Papers

We show how a counterexample to the Jacobian Conjecture can be found by elementary mathematics. The example we find is, apart from linear coordinate changes the example found by Levent Alpöge.

We generalize the theory of Wasserstein barycenters between N given measures from the Wasserstein distance to the weak optimal transport problem with barycentric cost. We find a multi-marginal formulation and its dual problem. The multi-marginal problem is new in itself; contrary to the classical multi-marginal optimal transport problem, the first M marginals of its competitors are not fixed but instead only need dominate the first M given measures in convex order, while the last N-M marginals of its competitors are dominated in convex order by the last N-M given measures.

We prove that first-order algorithms require $\Omega(\delta^{-1}\epsilon^{-3})$ gradient queries (in the worst case) to find a $(\delta,\epsilon)$-Goldstein stationary point of a Lipschitz function, at which there is a convex combination of gradients within distance $\delta$ whose norm is at most $\epsilon$. This lower bound is tight, matching known algorithms up to absolute constants, therefore resolving the complexity of convergence to stationarity in nonsmooth nonconvex optimization. We further prove a tight lower bound of $\Omega(\lambda^{1/2}\epsilon^{-7/2})$ for finding points satisfying the recently proposed relaxed notion of $(\lambda,\epsilon)$-stationarity, which allows combining further-away gradients. Our results reveal that convergence rates to nonsmooth stationarity are not affected by gradient stochasticity, in sharp contrast to smooth optimization.

Answering a question of Noga Alon, we generalize a result of Janzer and Yip to show that every $k$-edge-coloring of $K_{r^k+1}$ contains a monochromatic subgraph with chromatic number at least $r+1$ and diameter $O(k^{3/2}r^{k/2}\log r)$.

Downlink channel state information (CSI) feedback is a key bottleneck in frequency-division duplex (FDD) massive MIMO systems, as the user equipment (UE) must convey its estimated channel to the base station (BS) over a limited uplink (UL) budget. To improve CSI reconstruction accuracy under tight feedback constraints, we propose prior-aided masked vector quantization (PM-VQ), a learning-based separate source–channel coding (SSCC) feedback scheme conditioned on the average angle–delay power map—a compact representation of the channel second-order statistics available at both the UE and the BS. In PM-VQ, a prior-aided encoder maps the CSI to latent tokens, and a spatially-adaptive masking module (SAMM) scores and selects the most informative tokens within the feedback budget. The selected tokens are vector-quantized and fed back together with their positions, while an adaptive de-masking module (ADM) completes the latent representation at the BS before prior-conditioned decoding. To support variable-rate compression, a single model is trained over a range of selected-token counts, enabling operation across multiple feedback dimensions without retraining. We evaluate PM-VQ against three representative baselines on a Sionna-generated 3GPP TR~38.901 UMa dataset, focusing on the most challenging diffuse regime where channel energy is spread across many angle–delay coefficients. Simulation results show that PM-VQ achieves the lowest NMSE across all tested SNR levels and feedback dimensions in this regime. Moreover, the angle–delay power-map prior remains beneficial even when estimated from only a few channel realizations.

Many scientific problems require accurate modeling of complex physical phenomena, such as fluid dynamics or climate modeling. These phenomena often result in high-dimensional and thus computationally expensive computational models that can limit their application to real-time and multi-query problems. Model-order reduction (MOR) is an approach that seeks to approximate full-order models (FOMs) using reduced-order models (ROMs), trading a minor reduction in accuracy for a major reduction in computational cost. In this work, we propose a non-intrusive MOR method using generative machine learning by means of denoising diffusion probabilistic models (DDPMs) for generating solutions of the dynamical systems under different instances of their parameters. Unlike conventional DDPMs, which often operate purely in the spatial domain, we aim to generate spatio-temporal solutions to improve quality and temporal coherence. In addition, we embed the DDPM in a latent space obtained by sequentially applying proper orthogonal decomposition and an autoencoder to reduce the data dimensionality and the computational cost of the DDPM. We test our approach on a parametrized 2D fluid flow around an obstacle. The numerical experiments demonstrate that our latent DDPM can (i) produce accurate and temporally coherent solutions, (ii) achieve strong generalization capabilities to scenarios involving unseen parameter values, and (iii) extrapolate in time beyond the training horizon.

