Papers

We study maximal subgroups of topological full groups of irreducible one-sided shifts of finite type whose associated groupoids have Cantor unit space. For each wide open subgroupoid of such a groupoid, we construct a subgroupoid graph and identify the subgroupoid with a full compact open reduction of the corresponding countable graph groupoid. Every minimal wide open subgroupoid is either AF or purely infinite. A proper wide open subgroupoid is prime if it admits no proper intermediate wide open subgroupoid. We prove a trichotomy for prime subgroupoids: they are minimal, arise from a clopen bipartition, or have a unique nonempty proper closed invariant set, which is nowhere dense. For a prime subgroupoid, the normalizer of its topological full group is maximal in the ambient topological full group precisely when it maps onto the ambient abelianization. An analogous criterion holds in every subgroup containing the ambient commutator subgroup. For a minimal prime subgroupoid, its topological full group is maximal precisely when it equals its normalizer and the inclusion induces a surjection on first groupoid homology. We also describe all fully amplified maximal subgroups, namely those closed under local agreement. For Thompson's group $V$, the stabilizer of every simple type system is maximal, extending results of Belk, Bleak, Quick and Skipper for finite type systems and certain classes of infinite ones. Finally, every subgroup containing the ambient commutator subgroup has continuum many pairwise nonisomorphic maximal subgroups acting minimally on the unit space.

Metric spaces of negative type (NT) and strong negative type (SNT) are fundamental in wide applications across computer science and statistics. It is well-known that spheres equipped with geodesic distance are of NT, and subsets of a sphere that contain at most one pair of antipodal points (e.g. an open hemisphere) are of SNT. In this paper, we extend these results to a class of star-shaped manifolds. By constructing a metric $d$ based on layer-wise radial projections, we prove that every star-shaped manifold, equipped with $d$, is of NT. Moreover, we prove that a subset of a star-shaped manifold is of SNT under certain injectivity and directional asymmetry conditions. In particular, for a subset of a single layer, it is of SNT if it contains exactly one point from each preimage of an open hemisphere (a fully directional asymmetry); moreover, it is of SNT if and only if it contains at most one opposite-directional pair.

This paper studies popularity dynamics in social networks, where users' engagement with competing channels is driven by intrinsic content quality, social influence, and popularity feedback. We introduce a stochastic model in which users interact through a directed influence network and channel exposure depends jointly on these three mechanisms. We first characterize the resulting dynamics in the absence of intrinsic quality, showing that popularity concentrates asymptotically on a single channel, leading to a winner-takes-all regime. When all channels have positive intrinsic quality, we establish ergodicity of the popularity process and characterize its long-run behavior. We further show that, when intrinsic quality is small, the system exhibits metastable behavior: it can remain for long periods near configurations dominated by a single channel before transitioning to another. These results highlight how weak quality differences can coexist with strong path dependence induced by popularity feedback. We then investigate platform-level interventions and propose controls that compensate for network-induced exposure biases and mitigate excessive concentration. Analytical results are illustrated through numerical simulations and an empirical analysis of YouTube data, whose observed popularity dynamics are consistent with the proposed stochastic framework. The results provide a mathematical framework for understanding the interplay between quality, social influence, and reinforcement in online popularity dynamics.

It is folklore that passing "convergence" diagnostics does not guarantee that a Markov chain has converged. We prove this as an impossibility for any finite checklist of diagnostics. Running more chains, longer chains, and smarter diagnostics does not avoid this impossibility in certifying convergence. Under regularity assumptions, we determine how the worst-case undetected non-convergence decays as more diagnostics are added, and how to choose diagnostics optimally. Folklore suggests that the sampler's slowest mixing modes are the "dangerous" ones that must be monitored; we prove that the slow modes yield nearly optimal diagnostics as sampling proceeds. Given a source class, we also give an anytime-valid bound on the remaining discrepancy and show how undetected bias caps the effective sample size that a finite checklist can certify.

