantee the Frank-Wolfe gap $g(\mathbf{p})\le\varepsilon d$; through this dictionary the costs come apart. The $\varepsilon^{-1}$ is a certification artifact: the uniform average of the iterates, the certificate used throughout the line, has gap exactly $Θ(1/T)$, however cheap each iteration is made. Pointed instead at the last iterate the same oracle is fast: a warm-started accelerated method reaches the guarantee in $C(\mathbf{A})+O(\sqrtκ\log(1/\varepsilon))$ queries after an $\varepsilon$-independent setup $C(\mathbf{A})$, and once the optimal face is identified the facial problem is an unconstrained self-concordant minimization whose Hessian the oracle recovers exactly, so damped Newton needs only $O(\log\log(1/\varepsilon))$ steps, for a total of $C(\mathbf{A})+O(d^2\log\log(1/\varepsilon))$ queries. The accuracy dependence is thus doubly logarithmic after an $\varepsilon$-independent, condition-dependent setup; the open problem is the remaining identification cost (a condition-free bound on reaching the optimal face) and lower bounds. Accuracy is not the obstruction.

A class of pseudo-parabolic partial differential equations is considered. We derive a posteriori error bounds for approximations obtained by FEMs in space and a BDF formula in time. The analysis is based on the $C_0$ semigroup theory and an adaptation of the concept of elliptic reconstruction to pseudo-parabolic problems. The analysis is complemented with numerical experiments.

We introduce optimal coarse correlated equilibria for continuous-time mean field games. A coarse correlated equilibrium is a randomized recommendation scheme from which no player can gain by ignoring the recommendation and switching to an alternative strategy. The problem is as follows: a moderator selects, among all mean-field coarse correlated equilibria, one that optimizes a prescribed performance criterion, which may differ from the representative player's objective. After formulating the problem, we develop a linear programming (LP) formulation, prove the existence of optimal LP coarse correlated equilibria, and relate the LP characterization to the original probabilistic setting. Building on this characterization, we design a no-regret primal-dual algorithm, based on an equivalent Lagrangian formulation of the external-regret constraint, for learning such equilibria. We provide explicit convergence rates for the learning algorithm, and numerical examples illustrate the method.

The matrices arising from large scale $N$-body problems can be efficiently represented using hierarchical matrices, whose key idea is that the admissible off-diagonal sub-matrices can be well approximated by low-rank matrices across a hierarchy of matrix partitions. HODLR (Hierarchical Off-Diagonal Low-Rank) matrices are a subclass of hierarchical matrices in which all off-diagonal submatrices at every level of a recursive binary partition are low-rank. In this article, we present a neural network that learns the inverse operation of HODLR matrices based on the fast direct solver for HODLR matrices developed by Ambikasaran and Darve (2013). We further extend the architecture to learn nonlinear solution operators associated with PDEs by replacing some of the linear layers with deep sub-networks. We demonstrate the performance of the proposed architecture by performing a comprehensive set of experiments that include (i) solving a linear problem such as the Fredholm integral equation of the second kind, (ii) solving PDEs such as the nonlinear Schrödinger equation, Burgers' equation, and the steady-state Darcy's flow equation, (iii) generalization study across varying parameter values, (iv) comparing the inference time of the proposed network with the run time of a classical numerical solver, and (v) comparing the proposed network with some of the existing neural operator learning networks.

The stochastic theta method is proposed to approximate invariant measures of stochastic differential equations (SDEs), both of whose drift and diffusion coefficients may grow super-linearly. For the numerical solution generated by the stochastic theta method, we show the existence and uniqueness of the numerical invariant measure first. Then, we prove that the numerical invariant measure is convergent to the exact invariant measure of the underlying SDE. We also provide some numerical simulations to illustrate our theoretical results. This work could be regarded as an extension of the results in [Y. Jiang et al, Numer. Algorithms 83(4)(2020), pp. 1531-1553] to the case of super-linearly growing diffusion coefficient. As the backward Euler-Maruyama (EM) method is a special case of the stochastic theta method, the results derived in this work could also be regarded as a generalization of the results for the backward EM method in [W. Liu et al. Appl. Numer. Math. 184(2023), pp. 137-150] to the stochastic theta method.

This paper studies a class of distributed optimization problems in networked nonlinear systems (NNSs) subject to input delays and consensus constraints. It introduces input-delay tolerant semiglobal convergence (IDTSC), meaning that for any prescribed compact initial set there exists an admissible delay bound under which the optimal solution is computed within consensus constraints and all node states converge to the solution. Building on a hierarchical design and input-to-state stability analysis, a new semiglobal input-delay tolerant (SIDT) algorithm is developed that practically achieves IDTSC for distributed optimization under the coupling between input delays and nonlinear dynamics. Further, by relaxing strict convexity requirements through the Polyak-Łojasiewicz condition, the SIDT algorithm broadens its applicability to nonconvex optimization. Finally, numerical experiments corroborate the theory on NNSs with input delays.

