Papers

Let $f$ be a normalized level-one Hecke eigenform of even weight $k$, and let $Q_{f,m}$ be the derivative period polynomial formed from the critical values of the $m$-th derivative of its completed $L$-function. We study its odd part $Q^-_{f,m}(z)=(Q_{f,m}(z)-Q_{f,m}(-z))/2$. We prove that there is an absolute $K_0$ such that, for every even $k\ge K_0$, every normalized level-one Hecke eigenform $f$ of weight $k$, and every integer $m\ge0$, the nonzero zeros of $Q^-_{f,m}$ off the unit circle, if any, consist of four simple zeros forming a single real reciprocal quartet $\{\pm b,\pm b^{-1}\}$, where $0<b<1$. The occurrence and location of this possible quartet are governed by the critical derivative order $m_c(k)=(k-1)\log((k-1)/\pi)$. If $m/m_c(k)\to\theta\in(0,1)$, exactly one quartet occurs and $b\to(1+\theta)/2$; if $\theta>1$, every nonzero zero is eventually simple and lies on the unit circle. At $\theta=1$ the same real-or-unit-circle containment remains valid, and any quartet that is present consists of four simple zeros. For each fixed weight, all nonzero zeros are eventually simple and lie on the unit circle as $m\to\infty$. Consequently, the Diamantis–Rolen containment conjecture holds outside finitely many weight–derivative pairs. The proof combines an exact signed-reciprocal completion, uniform split-Mellin saddle estimates yielding a moving-sine model, and a winding count that transfers disk-zero information to the unit circle.

We extend Pommerenke's characterizations of boundary curves of conformal mappings in terms of the little Bloch and $\mathrm{VMOA}$ conditions from the unit disk $\mathbb{D}$ to the upper half-plane $\mathbb{H}$. Let $G\colon \mathbb{H}\to\Omega$ be a conformal mapping onto an unbounded quasidisk $\Omega$ with $G(\infty)=\infty$, and let $g\colon \mathbb{R}\to\Gamma=\partial\Omega$ be its boundary extension. In the non-compact setting, the Euclidean smallness on the boundary curve is not necessarily comparable to the smallness of the parameter on $\mathbb{R}$. To overcome this difficulty, we use relative versions of the asymptotic conformality and the asymptotic smoothness with respect to the parametrization $g$. We prove that $\log G'\in B_0(\mathbb{H})$ is equivalent to the asymptotic conformality of $\Gamma$ relative to $g$, and also to the asymptotic symmetry of the embedding $g$. We further prove that $\log G'\in \mathrm{VMOA}(\mathbb{H})$ is equivalent to the asymptotic smoothness of $\Gamma$ relative to $g$, and also to the asymptotic smoothness of $g$. These results provide half-plane analogues of Pommerenke's theorems and clarify the role of the parametrization in the unbounded case.

Offline change point detection tries to detect time points of distribution change in a given data sequence; and is now routinely used in signal processing, speech processing, climatology etc. Despite this broad applicability across economics, computer science, and planetary sciences, rigorous, nonparametric techniques for change point detection with non-independent and identically distributed (i.i.d.) datasets has remained elusive. This paper establishes such guarantees by proposing a non-parametric clustering algorithm which can accurately obtain the change points from a given Markovian dataset of length $n$. It does so by bridging together two different components of mathematical statistics; Rademacher complexities of Markov chains, and adaptive clustering via penalisation. Our first result uses recent advances in Rademacher complexities of regenerating Markov chains to derive a Dvoretzky Kiefer Wolfowitz (DKW) type inequality for the empirical distribution of the Markov chain. We then use this to show that an adaptive clustering algorithm recovers the correct change points for a Markovian sequence. We establish the tightness of our rates by showing that they essentially coincide with the best known rates for i.i.d. data. We end the paper by discussing the computational considerations of the problem.

