数学前沿

We prove that every wandering exposed Julia component of a rational map is a point. Here a Julia component is exposed if it meets the boundary of a Fatou domain. The theorem extends the corresponding result for polynomials to rational maps, without any assumption on critical recurrence. The proof combines an enhanced nest with a covering estimate for multiply connected pullback domains.

In this paper, we establish several deviation inequalities for the norms of vector-valued martingales, achieving bounds of the same quality as those available in the classical one-dimensional setting. Our results extend and refine previous work by Pinelis (1994) and Hayes (2005) in this area.

We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational curves with irreducible singularities (rational cuspidal curves) in the complex projective plane. We prove that every such curve is isotopic to a complex curve in degrees up to 5, and for curves with one singularity whose link is a torus knot. Classification results of symplectic isotopy classes rely on pseudo-holomorphic curves together with a symplectic version of birational geometry of log pairs and techniques from 4-dimensional topology.

Online feedback optimization (OFO) leverages real-time output measurements to optimize the operation of networked systems without requiring full knowledge of system dynamics or disturbances. We develop an OFO approach for constrained stochastic optimization problems in which the distribution of the system's random parameters shifts in response to the control actions. We propose a projected primal-dual algorithm where the true dual constraint sets are replaced by surrogate sets. Our main result is an upper bound on the mean-square tracking error, which decomposes into four interpretable terms reflecting: (i) the stochasticity of the problem, (ii) output measurement errors, (iii) time-variability of the problem, and (iv) the mismatch between surrogate and true dual constraint sets. The theory is illustrated in a numerical experiment for power grids with price-responsive assets.

G\"odelian incompleteness yields arithmetical sentences $A$ such that $\mathsf{PA}+A$ and $\mathsf{PA}+\neg A$ are both consistent. Are such extensions equally legitimate? I propose graded epistemic arithmetical non-pluralism: justification for choosing between them varies with the set-theoretic strength of $A$. I defend $\mathsf{PA}+\mathsf{Con}(\mathsf{PA})$ and show that Friedman's concrete incompleteness results pose a serious challenge to non-pluralism about first-order arithmetic. Resolving the selection problem for such sentences turns on justifying large cardinals. Defending graded non-pluralism thus engages G\"odel's programme and the justification of very large cardinals.

While district heating networks are a key component of the energy transition, their operational planning is challenging due to substantial uncertainty in heat demand. We study how unit-commitment schedules can account for this uncertainty and what reliability costs. By adapting chance-constrained programming (CC) and conditional value-at-risk (CVaR) optimization to the mixed-integer unit-commitment problem for district-heating networks, we can compute optimal unit-commitment schedules under demand uncertainty. We generate heat demand scenarios using a Bayesian model, which explicitly quantifies forecasting uncertainty and yields predictive distributions as input to the network optimization. To handle the inherent uncertainty in heat demand, we apply two stochastic optimization approaches that exploit this distribution: chance-constrained programming limits the probability of unmet demand, and in our scenario-based setting reduces to a deterministic MIP in which demand is replaced by an empirical quantile, while CVaR optimization penalizes severe shortfalls in the tail of the distribution. We evaluate the approaches both on real-world data from the Berlin district heating network and on a benchmark set of synthetic, realistically parameterized instances of varying sizes. The results show that the two methods are complementary: CC cheaply traces the high-reliability end of the cost--reliability trade-off and reveals the physical reliability limit of the network, whereas CVaR covers the low-cost, risk-tolerant regime at considerably higher computational effort.

This paper examines a single-server queueing system that serves both scheduled and strategic walk-in customers. The service discipline follows a first-come, first-served policy, with scheduled customers granted non-preemptive priority. Each walk-in customer strategically chooses their arrival time to minimise their expected waiting time, taking into account the reservation schedule and the decisions of other walk-in customers. We derive the Nash equilibrium arrival distribution for walk-in customers. We also study the case where early arrivals are allowed and analyse its impact on equilibrium arrival patterns. By analysing various appointment schedules, we assess their effects on equilibrium arrival behaviour, waiting times, and server idle time. Finally, we compare different scheduling policies and evaluate their impact on overall system performance.

Notation3 Logic was introduced as an extension of RDF allowing the user to express rules on top of triples and to make statements about graphs. The original proposal of the logic only came with an informal specification of its semantics which has been refined later on. While this refinement clarified the meaning of rules and implicitly quantified variables, it left the meaning of graphs used as terms - so-called graph terms - open. In this paper we aim to close this gap. We discuss in detail, how graph terms are used in Notation3, which formal requirements they need to meet, and provide examples to illustrate the difficulties when defining their semantics. We then introduce an abstract syntax for the logic and define its semantics in two steps: similarly to RDF's simple interpretations, we define interpretations and basic entailment for Notation3 which covers plain RDF triples but also triples containing graph terms. N3-entailment is then an extension of basic entailment which assigns meaning to the different special predicates of the logic like for example the implication symbol. N3-entailment is defined in a way that it can be extended whenever a new predicate is added to the logic. As Notation3 Logic is currently under development, we expect many predicates to be added. Our research provides and important step towards the full standardisation of the logic.

