Papers

This article studies the optimal common free-final time problem for steering the state covariance of a continuous-time stochastic linear system between prescribed initial and terminal covariance matrices. We first establish a deterministic reformulation of the SDE-constrained free-final time stochastic optimal control problem (SOCP). For the ensuing SOCP, we provide necessary conditions for optimality and establish sufficient conditions for the optimal common final time to be finite. Subsequently, we characterize the asymptotic behavior of the finite-horizon optimal solutions as the final time tends to infinity, the invariant sets and trajectories associated with this limiting regime, and derive the sensitivity of the Hamiltonian with respect to the common final time. Finally, leveraging these sensitivities, we develop a trust-region line-search algorithm together with an infinite-horizon case detection method, and demonstrate its performance on three different problems: (a) an illustrative covariance-steering example, (b) a demonstration of spacecraft maneuver using covariance control, and (c) Gaussian mixture to Gaussian mixture steering.

We study natural topologies on spaces of subobjects of a fixed object in a suitable category, with the aim of determining when these spaces are spectral. A central step in our approach, which is also of independent interest, is the comparison between the categorical notion of a $\lambda$-generated object and the order-theoretic notion of a $\lambda$-compact element in a lattice of subobjects. We prove that these notions coincide under some natural and mild assumptions . In the finitary case, this allows us to describe the finitely generated subobjects purely in order-theoretic terms and to construct, inside each interval of a subobject lattice, a canonical algebraic core. We study this construction abstractly for complete lattices and characterize it by a universal property. We then introduce the categorical Zariski topology on spaces of subobjects, relate it to the Scott topology, and obtain spectrality criteria for the whole subobject space and for its algebraic core.

Projection theorems in Euclidean space provide a fundamental link between the geometric structure of a set and the size of its lower-dimensional images. For 1-rectifiable sets in $\mathbb{R}^d$, a classical theorem of Federer shows that the 1-dimensional Hausdorff measure of such sets is controlled by the multiplicity-weighted lengths of finitely many linearly independent projections. We develop a framework for extending Federer's result into a diverse set of nonlinear problems. This technique yields a unified approach for studying sets through their lower-dimensional nonlinear images, as well as studying the exceptional sets which exhibit poor projective behavior. As illustrations of our technique, we show that (i) every 1-rectifiable set contains a pin whose pinned distance set has positive Lebesgue measure, and that the exceptional set of pins for which this fails is contained in a $(d-2)$-dimensional affine subspace; (ii) planar radial projections of a 1-rectifiable set can fail to have positive length from at most one vantage point unless the set is essentially linear; and finally (iii) unions of circles centered on a 1-rectifiable set have positive area under mild assumptions on the radius function.

This paper studies event-triggered gradient-based multivariable extremum seeking for nonlinear maps with polytopic Hessian uncertainty. Unlike existing event-triggered extremum-seeking methods, which first fix the controller (typically diagonal) and then design the triggering mechanism, the proposed approach jointly synthesizes the controller and the triggering mechanism through a co-design framework that admits both diagonal and full controller gain matrices. The co-design problem is formulated as a convex optimization problem with linear matrix inequality constraints. Its solution guarantees exponential convergence of the average closed-loop system with a prescribed decay rate while maximizing the admissible triggering threshold to reduce communication. Lyapunov and averaging analyses establish exponential stability of the event-triggered system, and Zeno-freeness is proved to guarantee implementability. Numerical results illustrate that diagonal gains cannot achieve the same triggering thresholds and decay rates as the full controller gain matrices, highlighting the benefits of exploiting Hessian coupling information.

For a centered convex body $K\subset\mathbb{R}^n$, let $\Psi_2(K)$ denote the symmetric convex body whose support function is given by the $\psi_2$-norms of linear functionals on $K$. The recent solution of Milman's problem on the existence of subgaussian directions by Letwin and Mikulincer naturally motivates the study of the geometry of this body. We prove that $\Psi_2(K)$ has bounded volume ratio with respect to the $L_2$-centroid body $Z_2(K)$. In the isotropic case, we also obtain sharp estimates for its mean width and the volume radii of its orthogonal projections, and derive consequences for the existence of subgaussian orthonormal bases. In particular, we construct orthonormal bases with quantitatively controlled subgaussian constants for every isotropic convex body.

