Papers

In 1967, Gerencser and Gyárfás determined the exact values of the two-colour Ramsey numbers of paths. In a footnote, they made the following observation: Every $2$-edge-coloured complete graph contains a Hamilton path with at most one colour change. Later, this led to a challenging and still wide open conjecture about covering edge-coloured complete graphs with monochromatic paths. Inspired by the original statement, we study paths and cycles with few colour changes in $3$-edge-coloured complete graphs. For this, we introduce a new Ramsey-type parameter: For $q,k \in \mathbb{N}$ and a graph $G$, let $R_q^k(G)$ denote the smallest $N \in \mathbb{N}$ such that every $q$-edge-coloured complete graph on $N$ vertices contains a copy of $G$ with at most $k$ vertices that are incident to edges in $G$ of different colours. For paths, we show that $R_3^1(P_n) = \frac{3n}{2} + O(1)$, and for even cycles, we show that $R_3^2(C_n) = \frac{3n}{2} + o(n)$.

In this paper we study a nonlinear Dirac equation with intrinsic Yukawa potential on compact manifolds. In dimension 2, and in dimension 2 after projection onto the positive-energy sector, we prove global well-posedness together with exponential upper bounds on the growth of higher-order Sobolev norms. The main ingredient that allows to extend the local solutions, which are constructed by standard contraction argument, is a commutator-based higher-order energy method, combined with Grönwall's lemma, which provides the necessary a priori Sobolev bounds on the solutions.

Market impact is defined as the difference between the observed price trajectory under a given execution strategy and the counterfactual trajectory that would have prevailed without it. Since this counterfactual is unobservable, estimating market impact requires simulating alternative paths under the same realized market randomness. We address this by studying the conditional simulation of point processes under perturbed intensities. Given an observed counting process whose intensity is determined by its own history, we characterize the conditional law of the latent Poisson random measure in a thinning representation. This yields an exact, event-driven algorithm that reconstructs counterfactual paths on a common randomness source, enabling rigorous pathwise market impact estimation for aggressive, passive, and mixed strategies.

The Norton inequality is one of the fundamental axioms in the theory of Majorana and axial algebras, yet its precise structural consequences have remained only partially understood. In this paper, we show that the Norton inequality alone forces the $0$- and $1$-eigenspace fusion rules for arbitrary idempotents in a commutative real algebra $A$ equipped with a Frobenius form. More precisely, if the Frobenius form is nondegenerate (as in Majorana algebras), we prove that the eigenspaces $A_0(e)$ and $A_1(e)$ of an arbitrary idempotent $e \in A$ are subalgebras and annihilate one another: \[ A_0(e)A_1(e)=\{0\}, \] while in the degenerate case the corresponding inclusions hold modulo the radical of the Frobenius form. This answers a question of T. M. Mudziiri Shumba and S. Shpectorov concerning the closure of the $0$-eigenspace $A_0(e)$.

We consider the question of convergence of particular series of integrals, which are labeled by rooted trees. Necessary and sufficient criteria for convergence are obtained, together with an explicit expression for the sum. The technique used is strongly reminiscent of the generating function approach of Galton and Watson to branching processes. The interest in these series derives from the Dyson series expansion for the perturbation of a free quantum dynamics by a local potential: the convergence of the series imlies that the perturbed dynamics exists and is unitarily equivalent with the free one.

We give a proof of Conjecture $1.3$ of \cite{strom2024maurer}.

We present a new quadratic model for the certification problem in adversarial robustness, which simultaneously accounts for all possible target classes. Building on this model, we propose a novel semidefinite programming (SDP) relaxation for incomplete verification. A key advantage of our approach is that it certifies robustness in a single optimization, avoiding the need for a separate resolution per class. This yields a significant computational speed-up and enables scalability to large datasets with many classes. To further improve efficiency, we also propose an effective pruning strategy of active neurons, thus reducing the problem dimensionality and accelerating convergence.