We study radially nonincreasing, positive blow-up solutions of a class of superlinear heat equations in the non-scale-invariant setting. This class contains nonlinearities with arbitrarily fast, super-exponential growth. Under suitable structural assumptions on the nonlinearity, we derive the sharp global blow-up profile in the original space-time variables. More precisely, for positive nonlinearities $f$ satisfying $$ \lim_{X\to \infty}f'(X)G(X)= q\ge1\quad \quad \hbox{ where }\quad\quad G(X)=\int_{X}^{\infty}\frac{ ds}{f(s)}<\infty\quad\forall\ X>0, $$ and assuming either the time monotonicity $u_t\ge0$ holds or $q>\frac{p_S}{p_S-1}$, we have $$ u(x,t)=(1+o(1))\,G^{-1}\bigg(T-t+\frac{|x|^2}{4q|\log |x|^2|}\bigg), \quad \hbox{as $(x,t)\to (0,T)$.} $$ This estimate in particular provides the sharp final space profile and the corresponding refined space-time profile. Moreover, our result recovers the universal structure of the global blow-up profile, given by the resolvent $G^{-1}$ of the associated ODE evaluated at a time-space building block, which only depends on the limit $q$.

We study unfoldings of foliated complex analytic structures defined by locally free subsheaves of cotangent sheaves. We prove theorem of existence for unfoldings of foliated complex analytic structures defined by locally free subsheaves of cotangent sheaves for one-parameter families. We prove theorem of completeness for unfoldings of foliated complex analytic structures defined by locally free subsheaves of cotangent sheaves. We also present unfoldings of foliated nonsingular varieties defined by locally free subsheaves of cotangent sheaves in the language of functors of Artin rings which is the algebraic version of unfoldings of foliated complex analytic structures. We identify first-order unfoldings and obstructions.

Let $E:y^2=x^3-x$. We prove that $\#E(\mathbf{F}_p)/8$ is squarefree and has at most two prime factors for $\gg X/(\log X)^2$ primes $p\le X$. The proof follows the Gaussian CM sieve framework of Iwaniec and Jiménez Urroz and incorporates the intersection terms identified by Xie. For a squarefree Gaussian ideal $\mathfrak{a}$, we separate its largest rational ideal factor and use the resulting divisibility of both integral coordinates to control the large-factor tail. This transfers an ideal Bombieri–Vinogradov estimate to rational norm divisibility at level $1/2$, with any fixed power of the divisor function as a weight. We prove the corresponding distribution estimate for the switched product sequence and apply the transfer to both sequences in Chen's weighted sieve. The primes counted may be restricted to $p\equiv5\pmod8$; their normalized orders are odd, squarefree, and supported on primes congruent to $1\pmod4$.

For all sufficiently large $n$, we prove that every Latin square of order $n$ can be made to avoid any prescribed partial Latin square of order $n$ by permuting only its rows and columns. The proof combines Hall's theorem with elementary estimates for a random permutation.

We study the minimax estimation of the pseudo-spectral gap $\gamma_{ps}$ of a Markov chain from the observation of a single trajectory. Extending earlier works on finite-state space chains by Wolfer and Kontorovich (2024), we provide an estimator in the case of a compact, continuous state-space. Our estimator appears to be minimax-optimal on the class of chains with a transition operator with Lipschitz density, up to logarithmic terms. Our estimator, combined with Paulin's version of Bernstein inequality, allows to derive empirical concentration inequalities for Markov chains. This unlocks many applications, from confidence intervals, to active learning and generalization bounds. In particular, for rested bandits with Markov rewards, we provide a version of UCB with a regret bound similar to the one in the iid case, without assuming any knowledge on the pseudo-spectral gap of the chains.

