A temporal graph is a graph in which every edge carries a non-empty set of time labels, and it is temporally connected if for every two vertices $u$ and $v$, there exists a $u$-$v$-path with non-decreasing time labels. A spanner is a subset of its edges preserving temporal connectivity. Unlike static graphs, temporally connected graphs need not admit sparse spanners; nonetheless, minimizing spanner size is a central and widely studied problem. A particularly intriguing question is whether temporal cliques admit spanners of linear size. Despite considerable effort over the past years, the best known upper bound remained $O(n \log n)$. We finally resolve this question, proving that every temporal clique on $n$ vertices admits a spanner of size $7n$. Moreover, such a spanner can be computed in polynomial time.

Formulating stable numerical methods for hyperbolic systems in curvilinear coordinate with singularities, e.g. spherical coordinates, is complicated by the presence of these singularities. We present a method for constructing high-order accurate, energy-stable finite difference operators satisfying the Summation-by-Parts (SBP) property on spherical domains, extending ideas presented by [C. Gundlach, J. M. Martín-García, and D. Garfinkle, CQG 30, 145003 (2013)]. We define discrete gradient and divergence operators that mirror the continuous integration-by-parts principle, even though there is a $1/r^p$ coordinate singularity present at the origin. We explicitly construct such operators up to order six. Our operators place a grid point directly on the origin. We also review how to construct stable SBP operators that straddle the origin. We analyze the accuracy and spectral radii of these operators, and we show example evolutions of the scalar wave equation to demonstrate the advantages of such operators.

We study the McKean--Vlasov free energy on the unit sphere associated with the unnormalized self-attention (USA) model for noisy transformer dynamics. We prove a sharp global-minimizer dichotomy in every dimension $d\ge2$. There is a unique $β_*^{(d)}>0$ such that \begin{equation*} \frac{I_{d/2+1}(β_*^{(d)})}{I_{d/2}(β_*^{(d)})}=\frac1d, \end{equation*} where $I_ν$ is the modified Bessel function of the first kind. For $0<β\le β_*^{(d)}$, the uniform density remains the unique global minimizer up to the linear-stability threshold \begin{equation*} K_\#^{(d)}(β)=\frac{β^{d/2}}{2^{d/2}Γ(d/2)I_{d/2}(β)}, \end{equation*} and the phase transition is continuous. For $β>β_*^{(d)}$, the uniform density is not globally minimizing at $K_\#^{(d)}(β)$, so the critical coupling satisfies $K_c<K_\#^{(d)}(β)$ and the transition is discontinuous. This result generalizes the authors' recent $d=2$ work arXiv:2604.16288 to arbitrary dimension. The proof uses the sharp Beckner--Onofri/logarithmic Hardy-Littlewood-Sobolev (HLS) inequality on the sphere, together with a Funk--Hecke/Bessel coefficient computation and a degree-two quartic obstruction.

We construct a stacky resolution of singularities for certain noncommutative spaces which can be viewed as `finite noncommutative extensions' of schemes. More precisely, we show that any order over a reduced separated finite type scheme over a field of characteristic zero can be resolved by an Azumaya algebra over a smooth Deligne--Mumford stack by a sequence of stacky blow-ups.

Koopman theory turns nonlinear dynamics into a linear spectral problem. In computation, however, everything depends on a hard finite-dimensional choice: the observables must be expressive, nearly invariant under the dynamics, and, ideally, compatible with composition. Deep Koopman methods learn flexible coordinates, whereas structure-preserving methods enforce operator identities on fixed dictionaries. We combine these ideas by introducing Deep Embedded Multiplicative Dynamic Mode Decomposition (DeepMDMD), a method that learns a latent space and a partition of it, while enforcing the Koopman product rule as an exact algebraic constraint. Training alternates between an exact multiplicative operator update and a differentiable latent-clustering step that promotes Koopman closure. The result is a finite transition map on learned latent cells. Its nonzero spectrum lies on the unit circle, its dictionary is shaped by the dynamics rather than by ambient geometry, and forecasts are made in latent coordinates before being decoded to physical space. Across Hamiltonian, chaotic, and fluid examples, DeepMDMD learns dictionaries that are far more compact and dynamically coherent than those produced by geometric MDMD partitions. It reduces spectral pollution, reveals richer continuous-spectrum structure, and gives stable forecasts under severe noise. In high-dimensional flows, including a 158,624-dimensional cylinder wake and a noisy $Re=20,000$ lid-driven cavity, it preserves coherent structures and long-time spectral statistics where state-space MDMD fails. These results suggest a practical rule for Koopman learning: learn the coordinates, constrain the algebra.

