Papers

This paper provides a finite-dimensional spectral characterisation of covariance existence for finite-state Markov-switching affine recurrences of the form \[ Z_{n+1}=A_{\Theta_{n+1}}Z_n+\zeta_{n+1}. \] While strict stationarity is typically governed by negativity of the top Lyapunov exponent of the random matrix products, this does not ensure finite variances, covariances, or Pearson correlations. We show that these second-order properties are governed instead by a Markov-switching Kronecker operator. Writing \(T_{ji}:=p_{ij}(A_j\otimes A_j)\) and denoting its spectral radius by \(\rho(T)\), we prove that, under a natural Perron-excitation condition on the innovation covariance, \(\rho(T)<1\) is necessary and sufficient for the stationary solution to be square-integrable. When this condition holds, the regime-weighted second moments solve a finite-dimensional linear system, yielding the covariance matrix explicitly. Thus, the mean-square spectral condition becomes an exact covariance-existence criterion for the stationary law. Examples illustrate the separation: the top Lyapunov exponent may be negative while the second-order spectral radius exceeds one, so strict stationarity can persist even when finite second moments, and hence population covariances and Pearson correlations, do not exist.

We study an infinite-horizon optimal stopping problem for a two-dimensional normally reflected diffusion in the quadrant with payoff \(G(x_1,x_2)=x_1\vee \alpha x_2\). The problem combines three features that complicate the usual free-boundary analysis: reflection on the coordinate axes, a genuinely two-dimensional stopping region, and a nonsmooth max-type reward. We formulate the associated reflected obstacle problem, prove a verification theorem under explicit Itô–Krylov–Tanaka admissibility and measure-superharmonicity assumptions, and derive a conditional epigraph structure for the stopping set. The main technical point is that the stopping-gain object \(\Gamma=c+rG-\mathcal LG\) is a signed measure rather than a function. Its diagonal component is $\Gamma^\Delta(dx) = -\frac{n^\top a(x)n}{2\sqrt{1+\alpha^2}}\sigma_\Delta(dx)$, $n=(1,-\alpha)$, which shows that pointwise stopping-gain sign conditions must be interpreted with care. We also prove that the correct potential representation is the killed-resolvent formula $V(x)=G(x)-R_r^{\mathcal C}\Gamma(x)$, rather than the unrestricted reflected resolvent. A constant-coefficient reflected Brownian example illustrates the diagonal singular term explicitly.

This paper is the first in a series following on our earlier work [arXiv:2205.14516, arXiv:2307.08180] studying the pair-of-pants product on fixed point Floer cohomology. In [arXiv:2205.14516, arXiv:2307.08180] we fully computed this product for Dehn twists on surfaces of genus greater or equal to 2, and used it to compute a version of the (small) "quantum cohomology" for nodal curves. In the present work, we develop tools for computing the fixed point Floer cohomology and the associated product in the case of Dehn twists in all higher dimensions: for iterated Dehn twists around a Lagrangian sphere in a Liouville domain, we show that the product and differential on the fixed point Floer cohomology split into local and Morse-theoretic contributions on the level of cochains, using some new confinement results for J-holomorphic curves. The local contributions are expected to recover a finite sector of the homology of (twisted) loop spaces of $S^n$ along with an associated Chas-Sullivan product, which we will examine in detail in future work. We also discuss some immediate applications and curiosities for future work.

The paper studies the problem of circular total positivity of the symmetric matrices with zero row sums. These matrices are exactly response matrices of the electrical networks. Alman, Lian and Tran described tests for circular total positivity in two related frameworks: the cluster algebra $\mathcal{CM}_n$ and the Laurent Phenomenon algebra $\mathcal{LM}_n$. Our first result is the construction of a seed in Scott's cluster algebra structure on the coordinate ring of the Grassmannian $\mathrm{Gr}(n-1,2n)$ that consists entirely of circular minors. We compare the cluster structure induced by this seed with $\mathcal{CM}_n$. In particular, for odd $n$ the cluster algebra structure $\mathcal{CM}_n$ is isomorphic to the cluster algebra structure on $\mathrm{Gr}(n-1,2n)$ subject to natural freezing and trivialization of certain cluster variables in their initial seeds. We use this isomorphism to relate circular total positivity to positivity in the Grassmannian. Our second result is that the Laurent Phenomenon algebra $\mathcal{LM}_n$ is isomorphic to the coordinate ring of the noncompactified space of electrical network, or equivalently, to a certain localization of the grove algebra.

