数学前沿

Learned multigrid components are often evaluated at a fixed iteration budget, but such comparisons do not distinguish local approximation gains from practical solver efficiency. We study this distinction for high-contrast elliptic problems on unstructured finite element meshes. A learned V-cycle uses operator-dependent transfers, Galerkin coarse operators, near-kernel deflation and an energy-norm safeguard; only its smoothers are trained. The architecture represents a specified class of fixed polynomial smoothers and is positively homogeneous in the load. On fresh conductive-channel problems, one learned pass reduces the energy error by factors of 5.7-6.1 relative to the same cycle with four-step Jacobi-preconditioned conjugate-gradient smoothing; the corresponding factors are 1.38-1.45 for insulating barriers. In a separate fixed-input test on channel problems, the fine-level learned smoother attains a smaller energy-norm error than the exact minimum over a five-dimensional Krylov space of its own input on 97-100 of 100 fresh inputs for each position and training initialization; this does not establish superiority over all polynomial-smoothed cycles. When the classical smoothing budget is increased, the paired geometric-mean accuracy crosses that of the learned cycle between 8 and 14 steps per smoothing call on channels and between 5 and 8 on barriers. The measured CPU implementation offers no computational advantage over the tested classical solvers, and load changes and floating islands further restrict its accuracy.

This paper establishes the analytic classification of smooth, simple, 3-fold flops, and in the process provides both an intrinsic characterisation and a classification of their contraction algebras.

We model the dynamics of the liquidity profile in Uniswap~V3 and other concentrated-liquidity automated market makers with a stochastic partial integro-differential equation (SPIDE). The profile, the aggregate of all provider positions across tick space, evolves under a degenerate Sturm--Liouville operator, which encodes mean-reverting concentration around the market price, coupled with a Hilbert--Schmidt integral operator, which captures nonlocal redistribution driven by provider repositioning. We establish existence and uniqueness of the mild solution in a weighted $L^2$ space built on the Gegenbauer eigensystem, and reduce the SPIDE to a coupled system of linear stochastic differential equations whose low-order modes describe the level, slope, and curvature of the profile. Pairing the SPIDE with a log-price process yields tractable expressions for pool reserves, pool value, and loss-versus-rebalancing under Ornstein--Uhlenbeck-type and Black--Scholes-type noise. A finite-rank calibration in a fixed reference frame, applied to three Uniswap pools, gives interpretable estimates and lowers active-band forecast error relative to a martingale benchmark on the two pools compatible with the stationary specification. On the third pool, out-of-sample diagnostics are consistent with nonstationarity, which motivates price-aware and time-varying extensions.

The classical Tur\'{a}n-type problem for graphs forbidding $k$ disjoint triangles has been resolved firstly by Erd\H{o}s [Arch. Math. 13 (1962) 222-227] for graphs of sufficiently large order and improved by Moon [Canad. J. Math. 20 (1968) 95-102], who established that the unique edge extremal graph is $K_{k-1}\vee T_2(n-k+1)$ for $n> \frac{19k-11}{2}$. In this paper, we determine the maximum signless Laplacian spectral radius among all $kK_3$-free graphs of order $n$ and characterize the unique signless Laplacian spectral extremal graph $K_{k-1}\vee T_{2}(n-k+1)$ for all $k\ge 2$ and sufficiently large $n$.

Blowup analysis is a central part of geometric analysis, and systems of PDEs such as Toda systems exhibit particularly rich blowup behavior. In [Nie25], the author constructed explicit families of radial Toda solutions whose blowup masses correspond to arbitrary elements of the Weyl group of the underlying Lie algebra. In this paper, we study these families within the framework of blowup analysis developed by C.-S. Lin and his coauthors. The key is an asymptotic analysis of the solution family that distinguishes Cartan elements in the interior of a Weyl chamber from those lying on its boundary faces. We prove a uniform upper bound for $u_i^{\lambda,H}(r)+2\log r$, so that the selection estimate holds with the center at the origin. We show that the successive bubbling scales for slow decay are determined by the intersections of the ray $H+tq_\mu$ with root hyperplanes. At each crossing, rescaling and the face asymptotics identify the limiting profiles as explicit solutions of the corresponding Levi-Toda subsystems. The masses of these successive bubbles, computed using [KLNW24], account for the entire blowup mass. Examples in type $A_2$ demonstrate this identification and exhibit both separated scalar bubbles and a simultaneous full-system bubble.

