Papers

We address fundamental challenges in representing and computing $\mathbb{R}^{d}$-valued predictable square-integrable processes over $[0,T]$, collected in the space $\mathcal{H}^2_T(\mathbb{R}^{d})$. These processes are central to continuous-time stochastic control, reinforcement learning, and mathematical finance. Although Wiener-chaos expansions offer strong theoretical tools, traditional computational methods are hindered by the need for large chaos dictionaries and high-order iterated integrals. To overcome these obstacles, we introduce NeuralChaos – a neural operator architecture that produces elements of $\mathcal{H}^2_T(\mathbb{R}^{d})$ using only finitely many evaluations of the driving Brownian motion, while preserving predictability and square-integrability. We prove that NeuralChaos is dense in $\mathcal{H}^2_T(\mathbb{R}^{d})$ and achieves the best $N$-term chaoslet approximation rates for compressible and Malliavin–Sobolev regular processes. Moreover, compressibility is shown to be typical for processes from $\mathcal{H}^2_T(\mathbb{R}^{d})$ under non-degenerate sub-Gaussian sampling. In contrast, we show that finite-dimensional Markovian neural SDE models constitute a meagre and Gaussian-null subset in $\mathcal{H}^2_T(\mathbb{R}^{d})$, regardless of discretization, whereas compressible processes are generic. Numerical experiments on a stochastic optimal control problem and dynamic hedging highlight the practical effectiveness of our approach. Our results enable more efficient and expressive modelling in stochastic analysis and mathematical finance.

In this paper, we establish that several natural topologies on the space of state-preserving unital completely positive maps coincide and that make it a Polish space. We then focus on the subspace of state-preserving conditional expectations and analyse its topology in detail, recovering the Haagerup-Winslow result that it aligns with the Effros-Maréchal topology on the space of von Neumann subalgebras. This correspondence is then applied to structural classes of subalgebras, including amenable, Haagerup and weakly amenable subalgebras. Among other consequences, we demonstrate the closedness of amenable subalgebras admitting state-preserving conditional expectations and analyze the semicontinuity and failure of continuity of the Cowling-Haagerup constant as a function on subalgebras. Finally, we investigate the space of von Neumann subalgebras that are not the image of state preserving conditional expectations for a fixed faithful normal state. For several important classes of von Neumann algebras, such as type ${\rm III}_\lambda$ factors with $0 \le \lambda < 1$ and type ${\rm III}_1$ factors with a state whose centraliser is infinite dimensional, we show that the subalgebras lacking state-preserving conditional expectations form an open and dense subset. Thus, in these settings, the generic subalgebra is not the range of state preserving conditional expectation.

The foundational work of Shannon (1948) identified the capacity of an additive noise channel under an average input power constraint as a mutual information maximization problem over input densities subject to a second moment constraint. However, a quantitative understanding of the channel capacity is significantly lacking even for very simple noise distributions beyond Gaussians. In particular, it is a long standing question to determine the capacity of channels with noise uniformly distributed over a centered interval. This paper settles this question by precisely characterizing the capacity and the corresponding capacity achieving input and output distributions of such channels. A key observation en route to these results is a certain periodization identity for the output density of a uniform noise channel which in turn allows for applications of Fourier analytic techniques.

We consider hyperbolic orthogonal ring patterns as introduced in arXiv:2409.06573 and focus on their characterization by uniformizing variables at the centers of the rings. Given a smooth solution of the sinh-Gordon equation $\Delta u -\sinh(2u)=0$, we restrict to a compact subset of its domain and discretize it by square grid lattices with edge length $\varepsilon$. Taking the values of $u$ as Dirichlet boundary conditions, we prove that the corresponding uniformizing variables $u^\varepsilon$ of the hyperbolic ring patterns converge to $u$ in $C^\infty$ with error of order $\varepsilon^2$, given that the pairs of rings suitably converge to circles. As a consequence we deduce that the hyperbolic orthogonal ring patterns converge to a harmonic map to the hyperbolic plane.