We formulate rate-independent elastic-ideally-plastic response directly in hereditary form and study its approximation by finite history surrogates. At the material-point level, the constitutive law is the vector play operator generated by metric projection onto a closed convex elastic domain in stress space. Starting from the closest-point return mapping for step inputs, we pass to absolutely continuous driving histories, for which the constitutive law admits a differential form: the stress remains in the elastic domain and the difference of rates belongs almost everywhere to the normal cone. In this $W^{1,1}$ setting, the hereditary law is causal, contracts variation, and satisfies a $BV$-to-$L^\infty$ stability estimate. We then approximate histories by right-continuous step surrogates with at most $N$ constant pieces. For absolutely continuous inputs, we prove a sharp minimax theorem for input approximation in $L^\infty$, normalized by the $BV$ norm: the optimal encoder is given by equal-variation sampling. For constitutive approximation, the correct vector-valued minimax statement is obtained by allowing the encoder to be material-law aware: it may compress the exact stress history $\mathcal P(\pi)$ rather than only the driving history $\pi$. The resulting stress-aware encoder, followed by the same discrete hereditary decoder, gives the sharp value $(2N)^{-1}$ under the natural nondegeneracy assumption $0\in\operatorname{int}C$. The scalar complementary-variable case is also recorded: there the input equal-variation encoder is sharp because the scalar stop/play operator is $L^\infty$-nonexpansive in the complementary variable. The results identify cumulative variation as the natural variable for sampling and compressing both driving and constitutive histories.

Extremal matroids are those that yield vertices in the polytope of matroids. We show how to get the point, in the appropriate polytope of matroids, for the free product $M\square M'$ of matroids $M$ and $M'$ from the points for $M$ and $M'$ in their polytopes of matroids. With this we show that if $M\square M'$ is extremal, then $M$ and $M'$ are extremal. The converse is false. We identify a large class of matroids that includes perfect matroid designs and sparse paving matroids, and we find sufficient conditions under which free products of extremal matroids in this class are extremal.

We consider bilevel optimization problems in which leader and follower jointly construct a feasible solution for an underlying combinatorial optimization problem. Response problems ask whether the leader can encourage – or, in the pessimistic setting, enforce – a reaction of the follower that includes a set of mandatory items while excluding a set of forbidden items. Our investigation focuses on tractability results for various cases which emerge from different combinations of the total number of mandatory, forbidden, and neutral items. After providing some results for response problems that hold for any underlying combinatorial optimization problem, we examine response problems over the maximum-weight matching problem and the minimum-weight perfect matching problem as illustrative and surprisingly varied examples. Among other results, we show that the response problem is hard for even a single given mandatory or forbidden edge. On the other hand, it is fixed-parameter tractable with respect to the total number of non-mandatory edges. If, however, each follower's edge is either mandatory or forbidden, the response problem for the perfect matching problem is solvable in polynomial time while it remains NP-hard for the maximum-weight matching problem.

Singular changes of variables in Hamiltonian systems, such as McGehee coordinates in celestial mechanics or renormalised variables in dispersive PDE blow-up, are designed to extend the equations of motion to a singular boundary. In contrast, it may be that near the boundary, the the induced symplectic two-form may rescale distinct geometric directions by distinct powers of the singular scale, and if the leading weighted part is degenerate, no single conformal factor makes the form extend as a smooth non-degenerate two-form. This anisotropic obstruction is complementary to the isotropic singularities studied in bm-symplectic geometry. We record it in an elementary finite-dimensional criterion and illustrate it with McGehee regularisation of homogeneous central-force collisions and, formally, with the modulation geometry of focusing nonlinear Schrödinger equation blow-up.

