Papers

We establish a Type I curvature bound for finite-time volume-collapsing Kähler–Ricci flows on compact Kähler surfaces. No symmetry assumption is imposed, and the initial Kähler class need not be rational. The proof combines local symplectic topology with the classification of gradient Kähler–Ricci shrinking solitons. Bamler's compactness and structure theory provides the shrinking limits used in the argument.

Let $T$ be a non-invertible surjective continuous endomorphism of the group $\ttf^d\,:= \rrf^d/\zzf^d$. Then $T$ preserves the Haar measure, is associated to some $d \times d$ matrix $A$ with integer coefficients and is $r$-to-one, where $r = |\det A| \ge 2$. In~\cite{Krzyzewski}, Krzyżewski gives a necessary and sufficient condition for $T$ to be exact, and shows that $T$ – viewed as a measure-preserving map – is isomorphic to the direct product of some (possibly trivial) toral automorphism by some exact toral endomorphism. We re-prove these results by more elementary and constructive methods, whereas Krzyżewski actually works with endomorphisms of compact Abelian groups and relies on non-constructive theorems. Krzyżewski's exactness condition and Mihailescu's papers ~\cite{Mihailescu 2012} and~\cite{Mihailescu 2013} show that $T$ is isomorphic to the one-sided uniform Bernoulli shift on $\odc 0,r-1 \fdc^{\zzf\_+}$ if and only if $A$ is expanding (i.e. the modulus of each eigenvalue is $>1$). When $A$ is not expanding and hyperbolic (i.e. all eigenvalues have modulus different from $1$), Mihailescu~\cite{Mihailescu 2012} shows that $T$ cannot have a generating Rokhlin partition. More generally, when $A$ is not expanding (whether hyperbolic or not), we show that the endomorphism $T$ cannot have a smooth finite generator (whether independent or not).

The next-generation matrix method is a powerful tool for computing the basic reproduction number in compartmental mathematical models of infectious diseases. The method has been recently extended to mathematical biochemistry, where it can be used to establish parameter regions of stability and instability for boundary steady states. Several significant challenges in its application remain, however, particularly around establishing conditions under which the method is mathematically valid, computationally tractable, and biologically meaningful in the biochemical setting. In this paper, we address these challenges by shifting the interpretation from new infections in the epidemiological setting to autocatalysis in the biochemical setting. We introduce a graph, called the ALT-graph (autocatalysis-leak-transition graph), which decomposes the contribution of each reaction to the Jacobian as autocatalytic, leak, or transition edges. We then present a systematic algorithm for splitting the ALT-graph, which is guaranteed to produce a valid reproduction number, $\rho(FV^{-1})$, while also decreasing computational complexity by lowering the rank of $FV^{-1}$. We apply the method to models of both biochemical reaction networks and infectious disease spread.

We study the Banach space and operator factorization structure of Janson-Sobolev and Janson-Sobolev-Hardy spaces. This new class of martingale spaces is determined by a $q$-adic filtration, a subspace $V\subset \mathbb{R}_0^{l\times q}$, and a rearrangement invariant function space $X$. Our main result shows that, for every bounded diagonal operator $D$, the operator $S = \sum_{t=1}^s \lambda_{\mathcal U}^{k_t}(D)Q_{\mathcal K_t}^{\mathcal B}$ determined by the linear functionals $\lambda_{\mathcal U}^{k_t}(D)$ and the canonical projections $Q_{\mathcal K_t}^{\mathcal B}$, almost projectionally factors through $D$ with constant $1^+$. As consequences, we obtain factorization results for diagonal operators both under a natural boundedness condition on the canonical projections and for all spaces equipped with the $L^1$-norm.

We construct the Buchholz algebra without a Fock representation, as the bounded inverse limit of intrinsic particle-cutoff quotients of the gauge-fixed resolvent algebra, with isometric particle-degree coordinates. Its finite matrix corners carry canonical traces, KMS states, and trace-norm product formulas, together with closed-path expansions that lay the groundwork for path-integral representations. The known Fock-space description and a proper quasilocal subalgebra arise as realizations.

