The generalized Krylov subspace (GKS) method is an effective projection technique for large-scale Tikhonov regularization with a general regularization matrix. As the subspace expands, however, two computational bottlenecks limit scalability: the thin QR factorizations of the tall projected matrices formed by the forward operator and the regularization matrix applied to the basis, and the full reorthogonalization of each new basis vector against all previous columns. We propose a sketched variant, named sGKS, that addresses both bottlenecks. The QR factorizations are performed on compressed matrices of much smaller row dimension, maintained incrementally via rank-one updates. Moreover, we observe that explicit reorthogonalization can be skipped entirely without compromising the quality of the approximation subspace, since no step of GKS relies intrinsically on the orthogonality of the basis. The resulting algorithm is independent of the choice of sketching operator and preserves the approximation quality of the original method: we show that, in the absence of sketching in the projected solve, sGKS produces iterates identical to those of standard GKS, and that the sketched projected solve delivers quasi-optimal residual norms controlled by the embedding quality. For more challenging problems where the loss of basis orthogonality becomes significant, we show that incorporating a small number of iterative refinement steps in the projected solve restores the spectral properties of the basis and recovers the full accuracy of the unsketched method. Numerical experiments on image deblurring, X-ray computerized tomography, seismic travel-time tomography, and dynamic computerized tomography demonstrate that sGKS matches the reconstruction quality of standard GKS while significantly reducing per-iteration costs and overall wall-clock time.

Let $G$ be a finite connected weighted graph and let $R$ be its effective-resistance matrix. For every nonempty vertex set $S$, we factor the cofactor sum and determinant of the principal resistance submatrix $R[S]$ into an enumerative term and a boundary potential-theoretic term. If $τ(G)$ is the weighted spanning tree enumerator and $κ_G(S)$ is the weighted enumerator of $S$-rooted spanning forests, then \[ \cof R[S]=(-2)^{|S|-1}κ_G(S)/τ(G). \] After Kron reduction to $S$, with reduced Laplacian $K=L^S$, $Q=K^+$, and $q=\diag(Q)$, the remaining normalized factor is \[ \det R[S]/\cof R[S] =\frac{2}{|S|}\tr Q+\frac12 q^{\mathsf T}Kq. \] Equivalently, this factor is the maximum of $u^{\mathsf T}R[S]u$ over all $u\in\R^S$ satisfying $\one^{\mathsf T}u=1$. This optimization viewpoint yields monotonicity under enlargement of $S$, an exact one-point update formula, and a support criterion for equality. Small star examples show that the resulting set function is neither submodular nor supermodular in general.

A foundational theorem of Ajtai, Komlós, Pintz, Spencer, and Szemerédi asserts that every $n$-vertex $k$-uniform uncrowded hypergraph with maximum degree $Δ$ contains an independent set of size $c_k n{\left(\frac{\log Δ}Δ\right)^{\frac{1}{k-1}}}$, for some constant $c_k>0$. Determining the optimal leading constant $c_k$ in this bound is a major open problem. A natural target is the so-called shattering-threshold constant $\left(\frac{1}{k-1}\right)^{\frac{1}{k-1}}$, which appears in the solution-space geometry of random constraint satisfaction problems, in average-case complexity theory, and in statistical physics, among other areas. We prove that uncrowded hypergraphs attain this threshold. More precisely, for every $ε>0$ and $k\geq 2$, every $n$-vertex $k$-uniform uncrowded hypergraph of sufficiently large maximum degree $Δ$ contains an independent set of size at least $(1-ε) n {\left(\frac{1}{k-1}\frac{\log Δ}Δ\right)^{\frac{1}{k-1}}}$. Consequently, we obtain the first pseudorandom class of hypergraphs whose guaranteed independence number matches the shattering threshold, resolving a folklore conjecture. Moreover, as another direct consequence, we resolve a conjecture of Verstraëte and Wilson by proving that there exists a constant $c_k=1-o_k(1)$ such that every $n$-vertex $k$-uniform linear hypergraph of maximum degree $Δ$ has independence number at least $c_k n\left(\frac{\log Δ}Δ\right)^{\frac{1}{k-1}}$. Our techniques are constructive yielding efficient algorithms for both static and distributed settings. Specifically, we provide an $\tilde O(nΔ)$-time randomized static algorithm and an $\tilde O(1)$-round randomized $\textsf{LOCAL}$ algorithm to find an independent set in uncrowded hypergraphs at the shattering threshold. These results extend seamlessly to the the setting of linear hypergraphs.