Fuzzy hypervector spaces provide a structural framework for modeling graded uncertainty in algebraic systems with multivalued operations. In this paper, we introduce and investigate admissible mappings as a constructive mechanism for generating fuzzy subhyperspaces. These mappings represent fuzzy substructures through families of fuzzy points and allow a systematic characterization of generators. We prove that maximal admissible mappings generate fuzzy subhyperspaces in a manner analogous to bases in classical linear theory, and we present examples over real and finite hypervector spaces to clarify the role of maximality. The proposed approach contributes to the structural foundations of fuzzy modeling in hyperalgebraic settings and may support further developments in soft computing environments involving graded and non-deterministic structures.

The first open case of Hadwiger's conjecture states that every $K_7$-minor-free graph is 6-colorable. We prove that this is the case for $K_7^=$-minor-free graphs, where $K_7^=$ denotes the graph obtained from $K_7$ by deleting two independent edges. The proof is based on an independently interesting density result: Every 5-connected $K_7^=$-minor-free graph with $n\ge 6$ vertices has at most $4n-8$ edges.

Muon updates matrix-valued neural-network parameters by orthogonalizing a gradient-based momentum matrix. Its reliance on derivatives limits its use when gradients are unavailable or unreliable. We develop a derivative-free framework that constructs Muon-style updates from structured finite differences. Four variants are considered: full entrywise recovery, random low-rank surrogates, basis-aligned rank-one probing, and direct structured search. Exhaustive basis-aligned probing is equivalent, up to positive scaling before ideal polar orthogonalization, to coordinate finite differences. Matrix-regression experiments show that random rank-one probing can reduce the number of function evaluations substantially, at the cost of less accurate updates. Controlled noisy-gradient experiments on regression and a neural network illustrate when accurate function values can compensate for an unreliable gradient oracle. A small CartPole study further examines orthogonal rank-one probes under a fixed episode budget. These results support structured probing as a practical option for selected black-box problems; they do not establish a general convergence guarantee or an advantage over accurate, inexpensive gradients.

Stochastic Gradient Langevin Dynamics (SGLD) reduces the cost of Langevin-based sampling by replacing full-dataset drift evaluations with mini-batch approximations, but the resulting subsampling error may offset this computational saving. We study this trade-off for stochastic differential equations with finite-sum drifts and compare the computational cost of SGLD with that of the Euler-Maruyama (EM) method. For a prescribed mean-square accuracy $\varepsilon^2$, we derive complexity estimates that explicitly track the dependence on the dataset size $m$, mini-batch size $s$, and accuracy parameter $\varepsilon$. The resulting comparison reveals distinct parameter regimes in which either method is preferable. In particular, EM can have lower leading-order cost only in a small-data, aggressive-subsampling regime, whereas SGLD is favoured over most of the remaining parameter space. In the practically relevant regime $s \ll m$, the transition between the two methods occurs at the scale $m \asymp \varepsilon^{-1}$. We complement the theoretical analysis with numerical experiments based on a Gaussian Bayesian inference model, which examine the predicted cost regimes together with the underlying discretisation error and variance estimates.

We define a diagonal-inversion statistic on labeled Dyck paths carrying both decorated rises and decorated contractible valleys. This gives an explicit candidate for a bivariate refinement of the univariate Theta conjecture of D'Adderio, Iraci, and Vanden Wyngaerd, recently proved by D'Adderio, Pagaria, and the authors of this work. The univariate conjecture can be recovered from our bivariate version by setting $q=1$. The new conjecture recovers the rise and valley versions of the Delta conjecture when either decoration parameter vanishes, and provides a combinatorial interpretation of the symmetric function $\Theta_{e_l} \Theta_{e_k}\nabla e_{n-k-l}$, which is also conjectured to be the Frobenius characteristic of a certain graded module of diagonal coinvariants with two sets of commuting variables and two sets of anticommuting variables. In support of the Theta conjecture and its touching refinement, we prove its Schröder case, that is, the scalar product of the symmetric function side with $e_{n-d} h_d$ matches the combinatorial side restricted to Schröder paths. The proof uses a finer combinatorial argument than the previously known cases, leveraging two Gaussian product identities.

