We introduce First-Order Trajectory Matching (FTM), a surrogate-modeling method that learns the first-order local transport of probability mass from trajectories of stochastic systems. By matching the symmetric first-order motion of trajectories, FTM learns the probability current velocity, whose flow preserves time marginals to match ensemble averages, while also capturing current-like trajectory quantities such as fluxes, circulations, and barrier-crossing currents. FTM learns the current velocity directly from trajectories, avoiding drift, diffusion, and score estimation. Our stability analysis separates discretization error from sampling variance and shows that the one-step simulation-free FTM loss is stable when temporal resolution and sample size are properly balanced. Across stochastic dynamical systems and PDE examples, we empirically demonstrate that FTM provides trajectory-aware ensemble predictions at low, deterministic-rollout cost.

The symmetric determinantal complexity sdc(f) of a polynomial f is the least m such that f = det(M) for an m x m symmetric matrix M of affine-linear forms. We prove, over the complex numbers, that sdc(sum_{i=1}^n x_i^n) >= (1/(2e) - o(1)) n^2. This is a symmetric companion to the author's non-symmetric polar-degree preprint (arXiv:7680505); the method parallels that work, but the proof here is self-contained and redoes the load-bearing local incidence analysis in the symmetric setting. The general theorem: if X = V(f) in P^{N-1} is a smooth degree-d hypersurface, N >= 3, and f = det(A_0 + sum x_i A_i) with all A_i symmetric of size m, then the top polar degree d(d-1)^{N-2} is at most 2^{N-2} C(m, N-1). The proof uses the symmetric rank-one kernel incidence M(z,x) u = 0. At a genuine polar point M has rank m-1, and a symmetric Schur-complement normal form eliminates the unique kernel line scheme-theoretically; on the resulting local graph the lifted conormal forms u^T A_i u are a common unit multiple of the partials d_i f, so the lifted polar equations cut the ordinary polar slice up to units and each genuine lifted polar point is a zero-dimensional isolated solution. Multihomogeneous Bezout on P^N x P^{m-1} then yields the bound 2^{N-2} C(m, N-1). For F_n = sum x_i^n this gives the constant 1/(2e). More generally, for F_{N,d} = sum_{i=1}^N x_i^d the same theorem gives sdc(F_{N,d}) >= (1/(2e) - o_N(1)) N(d-1) as N -> infinity. We give an explicit symmetric representation of F_{N,d} of size 2N(d+1)+1, so the diagonal bounds are non-vacuous and tight up to a constant. The result is for exact symmetric determinantal complexity in characteristic zero; it is not a border statement and not a uniform positive-characteristic theorem.

The bandwidth of a timed language characterizes the quantity of information per time unit (with a finite observation precision $\varepsilon$). The asymptotic behavior of the bandwidth as $\varepsilon \to 0$ classifies timed regular languages in three classes: meager, normal, and obese. Normal timed automata have a bounded frequency of events and some non-punctual transitions, and, up to now, were the only class of timed automata for which no algorithm was available for computing their bandwidth. In this article, we compute the bandwidth of any such automaton in the form $\approxα\log{1/\varepsilon}$. Our approach reduces this problem to computing the best reward-to-cost ratio in a weighted finite graph constructed from the given timed automaton.

Finding convenient spaces in which certain hypotheses regarding an assumed sparse structure of natural signals hold true has become a desirable result in recent research, its implications being reflected in areas such as data compression, noise reduction and feature extraction. While the extensively used analytical transforms, such as DFT or DCT, already provide efficient algorithms and robust sparse representations, they assume a fixed prior about the data, failing to accurately capture the specific structure of more restrictive classes of signals. To address this, the concept of a data-adaptive, learnt transform has been introduced in the literature, allowing for the reduction of a residual term in the transform domain. More recent studies have shown that the condition number serves as a good metric in this context, where the desired outcome alternates between a generalizing tendency and one that achieves minimal approximation error. Motivated by these considerations, we introduce the learning of a structured, explicitly conditioned transform formulated as the product of a fixed canonical matrix and a refining data-adaptive sparse component. This approach seeks to preserve the advantages of fast and stable analytical transforms, while introducing controllable adaptivity to the data. No references that concern this specific formulation have been identified so far, indicating its novelty. The proposed algorithm is motivated within the framework of inexact proximal methods, leveraging a newly derived closed-form projection operator. Empirical observations demonstrate state-of-the-art results on the doubly sparse transform learning problem and comparable performance with its dense variant at significantly lower computational costs and sometimes faster convergence and better avoidance of bad local minima.