The polynomial threshold degree (or threshold degree) of a Boolean function is the least degree of a real polynomial whose sign represents it. We prove that almost every Boolean function has polynomial threshold degree among the central one or two integers. More precisely, a uniformly chosen Boolean function on $n$ variables has threshold degree either $\lfloor n/2\rfloor$ or $\lceil n/2\rceil$, except with probability at most $\exp(-c2^{n/2}/n)$, for an absolute constant $c>0$ and all sufficiently large $n$. The paper proves a 1991 conjecture of Aspnes, Beigel, Furst and Rudich. It also proves a 1991 conjecture of Wang and Williams for even $n$ and its main concentration assertion for odd $n$. A parity example gives a lower bound on the exceptional probability whose exponent differs from ours by a factor of order $n$. The proof relates polynomial sign representation to recovery of Reed--Muller codes from random erasures, and combines sharp-threshold estimates with bounds on generalized Hamming weights and minimum distance.

Freed (arXiv:hep-th/9712042) formulated the theory of special K\"ahler structures. In particular, the regular part of the $\mathrm{SL}_2(\mathbb{C})$ Hitchin base $\mathcal{B}$ carries such a structure, and the associated metric $\omega_{\mathrm{SK}}$ develops singularities near the discriminant locus $\mathcal{D}\subset\mathcal{B}$. Baraglia--Huang (arXiv:1707.04975) showed that its Taylor expansion at regular points can be computed by topological recursion, and Hitchin (arXiv:1712.09928) later constructed integrable subsystems over smooth strata of $\mathcal{D}$ parametrizing spectral curves with only nodal singularities; these strata carry induced special K\"ahler structures. We show that near such strata, the special K\"ahler metric on $\mathcal{B}\setminus\mathcal{D}$ exhibits logarithmic asymptotics in transverse directions, whereas its tangential part converges to a metric on the stratum that agrees with the one induced by Hitchin's subsystems. We also prove that the induced metric on every radial complex line through the origin and a point of such a stratum is flat with cone angle $\pi$ at the origin.

The main purpose of this paper is to resolve the Hardy-Littlewood-type equivalence problem. In particular, we show that a Hardy-Littlewood-type theorem holds for arbitrary majorants in the case $p\in(1,\infty)$, and that the equivalence holds in the case $p=1$. For the proof, we employ new functional analytic approaches based on Fourier series, the Hilbert transform, and the uniform boundedness principle. Those proof methods are completely different from those of Dyakonov [Acta Math., 178 (1997), 143-167] and Pavlovi\'c [Acta Math., 183 (1999), 141-143]. In addition, we present several representative applications that improve a series of known results.

We give a construction of the previously announced family of three-dimensional noncommutative Fano minifolds of Euler degree $72$. We prove that their defining $\mathbb{Z}$-algebras are Koszul and Artin--Schelter regular, and compute the Hilbert series of their section algebras.

Uncrewed aerial vehicle (UAV) communication systems are increasingly used to provide temporary connectivity when ground communication infrastructure is damaged by disasters such as hurricanes and wildfires. However, the amount of aerial support required changes as ground networks recover and communication demand evolves. We focus on dynamically adjusting the number and allocation of actively deployed UAVs within a fixed available fleet. We develop a model predictive control (MPC) framework for dynamic fleet sizing of heterogeneous UAV relays during post-disaster communication recovery. The framework accounts for communication demand, ground network recovery, heterogeneous UAV operation, UAV energy availability, and backhaul capacity, which limits how much relayed traffic can be transmitted to the external communication network. The resulting formulation is a convex quadratic program that supports efficient online fleet sizing and deployment decisions. We evaluate the framework using seven historical disaster recovery case studies based on Federal Communications Commission reports, together with four backhaul scenarios and sensitivity analyses. Results show that required active UAV fleet size varies considerably throughout recovery, while backhaul capacity and fleet composition affect communication performance, highlighting the value of adapting active fleet size over time.

We prove the first quantitative bounds for the Gowers norms of the von Mangoldt function minus a Siegel corrected model function in short intervals. The result applies to intervals $(X,X+H]$ with $H\geq X^{5/8+\varepsilon}$ and gives quasipolynomial savings for the $U^k(X,X+H]$ norm. The main new ingredient is an efficient non-abelian Type II inverse theorem with explicit dependence on the nilsequence dimension. We prove this inverse theorem by applying Leng's efficient equidistribution theorem directly to an unweighted corner correlation in four variables, hence replacing the factorisation and subgroup argument in the non-abelian Type II proof of Matom\"aki, Shao, Tao and the author. In joint work with Florian Richter, we use this quantitative result as one ingredient in proving the existence of infinitely many sum-product patterns of the form $\{x,x+y,xy\}$ in the shifted primes $\mathbb{P}-1$.

