Papers

We calculate the first derivative of the Jacobi polynomials with respect to their order in explicit form. This derivative is not an elementary function, but contains elementary special cases. As an application, we use our result with a recently devised method for resolving a new class of indefinite integrals containing Jacobi polynomials.

Solving multi-agent optimal control problems in aerospace such as pursuit-evasion and contested space operations can be modeled as non-convex differential games for which, there are limited algorithms. In this work, a relaxation of generalized Nash Equilibrium problems (GNEPs) to exclude inter-agent control coupling in dynamics, which is representative of many multi-agent systems is introduced. The main contribution is an algorithm for solving a broad class of differential games named FALCON: Fast Augmented Lagrangian Convexification for Open-loop Nash equilibria is presented. Methodologically, sequential convex programming (SCP) is utilized to create tractable convex sub-games which can then be solved via standard convex programming methods involving a potential game reformulation. FALCON is demonstrated to have global convergence guarantees to an open-loop Nash equilibrium for non-convex differential games under mild assumptions. This is numerically shown through both cooperative and competitive differential games.

We construct a reduced planar convex body $R$ with thickness $\Delta(R)=1$ and \[\operatorname{area}(R)=0.786215\ldots>0.785398\ldots=\frac{\pi}{4}.\] Thus $R$ is a counterexample to Lassak's conjectured upper bound $\operatorname{area}\le(\pi/4)\Delta^2$ for planar reduced bodies. The construction is given by an explicit support function, and the proofs use only elementary support-function, width, area, and contact-point computations.

Cayley codes, introduced by Kaufman and Wigderson, are linear codes constructed from a Cayley graph and a smaller linear code. We explore general properties of the class of Cayley codes for finite groups. In particular we give a reduction to Cayley codes for connected Cayley graphs that maintains code properties such as rate, minimum distance and symmetry. Also, for a given Cayley code, we identify a family of symmetric Cayley codes, each associated with a normal edge-transitive Cayley graph, such that the given Cayley code embeds into the direct sum of the symmetric Cayley codes. We analyse several families of examples, in particular studying the behaviour of the Cayley code construction under forming direct products and cartesian products of Cayley graphs, and we pose a number of open questions.

Interval graphs are a special class of chordal graphs, and hence have connections to commutative algebra via Fröberg's theorem that characterizes linear resolutions of squarefree quadratic ideals. In recent years, several hypergraphic analogues of interval and chordal graphs have been proposed, in part as an effort to extend Fröberg's theorem to ideals generated in higher degree. In this paper, we study two such classes from the literature, cointerval hypergraphs and underclosed complexes, and show that they are in fact equivalent up to complementation. We then consider their place in the broader theory of higher-dimensional chordality, proving that an underclosed clutter is chordal in the sense of Woodroofe. As a consequence, we answer a question of Dochtermann and Engström by showing that the associated Alexander dual complexes are vertex decomposable, implying that the corresponding circuit ideals have linear quotients. We furthermore show that these dual complexes have shellings induced by their underclosed vertex orders.

We present a fast algorithm for evaluating conditionally convergent Coulomb lattice sums, governed by the Laplace equation with periodic boundary conditions on arbitrary unit cells (oblique in 2D, triclinic in 3D) and arbitrary particle distributions. The algorithm extends the dual-space multilevel kernel-splitting (DMK) framework to this context. The root of the adaptive tree is now a rectangular grid of cubes consisting of an inner block covering the unit cell and a surrounding halo of image cubes, rather than a single cube, and the smooth top-level periodic kernel – the only term that requires the consideration of conditional convergence issues – is evaluated by the “five-step procedure" used in fast Ewald summation: spreading, fast Fourier transform (FFT), diagonal scaling, inverse FFT, and interpolation. The resulting complexity is $O(N)$ for fixed cell shape. Benchmarked against the periodic fast multipole method on highly nonuniform source distributions, our 2D algorithm is roughly an order of magnitude faster across particle counts and target precisions; in three dimensions, it is often as fast as the free-space DMK on the same sources, even for triclinic cells with edge-length ratios up to roughly $17$.

