Papers

In this paper, we establish asymptotics for the small value probabilities of additive and derivative martingales in both supercritical branching Brownian motions and super Brownian motions, thereby extending the corresponding results for Galton–Watson processes and continuous-state branching processes. For the derivative martingale in branching Brownian motion, our result also agrees with the findings in the arXiv version of Arguin et al. [arXiv:1008.4386 v1] and with those of Hu [Ann. Inst. H. Poincaré Probab. Stat., 2016].

In high-dimensional hyperbolic space, concentration of a radial measure near a shell need not determine the pyramid limit: angular concentration and hyperbolic expansion alter separation. An effective radius is the scale on which positive-mass sets actually separate, which the raw radius need not give. With vanishing rescaling and radial fluctuations, radius convergence gives weak convergence to the pyramid of all metric measure spaces with the resulting diameter bound. At modal radial Gibbs shells, exponential decay of intrinsic shell curvature and dimension-normalized tangential Bakry-Émery Ricci curvature recovers the radius. The Gaussian intrinsic to hyperbolic volume and that obtained by wrapping a Euclidean Gaussian have different critical orders. They are Lévy below those orders, infinitely dissipate above them, and at criticality converge to the corresponding diameter-bounded pyramids.

Sampling representative nodes from large graphs is fundamental to graph signal processing and network analysis, yet existing methods require access to the full graph Laplacian, making them impractical at scale. We propose a simple and effective column-selective graph sampling algorithm based on a minimum inner product greedy selection rule. At each iteration, the algorithm accesses only a small random subset of Laplacian columns, requiring no eigendecomposition or global graph traversal, making it well-suited for large-scale graphs where the full Laplacian cannot be stored in memory. We analyze the algorithm under the stochastic block model and show that, when the degree distribution is balanced across nodes, the algorithm achieves sampling proportional to cluster size, and that the resulting mean estimate is controlled for band-limited graph signals in the Paley-Wiener space, with the error decaying as inter-cluster connectivity weakens. Numerical experiments on both synthetic and real-world data validate the effectiveness of the proposed method.

We show that every planar graph has a tree-decomposition with optimal width such that the subgraph induced by each bag has pathwidth at most 3. This bound is best possible, and for tree-decompositions that satisfy a certain minimality condition, we in fact give a precise description of the possible structures in each bag. Moreover, we show that the union of any $k$ bags has pathwidth $O(k)$. We also show that graphs excluding a fixed double-apex-forest minor have a tree-decomposition with optimal width such that the subgraph induced by each bag has bounded pathwidth. This includes graphs embeddable on any fixed surface. As a byproduct of our machinery, we give a new proof of the linear grid minor theorem for planar graphs.

Truncated shifted iYangians are a family of algebras expected to quantize certain components of affine Grassmannian islices. We introduce orientifold KLRW (oKLRW) algebras associated with quivers with involution and establish their faithful polynomial representations and diagrammatic bases. We also define KLR iYangians using double reflective KLR diagrams and construct diagrammatic realizations of the iGKLO homomorphisms. For integral parameters, we introduce interval oKLRW algebras and prove an equivalence between integral weight modules over truncated shifted iYangians and nilpotent modules over the corresponding interval oKLRW algebras.

In a previous paper, the second author proved that every supercuspidal representation of a rank-one p-adic group is induced from a compact-mod-center open subgroup. The method was geometric, localizing representations to obtain equivariant sheaves on trees. Here we provide two refinements. The first is a geometric description of the inducing data, via a geometrically minimal K-type. Second is a proof that the equivariant sheaves collapse onto injective sheaves. The two notions of geometrically minimal K-types and collapsibility generalize to higher rank groups, suggesting a pair of conjectures.

This article studies the conditional-density equation and its pathwise transformation in local stochastic rough volatility models, with rough Heston (rHeston) as the main explicit example. Under the stated common-filtration, measurability, predictability and spatial-regularity assumptions, we show that the Ito-Wentzell random-PDE reduction of the conditional density SPDE remains valid under local stochastic rough volatility. After fixing a common-environment realization and the associated stochastic flow, the transformed equation becomes a deterministic PDE with path-dependent coefficients. This yields a pathwise Fokker–Planck formulation that connects naturally with Rao–Blackwellized calibration. In the pure rough Heston case, the transformed coefficients simplify and the conditional density admits an explicit lognormal form.