We constructed a reciprocal graph-deviation functional for positive ratios on a finite weighted graph. Its scalar generator $J(x)=\tfrac12(x+x^{-1})-1$ was characterized previously by ratio-composition and unit log-curvature. Our principal new result was structural identifiability: already on a two-site, one-edge system, the complete minimizer-location map obtained by sweeping the boundary and site-to-edge weight ratio determines any reciprocal generator whose log-coordinate representation is $C^2$ with everywhere-positive curvature, up to a positive scale; unit-curvature calibration fixes that scale. For the calibrated reciprocal family~$J_p$ we obtained the equilibrium response in closed form. A synthetic $15$-condition sweep with seven simulated measurements per condition and log-scale noise $\sigma=0.05$ selected $J$ over~$J_{1.1}$ with probability~$0.997$ under exact boundary and weight parameters; a declared representation-uncertainty screen lowered that value to approximately~$0.88$. These probabilities characterize the simulated design under the declared data-generating model. The finite sweep discriminated prespecified generators, whereas nonparametric recovery required the complete response map. We also recorded the associated convex gradient-flow and Langevin dynamics.

Motivated by the classical Erd\H{o}s and Falconer distance problems and Magyar's work on distance sets in $\mathbb Z^d$, we study discrete analogues of the Erd\H{o}s-Falconer unpinned and pinned distance problems over the integers. We ask how the discrete Hausdorff dimension of a set $A\subseteq\mathbb Z^d$ influences the size of its squared-distance sets, measured by generalized upper natural density and upper discrete mass dimension. We propose conjectures for both problems and prove the unpinned conjecture in dimensions $d=1,2,3$ and the pinned conjecture in dimensions $d=1,2$. For the unpinned problem in higher dimensions, we obtain explicit lower bounds for the dimensions of distance sets using elementary integer sphere counting, results on the Erd\H{o}s distinct distances problem, and $\ell^p$-improving inequalities for discrete spherical averages. For the pinned problem, we obtain lower bounds for the dimensions of distance sets for some or every pin. We also establish more general results involving upper discrete mass dimension, upper discrete Banach dimension, and affine dimension, including bounds on the affine dimension of the exceptional set of pins. We construct examples showing that both conjectures are sharp in low dimensions. Further examples demonstrate how the distinctions among discrete Hausdorff dimension, upper discrete mass dimension, and upper discrete Banach dimension lead to different phenomena in the unpinned and pinned distance problems, including limitations on the nontrivial upper bounds these dimensions can provide for the dimensions of distance sets.

We study weighted power means on the Lie group $\operatorname{UT}_n$ of real upper triangular matrices with positive diagonal entries. For $t\in\mathbb R\setminus\{0\}$, the weighted power mean $P_t(\mathbf{w}; A_1,\ldots,A_m)$ is defined by $$X=\sum_{i=1}^m w_i(X\#_tA_i), \qquad X\#_tA:=X(X^{-1}A)^t.$$ Using divided differences, analytic matrix functions, and the Parlett recurrence, we prove existence, uniqueness, and real-analytic dependence of $P_t$ on the data and the weights. We derive finite entrywise and block-recursive algorithms, establish structural properties of the mean, and obtain explicit formulas for the cases $\operatorname{UT}_2$ and $\operatorname{UT}_3$. We prove that $P_t$ converges to the weighted Karcher mean as $t\to 0$. We also establish a Lie--Trotter formula and derive second-order asymptotic expansions for the power, Karcher, and log-Euclidean means, showing that the parameter $t$ first enters the local behavior at second order.

Let $S$ be a minimal surface of general type with $K_S^2=1$ and $p_g(S)=2$, called a $(1,2)$-surface. We give a geometric recovery and refinement of D. Wen's bound $|\mathrm{Aut}(S)|\leq 200$ using the canonical genus-two fibration and the double-cover model over the Hirzebruch surface $\Sigma_2$. Computing $\mathrm{Aut}(\Sigma_2)$ from its Cox ring separates the action on the base $\mathbb{P}^1$ from the vertical kernel. If $\overline K_0$ denotes the vertical image on $\Sigma_2$, then $\overline K_0$ is cyclic of order $5$, $4$, $2$, or $1$, and we prove the respective sharp bounds $|\mathrm{Aut}(S)|\leq 200, 192, 48, 80$. The proof combines singularity indices of genus-two fibrations with invariant divisors arising from the branch equation. We also construct admissible branch divisors attaining all four bounds and show that their full automorphism groups are, respectively, $(C_{10}\times C_{10})\rtimes C_2$, $(BO_{48}\times C_8)/\langle(-I,J^4)\rangle$, $(BT_{24}\times C_4)/\langle(-I,J^2)\rangle$, and $C_{40}\times C_2$. Here $BO_{48}$ and $BT_{24}$ denote the binary octahedral and binary tetrahedral groups, respectively, and $J$ denotes a generator of the corresponding cyclic factor.

