Papers

The goal of this work is to calculate the mapping class groups of solenoidal manifolds. We introduce the dynamical commensurator group associated to an odometer action, and develop methods to calculate the mapping class group using the dynamical commensurator. Our results generalize the previous results of Odden for the mapping class group of the universal solenoid over a Riemann surface, and of Bering and Studenmund for the pointed mapping class group of the universal solenoid over a compact aspherical manifold. We include a selection of examples.

In 2019, Liu proved that the cop number of any $(P_k, claw)$-free graph is at most $k-3$. In 2026, Clow and Meger improved and generalized the result of Liu by showing that every connected $(P_k, E)$-free graph has cop number at most $\lceil \frac{k-1}{2} \rceil + 3$, where the graph $E$ is obtained by subdividing two edges of a claw. In this paper, we show that every $(P_k, claw)$-free graph $G$ has cop number at most $\lceil \frac{k-1}{2} \rceil$. This improves the result of Clow and Meger on claw-free graphs. We further show that this bound can be improved to $\lceil \frac{k}{3}\rceil$, when $G$ contains a hole of length at least $k-1$. We show that the cop number of any $(P_k, fork)$-free graph is at most $\lceil \frac{2(k-1)}{3} \rceil$. This also improves the result of Clow and Meger on fork-free graphs for some values of $k$. We further show that every $(P_7, fork)$-free graph has cop number at most $3$, and this bound is tight.

Impulsive control, in the context of continuous-time Markov decision processes (CTMDPs), refers to actions that instantaneously update the state of the process, leading to theoretical complications. Previous work has overcome these complications, but at the expense of approachability and ease of implementation: those models are mathematically complex and incompatible with pre-existing solution methods. This paper presents an alternative formulation of impulsive control for CTMDPs that is easier to employ in the modelling of real-world problems and can be solved by pre-existing solution methods. This formulation is then extended to allow for multiple objectives and decomposability. A decomposition-based heuristic method is developed that leverages Dantzig-Wolfe decomposition to find approximate Pareto fronts for large multi-objective decomposable impulsive CTMDPs. The modelling framework and solution methodology are applied to a dynamic maintenance problem to demonstrate their computational efficiency, solution space coverage, and solution quality.

In this paper, we solve the direct monodromy problem for the Borel-summed $0$-parameter solution of Painlevé~I. Using the auxiliary time equation, we first construct formal WKB wave matrices whose amplitudes are regular at the double turning point. Rigorous Borel summation then yields exact WKB matrices in each Stokes region. Uniform estimates on moving action strips provide a common exact wave matrix in a full neighbourhood of the double point, proving that all four incident jumps are identities for consistent column normalisations. We formulate the resulting normalised Riemann–Hilbert problem and prove its existence and uniqueness on the Borel domains of the construction. Finally, after Painlevé scaling, the five Borel-summed solutions coincide with the five rotated tritronquée solutions of Painlevé~I.

We introduce a novel extension of reflection calculus, $\mathsf{RC_{BI}}$, to analyze a family of extensions of $\mathsf{ACA}_0$ generated by uniform $\omega$-model and syntactic reflection principles, together with several other related operators. The union of these theories coincides with the theory of Bar Induction ($\mathsf{BI}$). We prove that $\mathsf{RC_{BI}}$ is arithmetically sound, and that it is sound and complete with respect to a natural class of Kripke models. Leveraging these modal semantics, we demonstrate that the class of theories denoted by $\nabla$-free closed formulas is a pre-wellordering with respect to $\Pi^1_1$-soundness and consistency strength. This allows us to establish an ordinal analysis for $\mathsf{BI}$ and its fragments, analogous to Beklemishev's analysis of $\mathsf{PA}$. The structural properties of $\mathsf{RC_{BI}}$ naturally give rise to a novel ordinal notation system for the Bachmann-Howard ordinal which differs substantially from existing representations.