We study unconstrained bilinear zero-sum games, a fundamental model in online learning, adversarial optimization, and multi-agent decision-making. We introduce the implicit midpoint gradient descent rule, which we derive from continuous-time follow-the-regularized leader dynamics via symplectic integration methods. We prove that implicit midpoint gradient descent inherits several powerful properties from the continuous-time dynamics, including bounded orbits, fast ergodic convergence to Nash equilibria, and learning-rate-independent stability guarantees. This is the first traditional online optimization approach to simultaneously achieve these properties in unconstrained bilinear zero-sum games. Finally, computational experiments demonstrate that the proposed method significantly outperforms the standard methods, optimistic and alternating gradient descent.

The permanent of an $n \times n$ matrix is the coefficient of the fully mixed monomial $x_0 \cdots x_{n-1}$ in the product of its row forms $\prod_i\bigl(\sum_j b_{ij} x_j\bigr)$, and the classical exact algorithms of Ryser and Glynn compute it by summing $2^{n-1}$ evaluations of that product over the Boolean cube. We recast this as a single principle – an evaluation scheme together with an orthogonality criterion that decides, in one line, whether its weighted sum equals the permanent – and show that Ryser's formula, Glynn's formula, and a discrete Fourier transform are three instances of it: the $0$–$1$ and $\pm 1$ Hadamard systems, and a single cyclic transform of length $N$. The DFT scheme evaluates $\mathrm{per}\,B$ as one coefficient of a univariate polynomial modulo $x^N - 1$; it is exact exactly when the exponent set is valid mod $N$, runs in the same $\Theta(2^n n)$ operations as the classical formulas, and over a finite field is a number-theoretic transform that returns the exact integer permanent by the Chinese remainder theorem. Where the Hadamard schemes are orthogonal for free, the cyclic scheme trades this for an arithmetic condition on the exponents; the smallest length at which it can be met is $N = 2^n - 2^{\lfloor \log_2 n \rfloor}$, established in the companion paper arXiv:2607.08366. All three are character sums over a finite abelian group, and the criterion holds precisely for such groups.

We investigate the probabilistic counterpart of Yau's conjecture on the nodal volume of Laplace eigenfunctions on compact manifolds, by studying the high-frequency variance asymptotics of Riemannian random waves. Theorem A gives quantitative bounds on the fluctuations of their nodal volume for all regimes of spectral windows, including the monochromatic one, of spectral size 1. For every non-monochromatic window the variance has the exact order $1/N$, where $N$ is the dimension of the underlying space of eigenfunctions, on any compact manifold. In the monochromatic regime, on manifolds without conjugate points and in particular on negatively curved ones, we prove that the variance is $o(1/N)$: this improves the previous best bound, cf. Canzani and Hanin 2020, to its square divided by a logarithmic factor, and exhibits a form of Berry's cancellation, relative to the natural scale $1/N$, for monochromatic random waves on such chaotic manifolds. Our proofs rely on a local and global analysis combining the Kac-Rice formula, the Wiener–Itô chaos decomposition recently obtained by the authors Stecconi and Todino 2025, and a sharp analysis of the error in the pointwise Weyl law associated to an arbitrary spectral window (Theorem B). We also introduce a general machinery (Theorem C) that yields variance decay for the nodal volume of an arbitrary Gaussian field under broad geometric conditions, subject to correlation decay assumptions.

We study finite-horizon optimal control problems with scalar isoperimetric constraints from a function-space duality perspective. Controls are treated as elements of weakly compact subsets of \(L^\infty(0,T;\mathbb R^m)\), while the state equation induces a control-to-state map into \(W^{1,\infty}(0,T;\mathbb R^n)\). For linear dynamics, concave payoff, and an affine isoperimetric functional, we prove that the constrained value function has an interval domain, is concave, and admits a Fenchel–Moreau dual representation. This yields a superdifferential formula identifying the negative of the dual multiplier with the sensitivity of the value function with respect to the constraint level. A constraint qualification is then used to identify the dual multiplier with the normal Pontryagin multiplier of the augmented isoperimetric system. We also treat linear-quadratic problems with a single quadratic equality constraint by reducing them to quadratic forms on a Hilbert space. The resulting analysis separates the validity of the envelope formula from the regularity needed for Riccati synthesis, showing that sensitivity may persist even when the modified control-weight operator becomes singular.