Let $[0, 1]^{n} \subseteq \mathbb{R}^{n}$ be endowed with its pointwise order, and let $k$ be a positive integer. A subset $A$ of $[0, 1]^{n}$ is said to be a \emph{$k$-antichain} if $\operatorname{card}(A \cap C) \leq k$ for each chain $C \subseteq [0, 1]^{n}$. Letting $\mathcal{H}^{m}$ denote the $m$-dimensional Hausdorff outer measure, Pelekis and Vlasák [Publ.\ Math.\ Debrecen, 2020] conjectured that there exists a $k$-antichain $A \subseteq [0, 1]^{n}$ satisfying $\mathcal{H}^{n-1}(A) = k n$, and proved the special case of this conjecture for $n = 2$, whereas Janzer [Mathematika, 2020] proved the $k = 1$ case of Pelekis and Vlasák's conjecture. This conjecture is motivated by a result due to Erdős on $k$-antichains in $\{ 0, 1 \}^{n}$. We prove Pelekis and Vlasák's conjecture in full generality, thus establishing that their upper bound $\mathcal{H}^{n-1}(A) \leq k n$ is sharp for $k$-antichains $A$ in $[0, 1]^{n}$.

Packing problems form a central theme in graph theory, owing to their relevance in modeling conflict-free resource allocation, network design, and communication constraints. Motivated by applications in wireless networks where each device can participate in at most one communication at a time and simultaneous links must avoid interference we consider a generalization of induced matching known as edge open packing. Two edges of a graph are said to conflict if a third edge connects one endpoint of each; an edge open packing set is a set of edges containing no such conflicting pair. The largest cardinality of such a set is the edge open packing number of a graph. In this work, we study the computational complexity of the Maximum Edge Open Packing Problem. We give a polynomial-time algorithm for the problem in distance-hereditary graphs, exploiting their canonical decomposition via twin-set interactions. We further show that the problem remains polynomial-time solvable on biconvex bipartite graphs, thereby identifying a tractable subclass within bipartite graphs, in contrast to the known NP-hardness of the problem on Eulerian bipartite graphs. Finally, we initiate the parameterized complexity study of the problem and present a fixed-parameter tractable algorithm for chordal graphs, parameterized by the clique number $\omega$, running in $O(2^{\omega}\cdot\mathrm{poly}(n))$ time.

Median bias reduction of maximum likelihood estimators can substantially improve estimation and inference. Existing generally applicable methods are, however, implicit, requiring the solution of nonlinear systems of estimating equations for a specified parameterization. Their application to parameter transformations often involves tedious algebra and bespoke implementations. We develop an explicit median bias-corrected estimator for focus parameters that are smooth scalar transformations of a chosen reference parameterization. The estimator results from solving an equation derived from the Cornish-Fisher expansion of the centred and scaled maximum likelihood estimator of the focus parameter, and requires only the ML or an asymptotically equivalent estimator at the reference parameterization, the gradient and Hessian of the transformation, and expectations of products of log-likelihood derivatives. These expectations are available for many models in the bias reduction literature and can also be estimated by Monte Carlo simulation. The resulting estimators are third-order median unbiased and provide one-step approximations to estimators from implicit median bias reduction when the reference parameterization includes the focus parameter. The method can improve standard asymptotic inference and enables hull-based confidence procedures to produce intervals with near nominal finite-sample coverage under median bias control. We illustrate the framework through post-selection inference using the Focused Information Criterion, Mahalanobis distances, quantiles, and scalar focus parameters in regression, stratified, circular, and multiple-mediator models.

This paper constructs a framed real monopole Floer homology for three-manifolds with involutions, marked with multiple basepoints. The relative gradings of these Floer homologies depend on the framing information and the paper gives a sufficient condition for the existence of relative mod two gradings. Assuming orientability and choices of orientations, this paper also proposes a definition of $\mathbf{Z}$-valued framed real Seiberg–Witten invariants for 4-manifolds with involutions, marked with circles.

We prove a generalization for commuting $n$-tuples of unbounded self-adjoint operators and the Lorentz $(n,1)$ ideal,$n \ge 3$, of the Kato-Rosenblum theorem. The result is derived from earlier work for bounded operators [8]. Also, a very weak result for $n=2$ unbounded operators and other additional results are obtained.