Let $\mathbb{F}$ be a finite field and let $G$ be a group such that every finite subgroup of $G$ is cyclic. In this paper we classify the division gradings by cyclic groups on matrix algebras over $\mathbb{F}$. This yields the classification of the $G$-gradings on these algebras. Moreover if $G$ is abelian we prove that two matrix algebras over $\mathbb{F}$ with $G$-gradings satisfy the same graded identities if and only if they are isomorphic as graded algebras.

We study the representation of bilinear forms associated with Riesz fractional gradients by symmetric integral kernels. Motivated by [SS18, Open Problem 1.10], we give a complete description of the constant coefficient case. We prove that any symmetric representing kernel is uniquely determined and we derive its explicit formula in terms of the constant matrix $A$. This yields a sharp characterization of the matrices for which the kernel is either nonnegative or uniformly positive. In particular, [SS18, Open Problem 1.10] has a negative answer (even for constant matrices). We also identify a class of nonconstant antisymmetric divergence free perturbations that are invisible to the fractional bilinear form and hence leave the representing kernel unchanged. The general case of nonconstant matrices exhibits a substantially different behavior and will be the main subject of our forthcoming paper [DSV26].

We approximate the solution operator $(a,f)\mapsto u$ of $-\operatorname{div}(a\nabla u)=f$ by operator networks whose only primitives are heat-semigroup functions of a fixed reference operator and pointwise products. The main result is an emulation theorem on strongly local Dirichlet spaces with a spectral gap and a Bakry–Émery bound $\mathrm{BE}(-K,\infty)$: an explicit scheme reaches $\dot H^1$-error $\varepsilon$ uniformly over coefficients with bounded carré du champ and bounded generator image and over $L^2$ data, with parameters depending only on the coefficient bounds, the accuracy and $K$; on the torus this gives a $C^2$-type coefficient class with constants independent of the dimension. The torus version refines this to Wiener classes of fractional regularity $A^{1+\theta}$, with an explicit rate attained by the scheme's product-defect block. The scheme unrolls the Laplacian-preconditioned Richardson iteration, controls each inexact step in the operator norm of $\dot H^1$ without Galerkin truncation, and synthesises the coupling $\nabla a\cdot\nabla u$ from the product defect of one heat layer. All parameters are explicit; the results are approximation theorems for continuum operators.

Motivated by Mader's conjecture on connectivity keeping trees, we study the corresponding problem in triangle-free graphs. For positive integers $k,m$, we first prove that every $k$-connected triangle-free graph $G$ with $\delta(G)\geq k+\left \lceil \frac{m}{2} \right \rceil $ contains a path $P$ of order $m$ such that $G-V(P)$ remains $k$-connected. Moreover, if $\kappa(G)=k$, then for every end $F$ of $G$ and every vertex $v\in F$, the path $P$ can be chosen in $G[F]$ with $v$ as an endvertex. If $\kappa(G)\geq k+1$, then any vertex of $G$ can be chosen as an endvertex of $P$. This result confirms the conjecture of Luo, Tian and Wu on connectivity keeping trees in bipartite graphs for all paths, and it also sharpens Fujita's path theorem to the optimal minimum degree bound. For an integer $i$ with $1\leq i\leq m-1$, let $P_m^+(i)$ denote the tree obtained from a path of order $m-1$ by adding one pendant vertex adjacent to its $i$th vertex. Let $(X,Y)$ be the bipartition of $P_m^+(i)$. We further prove that every $k$-connected triangle-free graph $G$ with $\delta(G)\geq k+\max\{|X|,|Y|\}$ contains a subgraph $T\cong P_m^+(i)$ such that $G-V(T)$ remains $k$-connected. The minimum degree bound is sharp, and the result confirms the bipartite conjecture for all pendant extensions of paths.

We show that Payne's conjecture on the comparison between Dirichlet Laplacian eigenvalues and buckling eigenvalues does not hold on flat and hyperbolic cylindrical surfaces and on (round) hemispheres.