Recent research on Doeblin coefficients has shed light on their usefulness as a multi-way generalization of the Dobrushin contraction coefficient for TV distance, in a separate vein from their classic role in the theory of Markov chain ergodicity. However, strong conditions, such as being bounded away from 0, are typically necessary for Doeblin coefficients to establish the existence of information contraction. Building on recently formulated concepts of nonlinear information contraction, we aim to propose a finer-grained Doeblin-based characterization of multi-way contraction behavior which yields non-vacuous contraction guarantees even for channels whose Doeblin coefficient is 0. To this end, we introduce the notion of a Doeblin curve -- a nonlinear function which quantifies the contraction behavior of a Markov kernel on collections of input distributions at specific levels of divergence and power. Through the course of our analysis, we develop a new variational characterization of Doeblin coefficients, present several properties of Doeblin curves, define several versions of power-constrained Doeblin curves, and derive upper and lower bounds using our aforementioned variational characterization. We then utilize these results in diverse areas, including generalization bounds for noisy iterative optimization, error bounds for reliable computation with noisy circuits, and differential privacy guarantees for online iterative algorithms. In particular, we extend results in these areas to broader domains or group settings, leveraging Doeblin curves to reveal finer-grained contraction phenomena than Doeblin coefficients.

We introduce a second-order approximation to the compressible atmospheric Euler equations with gravity that is invariant domain preserving and well-balanced with respect to rest states. The approximation is built upon discrete auxiliary states derived from a hydrostatic reconstruction of the density. These auxiliary states, together with an affine shift of the numerical state, provide local bounds needed for maintaining well-balancing and invariant domain preserving properties of the method. The numerical method is then verified and validated with analytic solutions, well-balancing tests, and typical benchmark problems for atmospheric flows.

We prove an optimal dimension-free sparsification theorem for suprema of centered Gaussian processes. Given a bounded set $T\subseteq\mathbb{R}^n$, we show that the supremum of the canonical Gaussian process on $T$ can be $L^2$-approximated by the supremum of a shifted subprocess indexed by only $\exp(O(1/\varepsilon^2))$ points, with error at most $\varepsilon$ times the Gaussian width of $T$. In particular, the size of the approximating process is independent of both the ambient dimension and the cardinality of the original index set. This improves a recent sparsification theorem of De, Nadimpalli, O'Donnell, and Servedio (2026) by an exponential factor, and we show that the dependence on $\varepsilon$ is tight up to constants in the exponent. As consequences, we obtain an exponentially improved junta theorem for norms over Gaussian space and sharpen results on learning, property testing, and polyhedral approximation of convex sets under the Gaussian measure. The proof is based on an interpolation argument that combines Sudakov's minoration with the Brascamp--Lieb inequality.

We propose and analyze a structure-preserving numerical method for the $2\tfrac{1}{2}$-dimensional (2.5D) electron magnetohydrodynamics system with fractional dissipation on the periodic torus. The method works directly with the magnetic field components and combines this component formulation with the gradient recovery operator of [T. Chu, H. Guo, and Z. Zhang, SIAM J. Numer. Anal., 63 (2025), pp. 23--53]. We establish discrete energy stability for a semi-implicit structure-preserving formulation and use an explicit-Hall integrating-factor implementation for efficient computation on periodic grids. The fractional dissipation is treated exactly in Fourier space, and the in-plane divergence constraint is enforced by a spectral Hodge projection. Numerical experiments demonstrate second-order spatial convergence and stable Hall-driven dynamics across several benchmark tests.

Tight equalities between symmetric information and entropy production in driven steady states remain elusive. We show that they are forbidden by a parity selection rule for rotation-driven linear nonequilibrium steady states. Whenever the relaxation and diffusion matrices commute, the snapshot mutual information between two time slices is exactly even under drive reversal, and parity violation rises linearly in the commutator norm when alignment is broken. Full isotropy strengthens this to drive-independence, and the planar mutual information takes the closed-form value of about 0.145 nats. Under the same alignment, the entropy production is exactly quadratic in the drive, and its prefactor admits an explicit closed form in the traces and determinant of the two matrices. The orthogonality of even and odd sectors leaves only one-sided thermodynamic-uncertainty bounds. The rule rests on the rotational symmetry of the drift alone and survives heavy-tailed isotropic stable noise with tail index below two, where variance-based bounds become vacuous. A falsifiable test is proposed on an electrical Brownian gyrator augmented for independent drive control with circuit-level stable-noise injection.