In an asymmetric shelf shuffle, a deck of $n$ cards is dealt sequentially from the bottom and assigned one of the $m$ shelves uniformly at random. The card is placed at the top of the assigned shelf with probability $p$, and at the bottom of the assigned shelf with probability $(1-p)$. Analysis of the shelf shuffle has gained much attention recently, and the case $p=1/2$ was first treated by Diaconis--Fulman--Holmes [Ann. Appl. Prob. 23 (2013), no. 4, 1692--1720]. In this paper, we extend the analysis of the shelf shuffle to general $p\in (0, 1)$. In particular, we study the distribution of cycles, cycle lengths, number of descents, number of valleys, number of inversions, and the RSK shape of a permutation obtained from an asymmetric shelf shuffle. Our results extend the analysis of Diaconis--Fulman--Holmes to arbitrary $p$. Furthermore, our analysis of the distribution of descents and inversions is new even for $p=1/2$.

Let $a\ge2$, let $p\in\mathbb{Z}[n]$ be eventually increasing and eventually non-negative, and let $λ_n=a^{p(n)}+f(n)>0$, where $f(n)\in\mathbb{Z}$. We prove, using Bourgain's bounded entropy criterion, that if $\log(1+f(n)a^{-p(n)})$ is eventually non-zero and decays geometrically in absolute value, then $(λ_n)$ is neither $L^\infty$-Khintchin nor $L^1$-Khintchin. In particular, for every $c\in\mathbb{Z}\setminus\{0\}$, every positive tail of $(a^{p(n)}+c)_{n\ge1}$ is non-Khintchin. The same conclusion applies to the standard examples $a^n+c$, $a^n+b^n$, and, whenever eventually positive, $a^n-b^n$, with $a\neq b$. Thus these perturbations of geometric powers lie on the unstable side of the Khintchin problem. This gives a negative answer, in the translated-power case, to the question of Fan--Fan--Queffélec--Queffélec on the stability of translated powers.

A mechanical shuffler consists of $m$ shelves. A deck of $n$ cards, arranged in increasing order, is dealt from the bottom sequentially. Each card is assigned a shelf uniformly at random and placed on the top (bottom) of the existing pile with probability $p$ ($1-p$) independently. We refer to this as asymmetric shelf-shuffle. We find the law $ν_{n, m}^{(p)}$ of the permutation induced by the asymmetric shelf-shuffle and show that the pair consisting of the number of descents and the number of valleys is a sufficient statistic. This generalizes a result of Diaconis, Fulman, and Holmes (Ann. Appl. Prob., 2013) corresponding to the case $p=1/2$. For $p=1/2$, Chen and Ottolini (ECP, 2025) established the cutoff in the total variation distance near $\lfloor n^{5/4}\rfloor$. We establish the cutoff for the asymmetric shelf shuffle. Let $ν_n$ be the uniform measure on the set of all permutations $S_n$ of $\{1, \ldots, n\}$. For a fixed $p\neq 1/2$ and $c>0$, we show that \[\TV\left(ν_{n, \lfloor cn^{3/2}\rfloor }^{(p)}, ν_n\right)=1-2Φ\left(-\frac{|2p-1|}{4\sqrt{3}c}\right)+O_{c, p}(n^{-1/2})\;.\] We also establish the cutoff in the separation distance near $m\approx n^{2}$ and in the relative entropy near $m=n^{3/2}$. In both cases, we also obtain the cutoff profile explicitly.