We study the estimation of the interaction parameter in singular interacting particle systems from continuous and discrete observations over a fixed time horizon. The singularity of the interaction drift prevents the direct application of standard likelihood and contrast-based methods, which rely on regularity assumptions that fail in this setting. Our approach exploits an exact reduction of the particle dynamics to a one-dimensional stochastic differential equation for a quadratic empirical moment. By the symmetry of the interaction kernel, the singular terms cancel, yielding an explicit estimator in which the unknown parameter enters linearly. We establish consistency and asymptotic normality as the number of particles, \(N\), tends to infinity. Letting \(\varepsilon_N\) denote the microscopic noise scaling of the system, we obtain asymptotic normality at the rate \(\sqrt{N/\varepsilon_N}\) for strictly positive interaction kernels, under an additional condition on the limiting empirical information. We further extend these results to degenerate interaction kernels, achieving consistency and asymptotic normality at rate \(N\) under suitable structural assumptions and the canonical scaling \(\varepsilon_N=N^{-1}\). In both regimes, we provide conditions on the observation mesh ensuring that the statistical limits remain valid under discrete sampling, in high frequency. Finally, we illustrate the applicability of our methodology through Dyson Brownian motion, the Dyson Ornstein–Uhlenbeck process, and Wishart–Laguerre particle systems, alongside a numerical investigation of the estimators' finite-sample performance.

Let $\Gamma$ be a connected metric graph of first Betti number $b\ge 2$, and let $\Gamma_b$ be a trivalent graph of the same Betti number with all edges of unit length. We show that if $\text{length}(\Gamma) \le \lambda\, \text{length}(\Gamma_b)$, then the universal cover of $\Gamma$ contains a vertex around which, at every radius $R\ge 0$, the ball is at least as long as the ball of radius $R$ in the trivalent tree whose edges all have length $3\lambda$. In particular, for $\lambda\le\frac13$, this ball is at least as long as the ball of the same radius in the universal cover of $\Gamma_b$, with no multiplicative loss. This improves an earlier theorem of the author, which gives the weaker lower bound $(1-3\lambda)$ times the ball in the universal cover of $\Gamma_b$, and only for $\lambda\le\frac16$. The proof rests on an elementary comparison principle. Among trees whose vertices have degree at least three and whose edges have length at most $c$, the trivalent tree with all edges of length $c$ has the smallest balls at every radius. As an application, we double the exponential growth rate in the author's earlier uniform lower bound for the area of balls in the universal cover of small-area surfaces of hyperbolic type, from $\ln 2$ to $2\ln 2$, beyond the hyperbolic rate. We also obtain explicit constants in the positive answer to Guth's question for surfaces. Finally, we conjecture the sharp, all-scale version of the minimal volume entropy inequality of Kapovich–Nagnibeda and Lim, and show that it would be optimal.

We introduce binary cyclic gradient coding (BCGC), an exact gradient coding scheme for $n$ workers and up to $s$ stragglers. Its encoding matrix is binary, circulant when $(s+1)\mid n$ and nearly circulant otherwise, and the master decodes by a single summation. Like the fractional repetition coding scheme (FRC) and its generalized version (GFRC), BCGC splits the workers into $s+1$ classes that each cover the whole dataset, but it assigns the data through cyclic supports. Its encoding matrix has rank $n-s$, and $n-s-1$ workers keep the balanced load $s+1$, which is the largest possible number. We prove that encoding matrix of BCGC is not permutation-equivalent to those of FRC and GFRC schemes. Moreover, We also prove that a binary coded scheme with compression is equivalent to an uncoded compressed scheme on a coarser partition of the data. This explains why binary schemes remain stable under compression, whereas the large decoding coefficients of MDS codes amplify the compression error and lead to divergence in our experiments. For every contractive compressor, the standard convergence guarantee of compressed gradient descent holds on each coarse partition. FRC, GFRC and BCGC therefore share the same convergence guarantee and differ only in their partitions, with one partition for FRC, two for GFRC and $s+1$ distinct partitions for BCGC. Choosing among these partitions can recover the exact gradient in cases where FRC and GFRC cannot. Experiments on four datasets confirm the theory. BCGC matches GFRC, keeps its accuracy under top $\kappa$ sparsification while the MDS codes diverge, and on heterogeneous data its partition diversity reduces the compression error and improves the accuracy.

