Papers

We prove that stable degeneration, the canonical degeneration associated to the normalized volume minimizer of a Kawamata log terminal (klt) singularity, preserves non-degenerate reflexive differential forms. In particular, the stable degeneration of a symplectic singularity is again symplectic. Combining this with a deformation-theoretic rigidity result for symplectic degenerations, we confirm Kaledin's conjecture that the formal completion of any symplectic singularity is conical. As applications, we show that the natural base of any normalized nilpotent orbit closure is a K-semistable Fano variety, and that the normalized volume minimizer of a hypertoric singularity is induced by the standard dilation.

We investigate the stability of discrete structures comprising of woven elastic beams in a plane, exposing a connection between admissible over / under patterns and the first-order and static rigidity of an associated tensegrity that is related through a natural polarity transformation. The relationship between the Airy and Whiteley stress functions for the tensegrity and weaving structures is explored. Our results lead to an efficient method for finding such an over / under pattern. The method is illustrated through a worked example modelled on the complete bipartite graph $K_{4,4}$.

We show that any three-ball with mean convex boundary contains an embedded free boundary minimal disk. Moreover, when the three-ball is a strictly convex domain with nonnegative Ricci curvature (for instance, a compact convex domain in Euclidean three-space), we prove the existence of at least three embedded free boundary minimal disks. Our approach is based on a multiplicity-one theorem for the free boundary Simon-Smith min-max theory.

Permuted-basement Macdonald polynomials $E_\alpha^\sigma(\mathbf{x};q,t)$ are nonsymmetric generalizations of symmetric Macdonald polynomials that form a basis for the polynomial ring $\mathbb{Q}(q,t)[\mathbf{x}]$ for each fixed $\sigma$. There are combinatorial formulas for them as generating functions over composition-shaped non-attacking fillings. In this extended abstract, we bijectively prove identities for the relationship between $E_\alpha^\sigma$, $E_\alpha^{\sigma s_i}$, $E_{s_i\alpha}^\sigma$, and $E_{s_i\alpha}^{\sigma s_i}$. These identities correspond to two combinatorial operations on non-attacking fillings: (1) swapping adjacent entries in the basement, generalizing a result of Alexandersson (2019), and (2) swapping adjacent parts in the shape, which yields a straightening rule for expanding $E_\alpha^\sigma$ in the polynomials $\{E_{s_i\alpha}^\tau\}_\tau$.

Every knot $K \subset S^3$ that admits a symmetric union presentation bounds an immersed ribbon disk in $S^3$, while the converse is an open problem due to Christoph Lamm. The symmetric ribbon number $r_s(K)$ of $K$ is the minimum number of ribbon singularities in any symmetric ribbon disk bounded by $K$. In this paper, we undertake a systematic investigation of symmetric ribbon numbers of knots with at most 12 crossings. Along the way, we exhibit novel lower bounds for $r_s(K)$ arising from knot determinants, Alexander polynomials, Jones polynomials, and Kauffman polynomials.

An old conjecture of Burr and Erd\H os states that the Ramsey number of any $n$-vertex tree $T$ is at most $2n-2$. In 2012, Schelp asked whether a degree version of the Burr–Erdős conjecture holds. More precisely, Schelp asked if is it true that for any $\varepsilon>0$ and $\Delta\ge 2$, if $G$ is a graph on $N\ge (2+\varepsilon)n$ vertices and minimum degree $\delta(G)\ge \lfloor 3N/4\rfloor$, then every blue/red colouring of the edges of $G$ yields a monochromatic copy of each $n$-vertex tree with maximum degree at most $\Delta$. We prove this conjecture in a strong form, showing that it is true even if one removes the extra $\varepsilon n$ term in the size of the host graph.

In this study, a generalized operator-based mathematical conflict framework is presented to explicitly represent structural discrepancies between raw data and contextual data. The proposed structure treats conflict as a local, directional, and context-sensitive quantity, integrating components such as weighting, scale behavior, and output mapping under a unified abstract operator. Without being reduced to a specific learning algorithm or optimization method, the framework is defined as a general structure adaptable to different classes of problems. While existing approaches typically treat conflict merely as an implicit side effect embedded within the optimization process, the proposed framework considers conflict as an independent, operator-based, and component-level mathematical object.

We prove descendability for fppf covers of quasi-compact quasi-separated algebraic spaces. Consequently, we show that point-generatedness is detectable along fppf covers. Moreover, we classify $\otimes$-localizing subcategories of the derived category of complexes with quasi-coherent cohomology on point-generated algebraic spaces in terms of subsets of the underlying topology.