We give a novel proof of the fact that every coarsely separating family of subsets of the Euclidean space $\mathbb{R}^{d}$ must have asymptotic dimension at least $d-1$. The proof only uses singular homology/cohomology and standard facts from algebraic topology, such as Alexander duality. We do this by first reducing the problem to a finitary version of it. Using our approach, it follows immediately that every coarsely separating family of subsets of a $d$-dimensional Euclidean building or a product of $d$ geodesic, geodesically complete metric spaces has asymptotic dimension at least $d-1$. As a corollary, we obtain obstructions to coarse embeddings of Euclidean spaces into certain fundamental groups of graphs of groups.

This work focuses on the two-dimensional, nine-velocity (D2Q9) lattice Boltzmann model. First, we show that the D2Q9 scheme cannot achieve secondorder accuracy unless the cubic velocity terms are neglected, and we explain how some of these parasitic terms can be eliminated. Second, we demonstrate that the standard choice of the equilibrium distribution has no effect on the equivalent PDE at second order. Finally, we numerically investigate the effect of these cubic terms and study different choices of equilibrium distributions using a new benchmark called the Oblique Dipole Benchmark, which describes obliquely propagating 2D vortex dipoles with periodic boundary conditions.

We propose, and analyse, a spatially discrete evolving surface finite element method for the approximation of the enthalpy formulation of the two-phase Stefan problem posed on an evolving surface. Our approach does not rely on mass-lumping and discrete maximum principles. We prove this numerical method is numerically stable, and prove $\mathcal{O}(\sqrt{h})$ error bounds for the temperature in the $L^2_{L^2}$ norm under minimal regularity assumptions by introducing a new projection-type operator. We complement our analysis with discussion on the implementation of this numerical method, where we propose a novel implementation that avoids errors due to numerical quadrature and which has not previously been considered in the literature even in the stationary, flat setting. We also include numerical experiments and experimental order of convergence demonstrations.

World models – compressed latent representations of an environment that support action-conditioned prediction and planning – are typically presented as a product of modern self-supervised learning. This paper argues that the functional anatomy of a world model was independently developed, deployed, and formally analyzed decades earlier in the model-order-reduction (MOR) and control literature, under different names and for a different purpose: the real-time operation of physical systems. We trace the anatomy across three communities. Low-dimensional models of turbulence built on proper orthogonal decomposition (POD) supplied latent dynamics learned from data of a chaotic environment; eigenface methods in early computer vision supplied the encoder-decoder half, including a primitive runtime validity check; and measurement-based POD frameworks for facility thermal control assembled the complete loop – POD coefficients as latent state, parametric dependence on actuator setpoints as action conditioning, modal reconstruction as decoding, and, critically, a priori analytical error bounds as a verification layer that certified when the model's predictions could be trusted in closed loop. We then examine what each tradition possesses that the other lacks: MOR contributes verification, physical grounding, and extreme data efficiency; learned world models contribute nonlinear representation, transferability, and horizon. We argue that the outstanding obstacle to deploying world models in systems that cannot fail – power, thermal, process control – is not predictive fidelity but verifiability, and we outline a research agenda for physics-grounded, verifiable world models that unifies the two lineages.