The attention mechanism underpins modern large language models, yet their dynamics remain only partially understood. Recent perspectives interpret attention blocks as interacting particle systems, whose mean-field limits correspond to gradient flows of energy functionals on spaces of probability densities on $\mathbb R^d$ equipped with Wasserstein-type metrics. We extend this viewpoint by deriving accelerated attention blocks from inertial, Nesterov-type damped Hamiltonian dynamics on these density spaces, developing the geometry and differential calculus needed to make the construction rigorous. In particular, we introduce a local-mobility geometry for the Otto-Wasserstein and softmax-Wasserstein settings, and a quotient Stein geometry that resolves the degeneracy of the linear-attention kernel to yield a well-defined metric and differential calculus. We provide rigorous foundations when softmax attention admits a gradient-flow formulation and when its inertial counterpart admits a damped Hamiltonian on explicitly characterized domains. For linear attention, we obtain a finite-dimensional solution theory with sharp conditions for global existence versus finite-time second-moment blow-up. We further prove a quantitative link between continuous moment blow-up and the discrete-layer depth at which token norms cross a prescribed threshold. Finally, motivated by the finite-token geometry, we propose geometric corrections to token and momentum updates. In decoder-only, nanoGPT-style experiments on TinyStories and OpenWebText, these corrections lower validation negative log-likelihood on next-token prediction relative to momentum-only controls.

Using the Weil--Gel'fand--Zak transform of Faddeev's quantum dilogarithm, we construct a compact state integral for positively shaped triangulations of oriented pseudo three-manifolds. The integral is labelled by a circle-valued first cohomology class of the vertex complement (a ``sector'') and takes boundary values in a line bundle over compatible boundary cocycles. We prove absolute convergence, smoothness, invariance under shape gauge transformations and under compatible shaped $2$--$3$ moves, branching independence, and a single-valued global meromorphic continuation in all shape and boundary parameters for each fixed quantum parameter. Gluing is expressed by a conditional Haar integral which retains the final cohomology class; the ordinary trace of integrated cut amplitudes computes an average of the sectors over a torus of classes determined by the cut, and is a sewing formula for every cut of a closed oriented three-manifold, but not for arbitrary pseudo-manifolds. For closed admissible triangulations, the original Teichm\"uller partition function is the normalized average of the sectors; this yields its global meromorphic continuation. A translation law relates sectors differing by twice an extending class, with phases given by linking pairings; for closed three-manifolds the sectors are supported in a distinguished coset of twice the torsion group and are determined by a single amplitude, which a nonvanishing twisted quadratic Gauss sum recovers from the original partition function.

In this paper we introduce a compact stencil Runge-Kutta discontinuous Galerkin method for the Euler equations of compressible gas dynamics, which preserves a minimum entropy principle similar to that introduced by Tadmor for entropy solution of the Euler equations [4]. Compact stencil RKDG, introduced by Chen et al. in [15], is a variant of standard RKDG method which is known for its smaller stencil size and parallel efficiency. The main challenge is that the cRKDG method has to be expressed in Butcher form and not every Runge-Kutta method of this form is a convex combination of forward-Euler steps. Therefore, in order to establish this property we leverage the bound preserving framework of the cRKDG schemes [8], an entropy bounded limiter [6] and the classical positivity preserving limiter. In the last section numerical experiments are provided to demonstrate the robustness of the method.

We prove the following isoperimetric-type inequality: for every convex body $K$ in $\mathbb R^n$ and some $\sigma\subset[n]:=\{1,\dots,n\}$ there exists a suitable Hanner polytope $B_K$ with the same volume as $K$ and such that the volume of each of its orthogonal projections onto every subspace whose basis is formed by the canonical vectors $\{e_i:i\in\tau\cup([n]\setminus\sigma)\}$, for every $\tau\subseteq\sigma$, bounds from below the volume of the corresponding projections of $K$.