We introduce a new tool, Express, for converting a non-causal attention approximation into a causal approximation with matching approximation guarantees. When combined with the state-of-the-art Thinformer approximation, Express improves upon the best known causal attention guarantees, delivering $\log^{3/2}(n)/s$ approximation error with only $O(s)$ memory and $O(s^2 \log^2(n))$ compression overhead for a sequence of length $n$. We pair these developments with an efficient I/O-aware Triton implementation, demonstrate substantial speedups over FlashAttention 2, and use Express to overcome four resource bottlenecks in the language modeling pipeline: long-context prefill, KV cache compression, long-form memory-constrained decoding, and long-form compute-constrained decoding.

This paper introduces range regularization for federated learning with linear systematic components to enhance statistical accuracy and induce cross-client regularity conducive to quantization, coding, and resource efficiency. Our approach identifies features with shared weights across different clients and adaptively clusters the weights of personalized features at extreme values, a process we refer to as polar clustering. Theoretical analysis of the associated estimators poses significant challenges due to the seminorm nature and non-decomposability of the regularizer. We develop new proof techniques for the nonasymptotic analysis of statistical accuracy and faithful pattern recovery. Moreover, a fast optimization algorithm that leverages varying degrees of local strong convexity is proposed to reduce iteration complexity. Experiments support the efficacy and efficiency of the proposed approach.

Inverse Optimal Control (IOC) aims to recover the cost function that explains observed trajectories as solutions of an optimal control problem. Classical IOC formulations rely on bilevel optimization, which repeatedly solves a nested optimal control problem and quickly becomes computationally prohibitive for realistic systems. Recent projection-based approaches offer a promising alternative but suffer from numerical instability when solved with gradient-based methods due to violations of standard constraint qualifications. In this paper, we show that these difficulties stem from the geometric structure of the IOC feasible set. We demonstrate that the set of trajectories satisfying the optimality conditions naturally forms a manifold and reformulate IOC as an optimization problem on this manifold. Based on this insight, we propose a Riemannian Inverse Optimal Control (RIOC) method that projects observed trajectories onto the manifold of optimal solutions while preserving feasibility by construction. Experiments on real human arm trajectories show that the proposed method achieves comparable or better reconstruction accuracy than classical bilevel IOC while reducing computation time by about a factor of four. These results highlight the potential of geometric optimization methods to improve the scalability and reliability of IOC for robotics and human motion analysis.

The Euler Characteristic Curve (ECC) records the Euler characteristic of a linearly embedded cell complex as a function of filtration height in a given direction, and the Euler Characteristic Transform (ECT) is the injective shape descriptor obtained by collecting ECCs over many directions. How the ECT is encoded for a neural network is itself an inductive bias, conventionally fixed by discretizing each ECC. We introduce a continuous encoding: for each direction and each vertex it records the net Euler-characteristic change attributed to that vertex, producing a per-direction token sequence that a small transformer maps to a feature vector. We separate the resulting pipeline into two stages on orthogonal axes: an ECC encoder that acts within each direction, mapping its curve to a fixed-length vector, and an ECT representation that acts across directions, aggregating the per-direction vectors into one. We study six ECT representation architectures spanning a range of inductive biases, from a structure-agnostic feedforward baseline to convolutional and complex-valued models that preserve equivariance under planar rotations. Across six classification benchmarks covering point clouds, graphs, cubical complexes, and meshes, the continuous encoding improves accuracy on five of six datasets, and control experiments attribute the gain to the tokenization itself rather than to the added transformer capacity. The representation architecture matters less than the encoding, and the payoff from its inductive biases depends on the encoding: a feedforward network performs best under continuous encoding but is less robust under discretization than convolutional architectures.