We study a variant of the graph polytopes of a path and of a cycle where we replace the inequality $x_{i} + x_{i+1} \leq 1$ with the two inequalities $(1-\alpha) \cdot x_{i} + \alpha \cdot x_{i+1} \leq \alpha$ for $0 \leq x_{i} \leq \alpha$ and $\alpha \cdot x_{i} + (1-\alpha) \cdot x_{i+1} \leq \alpha$ for $\alpha \leq x_{i} \leq 1$. Using a self-adjoint operator and its eigenvalues we obtain convergent series for their volumes. As a corollary we obtain that the volumes of the set associated to a path on $n$ vertices and the set associated to a cycle on $n+1$ vertices are related by a constant factor of $\alpha$.

We prove, in the case $t=1$, the extension to the $m$-symmetric world of the original Macdonald positivity conjecture. This is achieved by giving a combinatorial interpretation of the Kostka coefficients $K_{\Omega \Lambda}(q,1)$ in terms of standard fillings of the diagram associated to the $m$-partition $\Omega$. This interpretation generalizes the one in the usual Macdonald case, which is given by a major index statistic on standard tableaux.

In this paper, a new model for multi-valued logic is presented based on the notion of a bundle of predicates. The degree of truth and various logical operations for predicates in our model of multi-valued logic are rigorously defined and investigated. Furthermore, the bundle of sets as a special type of predicate bundles is thoroughly investigated to provide a rigorous model for fuzzy sets and certain adjectives/adverbs in linguistics. In addition, solutions to necessity/possibility and the sorites paradox in modal logic are given.

We give a criterion for a fibration to be an associated fibre bundle. As an application, we show that, under suitable singularity and Cartier-index assumptions, any surjective fibration $X \rightarrow Y$ with nef $-K_{X/Y}$ is an associated fibre bundle. From this, we also deduce a weak Beauville–Bogomolov decomposition for varieties with nef anticanonical divisor.

The finite element immersed boundary method (FE IBM) with a distributed Lagrange multiplier is an attractive framework for fluid structure interaction (FSI) because it couples an Eulerian description of an incompressible fluid to a Lagrangian description of an immersed solid on two independent, nonconforming meshes, avoiding the costly remeshing required by body fitted arbitrary Lagrangian Eulerian methods. When the immersed solid is modelled as a fully incompressible hyperelastic material, however, a direct finite element discretization of the deviatoric solid stress fails to enforce the Lagrangian incompressibility constraint, and the computed structure exhibits spurious volumetric instabilities and locking. In this work we present a mixed formulation of the distributed Lagrange multiplier FEIBM that restores volumetric stability. Following the theory of nearly incompressible hyperelasticity, we augment the solid stress with a volumetric contribution derived from a dilatational strain energy and introduce an additional solid pressure field, enforced weakly, that plays the role of the Lagrange multiplier for the Lagrangian incompressibility constraint J = 1. The resulting formulation is discretized in space by finite elements and in time by an unconditionally stable semi implicit scheme, and is implemented as a reusable Immersed Boundary physics kernel within the GRINS multiphysics framework, built on the libMesh finite element library. The method is verified on three FSI benchmarks, an elliptically displaced thick ring returning to equilibrium, a radially stretched incompressible ring, and a disk falling under gravity in a viscous fluid, for which the mixed formulation removes the volumetric failure observed with the unstabilized formulation and reproduces the analytical terminal velocity of the falling disk to within 1%.