We study the relation between the exact-WKB Delabaere–Dillinger–Pham Stokes jumps and the multi-instanton expansion of a one-dimensional potential with $N$ equal-depth minima. For a single barrier this relation is the Bogomolny–Zinn-Justin cancellation between the instanton–anti-instanton ambiguity and the Borel ambiguity of perturbation theory, and we extend it to every consecutive barrier interval and to the integration contours themselves. In the dilute large-$T$ regime at fixed nonperturbative grade, we derive the quasi-zero mode integral over the instanton separations from the Euclidean path integral and reduce it to Gamma and Hankel periods. With the local Weber level function of each well and one normalization per well fixed by the local Weber connection matrix, these periods reproduce the two lateral connected exact-WKB coefficients. The two lateral contours of each instanton–anti-instanton separation differ by exactly one Hankel cycle, and this linear Picard–Lefschetz relation reproduces the Stokes jump of the connected coefficient. A numerical solution of the Schrödinger equation for a symmetric triple well confirms the resulting two-barrier level splitting. At higher fixed grade, the map from ordered event sequences to normalized scalar coefficient products is not one-to-one. At leading dilute order the scalar weight of a closed event sequence depends only on the numbers of barrier crossings, $\mathcal I\bar{\mathcal I}$ pairs, and $\mathcal I\mathcal I$/$\bar{\mathcal I}\bar{\mathcal I}$ pairs. We construct two inequivalent grade-six triple-well event sequences with the same product of local Weber amplitudes, which embed in every potential with $N\ge3$ wells. Our results hold within the regular dilute, one-loop, local Weber approximation.

Let $n\geq2$ and $1<p<n$. We study sharp Sobolev inequalities on the round sphere $\mathbb{S}^n$ for functions $u\in W^{1,p}(\mathbb{S}^n)$ satisfying $\int_{\mathbb{S}^n}x|u|^{np/(n-p)}\,d\sigma=0$. We prove that \[ \|u\|_{L^{np/(n-p)}(\mathbb{S}^n)}^p \leq 2^{-p/n}S_{n,p}^p\|\nabla u\|_{L^p(\mathbb{S}^n)}^p+B\|u\|_{L^p(\mathbb{S}^n)}^p \] holds for all such functions with some $B=B(n,p)<\infty$ if and only if $1<p\leq2$ or $2<p<n<3p-2$, where $S_{n,p}$ is the optimal Euclidean Sobolev constant. This determines the exact range in which Aubin's inequality holds with the sharp gradient coefficient. For every $2\leq p<n$, we also establish the corresponding inequality with squared norms and gradient coefficient $2^{-2/n}S_{n,p}^2$. The gradient coefficient in each inequality is optimal. The proof combines stereographic averaging with quantitative stability of the Euclidean Sobolev inequality to obtain local second moment estimates that allow absorption of the localization errors.

We study finite-maturity American puts in the Heston model without the Feller condition. At positive variance, the exercise boundary is smooth and strictly decreasing in remaining maturity and variance. At zero variance, the endpoint is locally Lipschitz in remaining maturity, the boundary moves with bounded speed in maturity uniformly down to zero variance, and it extends continuously to the axis. Each boundary slice at positive maturity has a regular $C^1$ closure tangent to the zero-variance axis, its stock threshold has variance derivative tending to minus infinity, and both partial derivatives of the inverse boundary graph vanish along the slice. The continuation premium at the axis contact stock is of order variance, and as a consequence smooth fit fails in the variance direction at the axis. At expiry the whole boundary, including its axis edge, converges to the strike. We give quantitative boundary gap bounds, an exact limiting weighted boundary Hessian, and vertical identities retaining the maturity term. Exact boundary gap equivalents remain open.

We give an elementary proof that if $f$ is a locally constant function on a topologically mixing subshift of finite type, then the equilibrium measures for $tf$ converge as $t\to\infty$, a result due to Brémont. The proof shows that the limit measure assigns an algebraic number to every cylinder.