Numerical simulation of the Vlasov-Poisson system faces fundamental challenges due to phase-space filamentation. Standard spectral methods rely on artificial filtering to suppress errors, which inadvertently degrades physical structures and accuracy over time. This paper proposes a dynamically adaptive Fourier spectral method combined with high-order time splitting to overcome these limitations. To prevent filamentation-induced aliasing, we dynamically expand the wavenumber domain via adaptive zero-padding whenever spectral tail monitoring detects emerging fine-scale structures. Acting as an exact trigonometric interpolation, it inherently avoids the artificial smoothing of traditional filters, preserving mass and effectively maintaining the $L^2$-norm within the dynamically resolved scales. Numerical experiments on Landau damping and various instability problems validate the robustness of our scheme.

For a graph $G$, let $\Delta(G)$, $\omega(G)$, and $\chi(G)$ denote the maximum degree, clique number, and chromatic number of $G$, respectively. Let $P_n$ and $C_n$ denote the chordless path and chordless cycle on $n$ vertices, respectively. In this paper, we prove that every $(P_6,C_4)$-free graph $G$ with $\Delta(G)\ge 9$ and $\omega(G)<\Delta(G)$ is $(\Delta(G)-1)$-colorable.

We study homogeneous diffusion martingales evolving in a bounded state space $D=[a,b]$, where $a$ and $b$ are zeros of the diffusion coefficient. We call a process of the form $Z_t=\mathbb{E}[B\mid\mathcal{F}_t]$, with $B$ a Bernoulli random variable, a Bernoulli-Doob martingale. Our main results establish a complete equivalence: every such diffusion martingale is a Bernoulli-Doob martingale (Theorem 2) and, conversely, every continuous time-homogeneous Markov Bernoulli-Doob martingale on a Brownian filtration arises from such a diffusion (Theorem 3). The intuitive reason is that a bounded martingale has constant expectation while accumulating variance, so it converges to the maximum-variance distribution with given mean and range, namely the Bernoulli. We further show that this Bernoulli limit is truly asymptotic: for any fixed finite horizon $T$, the probability of not yet having reached the boundary is strictly positive (Theorem 4), even when the individual boundaries are accessible. We clarify the relationship between Feller's boundary classification, the pathwise SDE framework, and the martingale constraint, showing that the martingale property forces absorption at any attainable boundary. The theory is illustrated with the $\Phi$-martingale, the Jacobi martingale, and applications to credit-risk modelling.

The cover number of a graph by a graph class $\mathcal P$ is the least number of $\mathcal P$-graphs necessary to cover its edges. A classical theorem of Harary, Hsu and Miller gives an exact formula for the cover number by the class of graphs with chromatic number at most $k$. We investigate analogous questions for the case of the fractional chromatic number $\chi_f$ and the local chromatic number $\psi$. We prove that an analogous formula cannot hold in the case of the cover number by graphs of fractional chromatic number at most $\beta$, and find a lower and an upper bound, that gives rise to interesting asymptotic questions. We also investigate this cover number for small specific graphs. In the case of the cover number by graphs with local chromatic number at most $k$, we find an upper bound in terms of $\psi$, and a lower bound in terms of $\omega$.

This paper presents a residual-certified adaptive method for tracking local solution manifolds in parametric dynamical systems. The method combines local POD reduction, full physical residual checks, state-distance snapshot forgetting, high-fidelity resampling, and a lightweight physics-informed neural correction. Instead of learning one global parameter-to-state map, the algorithm maintains the currently active local branch and updates it when the residual indicates loss of validity. The analysis explains why residual thresholds are meaningful on regular branches through local residual-error control, and why stricter local updates are needed near folds or other degenerate neighborhoods. Numerical studies on Ostwald ripening, a particle population-balance model, and the Bratu equation test the approach across low-dimensional dynamics, nonlinear nonlocal residual compensation, and near-fold model failure. The results show that residual-certified local model management can concentrate high-fidelity computation in difficult parameter regions while preserving an interpretable link between surrogate prediction, physical consistency, and active-branch tracking.