In this paper we apply Hirota's bilinear method to the scalar and coupled Ablowitz-Ladik systems in the defocusing dispersion regime under the assumption of a background amplitude $0<ρ<1$. We first establish, in the scalar case, the correspondence between the Hirota's parameters and the spectral parameters of the inverse scattering transform. Then we show that when the Hirota parameter associated to the discrete eigenvalue is chosen outside the range corresponding to a discrete dark soliton, novel solutions of the Ablowitz-Ladik system emerge. In general, these solutions are singular, but there exists a class of time-periodic solutions for which it is possible to choose the soliton parameters so that the breathers remain regular on the lattice for all times. We also discuss the interactions between a dark soliton and a regular breather, and between two regular breathers. For the coupled Ablowitz-Ladik system, by including in the background discrete, counter-propagating plane waves, we use Hirota's method to derive novel Akhmediev-type (i.e., space-periodic) discrete breathers which are regular for all times. Finally, taking the limit of the discrete Akhmediev breathers as the period approaches infinity (i.e., as the wavenumber approaches zero) we obtain novel rogue wave solutions of the coupled Ablowitz-Ladik system.

This paper is motivated by a broad question about AI-assisted mathematics: can an AI system help discover and certify an explicit bijection between two infinite sequences of complicated combinatorial sets already known to be equinumerous? The challenge is to find a reversible structure explaining that equality uniformly across the sequence. We give an affirmative test case in the setting of a partition problem. Andrews and Dhar introduced two partition families $\mathcal{C}_3(n)$ and $\mathcal{D}_3(n)$, and for "nonexceptional'' $n$, they asked for a bijective proof of their equality \[ |\mathcal{C}_3(n)|=\frac{|\mathcal{D}_3(n)|}{3}. \] We prove a residue-class equidistribution theorehm for $\mathcal{D}_3(n)$ that identifies a "canonical third'' subset $\mathcal{D}_3^{(0)}(n)\subseteq \mathcal{D}_3(n)$. Answering their question, we construct a bijection \[ ι_n:\mathcal{C}_3(n)\longrightarrow \mathcal{D}_3^{(0)}(n) \] as a highly structured composition of four maps. AxiomProver autonomously produced and Lean-verified the equidistribution theorem. The bijection was found through human--AxiomProver collaboration, and the theorem was autoformalized and verified by the system.

Tight bounds on quantum sample complexity and quantum query complexity have been known for various computational problems in the literature, whereas tight bounds on quantum time complexity (i.e., the size of quantum circuits) remain unresolved. In this paper, we provide a framework to establish lower bounds on the quantum time complexity for testing permutation-invariant properties of quantum states, via a reduction from quantum sample complexity. As an application, we obtain a series of matching lower bounds when given sample access to the input quantum states, including: 1. The SWAP test due to Buhrman, Cleve, Watrous, and de Wolf (Phys. Rev. Lett. 2001) is time-optimal to estimate the purity $\operatorname{tr}(ρ^2)$ and the inner product $\operatorname{tr}(ρσ)$. 2. The Shift test due to Ekert, Alves, Oi, Horodecki, Horodecki, and Kwek (Phys. Rev. Lett. 2002) is time-optimal to estimate the high-order functionals $\operatorname{tr}(ρ^k)$. 3. The productness tester for multipartite pure states due to Harrow and Montanaro (J. ACM 2013) is time-optimal. 4. The LMR protocol due to Lloyd, Mohseni, and Rebentrost (Nat. Phys. 2014) is time-optimal to implement the reflection operator about a pure state. 5. The samplizer due to Wang and Zhang (IEEE Trans. Inf. Theory 2025) is time-optimal for pure states. 6. The estimator for pure-state trace distance and fidelity due to Wang and Zhang (ICALP 2026) is time-optimal. To the best of our knowledge, this is the first method that allows us to systematically establish tight lower bounds on quantum time complexity.