The spherical code problem asks how to place $N$ points on a unit sphere in $D$ dimensions to maximize the smallest distance between two points. It is a non-smooth and non-convex max-min problem, where few problem sizes have known optimal solutions, some have constructions, but most still only have numerically obtained solutions. We use a simple solver, with seven parameters, that anneals a LogSumExp surrogate, minimizes with L-BFGS, applies a random perturbation between anneals, and repeats with no acceptance test until a fixed limit is reached. We search the solver parameters for every instance with $N \le 1024$, $3 \le D \le 32$, and $N > 2D$, a total of 29,670 instances, using 32 parameter trials. We then run the solver for 60 seconds with the best parameters for each instance, without utilizing any symmetries or existing solutions. After enforcing monotonicity of the results and numerically polishing the solutions, we improve the best known code in 4,577 instances. When a single set of solver parameters is used for all instances, with one 60-second episode, the number of improvements is still 3,649. From our records, we find 24 codes with an exact algebraic cosine. Among our ablations, we find that 1) the repeated anneal contributes most to improvements, 2) at our $32 \times 60$-second budget only fitting one parameter per-instance is better than fitting all seven, and 3) good parameters can be predicted from the instance size. We provide the solver code, the fitted parameters, and the codes.

Inspired by a pioneering work by Caffarelli, Gidas, and Spruck, we study positive solutions to the Yamabe equation with nonremovable isolated singularities. It is known that there is a formal one-to-one correspondence between the collection of the solutions to the Yamabe equation with nonremovable isolated singularities and the collection of harmonic functions through formal infinite series determined by the harmonic functions. In this paper, we prove that these formal infinite series yield actual solutions to the Yamabe equation with a nonremovable isolated singularity. These solutions to the Yamabe equation have designated asymptotic behaviors determined by the harmonic functions. In a broader context, we provide a general method of constructing all solutions near their isolated singular points and all solutions near infinity.

We develop a boundary value and potential theory for elliptic operators \[ L=-\Div(A\nabla)+\B\cdot\nabla \] with form-bounded drift in one-sided chord-arc domains. Assuming that the positive part of $\Div\B$ has sufficiently small form bound, we establish local regularity estimates and solvability of the Dirichlet problem, and construct the associated harmonic measure. Under the additional assumption $\Div\B\le0$, we construct a Green's function and prove pointwise and gradient estimates. For the characteristic distributional identity with a Dirac mass at the pole to hold, we will additionally assume either $\Div\B=0$ or $\B\in L^{n,1}_{\textup{loc}}(\Omega)$. Moreover, in the divergence-free case, we obtain a local lower bound for the Green's function, CFMS estimates, doubling of harmonic measure, a comparison principle for nonnegative solutions with zero trace, and a change of pole formula. Finally, using AI, we construct, in the upper half-space, an operator of the form $-\Delta+\B\cdot\nabla$ with $\B$ form-bounded and divergence-free, whose harmonic measure exhibits dimension drop.

We study a spherical perfectly matched layer (PML) method for the time-harmonic Maxwell scattering problem with large wavenumber. Using vector spherical harmonics and a modal transformation for the coupled transverse magnetic components, we establish the sharp wavenumber-explicit stability estimate for the truncated PML problem and prove that its inf-sup constant is of order $O(\kappa^{-1})$. We then analyze the first-order Nédélec edge element method and its continuous interior penalty counterpart. Under the condition that $\kappa^3h^2$ is sufficiently small, the preasymptotic errors in the energy norm and the $\kappa$-scaled $\mathbf{L}^2$ norm are of orders $O(\kappa h+\kappa^3h^2)$ and $O(\kappa^2h^2+\kappa^3h^2)$, respectively. Numerical experiments confirm the theoretical results.

Extremal constructions play a central role in the design and analysis of algorithms. Integrality-gap instances expose the limitations of relaxations, while hard configurations reveal structural sources of computational difficulty. Such constructions, however, are still typically discovered through a slow, manual process. This paper investigates how machine learning can accelerate their discovery to support algorithm design and analysis. We build on PatternBoost, a framework alternating local search with generation from a trained transformer, using its Axplorer implementation to investigate three problems: integrality gaps for maximum independent set of rectangles, integrality gaps for stabbing unit squares with axis-parallel lines, and hard instances for recursive guillotine cutting. By exploring candidate representations, local search operations, and surrogate models used to rank candidates before exact verification, we carefully adapt the framework to these combinatorial geometry problems.While naive representations and search strategies fail to find non-trivial instances, our tailored adaptation discovers numerous non-trivial constructions. For stabbing unit squares, the method reaches a gap of $20/13\approx 1.54$, improving on the literature's best known lower bound of $3/2$. Purely automated search converges rapidly and then plateaus, prompting the question of what is required to push the bound higher still. We find that initial instance diversity plays an important role: with modest expert input via a structured staircase construction, the system reaches an improved integrality gap of $14/9\approx 1.555$. These results demonstrate that machine learning and domain expertise can be combined to discover new extremal constructions that advance algorithmic lower bounds.