We study a mixed quantum Ising-$XY$ model on the semi-infinite rooted Cayley tree of order two. For every vertex $u$, the edge $\langle u,(u,1)\rangle$ carries an $XY$ interaction and the edge $\langle u,(u,2)\rangle$ carries an Ising interaction. Using the compatibility criterion for tree-indexed quantum Markov chains and consistently working with the normalized trace, we derive the translation-invariant boundary equation and compute explicitly the associated local transfer operator, namely the one-step partial-trace map which propagates successor boundary data to the parent vertex. We prove that the boundary equation has a unique positive translation-invariant solution for all $J_I,J_{XY}\in\mathbb R$ and $\beta>0$. Hence the model admits a unique translation-invariant quantum Markov chain generated by a positive translation-invariant boundary condition. We also show that the reduced boundary-law dynamics, i.e. the induced finite-dimensional recursion for the boundary-law parameters, has no admissible periodic points of period greater than one and compute the local two-site entanglement on the natural three-site cluster of the tree.

We calculate the asymptotics of the number of connected, labeled, genus $g$ Eulerian maps with an arbitrary degree sequence, in the limit as the total number of vertices tends to infinity. This asymptotic is universal, in the sense that the leading-order term depends on only three map characteristics, regardless of the choice of the degree sequence. The constant factor in this formula is related to the Painlevé I equation. Our methods combine for the first time the analysis of the recurrence coefficients associated to a particular family of orthogonal polynomials, and the theory of analytic combinatorics of several variables. We also derive an exact formula for the number of connected, labeled, genus $1$ Eulerian maps. These are the first results on this kind of enumeration problem for $g\geq 1$, non-regular (mixed-valence) maps.

The $p$-adic method for Dynamical Mordell–Lang often reduces a residue class of an orbit to the zero set of a locally analytic interpolating function. This paper assumes the standard interpolation, Strassmann, Mahler, and Weierstrass tools, and studies the effective local zero-bound problem that remains after interpolation: certifying the Strassmann index of the resulting one-variable analytic function. We give finite certificates for this index, including finite-data, finite-precision, refined-tail, adaptive residue-class, one-shot, and first-order escape criteria. Since the Strassmann index is a rigorous upper bound for zeros in $\Zp$ and, through Weierstrass preparation, a root count on the closed disc over $\Cp$, these certificates give checkable stopping criteria for local orbit-intersection computations. A residue-class zooming principle replaces a congruence class of times by the iterate $f^{p^h}$, gaining $h$ additional powers of $p$ in the certificate tails. We also introduce an arc-ideal viewpoint for target varieties defined by several equations, replacing a chosen hypersurface bound by the one-variable gcd of all defining equations along the interpolated orbit. In dimension one, the method identifies the certified Strassmann index with the corresponding local Weierstrass root count in the orbit ball. Applications include certified bounds for intersections of non-fixed power-map orbits with finite target sets, and root-of-unity avoidance for maps tangent to the identity at torsion units.

This article associates to each positive real number $x$ a dimer model on a snake graph with activity parameter $q>0$ whose structure is determined by the continued fraction expansion of $x$. When $x$ is rational, the model is finite and gives rise to a probability measure $\mu_{x,q}$ on perfect matchings. For irrational $x$, the model is infinite, and $\mu_{x,q}$ is defined as a limit over rational approximations to $x$; the main technical result of the paper shows that this limit is well defined and independent of the choice of rational approximants. $[[x]]_q$ denotes the odds that a $\mu_{x,q}$-random perfect matching includes a distinguished edge. When $x$ is rational, $[[x]]_q = q\:[x]_q$, where $[x]_q$ is the algebraic $q$-deformation introduced by Morier-Genoud and Ovsienko. This agreement, together with evidence from the irrational case, suggests a close connection between the probabilistic and algebraic constructions.