We derive the incompressible Navier-Stokes-Fourier limit from a nonlinear quantum Fokker-Planck equation with Bose-Einstein or Fermi-Dirac statistics. The model has a self-consistent collision structure, with the local density acting as the collision frequency and the bulk velocity and temperature determined by nonlinear quantum-weighted moments of the distribution. We work near a global quantum equilibrium under the diffusive scaling and keep the quantum parameter fixed. Uniform estimates with respect to the Knudsen number yield strong microscopic relaxation and identify the limiting infinitesimal quantum equilibrium. Using the local conservation laws, we prove the incompressibility condition, the Boussinesq relation, and strong compactness of the divergence-free velocity component and a quantum-adapted thermal mode, while the acoustic modes vanish locally by a dispersive estimate. The limiting viscous stress tensor and heat flux are identified by solving auxiliary equations for the linearized quantum Fokker-Planck operator and by expanding the local quantum equilibrium manifold. The resulting incompressible Navier-Stokes-Fourier system retains the effect of quantum statistics through its normalization constants and transport coefficients.

Let $\mathbb F_2$ be the finite field with two elements and let $\mathbb F_2^\omega$ denote the countably infinite-dimensional vector space over $\mathbb F_2$. We show that, unlike the case of polynomial maps $p:\mathbb Z\rightarrow\mathbb Z$ vanishing at zero which are always good for nice recurrence, there are polynomials $p:\mathbb F_2^\omega\rightarrow \mathbb F_2^\omega$ with $p(0_{\mathbb F_2^\omega})=0_{\mathbb F_2^\omega}$ which fail to have this property. This disproves a conjecture of Bergelson and McCutcheon (c. 2000), which predicted that for any countably infinite abelian groups $H$ and $G$, every polynomial map $p:H\to G$ with $p(0_H)=0_G$ is good for nice recurrence. Moreover, we develop a dynamical mechanism which shows that the magnitude of intersections along polynomial paths is degree-sensitive (even when one considers only weakly mixing systems). Among other things, we also show that the Furstenberg-Sarkozy theorem for $\mathbb F_2^\omega$-valued polynomials of degree at most $d$ vanishing at zero is equivalent to a $d$-dimensional symmetric-difference weakening of the density polynomial Hales-Jewett conjecture. Thus, as we explain in detail in this paper, our observations not only shed new light on the phenomenon of polynomial recurrence but also constrain possible strategies for proving or disproving the density polynomial Hales-Jewett conjecture.

We show that for a $2$-connected graph $G$ which is not a cycle, the mixed metric dimension of $G$ is at most $2c(G)$, where $c(G)$ is the cyclomatic number of $G$. As an immediate application, we prove a conjecture proposed by Sedlar and Škrekovski (2021).

New venture start-ups need to “survive” through multiple stages of reaching milestone targets. We investigate the strategies for start-ups in a milestone-oriented setting. We examine a model of an entrepreneurial start-up firm, where its state is captured by a diffusion process. The entrepreneur can choose between multiple activities (or controls), which incur different cost and determine the drift and the variance of the process. Depending on whether the process reaches a fixed upper boundary or a lower one, the start-up firm succeeds or fails. Continuous-time stochastic models with multiple ($\ge 3$) controls are typically very challenging to deal with. In this work, we are able to completely solve for the optimal policy and provide an explicit characterization of its structure. In particular, the optimal policy only uses controls from a set characterized by a so-called efficient frontier curve that orders the controls by two intuitive measures: riskiness (drift-to-volatility ratio) and cost-effectiveness (drift-to-cost ratio). A unique feature of our model is that depending on the model parameters, the efficient frontier curves can be of different types, resulting in qualitatively different structures of the optimal policy. As far as we know, this is the first study that analyzes a stochastic control model which admits efficient frontier curves of different types. Our work provides start-up firms with intuitive measures to evaluate their activities and offers valuable insights on how the optimal strategies in a milestone-oriented setting change qualitatively contingent upon the specific scenario. We believe the results provide a foundational block in the study of entrepreneurial decision-making.