We present a novel class of minimum energy estimation problems for positive linear systems with linear costs subject to coupled process and measurement disturbances, where the latter is bounded by the state and the former is bounded by the noise. Through dynamic programming we show that the problem admits explicit solutions. Remarkably, the Luenberger observer structure of the optimal estimator emerges from the optimization itself, with a switching gain that inherits the sparsity of the disturbance constraints, and it is obtained from a differential equation in the finite horizon case. In the infinite horizon case, an algebraic equation characterizes the limit of the finite-horizon solution and yields a stationary estimator with a static gain. We introduce two distinct notions of detectability to characterize when the static gain stabilizes the error dynamics while preserving their positivity, and analyze under which conditions the stationary solution can be formulated as a linear program. We further establish that a disturbance perturbation of finite L1-norm yields an estimation error of finite L1-norm. The framework is illustrated on a large-scale irrigation network.

Association schemes induced on the shells of the Grassmann scheme $Gr_q(N,a)$ form an important family in the study of multivariate $P$- and $Q$-polynomial structures. For $a<N/2$, the scheme induced on the furthest shell $X_a$ is the known bivariate $P$- and $Q$-polynomial scheme arising from attenuated spaces. In this paper, we prove that, for every $1\leq i\leq a-1$, the association scheme induced on the interior shell $X_i$ is both trivariate $P$-polynomial and trivariate $Q$-polynomial. The two structures share a common index set consisting of the lattice points of a bounded non-simplicial polyhedron. We establish the $P$-polynomial property through combinatorial interpretations of the intersection numbers. Using Dunkl's coordinate description of $X_i$ and the addition theorem, together with recurrence relations for the $q$-Hahn and $q$-Krawtchouk polynomials, we obtain the required vanishing and nonvanishing conditions for the Krein parameters, and hence the $Q$-polynomial structure. Motivated by this index set, we extend the notion of an $A_M$-Leonard pair from simplicial to polyhedral domains and show that every interior shell carries a polyhedral $A_3$-Leonard pair.

For integers $r\ge1$ and $n\ge2$, let $s_r(n)$ be the least $N$ such that every sequence of $N$ vectors in $\Z^r$, each nonzero modulo $n$, contains $n$ terms whose sum is zero modulo $n$ but nonzero modulo $n^2$. Let $w_r(n)$ be the corresponding least $N$ when any nonempty subsequence is allowed. These constants were introduced as zero-sum problems of new types by Z.-W. Sun. We prove $s_r(n)\ge(2^{r+1}-2)(n-1)+1$ by constructing an explicit sequence from a lattice polytope, disproving his conjecture $s_2(n)=4n+1$ for $n\ge4$. We obtain $w_r(2)=2r+1$ by classifying all sequences avoiding such nonempty subsequences modulo $4$. We also prove Sun's conjecture $w_2(n)=4n-3$ for $n=2^k3^\ell p$, where $p$ is a prime and $k,\ell\ge0$ are integers, by classifying the extremal sequences of length $4p-4$ modulo $p^2$ when $n=p$. For every fixed $n\ge2$, we prove $w_r(n)/s(\Z_n^r)=O_n(r^2/2^r)\to0$ as $r\to\infty$, where $s(\Z_n^r)$ is the classical Erdős-Ginzburg-Ziv constant, with an explicit threshold on $r$ for strict inequality. Thus, for every fixed modulus $n\ge2$, Sun's proposed equality fails for sufficiently large $r$.