We study $\mathrm{PD}^3$ groups which admit an unbounded quasimorphism to $\mathbb{R}$ with coarsely-connected quasikernel. We show that such a group must either arise as the fundamental group of a torus or Klein-bottle bundle over $S^1$, or be quasiisometric to a Riemannian manifold $(\mathbb{R}^3,g)$, with the quasikernel being coarsely equivalent to $\mathbb{H}^2$. If $G$ is moreover hyperbolic, it admits a faithful action on $S^1$ by quasisymmetric homeomorphisms. Our approach features a coarse generalisation of Shapiro's lemma, and the development of a theory of homological isoperimetric inequalities for metric spaces; these tools make use of Margolis's framework for coarse homological algebra.

Let $\mathbb{N}$ be the set of positive integers. For subsets $\mathcal{A},\mathcal{M}\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let $p(n,\mathcal{A},\mathcal{M})$ denote the number of representations of $n$ in the form $$ n=\sum_{a\in \mathcal{A}}m_a a, $$ where $m_a\in \mathcal{M}\cup \{0\}$ for all $a\in \mathcal{A}$, and only finitely many $m_a$ are nonzero. We prove that there exist two infinite sets $\mathcal{A}=\{a_n\}_{n=1}^{\infty}$ and $\mathcal{M}$ of positive integers such that $$ \lim_{n\to\infty}\frac{\log a_{n+1}-\log a_n}{\log n}=+\infty, $$ $p(n,\mathcal{A},\mathcal{M})>0$ for every $n\in\mathbb{N}$, and $p$ has polynomial growth. More generally, we prove a construction that associates restricted partition functions of polynomial growth with additive complements satisfying a simple counting condition. This answers a 2016 question of Dai and Chen in the affirmative.

The paper deals with new formulations of a Lagrange interpolant polynomial based on the nodes of the well-known anti-Gauss rule. A first representation is given in terms of the classical Christoffel-Darboux kernel appropriately modified. The second one closely follows the barycentric form of the classical Lagrange polynomial, while the third formulation represents the interpolant as a combination of an orthonormal family of polynomials with respect to the discrete anti-Gauss inner product. A numerical test shows the performance of the explored forms.

Whether the multi-dimensional Euler-Poisson system admits global smooth solutions remains a challenging open problem. In this paper, we construct a class of large-data global smooth solutions to the one-fluid Euler-Poisson system in $\mathbb{R}^d$ ($1\leq d\leq 5$) by using the relaxation dissipation mechanism. Precisely, assuming that the initial density is far from vacuum and $\varepsilon E_0$ is sufficiently small, where $E_0$ denotes the initial energy and $\varepsilon$ is the relaxation time, we establish the global well-posedness of smooth solutions to the Cauchy problem. In particular, the size of the initial perturbation may be arbitrarily large, provided that the relaxation time is sufficiently small. Furthermore, we introduce an effective unknown motivated by Darcy's law to derive quantitative error estimates at the rate $\mathcal O(e^{-λt}\varepsilon)$ between the rescaled Euler-Poisson system and the limiting drift-diffusion system for ill-prepared data. The new ingredient lies in developing the maximum principle for the nonlinear drift-diffusion system with nonlocal effect, which leads to the large-data global existence.

