Papers

In this paper we construct a recursive bijection between $4$-valent maps endowed with a Eulerian orientation – also known as the ice model – and a family of unary-binary trees. This answers a question of Bousquet-Mélou and Elvey Price, who established that both these families have the same generating function in arXiv:2503.15046 .

We consider $r$-refinement for the finite element discretisation of a Poisson problem. The goal of $r$-refinement is to reposition the nodes of a computational mesh in order to better approximate the finite element error. Since the mesh is being moved, it naturally becomes linked to shape optimisation methods. We propose a standard optimisation algorithm for this $r$-refinement procedure and show that if one seeks to minimise the actual error - which one cannot generally calculate - one has an algorithm which will terminate. The most novel aspect of this work is to apply shape optimisation techniques to the a standard residual error estimator, which is a functional that is differentiable with respect to the mesh, when considered for the Poisson equation in one dimension. To illustrate the approach, a number of numerical experiments are presented, which verify the efficacy of the method.

We study the canonical skew-binary representation of nonnegative integers, whose positional weights are the Mersenne numbers \(2^{k}-1\), and its decomposition into the binary and nonbinary sides. The nonbinary values, arranged in increasing order, form the OEIS sequence A055938, and their successor structure gives a direct proof of the relation with the OEIS sequence A080578 that is recorded as conjectural in the OEIS entry for A080578. The resulting formulas show that the Conolly sequence describes both sides of the decomposition: a shifted Conolly sequence is the counting function of the binary side, while the same sequence gives the increasing enumeration of the nonbinary side. We then develop the parent map, truncation blocks, truncation remainders, and the Mersenne tau function associated with this representation. The parent map is conjugate to deletion of the lowest digit. Differences of the Mersenne tau rows recover the parent iterates and the counting function, and they give formulas for digit reconstruction and for a diagonal tau defect. Finally, we revisit a known finite-depth one-soliton family of the finite-field BBS. The Mersenne combinatorics is used directly to reconstruct a global integer-valued traveling-wave profile and to prove an integer window-counting theorem. Reduction modulo \(3\) yields the corresponding finite-field traveling-wave solutions. We also construct an integer-valued traveling-wave tau function whose front values are given by the Mersenne tau rows. The resulting construction shows how the combinatorics of a number representation can enter directly into the reconstruction of solutions of an integrable system.

Score-based diffusion models have achieved remarkable empirical success in generative modeling, yet their approximation-theoretic foundations remain incomplete. In particular, although classical universal approximation theorems guarantee that neural networks can approximate score functions, it remains unclear whether such approximation guarantees translate into approximation of the probability distributions generated by reverse diffusion processes. In this paper, we establish a rigorous quantitative connection between these two notions. Specifically, we prove that if a neural network approximates the true score function sufficiently accurately, then the probability distribution generated by the corresponding reverse diffusion model is close to the target data distribution in Kullback-Leibler (KL) divergence, up to an irreducible mismatch between the terminal distribution of the forward diffusion process and the prior used to initialize the reverse process. More precisely, we derive an explicit upper bound on the distribution approximation error in terms of the score approximation error, the diffusion noise schedule, and the terminal prior mismatch. Our analysis combines Hornik's universal approximation theorem, Girsanov's theorem on path space, and the data processing inequality for relative entropy. Complementary to recent work that studies score approximation under finite-sample statistical settings and structural assumptions on the data distribution, our work develops an approximation-theoretic analysis based on classical neural network approximation theory. The resulting theorem provides a simple and explicit guarantee linking neural network approximation of score functions to approximation of the probability distributions generated by reverse diffusion models.