We establish a multilinear analogue of Zafran's interpolation theorem for operators acting on products of quasi-Banach spaces generated by the general real methods. We consider multilinear operators between quasi-Banach couples and obtain interpolation estimates expressed in terms of the fundamental functions of the underlying sequence spaces. As an application, we study the measure of noncompactness of interpolated multilinear operators in this setting and derive a corresponding interpolation estimate for this quantity. In particular, we obtain a one-sided compactness result for interpolated multilinear operators acting on spaces associated with the abstract real methods. These results recover known bilinear theorems as special cases and extend earlier results on multilinear interpolation and on the interpolation of the measure of noncompactness.

We classify, up to conjugacy, the subgroups of the Cremona group of the plane isomorphic to $(\mathbb{Z}/p\mathbb{Z})^r$, where $p$ is prime and $r \geq 2$, over an algebraically closed field $\mathbf{k}$ of characteristic not equal to $p$. In particular, we show that $r \leq 2$ if $p \geq 5$, $r \leq 3$ if $p=3$, and $r \leq 4$ if $p=2$. And hence reprove a well-known statement contained in the article of Arnaud Beauville "$p$-elementary subgroups of the Cremona group", 2007. The main contribution of this article is a concrete description of these groups. In fact, we give an explicit list of representatives via a set of 20 families consisting of subgroups of the de Jonquières group and subgroups of automorphisms of del Pezzo surfaces, and we study the possible conjugacies by birational maps between these families. Finally, we give some results on subgroups of the Cremona group of the plane isomorphic to $(\mathbb{Z}/p\mathbb{Z})^r$, where $p$ is prime and $r$ is an integer, over an algebraically closed field $\mathbf{k}$ of characteristic equal to $p$.

In this paper, we establish a geometric correspondence between the Lorentzian Raychaudhuri equation and Perelman's non-collapsing theorem for the Ricci flow. By interpreting the Raychaudhuri equation as a Lorentzian analogue of Ricci flow, we connect geodesic focusing in general relativity to the monotonicity and entropy functionals of geometric analysis. Using this correspondence, we derive a Lorentzian non-collapsing theorem and introduce a covariant entropy functional governing causal volume evolution. Finally, we propose the concept of geodesic entropy capacity, a curvature-bounded limit on the information that can be stored in spacetime regions, providing a unified geometric framework linking gravitation, thermodynamics, and information.

We study decentralized minimization of a sum of functions over a network where nodes may only leave, the remaining nodes stay connected, and the topology freezes between departures. Standard methods forget departed functions, causing a permanent bias proportional to the heterogeneity of the data. We propose Legacy Gradient Tracking (Legacy-GT): before leaving, a node compresses its function into a gradient-anchored quadratic legacy, bequeaths it to a neighbor, and provides a correction that exactly preserves the gradient-tracking invariant. We prove that the optimal legacy curvature is the average of the strong-convexity and smoothness constants, that legacies compose losslessly across chained departures, and that an adaptive anchor rule yields error bounds that decay geometrically after the last departure to a small residual–the minimum of the network's optimization error at departure and its heterogeneity radius. In contrast, the classic drop-and-forget baseline suffers a bias that never decays. Numerical experiments confirm the theory.

We investigate increasing sequences of integers with the property that the product of every two terms of the sequence is also a term of the sequence and the sum of every two terms is not a term of the sequence. We say that a sequence with the above properties is maximal if it is not a proper subsequence of another sequence with the above properties. We determine whether the sequences in three families are maximal or not.

We consider the three-dimensional stochastic Navier–Stokes equations with additive stochastic forcing. We study temporal discretizations based on a semi-implicit Euler as well as a Crank-Nicolson scheme. We prove that locally in time the former converges with rate 1 while the latter converges with rate 3/2. These theoretical results are confirmed by extensive numerical simulations. To the best of our knowledge this is the first time that numerical simulations for the three-dimensional stochastic Navier–Stokes equations have been performed.