Establishing stability guarantees for dynamical systems on Lie groups is a fundamental challenge, as classical Lyapunov methods developed for Euclidean spaces do not directly transfer to curved geometries. In this paper, we propose a framework for learning maximal Lyapunov functions for systems evolving on the special orthogonal group $\mathsf{SO}(n)$. Theoretically, we introduce a neural Lyapunov architecture based on the logarithmic map with proven approximation capabilities, and we formulate the learning problem via a Zubov-type characterization of the maximal region of attraction. A key technical contribution is the derivation of explicit, numerically tractable formulas for the derivative of the logarithmic map, enabling training through a two-phase algorithm that balances computational efficiency and accuracy. Empirically, we validate the approach on a low-dimensional nonlinear system.

The derivative nonlinear Schrödinger equation is a fundamental model for the propagation of nonlinear dispersive waves in, for example, plasma physics and nonlinear optics. In this work, we consider this model on the one-dimensional torus and study a filtered explicit Fourier integrator for the corresponding periodic problem. After applying a periodic gauge transformation, we consider a frequency-truncated model and its filtered exponential-Euler discretization. The main difficulty comes from the derivative cubic nonlinearity in the periodic setting, since local smoothing is unavailable and resonant interactions are stronger than in the non-periodic case. To address this issue, we develop a discrete Bourgain-space framework adapted to the gauge-transformed equation. For initial data $u_0 \in H^s(\mathbb{T})$ with $1/2 < s \le 5/2$, we prove that the numerical error is of order $\mathcal{O}(τ^{s/2-1/4})$ in $H^{1/2}(\mathbb{T})$, where $τ$ denotes the employed time step size. Numerical experiments confirm the predicted convergence behavior and demonstrate the effectiveness of the filtered scheme for rough solutions.

We introduce a new class of numerical integrators for the time integration of non-autonomous linear ordinary differential equations whose coefficient matrix is sparse and evolves within a quadratic matrix Lie group. In contrast to standard Lie group integrators, the proposed methods avoid the evaluation of matrix exponentials acting on vectors and instead rely on solving a sequence of linear systems with sparse coefficient matrices. Moreover, they are well suited for problems arising from unbounded operators, as they inherently produce bounded solutions. We construct optimised schemes of orders four and six and assess their performance on a representative numerical example, demonstrating clear advantages over existing Lie-group integrators.

We develop and analyze a Bregman-projected mirror iteration for low-order regularizations of stationary mean-field game (MFG) systems in their natural Banach space setting. For separable Hamiltonians of the form \(H(x,p,m)=H_0(x,p)-g(m)\), with quadratic or super-quadratic Hamiltonian growth and linear or super-linear density couplings, we formulate a low-order \(\barγ\)-Laplacian regularization of the stationary MFG system as a variational inequality on \(L^{\barβ}(\mathbb T^d)\times W^{1,\barγ}(\mathbb T^d)\). To approximate solutions of this regularized variational inequality, we introduce a Bregman geometry matched to the mixed Lebesgue--Sobolev exponents of the problem and analyze a constrained two-step mirror method with frozen operator evaluation. For the exact constrained iteration and each fixed regularization parameter \(\epsi>0\), we derive a one-step Bregman inequality and use it to prove that the constrained iteration converges strongly to the unique solution of the regularized variational inequality under natural summability conditions on the step sizes. Numerical experiments on one- and two-dimensional models, validated against exact test solutions, illustrate residual decay under mesh refinement and suggest improved practical performance of the two-step implementation in the tested discretizations.

We reduce the embracing exchange distance of bases of oriented matroids to the metric of oriented matroid polyhedra. This allows us to disprove recent conjectures of Caoduro, Khodamoradi, Paat, and Shepherd and of Bérczi and Nádor. On the other hand, we show that any two embracing bases of an oriented matroid of rank $r$ can be transformed into each other in at most $2r^{\log_2(r)+3}$ steps and in at most $r$ steps in a Lawrence oriented matroid, thus confirming the conjecture in this case.

This paper develops an accurate and effective combination of second order backward differentiation time discretization (BDF2) with modular, 2-step nudging-based data assimilation \begin{align} \text{Forecast step: } \quad &\frac{3\widetilde{v}^{n+2}-4v^{n+1}+v^n}{2Δt}+\widetilde{v}^{n+2} \cdot \nabla \widetilde{v}^{n+2} - νΔ\widetilde{v}^{n+2} + \nabla q^{n+2}=f(x) \notag \\ &\nabla \cdot \widetilde{v}^{n+2} = 0 \notag \\ \text{Analysis step: } \quad &\frac{3v^{n+2}-3\widetilde{v}^{n+2}}{2Δt}-χI_H(u(t^{n+2})-v^{n+2})=0. \notag \end{align} If $I_H=I_H^2$, the analysis step can be made explicit, taking the form \begin{align} v^{n+2}=\widetilde{v}^{n+2}+\frac{2Δtχ}{3+2Δtχ}I_H(u^{n+2}-\widetilde{v}^{n+2}). \notag \end{align} This implies the analysis step has the stability property of an implicit step and lower complexity than an explicit analysis step. Stability and error estimates for the BDF2 scheme are presented along with their proofs. Numerical experiments are conducted to assess the performance of BDF2 modular assimilation algorithm. The results of the experiments support the conclusion that modular data assimilation has comparable accuracy to standard, fully coupled data assimilation while greatly reducing computational complexity and cost.