We consider the Neumann initial boundary value problem associated to the chemotaxis system \begin{align}\label{prob:abstract}\tag{$\star$} \begin{cases} u_t = \big((u+1)^{m-1} u_x - u(u+1)^m v_x\big)_x & \text{in $(0, 1) \times (0, \infty)$}, \\ v_t = v_{xx} - v + u, &\text{in $(0, 1) \times (0, \infty)$}, \end{cases} \end{align} where $m \in \mathbb R$ is a given parameter. The relation between diffusion and taxis sensitivity is critical since the ratio $u(u+1)^m/(u+1)^{m-1}$ grows like $u^{2/n}$ for large $u$ with $n = \dim((0, 1)) = 1$. Nonetheless, we show that there is no critical mass phenomenon if $m \le -1$; that is, in that case all solutions emanating from suitably regular initial data are globally bounded. For certain parabolic-elliptic simplifications of \eqref{prob:abstract}, we obtain the same conclusion for all $m \in (-\infty, -1] \cup (0, \infty)$ and even for all $m \in \mathbb R$ if the initial datum is additionally assumed to be monotone. This stands in contrast to critical mass phenomena known to occur for critical quasilinear Keller-Segel systems considered in higher-dimensional domains. Accordingly, we make use of several special features of the one-dimensional setting such as the boundedness of the energy functional from below, the embedding $W^{1, n} \hookrightarrow L^\infty$, and the fact that the mass accumulation function solves a spatially non-degenerate parabolic equation.

In this paper, we study the Radon transform associated with the unipotent radical subgroups of $GL_n(\mathbb{F}_q)$. We analyze properties of the Radon transform with a specific emphasis on its eigenvalues. We provide a description of its eigenvalues for the cases of $GL_2(\mathbb{F}_q)$ and $GL_3(\mathbb{F}_q)$.

We determine the precise asymptotic number of unlabelled cubic planar graphs with $n$ vertices. Our approach blends generating series methods with computational bounds and probabilistic local large deviation theorems.

In this paper we develop a general framework for potentials on Lauritzen manifolds, namely smooth manifolds equipped with a pseudo-Riemannian metric and a pair of conjugate affine connections that may have non-vanishing torsion. We show how the theory of bi-forms accommodates torsion-full statistical structures and unifies contrast and pre-contrast functions in a cohomological framework. Within this formalism, we construct a canonical contrast bi-form on dually curvature-free Lauritzen manifolds and establish its principal structural properties. Several illustrative examples are analysed.

The inversion of a set $X$ of vertices in an oriented graph reverses every arc with both endpoints in $X$. The inversion graph $I(G)$ of a graph $G$ has the labelled orientations of $G$ as its vertices, two orientations being adjacent when a single inversion transforms one into the other, and the inversion diameter $\diam(I(G))$ is its diameter. Answering a question of Havet, Hörsch and Rambaud, we prove the bound in terms of edge number $\diam(I(G)) \le 2\sqrt{|E(G)|}$, and we complement it with a lower bound $\diam(I(G)) \ge \frac{|E(G)|}{|V(G)|}$ obtained by viewing $I(G)$ as a Cayley graph on $\F_2^{E(G)}$. We further refine the upper bound for bipartite graphs $G$ by showing $ \diam(I(G))\le \max\left\{ρ, \left\lceil\log_2\bigl(2+σ(2^{ρ-1}-1)\bigr)\right\rceil\right\}$ where the two parts of $G$ have maximum degrees $σ$ and $ρ$, respectively.

Communication-centric ISAC is a promising paradigm for future 6G networks, in which data payload signals are expected to be reused for sensing to enhance time-frequency resource efficiency. For random payload signals, existing studies have mainly characterized the expected sidelobe level (ESL) of the periodic auto-correlation function (P-ACF). However, ESL only captures the average sidelobe behavior and does not control large spurious sidelobe peaks in individual payload realizations, which may deteriorate weak-target detection performance. This motivates the design of information-bearing signals whose random P-ACF satisfies stringent peak sidelobe level (PSL) constraints. In this paper, we formulate a mutual information maximization problem under PSL constraints and a power budget. For quasi-static frequency-flat channels, we show that a continuous auto-correlation function (ACF)-domain uniform construction provides an asymptotically optimal high-SNR design principle. Motivated by this principle, we propose auto-correlation function keying (ACFK), a finite-constellation modulation architecture that embeds data symbols directly onto the ACF-domain sidelobes. ACFK enables exact control of the nominal P-ACF, which coincides with the actual P-ACF when a spectral non-negativity constraint is met. When this is not the case, we quantify the non-negativity violation probability and bound the resulting peak sidelobe level ratio (PSLR) degradation. We further provide a reference ISAC transceiver design for ACFK over quasi-static multipath channels, together with high-SNR approximate BER analysis. Numerical results validate the theoretical analysis and show that, compared with a generalized probabilistic amplitude shaping (PAS) baseline, ACFK provides substantially stronger PSLR control and improved weak-target detection performance under comparable sensing and communication settings.