We study a Lamé-Stokes coupled system arising as the incompressible-inclusion limit of periodic high-contrast elastic composites. The elastic matrix satisfies the Lamé equations, the fluid inclusions satisfy the Stokes equations with a local incompressibility constraint, and the two phases are coupled by continuity of displacement and traction across the interface. The main difficulty is that the incompressibility constraint is imposed only in the inclusions, while the interface structure prevents a direct application of standard elliptic homogenization theory. We develop a quantitative homogenization theory for this model. First, using the Babuška-Brezzi theory, we prove uniform well-posedness of the mixed variational problem, including inf-sup stability and a priori estimates independent of the microscale parameter. Second, by combining formal asymptotic expansions with two-scale convergence, we derive the homogenized effective elasticity equation and prove weak $H^1$ convergence of the microscopic displacement. The effective tensor is characterized by cell problems and is symmetric and strongly elliptic on symmetric matrices. Third, under smoothness assumptions on the interface, we establish piecewise higher-order Sobolev regularity and $L^\infty$ gradient bounds for the cell correctors. These estimates yield $O(\sqrt{\varepsilon})$ convergence rates for the displacement in $H^1$ and for the pressure in $L^2$ over the fluid region.

Photon loss in optical channels fundamentally limits long-range reliable quantum communication. A standard approach to overcoming this limitation is the use of quantum repeater nodes, which typically perform experimentally demanding non-Gaussian operations. However, whether Gaussian repeater protocols can enhance quantum communication rates over bosonic attenuation channels has remained open. In this work, we prove a no-go theorem for Gaussian quantum repeaters in a quantum network. Specifically, we show that any repeater chain composed of Gaussian operations, homodyne measurements, and arbitrary classical communication cannot enhance the quantum capacity of a pure-loss attenuation channel beyond that achievable by direct transmission. Our proof introduces a generalisation of $k$-extendibility to a notion of fractional extendibility for Gaussian states and establishes some of its useful properties, thereby providing a powerful framework for analysing Gaussian quantum networks.

We establish Diophantine approximation results for real symmetric matrices by collections of linearly independent integer vectors. For $X \in \mathrm{Sym}_d(\mathbb{R})$, we prove a Dirichlet-type theorem guaranteeing the existence of integral Lagrangian frames $(Q, P) \in \mathrm{Mat}_{d \times 2d}(\mathbb{Z})$ that satisfy $\lVert QX + P \rVert_{\mathrm{op}} \leq c_d/N$ and $\lVert Q \rVert_{\mathrm{op}} \leq N$ for any $N \geq 1$. Furthermore, we establish a Khintchine-type zero-one law, demonstrating that the size of the set of $ψ$-approximable symmetric matrices is determined by the convergence or divergence of the series $\sum_{q \geq 1} q^{ς- 1}ψ(q)^ς$, where $ς= d(d+1)/2$. The proofs rely on the reduction theory of the Siegel upper half-space, dynamical formulation over the space of symplectic lattices, and an analysis of the Siegel transform adapted to count Lagrangian frames instead of single lattice points.

For every $d \geq 3$, we construct infinitely many quasi-isometry classes of hyperbolic groups $G$ of cohomological dimension $d$ that algebraically fibre with finitely presented kernel. All our groups arise as finite-index subgroups of right-angled Coxeter groups. In many cases, the $L^2$-Betti numbers of the groups $G$ provide obstructions to higher finiteness properties of the kernel. Our groups therefore expand the list of subgroups of hyperbolic groups with exotic finiteness properties.