Let $(T,\Sigma,\mu)$ be a complete, $\sigma$-finite measure space, let $Y,Z$ be a separable Banach spaces, and let $S : T \rightrightarrows Z$ be a measurable multifunction with nonempty closed values. We study the relation between the Clarke tangent cone to the decomposable set $\mathsf{Sel}_p(S):=\{x \in L^p(T,Z) : x(t) \in S(t) \hspace{0.1cm} \text{a.e.}\}$ and the $L^p$-selections of the pointwise Clarke tangent cones. We prove that $$ \widehat{T}_{\mathsf{Sel}_p(S)}(x) = \{v \in L^p(T,Z) : v(t) \in \widehat{T}_{S(t)}(x(t)) \hspace{0.1cm} \text{a.e.} \} \hspace{0.2cm} \text{for any $p \in [1, \infty)$}, $$ More generally, we consider this problem in $L^p(T,Y) \times L^r(T,Z), p,r \in [1,\infty),$ and denote $S : T \rightrightarrows Y \times Z$ and $\mathsf{Sel}_{p,r}(S):=\{(x,y) \in L^p(T,Y) \times L^r(T,Z) : (x(t),y(t)) \in S(t) \hspace{0.1cm} \text{a.e.}\},$ The global-to-pointwise inclusion is shown to hold for all exponents unconditionally, whereas the reverse inclusion is obtained from an equi-integrable correction condition. Consequently, if $p,r$ are finite it holds $$\widehat{T}_{\mathsf{Sel}_{p,r}(S)}(x,y)=\mathsf{Sel}_{p,r}\left\{t \mapsto \widehat{T}_{S(t)}(x(t),y(t))\right\} \hspace{0.2cm} \text{for any $(x,y) \in \mathsf{Sel}_{p,r}(S).$}$$ Whereas, if $(p,r)=(1,\infty)$ the equality holds if $\mathsf{Sel}_{\infty,1}(S)$ satisfies a variational correction condition (V). Condition shown to hold whenever $S(t):=Gr F(t,\cdot),$ where $F : T \times Y \rightrightarrows Z$ is measurable in $t$ and $F(t,y) \subset F(t,x)+ \ell(t) ||y-x|| \overline{\textbf{B}}_{Z}.$ Applications are given to nonsmooth optimization with pointwise constraints, graphs of Nemytskii operators, and a multiplier rule for nonconvex integral programs is derived.

The classical Kolmogorov–Arnold representation theorem states that any continuous multivariate function can be exactly decomposed into a finite composition of univariate continuous functions and addition operations. This foundational result has recently inspired the development of Kolmogorov–Arnold Networks (KANs) in classical machine learning, as well as their extensions into the quantum domain (QKANs). In this paper, we establish two quantum analogues of the Kolmogorov–Arnold representation theorem for continuous unitary-valued maps of several variables within an open $1$-neighbourhood of the identity matrix \(O_1(\mathbf{I}) \subset \mathcal{U}(n)\). First, we prove a representation theorem that yields an exact additive decomposition inside the matrix exponent of anti-Hermitian-valued maps. Second, due to the non-commutative nature of quantum operators, we derive a factorised version expressing the target unitary map as a finite sequential product of univariate matrix exponentials. Finally, we provide a concrete topological counterexample based on the lifting property of \(\mathcal{SU}(2)\) to demonstrate that these local representation theorems cannot be globally extended to the entire unitary group \(\mathcal{U}(n)\) without encountering fundamental structural obstructions.

Random walks are commonly used to model movement throughout the sciences, from the motion of particles and molecules to the observed behaviour of animals and crowds. The correlated random walk, which assumes a level of persistence between movement directions, has become ubiquitous in the analysis and modelling of movement in recent times. Whilst many properties of the correlated random walk are known, there are still many which are not fully understood and, therefore, under utilised in movement data analysis. Here we consider the effect that sub-sampling has on the descriptive distributions of correlated random walks. Our work demonstrates the connection between the distributions of turning angles and step-lengths that characterise a correlated random walk, along with the resulting distributions found after sub-sampling. We provide examples for where this approach could aid in movement analysis as well as determining future ways the work could be extended.

Over the complex field, we show that the principally polarised abelian surfaces with a given refined Humbert invariant of rank 3 are all Galois conjugated.

This paper addresses the inverse optimal control (IOC) problem for stochastic linear systems subject to both Brownian motion and Poisson jumps, using an inverse reinforcement learning (IRL) framework. Given a target feedback gain from an expert, the objective is to identify an equivalent cost functional-specifically, the set of all cost weights-that yields this same gain. To solve this problem when system dynamics are unknown, we propose two model-free, off-policy IRL algorithms that operate entirely from data, circumventing the need to solve the generalized algebraic Riccati equation or compute the cost weights analytically. The first is an inverse Q-learning algorithm that constructs data-driven equations from expert demonstrations to compute the Q-function matrix, with equivalent cost weights updated algebraically and without requiring additional trajectory data. The second is a model-free off-policy inverse policy iteration algorithm that leverages data collected under an initial stabilizing policy, offering a complementary approach suited to different data availability scenarios. Crucially, by decoupling the data-collection behavior policies from the policies being iteratively updated, both algorithms can learn equivalent cost weights from sufficiently excited trajectories without identifying the system dynamics or jump intensity. Numerical simulations validate the effectiveness of the proposed methods.