We analyze Allen-Cahn functionals with stationary ergodic coefficients in the regime where the length scale $δ$ of the heterogeneities is much smaller (microscopic) than the interface width $ε$ (mesoscopic). In a companion paper, we show that if the ratio $ε^{-1} δ$ vanishes fast enough as $ε\to 0$, then the functionals converge to an effective surface energy where the energy density is determined by homogenization effects originating at microscopic scales. Here we prove that if the ratio $ε^{-1} δ$ vanishes too slowly, the limit of the functional may actually be smaller than this homogenized energy. We refer to this as the rare events regime. In the case of the random checkerboard in dimension one, we use large deviations techniques to give a complete description of the rare events regime, showing that the limiting energy depends in a nontrivial way on the limit of $ε^{-1} δ| \log ε|$. We further construct, in any dimension, examples of random media in which rare events become relevant at algebraic scales $δ\approx ε^{1 + α}$ for an arbitrary $α> 0$, as well as almost periodic examples in which atypical configurations play the same role as rare events.

For $n\ge 2$, let $\mathcal{M}(n)$ be the supremum of $\sum_{a\in A}1/(n-a)$ over pairwise coprime sets $A\subset [1,n)$. Erdős asked whether $\mathcal{M}(n)\le \sum_{p<n}1/p+O(1)$ uniformly in $n$. We prove the quantitative average-order formula $$ \sum_{n\le N}\mathcal{M}(n) = e^{-γ}N\log\log N+O(N). $$ The lower bound comes from the self-rough construction $\{n-d:P^{-}(n-d)>d\}$, while the upper bound uses bounded-cost dual certificates and Buchstab--de Bruijn estimates for rough numbers. We also prove that $$ \mathcal{M}(n)=(e^{-γ}+o(1))\log\log n $$ for almost all $n$, with a quantitative exceptional-set bound, and hence Erdős's inequality holds for almost all $n$. The almost-all proof uses a long-interval two-dimensional beta-sieve estimate for two moving forbidden residue classes, together with an exact finite singular-series cancellation. Finally, we prove the pointwise bound $\mathcal{M}(n)\le (2+\varepsilon)\log\log n+O_{\varepsilon}(1)$, explain the linear-sieve barrier behind the constant $2$, and record structural certificates, conditional window-packing reductions, numerical examples, and CRT sharpness constructions.

We develop a parsimonious stochastic growth model in which state-dependent saving behavior generates endogenous business-cycle-like dynamics. The model consists of three coupled equations: a Solow-type capital accumulation equation, a linear filtering equation for the saving rate, and a bounded stochastic adjustment process. Saving is modeled as a logistic function of deviations from a balanced growth path, introducing nonlinear feedback controlled by a gain parameter. In the deterministic limit, increasing feedback strength produces a supercritical pitchfork bifurcation, splitting the balanced-growth equilibrium into two locally attracting regimes corresponding to expansion and contraction. When stochastic perturbations are introduced, these equilibria become metastable states, and the economy undergoes rare noise-induced transitions between them. The resulting dynamics exhibit persistent regimes, bimodal stationary densities, and right-skewed dwell-time distributions with approximately exponential survival tails. A discrete-time approximation is estimated using U.S. real GDP data, and Monte Carlo simulations are used to compute stationary distributions and regime persistence statistics. The results demonstrate that nonlinear state dependence, bounded multiplicative noise, and time-scale separation are sufficient to generate realistic business-cycle behavior within a low-dimensional framework.