The main theme of approximation theory is to understand how well a general function $f$ can be approximated by a simpler function $g$ such as a polynomial or spline. In many applications, one wants $g$ to retain known properties of $f$ such as its inherent smoothness or a geometrical property such as monotonicity or convexity. Additional requirements on $g$ of this type are known as constraints. In this paper, we do a systematic study of constrained approximation to understand how the imposition of such constraints limits the efficiency of the approximation. We study constrained approximation in the setting of linear approximation where $g$ is to be taken from a finite dimensional linear space $V$ of a fixed dimension $n$. Kolmogorov widths describe how well one can approximate when using such linear spaces $V$. The first part of this paper introduces and studies several types of constrained widths, including the constrained Kolmogorov widths, and gives comparisons between them. The second part of the paper is restricted to classical settings where the constraint imposes a smoothness requirement on $g$. In this case, our results prove that the additional constraint can typically be imposed with no loss in the efficiency of the approximation.

In this work, we propose a robust Sparse Identification of Nonlinear Dynamics (SINDy) pipeline for handling datasets corrupted by noise and outliers. The method decouples outlier filtering from sparse regression by combining Iterative Least Trimmed Squares (ILTS) with Sequentially Thresholded Least Squares (STLS). Unlike standard approaches that treat all observations uniformly within a single regression stage, the proposed ILTS-SINDy framework first applies an ILTS procedure that iteratively minimizes the sum of the smallest squared residuals to identify the most reliable observations without prior knowledge of outliers, after which STLS is used to recover a parsimonious governing model. Extensive numerical experiments show that ILTS-SINDy can significantly outperform existing robust SINDy variants across a range of outlier contamination levels, with performance maintained even under settings with up to $20\%$ corrupted observations.

We consider solutions of the sinh-Gordon Painlevé III equation \[ u_{xx} + \frac{1}{x} u_x = \sinh u \] that are real on $(0,\infty)$. They are parametrized by the monodromy parameter $p\in\overline{\mathbb{C}}$, $|p|>1$, and an additional real parameter $s^{\mathbb{R}}$ when $p=\infty$. Our previous joint work with A. Its described the asymptotic behavior of these solutions as $x\to\infty$. Here, we describe the transition as $x, p\to \infty$, $2\Im(p)=-s^{\mathbb R}$, between singular solutions ($|p|<\infty$) and smooth solutions ($p=\infty$). In short, if we parametrize $|p|^2 = 1 + e^{2\varkappa x}$, then the smooth exponential asymptotics of the solutions extends to the region $\varkappa>1$, with a change of the leading order term at $\varkappa=2$; at $\varkappa=1$ the exponential behavior transitions into an elliptic asymptotics, which holds for all $0<\varkappa<1$; as $\varkappa$ decays to zero, elliptic asymptotics degenerates into trigonometric one, which holds for all $p$ fixed.

The Padovan sequence $\{P_{m}\}_{m\ge 0}$ is a ternary recurrence sequence with companion polynomial $X^{3}-X-1$ and initial conditions $P_{0}=P_{1}=P_{2}=1$. The Perrin sequence $\{R_{m}\}_{m\ge 0}$ is defined by the same companion polynomial as the Padovan sequence, but has initial values $R_{0}=3$, $R_{1}=0$, and $R_{2}=2$. We solve the Brocard-Ramanujan equation $n!+1=x^{2}$, where $n!$ is the factorial of $n$ and $x$ is a Padovan number or a Perrin number. In both cases, we prove that $(n,x)=(4,5)$ is the only solution.

We study a logarithmically hyperviscous Navier-Stokes model on the three-dimensional torus with Lie-transport noise, which includes both transport and stretching. We prove that, for noise of sufficiently large intensity and high frequency, the system admits a unique global smooth solution with probability arbitrarily close to one. Unlike previous works, this physically motivated noise does not preserve energy or enstrophy, but rather circulation. Global well-posedness is established through a probabilistic mechanism that produces effective dissipation via a scaling limit. Crucially, this approach bypasses the lack of conserved quantities and tames the singular nature of stochastic stretching.