We study the range of $\operatorname{Re}\{a_2 f(z)\}$ for normalized analytic functions $f$ in the unit disk belonging to several classes of conformal mappings. As our main contribution, we introduce the class $CC_\alpha$ of completely convex mappings of order $\alpha$, defined by a uniform two-point starlikeness condition, and we estimate the range of $\operatorname{Re}\{a_2f(z)\}$ in terms of $\alpha$, for all $f\in CC_\alpha$ and $z\in \mathbb{D}$, generalizing the classical result of Fournier–Ma–Ruscheweyh, which is recovered for $\alpha=0$. We also determine omitted value sets for convex functions of order $\alpha$, spherically convex mappings, uniformly starlike functions, and Nehari classes $\mathcal{N}_t$. The proofs rely primarily on the Schwarz–Pick lemma applied to auxiliary functions constructed from the two-point kernel $zf'(z)/(f(z)-f(x))$.

A numerical investigation of steady forced convection heat transfer from an isoflux circular cylinder immersed in the liquid metal Galinstan is presented. The governing streamfunction, vorticity, and energy equations are solved using a fourth-order compact finite difference scheme in cylindrical coordinates (FOCS–CC) coupled with a stable pseudo-time iteration (PTI) technique. The influence of the Reynolds number ($1 \leq Re \leq 600$) on the flow and heat transfer characteristics is systematically investigated for Galinstan with a Prandtl number of $Pr=0.025$. The performance and accuracy of the proposed scheme are first established through grid independence studies and validation against previously published numerical results for the average Nusselt number and total drag coefficient over a range of Reynolds numbers. Excellent agreement with the available literature confirms the reliability and robustness of the present formulation. The effects of Reynolds number ($1\leq Re\leq600$) on the flow and thermal fields are examined through streamline patterns, isotherm distributions, and local Nusselt number variations. The results reveal that increasing the Reynolds number promotes flow separation, enlarges the wake region, intensifies downstream thermal transport, and significantly enhances convective heat transfer from the cylinder surface. Furthermore, a new empirical correlation for the average Nusselt number is proposed for Galinstan fluid over the Reynolds number range $1\leq Re\leq600$, exhibiting excellent agreement with the numerical data with a coefficient of determination of $R^2=0.99939$.

Heterogeneous mean-field control (MFC) involves multiple interacting populations with distinct dynamics, control constraints, and coupling structures. We develop a heterogeneous occupation-measure mean-field control (OM-MFC) framework that reformulates the population-distribution control problem as an optimization problem over occupation measures. In this representation, potentially nonlinear dynamics enter through linear weak Liouville constraints, while nonlocal inter-population interactions become structured bilinear functionals of the occupation-measure marginals. Exploiting this structure, we derive a Frank–Wolfe (FW) method and show that its linear minimization oracle decomposes into independent population-wise optimal control problems once the current population distributions are fixed, enabling parallel computation without an a priori discretization of the measure space. We further establish a sufficient convexity condition based on the symmetrized matrix-valued interaction kernel, under which Frank–Wolfe with an exact oracle admits the standard $O(1/k)$ convergence rate. Numerical examples on heterogeneous UAV coordination and three-dimensional search-and-rescue illustrate the framework both within the certified convex regime and for directional interactions beyond that regime.

We study inverse-central-binomial series with finite reciprocal-square factors and harmonic weights involving $H_n$, $H_{2n}$, and $1/n$. An arcsine integral gives explicit odd-zeta evaluations and a classification of coefficientwise cancellation within a polynomial ansatz. The scalar examples include an evaluation of weight seven and a direct proof of Sun's Conjecture 3.10. An entire generating relation organizes a second family with finite rational coefficient formulas. We then derive that relation, a supplementary generating family, and the complete two-order construction from Bailey's cubic transformation, contiguity, Gauss's quadratic and connection formulas, Euler's transformation, and Clausen's identity. The derivation retains all three infinity branches and resolves the vanishing denominators in the lower contiguous value. A doubling relation with a unique analytic solution and an explicit algebraic identity complete the proof. Thus the evaluations have a proved classical origin; no independent new hypergeometric summation principle is claimed. An application of a known complete-monotonicity property gives strict signs and global Taylor bounds. All endpoint limits, parameter derivatives, and coefficient extractions are justified.