In this article, we prove a rigidity result for quasi-isometric embeddings from a random subset $D$ of the product $\mathbb{X}$ of two regular trees into $\mathbb{X}$ itself. This can be seen as an extension of Eskin's quasi-isometric rigidity of higher-rank nonuniform lattices to random subsets. As a consequence, we give a description of the self-quasi-isometric embeddings of a random sample. We also show that two independent samples are almost surely non-quasi-isometric, confirming that such a phenomenon occurs in the higher-rank setting, as suggested by Abért. This result contrasts with the result on quasi-isometric equivalence between random sequences by Basu and Sly.

Convex duality relations are a useful tool for deriving error estimates for challenging nonlinear and non-smooth variational problems. Applied at the continuous level they can deliver nonlinear analogues of the Prager-Synge a posteriori error identity, while at the discrete level they allow the derivation of minimal regularity a priori estimates. By leveraging elementary properties of Bregman divergences, we obtain three results on the error control via convex duality for a general class of problems: first, we prove a local efficiency bound for the duality gap error estimator, secondly, we derive a guaranteed a posteriori bound for non-conforming fields, and finally, we prove a minimal-regularity quasioptimal estimate for a Crouzeix-Raviart discretisation of the $φ$-Laplace problem.

This paper studies the problem of detecting a change in the distribution of a sequence of independent observations when the post-change distribution is unknown. We propose a novel change detection algorithm, termed Predictive-Mixture CuSum (PM-CuSum), which combines predictive distributions constructed from sliding windows of different lengths within a CuSum recursion. The predictive distributions are aggregated using adaptive weights based on their recent predictive performance. We show that PM-CuSum achieves first-order asymptotic optimality under mild conditions, and that its asymptotic delay bound has a smaller remainder order than what is achieved by any fixed (even oracle) window. Numerical simulations demonstrate that PM-CuSum performs well compared to existing methods. Moreover, it is demonstrated that forming likelihood ratios using full predictive distributions can substantially improve performance compared to plug-in likelihoods.

The weak Harnack inequality is a fundamental result in the theory of fully nonlinear parabolic partial differential equations. It has several important consequences such as the Hölder continuity of derivatives of solutions to certain fully nonlinear parabolic equations. Classical proofs of the weak Harnack inequality rely on decay of measure estimates, which require delicate localization and covering arguments. We give a new proof based on a detailed study of particular envelopes of solutions to parabolic partial differential equations. This allows us to completely bypass the need of covering arguments. We also prove a $W^{2,ε}$ estimate with an improved dependence of $ε$ on the ellipticity ratio compared to previous results.

We investigate the processes of local ordering for the partisan voter model on complex networks. In this model agents hold a binary opinion and a fixed preference that biases updates toward alignment with their preferred state. We first study the dynamics on uncorrelated random networks and derive a pair approximation that resolves the densities of links between different classes of agents. The analytical predictions are in excellent agreement with Monte Carlo simulations. In this setting, partisan bias leaves the total stationary density of active links, i.e. links connecting nodes in different states, unchanged and at the same value as in the standard voter model, but redistributes it among different categories of links. We then consider preference-dependent networks with homophilic and heterophilic attachment to analyze the competition between the global bias mechanism and the local effect of preference-based connectivity. In this case, structural correlations qualitatively modify the stationary state. We identify different regimes of local ordering in the space of parameters measuring the strength of the preference and the strength of the homophilic attachment. Our work clarifies the distinct roles of dynamical partisan bias and structural assortativity, and provides an analytical framework to study partisan opinion dynamics beyond mean-field theory.