We exhibit an explicit integer polynomial in five variables, of total degree $14$ and with constant Hessian determinant $128$, whose gradient is not injective. Consequently its formal Legendre transform is not a polynomial, and the Hessian conjecture $\HC_5$ is false. The counterexample is obtained from the six-variable doubling of Alpöge's 2026 Jacobian counterexample by a one-variable Schur descent—a partial Legendre transform in a single variable. Combined with de~Bondt's theorem that $\HC_n$ holds for $n\le3$, with the elementary doubling and stabilization bridges relating the Jacobian conjectures $\JC_n$ to the Hessian conjectures $\HC_n$, and with Alpöge's refutation of $\JC_3$, this decides the Hessian conjecture in every dimension except $n=4$: $\HC_n$ is true for $n\le3$, false for $n\ge5$, and open only at $n=4$. Exactly two statements of the two families remain unsettled, $\JC_2$ and $\HC_4$, linked by $\HC_4 \Rightarrow \JC_2$. Along the way we record, as a warm-up, an explicit six-variable counterexample to $\HC_6$ with constant Hessian determinant $-4$ and non-injective gradient. This note adds the five-variable counterexample to, and updates the status recorded in, the first author's earlier educational preprint \cite{MengRG2026}.

We study finiteness classes arising from Ramsey-theoretic principles in set theory without the Axiom of Choice. First, we answer a question of Brot, Cao, and Fernández-Bretón concerning Gowers' $\mathbf{FIN}_k$ theorem. The $\mathbf{FIN}_k$ operation gives rise to a finiteness notion for each $k\geq 1$, but we show that this notion is independent of the value of $k$: the resulting hierarchy collapses, and every level is equivalent to $H$-finiteness, i.e., to Dedekind-finiteness of $[X]^{<\omega}$. We then turn to the Rainbow Ramsey theorem and the Canonical Ramsey theorem. The corresponding failure classes $\operatorname{\mathbf{RRT-Fin}_n^m}$ and $\operatorname{\mathbf{CRT-Fin}}$ are genuine finiteness classes contained in $\operatorname{\mathbf{D-Fin}}$. We prove several inclusions placing them relative to standard Ramsey-theoretic finiteness classes of Brot–Cao–Fernández-Bretón, prove their independence from the collapsed Hindman–Gowers class, and analyze the parameter dependence of the rainbow classes through Fraenkel–Mostowski examples. The finer questions of strictness, and the full two-parameter structure of the rainbow classes, are left open where the present arguments do not settle them.

We present a novel leverage score-based sampling strategy for the randomized alternating least squares optimization (ALS) of the canonical polyadic decomposition (CPD-ALS). Unlike previous strategies, we determine row-wise samples for the CPD-ALS problem from the leverage scores of the target tensor which is being decomposed. We demonstrate that, when rows are sampled according to the leverage score distribution of the matricized target tensor, each least squares subproblem of the CPD-ALS problem achieves $(1+\epsilon)-$relative accuracy in the residual norm with probability at least $1-\delta$ using a sampling $s=\frac{R\gamma}{\beta} \max\left(\frac{4}{\delta \epsilon}, \frac{144\ln(2R/\delta)}{\epsilon_{0}^{2}}\right)$, where $\epsilon_{0}$ is a constant, $\beta$ is leverage score's approximation constant, $R$ is the target rank and $\gamma$ captures the coherence between the Khatri Rao product (KRP) of the CPD factor matrices and the exact KRP; $\gamma$ decreases as the ALS iterates converge. To efficiently approximate the leverage score distribution for each matricization of the target tensor without explicitly computing leverage scores we use a randomized strong rank-revealing QR (sRRQR) factorizations, SE-QRCS. By construction, this QR-based leverage score sampling method outperforms previously published schemes as it does not, in principle, require the resampling of the target tensor or recomputing the leverage scores of the KRP, minimizing the computational and storage overhead of the CPD-ALS procedure.

This paper presents an optimal control framework for achieving finite-time stabilization of linear systems. By introducing a suitably constructed integral cost function, we derive a new class of nonlinear controllers that guarantee finite-time stability through the application of the optimality principle. The relationship between the resulting optimal control law and the associated value function is analyzed, leading to the derivation of a Hamilton-Jacobi-Bellman (HJB) equation and the study of its regularity properties. Numerical simulations validate the theoretical findings and illustrate the effectiveness of the proposed method. Furthermore, a discussion on estimating the convergence time is provided.