Effective pricing and promotion planning constitutes a central pillar of strategic revenue management for firms operating in highly competitive and dynamic markets. These planning activities require the simultaneous consideration of demand elasticity, competitor actions, channel and market specific constraints, and financial objectives. As the dimensionality and interdependencies inherent in these problems increase, manual or traditional approaches become suboptimal and insufficient. In this context, Operations Research provides a robust methodological foundation for scalable, data-driven decision support systems that can optimize complex planning processes across large product and customer portfolios. This paper presents two large-scale optimization systems developed and deployed at PepsiCo to support Revenue Growth Management initiatives: PromoAI and PricingAI. PromoAI couples machine learning-based promotional forecasts with a mixed-integer linear programming model to optimize promotional calendars across trade channels, searching millions of product-promotion-timing combinations for the one that maximizes PepsiCo and retailer revenues subject to customizable business constraints. PricingAI optimizes base prices across product portfolios over multi-period horizons, using Bayesian hierarchical models to estimate own- and cross-price elasticities and competitive interactions, then feeding these into a nonlinear programming engine that recommends price changes aligned with revenue and margin targets under operational constraints. Together, these systems demonstrate the feasibility and scalability of advanced optimization in large-scale enterprise environments. They highlight the value of integrating statistical learning with mathematical programming to enable enterprise-level, automated decision-making that is both data-informed and aligned with strategic business objectives.

In joint source-channel coding (JSCC)-based semantic communication systems, achieving stable and reliable image semantic transmission under channel constraints remains a key challenge. In most channel adaptation modules, the signal-to-noise ratio (SNR) is often injected into each layer of a channel-adaptation model in an independent and layer-wise manner, which undermines global coordination across layers. Therefore, consistent noise-robust representations may fail to be learned throughout the model. To address this problem, we propose SA-RA-JSCC, a novel channel-adaptive JSCC model. SA-RA-JSCC maps SNR into a unified semantic vector in the feature space and then applies a one-shot global reweighting to the encoded features, thereby enabling globally consistent and learnable channel adaptation. Moreover, in order to further enhance the anti-channel capability of semantic information, a semantic-rate-aware module is introduced, enabling the adaptive policy to respond simultaneously to fluctuations in channel quality and changes in semantic-rate constraints, thereby enhancing global network coordination and channel adaptivity. Extensive experiment results across multiple channels and datasets demonstrate that SA-RA-JSCC significantly outperforms existing semantic communication models in terms of reconstruction metrics such as PSNR and MS-SSIM, exhibiting stronger robustness across a broad range of SNR regimes.

For PDEs naturally posed in the de Rham complex, structure-preserving mixed and saddle-point finite element discretizations typically produce indefinite linear systems. We propose a multigrid preconditioning framework that combines mass-lumped (explicitly invertible) FEEC mass matrices with transforming smoothers that map the operator to a block form with positive definite diagonal blocks, enabling Gauss-Seidel-type relaxation on the transformed system. Under mild h-uniform norm-equivalence assumptions (and for trivial topology), we prove stability of the mass-lumped systems, and by extension spectral equivalence between the mass-lumped and original FEEC operators, which motivates using multigrid cycles designed for the mass-lumped operators as preconditioners for the consistent FEEC systems. While our primary focus is on algorithmic design rather than formal convergence theory, extensive numerical experiments on the Hodge-Dirac operator, mixed Hodge-Laplacians, and a magnetostatics saddle-point system in 2D and 3D demonstrate the robustness of the approach.

Fang, Lan, and Xiao established the geometric Green's formula as a categorical isomorphism for arbitrary semisimple complexes. In this short note, we generalize their work to multi-step compositions. Specifically, we establish the iterated geometric Green's formulas for the composition of an $(n-1)$-fold restriction and an induction, as well as its dual.