We prove that the product of chains $\mathbf m\times\mathbf n$ is not Schur positive whenever $n\ge4$ and $m\ge3n-1$. Writing $m=n+k$, we exhibit a negative Schur coefficient indexed by $(2n+k-1,2n+k-3,\ldots,k+5,k,k,4)$ and evaluate it explicitly as $(n-2)!$ times a polynomial of degree three in $k$. Together with earlier results, this establishes the two-case interpretation of the conjecture of Li, Qiu, Yang, and Zhang: these products are not Schur positive when $n=2$ and $m\ge8$, or when $n\ge3$ and $m\ge n+5$.

Let $B_3=B(0,3)\subset\mathbb{R}^3$. For every sufficiently large integer $k$, we construct a nonzero real function $u_k\in C^2(\overline{B_3})$ and a real potential $V_k\in L^\infty(B_3)$ such that $\Delta u_k=V_ku_k$, $u_k|_{\partial B_3}=0$, $\mathrm{ord}_0u_k=k$, and $\|V_k\|_\infty\le Ck^{3/2}$. Consequently, for every sufficiently large $N$, there is such a Dirichlet solution with potential norm smaller than $N$ and vanishing order at least $cN^{2/3}$. The construction extends directly to higher dimensions and to spheres. Together with previous results, this example indicates that $C(1+\|V\|_\infty^{2/3})$ is the sharp bound for the vanishing order in the real-valued case. It also indicates that the bound conjectured independently by Kukavica \cite{Kukavica1998} and Kenig \cite{Kenig2006} is not attainable in general.

We develop a growth model with an R&D sector in which the elasticity of innovation with respect to existing knowledge can be negative. We investigate the existence and uniqueness of a balanced growth path (BGP) and derive closed-form growth factors, showing that productivity and output growth are semi-endogenous and population-driven. We also establish the global stability under a general production function. Under testable parameter restrictions, the economy converges to the BGP. However, when the knowledge elasticity is sufficiently low, a Jacobian-based condition implies instability, and an N-period innovation cycle can emerge. Comparative statics are conducted to explore the impact of several factors, including research efficiency, input elasticity, and population growth. Numerical simulations map out transitions and mark the points at which stability is lost. They make clear when innovation frictions endanger balanced growth and provide practical guidance for R&D and demographic policy.

We prove new structural results for algebraic generalized Ricci solitons on solvable Lie algebras and we discuss the existence of examples in low dimensions. We establish that all non-trivial solvable algebraic generalized Ricci solitons must be expanding and give necessary conditions for the existence of such solitons on one-dimensional extensions of abelian Lie algebras. Finally, we classify algebraic generalized Ricci solitons on three-dimensional Lie algebras, and algebraic generalized Ricci solitons on four-dimensional solvable unimodular Lie algebras, up to isometry and scaling.

Structural system theory studies which network topologies can sustain a prescribed system property such as controllability or stability. When the topology is itself random, the relevant question becomes probabilistic: how likely is a graph drawn from a stochastic model to sustain the property? Such probabilities measure the abundance and robustness of the property across topologies, and indicate whether systems requiring it can be reliably deployed in uncertain environments. We address this question for asymptotic stability of linear systems in the directed graphon setting. We consider two graph-theoretic properties. The first is $\mathcal N$, and it requires that for every $k\leq n$ some $k$-vertex induced subdigraph of $D$ admits a cycle cover, and the second is $\mathcal S$, which requires that these subdigraphs can be chosen so that their node sets form a nested sequence $V_1\subset\cdots\subset V_n=V(D)$ starting from a single vertex with a loop. We have shown that $\mathcal N$ is necessary and $\mathcal S$ is sufficient for structural stability. We sample $D$ from a directed step-graphon $W$. Our main results give necessary and sufficient conditions for $\Pr(\mathcal N)\to 1$ and $\Pr(\mathcal S)\to 1$ as $n\to\infty$. In more detail, to a step-graphon $W$ with skeleton digraph $S$ on $q$ nodes and concentration vector $x^*$ we associate a cycle polytope $\vec{\mathcal X}(S)\subseteq\Delta_q$. The conditions are then formulated in terms of the position of $x^*$ within $\vec{\mathcal X}(S)$, the dimension of the polytope, the loop density of $W$ and, for $\mathcal S$, an ordering condition on the cycles of the skeleton. Together these results identify, for directed step-graphons, the regime in which a sampled topology is overwhelmingly likely or unlikely to sustain stable dynamics.

