Papers

A trail is antidirected if its arcs alternate between forward and backward. A digraph $D$ is antistrong if, for every ordered pair of distinct vertices $x,y\in V(D)$, it contains a forward antidirected $(x,y)$-trail. Bang-Jensen, Bessy, Jackson and Kriesell [J. Combin. Theory Ser. B 122 (2017), 68–90] introduced antistrong connectivity and posed two problems concerning mixed arc-disjoint spanning subdigraphs. In the first problem, one seeks an antistrong spanning subdigraph and an arc-disjoint strong spanning subdigraph. In the second, strong connectivity is replaced by the requirement that the underlying graph of the second subdigraph be 2-edge-connected. Bang-Jensen et al. asked whether each of the two problems can be solved in polynomial time. We prove that the two associated decision problems are NP-complete. The first remains NP-complete for digraphs with maximum out-degree at most four and maximum in-degree at most five. The second remains NP-complete even for oriented digraphs that are strong and antistrong, whose underlying graphs are 3-vertex-connected, and in which all but at most two vertices have both in-degree and out-degree at most four. In particular, the latter hardness result does not rely on digons.

The domination integer root conjecture asserted that $0$ and $-2$ are the only integer roots of the domination polynomial $D(G, x)$ for any graph $G$. In this paper, we document a counterexample of order $n = 33$ possessing an integer domination root at $x = -4$. We provide the complete structural description of the graph $G_{33}$, present its exact domination polynomial $D(G_{33}, x)$, and demonstrate its exact rational factorization. Furthermore, we outline the structural gadget mechanism involving transfer matrices and $S$-unit branch cancellations that gives rise to non-trivial zero evaluation at $x = -4$.

In this paper, we consider warped product spaces $X\times_{\varphi}Y$, where $X$ is a complete geodesic Gromov hyperbolic space, $Y$ is a compact geodesic metric space, and the warping function $\varphi$ satisfies suitable exponential growth conditions. We prove that the warped product is Gromov hyperbolic and derive an explicit estimate for its hyperbolicity constant. We further establish a homeomorphism between its Gromov boundary and $\partial_GX\times Y$, with an explicit comparison formula for the visual metric on the Gromov boundary of $X\times_\varphi Y$ in terms of the visual metric $d_{\varepsilon, X}$ on $\partial_GX$ and $d_Y$ for suitable $\varepsilon>0$.

We give new lower bounds for the Shannon capacities of small odd cycles: $\Theta(C_7)\geq3.258805369885\ldots$, $\Theta(C_{11})\geq5.294502522149\ldots$, $\Theta(C_{13})\geq6.302455083464\ldots$, $\Theta(C_{15})\geq7.301600534487\ldots$, $\Theta(C_{19})\geq9.357192705918\ldots$, $\Theta(C_{21})\geq10.342455853338\ldots$, and $\Theta(C_{23})\geq11.328224257774\ldots$. The bounds are obtained by an iterative procedure due to Gao (2026) which is based on a method by Itty, Rosin, Carstensen and Reichman (2026). The bounds are fully formalised in Lean.

Density modes provide a localized and interpretable summary of multimodal distributions, but their estimation under rigorous differential privacy constraints remains largely unexplored. We study differentially private recovery of density modes for multivariate distributions under local smoothness, curvature, and separation conditions. We propose DP-GRAMS, a mean-shift inspired method that performs noisy ascent on a differentially private score estimator. Assuming the density belongs locally to a Hölder class with smoothness parameter $\beta > 2$, our score estimator uses bias-reducing higher-order kernels, and then enforces privacy in the gradient ascent steps via gradient clipping and calibrated Gaussian noise. A private initialization scheme combines a density-aware utility with a suppression rule and, with $k\asymp M\log n$ draws over a public $h_{\mathrm{DAP}}$-grid and suppression radius $\rho_{\mathrm{init}}\asymp (\log n)^{-1/d}$, achieves high-probability coverage of the modal basins by successively suppressing selected local neighborhoods in competitive regions, while correlated noise across multiple starts enables joint release under a single $(\varepsilon,\delta)$-differential privacy guarantee. We prove that all population modes are recovered with high probability and establish asymptotic error rates of the form $O\!\left((\tfrac{\log n}{n})^{\frac{2(\beta-1)}{d+2\beta}}\right) + O\!\left((\tfrac{\mathrm{polylog}(n,\delta)}{n^2\varepsilon^2})^{\frac{\beta-1}{d+\beta}}\right)$. We also provide minimax lower bounds for private mode estimation, and show that our estimators are nearly optimal, up to a logarithmic factor in the MSE. We present two natural extensions: DP-PMS, a private modal-regression method, and DP-GRAMS-C, a clustering pipeline. Extensive experiments on synthetic and real data demonstrate favorable privacy-utility trade-offs relative to common baselines.