Modern machine learning models are trained by optimizing high-dimensional non-convex empirical risk functions. Such cost functions can have a multitude of local optima and yet, gradient-based optimization appears to converge to near-global optima. Within a simple supervised learning setting, we develop a precise picture of which parts of the empirical risk landscape are accessible by polynomial-time algorithms. We are given i.i.d. pairs $\{(\boldsymbol{x}_i,y_i):\; 1 \le i\le n\}$ with $\boldsymbol{x}_i\in \mathbb{R}^d$ standard Gaussian feature vectors, and $y_i\in\mathbb{R}$ response variables that depend on $\boldsymbol{x}_i$ through their projections on an unknown $k$-dimensional subspace. We use empirical risk minimization to learn a model that depends on an $m$-dimensional projection of the data (e.g., an $m$-neurons neural network). We propose an incremental approximate message passing (IAMP) algorithm and precisely characterize the training error it achieves, as well as the relation between test and training error, in the high dimensional asymptotics $n,d\to\infty$, with $n/d\to\alpha \in (0, +\infty)$. Based on earlier work in related models, we expect that the performance achieved by our algorithm is optimal among polynomial-time algorithms.

The axiom of choice has divided the foundations of mathematics for over a century, but the distinction between classical and constructive proofs has remained a philosophical and methodological one. We use Lean 4's kernel-level tracking of axiom dependence to show that the axiom of choice has a measurable geometric correlate in proof space that obeys a one-parameter mixture law and has operational consequences for neural theorem provers. To do this, we partition $471{,}260$ declarations of Mathlib by transitive dependence on the axiom of choice and represent a filtered population of $42{,}355$ traced theorems by their sequences of tactic invocations. We use the constructive proofs in this dataset to train a self-supervised proof encoder and show that when using it to measure classical proofs, three complementary measurements (anomaly score, reconstruction loss, and density-superlevel containment) exhibit a common decline with the proof's distance from the axiom in the dependency graph, from sharp separation at the shallow boundary (AUC $0.847$ at distance $2$) to indistinguishability at distance~$9{+}$. Robustness controls show that the signature survives length, file, author, and topic controls, and replicates under full-source encoders trained on normalised proof source. Operationally, we show that on an evaluation sample of $251$ Mathlib theorems, Lean's aesop tactic solves constructive theorems at $13\times$ the rate of classical ones, and a neural-guided hybrid using the ReProver tactic generator compresses the gap to $5\times$. The geometric anomaly score predicts aesop failure beyond proof length, providing an operational link between the geometric signature and prover performance.

Mean-flow-based linear analyses of turbulent flows, such as resolvent analysis, provide valuable insight about flow structures and their dynamics that has been widely leveraged to model, control and understand the underlying flow physics. However, these analyses are computationally expensive for flows over complex geometries and require the use of specialized codes that are typically only available in research environments. On the other hand, data-driven modal decompositions, such as the dynamic mode decomposition (DMD), identify turbulent flow structures that, although statistically relevant, do not provide insight into the physical mechanisms driving their dynamics. Here we introduce a novel data-driven method – nonlinearity-subtracted DMD (NSDMD) – that leverages knowledge of the structure of the Navier–Stokes equations to ensure that the learned operator is a low-rank approximation of the underlying mean-flow-linearized dynamics. Specifically, the method uses snapshots of the nonlinear terms in the perturbation equations to explicitly account for the contribution of the nonlinear forcing to the dynamics. We demonstrate the use of NSDMD to perform data-driven resolvent analysis on direct numerical simulation (DNS) and large-eddy simulation (LES) datasets, starting with a minimal channel flow and scaling up to the flow over a full aircraft model. As a result, NSDMD allows performing linear analyses of turbulent flows as a post-processing step on simulation data obtained with any available high-fidelity computational fluid dynamics (CFD) code.