Long chains of decisions coupled only through a bounded cumulative state, such as the operation of an energy-storage device, form convex programs that generic solvers treat as arbitrary sparse models. We present an exact solver for this class that propagates the complete convex cost-to-reach function through restrictions and infimal convolutions, each evaluated by fast Legendre-Fenchel conjugation as an ordered merge of slopes: no state grid, no value approximation, and a function rather than a single primal-dual pair as output. Price-taking storage instances of 5,000 to 15,000 hourly steps solve 35-93 times faster than Gurobi. The value function is what pays: exact capacity marginals in one differentiated pass, recourse assembled rather than sampled on scenario trees, and a certified sizing loop whose cut count does not grow with the number of weather years. We then widen the carried state. Two storages at one node can be eliminated rather than coordinated by carrying the value function of the pair, and we prove that this object is combinatorially rigid: every breakline the recursion realizes is vertical, horizontal, or borne by a line through the origin, with no genericity assumption. For d co-located stores sharing one charge-discharge coefficient, every realized wall is a position hyperplane or an indifference hyperplane {s_i = s_j}. A catalogue law maps where speed holds: one-dimensional chains beat the frontal solve by 15-30x and growing, while the planar block loses by 13.5-107x on rich network curves. Those curves are themselves exact, traced by a parametric dual simplex. The open-source package FLToptim, every benchmark driver and all data accompany the paper.

We develop an unobservable M/M/1 queueing model to analyze advance selling and spot selling under uncertainty. Customers are heterogeneous in waiting costs, drawn from a uniform distribution. We disentangle two fundamental sources of uncertainty: customer valuation and service capacity. We derive the respective revenue functions for both advance and spot selling strategies and compare their performance. Our analysis reveals that advance selling is more profitable when customer valuations are uncertain. Conversely, spot selling emerges as the more profitable strategy when uncertainty originates from service capacity. When both sources are present, our numerical experiments suggest that the preferred regime depends on the relative dispersion of the two uncertainties: advance selling tends to be favored when valuation uncertainty is sufficiently pronounced relative to capacity uncertainty, whereas spot selling tends to be favored when capacity uncertainty is more pronounced. These results highlight how the origin and magnitude of uncertainty shape the optimal selling strategy in service queues and offer practical guidance for platforms and service providers deciding between advance and spot selling.

We extend Raynaud's theory of formal models from rigid-analytic spaces over a nonarchimedean field to uniform qcqs adic spaces $X$, with no finite-type assumptions, over an arbitrary Tate affinoid base $S$. The key new ingredient is the notion of a normalized formal blow-up which takes on the role played by admissible formal blow-ups in the classical theory.

We extend Mark Kac's kinetic framework from single-species Euclidean spaces to a two-species, non-linear interaction space on the open cone $(0, \infty)^2$ governed by the Matsumoto--Yor rational collision involution. We identify a non-linear conserved energy $E(x,y) = x + y + 1/x$ and an invariant conic metric measure $d\mu = \frac{dx \, dy}{x(x+y)}$. By establishing a two-species $H$-theorem, we prove that entropy production vanishes if and only if post-collision velocities are statistically independent ($I(U;V)=0$). In contrast to the transform-domain proof of Letac and Weso{\l}owski~\cite{letac2000}, which relies on joint Laplace transforms of general measures, our approach yields a direct physical derivation in state space via the transformation law for logarithmic gradients and separation of variables. This uncouples the functional-differential system to uniquely characterize the Generalized Inverse Gaussian (GIG) and Gamma distributions as the non-equilibrium thermodynamic equilibria.

For every normed space $X$ and integers $1\le k<n$, we construct a Lipschitz retraction from $X(n)$ onto $X(k)$ with constant at most $2+8n(n-1)$, independently of the dimension of $X$. Here $X(n)$ consists of the nonempty subsets of $X$ with at most $n$ points, equipped with the Hausdorff metric. For the consecutive case $k=n-1$, a separate construction gives the sharper asymptotic bound $O(n\log n)$. These results extend to barycentric metric spaces and settle some open problems on Lipschitz retractions.