We extend the multisymplectic reduction theory of Blacker – itself the extension of Marsden–Weinstein–Meyer reduction to $k$-plectic manifolds – to the setting of relative multisymplectic geometry, in which a smooth map $F\colon M\to N$ carries a closed nondegenerate relative $(k{+}1)$-form $\varpi=(\omega,\eta)$ in the mapping-cone complex of $F$. We introduce the Leibniz algebra of relative Hamiltonian pairs and the associated relative moment maps $\mu\in\Omega^{k-1}(F, \mathfrak{g}^{*})$, and prove a relative multisymplectic reduction theorem: for a closed equivariant level $\phi\in\Omega^{k-1}(F, \mathfrak{g}^{*})$, the level pair of $\mu$ descends to a reduced smooth map $F_\phi\colon M_\phi\to N_\phi$ carrying a unique reduced closed relative form $\varpi_\phi$; remarkably, the horizontality of the trivializing component is forced by the relative closedness of the level. We further prove relative analogues of the reduction of dynamics, of the structure theory of split moment maps $\mu=\nu\cdot\kappa$ – for which the splitting datum is a one-step cocycle in the relative Cartan model, so that split relative Hamiltonian $G$-spaces carry canonical relative homotopy moment maps – and of the Duistermaat–Heckman-type variation formula: with respect to suitable conjugate distributions, the reduced relative class satisfies $\partial_\lambda[\varpi_\psi]=\langle c, \lambda\rangle \cdot[\kappa_\psi]$, where $c$ is the Chern form of the target model bundle acting through the natural $\Omega(N)$-module structure of the mapping cone. Finally, we transport the exact stationary phase approximation to the target component and prove a genuinely relative localization constraint: the fixed-point contributions of the pulled-back split package on the source cancel identically.

With the growing demand for satellite sensing and communication, the limited wireless resources are difficult to support multiple satellite systems. Therefore, it is desired to investigate integrated sensing and communication (ISAC) in low Earth orbit (LEO) satellite systems to enable multi-functionality within a single satellite, thereby saving both spectrum and orbital resources. In this paper, a framework for ISAC in LEO satellite systems is established, where a satellite can simultaneously sense multiple targets and serve multiple communication users (CUs) over the same spectrum. Considering the limited onboard energy of satellite, a novel robust beamforming design algorithm is developed with the goal of minimizing total transmit power while satisfying the mean squared error (MSE) requirements for sensing and signal-to-interference-plus-noise ratio (SINR) requirements for communication in presence of channel phase uncertainty which exacerbates the cross-functional interference. According to theoretical analysis, the proposed algorithm for ISAC in LEO satellite systems is effective. Moreover, extensive simulations confirm the superiority of the proposed algorithm over baselines.

We investigate the $p$-torsional rigidity for the $p$-Laplacian, $1<p<\infty$, on compact connected metric graphs equipped with Dirichlet conditions on a nonempty set $\mathcal{V}^D$ of degree-one vertices and nonlinear Kirchhoff conditions at all remaining vertices. We establish the existence, uniqueness, and positivity of the $p$-torsion function, together with a variational characterization of the $p$-torsional rigidity. Our main contribution is the derivation of two sharp isoperimetric inequalities. We first prove a $p$-Saint-Venant inequality, showing that, among all compact metric graphs of prescribed total length, the $p$-torsional rigidity is maximized precisely by the interval with a single Dirichlet endpoint. We then derive a sharp $p$-Kohler–Jobin inequality, providing a scale-invariant lower bound for the first eigenvalue of the $p$-Laplacian in terms of the $p$-torsional rigidity. These results yield nonlinear counterparts, in the setting of compact metric graphs, of the classical Saint-Venant and Kohler–Jobin inequalities, and extend the linear theory, where $p=2$, developed by Mugnolo and Plümer to the full range $1<p<\infty$.