Moonshine is an autonomous agent whose central objective is to generate mathematical conjectures. Its core capability is to extract structure from classical problems, distill new concepts, and formulate conjectures of mathematical significance. Rather than treating the solution of a single proposition as its endpoint, Moonshine builds an extensible theoretical framework through conjecture generation, bridge building, and obstacle identification. This article uses Moonshine's exploration of the Jacobian conjecture as an example. It shows how the central logic of whether local nondegeneracy can force global injectivity is transferred to one-hidden-layer affine-ridge sigmoid networks. This leads to the formulation of the \emph{Neural Jacobian Conjecture} (NJC): if such a network has strictly positive Jacobian determinant on the whole space, then it must be globally injective. By invoking GPT-5.5-pro and DeepSeek-V4-pro separately, Moonshine obtained independent complete proofs for the case \(N=n+1\). In addition, with the assistance of ChatGPT through interactive use of its web interface with GPT-5.5-pro, a geometric-topological proof was developed. These results provide preliminary evidence for the plausibility of the conjecture. The general higher-width case \(N\ge n+2\), however, remains unresolved and is left for further investigation. This work illustrates Moonshine's ability to autonomously generate meaningful mathematical problems and make rigorous progress on them.

Max-policy iteration is an approach to computing precise numeric program invariants by successive attempts at resolving maximum operators and reduction to mathematical optimization. Mathematical optimization, though, may be expensive. Here, we show, for max-policy iteration on systems of equations over integers as well as over floating point numbers, that mathematical optimization can be replaced by plain value iteration -- which is still guaranteed to terminate. As an application, a precise bound analysis for integer or floating point variables is obtained, avoiding widening operators altogether. We also consider max-policy iteration over the rational numbers, where the right-hand sides are maxima of minima of affine combinations of unknowns. We propose min-policy iteration as an alternative to linear programming for solving the optimization problems posed by max-policy iteration. We prove that max-min policy iteration is guaranteed to return the least solution for bounded systems. We also show how to extend this algorithm to unbounded systems, and how to construct certificates of soundness as well as of optimality of the computed results.

We present NOVA, an autonomous symbolic regression framework that identifies interpretable car-following and lane-change structures from raw trajectory data with minimal behavioral priors. Applied to 4,765,788 active driving observations from the NGSIM I-80 and US-101 datasets, NOVA's deterministic Rust-powered search engine evaluates over 10,000 candidate algebraic structures and identifies a compact two-term acceleration model under a forward-shifted rolling-mean prediction target. Evaluated under two complementary preprocessing pipelines, NOVA achieves $RMSE = 1.376 m/s^2$ ($R^2 = 15.57\%$) on the intent-forecasting benchmark, outperforming the best recalibrated symbolic-regression baseline (SR-LLM, PNAS~2025) by 0.135 m/s$^2$ in RMSE under an identical evaluation protocol. Across eight independent experiments, a single dominant nonlinear term emerges as a robust backbone of human car-following; a residual-guided extension further links the selected structure to an established psychophysical theory of collision avoidance. The discovered feature operators transfer zero-shot between freeway sites with under 3 pp $R^2$ loss. Extended to lane-change modelling within a multinomial logit framework, NOVA achieves 67.4\% balanced accuracy under strict vehicle-ID holdout on 502 unseen drivers, surpassing existing lane-changing baselines by +29.8 percentage points on a three-class problem.

Tamed stochastic-gradient Langevin dynamics (SGLD) stabilizes large drifts by adding a denominator to the update. If this denominator uses the same stochastic-gradient sample as the update step, it can also change the conditional mean drift. We study deterministic denominators: the state-dependent envelope is fixed before the current oracle sample is drawn. The main question is how to design this envelope in practice. The design starts from an oracle score, builds a low-cost proxy score on pilot states, chooses activation thresholds by empirical quantiles, and then applies a small calibration layer. The analysis tracks three steps: proxy and threshold errors become envelope errors; envelope errors perturb one SGLD step; and the local residuals give stationary errors through a conditional perturbation bridge. Experiments show that the proxy-quantile denominators are close to oracle-score behavior, avoid the random-denominator mean-shift channel, and improve simple deterministic taming choices.