We study Fourier decay of natural invariant measures for real analytic skew products $T(x,y)=(f(x),g(x,y))$ on $\mathbb{S}^1 \times [-1,1]$, where $f$ is expanding and $0<|\partial_y g(x,y)|<1$. Let $\mathcal S$ denote the associated solenoidal attractor. We prove a dichotomy: either $\mathcal S$ is a single horizontal graph, or both the SRB measure and the measure of maximal entropy have polynomial Fourier decay. No transversality assumption is imposed.

A qKZ-type discrete flat connection with multiplicative step $p$ has monodromy operators when $p$ is a root of unity. The monodromy operators are commuting endomorphisms of the connection. We construct their eigensections and eigenvalues. The construction is based on the presentation of the discrete flat connection as a discrete Gauss-Manin connection.

Knot Floer homology associates to a knot in $S^3$ a bigraded chain complex over $\mathbb{F}[W,Z]$, from which many classical and concordance invariants can be extracted. Recent work shows that this algebraic object can equivalently be represented by a decorated immersed multicurve in a marked surface. This survey explains the immersed curve interpretation of knot Floer homology, aided by many examples, and shows how several invariants arising from the knot Floer complex can be extracted from the corresponding immersed curves. The decorated multicurve associated to a knot has a distinguished curve component $\gamma_0$ and a distinguished connected component $\Gamma_0$, both of which are concordance invariants of the knot. We pay particular attention to these components and various numerical concordance invariants that can be extracted from them. We introduce new generalizations of the $V_s$ invariants, and we give a new curve-based description of the Upsilon invariant by showing that it is determined by generalized $V_s$ invariants.

The McEliece cryptosystem based on algebraic geometry codes has been proposed as a way to reduce the key size of code-based cryptography, but several structural attacks have demonstrated the vulnerability of particular families of algebraic geometry codes. Despite this, until recently, there remained schemes and parameter sets that were not vulnerable to any known attack. We propose a new structural attack with “hints” that applies to elliptic codes with arbitrary effective divisors. In particular, we prove that, given the elliptic curve, the public generator matrix, and three points from the evaluation divisor, the entire divisor can be recovered in polynomial time, independently of the number of errors used in the cryptosystem. The attack requires $\mathcal{O}(k^2n^2+|\mathcal{E}(\mathbb{F}_q)|+n)$ operations in $\mathbb{F}_q$ and succeeds with overwhelming probability, after which the second divisor is recovered in $\mathcal{O}\!\left(k^2n^2 + (|\mathcal{E}(\mathbb{F}_q)|-n)n^2\right)$ operations. We further propose an optimized version of the attack that requires no additional information at all. Exploiting the action of the automorphisms of the curve, the three known points are replaced by the enumeration of a single pair of field elements, which yields an equivalent key on the given public curve in $\mathcal{O}\!\left(k^2n^2 + q^2 + (|\mathcal{E}(\mathbb{F}_q)|-n)n^2\right)$ operations on average.

We study the quasinormal-mode and quasibound-state spectra of a massless scalar field in weakly magnetized Ernst–Schwarzschild spacetime. In the separated $O(B_0^2)$ regime, the Klein–Gordon equation reduces to a radial confluent-Heun equation. We construct an endpoint-selected spectral Wronskian directly from the canonical confluent-Heun recurrence and its regular-to-irregular connection limit. Quasinormal modes and quasibound states are then obtained as zeros of the same analytic spectral function on the radiative and bound sheets, respectively, providing a unified treatment of the two spectral sectors within the separated weak-field problem. The resulting QNM and QBS frequencies agree with independent weak-field Ernst and massive-Schwarzschild benchmarks available in the literature. We further show, for representative cases, that the frequencies selected by the local Heun-series termination relation examined here do not satisfy the endpoint-selected quasibound-state condition. The construction therefore provides a direct horizon-to-infinity spectral condition for both radiative and bound states within the weak-field Ernst problem.