In this paper, we establish some curious identities involving Legendre polynomials and the first kind of Apéry-like numbers. As applications, many new supercongruences are deduced.

Next-generation low Earth orbit (LEO) satellite constellations face critical bottlenecks in spectral efficiency and onboard hardware complexity. To overcome these limitations, this paper introduces a novel architecture enabled by metasurface antennas (MAs) at the LEO satellites. In particular, MAs are metasurface-integrated feed antennas that perform high-precision beamforming directly in the wave domain, thereby effectively mitigating multi-user interference. Based on such an antenna architecture, a weighted sum rate (WSR) maximization problem is formulated by jointly optimizing the scheduling of feed antennas to terrestrial users (TUs) and the passive beamforming of the metasurface for system performance enhancement. To address this mixed-integer nonlinear programming (MINLP) challenge, an alternating optimization (AO)-based joint scheduling and beamforming algorithm is proposed. On the one hand, the proposed algorithm incorporates a polynomial-time minimum-cost maximum-flow (MCMF) method, which is dedicated to the optimal scheduling of feed antennas and TUs. On the other hand, it adopts a weighted minimum mean square error (WMMSE) method integrated with semidefinite relaxation (SDR) technique, which is tailored for metasurface beamforming design. Simulation results confirm the effectiveness of the proposed algorithm for MA-enabled LEO satellite constellation communications.

In this paper, we establish a precise relationship between CGW categories and pointed stable double Segal spaces, both of which were developed as general input for algebraic K-theory. In particular, we show that any CGW category can be regarded as a pointed stable double Segal space, and they can be identified using a classifying diagram construction for double categories with shared isomorphisms.

Interior-point methods (IPMs) are widely adopted due to their high practical efficiency in solving linear, convex, and nonconvex optimization problems. For convex optimization, this performance is theoretically well-supported: there are strong complexity guarantees for self-concordant barrier setups \cite{nesterov1994interior}, which cover linear and conic optimization. On the other hand, unconstrained convex optimization is well-studied. However, there is limited analysis of constrained convex optimization methods without the self-concordance assumption. We develop and analyze a regularized Newton method with line search applied to the log barrier function in the setting that the objective and constraints are thrice differentiable and have Lipschitz continuous first and second derivatives. Starting from a strictly feasible point, our method finds an $\epsilon$-approximately optimal solution in $\tilde{O}(\epsilon^{-2/3})$ iterations.

A simple algorithm for multiple class data classification is proposed, which is based on the infinit speed of propagation and the strong maximum principle for the heat equation. In this method, solutions of the heat equation whose initial data are the characteristic function of the training dataset are considered. The classification is established by the sign of the difference of two solutions in a very short time interval. A numerical algorithm for discrete training data in high-dimensional feature space is proposed. As an application of the proposed method, numerical results on the classification of handwritten digits database are presented in this paper.

We prove several new results about Lagrangian fibrations on holomorphic symplectic complex spaces, under the assumption that the total space is Kähler (but possibly non-compact or singular) and that the base is a complex manifold. First, we construct a meromorphic action by a family of meromorphic groups. Second, we use this structure, together with Hodge-theoretic methods, to prove a version of Ngô's support theorem for Lagrangian fibrations. Along the way, we prove a freeness theorem for the cohomology of compact Kähler spaces equipped with a meromorphic group action.

We propose a locking-free decoupling method for a mixed-dimensional poroelasticity model with fractures. By introducing the total pressure, the fractured Biot system is reformulated as a four-field formulation involving the displacement, total pressure, matrix pressure, and fracture pressure. We establish an energy dissipation law for the continuous model, which shows its consistency with the second law of thermodynamics. Based on this formulation, a time-decoupled scheme is developed. At the initial time step, a fully coupled scheme is employed, while for subsequent time steps, the flow problem is solved first, followed by the mechanics problem. A stabilization term is incorporated into the mechanical equation to help reduce the restrictions imposed on the model parameters in the stability analysis. Energy stability is established for the semi-discrete scheme. For the spatial discretization, the displacement and total pressure are approximated by the Taylor–Hood element, while Lagrange finite elements are used for the matrix and fracture pressures. A fully discrete decoupled scheme is then constructed. Energy stability and error estimates are derived for the fully discrete scheme, and the method is shown to be locking-free. Numerical experiments are presented to support the theoretical results.