Incorporating hysteresis and eddy currents into finite element simulations of laminated-core electrical machines is computationally challenging. Resolving the fields inside the laminations at each integration point and at every nonlinear iteration leads to computational costs several orders of magnitude higher than anhysteretic simulations, making such approaches impractical for design applications. Conversely, simplified models accounting only for magnetic saturation are becoming increasingly inadequate as electrical machine topologies and operating conditions grow in complexity. In this context, machine learning surrogate modeling has emerged as a promising alternative, offering efficient and accurate approximations of complex electromagnetic behaviors. In this paper, a recurrent neural network is trained as a surrogate of a laminated-core material model for an isotropic laminated core, and is integrated into realistic two-dimensional magnetodynamic finite element simulations based on a magnetic vector potential formulation. The proposed approach achieves excellent agreement with the reference laminated-core model while limiting the computational cost to about twice that of an anhysteretic simulation. By training the recurrent neural network on a sufficiently diverse set of artificially generated magnetic field sequences designed to mimic those encountered in electrical machine simulations, the proposed approach can be readily applied across a wide range of finite element simulations. Furthermore, the trained surrogate model is provided as a standalone component that can be easily incorporated into existing computational frameworks. It is publicly available at https://gitlab.onelab.info/getdp/lamnet.

We prove that every idempotent Schur multiplier is a finite signed sum of contractive idempotent Schur multipliers. This was conjectured by Katavolos and Paulsen in 2003 and previously known only for translation-invariant Schur multipliers, by the Cohen-Host idempotent theorem. Concretely, we show that any boolean matrix $A$ with Schur multiplier norm at most $\gamma$ (or equivalently $\lVert A\rVert_{\gamma_2} \le \gamma$) can be written as \[ A=\sum_{i=1}^{L}\sigma_i B_i,\] where $L\leq 2^{C\gamma^6}$ for an absolute constant $C$, $\sigma_i\in\{-1,1\}$ are signs, and each $B_i$ is a contractive idempotent Schur multiplier, that is, a boolean matrix whose $1$-entries form a union of all-one rectangular blocks, with no two blocks sharing a row or a column. As observed by Carenini, a key lemma in our work yields a new proof of the Cohen-Host theorem and gives a simple proof of the quantitative refinements of Green-Sanders and Sanders, with improved bounds. We include a self-contained exposition of these results in the case of finite groups.

We prove that $\mu$ is a semiclassical measure associated to a sequence of eigenfunctions of energy $E<0$ of the attractive Coulomb operator if and only if $\mu$ is a probability measure on the energy (hyper)surface $\Sigma_E$ invariant under the regularized Kepler flow due to Moser. The converse was shown in recent work by the author, and the present article proves the other direction (as well as an independent proof of the converse). We prove the main theorem for a general symbol class allowing certain non-decay at infinity, which implies that semiclassical measure mass entirely reflects off of the origin. In the special case of semiclassical measures of sequences of eigenfunctions of the exact Coulomb operator, this article solves an open problem posed by Keraani. The main tools include the celebrated Moser-Fock map along with a technical operator extension lemma in $\Psi_{\hbar}^0(\mathbb{S}^d)$, which utilizes standard eigenfunction concentration bounds.

After reviewing combinatorial and spectral definitions of expanders, we use precise lower bounds on Ramanujan graphs to investigate upper bounds – and, specifically, the lack thereof – on the eigenvalues of the Laplacian on metric graphs in terms of volume, diameter, girth, mean distance, and torsional rigidity, among others.

We study sparse threshold random geometric graphs generated by high-dimensional spherical or Gaussian latent vectors. Although each edge has marginal probability $p$, shared latent variables make the adjacency entries dependent. At the connectivity scale $np=\Omega(\log n)$, the spherical adjacency matrix satisfies, with high probability,$\|A-\mathbb E A\|_{\mathrm{op}}=O\left(\sqrt{np\log n}+np\tau\right)$, where $\tau$ is the cap threshold; an analogous estimate holds for Gaussian vectors after controlling radial fluctuations. This sharpens the spectral bound in Liu, Mohanty, Schramm, and Yang (2023) under weaker assumptions and strengthens the global-synchronization guarantee of Abdalla, Bandeira, and Invernizzi (2024) for the homogeneous Kuramoto model. The leading eigenspace also estimates the latent geometry. When $np\gg\log n$, vector and relative Gram-matrix errors vanish for$\log(1/p)\ll d\ll np\log(1/p)/\log n$ in the spherical model and $\log^2(1/p)\log n\ll d\ll np\log(1/p)/\log n$ in the Gaussian model, improving the recovery conditions of Li and Schramm (2023). For the Gaussian mixture block model introduced there, a polynomial-time semidefinite program gives, to our knowledge, the first exact-recovery guarantee at the connectivity scale in a moderate-separation regime. At much larger separation, fixed edge density creates isolated vertices and makes exact recovery impossible. Our reusable decoupling and matrix concentration framework avoids trace-moment methods and applies broadly to random graph models with latent vectors.