Stochastic recursive momentum (STORM) achieves fast convergence for nonconvex optimization via the variance reduction effect, but existing analyses rely on the strong average smoothness assumption. In this paper, we study the convergence of STORM for different objectives without average smoothness. We first revisit the results under average smoothness, obtaining the $O(T^{-1/3})$ bound for nonconvex objectives and the $O(\sigma^2/(\mu T))$ bound for last-iterate output under the $\mu$-Polyak–Łojasiewicz~(PL) condition. Without average smoothness, we design an auxiliary sequence and compare the STORM update with it in the analysis. With the help of this sequence, we prove that STORM still attains an $O(T^{-1/4})$ rate for nonconvex objectives, which is optimal under standard smoothness. For convex and $\lambda$-strongly convex objectives, we further prove averaged and last-iterate bounds with optimal rates of $O(\sigma R/\sqrt T)$ and $O(\sigma^2/(\lambda T))$, respectively. All the obtained results use the same STORM recursion with different hyperparameter choices.

A petal diagram of a knot is a projection with a single multi-crossing and no nested loops; it is encoded by a permutation of the heights of the strands through the multi-crossing. Colton, Glover, Hughes and Sandberg proved a Reidemeister-type theorem for petal diagrams: two petal permutations represent the same knot if and only if they are related by trivial petal additions and deletions and by crossing exchanges. We ask whether every petal diagram of the unknot can be reduced to the one-petal diagram without ever increasing the number of petals, in analogy with Dynnikov's monotonic simplification theorem for rectangular diagrams. By an exhaustive, certified computer search we show that this is true for diagrams with at most 7 petals and false for 9 petals. Of the 40320 petal diagrams with 9 petals, 24992 represent the unknot, and exactly 108 of them are hard: none of them admits a crossing exchange or a trivial petal deletion, even if two natural petal-number-preserving symmetries are allowed. Up to these symmetries and mirror image there are three hard diagrams. Two of them can be untangled by passing through 11 petals; the third cannot be untangled through diagrams with at most 11 petals, but can through 13. We explain why the phenomenon differs from the rectangular case: a petal diagram is an arc presentation whose cyclic order of pages is determined by the order of its vertices on the binding, and no elementary move of Cromwell and Dynnikov preserves this rigid structure.

This paper presents a cell-centred unstructured finite-volume method for time-harmonic two-phase acoustics. A volume-fraction representation supplies density and acoustic compressibility. Face fractions average available neighbouring planar interface cuts independently of velocity direction, retaining the established one-field face-density closure. Cartesian complex-coordinate stretching provides perfectly matched layers (PMLs). Real-imaginary splitting yields a real-valued block system, discretized with corrected non-orthogonal fluxes and consistent boundary contributions. Verification covers homogeneous waves, layered gas-liquid transmission with PML truncation, baffled-piston radiation with Kirchhoff far-field reconstruction, and resolved rigid-sphere radiation forces. Homogeneous-wave pressure converges at approximately second order on orthogonal meshes and on meshes with non-orthogonal interiors and orthogonal boundary cells. Reconstructed velocity converges at approximately second order on orthogonal meshes and with an observed order of about 1.7 on the latter mesh family. Under aligned refinement, the layered case's relative whole-domain complex-pressure $L_2$ error decreases to $2.657\times10^{-4}$. The resolved-sphere force differs from the Gorkov prediction by at most 1.5% over the tested Rayleigh size range. The results quantify the accuracy and current limitations of the heterogeneous Helmholtz-PML formulation.