In this paper, we study logarithmic Donaldson-Thomas invariants for local log Calabi-Yau $4$-folds with simple normal crossing divisors. Using the family version of shifted Lagrangian classes announced by Khan-Kinjo-Park-Safronov, we construct relative and family logarithmic $\mathsf{DT}_4$-theories for logarithmic Hilbert schemes of curves, and prove a degeneration formula. For logarithmic Hilbert schemes of points, we construct virtual classes and prove a degeneration formula in Chow groups; in particular, it is independent of shifted Lagrangian classes. Finally, combining our degeneration formula with logarithmic cobordism, we explicitly compute the zero-dimensional logarithmic $\mathsf{DT}_4$-invariants for local surface snc pairs.

The completeness properties of spaces of immersed curves endowed with Sobolev-type reparametrization-invariant Riemannian metrics have been studied extensively in recent years. Whereas constant-coefficient metrics of large enough order on the space of closed curves are known to be complete, this is not the case for open curves. In this work, we characterize the metric completion of the space of real-valued immersed open curves with respect to these metrics. We show that the set of additional limit points in the completion is homeomorphic to $H^n$-diffeomorphisms of a closed interval. This result answers a question of the structure of the completion, posed in (Bauer et al., Ann. Sc. Norm. Super. Pisa Cl. Sci. (2023)), in the negative, and suggests a more refined conjecture on the structure in the case of multi-dimensional ambient space.

We study an infinite-horizon stochastic control problem for the optimal exploitation of an exhaustible resource with unknown total reserves. Information is generated both endogenously through continued extraction without depletion and exogenously through an external information flow. This interaction makes the natural problem non-Markovian and time-inconsistent. We show that it nevertheless admits an equivalent time-consistent formulation with the same optimal controls. The associated value function is characterized as the unique viscosity solution of a stochastic Hamilton–Jacobi equation with random coefficients. We prove comparison on the infinite horizon through a Snell-envelope-based strict-contact argument and establish uniqueness by an independent Brownian regularization and a BSDE correction, avoiding piecewise Markovian approximations. Finally, we identify the deterministic benchmark and show that, under persistent reserve uncertainty, the rescaled stochastic value function converges to a pathwise deterministic control problem, with the long-run optimal extraction rate determined by the asymptotic hazard rate of the limiting reserve distribution.

By applying recent estimates on the number of integer solutions of ternary quadratic forms in boxes due to Browning and Heath-Brown, we obtain new estimates for the number of lattice points close to a curve. This leads to a new result on the distribution of squarefull numbers in short intervals.

The $\textit{Colored Knapsack Problem}$ (ColKP) generalizes the classical Knapsack Problem by partitioning the items into color classes and requiring the selected items to admit an ordering in which consecutive items have different colors. The problem is weakly $\mathcal{NP}$-hard and admits two pseudo-polynomial dynamic programming (DP) algorithms proposed in the literature. These two DP algorithms have worst-case running times $O(b \, n^4)$ and $O(b^2 \, n^3)$, respectively, where $b$ is the knapsack capacity and $n$ is the number of items. We develop the first approximation algorithm for the ColKP. By rounding an optimal basic solution of the linear programming relaxation of its natural integer programming formulation and repairing color feasibility, we obtain a linear-time approximation-algorithm whose worst-case performance ratio is $1/3$. We then reformulate both DP algorithms so that profit, rather than knapsack capacity, indexes their pseudo-polynomial dimension, and combine them with profit scaling to obtain two fully polynomial-time approximation schemes (FPTASs). The first FPTAS runs in $O(n^5/\varepsilon)$ time for nonnegative profits and in $O(n^6/\varepsilon)$ time for arbitrary integer profits. The second FPTAS runs instead in $O(n^5/\varepsilon^2)$ and $O(n^7/\varepsilon^2)$ time, respectively. The approximation guarantee, along with new structural insights, provides the bounds needed to control the scaled profit range and establish these running times.