We consider a reaction–diffusion equation with Kolmogorov–Petrovskii–Piskunov (KPP) nonlinearity and Neumann boundary conditions in a narrow tube $G^{\varepsilon,\delta}\subset\mathbb{R}\times\mathbb{R}^{n}$ of width of order $\varepsilon$. The tube is chosen from a family indexed by a second parameter $\delta>0$, whose boundary $\partial G^{\delta}$ is smooth for each $\delta>0$ but degenerates as $\delta\downarrow0$. In the iterated limit $\varepsilon\downarrow0$ followed by $\delta\downarrow0$ the problem becomes a Cauchy problem on the line whose spatial operator is a generalized second order differential operator $D_{v}D_{u}$ in the sense of Feller, rather than a classical second order elliptic operator. Solutions of the limiting problem need not be differentiable in $x$, so they must be understood in a generalized sense. Using the Feynman–Kac representation, we prove that the limiting problem has a unique generalized solution. When the cross sectional volume is bounded above and below, we also prove that this solution is Hölder continuous in $t$ with exponent $1/2$, locally uniformly in $x$, whenever it is Lipschitz continuous in $x$. Under the same volume bounds, this Lipschitz condition holds when the initial function and the nonlinearity are Lipschitz continuous in $x$. Finally we prove that the solutions in the narrow tubes converge to the generalized solution uniformly on compact time intervals in the same iterated limit if the tube, the initial function, and the nonlinearity are periodic in $x$.

This work focuses on planning daily routes for a group of employees of a condominium company. Given a set of employees and a set of customers with pre-scheduled visits, we want to build doubly-open routes to guarantee visits to all customers, ensuring that each customer is visited by an employee with the appropriate skills to carry out the required activity. The travel times between customers and the service time in each customer are known. The problem includes time windows for visiting customers, each employee has a maximum working time, and the daily working period is between 9 AM and 6:30 PM. The routes respect the lunch break period if they extend from morning to afternoon, which means that lunch break can be avoided if the route is fully completed in the morning or in the afternoon. The aim is to minimize the distance covered by all routes with a penalty for including lunch breaks. The distances and times between customers are collected via an API to Microsoft Azure; and the solutions are then displayed on Bing Maps. The results obtained show the operational advantages of an efficient and effective vehicle routing planning in a specific condominium management company. The model can be extended to other service provider companies.

Network coding allows intermediate nodes to combine messages, whereas routing only forwards them. The Li-Li conjecture asserts that network coding and fractional routing have the same capacity region in undirected multiple-unicast networks. We give a counterexample to this conjecture and apply it to external-memory integer sorting. Since Aggarwal and Vitter posed the question in 1988, whether integer sorting can beat the external-memory comparison bound has remained a central and extensively studied open problem. We give a deterministic sorting algorithm: for $N$ integers of $\Theta(\log N)$ bits, a memory holding $M$ integers, and block transfers of $B$ integers, where $2B\le M\le N$, the number of block transfers is $O\left(\frac{N}{B}\left[1+\frac{\log N}{(\log(M/B))^{1-\varepsilon}(\log M)^\varepsilon}\right]\right)$, where $\varepsilon>0$ is an absolute constant. In a range of parameter regimes, this bound is asymptotically smaller than the classical comparison-sorting bound. We also use Lean 4 to verify the sorting algorithm's correctness and complexity bounds, and the strict advantage of coding over routing.

Let $G$ be a group of permutations of a set $\Omega$. The relational complexity $\mathrm{RC}(G)$ is the least integer $r\geq 2$ such that the $G$-orbit of every finite tuple is determined by the $G$-orbits of its $r$-subtuples; there is currently only a small collection of groups $G$ for which $\mathrm{RC}(G)$ is known precisely. In this paper we study the relational complexity of finite affine groups acting naturally on a $d$-dimensional vector space over $\mathbb{F}_q$. We prove that every such group has relational complexity at most $d+2$ when $q>2$ and at most $d+1$ when $q=2$; we produce a complete classification of the primitive such groups for which these bounds are attained.