We develop compact coefficient-extraction formulae for several families of hyperbolic, logarithmic tangent, and Malmsten-type integrals whose values are finite linear combinations of odd zeta values and even Dirichlet beta values. The principal advantage of these formulae is that coefficients previously encoded by recursive arrays or nested finite sums are replaced by a single coefficient of an explicit elementary expression. This makes the coefficients easier to compute, keeps the dependence on the parameters visible, and reveals structural features—such as vanishing ranges, extremal coefficients, and sign patterns—without hidden cancellations. For shifted hyperbolic integrals with numerator $\sinh((2k+1)x)$, the coefficients are expressed through Chebyshev–arcsine extractions. The same mechanism yields Laurent coefficient formulae for logarithmic tangent integrals and leads to direct proofs of simple initial and terminal coefficients, including a parity-free terminal identity. For $m,n\geq1$, $m\geq n$, and $m+n$ even, we prove \[ \int_0^\infty\frac{\tanh^{m+1}x}{x^{n+1}}\,dx = (-1)^{(m-n)/2} \sum_{p=\lceil n/2\rceil}^{(m+n)/2} \binom{2p}{n} (2^{2p+1}-1) \frac{\zeta(2p+1)}{\pi^{2p}} [u^{m+n-2p}](u\cot u)^{m+1}. \] \[\int_0^\infty \frac{\tanh^m x}{x^n\cosh x}\,dx = (-1)^{(m-n)/2} \sum_{p=\lceil n/2\rceil}^{(m+n)/2} 2^{2p}\binom{2p-1}{n-1} \frac{\beta(2p)}{\pi^{2p-1}} [u^{m+n-2p}] \frac{u}{\sin u}(u\cot u)^m \] In the diagonal zeta case, the formula gives the family $\int_0^\infty(\tanh x/x)^N\,dx$ in a direct, non-recursive form and makes the disappearance of the initial zeta values immediate.

We study the $\pi_2$-systole of positive scalar curvature $3$-manifolds via a local-to-global approach. The tessellated local building blocks are capillary prisms $M$: Riemannian cylinders enclosed by mean convex surfaces meeting at prescribed dihedral angles. For a class of such prisms, called energy-essential prisms, we obtain angle-sensitive relative $\pi_2$-systole estimates. In particular, if the upper capillary angle is at most $\alpha \in (0,\tfrac{\pi}{2}]$, then \[\mathrm{sys}_2(M,\partial_0 M, g)\cdot \inf R_g \lesssim |\log \alpha|^{-1},\qquad \alpha\to 0.\] The proof relies on the monotonicity of a new quasi-local mass. The local-to-global step is achieved via Coxeter gluing of copies of capillary prisms. We prove a general Coxeter gluing and smoothing theorem for scalar curvature: under Coxeter compatibility of the corner strata and nonnegative mean curvature jump conditions along the glued facets, we prove that the resulting piecewise smooth manifold can be smoothened while preserving the scalar curvature lower bound. This theorem gives, to our knowledge, the first general rigorous formulation and proof of the Coxeter-polyhedral smoothing principle for scalar curvature lower bounds, a principle that has long been used heuristically in positive scalar curvature geometry. As an application, we construct smooth positive scalar curvature metrics with large $\pi_2$-systole on connected sums of copies of $S^2\times S^1$ and lens spaces that are not covered by $S^2\times R$. These examples give the first explicit metrics with quantitatively large $\pi_2$-systole on these topologies. We also discuss rigidity and non-rigidity phenomena for the local $\pi_2$-systolic estimates.