Entropy regularization is widely used in continuous-time reinforcement learning (RL) to reduce sensitivity to environmental perturbations, yet its robustness benefits lack a rigorous theoretical foundation. This paper establishes the first robustness guarantees for entropy-regularized continuous-time Markov decision processes. We show that maximizing an entropy-regularized objective yields a lower bound on a worst-case robust RL problem with joint reward and transition perturbations. We analytically characterize the induced robust sets and prove that they expand monotonically with the regularization strength, justifying the empirical observation that stronger entropy improves robustness. In contrast to prior discrete-time analyses, our results remove the intractable state-distribution entropy term and provide guarantees invariant to action frequency. Experiments on queueing network control and market making confirm our theory, showing that entropy-regularized policies outperform greedy and $\epsilon$-greedy baselines under dynamics perturbations.

We introduce a polyhedral realizability problem for complete bipartite preference structures, the two-sided strict preference profiles that underlie the stable marriage problem. Given two parts M and W of size k, a strong polyhedral realization asks for a bipartite polyhedral graph with bipartition classes of size k in which every vertex is adjacent to all but one vertex on the opposite side. This condition attempts to represent complete opposite-side preference capacity by direct polyhedral adjacency, with one geometrically exceptional opposite-side vertex for each agent. We prove an Euler-type criticality theorem: such a strong polyhedral realization exists if and only if k=4, and in that case the underlying graph is the cube graph Q_3, equivalently K_{4,4} minus a perfect matching. The proof is the collision of the strong edge requirement E=k(k-1) with the bipartite planar bound E<=4k-4 and the minimum-degree constraint for polyhedral graphs. We then distinguish weak realizations. A family of (k-1)-gonal trapezohedra gives balanced bipartite polyhedral graphs with 2k vertices and 4k-4 edges for all k>=4, showing that weak maximal realizability persists beyond the cubic critical case. However, the Euler interval 3k<=E<=4k-4 is not fully realizable: we give a direct proof that no bipartite polyhedral graph exists with V=10 and E=15, equivalently that there is no even triangulation of the sphere on seven vertices. Finally, in the cube case, we show that cube-distance compatibility of a size-four preference profile is equivalent to the existence of a perfect matching of mutually last-ranked pairs. This gives a first preference-theoretic manifestation of the cubic criticality.

For each integer $K\geq2$ when $n\geq4$, and for $K=2,3,4$ when $n=3$, we construct an almost-calibrated Lagrangian mean curvature flow $L_K(t)$ in $\mathbb{C}^{n}$, starting from initial data arbitrarily close to being special Lagrangian, which develops a finite-time Type II singularity at time $T$ with the explicit curvature blow up rate \[ \sup_{L_{K}(t)} |\mathbf{A}_{L_{K}(t)}| \sim (T-t)^{-K/2} \qquad \text{as } t\nearrow T . \] The tangent flow at the singularity is a transverse pair of cohomogeneity-one special Lagrangian cones, while the Type II blow-up limit is a smooth cohomogeneity-one special Lagrangian desingularization. This gives a quantitative construction of Type II blow-up for a fully nonlinear parabolic PDE arising from cohomogeneity-one Lagrangian mean curvature flow. Our construction is based on a modulation analysis around a shrinking family of cohomogeneity-one special Lagrangian desingularizations, using the perturbative spectral theory developed in the companion paper.