SignMuon compresses the Muon update to one bit per parameter by taking its elementwise sign, providing the most direct way to run a matrix-aware optimizer under an extremely low communication budget. It outperforms SignSGD in practice, yet it can ascend even on a linear function. Signing the gradient before the Linear Minimization Oracle (LMO), rather than after, does not repair this: we construct a small explicit instance on which sign-before (MuonUSign) and sign-on-both-sides (MuonSign) ascend as well, so no placement of the sign around the oracle descends in general. Error feedback, the standard remedy for a biased compressor, does not rescue SignMuon: when applied to Muon's output, error feedback can fail for every smoothness constant, step size, and momentum. Applied to the gradient, error feedback does work, and EF21-MuonUSign and EF21-MuonSign attain the standard $\mathcal{O}(T^{-1/2})$ rate for the squared gradient norm on smooth nonconvex problems, the latter at one bit in each direction. Experiments then reverse the ordering: across centralized CIFAR-10, federated CIFAR-10, and the nanoGPT speedrun, the strongest compressed method is consistently sign-after-the-LMO, precisely the placement we prove divergent, with the provably convergent variants trailing it. Compressing after the LMO, a heuristic, matters more at these scales than the guarantee does.

This paper presents a Matlab implementation of an arc-search infeasible interior point algorithm for linear programming (LP), which has a proven polynomial bound of $\mathcal{O}(\sqrt{n}L)$, the best among all interior-point algorithms for LP. Software architecture and major functions are discussed. Its ease of use is described by a simple example. Crucial strategies are summarized. Quality of the software is assured because this software has been extensively tested on both PC and Linux for the widely used Netlib benchmark linear programming problems in standard form. Some benchmark test problems involve tens of thousands of constraints and hundreds of thousands of variables. For all tested problems, the code found the optimal solution. The numerical results have been compared to those obtained by the popular Mehrotra's predictor-corrector algorithm. We conclude that the implemented algorithm not only has the best polynomial bound but also is computationally competitive compared to the popular Mehrotra's predictor-corrector algorithm.

We show that the motivic Lie algebra of mixed Tate motives over $\mathbb{Z}$ embeds canonically into the unstable compactly-supported cohomology of locally symmetric spaces for $\mathrm{GL}_g(\mathbb{Z})$, and into the weight-zero compactly-supported cohomology of $\mathcal{A}_g$, the moduli space of principally polarized abelian varieties. Our construction passes through tropical geometry and graph complexes: compactly-supported analogues of the Borel classes pull back via the tropical Torelli map to canonical graph cocycles. The latter were recently identified with cocycles studied previously by Rossi and Willwacher. We combine their results with the theory of single-valued periods to conclude that the cocycles map to generators of the motivic Lie algebra. We also compute the canonical graph cocycles for all graphs with $\leq14$ edges.

We prove a Skoda division theorem on a compact Kähler manifold for holomorphic sections of line bundles with singular hermitian metrics. As a corollary, we prove a Siu-type division theorem on compact Kähler manifolds for holomorphic line bundles endowed with singular hermitian metrics.