We study the average-case matrix discrepancy problem: given independent normalized $d\times d$ Gaussian orthogonal ensemble matrices $A_1,\dots,A_N$ and a fixed margin $\kappa>0$, find signs $\sigma_1,\dots,\sigma_N\in\{-1,1\}$ such that the operator norm of $\sum_{i=1}^N \sigma_i A_i$ is at most $\kappa\sqrt{N}$. Focusing on the proportional regime $N/d^2\to \tau\in(0,\infty)$ as $d\to\infty$ followed by the small-margin limit $\kappa\downarrow 0$, we characterize the density required by stable offline algorithms and by online algorithms. In the offline setting, we construct a polynomial-time recenter-and-round algorithm that is noise-stable and succeeds whenever $\tau=\Omega(\frac{1}{\kappa^2\log(1/\kappa)})$, along with a matching lower bound for all stable algorithms. In the online setting where each sign must be chosen irrevocably upon observing the corresponding matrix, we determine the exact limiting performance of the Frobenius-greedy algorithm, establishing that it succeeds when $\tau>\tau_{\rm FG}(\kappa)\sim \frac{\pi}{4\kappa^2}$, as well as a matching lower bound for all online algorithms by conditioning on a revealed prefix. At the core of our algorithms lies rotational symmetry, which enables us to transfer Frobenius norm control into operator norm guarantees. Together, our results identify the algorithmic phase transition points for random matrix discrepancy: $\Theta(\frac{1}{\kappa^2\log(1/\kappa)})$ for stable offline algorithms and $\Theta(\frac{1}{\kappa^2})$ for online algorithms. Both thresholds lie far above the satisfiability scale $\Theta(\log(1/\kappa))$, as shown by Maillard~\cite{maillard2025}.

We study finite-time blow-up and global existence for semilinear heat equations with time-dependent reaction coefficients on metric measure spaces. For convex nonlinearities, we establish a blow-up criterion that retains the dependence on the initial datum through its linear evolution and does not require stochastic completeness. Under additional conservativity and heat kernel assumptions, we also prove blow-up at the critical Fujita exponent associated with polynomial volume growth. Our main global existence result is based on a heat-kernel-weighted functional space in which the mild solution map becomes a contraction. This yields a unified construction covering both polynomial and exponential heat kernel decay, general locally Lipschitz nonlinearities, and time-dependent reaction coefficients, without convexity or monotonicity assumptions. In particular, the global existence criterion only uses the behaviour of the nonlinearity on the range explored by the solution. Applications include Riemannian manifolds, metric graphs, and, within a separate discrete framework, weighted graphs. The arguments, especially the functional construction underlying global existence, are new and lead to new results even for the classical underlying spaces considered above.

We consider the distribution-constrained optimal stopping problem $\sup_{\tau\sim \mu} \mathbb E[B^*_\tau]$, where $\mu$ is a probability distribution on $\mathbb R_+$, and $(B^*_t)$ denotes the running maximum of a standard Brownian motion. This problem was introduced in Beiglbock et al. (PTRF, 2018), where a monotonicity principle is used to establish the optimal stopping time as the hitting time of a specific boundary. In this paper, we characterize this boundary by a variational inequality. In the spirit of Cox et al. (PTRF, 2019), we provide a novel probabilistic representation for the variational inequality as a time-reversed optimal stopping problem. A key ingredient for proving the viscosity solution property and comparison principle is a quantitative estimate near the singular corner of the time-space domain, where the initial and boundary conditions are incompatible. We then prove the optimality of the resulting hitting time through a discrete-time Snell envelope construction and a stability argument for the associated stopping times.