We consider the big finitistic projective and flat dimensions of a commutative ring and of its formal power series ring. When the ring is Noetherian, both invariants increase by one. Coherence of the power series ring does not force either equality to hold. For every \(n\ge0\) and every \(d\) with \(n+1\le d\le\infty\), the flat pair \((n,d)\) can be realized. The finite projective pairs are realized when \(n\ge1\), and also for \((0,1)\). For every \(n\ge0\), one family realizes \((n,\infty)\) for both invariants at the same time. The seed is a zero-dimensional perfect local ring. Its power series ring has countably presented modules whose flat dimension is \(r\) and whose projective dimension lies between \(r\) and \(r+1\), for every \(r\ge1\). The remaining finite projective case is controlled by a dichotomy for perfect coefficient rings. If the Jacobson radical is nilpotent, the power series value is one; if it is not nilpotent, there is a module of projective dimension exactly three. In particular, the pair \((0,2)\) is impossible. For one fixed countable perfect ring, the power series ring has finitistic flat dimension one. Its finitistic projective dimension is \(m+2\) when \(2^{\aleph_0}=\aleph_m\) with \(1\le m<\infty\), and is infinite when \(2^{\aleph_0}>\aleph_\omega\).

OpenAI (using a non-public "internal frontier model”) has recently (6 October 2026) made a number of remarkable claims, including some claims regarding the quasi-Riemann hypothesis (QRH). While there are good reasons to be cautious regarding these claims, one can nevertheless ask what (if true) their implications would be. While there is widespread agreement that any version of the QRH will provide various improved bounds on the location of the prime numbers, I feel that there is some value in making these improvements explicit. Specifically, if we assume validity of the claimed 7/8 version of the QRH then I shall demonstrate (for all $n\in \mathbb{Z^+}$): 1) the existence of primes between eighth powers, in the interval $(n^8,(n+1)^8)$; 2) the existence of primes in the shorter interval $(n^8,n^8+8n^7)$; 3) the existence of primes in the even shorter interval $(n^8,n^8+n^7).$ Similarly for reals greater than unity $x>1$ one can demonstrate: 4) the existence of primes in the Hoheisel-style interval $(x,x+x^{7/8})$. All of this is contingent on the validity of the claimed 7/8 version of the QRH.

The horizontal mean curvature flow is a geometric evolution associated with the horizontal perimeter in sub-Riemannian spaces. In this paper, we introduce a minimizing movement scheme for this flow in the first Heisenberg group $\mathbb{H}$, and prove the existence of a flat flow which is locally $1/2$-Hölder continuous in time, with respect to the $L^1$-distance. The construction applies to arbitrary bounded sets of finite horizontal perimeter and does not require smoothness or symmetry of the initial datum.

We study equilibrium fluctuations for the symmetric partial exclusion process (SPEP) on the one-dimensional integer lattice $\mathbb{Z}$, an interacting particle system in which each site is occupied by at most $\alpha$ particles. We assume that the jump rate is decomposable and that the instantaneous current admits a gradient form. Our aim is to study the scaling limit of the density fluctuation field of the SPEP starting from an invariant measure and to derive the corresponding stochastic partial differential equation (SPDE). A distinctive feature of our setting is that the maximum number of particles per site depends on the scaling parameter $n$ as $\alpha=\alpha_n \propto n^a$ for some $a>0$, so that the model interpolates between the partial exclusion process ($\alpha <\infty$) and the symmetric zero range process (SZRP, $\alpha = \infty$). Under the diffusive rescaling in which the lattice spacing is $1/n$ and time is accelerated by a factor $n^2$, we prove that the density fluctuation field converges to the solution of the stochastic heat equation. The limiting SPDE coincides with the one obtained for the SZRP. A key ingredient in the proof is the convergence, under a suitable rescaling, of the partition function of the reversible measure of the SPEP to that of the SZRP. This convergence implies that expectations of a wide class of local functions under the reversible measure of the SPEP converge to the corresponding expectations under the reversible measure of the SZRP, and provides a rigorous sense in which the SPEP approximates the SZRP as $n\to\infty$.