Regime-conditioned transition probabilities and moments are basic inputs for prediction and decision making in hybrid systems, but their short-time infinitesimal structure is not directly observable. We study nonparametric estimation of regime-indexed semigroup blocks for switching diffusions observed together with their regimes. A Dynkin–Taylor expansion identifies the first two block-generator coefficients, and localized probes recover switching intensities, drift, and diffusion. Under shrinking meshes, a common-design difference cancels the localized level and yields a martingale array; nonoverlapping second differences preserve within-block covariance and produce the variance factor two. The resulting first- and second-order estimators are asymptotically normal, admit feasible studentization, and support short-horizon approximation and specification checks for interactions among primitive coefficients. Fixed-mesh local-polynomial theory and a numerical study complement the recovery results.

We compute the chain-level change of local Floer complexes across the generic bifurcations of periodic orbits in four-dimensional Hamiltonian systems. Although local Floer homology is invariant across a bifurcation, the underlying chain complex mutates: periodic orbits and their Conley–Zehnder indices change discontinuously, and the Floer differential must change to compensate. Two tools make this mutation computable. First, we prove a Floer cascade counting theorem for Floer cylinders between multiply covered orbits of an autonomous system, which is not specific to the bifurcation analysis carried out here. Second, for each bifurcation type in Meyer's classification, and for two additional $\mathbb{Z}_2$-symmetric types, we construct a model Hamiltonian whose Floer cylinders are lifts of planar gradient trajectories, which we call the gradient revolutions, reducing the count to finite-dimensional Morse theory and cascade counting. We determine all seven mutations explicitly.

In addition to reflection and refraction, another form of wave deviation is defined as diffraction. Notably, when incident waves strike a corner, diffracted waves emanate from the corner tip and propagate omnidirectionally. This paper proposes a novel framework for detecting rigid cornered obstacles using measured diffracted wave data. The framework first transforms the underlying initial-boundary value problems into initial value problems on conic manifolds via the method of images. Subsequently, the retrieval of obstacle information is achieved through Cheeger–Taylor functional calculus and microlocal analysis on conic manifolds. Specifically, we prove that for a given pulse, measurements of the resulting diffracted waves captured by a curve receiver uniquely determine both the location and shape of the visible portion of a polygonal obstacle. The proof is constructive, explicitly formulating the corresponding recovery scheme. This methodology offers two key advantages: first, the size and placement of the receiver can be arbitrary; second, the inversion only requires measurements of diffracted waves and obviates the need to solve wave equations within the cornered domain as in conventional methods.

In recent years, significant advancements have been made in understanding the spectral theory of hyperbolic spaces, particularly in the context of twisting representations, both unitary as well as non-unitary. We survey some results that we obtained within the framework of the SPP 2026 "Geometry at Infinity" as well as some closely related results.

Let $\Lambda$ be a finitely generated abelian group and $\Gamma=\mathbb Z^{*d}$, we study the Cayley graph of the wreath product $G=\Lambda\wr\Gamma$ with natural set of generators and their inverse $S$. First, we establish a random matrix model $X_N=\sum_s X^{(s)}_N$ where the sum is indexed by the set $S$. As the size $N$ of the matrices goes to infinity, the traffic distribution of the $X^{(s)}_N$'s converges to that of the image of these generators in the reduced $C^*$-algebra of $G$. In particular, the spectral measure of $X_N$ converges toward that of the Cayley graph of $G$ with generators $S$. Moreover, in the case $\Gamma=\mathbb Z$, we establish a central limit theorem for linear statistics of this random matrix model. Then, we exhibit a formula for the asymptotic $R$-transform and derive the second-order distribution of the limit of $X_N$ in terms of its limiting first-order distribution.

In this paper, we study a prototype hyperbolic system arising from the coupling of Generalized-Lagrangian-Multiplier (GLM) curl-cleaning with linear acoustics. We first show that, in the absence of dissipation, the equations can be rigorously derived from an underlying variational principle and that the resulting system is symmetric-hyperbolic. Further analysis shows that it not only conserves the total energy but also admits a set of invariants consisting of quadratic combinations of differential operators applied to the state variables. The addition of a stiff relaxation source term allows the system to be extended to dissipative dynamics, describing, for example, Cattaneo-type heat transfer processes that are asymptotically compatible with the Fourier law in the stiff relaxation limit; in this case, the total energy and invariants are dissipated accordingly. A new semi-implicit compatible numerical scheme on staggered grids is developed to solve this system while exactly preserving its key properties at the discrete level. In particular, we prove that the scheme is asymptotic-preserving in the Fourier limit, with a convergence rate depending on the initial data. We also show that the scheme conserves exactly the total energy as well as all the invariants in the absence of relaxation, and dissipates them consistently in its presence. These findings are demonstrated on a set of representative test cases.