This paper addresses the backstepping boundary stabilization of coupled multidimensional first-order hyperbolic systems. We consider systems whose transport velocity fields are collinear, meaning that each velocity field is a scalar multiple of a common base velocity field. Building upon a recent framework developed for scalar multidimensional first-order hyperbolic equations, we introduce a change of variables, based on characteristic curves defined entirely in the spatial domain, that converts the original multidimensional system into a continuum of coupled one-dimensional first-order hyperbolic systems. By designing a backstepping controller for each system in the continuum representation, and assuming that the transit times of the characteristic curves are uniformly bounded, we achieve finite-time stabilization of the multidimensional system.

In the implicit version of a propositional proof system Q, we work with Q-proofs that are not written down directly, but are succinctly encoded by circuits. Thus implicit Q-proofs are potentially exponentially shorter than usual Q-proofs. We study narrow implicit proofs, a restricted version of this notion, in which lines in the encoded proof can only have polynomial size. We use a cut-elimination construction to show that G_{i+1} is equivalent to narrow implicit G_i, for i >= 1, where G_i is the extension of Frege allowing reasoning with Sigma^q_i quantified propositional formulas. We show that G_1 is equivalent to implicit resolution.

``Block what you can and randomize what you cannot'' is the core principle for treatment effect estimation in randomized controlled trials. Although a wealth of allocation strategies has been developed, an explicit trade-off between the covariate balance achieved by blocking and the robustness guaranteed by randomization is seldom quantified. Motivated by the second law of thermodynamics, this work posits a new criterion that lowers the covariate imbalance while maximizing the entropy that quantifies contrast and allocation diversity. The resulting optimal strategy, termed the minimum free energy randomized design, is then derived, thereby formally achieving such a trade-off. To facilitate practical implementation, we further develop a computationally efficient dynamic allocation algorithm with theoretical guarantees. Using a finite-sample variance decomposition, the proposed randomization strategy is shown to control covariate imbalance while preventing unobserved heterogeneity from dominating the mean squared error, thus retaining minimax efficiency under the prescribed design constraints. Extensive numerical simulations demonstrate that our method achieves superior statistical efficiency and greater robustness than existing approaches.

Neural combinatorial optimization (NCO) trains autoregressive policies to solve routing problems. The standard training algorithm, REINFORCE with a rollout baseline, requires maintaining and periodically updating a frozen copy of the policy for variance reduction. This baseline introduces a structural vulnerability: on harder instances, a poor baseline produces noisy gradient estimates that can destabilize training. We evaluate Group Relative Policy Optimization (GRPO), an algorithm from large language model alignment that eliminates the baseline entirely by normalizing advantages within groups of sampled trajectories. In a controlled comparison of five RL algorithms on TSP and CVRP benchmarks within the RL4CO framework, we find that: (i) GRPO avoids the training collapse observed with REINFORCE on TSP-100, where performance degrades from cost 9.8 to 52.1 immediately after the warmup phase and does not recover under extended training; (ii) at matched gradient updates, GRPO achieves solution quality within 2% of POMO, a strong AM-based multi-start baseline, while requiring no external baseline; and (iii) P3O, a pairwise preference algorithm also from the alignment literature, is competitive on TSP but shows higher variability on CVRP. These results identify GRPO as a promising baseline-free alternative for NCO, particularly in settings where baseline-dependent training becomes fragile.

The Gromov--Wasserstein (GW) distance provides a framework for comparing metric measure spaces, regardless of their underlying structure or geometry. For network-based data, it enables direct comparisons of graphs with different numbers of nodes, without requiring an embedding or other abstraction. Furthermore, through a variant of GW known as fused Gromov--Wasserstein (fGW), it is also possible to incorporate node features in addition to graph structure. In this work, we implement $k$-nearest neighbors ($k$-NN) classification using the GW and fGW distances. We prove the universal consistency of the GW-$k$-NN classifier on the space of equivalence classes of metric measure spaces with finite support and uniform probability measure. By viewing graphs as finitely supported metric measure spaces equipped with the pairwise distance metric and a uniform probability measure on the nodes, we obtain universal consistency of GW-$k$-NN for the space of graphs. Likewise for fGW-$k$-NN, we prove universal consistency on the space of weak isomorphism classes of structured objects consisting of metric measure spaces with finite support and uniform probability measure and feature maps into Euclidean space, thus establishing universal consistency on the space of node-attributed graphs. Our numerical experiments show that GW-$k$-NN and fGW-$k$-NN consistently perform well across multiple graph datasets, suggesting that metric classifiers such as $k$-NN work well in the GW framework.