We show that every vineyard can be decomposed into vines, that is, every path in the space of persistence diagrams $\mathcal{D}$ can be decomposed into a sequence of paths in $\mathbb R^2$. Moreover, we characterize which spaces $X$ have the property that every continuous map $X\to \mathcal{D}$ has such a decomposition. Our results hold not just for $\mathcal{D}$, but also for many other spaces of unordered tuples.

To any compact holomorphically symplectic manifold and a nef BBF-isotropic class we associate a quasi-projective variety called the Markman–Markushevich space which parametrises the Lagrangian fibrations with this isotropic class modulo Tate–Shafarevich twists. We give a systematic interpretation of known results about Tate–Shafarevich twists of Lagrangian fibrations using this space, employ it to obtain a conjectural result for the last deformation type for which such a result is not yet proposed, and make some first steps towards its proof. As a byproduct, we generalise the results of Kumar on existence Shioda–Inose partners for abelian surfaces to a slightly more general class of K3 surfaces, and associate a canonical K3 surface with transcendental lattice isometric to $U^2\oplus\langle-2\rangle^2$ to a smooth cubic surface with a choice of a Schläfli double-six. We also review some other approaches towards the construction of complete families of O'Grady six-manifolds with Lagrangian fibrations, and show how they may still be useful even though they do not quite work.

We fix a spacelike two-dimensional left-invariant distribution and its associated almost contact structure on a three-dimensional Lorentzian Lie group, and investigate the star-Ricci contraction of the curvature of the Bott connection. We compute the complete tensor for seven standard bracket families and determine all real parameter solutions of the general algebraic equation and its self-adjoint subclass. Using the symmetric part of the tensor, we also define a Bott affine star-Ricci soliton equation and classify all left-invariant vector field solutions. Every affine solution is steady, while two non-unimodular families admit branches with a nonzero symmetrized tensor. The proofs provide complete component equations, treat all degenerate parameter branches, and establish sufficiency by explicit substitution. We also distinguish almost contact structures from contact metric structures.

Define $\nabla(n)$ as the least possible number of cells that must be removed from the $n\times n$ square board in order to leave no cycles. Put $L_n=\left\lceil\left((n-1)^2+1\right)/3\right\rceil$. The results of Luccio, Madelaine-Stewart, and Lien-Fu-Shih imply that $\nabla(n)=L_n$ in case $n\equiv 1\pmod 6$ and $n\equiv 4\pmod 6$, while for the remaining $4$ residue classes $\nabla(n)=L_n$ or $\nabla(n)=L_n+1$. In this paper, we provide optimal constructions in the odd residue classes $n\equiv3,5\pmod6$, (proving $\nabla(n)=L_n$) and a geometric obstruction showing that equality in lower bound is impossible when $n\equiv0,2\pmod6$ (yielding $\nabla(n)=L_n+1$). Starting with $n=3$, the resulting sequence is thus given by \[ \nabla(n)=\bigg{\lceil}\frac{n^2}{3}-\frac{n}{6}+\frac{4}{3}\bigg{\rceil}-\bigg{\lceil}\frac{n}{2}\bigg{\rceil}. \]

The Fomin-Kirillov algebras $\mathcal{FK}_n$ are shown to be infinite-dimensional for $n\geq 6$. By computer calculations, we find an explicit nonzero primitive element of degree 14 in the identity component of $\mathcal{FK}_6$. For every $n\geq 6$, the existence of such an element implies that $\mathcal{FK}_n$ contains a polynomial subalgebra and is therefore infinite-dimensional, and that the canonical homomorphism from $\mathcal{FK}_n$ onto its associated Nichols algebra is not an isomorphism.