We study the differential Poisson sigma model (DPSM) in the symplectic case and show that its classical reduction defines a distinguished class of A-type models on symplectic targets, not necessarily Kähler. The DPSM is a covariant first-order sigma model whose graded target is the parity-shifted tangent bundle $T[1]M$ of a Poisson manifold $M$. Its graded Poisson tensor encodes a differential Poisson bracket on $C(T[1]M)\cong\Omega^\bullet(M)$, written covariantly in terms of a connection $\Gamma$ and its transpose $\widetilde\Gamma$. In the nondegenerate case, the Jacobi identities force $\Gamma$ to be flat, while the quartic coupling of the reduced action is given by the curvature of $\widetilde\Gamma$, induced by the torsion of $\Gamma$. Thus, the DPSM selects a symplectic class in which the A-model curvature coupling acquires a first-order Poisson origin. We describe this class through examples and obstructions; $\mathbb{CP}^n$ and K3 surfaces are excluded, while affine symplectic targets, symplectic tori, and the Kodaira–Thurston manifold furnish explicit examples. The graded parent geometry on $T[1]M$ equips $\Omega^\bullet(M)$ with a differential graded Poisson algebra structure; in particular, the underlying differential graded Lie algebra defines a strict $L_\infty$-algebra on the observable complex. This chain-level structure is not manifest in the usual Kähler formulation of the A-model.

Let $(A_i,\Phi_i)$ be finite energy $\mathrm{SU}(2)$ monopoles of charge $k>0$ on an asymptotically conical $3$-manifold with one end, with masses $m_i\to\infty$. After passing to a subsequence, the mass-renormalized energy measures concentrate at finitely many points $x_a$ with concentration weights $4\pi K_a$, where $K_a$ is the total charge of the complete finite cluster of mass-one Euclidean monopoles lying over $x_a$. We prove that, on the complement $M$ of these points, the fields abelianize exponentially. After translating the Higgs fields by their masses along the unit Higgs directions and applying gauge transformations, the translated pairs converge smoothly locally to a reducible monopole $(A_\infty,\Phi_\infty)$ of the form \[ \Phi_\infty=-u\Psi_\infty, \qquad F_{A_\infty}=-*du\,\Psi_\infty, \qquad u=4\pi\sum_aK_aG(\,\cdot\,,x_a), \] where $\Psi_\infty$ is a parallel unit section and $G$ is the minimal positive Green function. Consequently, $x_a$ is a Dirac singularity of charge $K_a$. The singular part of the residual limit is determined by the weighted $0$-cycle of concentration points and total cluster charges, and does not retain the individual Euclidean profiles or their separation hierarchy. We also show that $k-\sum_aK_a$ is exactly the charge escaping through the asymptotically conical end, and describe the residual flat abelian ambiguity.

PDE discovery from sparse observations requires reconstructing a continuous field and selecting the correct differential terms. Our analysis of optimization paths in coupled neural PDE discovery reveals three behaviors: the exact support can persist to the end of training, appear only transiently, or fail to emerge. To decouple equation selection from neural optimization, we develop a freeze-then-select method combining a structured field adapter with Stability-Validated Weak Selection (SVWS). Trained from observations without a PDE residual, the adapter factorizes the field into learned spatial features and temporal coefficients represented by cubic splines. After freezing the field, SVWS identifies recurrent terms across independent weak-form systems, refits candidate supports, and selects the final equation on held-out weak-form systems. Beyond fixed libraries, we apply the same principle to expressions generated by genetic programming and recover the power-law form of an unknown nonlinear diffusion function from sparse, noisy observations. Across all six sparse MDBench regimes, our method attains the highest exact support recovery rate, with its clearest gains over classical and neural baselines on challenging Kuramoto-Sivashinsky dynamics.

This paper presents a method for computing inner polytopic approximations of admissible sets for continuous-time linear control systems subject to multiple affine state constraints, with a particular concern on computational tractability for large dimensional problems. In place of globally computing the admissible set and the part of its boundary called the barrier, we compute the so-called individual admissible sets and the corresponding barriers for each single constraint. We then use the exact sampling of linear systems and generate polytopes in half-space representation that provide an approximation of the individual admissible sets, to finally intersect them. We provide a complexity analysis of the whole procedure to evaluate its efficiency. The approach is illustrated by two examples -a triple integrator and a mass-spring-damper chain considered in 4, 6, 8, and 10 dimensions- with corresponding runtimes evaluated for both.

In dimension one, we introduce a one-parameter family of nonlocal energies, including subcritical Gagliardo seminorms and Riesz functionals, acting on scalar functions whose derivative is given by a periodic configuration of screened Dirac masses. We prove that the global minimizers are given by the equi-spaced configurations of particles. This result is achieved by representing the energy functionals as interaction potentials depending on the geodesic distances between the particles, exploiting the complete monotonicity properties of such potentials and invoking the celebrated Cohn-Kumar Universality Theorem.