The Cifarelli–Regazzini identity lies at the foundation of the theory of the Dirichlet process and has played a central role in Bayesian nonparametric statistics. By providing the generalized Cauchy–Stieltjes transform of linear functionals of the Dirichlet process, together with an analytic inversion, it yields explicit representations for the distribution function and density of the Dirichlet mean. This transform identity was subsequently extended to the Pitman–Yor process, a generalization of the Dirichlet process, leading to an analogous distributional theory for its linear functionals. Nonlinear functionals, by contrast, remain less understood, with results largely confined to a few examples for the Dirichlet process. In this paper, we move beyond linear functionals by considering the variance of the Pitman–Yor process, a genuinely quadratic functional. In particular, we establish a Cifarelli–Regazzini-type identity providing the generalized Cauchy–Stieltjes transform of the Pitman–Yor variance for general base probability measures under suitable integrability assumptions. We then analytically invert this transform under general conditions on the base probability measure, thus obtaining explicit representations for both the distribution function and density. Corresponding formulas are also obtained for the Dirichlet variance, yielding more explicit representations than those currently available in the literature. We also establish a distributional representation of the Pitman–Yor variance in terms of a Dirichlet process whose base probability measure is the joint distribution of the mean and variance of a suitably specified Pitman–Yor process, mirroring the classical within–between decomposition of variance and providing a quadratic counterpart to a known distributional identity for Pitman–Yor means.

The integration of weather-dependent renewable generation increases the volatility of residual demand and raises the value of dispatchable low-carbon flexibility. This paper studies the optimal operation of a load-following nuclear power plant owned by a producer that must balance stochastic residual demand while accounting for ramping limits and costly changes in operating regimes. Nuclear output can be increased, decreased, or kept constant, and the production decision is formulated as a finite-horizon optimal switching problem. We analyze both a closed-economy benchmark, where excess production cannot be sold and shortages require costly back-up generation, and an open-economy setting, where the producer can trade electricity at prices driven by aggregate market residual demand. The value functions are characterized as viscosity solutions of a system of Hamilton-Jacobi-Bellman quasi-variational inequalities, and optimal policies are computed using a monotone semi-Lagrangian scheme. The numerical results show how shortage costs, switching costs, ramping capability, and market access shape optimal nuclear load following. The analysis highlights the economic value of controllable low-carbon capacity in renewable-intensive systems and provides implications for flexibility remuneration, balancing-market design, and interconnection policy.

We prove a necessary and sufficient Hall condition for a family $A=(A_e)_{e\in E(G)}$ of hypergraphs indexed by the edges of a forest \(G\). This restriction on the index graph is sharp. The loop-only case recovers the classical Hall's Theorem with multiplicities for arbitrary finite set systems, while the loopless case shows that full rainbow matching is polynomial time solvable under this forest structure, although the problem is NP-complete in general. As another application, we prove every $5$-tough chordal graph is Hamilton-connected, improving toughness bounds of $18$ for Hamiltonicity (1998) and $10$ for Hamilton-connectedness (2017).

We develop a theory of integration with respect to quantum instruments through two complementary approaches: a vector measure formulation based on Bartle's integration theory and a tensor product construction. As a principal application, we establish a CP map-valued Lyapunov theorem by characterizing the convexity of the range of non-atomic completely positive instruments via the associated integration map. This extends the work of Plosker and Ramsey~\cite{Plosker_Ramsey} from POVMs to completely positive instruments and gives a general characterization of the phenomenon exhibited in their setting. We also establish a correspondence between completely positive instruments and completely positive maps, prove a Krein–Milman type theorem for the $C^*$-convex set of unital completely positive instruments, and show that the tensor product construction connects the theory of CP instruments with the integration theory for POVMs due to Farenick et al(\cite{douglus_plosker_ramsey_povmintegration_1}).

This paper analyzes mixing time lower bounds for random walks on sparse, heavy-tailed Random Intersection Graphs. In sparse feature regimes, heavy-tailed feature distributions lead to the formation of peripheral trap – chains of overlapping low-weight feature cliques attached to high-weight hub nodes within the graph's giant component. By modeling escape trajectories from these traps as continuous limit hitting times for reflected Brownian motion, the analysis demonstrates that random walks experience logarithmic squared delays. Consequently, the mixing time is bounded below by $\Omega(\log^2 n)$, and the local total variation distance exhibits non-concentrated decay, formally preventing a sharp cutoff phenomenon.