Let $T(n)=n+τ(n)$, where $τ$ is the divisor function. We study the Erdos-Graham coalescence problem by encoding finite-level obstructions in the divisor-successor graph and in square-annular transfer maps. Coalescence is equivalent both to connectedness of this graph and to synchronization along an infinite non-autonomous sequence of finite annular systems. The basic identities are \[ \operatorname{im}(\mathcal A_k)=E_{k+1}, \qquad \mathcal F_{k^2}=k^2+E_k, \] where $E_k$ is the set of square-crossing overshoots from below $k^2$. We prove a transfer parity law, dynamic frontier bounds for the widths $W_{k,s}$, and the criterion that $\liminf_k|\mathcal A_k(E_k)|=1$ would imply connectedness. Unconditionally, \[ R(X)\le \log X+2γ+O(X^{-1/4}), \] and the exit sets are residue-universal, satisfy $|E_k|\le k^{o(1)}$, and obey \[ \frac94K+O(1)\le \sum_{k\le K}|E_k|\ll K(\log K)^3. \] Using the shifted-square estimate HST, obtained from the corrected Henriot--Nair--Tenenbaum theorem in the specialized form of Proposition 8.4 and from separate square-shift estimates, we obtain fixed-moment bounds \[ \sum_{k\le K}|E_k|^m\ll_m K(\log K)^{C_m}\quad(m\ge2). \] A further first-moment refinement to $K(\log K)^2$ is conditional on the additional, currently unproved, uniform quadratic Euler-product mean-value hypothesis HQE. We also prove quantitative large-jump and lower-runner race theorems, isolate interval filling, and formulate a square-gated two-branch criterion. No proof of the full Erdos-Graham problem is claimed.

In this short note, we prove that if $A \in \mathbb C^{N \times M}$ with $N=4M-5$ has i.i.d.\ standard complex Gaussian entries, then the probability that the phase retrieval map generated by $A$ is not injective is positive. This proves Part (1) of a conjecture of Cynthia Vinzant, which was later restated by Afonso S. Bandeira in \cite{BDL+26}. The main result of this paper was obtained using generative AI, in particular the Rethlas system.

We develop and analyze a conforming Virtual Element Method (VEM) for the fourth-order Poisson-Nernst-Planck-Navier-Stokes (PNP-NS) system. The proposed scheme is based on compatible discretizations of each component: an \(H^2\)-conforming VEM for a fourth-order electrostatic potential equation, an \(H^1\)-conforming VEM for the Nernst--Planck equations, and a \textit{divergence-free} and \textit{pressure-robust} VEM for the Navier--Stokes equations. Time integration is performed using a backward Euler scheme to ensure stability. We establish the well-posedness of the continuous problem up to three-dimensions and also establish existence and uniqueness of the fully discrete solution via a fixed-point argument. Further, we derive a priori error estimates showing that the electrostatic potential, concentration and velocity converge optimally in Bochner norms \(L^\infty\bigl(0,T; H^2(Ω)\bigr)\), \(L^2\bigl(0,T; H^1(Ω)\bigr)\), and \(L^2\bigl(0,T; \bm{H}^1(Ω)\bigr)\), respectively. The analysis requires a sophisticated argument to avoid any restrictive assumptions on the coercivity and continuity constants and to handle the trilinear form involving three different variables. The pressure-robust design permits the use of lowest-order pressure approximations without compromising convergence rates of the other variables, reducing computational cost. Numerical experiments confirm the theoretical convergence rates for various polynomial orders and demonstrate the scheme's robustness, including in low-viscosity regimes.