For $d\geq 4$, we consider incompressible Euler flows in $\mathbb{R}^{d}$ with bi-rotational symmetry and without swirl. Our first result gives the local wellposedness of the Yudovich-type solution. The second result provides global wellposedness up to $d\leq 6$. In particular, it shows that the rate of growth of the vorticity maximum coincides with the rate from axisymmetric flows without swirl, which was obtained in the paper by the second author and Jeong (Arch. Ration. Mech. Anal. 249(3):32, 2025) and Shao–Wei–Zhang (Acta Math. Sin. (Engl. Ser.), 42(3):663-679, 2026).

The classification of perfect $e$-codes over an arbitrary alphabet of size $q$ is complete for $e > 2$. In the case of non prime power $q$, it is conjectured that no perfect $2$-codes exist. We confirm this conjecture in a number of situations, including the case where $q=2^\alpha p^\beta$ with $p$ prime, $\alpha$ and $\beta$ positive integers, and either $\alpha \leq 20$, or $\alpha$ sufficiently large.

We introduce puncture-forgetting maps for measured foliations and investigate their relations with mapping class groups, Teichmüller spaces, and extremal length. To this end, we develop the notions of cube complexes of pre-homotopic multicurves and tree coordinate systems on CAT(0) cube complexes. As an application to the dynamics of post-critically finite rational maps on the Riemann sphere, we obtain a partial result toward the finite curve attractor conjecture posed by Kevin Pilgrim. We also apply these methods to uncover a relation between horospheres and geodesic flows in the universal curve over Teichmüller space.

Grothendieck cofibrations describe transport in a category varying over a base, while factorization systems organize the arrows of a category into two complementary classes. We give a fully 2-categorical account of the passage from the former structure to the latter. The global comma 2-monad on the arrow 2-category encodes Grothendieck transport, whereas the squaring 2-monad encodes factorizations. We prove that split cofibrations are precisely the strict algebras for the comma 2-monad, including their 1-cells and 2-cells, and that normally cloven cofibrations are precisely its normal pseudoalgebras. A canonical colax morphism from the comma 2-monad to the squaring 2-monad then turns cocartesian transport into the cocartesian-vertical factorization of arrows in the total category. At the strict level, this yields the strict factorization system of designated cocartesian and vertical arrows; at the coherent level, it yields the orthogonal factorization system whose left class consists of all cocartesian arrows and whose right class consists of the arrows sent to isomorphisms in the base. We also separate unrestricted global pseudoalgebras, which retain a coherently trivial base action, from fixed-base pseudoalgebras, which correspond to arbitrary cleavages, and record the dual strict result for fibrations. This places the classical cofibration-factorization interaction, in all these variants, within a single change-of-2-monads construction and relates it directly to the existing fibrational and factorization literature.

Fix a prime power $q$. Let $D_q(n)$ be the largest possible least degree of a polynomial omitted by a non-covering family of $n$ congruence classes in $\mathbb F_q[x]$. Assuming the known theorem that every non-covering family of $n$ classes omits a polynomial of degree less than $n$, we prove \[ D_q(n)=\frac{n}{q-1}+O_q(1). \] The upper bound combines a minimal-counterexample reduction to irreducible moduli with a truncated inclusion–exclusion (Brun sieve) argument. A nested-modulus construction gives the matching lower bound. This is a follow-up to the author's 2025 work.