A uniformly random $m$-subset of $[n]=\{0,\ldots,n-1\}$ has entropy $\log_2\binom{n}{m}$. Standard without-replacement procedures often expose an additional ordering coordinate that is absent from the returned set. We show that Floyd's subset sampler admits an exact round-local factorization of this coordinate. In round $r$, let $S$ be an $(r-1)$-subset of $[j]$, let $T\sim\operatorname{Unif}([j+1])$, and let $S'$ be the result of Floyd's transition. If $D$ is the zero-based rank of the original draw $T$ in $S'$, then $(S,T)\leftrightarrow(S',D)$ is a bijection between $\binom{[j]}{r-1}\times[j+1]$ and $\binom{[j+1]}{r}\times[r]$. Consequently, $S'$ and $D$ are independent and uniform on their respective spaces. The digit $D$ can therefore be merged immediately into a residual uniform random state; an induction shows that the partial subset remains independent of that state after every round. For $k=\min(m,n-m)$, the sampling phase uses $O(k\log k)$ time and $O(k)$ auxiliary space with an order-statistic tree; explicitly materializing a complement incurs the unavoidable output cost. The combinatorial layer avoids binomial-coefficient arithmetic and recovers the complete $k!$ state-space factor exactly. We also give a finite counterexample showing that analogous immediate rank recycling in a partial Fisher-Yates array is invalid because the unselected suffix retains a correlated ordering. A 64-bit Rust implementation is checked by exhaustive state-space enumeration for all $n\leq 8$ and by an entropy-accounting trace for choosing $20{,}000$ of $30{,}000$ items. We make no claim of runtime superiority over existing subset samplers.

We establish a bounded derived Koszul duality for infinite-dimensional Koszul algebras and derive the corresponding singular Koszul duality. We then specialize this framework to two classes of Koszul algebras, namely quadratic monomial algebras and absolutely Koszul algebras satisfying an additional homological condition, for which the resulting dualities admit particularly well-behaved forms. As an application to algebraic geometry, let \(X\subseteq \mathbb{P}_k^n\) be an arbitrary closed projective subscheme. We obtain a Koszul-dual description of the bounded derived category \(\mathsf{D}^{b}\!\bigl(\operatorname{coh}(X)\bigr)\), thereby yielding a BGG-type correspondence for arbitrary closed projective subschemes in projective space. As a second application, in noncommutative projective geometry, we consider generalized Artin–Schelter regular Koszul algebras \(\Lambda^{!}\) arising as Koszul duals of finite-dimensional self-injective Koszul algebras \(\Lambda\). We show that \(\mathsf{D}^{b}\!\bigl(\operatorname{qgr}(\Lambda^{!})\bigr)\) is triangulated equivalent to the bounded derived category of finite-dimensional modules over a finite-dimensional Koszul algebra of finite global dimension. This yields a Beilinson-type description of \(\mathsf{D}^{b}\!\bigl(\operatorname{qgr}(\Lambda^{!})\bigr)\), extending the classical description of coherent sheaves on projective space to this noncommutative setting.

In this paper, we present a structure-preserving method for the compressible resistive Hall-magnetohydrodynamics (MHD) model. The differential operator is split into two parts: a hydrodynamic part consisting of the compressible Euler equations, and a magnetic part consisting of a system coupling the Lorentz force and the induction equation. The method uses continuous Lagrange elements for the Euler part and a curl-conforming finite element space for the magnetic part. The hydrodynamic part preserves the positivity of the density and internal energy, the conservation of total energy, and the minimum principle for the specific entropy. Owing to the choice of finite elements, the magnetic part preserves the divergence involution constraint. The fluid part is solved using explicit strong-stability-preserving Runge-Kutta (SSP-RK) methods, whereas the magnetic part is solved by Crank-Nicholson method, which requires using Newton's method. Coercivity estimates for the Jacobian of the corresponding Newton iteration are presented. We introduce a high-order artificial resistivity to improve the conditioning of the nonlinear residual and the invertibility of the Jacobian. Several challenging benchmarks, including a smooth whistler wave, the Orszag-Tang vortex for comparing resistive MHD with resistive Hall-MHD, and a magnetic reconnection problem, are solved to validate the robustness and accuracy of the method.