We develop a regular CW analogue of the classical assembly formalism for chain complexes appearing in algebraic surgery theory. From the cell poset, we construct combinatorial path and loop objects using fences of comparable cells and prove that their classifying spaces recover the homotopy types of the ordinary based path and loop spaces. The resulting loop object carries a natural monoid structure, giving rise to a DG algebra defined directly from the cellular structure. For complexes of cellular cosheaves, we introduce a universal assembly functor to modules over the group ring of the fundamental group and study the localization determined by global equivalences. The associated homotopy category is identified with a Verdier quotient of the derived category of cellular cosheaves, and its fibrant objects are precisely the homotopy locally constant complexes. A single elementary cosheaf becomes a compact generator after localization, and its derived endomorphism DG algebra is identified with singular chains on the cellular loop space. Consequently, the localized theory admits a Morita description in terms of DG modules over the loop DG algebra. The formalism provides a regular CW counterpart of the classical delta-set approach to assembly in algebraic surgery theory due to Ranicki and Weiss.

We present in this paper a framework for the identification of control-affine reduced-order models (ROMs). The proposed method utilizes autoencoders (AEs) to transform the high-dimensional states, and potentially the high-dimensional inputs, into reduced latent ones suitable for control-affine state-space dynamics. This is achieved by simultaneous training of the AE and the state-space model. In addition, we extend the discrete ROM formulation to a sequence-based model, which processes state and input histories to improve prediction accuracy while preserving the control-affine structure. We motivate our framework by applying feedback linearization to the derived models, and we present guidelines for its efficient use. The proposed framework is assessed on two numerical examples and its performance is compared to a baseline model, where the AE identifies a latent space with linear state-space dynamics. The assessment involves evaluating the prediction accuracy of the ROM on test data and its effectiveness in controlling the system to a desired state or trajectory.

Given a discrete-time, transient Markov chain, we establish explicit total variation error bounds in the approximation of the number of visits to its starting state within the first $n$ time steps by a geometric distribution. Our error bounds are expressed in terms of the heat kernel of the Markov chain. As applications, we consider Galton--Watson processes with a geometric offspring distribution, and random walks on the one- and three-dimensional integer lattices, regular trees, supercritical percolation clusters, and groups with polynomial growth.

This paper proposes a novel approach for the synchronization problem of nonlinear dynamical systems, integrating dual Lyapunov stability analysis with polynomial optimization. A comprehensive review of the relevant scientific literature on synchronization methods is conducted, with a particular focus on classical Lyapunov-based methods for chaotic systems. In this study, the Rössler system is synchronized by employing dual Lyapunov-based closed-loop synchronization method. This method uses semidefinite programming and sum-of-squares polynomials to compute a nonlinear state feedback function which synchronize a chaotic system to a selected reference model. It is aimed that chaotic behavior is destroyed and, instead, a limit cycle becomes attracting. Simulation works are performed for randomly selected 100 different initial conditions to show that synchronization process is successfully performed. Furthermore, bifurcation diagrams and phase portraits are evaluated to analyze the system dynamics. The paper discusses results and how new constraints should be employed and adapted to more complex systems.

A set in $\mathbb{R}^n$ is median porous if the logarithm of its distance function has bounded mean oscillation. We show that this property is preserved under quasiconformal mappings. In particular, median porosity is quasiconformally invariant. We also show that the stronger notion of weak porosity, by contrast, is not quasiconformally invariant.

We establish a single-letter and efficiently computable strong converse bound on the classical identification capacity of quantum channels. By equipping the $n$-fold channel output space with a product state-weighted $σ$-Euclidean geometry, we allow trace-distance separation constraints for identification codes to be controlled by Euclidean covering estimates. Using Sudakov's inequality, we bound the covering numbers of the $n$-fold channel outputs via their Gaussian mean widths in the weighted geometry, whose exponential growth in $n$ is governed by the operator norm of a single-letter positive operator. Upon optimizing over all weighing states $σ$, this yields a strong converse bound on the identification capacity of the channel, which also admits a semidefinite representation. Our method improves the best known converse bounds on the identification capacity of several important examples, such as depolarizing, Pauli, erasure, and amplitude damping channels. We also discuss extensions of this method to more general Euclidean geometries on the output space.