We prove a universality theorem for random persistent homology over a class of triangulable spaces. More precisely, let $M \subset \mathbb{R}^D$ be a compact $C^2$-triangulable space satisfying a geometric quality condition and let $f: M \to \mathbb{R}$ be a probability density. Then the expected persistence ratio measure computed from the Čech or Vietoris-Rips complex of a Poisson point process with intensity $nf$ has a universal limit independent of $(M, f)$. Since smooth manifolds, algebraic varieties, semialgebraic sets and Whitney stratified spaces are all triangulable spaces, our theorem applies to a large class of non-Euclidean spaces. Beyond persistent homology, our proof covers a general class of scale-invariant functionals. It relies on a geometric transfer method that adapts constructions in Euclidean space to triangulable spaces through successive approximations by Freudenthal-Kuhn triangulations, and control of interference across singular strata.

We answer a question of Bergelson and Richter about the probability of the relation $\gcd(n,\lfloor\alpha_1n\rfloor,\lfloor\alpha_2n^2\rfloor,...,\lfloor\alpha_kn^k\rfloor)=1$ for $n\in\mathbb{N}$ when $\alpha_1,...,\alpha_k$ are fixed irrational numbers. In particular, we avoid the use of exponential sums, as they are difficult to control when one of $\alpha_1,...,\alpha_k$ admits very efficient rational approximations. Instead, we use an elementary method similar to that of Erdős and Lorentz, together with some general bounds on those $n$'s which share a large prime divisor with $\lfloor\alpha_1n\rfloor$.

We study the problem of routing multiple commodities through a logistic transport network so as to minimise the joint cost of vehicle movements and of transshipment (reload) operations at intermediate storages. The problem combines two features that are common in practice but rarely treated together: edge costs that are nonlinear in the transported volume, because freight is carried by vehicles of fixed capacity, and hard upper bounds on the volume that can be reloaded at each node. We formulate the problem as a mixed integer linear program in both an arc-based and a path-based form, derive its linear relaxation, and propose a metaheuristic solver based on General Variable Neighborhood Search (GVNS). The method works in the path space and relies on three problem-specific neighborhoods that generate new candidate paths by (i) shortest paths under a reload-aware metric, (ii) edges carrying partially loaded vehicles, and (iii) detours around congested storages. We evaluate the method on our open benchmark for this problem (arXiv:2606.20653): a corpus of realistic logistic networks calibrated on real road graphs of nine European countries, five U.S. states and the European part of Russia, with near-planar topology (Zipf city sizes, a spatial-network budget model, and a doubly constrained gravity demand model). On 35 synthetic and 15 real-geometry instances with up to 150 storages, GVNS returns solutions within about one percent of the best dual bound on small networks – a branch-and-bound lower bound substantially tighter than the LP relaxation – is stable across random seeds, and for a comparable wall-clock budget is up to 55% better than a time-limited exact solver on the medium instances.

This paper addresses the formal definition of average functions for interval data, extending classical aggregation theory beyond precise numerical values to the more general setting of real-valued intervals. Recognizing that interval data nat- urally arise in contexts involving imprecision or uncertainty, we examine how established notions of averages and aggregation functions can be generalized when the underlying domain is no longer restricted to the unit interval, but instead encompasses the entire real line. We analyze the impact of different admissible orders on intervals, particularly focusing on the lattice order and several total orders, and provide rigorous definitions of idempotency and mono- tonicity in this context. Furthermore, we adapt and study classical families of aggregation operators such as power means, minimum, maximum, and median to the interval setting, characterizing when these operators satisfy the properties required of average functions for intervals. Our results highlight that the choice of order plays a crucial role in the behavior and suitability of these operators. This work lays a foundation for the systematic aggregation of interval-valued information, with potential applications in statistics, decision making, and fields where data imprecision is inherent.

