Papers

We define and study the $F$-signature of big divisors on projective varieties in positive characteristics. For a normal projective variety $X$, we show that the $F$-signature function defines a continuous function on the big cone of $X$, extending our previous results on the ample cone. By studying the the Frobenius-alpha invariant on the big cone, we prove that the positivity of the $F$-signature of any big $\mathbb{R}$-divisor on $X$ is equivalent to the global $F$-regularity of $X$. As applications, we prove a birational transformation rule for the $F$-signature in terms of the $F$-signature of certain graded ideal sequences. We also prove that in a reduction modulo $p$ situation, a characteristic independent lower bound for the $F$-signature of $X$ with respect to any ample divisor $L$ spreads out to a similar bound for arbitrary big divisors on $X$.

We prove that if a minimal ordered structure $(M, <, \ldots)$ with infinite chains in an arbitrary language extending the language of strict orders interprets (in some power $M^n$) the linear order $(\omega, <)$, then this fact can be witnessed for $n=1$ via the incomparability relation of an $<$-definable strict order $R$; that $R$ is unique up to “almost equality”, i.e., finite rearrangements of elements of $M$; and that $R$ can be defined from $<$ in a constructive way. We also show that several variants of the question whether a minimal ordered structure interprets an infinite linear order are all equivalent.

This article is aimed at introducing a new procedure to define reference solutions to which the numerical approximations of anisotropic problems arising in plasma physics may be compared. In this context, the main issue is related to the construction of functions with no gradients along the vector field defining the anisotropy direction. The so-called solution manufacturing, implemented so far in the literature, only applies to frame- work where all the computations can be carried out analytically. In the present paper, we propose a new method in which this requirement is not mandatory. Indeed, via a Riemannian metric, we use a propagation of the boundary value by the flows related to the vector field defining the anisotropy direction. This permits to extend the application of the solution manufacturing to a broader range of contexts.

We extend Rainwater's theorem that, for Baire measures on a compact Hausdorff space,a measure singular with respect to any measure from a $w^*$-compact convex set $M$ of probability measures sits on a Baire set which is $\mu$-null for every $\mu\in M$. Our main result shows that this statement holds under the weaker assumption of convex analyticity of $M$. In the standard way, this extends the Glicksberg–König–Seever decomposition theorem. We also give conditions for our results to hold for sets of signed measures.

We expand on the concept of admissible pairs from \cite{Levin:GCD} and explore its relation with the main Diophantine arithmetic inequalities, with discriminant term and for points of bounded degree, that have been predicted by Vojta \cite{Vojta:1998}. In this context, among other new results, we prove a harder implication which is in the spirit of Vojta's approach to the abc Conjecture (from \cite{Vojta:1998}). As our main result, and application of our viewpoint here, we deduce for the case of certain general type nonsingular Deligne-Mumford stacks, with projective course moduli space, a form of the Bombieri-Lang Conjecture for $(D_0,S)$-integral points of bounded degree. A key input for this is a slicing theorem, for Deligne-Mumford stacks, that was obtained by Abramovich and Várilly-Alvarado, \cite{Abramovich:VarillyAlvarado:Pera:2017}, and building on earlier work of Kresch and Vistoli \cite{Kresch:Vistoli:2004}. Another important ingredient is an inequality of Silverman, from \cite{Silverman:1984}, which bounds the discriminant of points in projective space in terms of their heights. As an illustration of our results, we discuss them within the context of the interesting work of Abramovich and Harris \cite{Abramovich:Harris:1991} and others.

We study the delayed one-dimensional swarmalator model when all units share a common intrinsic frequency {\omega}. Unlike the zero-frequency case, {\omega} cannot be removed by a rotating-frame transformation because delayed interactions retain the phase accumulated over the lag. The result is a two-order-parameter analogue of the delayed Kuramoto model: the asynchronous state acquires incoherence lobes, the synchronized state becomes a rotating branch stable even when both the spatial and phase couplings are repulsive, and the phase wave develops narrow resonant stability bands. These delay-selected windows overlap to produce broad branch coexistence, including a narrow window near K=-J where all three canonical states-asynchronous, phase wave, and synchronized-are simultaneously stable. In regions where no canonical branch is stable, the order parameters oscillate persistently.