Using the construction of Bachiller, Cedó and Jespers, we give a complete classification of the indecomposable involutive set-theoretic solutions to the Yang-Baxter equation whose permutation brace has size $p^3$, where p is an odd prime. We also give an algorithm for systematically producing all indecomposable involutive solutions with a given permutation brace, and use this to enumerate these with the permutation brace of size up to 107.

We investigate Dirichlet boundary value problems for a class of second-order parabolic equations in divergence-form with coefficient matrices that exhibit singular and degenerate behaviors characterized by a Muckenhoupt weight class. This framework serves as the parabolic analogue to the singular-degenerate elliptic equations pioneered by Fabes, Kenig, and Seraponi. Under a smallness assumption on the partially weighted mean oscillation of the coefficients, we establish the existence, uniqueness, and local interior and boundary regularity estimates for weak solutions within appropriately defined weighted Sobolev spaces. The proofs rely on the freezing coefficient technique alongside the level-set method introduced by Caffarelli and Peral. Additionally, we develop the necessary weighted Sobolev space framework and related weighted inequalities. Finally, a compactness argument is utilized to demonstrate that solutions to these equations remain locally close, in the weighted Sobolev norm, to their frozen-coefficient counterparts.

Prophet inequalities have become a central tool for analyzing the performance of online algorithms. However, most existing results assume that input random variables are independent, which limits their applicability. Motivated by this gap, we study prophet inequalities under two correlation models induced by a latent state of the world variable $Z$. In the common-base model, the algorithm observes the sequence $Z+X_1,\dots,Z+X_n$. We analyze single-threshold algorithms with the constraint that they always accept the final item, guaranteeing a reward of at least $Z$. When $Z$ is chosen adversarially, we characterize the optimal deterministic algorithm of this form, achieving a competitive ratio of $0.381$. We then show that randomizing improves the guarantee to $0.4$. By a minimax argument, the same ratio is achievable when $Z$ is random. We depart from standard techniques by establishing a stronger lower bound of $0.41$ and an upper bound of $0.475$, ruling out the possibility that this class of algorithms attains the $1/2$ ratio known for independent inputs. The core technical contribution is a new analytical framework that captures the reward dynamics of single-threshold algorithms. We introduce a differential equation characterizing the expected reward of a threshold in the worst-case instance, parameterized by the distribution of the maximum. This equation admits a closed-form and unifies known single-threshold prophet inequalities, yielding a simple threshold-optimality condition applicable to the common-base model. Finally, we study the common-scale model, where inputs take the form $Z\cdot X_1,\dots,Z\cdot X_n$. We show that this minimal multiplicative correlation yields strong impossibility results: no algorithm can achieve a competitive ratio exceeding $1/n$.

We show that solving the abelian part of the local anomaly cancellation conditions for a 4-d gauge theory whose gauge Lie algebra has an abelian summand with rank $K \geq 1$ is equivalent to the problem in algebraic/arithmetic geometry of finding $(K-1)$-dimensional projective linear subspaces of a cubic hypersurface over the rational numbers, where the cubic is determined by the data of the semisimple summand and the representation thereof carried by the Weyl fermions. We then use this reformulation to solve a variety of examples with rank 2. The simplest non-trivial example from physics, namely gauge Lie group $U(1)^2$ and six fermions, nevertheless has a rich (and well-studied) geometry: it corresponds to the Fano variety of lines in the Segre cubic primal threefold. This is a surface with 15 irreducible components that are planes (which correspond to lines lying in the 15 planes in the Segre cubic primal and which give rise to non-chiral fermions) and 6 components that are split del Pezzo surfaces of degree 5 (which give rise to chiral fermions). These components are all rational varieties, enabling all solutions to the anomaly cancellation conditions to be parametrized and their collective properties (e.g. their topology and asymptotic distribution) to be described.