We discuss an interesting distinction between the incompressible Navier-Stokes equations (the velocity equations) and its vorticity form in whole space. We show that if the initial vorticity has a Gaussian bound then the bound is inherited up to the maximal lifespan of the strong solution. However, it turns out that the velocity equations don not share the same property. In fact, $L^p$-strong solutions to the velocity equations arising from “well-localized" initial value generally behave at infinity like derivatives (of order $\geq3$) of the fundamental solution of Laplacian. To show this, a clean expansion up to maximal lifespan is derived : \begin{align} u(x,t)=-\nabla\sum_{|\alpha|=0}^{d-1}\frac{(-1)^{|\alpha|}}{\alpha !}\partial^\alpha\partial_{i,j}^2\Gamma(x)\int_0^tM_{\alpha}^{i,j}(s)ds+O\big(|x|^{-2d-1}\big)\nonumber \end{align} where ${\rm M}_{\alpha}^{i,j}(t):=\int_{\mathbb R^d}y^\alpha u^i(y,t)u^j(y,t){\rm d}y$ and $\Gamma$ is the fundamental solution of Laplacian. This improves the first order expansion given by L. Brandolese and F. Vigneron \cite{BV07}.

We give new proofs computing the asymptotic densities for orbits of the arithmetic Hecke triangle groups $\Gamma_q$ when $q=4$ and $q=6$. We use elementary number theory techniques along with basic properties of the Möbius function and Riemann zeta function with additional congruence conditions. This note actually came about by observing that, in the $q=4$ case, the underlying congruence condition partitions the set of coprime integer pairs into three classes of equal density.

Binary cyclic codes are an important class of linear codes that admit highly efficient encoding and decoding algorithms. Constructing binary cyclic codes with favorable parameters has been an active research topic for several decades. In recent years, substantial progress has been made in the study of binary $[n,k,d]$ cyclic codes with $k$ close to $n/2$ and $d\geq \sqrt{n}$. Nevertheless, among the known codes of this type, only a few families simultaneously possess both large minimum distances and large dual distances. In this paper, for even $s$, we focus on new constructions of $[2^{s}-1,k]$ binary cyclic codes with rate approximately $1/2$, for which both the minimum distance $d$ and the dual distance $d^\perp$ are simultaneously large. Some of the constructions yield binary cyclic codes with parameters $[2^s-1,2^{s-1}\pm 1,d]$ such that the lower bounds of $d\cdot d^\perp$ are close to $n$ or $2n$. The other constructions yield binary cyclic codes with parameters $[2^s -1, 2^{s-1} + c, d]$ for some $ 1- \frac{5s}{2} \leq c \leq \frac{5s}{2} -1$ such that the lower bounds of $d\cdot d^\perp$ are close to $n$ or $2n$. Compared with the known binary cyclic codes of the same length and dimension, the majority of the binary cyclic codes we construct exhibit larger minimum distances. In particular, we obtain several codes with excellent parameters, which are verified against the Code Tables available at http://www.codetables.de/.

We focus on a stochastic system that models a collection of $N$ identical oscillators with unidirectional coupling, perturbed by white noise. The unidirectional coupling breaks the symmetry of mutual dependence among the oscillators. It is shown that the associated random dynamical system $\phi$ admits a dynamical order, meaning that $\phi$ admits a simple asymptotic one-dimensional structure that is totally ordered with respect to the standard order in $\mathbb{R}^N$. Our approach takes a novel geometric perspective: we introduce a random dynamical system $\Phi$ on a smooth Riemannian manifold $M$, diffeomorphic to $\mathbb{S}^1 \times \mathbb{R}^{N-1}$, by “wrapping” the random system $\phi$ from $\mathbb{R}^N$ onto $M$. By choosing an appropriate random cone field $\mathcal{C}_M$ on $M$, we show that $\Phi$ is a differentially positive random system on $M$, which is a random counterpart of the differentially positive systems introduced by Forni and Sepulchre. We further demonstrate that the traditional well-known horizontal curves can be identified as the conal curves on $M$, thereby providing a crucial tool for establishing the dynamical order of the random system $\phi$.