Motivated by networked systems in which equilibrium conditions apply to a regulated state rather than directly to the decision variable, we study transformed-state general variational inequalities. Using a projection-residual reformulation, we develop residual feedback methods that operate in the decision space without inverting the state mapping $H$. For a one-step method, we establish an $\mathcal{O}(T^{-1/2})$ best-iterate residual rate under solution-restricted cocoercivity and linear convergence under solution-restricted strong monotonicity and residual Lipschitz continuity. We also obtain an $\mathcal{O}(T^{-1/2})$ best-iterate residual rate for a predictor-corrector method applied to the induced GVI residual under monotonicity and Lipschitz continuity. The assumptions are imposed directly on the computable residual and are illustrated by affine network and rank-deficient examples. Finally, we extend the residual framework to generalized quasi-variational inequalities with decision-dependent feasible sets.

This study develops a projection-based descent method for minimizing weakly semismooth functions over polyhedral sets, taking an initial step toward extending projection-based descent algorithms to nonsmooth, nonconvex optimization problems. At each iteration, the method considers an inner approximation of the Clarke $\varepsilon$-subdifferential at the current iterate over the feasible region and uses its least-norm element to generate a search direction. An exponential limited backtracking line search is then developed to assess the quality of the generated direction, while a projected variant of Mifflin's line search is proposed to identify nonredundant subgradients that enrich the Clarke $\varepsilon$-subdifferential approximation when necessary. To quantify stationarity, a characterization of stationary points based on the projection operator yields a computable optimality measure for the proposed method. The convergence properties of the proposed method are established under mild assumptions. If the algorithm generates infinitely many serious steps, we identify two subsequences such that every cluster point of each subsequence is stationary. If the number of serious steps is finite, we show that the final serious step generates a stationary point. Numerical experiments demonstrate the efficiency and broad applicability of the proposed method across a wide range of test problems, including applications in image denoising, multiobjective optimization, and data clustering. To extend the applicability of the proposed method to problems with general smooth nonlinear constraints, we propose a heuristic approach based on sequential linearization.

The shallow embedding of higher-order modal logic in classical higher-order logic, used in Benzmüller and Scott's Notes on Gödel's and Scott's variants of the ontological argument, reaches beyond the modal object language: its quantifiers range over domains containing the nominals and satisfaction operators of hybrid logic. A proof using one proves a theorem of the embedding, not necessarily of the modal logic. Whether a result is one of the modal logic can be settled by replaying it in an explicit calculus, or, more cheaply and for all results at once, by analysing what the embedding's proofs substitute. Every statement the Notes prove has a proof in the embedding substituting nothing outside the object language, 294 of them explicit. The Notes' own proofs substitute none either; what it flags are terms a prover introduced. A soundness theorem over general models closed under the object language, and a sentence of that language the embedding proves but which fails in one of them, fix what the certificate says. The Notes' three open questions are settled too, without nominals, but by the empty instance of the conjunction axiom, too cheaply to convince. Generalised to Gödel's "any number of summands", the axiom covers the empty conjunction and the singleton. This article restricts it to at least two different conjuncts, a non-degenerate reading of Gödel's footnote, and settles them again without nominals. In effect only the empty conjunction is excluded; the singleton comes back with the other axioms. In one variant the restriction refutes the possible existence of a God-like being; the answers return, with no side condition, once the Notes' non-emptiness clause is confined to Gödel's essence. Every theorem here is verified in Isabelle/HOL and independently in Lean 4, the separating model in its propositional part; the countermodels are Nitpick's, checked by the build.