Let $x_1,\ldots,x_n$ be independent standard Gaussian vectors in $\mathbb{R}^d$. An ellipsoid fit is a matrix $S \succeq 0$ such that $x_i^\top S x_i =d$ for every $i$, so that all the points lie on the boundary of the centered ellipsoid $\{ x : x^\top S x = d\}$. Saunderson, Parrilo and Willsky conjectured that, as $n,d \to \infty$, this semidefinite feasibility problem undergoes a sharp transition at $n \sim d^2/4$. We prove this conjecture. If $\lim \sup n/d^2 = \alpha^* <1/4$, then, with probability tending to one, an ellipsoid fit exists; moreover, one can choose $S$ with all eigenvalues in a fixed interval $[\lambda_- , \lambda_+] \subset (0,\infty)$ depending only on $\alpha^*$. Conversely, if $\lim \inf n/d^2 > 1/4$, then, with probability tending to one, no ellipsoid fit exists, without any spectral restriction. Our proof builds on the Gaussian-equivalence framework developed by Bandeira and Maillard (2025) and closes the two gaps left open in their work: establishing exact fitting and removing the operator-norm constraint. On the satisfiable side, the new ingredients are a head-tail decomposition of the dual vector, exact correction of the sparse head constraints, and a Gaussian comparison principle for the low-influence tail. On the unsatisfiable side, we split a candidate into a low-rank spectral head and a Schatten-3 diffuse bulk, Gaussianize the bulk conditionally on the head, and apply a projected Gordon escape argument. The threshold is governed by the statistical dimension $d(d+1)/4$ of the positive semidefinite cone.

We introduce an infinite-dimensional Galves–Löcherbach system with bounded Elephant-type reinforced synaptic interactions. The reinforcement mechanism modifies the probabilities of excitatory and inhibitory synaptic updates while keeping their amplitudes uniformly bounded. We prove non-explosion by means of a Lyapunov estimate and establish a conditional Wasserstein contraction for the membrane-potential dynamics when the coupled systems share the same reinforcement profile. Finally, we derive the corresponding replica mean-field equation under the Poisson hypothesis.

A tensor has border rank at most $r$ if it can be written as $T=\lim_{\varepsilon \rightarrow 0} T(\varepsilon)$ where $T(\varepsilon)$ has rank at most $r$ for all sufficiently small $\varepsilon$. It is known that the map $\varepsilon \mapsto T(\varepsilon)$ can be assumed to be a (tensor valued) polynomial in $\varepsilon$. The smallest possible degree of such a map is called the error degree of $T$. The error degree and the related notion of order of degeneration are the two key quantities that we study in this paper. One motivation comes from debordering: by polynomial interpolation on the map $\varepsilon \mapsto T(\varepsilon)$ we can upper bound the tensor rank of $T$. For order 3 tensors, exponential upper bounds on the error degree and degeneration order were given almost 40 years ago in (Lehmkuhl Lickteig, 1989) and were not improved ever since. In this paper we give bounds that apply to a wide class of tensors, exponentially improving on (Lehmkuhl Lickteig, 1989). Our results are most general for tensors with 3 slices (format $m \times n \times 3$). In this case, our main assumption is on the rank of the matrix slices. We also give bounds that apply to arbitrary rectangular formats ($m \times n \times p$). In this case, we need an additional 1-regularity assumption on one of the slices of the tensor (recall that a matrix is said to be 1-regular if its eigenspaces are 1-dimensional). Under these assumptions we show that the error degree is at most 1, which yields a nontrivial debordering result (tensor rank at most $2r$ for border rank $r$). The results in (Lehmkuhl Lickteig, 1989) rely on an upper bound on the degree of the variety of tensors of border rank at most $r$. We rely instead on more specific properties of this algebraic variety, and in particular on commutativity properties of certain matrices derived from the tensor slices.