We introduce several notions of resonance for systems exhibiting weak or partial dissipation: $\dot u=Au+f(t)$, with $A$ the generator of a strongly stable semigroup $S(t)$ on a Hilbert space $H$. We refer to resonance as the phenomenon where a time-periodic $f$ may yield an unbounded $u$, classically occurring when $A$ has imaginary eigenvalues. In infinite dimensions, however, unbounded growth may depend on topological choices, and several “resonances" may occur. Classically, existence of $T$-periodic $u$ under $T$-periodic $f$ is equivalent to a range condition: $\mathcal{R}(I-S(T))=H$. Consequently, its failure permits unbounded solution growth and we elucidate that connection through properties of $S(t)$. We describe a hierarchy of range failures, and provide an associated resonance taxonomy. The growth rate of ~$||(A-\lambda I)^{-1}||_H$ on ~$i\R$ dictates a regularity gap between $u$ and $f$, and, for rapid growth in $\lambda$, that gap can be infinite. In this case, we demonstrate implications for periodic solvability, and a mechanism for constructing smooth resonant forces. The theory developed here is motivated by (and demonstrated for) a hyperbolic-parabolic system, where the latter component provides the only system dissipation. Such dynamics are a simplification of fluid-structure phenomena, which demonstrate strong stability but do not provide unconditional periodic well-posedness. Though point-spectral resonance is ruled out, we note that $\mathcal R(I-S(T))\neq H$. Our main result here shows that, for a carefully constructed geometry, a classical heat-wave system possesses periods $T$ which yield infinite derivative loss and, via supporting results, yield resonance with smooth forcing.

In the moduli space of abelian varieties, Prym loci parametrise Prym varieties associated to covers of curves with fixed degree and monodromy data. We derive and implement an algorithm computing the tautological projections of classes of Prym loci. The key input is a comparison formula relating three different Hodge bundles on moduli spaces of admissible covers. We present computer and hand calculations for double and triple covers, with the locus of hyperelliptic Jacobians as a special case. Two applications follow. First, we calculate the degrees of the Prym maps in the codimension-zero cases, unifying a series of previously ad hoc results. Second, we prove in low genera the non-divisibility of the Jacobian class by the Prym class.

We present an integer-coordinate realization of a genus-3 polyhedral surface with eight planar simple nonagonal faces. It has 24 vertices and 36 edges, and each vertex is incident with three faces. Every pair of faces shares an edge: 20 pairs share one edge and 8 pairs share two. Vertex coordinates, face walks, and plane equations are given. Exact verification confirms that all faces are planar, the surface is closed and orientable, no unintended intersections occur, and the realization has $C_4$ symmetry. The construction is based on an earlier construction and preserves its incidence structure.

We introduce bicomplex fiber bundles and develop a framework for bicomplex characteristic classes and almost bicomplex structures on smooth manifolds. In the first part, we present the basic definitions and properties of these bundles, including their fundamental idempotent decomposition and the correspondence between their isomorphism classes and homotopy classes of maps into the bicomplex infinite Grassmannian. We then define bicomplex characteristic classes and show that, in general, they are not determined by the Chern classes of the underlying complex bundle. We also construct the bicomplex Chern character and discuss its algebraic properties. Finally, we apply the resulting theory to almost bicomplex manifolds, providing examples and obstructions to the existence of such structures.

We present a framework for the modeling and analysis of geometric hybrid dynamical systems as hybrid inclusions on $C^1$-manifolds. Using tools from nonsmooth and set-valued analysis on manifolds, we derive coordinate-independent sufficient conditions for the existence of nontrivial solutions to this type of systems. We present geometric notions of uniform stability and attractivity of compact sets, and establish their equivalence to metric-based stability notions when the manifold is endowed with a Riemannian structure. We also introduce nonsmooth Lyapunov functions and a hybrid Lyapunov theorem providing sufficient conditions for uniform global asymptotic stability of compact sets. Finally, by characterizing $\omega$-limit sets of precompact solutions, we derive a hybrid invariance principle for geometric hybrid dynamical systems. The results are demonstrated through several running examples.

We study the representation theory of the three-dimensional extended corner symmetry group associated with a circular corner. Using induced representation techniques, we construct families of representations of the infinite-dimensional corner symmetry group and analyze their unitarity and irreducibility. We also determine the maximal central extension of the symmetry group and construct the corresponding projective representations. The analysis is motivated by the role that corner symmetries play in quantum gravity, where boundary degrees of freedom — acted upon by the corner transformations — are expected to encode aspects of the quantum structure of spacetime. In three dimensions, the representation-theoretic structures that emerge are closely related to symmetry groups and quantum theories that already appeared as central to the study of gravity. This work provides a first step toward a systematic description of quantum gravitational corner degrees of freedom in three dimensions.