We study the asymptotic enumeration of labelled connected multitype graphs in the sparse regime, where both the number of vertices and edges grow linearly and the excess is proportional to the size of the graph. Extending the classical theory of connected graph enumeration to the multitype setting, we consider graphs with prescribed numbers of vertices of each type and prescribed edge counts between each pair of types. Our approach is probabilistic and relies on the theory of inhomogeneous random graphs. In particular, we exploit large-deviation principles and asymptotic estimates for connectedness probabilities to relate the counting problem to the emergence of giant components in suitably tuned supercritical random graphs. From large deviation asymptotics of connected components of inhomogeneous random graphs, we recognize that a connected graph with a given edge statistics corresponds to the (unique) giant component of larger inhomogeneous random graph with a suitably chosen connection kernel. This correspondence allows us to derive the leading exponential asymptotics for the number of connected multitype graphs with fixed type profile and edge matrix. The resulting formula generalizes the asymptotic enumeration results of Bender, Canfield, and McKay for connected sparse graphs to the multitype framework. More broadly, the paper illustrates how probabilistic techniques can provide transparent and effective tools for addressing new combinatorial enumeration problems.

For a graph $G$ put $h_r(G)=\max{χ(H):H\subseteq G,\operatorname{girth}(H)\ge r}.$ Erdős and Hajnal asked whether $h_r(G)\to\infty$ as $χ(G)\to\infty$, for every fixed $r\ge4$. We prove this in every fixed polynomial edge-density regime: for all $r\ge4$, $k\ge2$, $P,C>0$, there is $M=M_{r,k}(P,C)$ such that $χ(G)\ge M,\ e(G)\le Cχ(G)^P\Longrightarrow h_r(G)\ge k.$ Quantitatively, after replacing $P$ by $P\vee2$ and $C$ by $C\vee2$, $M_{r,k}(P,C)\le \exp!\left(O_{r,k}\bigl((P+2+\log(C\vee2))^2\bigr)\right),$ and consequently the same conclusion holds throughout the quasi-polynomial range $e(G)\le \exp\bigl(C_0(\logχ(G))^a\bigr),\ 1 < a < 3/2,$ for all sufficiently large $χ(G)$. In each fixed polynomial-density regime we also obtain $f_{P,C}(k,r)\le k^{O_{r,P,C}(1)}.$ The proof combines a chromatic-defect random extraction lemma, compact and near-quadratic sparse-core bases, and a peeling/thinning bootstrap increasing the admissible edge exponent by $1/(r-1)$. We also prove structural saturation results for possible counterexamples, including Moore-strength exact-cycle packings and quadratic saturation in projected colour-pair space. Finally, writing $h_r^{\mathrm f}(G)=\max{χ_{\mathrm f}(H):H\subseteq G,\operatorname{girth}(H)\ge r},$ we develop a fractional random-extraction framework based on Mohar-Wu preservation. We prove sufficient cheap-cycle-killing criteria and verify them for several structured families, including clique-organised families, line graphs of incidence graphs of equal-order generalized quadrangles and generalized hexagons, and the Bohman-Keevash tracking-time triangle-free-process graph. We also isolate a density-free obstruction that any proof using this fractional surgery route must overcome.

Let us consider a random infinite lower triangular matrix, where the entries on and below the diagonal are i.i.d. uniform random elements of a fixed finite field. We investigate the evolution of the span of the first $n$ rows of this matrix as $n$ grows. Many properties of this evolving subspace can be captured with the help of the verbose persistence diagram, which is a standard tool in stochastic topology and topological data analysis. We give an explicit formula for the distribution of the persistence diagram. We prove a law of large numbers for the distribution of lifetimes. We also describe the fluctuations of the persistent Betti numbers.

A cyclic square matrix (and its conjugacy class) C over a field K is called (m,k)-cyclic if it has a decomposition $C = A \oplus B$, where $\dim A = m, \dim B = k$ and $m, k \ne 0$. It is shown that the product of two nonsingular (m,k)-cyclic conjugacy classes $Ω$ and $Ψ$ of GL(m+k,K) contains all nonscalar matrices P in GL(m+k,K) with determinant $\det P = \det ΩΨ$.