We propose an algorithmic framework for solving adaptive robust optimization with provable guarantees on both tractability and solution accuracy. The framework introduces soft separation, a probabilistic mechanism for identifying worst-case uncertainty realizations via a time-inhomogeneous Markov chain. Rather than solving an exact separation problem in each iteration, which is intractable in general, the chain carries adversarial information across iterations, co-evolves with the optimization iterates, and recovers exact separation at terminal iterates with high probability. For continuous first-stage (here-and-now) decisions, we design a first-order method that uses soft separation to produce adaptive gradient estimates. Notably, we prove polynomial-time convergence in expectation to the global optimum. For mixed-integer here-and-now decisions, we embed soft separation within a branch-and-cut framework to generate valid cuts for the robust objective and obtain a high-probability certificate of global optimality. Numerical experiments demonstrate that the proposed methods scale favorably with problem dimension and scenario size relative to state-of-the-art approaches. The instances and code are available online at https://github.com/xuqy2002/SoftSeparationARO.

Disk-level scattering amplitudes involving one Ramond–Ramond and two NSNS states on a D-brane obey nontrivial identities among scalar world-sheet integrals imposed by gauge and T-duality Ward identities. We investigate the cohomological organization of these relations using twisted de Rham cohomology and show that the Ward identities admit twisted-exact representatives. In the $I/J$ sector, the unique generic linear dependence among the Ward identities is realized one degree lower in the same twisted de Rham complex, revealing a common cohomological origin for the identities and their syzygy. We also identify independent pointwise algebraic relations among the integrands. An independent cohomological rank computation then establishes that the physically selected $I/J$ family spans a five-dimensional subspace of the top twisted cohomology at generic complexified kinematics. Extending the construction to the $K$-family, including the exceptional double-pole representative $K_3$, introduces no additional cohomology direction. Hence the complete $I/J/K$ family considered here spans the same five-dimensional subspace.

We study one-sided, two-sided and reflected exit problems for a finite-state Markov-modulated additive-increase and multiplicative-decrease process. For upward passage, a first-jump decomposition and a spatial Laplace transform lead to a recursion indexed by finite words of phases and to a convergent Green series. For downward passage, the first-jump operator is a strict contraction and the known contribution from $[p_i b,b]$ becomes an explicit boundary forcing term in the same spatial recursion. These allow to derive the corresponding Green representation of the first passage matrix and its finite-dimensional compatibility condition. The derivation of the two-sided exit identity is based on the Markov property, which in this context is equivalent to the cocycle property of the Laplace transform, together with the recursive equation at the upper level. For reflection at the running infimum, a finite $\bar p$-geometric method of steps realizes a certain Picard map that allows to find the exit matrix. As a result, we construct a Doob martingale related to the first passage time of the reflected Markov-modulated AIMD.

Hedging a derivative by trading the underlying asset changes the payoff that the hedging intended to replicate. We study this phenomenon when trading generates price impact and execution costs. In a binomial model, we characterize replication through a fixed-point equation. In continuous time, we derive a nonlinear pricing PDE whose implicit terminal condition captures the nature of the moving target problem of the hedger. For monotone convex Lipschitz payoffs (such as calls and puts) we establish exact replication under midpoint execution costs. Numerical experiments illustrate: (i) how price impact shifts the effective strike, (ii) the non-linear dependence of the option price on the number of contracts, (iii) how execution costs smooth terminal holdings, (iv) the extent to which the hedger's own trading can bring an otherwise worthless option into the money, and (v) we explain the spread and the shape of the limit order book in the options market based on the price impact and the shape of the limit order book of the underlying.

We exhibit two vector fields on $S^2$, which generate the entire Lie algebra of algebraic vector fields. Compositions of the flows associated with these two vector fields enable us to approximate every diffeomorphism of $S^2$ isotopic to the identity in the $\mathrm{C}^r$-sense for any $r\geq 0$.

We introduce a new family of recursively constructed sparse matrices, termed Fractus matrices, and investigate their use in constructing low-density parity-check (LDPC) codes. Generated through a self-similar recursive process, these matrices yield regular sparse parity-check matrices while preserving key structural properties across successive iterations. This recursive structure enables an efficient encoding algorithm with computational complexity that is nearly linear in the block length. Decoding is performed using standard iterative message-passing algorithms, thereby retaining the low-complexity decoding characteristic of LDPC codes. The proposed construction produces Tanner graphs with girth six and guarantees a minimum Hamming distance of at least $\ell+1$. We establish several algebraic properties of Fractus matrices, including sparsity, regularity, recursive decomposition, and symmetry under the flip-transpose operation. In addition, we show that the family of Fractus matrices admits a natural lattice structure and that the associated LDPC codes inherit corresponding lattice-theoretic properties. These results establish a connection between order theory and coding theory. Overall, the proposed framework integrates recursive matrix constructions, efficient encoding, graph-theoretic analysis, and lattice theory into a unified algebraic approach to the design and analysis of scalable LDPC codes.