We study the linear energetic stability of non-Abelian vortices in an $SO(3)$ gauge-Higgs theory with residual gauge group $A_4$. In our previous work, we identified two families of such vortices associated with order-two and order-three elements of $A_4$, and found Bogomol'nyi–Prasad–Sommerfield-like behavior at special ratios of the Higgs and gauge-boson mass scales. Here, we investigate their stability against $SO(3)$ gauge and Higgs perturbations by analyzing the second order variation of the vortex tension using analytical and numerical methods. In particular, when all relevant mass scales are equal, we prove linear stability for all vortices in the order-three family. By contrast, at the same mass point, we prove that a vortex in the order-two family is unstable. Thus, although both vortex families exhibit Bogomol'nyi–Prasad–Sommerfield-like behavior in their reduced equations at this mass point, their stability properties in the full $SO(3)$ theory are qualitatively different.

Let $A_q(n,d)$ denote the largest size of a code of length $n$ and minimum Hamming distance $d$ over an alphabet of $q$ symbols. For fixed $q\ge2$ and $d\ge4$ we construct codes with redundancy $n-\log_q A_q(n,d)\le(d-3)\log_q n+O(\sqrt{\log n})$ for all large $n$. No algebraic structure is imposed on the alphabet, and the codes are not linear. For $d=4$ and $d=5$ the coefficient of $\log_q n$ matches the Hamming bound, so the redundancy is determined up to lower-order terms. At distance four the codes are provably better than every linear code: by the polynomial-method bound for cap sets, $\mathbb{F}_q$-linear codes with $q>2$ require a strictly larger coefficient in front of $\log_q n$. The construction imposes moment checksums over an auxiliary finite field, as BCH codes do, but one fewer than the distance would normally require. The affine-line theorem from prior work describes the dependencies left by these checks. We exclude them by placing the code positions inside a specially structured subset of the field, built from Behrend's sphere construction with the help of algebraic number theory.

In this paper, we evaluate two companion $q$-series recently suggested for investigation by Andrews and El Bachraoui in the concluding remarks of their work. From these evaluations, we derive the corresponding partition identities and give combinatorial proofs.

We investigate the existence of (metrically) expansive operators on Fréchet spaces without continuous norm. We first solve a problem posed by Bernardes, Caraballo, Darji, Fávaro, and Peris in 2025, by showing that there is no expansive operator on the Fréchet space $\omega$ of all sequences. We prove this fact by investigating the existence of orbits remaining cofinitely inside some fixed closed subspace of finite codimension. We then give examples of expansive operators on some other Fréchet spaces without continuous norm.

We study the discrete Huygens phase map on the ring $C_n$, the simplest oscillator network in which the coupling can be frustrated. Reciprocal coupling favours phase opposition between neighbours, while the edge phase differences must sum to zero modulo $2\pi$ around the cycle; for odd $n$ these two requirements are incompatible. The parity of $n$ determines the absolute minima of the potential. When $4\mid n$, the critical set also contains circles of critical points, all of which meet at two completely degenerate configurations. After quotienting by the common phase, the map is a gradient step for an intrinsic potential on an $(n-1)$-torus, and the cycle spectrum gives an explicit interval of interaction strengths on which every orbit converges to a critical point. Our main result is that, on this whole interval and for every $n\ge3$, almost every orbit converges to a uniform twisted minimum, including at interaction strengths for which the map is not invertible. When $4\mid n$, the non-minimal points of the critical circles are treated by a centre-stable manifold argument. At the two degenerate intersections, a local blow-up reduces the problem to the cubic leading term of the potential and its restriction to the unit sphere. Orbits approaching along directions where this restriction is positive form a null set, because the cubic is harmonic in the intrinsic metric. No orbit approaches along its zero level, where the critical points are non-degenerate saddles on invariant rays. Hence the degenerate equilibria also have basins of zero measure.