In this paper, we introduce and study a novel family of generalized Chebyshev polynomials defined via a rational generating function. We demonstrate that the classical Chebyshev polynomials of the first, second, third, and fourth kinds naturally emerge as special cases of this framework. Furthermore, we derive comprehensive recurrence relations, tridiagonal determinant representations, and explicit closed-form expressions for these polynomials. We also establish some connections between the generalized Chebyshev polynomials and other classical sequences, including Morgan-Voyce polynomials, Fibonacci polynomials, and Fubini polynomials via Stirling numbers of the second kind. Finally, we examine the associated Euler-Seidel matrix and provide a continued fraction expansion for the quotient of consecutive polynomial terms.

Quantum integrability is a cornerstone of the exact theory of interacting quantum spin chains. In its standard formulation, however, one starts from R-matrices satisfying the Yang–Baxter equation, rather than from the Hamiltonian itself. It has therefore remained unclear how Yang–Baxter solvability can be characterized directly at the Hamiltonian level, and how it is related to the existence of local conservation laws. Here we prove that, in a broad standard setting, the Reshetikhin condition is not only necessary but also sufficient for Yang–Baxter integrability, thereby reducing the hidden algebraic structure of integrability to a Hamiltonian-level conservation law. Since the Reshetikhin condition is equivalent to conservation of the total energy current, this Hamiltonian-level criterion is also experimentally accessible. This result establishes a quantum counterpart of the Liouville–Arnold theorem for isotropic spin chains, stating that Yang–Baxter solvability is equivalent to an infinite hierarchy of local conserved quantities. Our result also simplifies substantially the search for integrable spin chains by replacing the search for R-matrices with a direct criterion on local Hamiltonians.

We prove that the Lipschitz-free space $\mathcal{F}(M)$ contains a complemented copy of $\mathcal{F}(\mathbb{Z}^n)$ whenever $M\subset\mathbb{R}^n$ is not porous. Consequently, if $M\subset\mathbb{R}^n$ is uniformly discrete and not porous then $\mathcal{F}(M)$ is isomorphic to $\mathcal{F}(\mathbb{Z}^n)$.

Elastic curves are the mathematical shapes of thin elastic rods in equilibrium, with deep connections to mechanics, geometry, and computer graphics. Traditionally described as stationary points of bending energy under length and torsion constraints, their rich theory admits many equivalent characterizations. We develop a new one from the viewpoint of geometric mechanics. Our main contribution relies on a lesser-known isoperimetric characterization: a curve is elastic if and only if it is a critical point of the length functional under fixed area and volume vectors. We show that these constraints transform naturally under orientation-preserving rigid body motions, identifying them as momentum variables for these symmetries. This structure suggests a new discrete theory. We show that the low-order integral quantities length, area, and volume vectors are all naturally defined for polygonal curves, leaving the same transformation laws exactly satisfied. The resulting definition of discrete elastic curves in terms of the isoperimetric characterization restricted to discrete polygonal curves is variational, structure-preserving, and requires no auxiliary discretizations of curvature or material frames. Finally, the same structure carries the Marsden–Weinstein form, a canonical (pre-)symplectic structure on the space of curves, to polygonal curves. This yields novel approaches to Hamiltonian dynamics on discrete space curves, including tangent, vortex-filament, and modified Korteweg–de Vries flows.

We obtain a tensionless limit of the M-theory 5-brane action and discuss its properties. The consistency of this limit requires that the field strength $H_3$ of the 2-form gauge field on the M5 brane worldvolume is non-vanishing and non-degenerate, so that one can also regard it as a strong $H_3$-field limit. The induced $6d$ worldvolume metric of this theory is non-degenerate in contrast to conventional tensionless (null) $p$-branes. We find that the tensionless M5-brane action is invariant under a super-Weyl symmetry acting on the bulk supergeometry, while in the purely bosonic case the action is invariant under an 11D target-space conformal symmetry, a property which is characteristic for the tensionless bosonic branes. Upon imposing a static gauge, which fixes worldvolume diffeomorphisms, this 11D conformal symmetry becomes a deformed 6d worldvolume conformal symmetry whose action on the worldvolume coordinates involves field dependent terms. When the transversal fluctuations of the 5-brane in the bulk are zero, the model reduces to the conformal non-linear chiral 2-form electrodynamics considered earlier by Gibbons and West, and by Townsend.