This paper proposes a unified modeling framework centered on "state-cost-extremal evolution," referred to as the SMFCE paradigm. The paradigm represents the effective state of a system as a point in a state space, characterizes state cost by a single-valued scalar field, and describes system evolution through local cost descent or local extremum selection. Its mathematical core employs classical tools from differential geometry, gradient systems, and dynamical systems. The contribution of this paper is not to reintroduce these foundational results, but to organize them into two operational modeling principles. On finite-dimensional smooth Riemannian manifolds under appropriate regularity conditions, the path independence of cost differences and the cost monotonicity of negative gradient flows follow directly from classical results, while the descent property of discrete local-minimum transitions follows directly from the definition of a minimum. For same-level iterative emergence and present-centeredness, the paper provides conditional mathematical formulations using, respectively, parameterized gradient systems and bifurcation theory, and Markov processes and receding-horizon optimization. The paper further introduces three layered metrics: cost quantity, directed coupling degree, and commitment fulfillment rate, at the state, transition, path, and system levels. Finally, the paper discusses semantic modeling of large language model training and inference within this framework, explicitly distinguishes classical mathematical results, conditional propositions, and empirical adaptation models, and provides reproducible empirical testing procedures for the two modeling principles.

The class $M_\alpha$ of $\alpha$-convex functions, introduced by Mocanu in 1969, interpolates between starlike ($\alpha=0$) and convex ($\alpha=1$) mappings and has traditionally been studied only for $\alpha\geq0$. We show that several of its sharp invariant estimates in fact extend to the entire real line. Our main result characterizes $M_\alpha$ for every real $\alpha\neq0$ through an explicit pointwise inequality, generalizing a theorem of Chuaqui, Duren and Osgood from the convex case, and we determine the sharp value of $\beta$ for which $M_\alpha\subset C_\beta$ when $\alpha\geq1$; we also show that the reciprocal inclusion $C_\beta\subset M_\alpha$ fails for every $\alpha>1$. Building on this, we obtain a sharp Fekete--Szeg\H{o} inequality, valid for every $\alpha\notin\{0,-1,-1/2\}$, and sharp bounds for the first two logarithmic coefficients of $M_\alpha$, which recover the classical values for starlike and convex functions as special cases. We also determine the order of $M_\alpha$, establish a piecewise sharp bound for its Schwarzian norm together with an explicit quasiconformal extension criterion via the Ahlfors--Weill theorem, and compute the Schwarzian norm of the $\alpha$-Koebe function explicitly for $\alpha=1/n$, $n\in\mathbb N$, verified for $n\leq9$ and numerically supported for $n<50$, leading to a conjecture for the general case. Finally, we propose a conjectural sharp bound for the second Hankel determinant of $M_\alpha$, consistent with the known values $1$ and $1/8$ for the starlike and convex subclasses.

Multi-robot coordination of nonlinear systems is naturally formulated as joint-space optimal control subject to constraints over groups of robots. Existing methods that approach joint optimality remain largely centralized, offline, and confined to collision-free motion planning, while decentralized, real-time counterparts target safe navigation without pursuing the joint optimum. This paper develops a decentralized, real-time planner that approaches the joint optimum as an anytime process. We extend a recent single-robot framework that searches dispersive trees of forward-propagated commands for differentially flat systems to coordinate a multi-robot team near-optimally. We first show that the inter-robot coupling can be effectively replaced by some local constraints, under which locally grown dispersive trees contain a near-optimal agreement. It is this structure that the robots pursue by decentralized column generation, negotiating rents on the shared space that guide the search of each robot's tree. The negotiation settles on an agreement whose near-optimality is certified at quiescence. Our protocol, WarpWeaver, implements this negotiation as an asynchronous real-time loop over a receding horizon, and demonstrates effective closure to quiesced agreements even when real-time deadlines cut the negotiation short. In simulation, WarpWeaver reduces total travel time by up to 22% against centralized offline planners on the Dynobench suite and by up to 37% against a decentralized real-time baseline, and nearly doubles the success rate of a task-specific planner on graph connectivity. Hardware experiments on six robots planning onboard demonstrate real-world implementability.

We bound fractional chromatic numbers by establishing a connection with low-weight odd-cycle transversals. Let $\ell\geq 3$ be an integer and let $G$ be a graph of odd girth at least $\ell$ whose odd-cycle covering polyhedron is integral. For every nonnegative weight vector $w=(w_v)_{v\in V(G)}\in \mathbb{R}_{\geq 0}^{V(G)}$, we obtain a set $Z\subseteq V(G)$ such that $G-Z$ is bipartite and $w(Z)\leq\frac{mad(G)}{2\ell}w(V(G))$. Here, $w(S)=\sum_{v\in S}w_v$ for any $S\subseteq V(G)$, and $mad(G)$ denotes the maximum average degree of $G$. Consequently, if $mad(G)<2\ell$, then $G$ has fractional chromatic number at most $\frac{4\ell}{2\ell-mad(G)}$. As applications, we prove that, for every positive integer $k$, every $n$-vertex planar graph of girth at least $2k+3$ has fractional chromatic number at most $\frac{2k+1}{k}(1-\frac{1}{kn+1})$, establishing the girth version of Naserasr's fractional coloring conjecture with a strict inequality. For $k=1$, we furthermore prove that the upper bound $\frac{3n}{n+1}$ holds for every $n$-vertex triangle-free planar graph and this is attained for infinitely many values of $n$.