For a regular higher order Markov chain, its reduced first order chain may be reducible. When the reduced first order chain is reducible, it does not have a unique stationary distribution. However, it has been shown that the probability distribution of the current state of a regular higher order chain converges to a unique limiting probability distribution. In this paper, we interpret this limiting distribution as a marginal stationary distribution and prove some properties of marginal stationary distributions. We then establish some convergence rate results for such type of convergence. Finally, we introduce two types of marginal mixing times, which extend the notions of mixing times for first order Markov chains to higher order chains.

We prove an effective version of Solomon's first conjecture for lattices over orders in finite-dimensional semisimple algebras over nonarchimedean local fields. We express the quotient of a partial Solomon zeta function by the corresponding maximal-order zeta function as a finite sum whose terms are determined by finite module-theoretic data and weighted by polynomials defined using the Möbius function of finite submodule posets. The resulting expression is independent of the chosen maximal overorder. Our proof is purely algebraic and is first formulated for the refined Bushnell–Reiner zeta functions. As an application, we obtain explicit formulas for the Solomon zeta functions of all lattices over $\mathbb{Z}_p[\mathbb{Z}/p\mathbb{Z}]$, including non-projective lattices.

For a fixed graph $G$, an $r$-uniform hypergraph is said to contain a Berge-$G$ if there exists a bijection $f\colon E(G)\to E(\mathcal{H})$ for some subhypergraph $\mathcal{H}$ such that $e\subseteq f(e)$ for every $e\in E(G)$. Motivated by Alon and Frankl's study of Turán problems under bounded matching constraints, we investigate the maximum number of edges in $r$-uniform Berge-$K_3$-free hypergraphs with matching number at most~$s$. We determine the exact Turán numbers for the cases $r=3$ and $r=4$. For $r=3$ and $n \geq 3 s$, we prove that every $n$-vertex Berge- $K_3$-free 3-graph with matching number $s$ has at most $s(n-2 s)$ edges, and we characterize the unique extremal hypergraph attaining equality. For $r=4$ and $n \geq 4 s$, the maximum number of edges is $s\lfloor(n-2 s) / 2\rfloor$, except for the exceptional case $s=1$ and $n \equiv 1(\bmod 4)$, in which the bound is $(n-1) / 2$. As a corollary, our results recover the classical theorem of Győri on Berge-$K_3$-free hypergraphs.

It is shown that the asymptotics of the Ursell trapping modes on a sloping beach of small slope angle alpha coincide with the inverse Fourier transform of standard WKB exponentials, i.e., the Maslov canonical operator, as alpha tends to 0.

Nearest structured approximation and best structured preconditioning solve different matrix optimization problems. We determine their exact relation for Kronecker positive-definite matrices under the affine-invariant Riemannian metric. The Kronecker family is closed and geodesically convex, so every full matrix has a unique affine-invariant projection. Its logarithmic residual satisfies partial-trace normal equations and yields certified point and objective errors for an Armijo projection solver. Our central result shows that this unique projection is also a minimizer of the Hessian-relative condition number if and only if the extreme spectral states admit identical tensor marginals. A computable marginal-mismatch residual either vanishes at a condition-optimal projection or produces a strict descent direction. Two relative spectral levels always force projection optimality; more strongly, every $2\times 2$ Kronecker projection is condition-optimal. An explicit $2\times 3$ construction is therefore a dimension-minimal strict separation. Residual-calibrated bounds further bracket the best attainable Kronecker condition number and the suboptimality of the projection. Supporting results place classical diagonal and block Loewner sandwiches, fixed-basis primal–dual obstructions, and general log-spectral targets in the same certificate language. Given validated numerical enclosures and outward-rounded comparisons, an interval-safe corollary preserves the soundness of the full Kronecker tests. Deterministic small-matrix checks, including a multistart generic log-factor oracle independent of the partial-trace solver, verify the stated identities and bounds.