We prove a dimensional strengthening of the colorful Helly theorem. Let $\mathcal{C}_1,\dots,\mathcal{C}_{d+1}$ be finite nonempty families of convex sets in $\mathbb{R}^d$. If $C_1\cap\cdots\cap C_{d+1}\neq\varnothing$ for every choice of $C_i\in\mathcal{C}_i$, then $\sum_{i=1}^{d+1}\dim(\bigcap\mathcal{C}_i)\geq0$, where $\dim\varnothing=-1$. We also obtain a dimensional strengthening of a theorem of Kim and Lew, in which the intersections are taken over unions of color classes. For simplicial complexes, we introduce the link-Leray dimension, defined as the Leray number of the link for a face and as $-1$ for a nonface. Using this invariant, we prove a matroidal extension of the dimensional colorful Helly theorem for $d$-Leray complexes. We further establish a corresponding strengthening of the topological Kim–Lew theorem in two ranges of parameters. As a special case, we recover the topological colorful Helly theorem of Kalai and Meshulam. Finally, we construct counterexamples to the unrestricted topological extension, even under a stronger local condition.

Does representation learning stop when the training loss stops improving? We study this question for matrix Muon, whose polar-normalised updates have a step length set by the gradient's rank rather than its norm. Near the edge of stability, full-batch Muon on teacher-student problems enters approximately period-2 loss oscillations that persist for thousands of steps: the cycle-mean loss stays flat or rises, yet the weights keep moving and the learned features continue to align with the teacher subspace. For linear teacher-student learning toys, we derive explicit cycle and alignment formulas and conditional plateau and decay bounds. For a population mean-field ReLU model, we prove that, under stated dimension, initialisation and small-head conditions, the leading eigenspace of the average gradient outer product (AGOP) recovers the teacher subspace exactly during a loss plateau, before the loss later drops. In all 33 ReLU, GELU and SiLU teacher configurations we study, direction-only alignment metrics show the student AGOP aligned with, or still aligning to, the teacher subspace during the period-2 oscillations; projected head refitting on selected configurations shows that the learned directions are useful for prediction, and further measurements distinguish AGOP alignment from weight-mass concentration. In deep residual ReLU students, freezing the downstream layers while the first layer trains with full-batch exact polar updates recreates a nearly flat cycle-mean loss with improving input-AGOP alignment; freezing and unfreezing switch between this plateau and loss decrease, and the effect is sensitive to momentum and to the choice of orthogonaliser.

Many learning tasks require stability to small transformations while retaining sensitivity to larger ones. We introduce transversal pooling neural networks, which generalize spatial max pooling to affine group actions. We establish equivariance to a chosen subgroup and derive explicit stability bounds for individual pooled wavelet coefficients under affine perturbations of the input. Experimentally, we demonstrate the utility of our networks in low-data environments and for predicting tropical cyclone intensification.

Let $X\subset\mathbb P^N$ be a smooth complete intersection of two quadrics. We find formulas in the Grothendieck ring of varieties for the class of $X$ and for the class of its Fano scheme of lines, thereby proving the first two cases of a conjecture of Belmans et al. We also find a formula for the class of the relative Fano schemes of linear subspaces (of any dimension) in the fibers of quadric fibrations over curves, under a simple degeneration assumption. As consequences, we compute the rational Chow motive of the relative Fano scheme, and we provide evidence for a conjecture of Shah on a residual category in a semiorthogonal decomposition of the relative Fano scheme of lines.