Let $F$ be a field of exponential characteristic $e$. We compute the slices of $\mathbf{MSL}[e^{-1}]$, where $\mathbf{MSL}$ is the special linear algebraic cobordism spectrum defined by Panin and Walter. The answer is expressed in terms of the second page of the Adams-Novikov spectral sequence for the special unitary cobordism spectrum, which was explicitly determined by Novikov. Its applicability is demonstrated by computations with the slice spectral sequence for $\mathbf{MSL}$, which determine the first few Milnor-Witt stems of its homotopy groups (up to the third) in terms of very effective hermitian $K$-theory. We also establish a decomposition of the rational special linear algebraic cobordism spectrum over an arbitrary qcqs scheme.

We consider the Schrödinger equation associated with the harmonic oscillator and show that if the initial data and its Fourier transform are dominated by Gaussian functions of widths $a>0$ and $b>0$, respectively, satisfying $ab<1$, then the evolved solution and its Fourier transform are dominated by a Gaussian of width $\frac{1}{2}\left(\frac{1}{a}+\frac{1}{b}- \sqrt{\left(\frac{1}{a}+\frac{1}{b}\right)^2-4}\right),$ for all times except for a discrete set, and for all times in one dimension. In the one-dimensional case, we prove that these estimates are sharp. Moreover, for a more restrictive class of initial data, we establish sharper time-dependent Gaussian bounds.

We prove that every simply connected nilpotent Lie group endowed with a left-invariant nilpotent complex structure is biholomorphic to $\mathbb{C}^n$. Moreover, we construct such a biholomorphism explicitly by polynomial maps in exponential coordinates. As a consequence, every lattice in such a Lie group admits a free, properly discontinuous and cocompact action on $\mathbb{C}^n$ by holomorphic polynomial automorphisms. We interpret this consequence as a holomorphic analogue of polynomial crystallographic actions introduced by Dekimpe, Igodt, and Lee.

We study Haantjes operators, that is, (1,1)-tensor fields with vanishing Haantjes torsion. Our main result is a complete local description of gl-regular Haantjes operators. Additional results include a splitting theorem for general (not necessarily gl-regular) Haantjes operators and, more generally, for operators with vanishing generalised Nijenhuis torsion of an arbitrary level, as well as a complete treatment and understanding of the case when the eigenvalues of a Haantjes operator are complex; the latter case was ignored in many previous papers on this and related topics.

We give an overview of a recent conjecture of the authors on the leading constant in Malle's conjecture on number fields of bounded discriminant. This comes from applying the philosophy from Manin's conjecture on rational points of bounded height on Fano varieties to classifying stacks. To make these ideas more accessible we assume no background in algebraic geometry, which requires some new perspectives and alternative approaches to the theory. We also give some new conjectures on multi-heights and Bhargava's heuristics on counting with local conditions imposed.

This paper analyzes the convergence of parallel overlapping domain-decomposition methods with impedance boundary conditions for the time-harmonic Maxwell equations in heterogeneous media. We prove that the parallel iterative method is well-posed in an appropriate function space, and characterize the error propagation operator through impedance-to-impedance maps that describe interactions between neighboring subdomains. For strip domain decompositions, we derive explicit convergence estimates in terms of the norms of the impedance-to-impedance maps. At the discrete level, we develop the finite-element counterpart of these results based on Nédélec-element discretisations. Under the assumption that the discrete impedance-to-impedance maps approximate their continuous counterparts as the mesh is refined, we show that the discrete method inherits the convergence behavior of the continuous method. We illustrate this theory with numerical experiments for strip domain decompositions, and also present numerical experiments for checkerboard domain decompositions that go beyond our theory.

We establish a homological mirror theorem for the 4-manifolds arising as moduli of (irregular) rank two local systems on the projective line. Specifically, we prove that the Fukaya category of a moduli of such local systems with generic microlocal monodromy at punctures is equivalent to the category of coherent sheaves on the minimal resolution of the corresponding moduli of local systems with trivial microlocal monodromy.