Erdős problem 364 asks whether three consecutive powerful numbers exist, where $n$ is powerful if $p \mid n$ implies $p^2 \mid n$. Erdős (1976) and, independently, Mollin and Walsh (1986) conjectured that none do; the $abc$ conjecture implies at most finitely many. The conjecture remains open. We present the first verification of the conjecture at any finite bound that is checked end to end by a proof kernel: machine-checked theorems in Lean 4 establishing that no triple of consecutive powerful numbers exists below $10^{12}$ and below $10^{14}$, with the axiom footprint of both theorems being exactly {propext, Classical.choice, Quot.sound} – no sorry, no native_decide, no trusted external computation. The statements are phrased in the byte-identical vocabulary of the google-deepmind/formal-conjectures formalization of the problem, and we prove abstractly that the open conjecture implies each bounded form, pinning the statement correspondence. The proof reduces the search to odd numbers via a mod-4 argument, represents every odd powerful number as $a^2 b^3$ with $a, b$ odd and $b$ squarefree, enumerates all odd powerful numbers by a fueled, kernel-reducible generator whose completeness is proved once and instantiated across 3,524 per-interval Boolean certificates, and eliminates the seven surviving distance-2 pairs (the members of OEIS A076445 below $10^{14}$) by explicit non-powerfulness witnesses. Every certificate's expected values are computed independently by a Python engine, so each kernel-checked equality doubles as a cross-implementation agreement. Total certified kernel time is roughly 46 CPU-hours. Larger uncertified computations exist (exhaustive to $10^{22}$; conditionally to about $7.38 \times 10^{28}$); our contribution is not a computational record but the elimination of trusted enumeration code from the evidence chain.

In this article, we propose two other DLRA-type dynamics for stochastic differential equations (SDEs) than the one studied in arXiv:2308.11581, derived from a minimization of functionals and (informally) from a Stratonovich formulation, respectively. The former approach resembles the DLRA for SDE system proposed in arXiv:1803.00499. Providing the differentiability of the diffusion, the latter procedure registers an additional term in the drift. Indeed, its derivation exploits the Stratonovich formulation to write stochastic processes on manifold, and, hence, possesses a term that depends on the geometry of the manifold itself. These developments open the debate on which formalism is more suitable and what DLRA for SDEs really is.

We prove that the full spectrum of the graph $p$-Laplacian is finite for every finite graph and every real $p>1$. This resolves the problem of finiteness of the full graph spectrum posed by Amghibech (2003). More generally, for signed weighted graph $p$-Schrödinger operators with arbitrary real potentials, we obtain bounds in terms of the numbers of vertices and positive-weight edges, uniform in all coefficients and in $p$. On $n$ vertices with $m$ positive-weight edges, the logarithms of the spectral cardinality and of the total number of connected components of the normalised eigenvector sets are $O((n+m)^2)$; the sum over the full spectrum of their rational Betti numbers is at most $\exp(C(n+m)^2)$ for an absolute constant $C$. These topological bounds extend to homogeneous eigenproblems built from arbitrary finite families of linear forms, including generalised $p$-eigenvalue problems for matrix pairs. For the uniformly weighted $K_n$ and $p\ne2$, we identify the nonconstant eigenlines with the barycentres of the coordinate-hyperplane cell decomposition of $\mathbb RP^{n-2}$ and determine their local Morse data on either side of $p=2$. A renormalised logarithmic limit describes the transition at $p=2$. Positive integer weights then separate these critical values, giving at least $(3^n-2^{n+1}+3)/2$ distinct spectral values for every fixed $p\ne2$ and every $n\geq2$. By contrast, the maximal spectral cardinality at $p=2$ is $n$; at $p=4$, the maximal spectral cardinalities in the unsigned and general classes both have exponential growth rate exactly $3$. For $n\geq3$ the same examples resolve Amghibech's extremal question. The proof combines o-minimal and Pfaffian geometry with projective $L^p$-duality, Morse theory, critical groups, and tensor eigenvalue bounds.

In this paper, we study a Ma–Minda type subclass of starlike functions associated with the normalized arcsine function $\varphi(z)=1+\frac{2}{\pi}\arcsin z.$ Using the Carathéodory function approach together with sharp coefficient estimates for Schwarz functions, we obtain explicit sharp bounds for the initial Taylor coefficients and logarithmic coefficients of functions in this class. We also derive a sharp estimate for the second Hankel determinant \(H_{2,2}(f)\), as well as for the Hankel determinants \(H_{2,1}(F_f/2)\) and \(H_{2,1}(F_{f^{-1}}/2)\) associated with the logarithmic coefficients of a function and its inverse. In addition, we establish a sharp bound for a difference involving logarithmic coefficients. The extremal functions for all the main results are identified, confirming the sharpness of the estimates.