Our recent research on Langevin dynamics with self-harmonic drift and classical spins has revealed a strong connection to “Stochastic Localization," a concept attracting attention in the fields of mathematical geometry and data science. Roughly speaking, the former can be viewed as a version of the latter in which the $n$-dimensional Euclidean space - the domain of the probability distribution function appearing in the latter - is “compactified” to the surface of an $n$-dimensional sphere. By comparing the two frameworks, we argue that this compactification is a consequence of requiring the drift to be autonomous. The relationship between the two approaches may expand the scope of Stochastic Localization.

We study the dynamical behavior of tokens in transformers from a control-theoretic perspective. Our model includes the feed-forward layer present after the self-attention mechanism, with the self-attention mechanism interpreted as an interacting particle system and the feed-forward layer as an independent control. Our main theoretical result establishes that the feed-forward network can steer the tokens arbitrarily close to consensus regardless of the key, query, and value matrices. Our result are easily extended to convergence to many clusters and to multi-head attention. We conduct numerical experiments to verify our results, and compare thresholding behavior from our theory to real-world LLMs.

We prove Tachibana-type rigidity theorems for closed manifolds under weakened partial-sum conditions on the eigenvalues of curvature operators. Specifically, an Einstein manifold of dimension $n\geq4$ is locally symmetric if its curvature operator is $\frac{2(n-1)}{3}$-nonnegative. The corresponding thresholds are $\frac{2(n+1)}{3}$ for the primitive curvature operator of a Kähler-Einstein manifold of complex dimension $n\geq2$, and $\frac{2(n+2)}{3}$ for the curvature operator of a quaternionic-Kähler manifold of real dimension $4n\geq8$.

The classical Hadwiger-Nelson problem asks for the chromatic number of the unit distance graph of the Euclidean plane. Over the years, this problem has been considered for a variety of other normed spaces, with higher-dimensional Euclidean space $\mathbb{R}^d$ being perhaps the most natural and well-studied. Alon, Bucić, and Sauermann proved that, for a typical norm on $\mathbb{R}^d$, the chromatic number of the unit distance graph is at most $2^d$. We improve this exponential bound to a linear one by showing that the chromatic number of a typical norm on $\mathbb{R}^d$ is at most $2d$ and that this is tight. This shows a stark difference in the behavior compared to the Euclidean case, where there is an exponential lower bound. One of the key ingredients is a certain high-dimensional, matrix generalization of the Lonely runner conjecture, which also allows us to completely settle the so-called view-obstruction conjecture of Schoenberg from 1978 and a more recent covering-radius conjecture of Henze and Malikiosis.

Neural operators are typically trained in a supervised fashion, which requires a dataset to be generated with a classical solver. Training them physics-informed, i.e., purely from the governing equations, removes this large offline cost and allows fresh samples to be drawn at every optimization step, but has so far been limited to simplified problems and trails supervised training in accuracy. The obstacle is the ill-conditioning of physics-informed losses, which differential operators induce and which worsens as the discretization is refined. We therefore propose a preconditioned residual loss function and show mesh-independent conditioning for elliptic problems and greatly improved conditioning for saddle point problems. Realized through geometric and algebraic multigrid, the construction applies to linear and nonlinear equations, steady or time-dependent, on structured and unstructured meshes, is agnostic to the neural operator architecture, and adds no cost at inference. On the Poisson, Allen-Cahn and stationary Stokes equations, the resulting label-free training matches supervised training and is four to twenty-five times more accurate than previous physics-informed operator learning methods.

Inverse optimal control provides optimality and robustness guarantees without solving the Hamilton–Jacobi–Bellman (HJB) equation, but its application to 3D nonholonomic vehicles has been precluded by the absence of strict control Lyapunov functions (CLFs). Recently, such strict CLFs were constructed for the 3D nonholonomic vehicle in spherical coordinates, which circumvent Brockett's obstruction, on the largest possible domain. Building on this construction, this paper develops a general inverse optimal stabilizing controller for the 3D nonholonomic vehicle actuated by surge velocity, pitch rate, and yaw rate. The design is a damping $L_gV$ feedback that minimizes a meaningful cost and inherits stability margins. Its flexibility is demonstrated through two choices of penalty function: a quadratic running cost on the state, recovering a near-classical optimal control formulation, and stabilization under user-defined input constraints. Numerical simulations illustrate stabilization performance, control-effort tradeoffs, and the effect of input constraints.