We prove that a graph $G$ is quasi-planar - i.e. quasi-isometric to a planar graph - if and only if it can be obtained by iterating the following two operations a bounded number of times: a) subdividing each edge into a path of bounded length, and b) taking the intersection graph of a family of connected subgraphs covering $G$. This applies both to infinite graphs, and to families of finite graphs with uniform constants. The backward implication relies on, and generalises, a deep result of Davies, partly proved independently by Chang, Conroy, Tan & Zheng, saying that every string graph is quasi-planar. The forward implication requires new ideas. As a byproduct of our proofs, we deduce that every contraction minor of a quasi-planar graph is quasi-planar. Moreover, if $G$ admits a tree-decomposition with adhesions of bounded diameter and quasi-planar induced bags, then $G$ is itself quasi-planar. Our results apply to other graph classes as well, and we offer various tools for understanding quasi-isometries as well as bi-Lipschitz equivalences between graphs.

Inspired by recent developments in dissipativity-based control, this work proposes new sufficient conditions for asymptotic stabilization of nonlinear systems using state feedback. We prove that this new framework is well suited for data-driven control of polynomial systems using noisy measurements, a topic that has lately attracted considerable attention. Two iterative procedures for data-based state feedback design are provided using the proposed framework, that use sum-of-squares (SOS) optimization. Typical limitations of conventional SOS methods based on alternating (D-K) procedures for controller design, such as the need to provide an initialization for a control-Lyapunov function (CLF), are overcome in this paper. Numerical examples demonstrate the applicability and the advantages of the new strategies.

Powerful black-box predictive models have motivated many proposals for combining observed data with predictions to perform valid statistical inference. Despite this progress, the field lacks a unifying principle that explains how hypothesis tests should integrate data and predictions in a way that is both valid and efficient. In this work, we address this gap in the high-dimensional Gaussian sequence model. We characterize the information-theoretic limits of inference with black-box predictions when their accuracies are known and, for orthogonal predictions, when they are unknown. Building on these characterizations, we develop practical hypothesis tests that adapt to the unknown accuracies of the predictions while benefiting from strong alignment among them.

For every nontrivial finite group, we prove that its quasirandom degree gives a polynomial lower bound on its uniform twin-width, whereas its minimum faithful complex representation degree gives a linear upper bound. For nonabelian finite simple groups, these two parameters coincide, so uniform twin-width is polynomially equivalent to quasirandomness in that class, yielding a new definition of quasirandomness in the sense of Gowers. We use the lower bound to prove that uniform twin-width is unbounded over finite groups, which helps us construct finitely presented groups with finite twin-width but infinite uniform twin-width. This answers a question of Bonnet, Geniet, Tessera and Thomasse. Finally, we determine the uniform twin-width of all three Thompson groups.

Reinfection and relapse are known to induce backward bifurcation (a subcritical pitchfork bifurcation) in many infectious disease models, particularly tuberculosis. This phenomenon implies the coexistence of multiple equilibria when the basic reproduction number is below one: a stable disease-free equilibrium and two endemic equilibria, one stable and one unstable. As a result, hysteresis may arise, meaning that reducing the reproduction number below one may not eliminate the disease given that the system remains in the basin of attraction of the endemic state. Another relevant phenomenon, often associated with intervention delays, is the occurrence of supercritical Hopf bifurcations. In this case, the endemic equilibrium loses stability for reproduction numbers above one, leading to sustained oscillations in the form of stable limit cycles. Although less explored in the tuberculosis literature, this behavior has important implications for disease control. In this work, we analyze a tuberculosis model to investigate these dynamical effects and show that Hopf bifurcation can arise even without delays or saturation effects. The bifurcation analysis focuses on two key public health interventions -case finding and case holding- and their combined effect on disease dynamics. Numerical simulations, based on epidemiological data from Salta, Argentina -a province with high TB burden- illustrate the challenges these dynamics pose for the design and implementation of public health policies.