This article treats kernel approximation and interpolation on embedded manifolds of $\mathbb{R}^N$using restrictions of positive and conditionally positive definite kernels. The main challenge is to develop an approximation theory that treats error measured in highly regular smoothness spaces relative to the kernel. This means that the order of smoothness is higher than that of the kernel's associated native space (in the positive definite case, the reproducing kernel Hilbert space generated by the kernel). This prevents the use of standard techniques for controlling error in this setting, especially RKHS space arguments like orthogonality of the interpolation projector, or bounds using the power function. To address this challenge, we extend methods for treating target functions given as potentials introduced by DeVore and Ron, and give conditions guaranteeing that restrictions of such functions span Sobolev smoothness spaces on the manifolds. Together with kernel-based Bernstein inequalities for embedded manifolds, these results give a comprehensive theory for interpolation error. As an application of the theory, we derive new error estimates for approximating the eigenspace of certain differential operators arising from kernel-based methods for partial differential equations on manifolds.

Systems defined by linear and complementarity constraints (SLCCs) arise frequently in engineering, economics, and other related fields. They also appear in the optimality conditions of many challenging optimization models, such as bilinear optimization and linearly constrained quadratic optimization. It is known that finding a feasible solution to an SLCC is NP-hard in general. In this paper, we study the feasibility problem for a given SLCC: either find a feasible solution or determine that the system is infeasible. To this end, we introduce a universal relaxation theory (URT), which reformulates SLCC feasibility as an equivalent bilinear optimization problem with linear constraints in a lifted space. We then analyze the resulting bilinear model and derive necessary and sufficient optimality conditions for its global solutions. Based on these theoretical insights, we introduce a relaxation-rectification (ReCR) framework for finding a feasible solution to a given SLCC instance or certifying infeasibility. We develop several ReCR methods that differ in their working spaces and subproblem formulations and analyze their convergence properties. We also develop a numerical procedure for obtaining an infeasibility certificate when the ReCR methods do not find a feasible solution. We conduct numerical experiments to evaluate the reliability, robustness, and scalability of the proposed ReCR methods and compare them with existing SLCC solvers. On the tested small- and large-scale LCP instances from the literature, the proposed ReCR methods typically find feasible solutions in a few iterations. We also extend the benchmark with more challenging medium-scale SLCC instances, on which the proposed hybrid ReCR (H-ReCR) method exhibits promising performance.

In this article, we introduce a new noise model for data-driven control. The model can be interpreted as a generalization of a Frobenius norm bound on the matrix of noise samples. For instantaneously bounded noise, the proposed model provides a less conservative overapproximation than an existing noise model based on a quadratic matrix inequality (QMI). Using the new model, we derive necessary and sufficient conditions for data-driven control. The framework covers a broad class of design problems, including quadratic stabilization, $\mathcal{H}_2$ control and $\mathcal{H}_{\infty}$ control, and is further extended to cover data-driven analysis problems, ranging for stabilizability to dissipativity. A key technical contribution is a new type of S-lemma that offers necessary and sufficient conditions under which a quadratic matrix inequality is implied by a quadratic inequality in vectorized variables.

In this paper we study the structure of complex nilmanifolds $X$ admitting some special classes of Hermitian metrics, namely, astheno-Kähler, strongly Gauduchon and balanced metrics. We prove that, in complex dimension 4, the existence of a (non necessarily invariant) astheno-Kähler metric on $X$ implies that the nilmanifold is at most $2$-step and it has first Betti number $\geq 6$. Moreover, the complex structure has a very specific form, sometimes called "of special type" in the literature. We also study the interplay between the existence of astheno-Kähler metrics and that of strongly Gauduchon or balanced metrics. A key result is the use of some obstructions that are preserved by what we call $\mathfrak{b}$-extensions. This allows us to study the existence of these metrics on important classes of complex nilmanifolds, such as almost abelian, those having maximal nilpotent complex structures, and 8-dimensional nilmanifolds with non-nilpotent complex structures. We also construct, in every complex dimension $n\geq 4$, complex nilmanifolds admitting both an astheno-Kähler metric (possibly also being strongly Gauduchon) and another metric that is balanced. As an application, astheno-Kähler nilmanifolds with balanced metrics and with Frölicher spectral sequence not degenerating at the second or third pages are found. To our knowledge, these are the first compact astheno-Kähler manifolds with such properties.