We prove unique continuation for the real periodic Benjamin–Bona–Mahony equation in the energy space: a solution that equals a constant on a nonempty space–time strip is that constant everywhere. In particular, Rosier's unique continuation conjecture [23] holds without any smallness or mean assumption. It even suffices to observe, at one point $x_0$ and for $t$ in an interval, the solution and the potential $\psi=(1-\partial_x^2)^{-1}(u+u^2/2)$ of its flux: $u(x_0,t)=C$ and $\psi(x_0,t)=C+C^2/2$ imply $u\equiv C$, and the second condition cannot be dropped when $C=-1$. The proof uses an exponentially weighted Hamiltonian of indefinite sign with a coercive dissipation identity, and it applies to every nonconstant real polynomial flux. Consequently, Rosier's stabilization results for two localized feedback laws and for a boundary feedback hold unconditionally, for arbitrary data. Stationary step profiles show that the theorem fails in $H^s$ for $0\leq s<1/2$. We also give a generating-series proof of single-point unique continuation for the linearized equation and discuss related control results.

This short note is a concise yet complete overview, intended for students, researchers, and AIs, bringing together in a unified notation the closed-form and reciprocity identities arising from weighted mean squares of the Riemann zeta function and Dirichlet $L$-functions on the critical line. In addition, we prove a new reciprocity identity: the exponentially weighted counterpart of a formula of Lewis and Zagier for the primitive character of modulus $4$.

When a free energy admits several local minimisers, the associated mean-field particle system exhibits a metastable behavior, undergoing random transitions between these minimisers. However, when the number of particles is large, these transitions are rare events and typically occur at a time-scale which is out of reach for simulations. In this work, we introduce a system of particles coupled with a slow macroscopic external variable which approximately follows a Langevin dynamics associated to a coarse-grained free energy. The empirical distribution of the particle system thus samples all local minimisers of the initial free energy, with transition times which are now independent from the number of particles. The construction is inspired by a theoretical tool: the two-scale approach for proving log-Sobolev inequalities for Gibbs measures.

We study feasibility correction for linear programs. Suppose an initial point satisfies most of the constraints. Can it be moved into the feasible region without changing the objective value? Our main deterministic result says that this is possible if the constraint matrix satisfies a restricted isometry property (RIP). We apply the correction theorem to obtain lower bounds for random linear programs with finite-moment or sub-Weibull entries. We complement these with matching-order upper bounds under a uniform $p$th-moment bound for some $p>2$ and a delocalization condition on the objective direction. As an application of our techniques, we also derive lower bounds for the Gaussian width of feasible regions under RIP assumptions.

Let $k$ be a finite field of characteristic $p\neq 3$, let $E/k$ be the Fermat cubic curve, and let $\ell\neq p$ be a prime. If $p\equiv1\pmod3$, assume moreover that $\ell>3$. Then $H^3_{\mathrm{nr}}(k(E^3)/k,\mathbb{Q}_\ell/\mathbb{Z}_\ell(2))=0$ and the cycle map \[CH^2(E^3)_{\mathbb{Z}_\ell}\longrightarrow H^4(E^3,\mathbb{Z}_\ell(2))\] is surjective. In particular, $H^3_{\mathrm{nr}}(\overline{k}(E^3)/\overline{k},\mathbb{Q}_\ell/\mathbb{Z}_\ell(2))=0$. Assuming the Tate conjecture for surfaces over finite fields, we prove an analogous surjectivity result, for all but finitely many primes $\ell\neq p$, for the integral cycle maps for $1$-cycles on every smooth projective variety of dimension $d$ over a finite field of characteristic different from $2$ which admits a smooth projective lift to the ring of Witt vectors. We apply our results to a conjecture of Colliot-Thélène on the local–global principle for zero-cycles over global function fields. To further illustrate these results, we exhibit examples showing that vanishing of degree-$3$ unramified cohomology over the algebraic closure of the ground field does not imply vanishing over any finite subextension, not even for Fano varieties.