Let $G_{n,m}=\overline{\C_n^{[m]}}$, where $\C_n^{[m]}$ denotes the closed $m$th power of the $n$-cycle. We study the graded Betti numbers and homological invariants of the edge ring of $G_{n,m}$ in the range $n\geq 3m+1$. These graphs form a natural family for the study of edge rings whose regularity can be compared explicitly with the induced matching number. In particular, for $n\geq4m+1$, the graph $G_{n,m}$ has induced matching number one, whereas its edge ring has regularity two. Our approach is based on a characterization of the homology of the induced subcomplexes of the independence complex $\Delta(G_{n,m})$. We introduce a family $\mathcal{S}_V(k,m)$ of vertex subsets characterized by their successive gaps around the cycle and show that, for $W\in\mathcal{S}_V(k,m)$, the induced subcomplex $\Delta[W]$ has the homotopy type of $\mathbb{S}^1$, whereas for $W\notin\mathcal{S}_V(k,m)$ all its positive-dimensional reduced homology groups vanish. Combining this characterization with Hochster's formula and an explicit enumeration of $\mathcal{S}_V(k,m)$, we obtain a closed formula for the graded Betti numbers in the second strand. We further determine the extremal Betti number, regularity, and projective dimension of the edge ring of $G_{n,m}$. Finally, we compute the $f$- and $h$-vectors of the independence complex and use the Hilbert series to determine the graded Betti numbers in the linear strand. The case $m=2$ recovers the corresponding results for complements of squares of cycles obtained in~\cite{RatherSquare}.

The finite-time blow-up construction for the forced Navier-Stokes equations released by OpenAI in Sep. 2026 (OpenAI 2026) has as its object an axisymmetric swirl core in anisotropic similarity variables. Its geometry is that of an exact steady swirl between rotating porous cylinders found by Gumerov and Duraiswami in 1998 (GD1998). We find that the GD1998 boundary value problem does not recast into the similarity variables and that its generalization is a two-dimensional profile problem between porous walls held at fixed similarity radii, second order in the radial variable X and first order in the time-like axial variable $\eta$. We solve it by tensor Chebyshev collocation with a pinned pressure gauge, complex-step Newton and arclength continuation, verify it three ways, and sweep the radial Reynolds number, the wall swirl and a symmetry-breaking datum. Below inflow 9 the solution is one smooth branch; above it the symmetric branch folds at a critical swirl $F^*=0.41V_0^2$. The Dirichlet axis problem has no resolution-stable solution; the core must be a Cauchy problem from the axis. Its moment identities cannot be met by a core symmetric about the dividing plane: the core is an axial through-flow, as its authors chose it; a non-symmetric core meets them to 0.2%. The cone condition is Rayleigh's criterion with axial shear and requires radii of order $10^{20}$. Dynamic rescaling shows the profile to be an attractor at weak inflow and toward the fold, a saddle-node; at moderate inflow and strong swirl the spectrum does not converge in the axial resolution and is left open. A real fluid cavitates (water) or shocks (air) first; the GD1998 chamber is proposed as the experiment. Nothing found suggests the mechanism is reachable in a flow one computes or builds; the forced theorem, and this study, leave the unforced equations of practice as they were. Code, tests and log accompany the paper.

Let $P\in \mathbb Z[X]\setminus\{0\}$ be of degree $\delta\ge 1$ and usual height $H\ge 1$, and let $\alpha\in \overline{\mathbb Q}^*$ be of degree $d\ge 2$. As a consequence of general result due to Lang and Galochkin, we have the following transcendence measure: for any $\varepsilon>0$, there exists $c>0$ such that $\vert P(J_0'(\alpha)/J_0(\alpha))\vert>c/H^{4d^2\delta+\varepsilon}$ where $J_0$ is the Bessel function. In this paper, we prove that the exponent $4d^2\delta$ can be replaced by a smaller (explicit) quantity $\mu(d,\delta)\le 4d^2\delta-2d\delta-1$. A similar improvement holds more generally for the logarithmic derivative of any $E$-function $f$ of differential order 2 and such that $f$ and $f'$ are homogeneously algebraically independent. Our method rests upon the optimization of the size of a determinant that appears naturally in the classical Siegel-Shidlovskii method, following the steps of our previous improvement of the transcendence measure of the value $e^\alpha$ for any $\alpha\in \overline{\mathbb Q}^*$.