Bluetooth Low Energy (BLE) direction-finding is promising for indoor industrial localization, but its accuracy degrades in multipath environments where reflections and scattering bias angle estimates. Although line-of-sight (LOS) and non-line-of-sight (NLOS) detection is well studied for wide-band radios, BLE direction-finding still lacks narrow-band channel-feature representations, scalable kernel-based feature transformations, and dedicated datasets for data-driven, lightweight channel classification. To address this gap, the work introduces a controlled BLE measurement setup that generates labeled LOS/NLOS data in two distinct propagation environments. A quality-driven machine learning (ML)-based pipeline is then developed for BLE Constant Tone Extension (CTE) In-phase-Quadrature (IQ) features. First, robust quantile-based standardization is applied to reduce the influence of outliers and heavy-tailed effects. The standardized features are then analyzed using Principal Component Analysis (PCA) and Adaptive Kernel Density Estimation (AKDE) to verify scenario-dependent statistics and reveal LOS/NLOS separability. Next, Nyström Kernel Approximation (NKA) constructs low-rank nonlinear feature maps followed by a lightweight Support Vector Classifier (SVC) head for LOS/NLOS detection. This classifier is compared with Random Forest (RF) and Multilayer Perceptron (MLP) models. Results show that NKA improves accuracy by about 7-14% relative to the raw baseline. Although the MLP achieves higher absolute accuracy, the Nyström--SVC approach offers a more favorable trade-off between training complexity, inference cost, and memory footprint. Finally, several pipeline-calibrated posterior probabilities are utilized for cost-aware threshold selection and efficient real-time LOS/NLOS detection in resource-constrained localization systems.

We give a new and elementary construction of primitive positive decomposition of higher arity relations into binary relations on finite domains. Such decompositions come up in applications to constraint satisfaction problems, clone theory and relational databases. The construction exploits functional completeness of 2-input functions in many-valued logic by interpreting relations as graphs of partially defined multivalued 'functions'. The 'functions' are then composed from ordinary functions in the usual sense. The construction is computationally effective and relies on well-developed methods of functional decomposition, but reduces relations only to ternary relations. An additional construction then decomposes ternary into binary relations, also effectively, by converting certain disjunctions into existential quantifications. The result gives a uniform proof of Peirce's reduction thesis on finite domains, and shows that the graph of any Sheffer function composes all relations there.

In this work we investigate the role of neural architectures as implicit functional priors in control problems governed by ordinary differential equations. Rather than focusing on highly complex problems, our objective is to investigate architecture-dependent effects in controlled dynamical systems within the simplest physically interpretable settings possible. In particular, we study a controlled linear RLC electrical circuit and a nonlinear Duffing-type dynamical system. Both systems are analyzed first through classical optimal-control formulations and later through PINN-based approaches. We compare different combinations of multilayer perceptrons (MLPs) and Fourier-based KAN-like architectures, and analyze their influence on the resulting controls. The numerical experiments suggest that different architectural choices systematically generate qualitatively distinct controls, even under identical governing equations, loss functionals, initial and target states, training parameters and physical constraints. Significant differences appear in the spectral structure, smoothness, energy distribution, and phase-space behavior of the learned solutions. A central observation of this work is the emergence of a functional specialization phenomenon when the neural architectures are allowed sufficient freedom to shape the structure of the learned controls. More specifically, in the systems considered here, Fourier-based architectures tend to produce trajectories with richer oscillatory content, whereas smoother low-frequency-biased architectures tend to generate more regular and energetically efficient controls. This suggests that different functional components of the control problem may be handled more efficiently by different neural architectures, leading to an implicit specialization between state representation and control generation.

We introduce the Frontier decoder, a pruned dynamic-programming decoder for sparse quantum decoding problems. Frontier processes error variables in a chosen order, merges prefixes with the same residual syndrome and logical label, and approximates logical-coset posterior masses by retaining only a narrow scored frontier. Without pruning, the recursion is exact ordered inference with exponential complexity. In the code-capacity setting, the decoder reaches thresholds close to optimal for the surface code and the color code. In the circuit-level noise model, it achieves state-of-the-art performance with a very small average retained list size: less than 100 for the gross code $[[144,12,12]]$ at a physical error rate of $0.001$. When the list size is constant, the decoder has linear complexity, suggesting the possibility of low-latency implementations.