Let $G_n$ be the $n\times n$ square grid, and let $k\geq 2$. A set $D\subseteq V(G_n)$ is an \emph{exact-distance $k$-dominating set} if every vertex $v\in V(G_n)\setminus D$ has a vertex $u\in D$ with $d(u,v)=k$. We write $D_{\mathrm{opt}}^{(k)}(G_n)$ for the minimum cardinality of such a set. For every fixed $k$, consider the limit $ \delta_k= \lim_{n\to\infty} \frac{D_{\mathrm{opt}}^{(k)}(G_n)}{n^2}. $ We prove that, for every fixed \(k\geq 3\), $ \frac{1}{4k} \leq \delta_k \leq \frac{k-1}{3k^2-k-1}. $ For $k=2$, the exact value $\delta_2=1/9$ follows directly.

We study $g$-dimonoids, that is, algebraic structures endowed with two associative binary operations satisfying a specified system of axioms. The paper investigates duality and isomor\-phisms of $g$-dimonoids and provides a complete characterization of all rectangular commuta\-tive $g$-dimonoids. Several new examples are constructed, including commutative iso-dual $g$-dimonoids that are not abelian, and noncommutative nonabelian rectangular $g$-dimonoids that are not dimonoids. A complete classification of all two-element $g$-dimonoids is obtained. Finally, we present the results of computer computations determining the numbers of all pairwise non\-isomorphic $g$-dimonoids of orders up to~$5$, as well as all pairwise nonisomorphic commutative, abelian, and rectangular $g$-dimonoids of orders up to~$6$, obtained using GAP, Python, and C++.

We establish a sharp bound on the Hausdorff dimension of the non-branch singular set of branched stable minimal immersed hypersurfaces whose singular sets have locally finite $\mathcal H^{n-2}$-measure: the non-branch singular set is empty when $n=2$, discrete when $n=3$, and has Hausdorff dimension at most $n-3$ when $n\geq4$. We also construct a non-flat stable minimal cone in $\mathbb R^4$ arising from a branched minimal immersion whose vertex is a non-branch singularity. Taking products with Euclidean factors yields examples whose non-branch singular sets have Hausdorff dimension exactly $n-3$, showing that our regularity bound is sharp in every dimension $n\geq3$. The main ingredients in our proof are a generalized Schoen inequality and a corresponding branched sheeting theorem near stationary classical cones and unions of hyperplanes.

For a finite CW pair $(K,L)$, we show how to construct a Poincaré triad $(P;\partial_0 P,\partial_1 P)$ and a weak homotopy equivalence $(K,L) \overset{{}_\sim}\to (P,\partial_0 P)$. Furthermore, the Poincaré pair $(P,\partial P)$ has a trivial Spivak normal fibration. The proof is homotopy-theoretic. We also prove a uniqueness result.

We determine the asymptotically best possible minimum vertex degree condition forcing a two-coloured $3$-graph to contain a colour-biased tight Hamilton cycle. This confirms a conjecture of Hàn, Lang, Marciano, Pavez-Signé, Sanhueza-Matamala, Treglown and Zárate-Guerén.

We study joint parameter estimation from a single observation in high-dimensional Gibbs measures with cubic tensor interactions, motivated by dense ERGMs, arithmetic-progression models, and inhomogeneous random hypergraphs. Focusing on the maximum pseudolikelihood estimator, we give checkable conditions for joint consistency and asymptotic ill-conditioning. For the edge-triangle ERGM, pseudolikelihood is ill-conditioned in the ferromagnetic regime with nonnegative field, but consistent in a sufficiently strong antiferromagnetic regime. For the edge-three-star ERGM, it is ill-conditioned for all inverse temperatures and external fields. We also study consistency for arithmetic-progression, and inhomogeneous hypergraph models. Our proofs develop nonlinear large-deviation and mean-field approximation tools for cubic tensor Gibbs measures, which have scope for broad applications.