A dominating set $D$ of a graph $G$ is called $k$-limited if every vertex of $D$ has at most $k$ neighbors outside $D$. The minimum cardinality among all $k$-limited dominating sets of $G$ is the $k$-limited domination number, denoted by $\gamma_k^L(G)$. In this paper, we prove that the $1$-Limited Dominating Set problem is $\mathsf{NP}$-complete, answering an open question posed in the literature. We further show that, for every positive integer $k$, the $k$-Limited Dominating Set problem is $\mathsf{NP}$-complete even when restricted to planar graphs. In addition, we study the $k$-limited domination number of Sierpiński graphs. We determine the exact value of $\gamma_1^L(S(n,m))$ for all integers $n\ge 1$ and $m\ge 2$, and obtain the $k$-limited domination number for the planar family $S(n,3)$.

We establish viability conditions for a broad general class of stochastic differential equations formulated within the variational approach,% \[ dX_{t}=A\left( X_{t}\right) dt+B\left( X_{t}\right) dW_{t}. \] The underlying functional space setup is given by a general Gelfand triple $V\subset H\subset V^{\ast},$ where $H$ is a separable Hilbert space in which the stochastic evolution equation (SEE) is considered. Our approach to deriving viability conditions is inspired by the techniques developed by Aubin and Da Prato for finite-dimensional forward stochastic differential equations. We establish necessary and sufficient conditions for the viability of closed random constraint sets in terms of adapted variational tangent (and contingent) sets. Furthermore, we illustrate the applicability of the proposed framework by discussing several important classes of stochastic differential equations that fit naturally within our variational setting.

This paper investigates stochastic multi-level optimization where the objective is a nested composition of several smooth non-convex functions. We assume that only stochastic estimates of the gradient and function values for each level are accessible. Consequently, obtaining an accurate estimate of the overall gradient is challenging due to the nested structure. To address this, we employ a momentum-based estimator with mini-batches to track the function values of each level, which are subsequently used to construct momentum gradient estimators. We establish an optimal sample complexity of $\mathcal{O}(\epsilon^{-4})$ for finding an $\epsilon$-stationary point, avoiding the stronger average smoothness assumption commonly relied upon in prior literature. Furthermore, by employing a normalization technique, we attain the same rate without requiring problem-dependent constants to set hyperparameters. To achieve the optimal rate without mini-batches, we further develop a batch-free method that incorporates a first-order approximation and a clipping technique for function value estimation. Finally, we validate the effectiveness of our proposed methods through experiments on risk-averse portfolio optimization and hierarchical tilted empirical risk minimization.

Large-cap or large-ensemble decoupling inequalities were initially studied by Demeter. These are intimately connected to the reverse square-function conjecture and restriction conjecture in Fourier restriction theory. By strips, we mean a partition of the $R^{-1}$-neighborhood of the paraboloid into $1\times R^{-1/2}\times R^{-1}$ curved large-caps. In this article, we prove the sharp $\ell^2(L^{10/3})$ decoupling estimate for strips in $\mathbb{R}^3$. We reduce the strip decoupling inequalities to a planar incidence problem, which is dealt with via two-ends Furstenberg incidence estimates. Our method is inspired by Wang–Wu on the restriction conjecture.

We develop a conservative micro-continuum-cellular automaton method for simulating multispecies biofilm dynamics in complex flows. The proposed method couples the Darcy-Brinkman-Stokes equations, reactive transport, suspended bacteria, and biofilm dynamics with a two-stage cellular automaton algorithm for biofilm redistribution and interface evolution. While treating biofilms as evolving porous media, we ensure conservative redistribution of multispecies biomass across partially occupied cells. Donor and recipient cell volumes are explicitly accounted for to conserve biomass and preserve species composition on non-uniform meshes. The proposed method is assessed against diffusion-dominated benchmark cases, including single-species fingering and multispecies stratification, and is further evaluated through a mesh-convergence study for flow and growth over a rectangular bump. The framework is then applied to counter-diffusional biofilms in a membrane-aerated biofilm reactor as a canonical example. The results demonstrate that the framework captures the expected biofilm morphology and stratification in systems involving coupled flow, substrate transport, and biofilm dynamics. The proposed method provides a flexible computational approach for simulating multispecies biofilm dynamics in complex flows and geometries.