Flow matching has emerged as an effective framework for learning complex data distributions, but adapting pretrained flow models to new tasks often requires computationally expensive retraining. Post-training guidance provides a more efficient alternative, but existing methods are largely heuristic and offer no explicit stability guarantees. We address this limitation by proposing LyaGuide, a unified Lyapunov-guided framework that formulates flow guidance as a Lyapunov control problem. Our main theoretical result establishes an equivalence between guided flow matching and Lyapunov control, thereby unifying common guidance strategies, such as classifier guidance, reward guidance, and energy-based guidance, within a single control-theoretic framework. To enforce the Lyapunov condition, we introduce a pseudo-projection operator with a closed-form expression that endows learned or heuristic guidance terms with explicit stability guarantees. LyaGuide supports two practical settings: a model-driven setting, where the target guidance distribution is specified through a known Lyapunov function, and a data-driven setting, where the guidance is adapted from task-specific downstream data. LyaGuide is compatible with existing guidance methods, introduces minimal additional computational overhead, and is straightforward to integrate in practice. Extensive experiments on synthetic benchmarks, image inverse problems, reinforcement learning planning, and energy-based modeling demonstrate consistent improvements in sample quality, guidance fidelity, and robustness, while maintaining computational efficiency.

It is proved that if $\mathbb{F}$ is a field of positive characteristic $p,$ and if $m$ and $n$ are positive integers such that $m\geq2,$ and $4\leq n\leq p\leq mn-1,$ for every $n\times n$ nonderogatory matrix $A\in \mathbb{M}_n(\mathbb{F}),$ with trace in $\{k.1_{\mathbb{F}}\mid k\in \{0,1,\dots,p-1\}\},$ there exist $m$ idempotent matrices $E_1, E_2,\dots, E_m,$ and a nilpotent matrix $N$, such that $A=E_1+E_2+\dots+E_m+N,$ with $N^k=0,$ where $k=n$ if $p\in \{nm-1,nm-2\},$ $k=n-1$ if $p=nm-3,$ otherwise $k=\mathrm{max}(2,1+\lfloor\frac{n-1}{r}\rfloor),$ if $n$ is even, and $k=\mathrm{max}(3,1+\lfloor\frac{n-1}{r}\rfloor),$ if $n$ is odd, where $r:=\lfloor\frac{nm-p}{2}\rfloor.$ Moreover, for $4\leq n>p,$ $A$ is the sum of two idempotent matrices, and a square zero one, if $n$ is even, and it is sum of two idempotent matrices and one whose third power is zero, if $n$ is odd.

Finite-time blow-up is one of the ways in which a dynamical model can become singular, often signaling the breakdown of either the modeled physical system or the model itself. Determining whether blow-up occurs, and for which parameter values and initial conditions, is therefore a fundamental problem in the analysis of nonlinear dynamical systems. We develop a numerical framework for identifying regions of parameter space in which a dynamical system governed by a system of first-order ordinary differential equations with polynomial right-hand sides exhibits finite-time blow-up for at least one initial condition. The approach combines compactification of the phase space with computational algebraic techniques, producing partitioned parameter landscapes that reveal blow-up and non-blow-up regimes. Through several examples, we show that the method replaces problem-specific hand calculations with an automated computational tool for analyzing blow-up regions in parameter-dependent dynamical systems.