Contact-rich robotic insertion is challenging because uncertainty in alignment, friction, and contact conditions can lead to unsuccessful assembly, excessive contact forces, and jamming. Designing a reliable controller therefore requires balancing insertion performance, robustness, and recovery from unexpected contact. In this paper, we propose PSO+TR, a derivative-free optimization (DFO) method that combines the exploratory search of particle swarm optimization (PSO) with a trust-region (TR) procedure for local refinement to improve the search process when progress stagnates. We apply the method to derivative-free policy optimization of an interpretable robotic insertion controller, jointly tuning its motion, feedback, and recovery parameters while preserving its predefined control structure. The proposed approach is evaluated on 93 eligible benchmark problems from the bound-constrained programs (BCP) collection and a synthetic contact-rich planar insertion task under uncertain operating conditions. Three random seeds per benchmark problem yield 279 matched problem–seed instances per method. PSO+TR reaches the prescribed target on 228 of these instances (81.72%), the highest observed target-attainment rate among the five compared methods, which also include PSO, randomized black-box optimization (VRBBO), matrix adaptation evolution strategy (MAES), and differential evolution (DE). Across eight robotic optimizer seeds, the deployed controller-selection pipeline achieves 97.78% completion under development conditions and 58.00% under withheld stress conditions, compared with 91.22% and 31.12%, respectively, for the fixed staged controller. The stress violation rate is reduced from 20.69% to 2.19%. These findings support combining global and local search for DFO and structured contact-rich controller tuning. The synthetic planar results require validation on physical robots.

In this paper, we study the background stability problem for three-dimensional compressible adiabatic flows through porous media. For two given equilibrium states $(\bar{P}^k,\vec{0},\bar{S}^k)$ ($k=0,1$), under the assumption of small initial perturbations around each equilibrium state, pointwise estimates of solutions for individual perturbation problems are obtained. Based on these pointwise structures, a comparison between two individual solutions could be applied and it is proved that the difference between two solutions depends continuously on the equilibrium distance $|\bar{P}^0-\bar{P}^1|+|\bar{S}^0-\bar{S}^1|$ as well as the initial perturbation difference, with additional decay rates, thereby establishing a nonlinear comparison principle for this system. Compared with previous works on background stability problems, it is the first time to obtain stabilities in the pointwise sense. Our main tools are Green's functions, a priori energy estimates, and refined decay estimates for higher-order derivatives.

A finite Gödel chain generates not one integer sequence but a large family of them. The combinatorial mechanism is developed in a companion paper; here the sequences themselves are the primary objects. For $V_m=\{v_0<\cdots<v_{m-1}\}$, let $g_{j,n}^{(m)}$ count outputs equal to $v_j$, aggregated over all full bracketings on $n$ variables and all $m^n$ valuations. The root-pair refinement produces $N_{i,j}^{(m)}(n)=\sum_{r=1}^{n-1}g_{i,r}^{(m)}g_{j,n-r}^{(m)}$, with generating functions $N_{i,j}^{(m)}(x)=G_i^{(m)}(x)G_j^{(m)}(x)$. We prove that the initial coefficients of these products uniquely identify the unordered root-value pair. Consequently, for each fixed $m$, there are exactly $\binom{m+1}{2}$ distinct root-pair generating functions. We catalogue the factories for $m=4,5,\ldots,10$, giving the first ten terms of every base sequence and every symmetric root-pair sequence. This range contains 49 base sequences and 210 symmetric pair classes, yielding 259 indexed sequence families before additional aggregates. The catalogue provides reproducible data for integer-sequence analysis and comparison with the On-Line Encyclopedia of Integer Sequences (OEIS).

In 1988 Broughton proved that if a complex polynomial function on ${\mathbb C}^n$ is tame at infinity its fibers are homotopy equivalent to bouquets of some spheres. Since then several basic properties of polynomial functions were studied by many mathematicians in various situations. In this paper, we show some evidence for the possibility of extending Broughton's result to rational functions satisfying some tameness condition at infinity and the one at their points of indeterminacy. Specifically, assuming some conditions on a rational function on ${\mathbb C}^n$ we prove that the cohomology groups of a generic fiber of it are isomorphic to those of a bouquet of $S^1$ and some spheres. In the course of the proof, the theory of the nearby and vanishing cycle sheaves for meromorphic functions will play an essential role.