In perfect conductivity and linear elasticity problems, the electric field and stress always become highly concentrated within narrow regions between adjacent perfect (rigid) inclusions, and blow up as the distance between inclusions approaches zero. The design of high-order numerical methods with rigorous error analysis for such concentration problems remains open. In this paper, we present the first high-order finite element method for solving these problems. Our approach is based on asymptotic estimates of high-order derivatives of solutions, employing a graded mesh and auxiliary basis functions specifically designed from these derivative estimates. We prove that the proposed method converges with an $H^1$-error bound of $O(h^p)$ and the error bound is independent of the distance (possibly approaching zero) between inclusions, where $h$ is the mesh size and $p$ is the degree of finite elements. Numerical examples in both two and three dimensions are presented to demonstrate the convergence rates of the numerical solutions. In particular, the blow-up behaviors of the gradients of solutions are demonstrated when the inclusions approach each other.

Let $I$ be an ideal of a polynomial ring $S$ over a field $K$ for which $I$ is minimally generated by at least two quadratic binomials. Suppose that (i) $I$ is prime, (ii) $S/I$ is Cohen–Macaulay and (iii) $I$ has linear resolution. The question whether $I$ is equal to the ideal of $2$-minors of a $(2 \times n)$-matrix of variables is mainly studied.

We give self contained proofs of the completeness of countable $\sigma$-coherent theories, for both propositional and predicate logic. By $\sigma$-coherent logic we mean the fragment of geometric logic that only allows countable signatures and countably infinite disjunctions. In the propositional case, each theory determines a quasi-Polish space of models, and formulas (up to provable equivalence) are interpreted as continuous functions from the space of models to the Sierpinski space. In the predicate case, each theory determines a quasi-Polish category of models, and formulas (up to provable equivalence) are interpreted as continuous functors from the category of models to the quasi-Polish category of overt discrete quasi-Polish spaces. We further extend this analogy by investigating a notion of “sobriety” for certain topological categories.

Let $Q$ be a Dynkin quiver over $k=\mathbb F_q$, let $A=kQ$, and let $\Ktwo(\cP)$ be the extriangulated category of two-term complexes of projective $A$-modules. We study the square-root normalized Hall algebra of $\Ktwo(\cP)$. We first establish a PBW-type vector-space factorization into the Ringel–Hall part and the shifted-projective part, and derive an explicit mixed multiplication formula. For the indecomposable projectives $P_i$, let $p_i$ and $z_i$ be the normalized Hall classes of $(0\to P_i)$ and $(P_i\to0)$, respectively. We prove \[ z_ip_i=q^{-1}p_iz_i+1, \] and determine all off-diagonal products $z_ip_j$ in terms of kernels and cokernels of maps $P_i\to P_j$. Hence each pair $(p_i,z_i)$ generates a rank-one quantum Weyl algebra, while every pairwise Hom-orthogonal family of projectives generates a higher-rank quantum Weyl subalgebra. For these subalgebras we construct explicit Hall–Fock modules, on which the $p_i$ act as creation operators and the $z_i$ act as $q$-annihilation operators. This provides a finite-field Hall model parallel to categorical Hall-type Weyl actions in Donaldson–Thomas theory. Finally, we show that BGP reflection transports the corresponding reflection subalgebras and preserves the mixed Hall coefficients.