Membership problems in the completely monotone–Stieltjes–Bernstein hierarchy often become explicit after passage to a determinate representation. We organize that passage as a witness calculus. For the Laplace, Stieltjes, and Hausdorff transforms used here, W f denotes the unique admissible signed representing datum; a certificate either proves its positivity or exhibits an obstruction. A concrete dictionary for completely monotone, logarithmically completely monotone, Bernstein, complete Bernstein, and Stieltjes functions and for Hausdorff moment sequences is combined with transport rules for products, shifts, sampling, and Bernstein increments. The applications are consequences of this common calculation. A triple-product identity for a digamma bridge controls ordinary Bernstein and logarithmic complete monotonicity ranges for gamma ratios and gives the strict power cutoff in Szabo's problem. For integer gamma quotients, the inverse Laplace density is a damped Jacobi polynomial, so classical zero geometry decides broad parameter regions. Boundary signs distinguish Bernstein from complete Bernstein behavior and separate completely monotone functions from the Stieltjes class. On the discrete side, signed atoms and finite differences settle interpolation and coefficient questions. The Ramanujan integral and a fractional Volterra symbol provide positive recognition examples. Across the applications, the workflow is the same: normalize the class question, compute or transport its witness, and decide one explicit sign problem. Every exclusion is accompanied by an exact certificate.

We study the mixing time of the systematic scan dynamics for the $q$-state ferromagnetic Potts model on the $n$-vertex complete graph, known as the mean-field model. This Markov chain updates vertices sequentially according to a fixed predetermined order, in contrast to the Glauber dynamics which updates a uniformly random vertex at each step. Systematic scan dynamics are attractive in practice as they often demonstrate strong empirical performance. However, their theoretical analysis remains far less developed than that of the Glauber dynamics. We take a step toward addressing this imbalance by showing that for every $q\ge 2$ and $\beta<\beta_s$, where $\beta_s$ is the metastability threshold associated with the onset of slow mixing for the Glauber dynamics, the systematic scan dynamics for the ferromagnetic mean-field Potts model mixes in $\Theta(\log n)$ scans or, equivalently, in $\Theta(n\log n)$ single site updates. We in fact prove a sharper result; namely, that there exists a constant $c(\beta,q) > 0$ such that the mixing time is $c(\beta,q)\log n + \Theta(1),$ which implies that the Markov chain exhibits the cutoff phenomenon, with the total variation distance to the stationary distribution dropping abruptly from nearly 1 to nearly 0 within a narrow $\Theta(1)$ time window. This result is tight in $\beta$ as well since the dynamics mixes exponentially slowly for $\beta > \beta_s$. To the best of our knowledge, this is the first general cutoff result for the systematic scan dynamics in the context of spin systems. The result may also be of independent interest in the theory of Markov chains, since the systematic scan dynamics is both global and non-reversible, two settings in which cutoff remains poorly understood.

We construct an exact cluster $F\subseteq\mathbb{Z}^2$ of cardinality $8$ with full affine span, together with an $F$-tiling $T$, such that the orbit closure of $T$ in $\{0,1\}^{\mathbb{Z}^2}$ does not contain a $1$-periodic $F$-tiling. Since every $F$-tiling is a low-complexity configuration with respect to the window $\bar F := \{-a : a \in F\}$, this supplies a "non-degenerate" counterexample, in a strong sense, to Nivat's conjecture for non-convex windows. This answers, in the negative, a question of Kari and Moutot (2023) whether every such counterexample must be degenerate, in the sense that the probing window is contained in a coset of a proper finite-index sublattice. We complement this with a positive result: for every exact cluster $F$ of full affine span whose cardinality is the square of a prime, every $F$-tiling has a $1$-periodic $F$-tiling in its orbit closure. Together with Szegedy's theorem that every tiling by a cluster of prime cardinality is $1$-periodic, this shows that no cluster of fewer than $8$ cells can exhibit the phenomenon, with the possible exception of cardinality $6$, which we leave open.

This paper is a continuation of our earlier work, in which we described a Las Vegas algorithm to solve the elliptic curve discrete logarithm problem. The Las Vegas algorithm reduces the elliptic curve discrete logarithm problem to finding a zero minor in a matrix. Using the intersection poset of a hyperplane arrangement, we develop an algorithm to find a zero minor in a rectangular matrix. Our methods are elementary. We discuss the complexity and success probability of our algorithm and provide implementation details. Finding a zero minor in a matrix is also of independent interest.