Let $\{X_i\}_{i=1}^n$ be independent, identically distributed lattice random variables with zero mean, unit variance, and maximal span $h>0$. Let $S_n=(X_1+\cdots+X_n)/\sqrt n$. Let $Z_n$ denote the quantization of a standard Gaussian random variable on the support lattice of $S_n$. We show that the R\'enyi divergence $D_\alpha(S_n\|Z_n)$ of order $1<\alpha<\infty$ tends to 0 as $n\to\infty$ if and only if $D_\alpha(S_{n_0}\| Z_{n_0})<\infty$ for some integer $n_0$ and a sub-Gaussian condition holds. Furthermore, we derive an Edgeworth-type asymptotic expansion of the divergence to arbitrary order. We also show that the R\'enyi divergence $D_\infty(S_n\|Z_n)$ of infinite order tends to 0 as $n\to\infty$ if and only if a sub-Gaussian condition, a curvature condition, and a uniform separation condition hold. These results constitute lattice counterparts of R\'enyi entropic central limit theorems for continuous random variables obtained by Bobkov, Chistyakov and G\"{o}tze (\emph{Ann. Probab.} \textbf{47} (2019), 270--323) and Bobkov and G\"{o}tze (\emph{Ann. Probab.} \textbf{53} (2025), 453--477). Uniform separation is an additional assumption beyond the sub-Gaussian and curvature conditions used in the continuous setting. The failure of uniform separation may cause an exponentially tilted version of $X_1$ to concentrate on a sublattice of max span $kh$ for some integer $k>1$, which yields the lower bound $D_\infty(S_n\|Z_n)\ge\log k$, and hence a failure of the entropic central limit theorem.

For a $t$-graph $F$ and an integer $r\ge t$, the \emph{$r$-augmentation} $F(r)$ is the $r$-graph obtained by adding the same set of $r-t$ new vertices to every edge of $F$. We investigate the behaviour of the \emph{Tur\'an density $\pi(F(r))$}, which is the asymptotically maximum edge density of a large $F(r)$-free $r$-graph, as a function of $r\to\infty$. Let $H_k^r$ be the $r$-augmentation of the complete $(k-1)$-graph on $k$ vertices; equivalently, $H_k^r$ is the (unique up to isomorphism) $r$-graph with $r+1$ vertices and $k$ edges. This paper determines the Tur\'an density of each $r$-graph $H_k^r$ within a constant factor and estimates the lower bounds on $\pi(H_k^r)$ coming from the circular construction of Sidorenko. In particular, it is shown that if $k/\log r\to\infty$ then $\pi(H_k^r)=(1+o(1))k/r$. Also, we consider augmentations of graphs (that is, the case $t=2$) and prove that $\pi(F(r))\ge(\log r)^{-1-o(1)}/r$ for every 3-chromatic graph $F$, which comes close to the known upper bound $1/r$. Note that, for all other graphs $F$, the function $\pi(F(r))$ is known to be either identically 0 or uniformly bounded away from 0.

We prove Gamma conjectures I and II of Galkin, Golyshev and Iritani for Milnor hypersurfaces, the smooth hypersurfaces of bidegree $(1,1)$ in the product of two projective spaces. We provide a criterion for Gamma conjecture II for general Fano manifolds, generalizing the criterion by Hu and Ke. Using the mirror symmetry technique of Galkin and Iritani, we establish global Gamma-I for Milnor hypersurfaces, thereby proving Gamma conjecture I as a special case. Using it, together with the Galois action and our criterion, we verify Gamma conjecture II for Milnor hypersurfaces.

This work presents a finite element formulation for coupled chemo-mechano-thermodynamical large deformation contact. The formulation is based on the contact theory of Sauer et al. (2022) that contains six coupled (but separate) fields: the deformation and temperature of the two contacting bodies, as well as an interfacial bonding field and interfacial temperature. The latter is governed by the chemical and mechanical energy dissipation at the interface. Here the focus is placed on the evolution of bonding and debonding, and how it is coupled to the mechanical and thermal contact state. Several elementary models are proposed for this based on a quadratic contact potential. The resulting contact formulation becomes very general and versatile, which is illustrated by several challenging examples. They include pressure- and gap- dependent bonding, exothermic bonding reactions, thermal hardening and thermal expansion, as well as simultaneous bonding and debonding. They are based on a monolithic finite element implementation using classical and isogeometric shape functions together with implicit time integration. Its full linearization, required for the Newton-Raphson solution method, is also provided for the six-field problem, as well as for several of its simplifications. If bonding sites are material points, the bonding variable can be condensed-out locally.