Several recent papers have demonstrated the utility of using Lie groups within estimation problems, yielding improved accuracy and consistency. This paper introduces a new tool for describing operations with matrix Lie groups: tensors and the Einstein summation notation. While tensors and Einstein notation are well-known in other research fields, applying this mathematical notation to represent and compute matrix Lie derivatives is novel. More importantly, this new notation greatly clarifies the derivatives and operations necessary to work with matrix Lie Groups in (gradient-based) estimation frameworks. Therefore, the main contribution of this paper is not a new capability, but a more perspicuous mathematical notation for working with matrix Lie groups.

Current-status data arise when an event time is observed only through an indicator of whether it occurred before an examination time. This paper studies a nonparametric neural-network sieve maximum likelihood estimator of the conditional cumulative distribution function of the event time. Under Hölder smoothness assumptions, we establish an explicit convergence rate by combining approximation theory for rectified linear unit neural networks with empirical-process arguments. This result provides theoretical support for neural-network estimation and subsequent inference under current-status observation.

Matrix-valued time series arise in a wide range of applications, such as spatio-temporal data from medical imaging and geophysics. Existing methods are mainly designed for static settings and lack adaptability to streaming and time-varying environments. Adaptive filtering techniques have also been largely limited to data with scalar or vector values, leaving adaptive forecasting for matrix-valued time series inadequately understood. To bridge these gaps, we develop an adaptive tensor regression framework that includes Matrix-on-Matrix (MoM) and Tensor-on-Matrix (ToM) formulations for streaming matrix-valued prediction. The two formulations differ in whether to directly model matrix-valued outputs or to exploit temporal structure via higher-order tensor representations. For the proposed tensor regression framework, we develop stochastic gradient descent (SGD) algorithms for online learning. We show that stacking multiple responses across time into higher-order tensors improves performance; in particular, the ToM achieves lower steady-state error and stronger denoising capability than MoM, motivating our focus on the ToM model. We further characterize the tracking behavior of SGD under time-varying dynamics. From a statistical perspective, we establish fixed-time recovery guarantees for ToM under general low-dimensional structures, including sparsity, low-rankness, and their joint sparselow-rank models.

In this paper, we consider the data-driven discovery of Lyapunov functions for autonomous dynamical systems. We represent the Lyapunov function as an expression tree of fixed depth and formulate the Lyapunov discovery task as a constrained self-supervised symbolic regression problem. The constraints model the output of the Lyapunov function for a given input as well as the Lyapunov stability conditions. This modeling approach makes no a priori assumptions about the functional form of the Lyapunov function, is inherently interpretable since the function is obtained in a symbolic form, and, in principle, can be applied to any continuous dynamical system. We also develop a tailored branch-and-bound-and-check solution approach to efficiently solve the resulting learning task. Applications to several case studies show the ability of the proposed approach to discover Lyapunov functions.

A central objective in multimodal learning is to capture synergy: task-relevant information that arises only from the joint use of multiple modalities, and is not available from any single modality alone. While most approaches operate at the architectural level through larger or more complex fusion models, we propose a complementary axis: shaping the training objective itself. Standard training often emphasizes unimodal or redundant information, falling short on examples that require cross-modal reasoning. We formalize multimodal synergy through information theory and introduce the Synergistic Information Bottleneck (SynIB), a scalable objective that targets synergy directly. To prioritize learning synergy, SynIB motivates the model to predict accurately from all modalities while penalizing confidence when information from any modality is withheld. Alongside the standard task loss, the model runs forward passes with one modality masked at a time and is penalized for remaining confident, which would indicate reliance on unimodal cues rather than cross-modal interactions. We validate SynIB in two regimes. On synthetic XOR tasks where the ground-truth synergy is known by construction, standard training fails to recover it while SynIB does. On five real-world benchmarks, including three MultiBench affective tasks, Hateful Memes with CLIP-ViT and DeBERTa backbones, and a controllable irony extension of CREMA-D we introduce, SynIB improves accuracy on synergy-dependent examples by up to 7.8% and overall accuracy by up to 3.8%.