The conjugate gradient method (CG) solves a symmetric positive definite (SPD) system $Ax=b$, but does not respect $x\geq0$. We develop non-negative conjugate gradients, a solver for the bound-constrained quadratic $\min_x \tfrac12 x^\top A x - b^\top x$ subject to $x\geq0$, $Bx=c$, $A\succ0$, with $Bx=c$ present only in the equality-augmented variant. The solver wraps CG in a primal-dual active-set loop: each step solves the unconstrained system over the free variables, drops variables that turn negative, re-admits any bound variable with negative reduced gradient, and restarts CG. These toggles are the principal pivots of the linear complementarity problem $(A,-b)$, well defined since $A$ is a $P$-matrix. Termination uses the block-principal-pivoting construction of Júdice and Pires, with a least-index Bland fallback, reaching the unique global minimiser in finitely many outer steps. Our contribution carries this guarantee to the inexact, matrix-free regime the Krylov inner solve requires: once CG's residual is tied to the decision margin, the approximate loop makes the same primal and dual sign decisions as the exact one, and a direct factorisation may replace CG unchanged. Each inner solve is matrix-free: for $A=M^\top M$, the operator $v\mapsto M^\top(Mv)$ needs two products with $M$, no $O(n^2)$ storage, and retains CG's $O(\sqrt\kappa)$ rate, $\kappa=\kappa(A)$. Regularisation or preconditioning that compresses the spectrum lowers the iteration count; a ridge/Tikhonov split $A\mapsto(1-\alpha)A+\alpha R^\top R$ also secures the needed $P$-matrix property. A loop-free projected-gradient reference method, analysed alongside, converges at the slower $O(\kappa)$ rate. On synthetic SPD problems the loop reaches the planted optimum in at most 6 outer steps, and with a direct free-set solve outpaces Lawson–Hanson and an interior-point solver by an order of magnitude on dense instances.

The dissipative barrier method suppresses spectral pollution, but whether it can itself conceal genuine spectral points has remained open. Known as the graveyard problem in computational spectral theory, the higher-dimensional case has remained unresolved for more than a decade. We resolve it for Schrödinger operators in dimensions $d\geq2$; together with the known one-dimensional theorem, this settles the no-invisibility problem in all dimensions. Let $A=-\Delta+V$ be a Dirichlet Schrödinger operator on a connected open set $\Omega\subseteq\mathbb R^d$ with $V\in L^1_{\mathrm{loc}}(\Omega)$ bounded below, and set $H=A+iS$, where $S\geq0$ and $S\in L^p(\Omega)$, with $1<p<\infty$ for $d=2$ and $d/2\leq p<\infty$ for $d\geq3$. Let $(\Omega_R)_{R>0}$ be a nested family of nonempty connected bounded open sets such that $\Omega_R\nearrow\Omega$ as $R\nearrow+\infty$. Denote by $H_R$ the Dirichlet truncation of $H$ to $\Omega_R$. We prove that every spectral point of $H$ is detected by the truncations: for every $\lambda\in\sigma(H)$ and every neighborhood $U$ of $\lambda$, $\sigma(H_R)\cap U\neq\emptyset$ for sufficiently large $R$. Equivalently, $\sigma(H)\subseteq\liminf_{R\to\infty}\sigma(H_R)$. Thus the barrier method does not trade suppression of spectral pollution for spectral invisibility. No regularity of $\partial\Omega$ is required, and the assumptions reach the critical Sobolev scale. The proof combines compactness of the dissipative form perturbation, Cwikel-type Schatten estimates for Birman–Schwinger operators, generalized strong resolvent convergence, and a reverse Hansmann–Weyl spectral-variation inequality due to Gil'. A two-dimensional numerical example illustrates the absence of spectral invisibility for the truncated dissipative operators.