Scalar objective functions are required when a multi-criteria optimization problem must yield a single preferred design rather than only a Pareto set. The choice of scalarization influences which compromise is selected, how preference parameters are interpreted, and whether non-supported Pareto regions can be reached. This paper compares four formulations for normalized bi-criteria minimization: weighted sums, achievement scalarizing functions, desirability functions, and a fuzzy-logic-based formulation. Two analytically defined Pareto fronts, one convex and one concave, isolate the effect of the objective formulation from numerical optimizer behavior. The comparison focuses on reachable Pareto regions, parameter-induced selection density, compensation between criteria, sensitivity, and interpretability. Results show that weighted sums are simple but structurally limited on concave fronts, while achievement, desirability, and fuzzy formulations reach interior non-supported regions through different mechanisms. Desirability functions introduce nonlinear single-criterion preference mappings, whereas fuzzy rules express nonseparable and reference-dependent engineering preferences.

Maranca and Rosenberg (2024) devised a ranking scheme for unlabeled multifurcating rooted trees, in which the trees are bijectively associated with the positive integers. Here, generalizing earlier results for bifurcating trees, we determine, for trees with a fixed number of leaves, which multifurcating trees obtain the maximal and minimal ranks. We identify these maximizing and minimizing trees for each of two sets of unlabeled multifurcating rooted trees: strictly $k$-furcating trees, in which each internal node possesses exactly $k$ descendants, and at-most-$k$-furcating trees, in which internal nodes possess at least 2 and at most $k$ descendants. In both scenarios, we find that a tree that can be regarded as maximally balanced attains the minimal rank, and a minimally balanced tree attains the maximal rank. We deduce recurrences for the maximal and minimal rank for trees with fixed numbers of leaves in both the strictly $k$-furcating and at-most-$k$-furcating cases. The maximal rank on $(n-1)(k-1)+1$ leaves grows with $(k!)^{\frac{1}{k-1}} \beta_k^{(k^n)}$ in the strictly $k$-furcating case, and the maximal rank on $n$ leaves grows with $(k!)^{\frac{1}{k-1}} \gamma_k^{(k^n)}$ in the at-most-$k$-furcating case, where $\beta_k > 1$ and $\gamma_k > 1$ are constants that depend on the value of $k$. We show that $\beta_k$ decreases as the value of $k$ increases, and that $\gamma_k > \beta_k$ for $k \geq 3$. The results contribute to the use of tree encodings for empirical characterization of phylogenies and measurement of tree balance.

Combinatorial optimization is a leading target for quantum algorithms, but finite-depth QAOA can suffer from strong diabatic errors when the interpolation Hamiltonian has small, or exponentially small, spectral gaps. We propose a Pauli-sparse counterdiabatic extension of linear-ramp QAOA based on the regularised adiabatic gauge potential \[ \bigl(\mathcal L_H^2+\eta I\bigr)A_\lambda^{(\eta)} = -\mathrm{i}\mathcal L_H(\partial_\lambda H), \qquad \mathcal L_H(X)=[H,X]. \] Instead of computing a dense AGP, we solve this equation approximately by an inexact conjugate-gradient method in Pauli coordinates, truncating the Pauli expansion during the iteration to obtain a gate-budget-aware set of implementable rotations. The selected support is then improved by a Galerkin refit and certified by an a posteriori residual bound. The regularization parameter \(\eta\) acts as an energy-resolution scale: it suppresses transitions below \(\sqrt{\eta}\) while retaining larger-gap transitions. Thus, the method can avoid resolving exponentially small splittings inside a low-energy solution manifold while reducing leakage away from it. Numerical experiments on Ferromagnetic Chain (FC) and perturbed FC–MaxCut/MarketSplit instances show that the resulting LR-CD-QAOA ansatz improves approximation ratios over the uncorrected linear ramp, especially in regimes where LR-QAOA remains far from the optimum. Overall, the proposed regularized LR-CD-QAOA framework substantially broadens the practical applicability of QAOA to QUBO optimization by improving its robustness across heterogeneous problem landscapes, including instances with near-degenerate low-energy structures and small spectral gaps.