The proposal is to probe Majorana modes in the system when its parameters are tuned to bring it into a disentangled ground state, which occur on the parametric curves known as disorder lines (DL). In such state certain correlation functions do not depend on separation. The exact results are presented for the XY spin chain in transverse field, also called the Kitaev chain in the Majorana representation. The single Majorana modes are shown to be localized near the ends of the chain, as in the states off the DL, while the disentangled $n$-particle Majorana modes ($n \geq 2$) penetrate into the bulk without attenuation. The predicted bulk-edge effects can be detected in the specially engineered optical chains and lattices.

We introduce a new class of Riordan matrices corresponding to lattice walks with colored steps. We show that this class corresponds to a class of pseudo-involutions and then find both generating functions and explicit combinatorial formulas for entries of these Riordan matrices. We use these Riordan matrices to find an explicit combinatorial formula for the occurences of an arbitrary pattern in our lattice walks. Finally, we define a distributive lattice on our lattice walks and connect our new RNA matrices to Whitney numbers of lattices.

A recently derived nonlinear velocity theory of extreme type-II superconductors is shown to possess an exactly solvable one-dimensional sector. Its solutions include nonlinear Meissner states exhibiting universal tails, vortex sheets, and periodic laminar structures. The thermodynamic critical field emerges from a marginal Meissner profile equivalent to the normal-superconducting boundary and may be viewed as a half-soliton. The vortex sheet is a soliton characterized by a discontinuity of the velocity field, a continuous and localized magnetic field and vanishing superconducting density at its center, and may be interpreted as the coarse-grained limit of a dense row of Abrikosov vortices. Periodic solutions describe laminar states that may be interpreted as coarse-grained rectangular vortex lattices.

In multifidelity Monte Carlo (MFMC), optimal sample allocations are typically derived from a continuous relaxation of a variance minimization problem, with integer solutions obtained through post hoc rounding. Such rounding procedures may fail to fully exploit the available cost budget, particularly under tight cost budgets or when model costs vary significantly. In this work, we reformulate the MFMC allocation problem as a variance-constrained cost minimization problem that is equivalent to the standard budget-constrained formulation at the continuous level. This reformulation admits a recursive structure that enables the construction of an integer allocation strategy based on Bellman's principle of optimality. Since the resulting MFMC allocation has a mathematical structure similar to the optimal multilevel Monte Carlo (MLMC) allocation, the proposed strategy naturally extends to MLMC. The resulting algorithm constructs integer-valued sample allocations that more closely follow the continuous variance–cost tradeoff while using the prescribed variance tolerance more efficiently. Numerical experiments demonstrate that the proposed approach satisfies the prescribed variance tolerance with less computational overhead than standard rounding strategies.

We study nonlinear reactive transport in a two-dimensional bulk domain containing a thin branched fracture composed of three narrow branches of thickness epsilon connected through a junction node of diameter O(epsilon). The microscopic model couples nonlinear parabolic reaction-diffusion equations in the bulk with an advection-diffusion equation in the fracture, where the longitudinal Peclet number is of order epsilon^{-1}, leading to advection-dominated transport along the branches. Nonlinear side-dependent flux conditions describe the coupling between the bulk and the fracture. As epsilon tends to zero, the fracture collapses to a one-dimensional graph, and we derive a recurrent structure of effective limit problems: a first-order hyperbolic problem on the graph satisfying the classical Kirchhoff transmission condition at the node, and reaction-diffusion equations in the bulk with nonlinear Robin conditions on the graph edges involving the graph solution. The node boundary conditions of the microscopic model do not affect these leading-order limits. To capture the influence of the node geometry, we construct node-layer and corner-layer correctors and determine subsequent terms of the asymptotic expansion. Their coefficients solve auxiliary boundary-value problems in unbounded domains with outlets at infinity and in corner-type geometries. We assemble a complete multiscale approximation combining bulk, branch, node-layer, and corner-layer contributions, and establish quantitative error estimates in appropriate energy norms. These estimates demonstrate the accuracy of the approximation relative to epsilon and depend explicitly on the corner angles of the limiting graph, reflecting the geometric complexity of the fracture network.