Every solution of the hyperbolic Toda equation determines a classical Abelian monopole on a (2+1)-dimensional Einstein–Weyl space. We determine when its connection is also a source-free electromagnetic potential on a curved Lorentzian 3+1 spacetime with a spacelike Killing field. On the generic branch, Maxwell's equation classifies the admissible Killing-fibre scale and horizontal twist through one characteristic of the Toda field. A separated Liouville branch gives purely magnetic fields with considerably freer fibre geometry. The round spherical solution yields a monopole of unit charge with conserved flux on a contracting screen. The background metric here is neither assumed to be Einstein nor self-dual.

We construct an explicit symmetric Banach norm on a function space over $(0,\infty)$ which is monotone with respect to logarithmic submajorization but is not strongly symmetric, and hence is not fully symmetric. This answers a question raised by Dodds, Dodds, Sukochev, and Zanin. We also construct a strongly symmetric Banach function space over $(0,\infty)$ which is monotone with respect to logarithmic submajorization and admits no equivalent fully symmetric norm.

We study boundary free energies in disordered Edwards–Anderson Ising models. For rectangular strips of fixed width, we prove that the expected free-to-fixed boundary correction, divided by the tangential length, converges for every finite inverse temperature and at zero temperature. The proof combines a uniform almost-additivity estimate with product-measure concentration, yielding self-averaging along tangential intervals. We derive an exact finite-volume Gaussian interpolation identity and prove that a uniform exponential boundary-mixing condition implies boundary localization with an explicit exponential rate. The tangential pressure theorem extends to all dimensions and symmetric coupling laws with finite first moment; under a product-concentration hypothesis, self-averaging is obtained along tangential cubes. Finally, a subcritical open-bond criterion, verified explicitly for sparse signed couplings, gives low-temperature localization without assuming an infinite-volume Gibbs-state property.

This paper addresses the problem of designing distributed optimization algorithms. Different to existing methods that usually give sufficient conditions and conservative convergence rates, we propose a framework for determining the exact worst-case convergence rate. The worst-case convergence is defined over the set of $\mu$-strongly convex and $L$-Lipschitz objective functions and the set of connected graphs with given algebraic connectivity. Focusing on the Distributed Inexact Gradient Tracking (DIGing)–an algorithm widely deployed in applications and extensively studied in recent years, we develop a systematic design methodology by integrating techniques from graph signal processing and robust control theory. First, we present a novel decomposition structure that enables partial subsystem decoupling. Then we derive explicit formulas for the exact worst-case convergence rate and the corresponding parameters. Numerical experiments validate the correctness and effectiveness of our theoretical results.

Customer choice models are central to revenue management in attended home delivery, yet the field usually picks one without much thought. Does the choice model matter? We believe it does, and this paper shows that revenue management policies judged by their own assumed model, or trusted under a single set of parameters, can be worse than they look.

Assembly theory (AT) quantifies complexity through the assembly index (ASI) – a metric proven equivalent to the size of the smallest straight-line program (SLP). While this equivalence links AT to data compression – a field shaped by decades of research – the practical efficacy of specific algorithms as ASI approximators remains largely unexplored. This study provides an empirical evaluation of eight compression algorithms, spanning grammar-based and dictionary schemes (CAs) across a dataset of 408 strings – comprising 368 synthetic strings (including max-complexity strings and strings with varying levels of symbol distribution balance, quantified by Shannon entropy) and 40 natural biological sequences (genomic and proteomic). We introduce Re-Pair T-NDR, a branch-and-bound tie-resolving variant of Re-Pair, providing a tighter upper bound on the ASI than the other CAs researched in this study. Increasing alphabet size extends the regime in which CAs closely track the ASI, delaying a “divergence point”, where a CA deviates from the optimal assembly path. The results establish compression algorithms as practical tools for ASI estimation.