Slowness surfaces are algebraic varieties arising from propagation of elastic waves. In dimensions $2$, we completely classify the types of singularities slowness surfaces can have. The two types of possible singularities are a transversal self-intersection and a tangential singularity produced by a concentric circle and ellipse that are tangent to each other. To interpret these results analytically, in the case that the slowness surface has transversal self-intersections, we show that the principal symbol of the elastic wave operator is locally smoothly diagonalizable.

Accurate and efficient parameter estimation is essential for applying electrochemical battery models in simulation, state estimation, control, and repeated model updating. However, conventional optimization methods, such as particle swarm optimization (PSO) and genetic algorithms (GA), often require many model evaluations and show considerable run-to-run variability, limiting their use in time-sensitive calibration scenarios. This study proposes a Batch-Optimized Local-to-Global Technique (BOLT) for rapid and robust parameter estimation of electrochemical battery models. BOLT combines diversified candidate initialization, batch-parallel trust-region reflective (TRF) local refinement, JIT-accelerated model evaluation, and multi-condition consistency screening within a unified calibration workflow. Comparative experiments based on a grouped single-particle model and measured data from a commercial 18650 NMC lithium-ion cell show that BOLT achieves a favorable trade-off among voltage-response accuracy, computational efficiency, and repeated-run stability. BOLT(32) achieves an average mean absolute error of \(12.4 \pm 0.1\) mV over five operating conditions, requiring only \(20636 \pm 3081\) model calls and \(8.97 \pm 1.20\) s per run. Synthetic-data validation with a known parameter vector in the grouped SPM formulation further shows that BOLT recovers the reference parameter vector under model-consistent conditions and remains robust under 1–3 mV voltage-noise perturbations, with the mean parameter absolute relative error below \(0.6\%\). These results indicate that BOLT provides a practical calibration framework for BMS parameter updating, control-oriented battery digital twins, and second-life battery screening.

Quantum impurity models provide a paradigmatic setting for studying Kondo screening, boundary criticality, and impurity entropy. While these phenomena are well understood in unitary systems, their fate in non-Hermitian many-body settings remains largely unexplored. We study a $\mathscr{PT}$-symmetric quantum impurity model consisting of a unitary SU(2)$_1$ Wess–Zumino–Witten bulk attached to two impurity spins through complex-conjugate Kondo couplings. Using an integrable lattice realization solved by the Bethe Ansatz and benchmarked against finite-temperature matrix-product-state calculations, we determine the impurity free energy, entropy, and the corresponding $g$-function. We identify three distinct impurity phases. In the Kondo-screened phase, the spectrum remains real and the impurity entropy decreases monotonically from $\ln 4$ in the ultraviolet to $0$. Remarkably, this monotonic flow persists despite the nonunitary boundary interaction, beyond the standard domain of the $g$-theorem. The Yu–Shiba–Rusinov phase instead exhibits spontaneous $\mathscr{PT}$-symmetry breaking and loss of boundary conformal invariance, while the local-moment phase displays cyclic renormalization-group flow, with identical impurity entropy and $g$-value in the ultraviolet and infrared limits.

In the full transformation semigroup $T_n$ on a finite chain $X_n$, let $D_n=\{\alpha \in T_n:(\forall x \in X_n) \ x\alpha \leq x\}$ be the subsemigroup of all order-decreasing maps of $T_n$, and let $O_n=\{\alpha \in T_n:(\forall x ,y\in X_n) \ x \leq y \Rightarrow x\alpha \leq y\alpha\}$ be the subsemigroup of all order-preserving maps of $T_n$. The Catalan monoid $C_n$ is a semigroup of all order-decreasing and order-preserving full transformations of $X_n$. In this paper, it is shown that $O_n$ is closed in $T_n$. Also, the dominion of $D_n$ and the dominion of $C_n$ in $T_n$, denoted by $Dom_{T_n}(D_n)$ and $Dom_{T_n}(C_n)$, are characterized, and it is shown that they are regular idempotent-generated subsemigroups of $T_n$. Moreover, a formula for the number of their elements and their idempotents is given.