This paper gives the first convergence analysis and proof of acceleration for nonlinear GMRES (NGMRES) applied to contractive and noncontractive fixed point iterations (FPIs) for solving general nonlinear systems. Our main results are that in both the contractive and noncontractive cases, the ratio gain of the optimization problem is the mechanism responsible for accelerating (or enabling) convergence. Our analysis also reveals a second important quantity related to the optimization problem, which directly predicts the linear convergence rate at each iteration and proves it is at most 1; hence only higher order terms are responsible for NGMRES non-convergence. Numerical results for several challenging nonlinear test problems are given that illustrate the theory, show how the acceleration improves convergence, show that the quantity predicting the linear convergence rate is remarkably accurate and moreover can be useful for adaptively choosing NGMRES depth, show how restarts can improve convergence in noncontractive iterations, show how NGMRES is naturally suited for finding distinct solutions of a multi-solution PDE, and that NGMRES can perform better than Anderson acceleration when applied to superlinear FPIs.

We develop a rigorous framework for the swapping mechanism for two interacting jump-diffusion processes: each particle evolves independently but their positions are exchanged at random times, providing a continuous-time description of the replica-exchange Monte Carlo. Under mild hypotheses we establish strong existence, pathwise uniqueness, and a forward Kolmogorov equation for the transition density. A symmetrisation identity links swapping to the more standard regime-switching: swapping densities are relabelled sums of switching sectors, so permutation-invariant observables of these processes coincide, while antisymmetric ones differ. We prove that under detailed balance and reversibility the swapping process never degrades the spectral gap, and that, under label invariance, the antisymmetric gap improves compared to the decoupled process. We also show that, for $N$ particles, swapping remains Markov on $\mathbb{R}^N$ with polynomial moment closures, whereas switching requires tracking the full symmetric group, highlighting a fundamental complexity difference. We conclude the work presenting an exactly solvable benchmark of two Brownian motions with constant drifts and swap rate. For this model we obtain closed-form transition densities in terms of modified Bessel functions, exact moments, and two-time correlations. The asymptotic analysis reveals persistent asymmetries, an effective diffusivity correction, and a convergence rate of order $s^{-1/2}$ in the fast-swap limit (where $s>0$ is the constant swap rate).

We construct a domain $D$ in the plane whose Hardy and Bergman numbers satisfy $0<h(D)<b(D)<+\infty$. We also calculate the Bergman number for certain classes of domains having countable complement in the plane. Finally, we investigate some of the implications of our analysis in the theory of iteration of holomorphic self-maps of the unit disk. Our methods rely on estimates for the hyperbolic metric and a recent related result that connects the Bergman number with the hyperbolic metric.

Wyman and Xi (2023) proved a Kuznetsov-type theorem for the submanifold counting function associated with Laplace–Beltrami eigenvalues. We refine the results of Wyman and Xi, improving their asymptotic: we remove a constant in the codimension-one case, and we prove a new asymptotic in the case that the set of looping times has measure zero.

Solutions of the slightly compressible shallow water equations on a rapidly rotating sphere are shown to be bounded uniformly when ratio of the Froude number to the Rossby number is bounded. Moreover, in the singular limit in which the ratio of those parameters remains fixed while they both tend to zero, solutions with well-prepared initial data tend to corresponding solutions of limit equations. A convergence result is also obtained for the three-scale singular limit in which the Froude number tends to zero faster than the Rossby number.

We introduce notions of (weak) shuffle-compatibility for statistics on words, parking functions, and set partitions, generalizing Gessel and Zhuang's shuffle-compatibility for statistics on permutations. For parking functions and set partitions, we perform a systematic review of statistics that appear in the FindStat database (as well as the literature). We further define (shifted) shuffle algebras of (weakly) shuffle-compatible statistics on the equivalence classes induced by the statistics. These algebras relate closely to various combinatorial Hopf algebras such as QSym, FQSym, PQSym, and NCSym. These constructions yield new combinatorial interpretations of various Hopf algebra bases and, in some cases, new bases entirely.