The informativeness of a training set is as consequential as its size, yet most sampling strategies remain agnostic to the intrinsic geometry of the data distribution. We introduce CuBAS (Curvature-Based Adaptive Sampling), an information-geometric framework for adaptive data selection in supervised classification, grounded in the q-state Potts Markov random field (MRF) model. The central insight is that a labeled dataset can be viewed as a statistical manifold, on which local curvature, estimated via the ratio of second to first-order observed Fisher information, faithfully encodes the geometric complexity of the data distribution. We construct a k-nearest-neighbor graph over the labeled data and derive a closed-form curvature score at each vertex from the Potts sufficient statistics. This curvature signal partitions the graph into two complementary regimes: low-curvature regions, corresponding to smooth, homogeneous clusters, and high-curvature regions, concentrated around decision boundaries that are disproportionately informative for classification. By selecting nodes from both regimes, CuBAS constructs compact yet maximally informative training subsets. Empirical evaluation across more than 60 benchmark datasets demonstrates consistent and statistically significant improvements over random sampling and uncertainty-based baselines, across a wide range of labeling budgets and classifier architectures. CuBAS is computationally efficient (linear in the number of k-NN graph edges), theoretically grounded in the differential geometry of statistical manifolds, and interpretable in terms of the local shape operator of the data manifold.

We deal with a Kirchhoff problem on a smooth bounded domain of $\mathbb{R}^4$ with competing critical and concave terms. By using new approximation techniques and the Nehari manifold analysis, we derive several existence results, complementing earlier ones obtained in the critical-convex case. Compared to similar results in the literature, we provide explicit bounds of the range of the parameters leading to the existence of solutions.

We study the homogenization, in the critical scaling regime, of a boundary value problem with a nonlinear dynamic Signorini-type condition posed on a rapidly oscillating portion of the boundary. The source term is time-periodic and we look for time-periodic solutions. Using the method of oscillating test functions (Tartar), compactness, and monotonicity arguments, we identify the homogenized problem and the effective nonlinear boundary operator. In contrast with the evolutionary (initial value) setting, the periodic framework eliminates memory effects and yields an instantaneous time-periodic operator defined through a periodic-in-time cell problem.

The two-boost problem in space mission design asks whether two points of phase space can be connected with the help of two boosts of given energy. In this article we provide a positive answer for the rotating Kepler problem by generalizing the definition of Lagrangian Rabinowitz Floer homology from the class introduced in [4] to a strictly broader class, and computing the corresponding homology in this more general setting. The principal technical challenge is the non-compactness of the energy hypersurface both near the collision singularity and in the infinity.

We construct the Jiang-Su algebra Z as an inductive limit of dimension drop algebras, describing the latter as entangled matrix cones to explicitly define the connecting *-homomorphisms. This construction gives rise to a C*-diagonal in Z with one-dimensional spectrum which is not locally connected. Along the way, we give a new characterisation of normalisers in Cartan pairs in terms of state excision.

We study preimage regions on the open probability simplex associated with symmetric separable functionally generated maps. The problem is a finite-dimensional geometric question about convexity and barycentric star-shapedness of these regions. In the portfolio interpretation, the regions consist of the points whose generated portfolio has no negative coordinate. For symmetric separable generators, the defining first-order inequalities split into a coordinate term and a symmetric aggregation term. This coordinate–aggregation decomposition is the main organizing device of the paper. We show that the aggregation term may destroy convexity, and may even destroy barycentric star-shapedness. In particular, moving closer to the barycenter need not preserve the long-only property. We then give a necessary and sufficient threshold criterion for barycentric star-shapedness and derive sufficient conditions that recover it. These conditions are expressed in terms of concavity and second-derivative domination for the aggregation function. The entropy case is the affine aggregation case, in which the long-only constraints reduce to coordinate thresholds.

We establish the equivalence between weak and viscosity solutions for a broad class of nonlocal $p$-Laplace type equations on the Heisenberg group whose kernels satisfy standard symmetry, ellipticity, and left-translation invariance assumptions. As a particular case, our results apply to the fractional Heisenberg $p$-Laplacian. The proof combines intrinsic approximation by modified infimal convolutions, Heisenberg mollification, and a nonlocal integration-by-parts argument adapted to the sub-Riemannian geometry. The main analytical difficulty stems from the noncommutative structure of the Heisenberg group, which prevents a direct extension of the Euclidean theory and requires new localization and approximation techniques.