Nonconvex optimization on Stiefel manifolds underlies many problems in machine learning and deep learning. However, existing stochastic Riemannian retraction methods often rely on stepsizes that require prior knowledge of the gradient Lipschitz constant, a prescribed iteration horizon, or restrictive batch-size choices. We develop RGR, a recursive gradient retraction method for stochastic nonconvex optimization on Stiefel manifolds. Instead of recursively updating gradient estimators in changing tangent spaces, RGR constructs them in the Euclidean space and subsequently maps them to tangent space, thereby avoiding vector transport. The method comprises three variants: RGR-SPIDER for finite-sum optimization and RGR-SRM and RGR-EMA for online stochastic optimization. RGR-SPIDER is Lipschitz-free and horizon-free, as its adaptive stepsize requires neither prior knowledge of the gradient Lipschitz constant nor a prescribed iteration horizon. RGR-SRM and RGR-EMA are mini-batch and horizon-free methods whose convergence guarantees do not rely on prescribed large batch sizes and whose parameters can be chosen without prior knowledge of the total iteration budget. We show that RGR-SPIDER reaches an $\varepsilon$-critical point using $\mathcal{O}(\varepsilon^{-2})$ iterations and $\mathcal{O}(N+\sqrt{N}\varepsilon^{-2})$ component-gradient evaluations, where $N$ denotes the number of component functions. For the online setting, RGR-SRM and RGR-EMA achieve iteration complexities of $\tilde{\mathcal{O}}(\varepsilon^{-3})$ and $\tilde{\mathcal{O}}(\varepsilon^{-4})$, respectively. Experiments on principal component analysis, low-rank matrix completion, and orthogonal convolutional neural networks demonstrate faster empirical convergence than methods that construct recursive gradient estimators directly in Riemannian tangent spaces.

We construct countable, stable, $\omega$-categorical structures with at most two binary relations whose CSPs exhibit three complexity regimes: NP-complete, decidable but not in NP, and undecidable. The templates arise from a controlled Hrushovski construction of stable $\omega$-categorical graphs $M_\alpha$ for suitable irrational parameters $\alpha>1$. Some of these graphs have NP-complete CSPs. Expanding the graphs $M_\alpha$ by the non-edge relation turns age membership into a restriction of the CSP. For every time-constructible bound $t$, there is such an expansion whose CSP is decidable but not in DTIME$(t)$. For an explicit parameter $\alpha$, the restriction to complete instances belongs to $\mathrm{coNP}$ and the full CSP belongs to $\Sigma_2^{\mathrm P}$. Separating the ages of the $M_\alpha$ using graphs with large girth and prescribed density yields $2^{\aleph_0}$ distinct CSPs of these expansions, all but countably many of which are undecidable.

Two theorems of Loureiro and Van Assche on ratio and zero limits provided part of the motivation for this note. Their elementary nature permits direct proofs; Appendix B gives complete proofs of both for the canonical initial conditions, using earlier continued-fraction identities. We prove ratio limits and weak limits of normalized zero measures for varying higher-order three-term recurrences with arbitrary coefficient periods and independent limiting functions for their phases. Joint local coefficient limits give ratio convergence without uniform coefficient bounds. For zero measures, convergence in measure to finite measurable profiles suffices, without boundedness or integrability; the limiting measures may have noncompact support. Coefficients may vanish, and real diagonal terms are allowed for ordinary three-term recurrences. The proofs use interlacing, partial fractions with nonnegative residues, local computations of moments, and stability of zero counts under changes of recurrence coefficients. Truncation removes the coefficient bounds from the zero theorem, whose proof is independent of ratio asymptotics. Uniform bounds give the version with compact support. Power-law limits and their explicit transform formula follow as a special case.

We study continuous-time primal-dual dynamics for tracking solutions of time-dependent constrained optimization problems. First, for uniformly strongly convex-strongly concave saddle functions, we establish explicit finite-time tracking and additive-disturbance bounds. For convex problems, we then introduce a directional primal regularization, combined with dual Tikhonov regularization, and quantify the resulting optimization bias and constraint violation. For smooth nonconvex problems, we couple a proximal local convexification with a two-time-scale primal-dual flow and obtain a local tracking result under constraint qualification, second-order sufficiency, and temporal regularity. Numerical traffic-signal experiments at a single intersection evaluate the proposed method under vehicle-throughput and pedestrian-aware objectives, comparing it with standard signal-control schemes.