We identify the graded metabelian locus inside the deformation theory of positively graded Lie algebras. Let $M(U)$ be the free metabelian Lie algebra on $U$ and $B(U)=M(U)'$. For every finite graded rank vector $h$, we prove that the moduli stack of such algebras is equivalent to the quotient stack $[\mathrm{Quot}^{\mathrm{gr}}h(B(U))/\mathrm{GL}(U)]$. At a quotient $B(U)\to C$ with kernel $N$, its tangent complex is the two-term complex from $\mathrm{End}(U)$ to $\mathrm{Hom}{\mathrm{Sym}(U)}(N,C)_0$. For every algebra $\mathfrak{g}$ in this stack, restriction to $\Lambda^2\mathfrak{g}'$ induces a defect map on $H^2_0(\mathfrak{g};\mathfrak{g})$, and we prove that its kernel is $H^0$ of the Quot-stack tangent complex. Thus a first-order deformation is tangent to the metabelian locus exactly when its derived–derived restriction vanishes. We also recover the inverse-system module degree by degree from intrinsic lower-central tensors; on the level locus its terminal tensor suffices. For the $14$-dimensional algebra attached to a regular pencil of binary quartics, exact computation gives $\dim H^2_0=11$ and $H^3_0=0$. Its effective miniversal graded deformation germ is formally smooth of dimension $11$, while its metabelian subgerm is formally smooth of dimension $3$. Terminal restriction identifies the $8$-dimensional normal space with derived–derived brackets. Using the pencil's classical $V_4$-symmetry, we determine its induced representation on the graded tangent space and show that four negative-weight primary obstruction maps are surjective.

We establish the existence of a weak solution to the Voltage-Conductance kinetic equation for any conductivity parameter and any reasonable initial datum, and in particular without smallness condition as it was the case in [9]. The proof is based on some a priori estimates mainly drawn from [15], some trace results adapted from [14] and a DiPerna-Lions weak stability argument.

In 2024, Cisto et al. introduced polyocollections and polyocollection ideals, which generalize the notions of collections of cells and inner $2$-minors ideals, respectively. In this paper, we naturally extend the notions and results about zig-zag walk, zero-sum condition, rook number, and switching polynomial from collections of cells to polyocollections, and we define a class of polyocollections, called natural polyocollections, whose structure is similar to that of collections of cells. We focus on natural polyocollections whose polyocollection ideals admit quadratic Gröbner bases with respect to lexicographic order induced by some specific orders on vertices. Using the zero-sum condition, we prove the primality of this class of polyocollections, which shows that the zig-zag conjecture proposed by Mascia et al. holds true for collections of cells satisfying the aforementioned properties. Moreover, we show that $h$-polynomials and regularities of the coordinate rings of this class of polyocollections are equal to their switching rook polynomials and rook numbers, respectively. These results give positive answers to the switching rook polynomial conjecture proposed by Jahangir and Navarra, and the rook number open problem proposed by Rinaldo and Romeo for collections of cells satisfying the aforementioned properties.

In this paper, we continue investigate the structure of the weak $\sigma$-attractor $\mathcal{A}$ for the 2D Boussinesq equations with viscosity and without heat diffusion. Specifically, we show that the projection of the attractor $\mathcal{A}$ onto the temperature component is surjective, i.e., $P_{\theta}\mathcal{A}=L^2$; and, for any $r>0$, the projection of $\mathcal{A}_r$ onto the velocity component contains a infinite-dimensional Hilbert ellipsoid, which implies $P_u\mathcal{A}_r$ is infinite dimensional immediately.

We investigate the permanence of local stability for one-parameter families $X\to C$ under finite flat base-changes, when the base-field has positive characteristic $p>0$. Building on previous work of Hu–Zong, we show that it suffices to consider base-changes by Frobenius morphisms. In that case, we show that the situation is governed by the discrepancies of the pairs $(X,X_c)$, together with some differential invariants of the vertical divisors whose multiplicity in their fiber is divisible by $p$. While the behaviour of these invariants remains in general mysterious, we establish upper bounds under some $F$-splitting assumptions.

Grouping species according to their phylogenetic relationships often results in different groups than grouping them according to their shared traits. Monophyletic groups play an important role in this regard, as they are groups of species sharing the same trait and being uniquely defined by a joint phylogenetic subtree. This immediately leads to the question of how to identify possible monophyletic groups in characters, which assign each present-day species a certain trait and which are typically used for phylogenetic tree reconstruction. In our manuscript, we provide a general formula to quantify how many different characters are monophyletic on any given tree and provide simple formulae for binary characters and for certain tree shapes. We also investigate relations between monophyly and the well-known phylogenetic tree reconstruction criterion maximum parsimony by providing a linear-time algorithm which determines the parsimony score together with the monophyly type of a character on a tree.