Universal coding of integers (UCI) provides binary codewords for positive integers such that, for every nonincreasing source distribution $P$, the average codeword length stays within $K$ times $\max\{1,H(P)\}$. The smallest constant $K$ is called the minimum expansion factor of UCI $\mathcal{C}$, denoted $C_{\mathcal{C}}^{*}$. The optimal minimum expansion factor $C^*=\inf\{C_{\mathcal{C}}^{*}\}$ is the minimum expansion factor corresponding to the optimal UCI. The optimal minimum expansion factor is currently known to lie in the range $2\le C^*\le 2.0386$. In this paper, we construct a family of one-point plus uniform-tail distributions and prove that, for every universal code, the worst-case ratio is attained by a distribution in this family, so that the family is least favorable for the UCI problem. We further establish an inequality, called the UCI inequality, which plays the same role for UCI as the Kraft inequality does for prefix codes: for any real number $B$, it decides whether $B$ lies below or above $C^*$. Through the UCI inequality, we obtain an equivalent definition of $C^*$. By numerical computation, we determine $C^*=2.000124757036101\cdots$, the first fifteen decimal digits being certified. Once $C^*$ is known, we can theoretically construct the optimal UCI.

We establish a class of sharp higher-order inequalities related to the uncertainty principle. First, by means of the expanding the squares method, we give a concise proof of sharp higher-order Heisenberg uncertainty principles. We provide explicit expressions for the optimal constants and establish the existence of extremal functions. Furthermore, we obtain sharp higher-order Heisenberg uncertainty principles for curl-free vector fields. In particular, when $N=2$, we give an answer to the higher-order version of Open Problem 9 raised by Maz'ya in [Integr. Equ. Oper. Theory (2018) 90:25]. Second, we establish sharp higher-order Heisenberg uncertainty principles that do not involve higher-order tensors. We also prove sharp higher-order Heisenberg uncertainty principles and higher-order hydrogen uncertainty principles for radially symmetric functions.

We prove a factorization theorem for ordinary topological recursion near a regular nodal degeneration of a spectral curve, allowing logarithmic spectral coordinates. The partition function for closed genera $g\ge2$ factors into the partition function of the normalization, a universal Gaussian vacuum, and the exponential of a connected graph sum. This graph sum has strictly positive order in the vanishing period, and thus the factorization recovers the conifold gap, identifies the term of degree zero in the vanishing period with the free energy of the normalization, and gives finite graph formulas for the coefficients of positive powers of the period. At fixed genus, the regular series converges jointly in the period and the parameters along the nodal locus. The Gaussian neck tensors can be expressed in terms of relative Gromov–Witten invariants of a parametrized $\mathbb P^1$. As examples, we use remodeling to obtain Gromov–Witten factorizations for local $\mathbb P^2$ and local $\mathbb F_0$ in the conifold frame, with backgrounds $\mathbb C^3$ and the resolved conifold, respectively. The analogous local $\mathbb F_1$ factorization has background $\mathcal O(1)\oplus\mathcal O(-3)$ over $\mathbb P^1$.

This article provides a complete algebraic description of $\frac{1}{2}$-derivations, local $\frac{1}{2}$-derivations, and 2-local $\frac{1}{2}$-derivations on $n$-dimensional totally graded complex filiform Lie algebras of maximum length. Based on the foundational classification framework established by Janez Bernik (2020), we systematically determine the vector spaces of $\frac{1}{2}$-derivations for the six infinite structural sequences ($m_0(n)$, $m_2(n)$, $W^+(n)$, $m_{0,1}(n)$, $m_{0,2}(n)$, $m_{0,3}(n)$) and the five exceptional one-parameter families ($g_{7,\alpha}$ through $g_{11,\alpha}$). By analyzing the pointwise local evaluation equations via parametric matrix systems, we establish the structural linearity and rigidity of local $\frac{1}{2}$-derivations. In contrast, we demonstrate that the independent parameters residing in the boundary rows of the $\frac{1}{2}$-derivation matrices provide sufficient degrees of freedom to bypass linearity constraints. Exploiting these boundary configurations, we explicitly construct pure non-linear and non-additive 2-local $\frac{1}{2}$-derivations leveraging the homogeneous function of degree one, $f(z_1, z_2) = z_1^3 / (z_1^2 + z_2^2)$, thereby defining the exact boundary where local rigidity fails.