We present a new proof that an integer rectifiable current with finite mass, and whose boundary has also finite mass, is integral. We deduce the result from De Giorgi's structure theorem for integer-valued $BV$ functions and a cylindrical projection argument. As a consequence, we also give a new proof of the compactness of integral currents that is ultimately based on the $BV$ theory.

We initiate the study of nonparametric empirical Bayes denoising methods in the setting where both the latent variables and their measurements lie on a compact Riemannian manifold, and where the likelihood is a Riemannian Gaussian distribution. Our starting point is a novel Tweedie-Eddington formula for Riemannian Gaussian mixture models which identifies a certain surrogate oracle denoiser in terms of the marginal distribution of the measurements; it avoids the explicit computation of the posterior Fréchet mean (as required by the Bayes denoiser) via a first-order approximation, hence we refer to it as the "tangential" Bayes denoiser. We show that this surrogate oracle achieves nearly the Bayes risk in a low-noise regime, we construct a fully data-driven approximation of it using the spectral theory of the Laplace-Beltrami operator, and we establish finite-sample rates of convergence for the distance between the the surrogate oracle and its approximation. Contrasting the nearly-parametric rates from the Euclidean setting, the rates in the Riemannian setting are slower due to the singularities of the Riemannian Gaussian density at the cut locus of its Fréchet mean; in the special case of the circle we establish matching lower bounds which show that our proposed denoiser is minimax-optimal, and that the denoising problem exhibits a genuinely nonparametric rate of convergence. Lastly, we implement our methodology in two scientific applications: in astronomy, the sphere-valued problem of denoising the locations of gamma ray bursts; in structural biology, the torus-valued problem of denoising pairs of torsion angles of adjacent amino acids in a protein (i.e., the Ramachandran plot).

Prior work has shown that shifted contact derived Artin stacks admit smooth Darboux atlases. However, establishing enumerative invariants and linearizing these categorical structures requires equivariant local models. We establish an equivariant Darboux theorem for $-1$-shifted contact derived Artin stacks. We prove that, in the smooth topology, these stacks admit smooth atlases by the derived contact Darboux scheme $Δ\mathrm{loc}(s)$ associated to the derived discriminant locus of a relative section $s$. In the presence of reductive stabilizers $G$, this refines to the equivariant geometric quotient stack $[Δ\mathrm{loc}(s)/G]$. By applying the BBDJS minimal model to the derived symplectification and descending algebraically along the structural free $\mathbb{G}_m$-action, we construct an $\ell$-adic perverse sheaf on any oriented $-1$-shifted contact stack. We utilize Verdier's specialization equivalence for monodromic sheaves to equip this perverse sheaf with a tame geometric monodromy automorphism $T$. This structure allows for the extraction of derived enumerative invariants via the $\ell$-adic Grothendieck-Lefschetz trace, thereby resolving the issue of generic topological acyclicity. The content of the other main results in this paper relies on a prior work, in which we have shown that derived intersections of $n$-shifted Legendrians yield $(n-1)$-shifted contact stacks and formulated the non-linear 2-categories of Legendrians $\mathcal{F}_c(X)$ and $Leg_n$. Using this geometric setup, we formulate in this paper a contact analogue of Joyce's conjecture to linearize these structures. We then construct the categorified Legendrian 2-categories $\mathfrak{L}\mathcal{F}c(X)$ and $LLeg_0$ via $\ell$-adic Fourier-Mukai pull-push functors, connecting the study of derived contact moduli spaces to microlocal sheaf theory.