Understanding the geometry of complex networks is critical for effective modeling and analysis across domains. While discrete notions of Ricci curvature have emerged as powerful tools for characterizing both local and global network structure, existing formulations are largely confined to undirected networks with real-valued weights. This limits the use of curvature-based analysis of directional and complex-weighted relations that arise naturally in many applications, from social and biological systems to quantum and signal-processing networks. In this work, we introduce a principled extension of Ollivier's Ricci curvature to complex-weighted graphs, which encompasses directed graphs as a special case. We establish fundamental theoretical properties of this new notion, including relations to the magnetic Laplacian and combinatorial upper and lower bounds that relate curvature to cycle structure in local neighborhoods. We further develop computational methods for curvature estimation and demonstrate their utility in community detection on directed networks.

Let $\gamma_1$ and $ \gamma_2$ be regular $C^1$-smooth curves in the plane and $\gamma$ be a regular $C^2$-smooth curve in the same plane. Consider all triples of points $(A, A_1, A_2)$, $A\in \gamma$, $A_1\in \gamma_1$, $A_2\in \gamma_2$, such that $A_1\neq A_2$, and the line $A_1 A_2$ is the normal to $\gamma$ at $A$. We show that, if $\gamma$ has non-vanishing curvature and the triple $(A^0, A_1^0,A_2^0 )$ is a local maximum or a local minimum for the distance $|A_1A_2|$ between the points $A_1$ and $A_2$, then the following three lines either meet at a single point or are parallel: the normal to $\gamma_1$ at $A_1^0$, the normal to $\gamma_2$ at $A_2^0$ and the line which is perpendicular to $A_1^0A_2^0$, and passing through the center of curvature of $\gamma$ at $A^0$. The particular case of this optimization problem, when $\gamma$ is a circle with a given center $O$, coincides with the already partially studied problem of finding the locally shortest (or the locally longest) non-degenerate segments $[A_1A_2]$ such that $A_1\in\gamma_1$, $A_2\in\gamma_2$ and $O\in A_1A_2$. We also show that the seemingly different problem of finding the locally shortest (or the locally longest) non-degenerate segments $[A_1A_2]$, such that $A_1 \in \gamma_1$, $A_2 \in \gamma_2$, and the line $A_1A_2$ is tangent to $\gamma$ is also, in essence, a particular case of the above optimization problem. We consider in detail the “degenerate cases” naturally appearing in this setting (when, for instance, $\gamma_1$ or $\gamma_2$ coincide with $\gamma$, or when the optimal line $A_1^0A_2^0$ is tangent to at least one of $\gamma_1$ or $\gamma_2$).

We classify Fourier summation identities in which the measure on the Fourier side is supported in a horizontal strip of $\mathbb{C}$. Let $\mu=\nu+\eta$, where $\nu$ is a strongly tempered measure on $\mathbb{R}$, $\eta=\sum_{m\geq1} b(\gamma_m)\delta_{\gamma_m}$ is a strongly tempered pure point measure supported off the real line, and $a:\Lambda\rightarrow\mathbb{C}$, with $\Lambda=\{\lambda_n\}_{n\geq1}\subset\mathbb{R}$, has finite exponential growth. Under natural real-antipodal and conjugation-symmetry assumptions, we characterize summation identities of the form \begin{align} \sum_{n\geq1} a(\lambda_n)\varphi(\lambda_n)=\int_{\mathbb{R}} \widehat{\varphi}(t)\mathrm{d}\nu(t)+\sum_{m\geq1} b(\gamma_m)\widehat{\varphi}(\gamma_m), \end{align} valid for every $\varphi\in C^\infty_c(\mathbb{R})$, where $\widehat{\varphi}$ denotes the Fourier transform. We prove that each such identity determines a unique generating function $F$ that is holomorphic and almost periodic in the half-plane above the strip and admits a meromorphic continuation to $\mathbb{C}^+$. The measure $\eta$ encodes the poles and residues of $F$, while $\nu$ describes the boundary behavior of its regular part through a generalized Nevanlinna representation, and $a$ determines its Fourier coefficients. Conversely, every function in the corresponding meromorphic class whose Fourier coefficients satisfy a local summability condition determines a unique summation identity of this form. The proof combines a strip version of the Bridge Lemma with a Cauchy-transform argument that accounts for the off-real poles. As an application, we show that the Guinand-Weil explicit formula for every member of the Selberg class, including non-self-dual members, fits into our framework, and we identify its associated generating function.