We evaluate exactly the radial metric factor $h(\zeta)$ generated by Schwarzian averaging in the AdS$_2$ throat of an extremal Reissner–Nordström AdS$_5$ black brane. The result is a Gaussian integral against $\coth(\pi y)$, valid at all radial depths, which Mordell's identity turns into an exact Appell–Lerch $q/q^\ast$-series representation. The dual series identifies the nonperturbative scale $e^{-\pi^2 C/\zeta}$ missed by any finite near-boundary truncation. The third parametric derivative required by the evaluation generates the quasimodular Eisenstein series $E_2$, absent from the classical Mordell identity. From the integral representation we prove that $\mathcal{G}_0(\zeta):=h(\zeta)/\zeta^2$ is completely monotone and hence has no interior minimum, so any confining minimum produced by a finite near-boundary truncation is an artifact. We also compute the exact relative variance of the Schwarzian kernel, which makes the averaged-metric approximation error quantitative and shows that the absence of a confining minimum is robust across moment-based effective geometries. Applied to the temporal rectangular Wilson loop, the exact throat gives algebraic screening, $E(L)\sim -\kappa_{\rm IR}/L^2$, the Wilson-loop diagnostic of the semi-local quantum-liquid IR of the extremal RN brane. A numerical check in a simple matched geometry confirms that the screened saddle is the dominant string configuration, and an exact-versus-truncated force comparison shows that the apparent constant-force regime of the fourth-order truncation is not a feature of the exact geometry.

We prove a multivariate Lam–Postnikov–Pylyavskyy type inequality for hybrid Grothendieck polynomials, unifying and refining results for stable and dual stable Grothendieck polynomials established by Chan–Chen–Pak–Soskin. We also conjecture extensions of the Lam–Postnikov–Pylyavskyy inequality and a conjecture by Thomas–Yong to the (equivariant) Schubert and Grothendieck polynomial setting.

We introduce the beta-Gevrey lp-rapid decay property (GRD){beta,p}, for 0 < beta <= 1 and 1 <= p < infinity, for countable discrete groups. This property is a subexponential analogue of classical rapid decay, in which polynomial control is replaced by logarithmic subexponential control of order o(R^beta). We establish basic results for (GRD){beta,p}. We then apply this framework to compact quantum metric structures on reduced Lp-group algebras. We introduce strongly dense-core beta-Gevrey regular lp-spectral triples and give two classes of examples. For countable discrete groups satisfying (GRD)_{beta,p}, we prove, using Rieffel's criterion, that the corresponding Gevrey seminorms induce metrics on the Banach-algebra state space which metrize the weak-* topology. This yields compact quantum metric space structures in settings beyond classical rapid decay, including groups of intermediate growth such as the first Grigorchuk group.

The Hamming distance between two equal length words $\alpha, \beta$ is the number of positions where $\alpha$ and $\beta$ differ. For $x, y \in \Sigma^*$ and antimorphic involution $\theta$, $x$ $\theta$-commutes with $y$, if the Hamming distance between $xy$ and $\theta(y)x$ is zero. When the Hamming distance between $xy$ and $\theta(y)x$ attains its minimum non-zero value of one, then we say $x$ nearly $\theta$-commutes with $y$. This manuscript investigates properties of $x$ and $y$ such that the Hamming distance between $xy$ and $\theta(y)x$ is one. We provide a complete characterization of such $x$ and $y$. We introduce a binary relation $R_\theta$ on $\Sigma^*$, where $x R_\theta y$ holds if and only if $x$ nearly $\theta$-commutes with $y$. Finally, for a given $y$, we collect all $x$ such that $x$ nearly $\theta$-commutes with $y$, and discuss various properties of this set.