Incremental redundancy (IR) can reduce error rates by spreading coded bits across multiple transmission attempts. However, conventional stop-and-wait operation with coarse feedback often over-provisions retransmissions, triggers unnecessary decoding attempts, and increases end-to-end latency. This paper develops enhanced feedback and scheduling mechanisms that predict the additional redundancy needed for successful decoding and allocate only the required resources. We study two complementary strategies. First, using channel statistics, we learn a one- or two-shot mapping from channel quality to the minimum redundancy budget. As a byproduct, we derive an achievable reliability lower bound on the error probability of hybrid automatic repeat request (HARQ) systems. Numerical results with polar-coded IR-HARQ scheme show that the bound can be closely approached by appropriately selecting the second-transmission redundancy over a wide SNR range with savings up to 60% in retransmission size. Second, we propose a realization-aware early-feedback mechanism that uses first-transmission reliability information to make per-codeword decisions before decoding: whether the codeword is already decodable, if not, how many additional redundancy versions are needed, or whether decoding is unlikely and rate adaptation is preferable. Link-level simulations with 5G NR LDPC codes show that both predictors achieve high accuracy (about 96% in our study), increasing the probability of successful decoding within at most two transmission occasions.

We display 4-dimensional counter-examples to a restricted form of Rosenberg's S^1-Stability Conjecture recently proposed by Jie Xu. Namely, we show that there are smooth compact 4-manifolds M with Euler characteristic zero that do not admit positive-scalar-curvature Riemannian metrics, but for which the corresponding Cartesian products M x S^1 nevertheless do admit such metrics.

The celebrated (offline) Beck–Fiala conjecture asserts that every matrix $A\in\{0,1\}^{n\times T}$ with at most $d$ nonzero entries in each column has discrepancy $O(\sqrt d)$. In this note, we consider the online version of the conjecture. For every fixed $\eta>0$, we give a randomized online algorithm that achieves $O_\eta(\sqrt d)$ prefix discrepancy for every fixed sequence of $d$-sparse vectors $a_t\in[-1,1]^n$, except with probability at most \[ C_\eta T\exp\left[-\frac{c_\eta d}{\log^{2+\eta}(ed)}\right]. \] In particular, this gives an optimal $O(\sqrt d)$ discrepancy upper bound for $d\geq C_\eta(\log T)(\log\log T)^{2+\eta}$. This range of sparsity is optimal up to $\log\log T$ factors. The result also resolves in strong form the online Spencer problem, corresponding to the case $d = n$. The algorithm is based on a compactly supported Metropolis walk, constructed by combining ideas from several recent works on online vector balancing. The proof was generated in conversation with ChatGPT 5.6 Pro; the authors provided high-level guidance in several rounds of prompting, followed by manual checking and rewriting of the proof.

Constrained combinatorial optimization underlies many industrial and technological decision problems. We develop a spectral theory that unifies many quantum optimization algorithms. We show that computational slowdown is driven by entanglement restructuring: the creation, redistribution, and destruction of entanglement during system evolution. The severity of the slowdown depends on how much entanglement must be changed. We show that algebraic properties of constraints induce such restructuring, and that constraint-aware dynamics reduce the associated slowdown by avoiding unnecessary restructuring. This framework explains why constraint-aware quantum methods can outperform generic penalty-based approaches. The theory connects constrained optimization, computational complexity, entanglement dynamics, and Hamiltonian spectral structure across continuous-time and circuit-based quantum optimization paradigms.

Magic and entanglement are independent quantum resources, yet their exact relation in many-body dynamics has remained elusive. We uncover two structural principles. First, at any stabilizer state, the curvature of the second stabilizer Rényi entropy under an arbitrary Hermitian generator equals the quantum Fisher information up to a fixed normalization, creating a bidirectional bridge between computational and metrological resources. Second, for commuting Ising evolution on any forest graph, a Clifford pruning circuit yields the full stabilizer-Rényi family at arbitrary size and in any spatial embedding, thereby furnishing a graph-theoretic construction of families with finite magic density and vanishing entanglement density in the thermodynamic limit. We solve two paradigmatic one-dimensional realizations central to quantum simulation – an Ising quench and a kicked Floquet chain – exactly for arbitrary system size and directly in the thermodynamic limit, revealing finite magic density with vanishing entanglement density, distinct magic and entanglement revival periods, and Clifford points with zero magic but finite bipartite entanglement. The same tangent geometry fixes initial growth, perturbative revival lifting, and stability of thermodynamic magic minima. Large-scale Pauli-basis matrix-product-state calculations verify all predictions, and the tangent bridge yields a concrete protocol for detecting magic through established quantum-Fisher-information measurements.