We provide a fairly simple list of conditions on a group $G$ acting acylindrically on a $\delta$-hyperbolic metric space sufficient for the group to admit an action on a set that is $k$-sharp, transitive on $k$-sets and has the property that for any $k$-set $A$ the setwise stabilizer of $A$ acts on $A$ as prescribed by some fixed subgroup of the group of permutations of $k$ elements. In particular, this yields a large class of non-split sharply $2$ and $3$-transitive actions containing many finitely presented groups, and allows us to characterize sharp $2$-transitivity within the class of hyperbolic groups that do not contain a copy of $C_{2}\times C_{2}$.

We prove that compact hyper-Kähler manifolds with a Lagrangian fibration over a projective space have no multiple fibers in codimension one. This has several consequences for the structure of Lagrangian fibrations, including progress on Sawon's conjecture on general singular fibers, Kamenova–Lu anti-hyperbolicity, and the extension of the Néron model action to a big open subset of the base. We prove the same result for Calabi–Yau fibrations on simply-connected K-trivial varieties, including elliptic fibrations, with a single exceptional case: an odd-dimensional K-trivial variety with a single fiber of multiplicity 2 over the projective line. This exceptional case is realized by examples of Borisov–Nuer and their generalizations.

Thread scaling in Hexaly on the Time-Dependent Vehicle Routing Problem with Time Windows (TDVRPTW) depends on modeling, and on the language binding through which the model reaches the solver. We compare two encodings, both reaching the solver through its C++ binding: one evaluates continuous travel-time functions exactly through external callbacks, which that binding evaluates concurrently, while the other approximates them with time slices evaluated natively by the solver. Independent CPU accounting confirms that both use the cores they are allocated, but they respond differently to width. The external-function encoding stabilizes, reducing its seed dispersion by about 30% up to 8 threads before widening again, yet its pooled quality barely moves, because nine of the ten instances improve while the hardest one degrades by enough to cancel them. For the time-sliced encoding we first select a discretization on feasibility under the original travel-time functions rather than on approximation error, and the retained setting then improves monotonically with the thread count, gaining about a quarter of its final gap and reducing the seed dispersion by nearly half between 1 and 16 threads. At equal single-threaded budget the two encodings are close, and the small pooled edge of exact evaluation comes from that same hardest instance, so the practical advantage of the discretization is thread scaling rather than fidelity traded for speed. These descriptive results support multi-threading as a way to improve solution quality and stability when the model can exploit it, and they show that binding-level constraints on external evaluation must be measured, not assumed.

We study the recursion coefficients of orthogonal polynomials and their associated Krylov dynamics in random matrix models with high-degree and possibly asymmetric polynomial potentials. We develop a moment recursion method that, when combined with the recursive algorithm, provides an efficient construction of the recursion coefficients. We also obtain their large-$n$ asymptotic behavior for general asymmetric potentials; for $Nw_d=1$, the leading asymptotic form of $R_n$ reproduces Freud's conjecture. We apply this framework to an asymmetric quartic potential and to the double-scaled Sachdev-Ye-Kitaev (DSSYK) model. In both models, the recursion functions capture the overall qualitative behavior of the recursion coefficients, and the gradient catastrophes of the recursion functions are associated with “chaotic” transition regions in the recursion coefficients. For the quartic potential, such regions can occur in both $R_n$ and $S_n$, whereas the DSSYK model can exhibit multiple transition regions in $R_n$, with the recursion function remaining accurate in the smooth intervals between them. Finally, we compute the corresponding spread complexity and find that transition regions do not qualitatively modify its behavior, while a two branch structure produces early time oscillations followed by monotonic growth.