We construct two stable limit representations of the double affine Hecke algebra of type $(C^\vee,C)$ on the space of almost symmetric Laurent polynomials, namely the positive and negative stable limit representations. Starting from the standard polynomial representation of the finite rank DAHA of type $(C_n^\vee,C_n)$, we study the asymptotic behavior of the Cherednik operators under the two natural rescalings by positive and negative powers of the parameter $t$. We prove that these rescaled Cherednik operators admit well-defined limits on the ring of almost symmetric Laurent polynomials. This yields stable positive and negative actions of a common stable limit DAHA. The action of the limit Cherednik operators is also proven to be triangular on a natural basis of almost symmetric Laurent polynomials labeled by tuple-partition symbols with respect to the induced Bruhat order. We further construct for each of the two stable limit representations a set of simultaneous eigenfunctions of the limit Cherednik operators using the partial symmetrization operators acting on the non-symmetric Koornwinder polynomials. We show that each of the two sets of the eigenfunctions form a basis of the space of almost symmetric Laurent polynomials, and denote them by the positive and negative stable limit Koornwinder polynomials.

This paper studies the effective stability of nearly integrable Hamiltonian PDEs with asymptotically vanishing spectral gaps ($0 < \alpha < 1$). Under a unified high-low frequency decomposition, we construct a modified block clustering partition based on Bourgain's ideas. By leveraging the vanishing of spectral gaps to suppress high-frequency resonant contributions, the overall non-resonance property is maintained under high-regularity weights. This framework is applied to space fractional and fully dispersive Whitham-Schrödinger equations, uniformly yielding explicit stability estimates in Gevrey, logarithmic ultra-differentiable, and Sobolev spaces.

We show that a random sample from an arbitrary probability measure on $\mathbb{R}^d$ is close to a low-discrepancy point set. Namely, after moving only a small fraction of the sample points in expectation, one obtains an $n$-point set with star discrepancy $\operatorname{polylog}(n)/n$ with respect to the original measure.

Hessenberg varieties are subvarieties of the flag variety, defined by containment conditions on flags with respect to a linear operator. The study of these varieties lies in the intersection of algebraic geometry, combinatorics, and representation theory. In this paper, we develop an algebro-geometric procedure for determining the closed subvariety structure of a Hessenberg variety $\mathcal{H}(X,h)$ in the flag variety for any linear operator $X$ and Hessenberg function $h$, by imposing a partial order on the Hessenberg functions and analyzing the relation of the corresponding Hessenberg varieties. In particular, we give a concrete recursive formula for determining all equations cutting out a given Hessenberg variety in each Schubert cell. As an application, we provide an alternative geometric proof of Tymoczko's results on the existence of affine pavings of a given Hessenberg variety and on the dimension count of its cells.

We develop a contour-count indicator method for visible pole clusters in outward meromorphic continuation from circular boundary data. The method starts from determinant characteristics built from positive Fourier coefficients. In the pure finite-pole model, the correct determinant characteristic factors into a polynomial whose zeros are the reciprocals of the exterior poles. In the presence of a holomorphic background, finite sampling, and noise, roots of individual determinants are unstable and are used only as local evidence. We aggregate this evidence into a scalar indicator field on the reciprocal pole plane: at each sampling point, the field records the fraction of determinant orders and shifts for which a small contour centered at that point encloses exactly one empirical determinant zero. The resulting field plays the role of a sampling-type imaging functional for pole visibility. Fixed superlevel sets give visible-pole clusters, while zero-dimensional persistent homology is used only as a threshold-robust post-processing step. We prove deterministic results linking pure-pole contour counts, Rouché stability, indicator-field contrast, fixed-threshold component recovery, and persistence-gap stability. These results explain why isolated poles with sufficient residue and separation generate stable high-value components, whereas weak, close, boundary-near, or noise-dominated poles may give low, short-lived, or merged components. The framework is a cluster-certification and imaging method, not an unconditional all-pole recovery procedure.