A central open question in extremal design theory is Nash-Williams' Conjecture from 1970 that every triangle-divisible graph on $n$ vertices (for $n$ large enough) with minimum degree at least $0.75 n$ has a triangle decomposition. In this paper, we prove this conjecture in full. In 2016, Barber, Kühn, Lo, and Osthus proved that if the fractional relaxation of Nash-Williams' Conjecture holds for minimum degree $cn$ for some constant $c\ge 0.75$, then Nash-Williams' Conjecture holds for any constant $c' > c$. The previously best-known bound on the fractional relaxation was due to Delcourt and Postle from 2021 with $c= \frac{7+\sqrt{21}}{14} \approx 0.82733$. This bound on the fractional relaxation has grown in importance over the years as it has been directly tied to bounds for a number of other problems in extremal design theory. This paper consists of three parts. In Part I, our first main result is a proof of the Fractional Nash-Williams' Conjecture: if $G$ is a graph on $n$ vertices with minimum degree at least $\frac{3n}{4}$, then $G$ has a fractional triangle decomposition. In Part II, our second main result is a Fractional Stability Theorem for Nash-Williams' Conjecture: if a graph $G$ on $n$ vertices has minimum degree close to $\frac{3n}{4}$ but no fractional $K_3$-decomposition, then $G$ is close (in edit distance) to the join of two $\frac{n}{4}$-regular graphs each on $\frac{n}{2}$ vertices. We use this to prove that if a triangle-divisible graph $G$ on $n$ vertices has minimum degree close to $\frac{3n}{4}$ but no $K_3$-decomposition, then $G$ is close (in edit distance) to the join of two $\frac{n}{4}$-regular graphs each on $\frac{n}{2}$ vertices. In Part III, our final main result is a proof of Nash-Williams' Conjecture in full.

In this paper, we prove that for every $k$ and every graph $H$ with $m$ edges and no isolated vertices, the Ramsey number $R(C_k,H)$ is at most $(k-1)m+1\le km$. This settles a problem of Erdős, Faudree, Rousseau and Schelp, which is listed as problem 34 in the graph theory collection.

Using the theory of complex spinorial forms, we prove that an odd-dimensional Riemannian manifold admits a pure spin-c Killing spinor with a real Killing constant $α\in\mathbb{R}^{\ast}$ if and only if it is $α$-Sasakian, thereby obtaining an extension of a well-known result by A. Moroianu that, under the purity assumption, does not require simple connectivity or completeness.

We study the pressureless Euler Alignment system with unidirectional velocity (u,0,...,0). By re-casting the system as a family of coupled scalar balance laws, one for each horizontal slice of R^d, we are able to prove existence, uniqueness, and stability within the class of unidirectional solutions, under the assumption of a bounded Lipschitz communication protocol. The nonlocal coupling between horizontal slices provides the system with a rich structure that is absent from the 1D setting and also constitutes the main technical difficulty of the present work. We construct our solutions as limits of sticky particle Cucker-Smale dynamics, discretizing first transverse to the flow and then along it. The transverse discretization depends crucially on the more subtle of our two complementary stability estimates, which relies only on L^1-L^infty control of the flux difference. This low-regularity estimate is essential since our discretized fluxes cannot in general be expected to converge in (for instance) Lipschitz seminorm. The estimate itself is inspired by work of Bouchut and Perthame and adapted here through a careful additional analysis of the nonlocality. In order to compare the dynamics along different horizontal slices, both stability estimates are most naturally formulated in terms of quantities involving the optimal coupling between the projections of two density profiles onto R^{d-1}. We also investigate the long-time behavior of unidirectional solutions. In addition to treating the standard heavy-tailed regime, we make the simple observation that the unidirectional geometry allows for flocking (with a rate independent of the number of agents) even when the communication protocol vanishes in a cylindrical neighborhood of the axis parallel to the flow. This demonstrates that direct communication along the direction of motion is not necessary for flocking to occur.

Given a family of semi-stable curves together with two degree 0 line bundles, the double double ramification cycle measures the locus where both line bundles are trivial on the fibers. When the two line bundles come equipped with natural roots, we provide a refinement of the DDR-class using the Weil pairing of the roots. We prove that the refined classes satisfy a multiple cover formula analogous to the one for correlated invariants of projective bundles on elliptic curves proved in [BC25b]. As a consequence, we prove that log-GW invariants of toric surfaces can be refined taking into account the position of the points mapped to the toric boundary, and that these refined invariants also satisfy a multiple cover formula; the latter is as a variation of the N. Takahashi conjecture for genus zero maximal contact curves for P2 relative a smooth elliptic curve E.