Among Archimedes' many celebrated discoveries is a striking fact: the region of the unit sphere between two parallel planes, both meeting the sphere and separated by a distance $h$, has area $2\pi h$. We prove that, among connected smooth closed embedded surfaces in $\mathbb R^3$, this property characterizes the unit sphere for a single fixed width.

We study the minimal tense logic $K_t$ from an algebraic and proof-theoretic perspective. We introduce the degree-of-truth-preserving logic associated with the class of tense Boolean algebras. We then introduce a sequent calculus for this logic and establish its soundness and completeness with respect to tense Boolean algebras by means of an adaptation of the Lindenbaum–Tarski construction. Consequently, this calculus provides an additional syntactic presentation of the minimal tense logic $K_t$. We also provide a second, purely syntactic proof of completeness. Furthermore, we establish an algebraic soundness and completeness theorem for Kashima's Gentzen-style calculus. To this end, we develop an adaptation of the Lindenbaum–Tarski construction to Kashima's nested sequent framework, which allows us to construct the algebraic semantics directly from the proof-theoretic system. This yields a direct algebraic completeness proof for Kashima's calculus and connects its nested-sequent formulation with the algebraic semantics of tense Boolean algebras.

Primal-dual hybrid gradient (PDHG) methods solve large-scale linear programs (LPs) using GPU-friendly matrix-vector products and projections, but their practical performance depends on coordinating algorithm parameters, acceleration, and restarts. We introduce GALLOP, which uses reinforcement learning to jointly learn continuous algorithm parameters and discrete restart decisions without differentiating through the solver. Its generalized accelerated PDHG update combines separate primal and dual extrapolation, history corrections, and restart anchoring with independently adjustable coefficients. We train a dimension-agnostic feedback policy using a groupwise proximal policy optimization objective that clips likelihood ratios separately for different control groups and excludes inactive acceleration controls on restart transitions. We evaluate GALLOP on six LP families and a public item-placement benchmark. On the main evaluation settings across the six families, GALLOP reduces iteration counts by factors of $1.9$-$5.6$ and achieves up to a $16.0\times$ speedup in algorithm wall-clock time over MPAX. With one policy trained per family, the learned policies generalize without retraining to within-family LPs $3\times$-$400\times$ larger than the largest training instances, including Transport LPs with $10.24$ million variables.

We study a mixed regular–singular control problem for stochastic evolution equations in a Hilbert space with possibly unbounded random linear operators, a nonconvex regular-control domain, and a state-dependent singular coefficient. The singular control is an adapted nondecreasing càdlàg process whose terminal value need not be bounded. We prove well-posedness and weighted moment estimates for the forward and backward equations, and characterize the second-order adjoint by a conditionally expected operator-valued backward stochastic integral equation. An Itô-type formula for the quadratic form of this adjoint, together with spike and convex variations, yields a second-order Hamiltonian condition for the regular control, as well as nonnegativity and a contact condition for the optional singular Hamiltonian.

We construct a groupoid model for Rordam's example of a simple, nuclear, separable $C^*$-algebra containing a finite and an infinite projection. This groupoid model arises from a (crossed product of a) carefully chosen model for the inductive limit construction in Rordam's original approach, which we show to yield a Cartan subalgebra of the limit. In particular, this procedure yields a locally compact, second countable, Hausdorff, étale, amenable, minimal and topologically principal groupoid whose unit space is compact, and whose $C^*$-algebra contains an infinite and a non-zero finite projection. Hence, this groupoid is amenable, and yet does not have comparison.