Microwave linear analog computers (MiLACs) offer a transformative paradigm for future multiple-input multiple-output (MIMO) systems by shifting complex signal processing into the analog domain, thereby significantly reducing computational complexity, radio-frequency chains, and analog-digital converters, while speeding up computation. However, the practical deployment of MiLACs is severely constrained by the inherent hardware losses of the tunable admittance components (TACs) interconnecting MiLAC ports, which introduce severe inter-stream interference and fundamentally limit the spectral efficiency (SE) of the system. In addition, while denser architectures offer greater spatial degrees of freedom to mitigate inter-stream interference, the cumulative hardware losses and power consumption of massive TACs severely degrade the system's energy efficiency (EE). Consequently, designing architectures for lossy MiLACs emerges as a critical yet unresolved challenge, as it necessitates striking a delicate tradeoff between interference suppression and cumulative hardware losses/power consumption. To address this challenge, this paper investigates the joint MiLAC architecture design and performance (SE/EE) maximization in lossy MiLAC-aided MIMO systems. We propose a novel learning-based joint architecture and performance optimization framework (LJAPOF) that unifies the design of MiLAC architectures and analog beamforming configurations for lossy MiLACs under both SE- and EE-oriented objectives. Numerical results demonstrate that by intelligently navigating the fundamental tradeoff between interference suppression and hardware/power consumption, the proposed LJAPOF can design optimal MiLAC architectures that consistently outperform stem-connected and fully-connected MiLACs in maximizing the system's SE and EE.

Let $m,l\in\N$ be relatively prime and let \[ \Omega_{m/l}^{n+1}=\bigl\{(z,w)\in\C^n\times\C:\ \norm{z}^{m}<\abs{w}^{l}<1\bigr\} \] be the rational generalized Hartogs triangle of exponent $m/l$ in $\C^{n+1}$. In this paper, we study the Berezin transform $\BB_{m/l,n}$ associated with the Bergman kernel of $\Omega_{m/l}^{n+1}$, and prove that $\BB_{m/l,n}$ is bounded on $L^p(\Omega_{m/l}^{n+1})$ if and only if $p>m+nl$.

Burr and Erdős conjectured in 1976 that for all integers $k>\ell\geq 0$ such that $k\mathbb{Z}+\ell$ contains an even integer, every $n$-vertex graph without cycles of length $\ell$ modulo $k$ has at most a linear number of edges in $n$. Bollobás confirmed the conjecture in 1977, and Erdős further asked for the exact extremal number. To the best of our knowledge, this problem has been solved only for all residues when $k\leq 4$, and for $\ell\in \{0,2\}$ when $k\geq 5$ is odd. In particular, Bai et al. [arXiv:2503.03504] proved that if $G$ is an $n$-vertex graph with no cycles of length $1$ modulo $3$, then $e(G)\le \frac{5}{3}(n-1)$, and when $9\mid (n-1)$ the equality holds if and only if each block of $G$ is isomorphic to the Petersen graph. Note that for $n> 18$ every extremal graph contains a cut-vertex. In this paper, we investigate the 2-connected setting and determine the maximum number of edges in a 2-connected graph with no cycles of length $1$ modulo $3$. Our results provide a sharp extremal bound and a complete characterization of the extremal graphs, revealing structural differences from the general case. Combining this with the result of Bai et al., we also obtain a complete characterization of all extremal graphs in the general setting, including the cases where $9\nmid (n-1)$. Finally, we determine the maximum number of edges in a $2$-connected graph with no cycles of length $2$ modulo $4$, whose extremal graphs differ substantially from those in the general setting. Consequently, the extremal numbers for $2$-connected graphs with no cycle of a fixed length modulo $k$ are now determined for all $k\leq 4$.

Paired-comparison data are naturally represented by tournaments, where transitivity corresponds to the existence of a global ranking consistent with all pairwise outcomes. Accordingly, the classical Kendall-Smith coefficient of consistency measures deviations from transitivity in a tournament by counting the number of circular triads (directed $3$-cycles). In this paper, we characterize the fluctuations of the number of circular triads in inhomogeneous random tournaments and develop an inferential framework for the consistency coefficient. Specifically, we consider the $W$-random tournament model, where the comparison probabilities are determined by a tournamenton $W$, the analogue of a graphon in the tournament setting. We show that, for a $W$-random tournament on $n$ vertices, the number of circular triads exhibits three different fluctuation regimes, determined by suitable notions of regularity and uniformity of $W$. We further develop a novel tournamenton multiplier bootstrap that consistently approximates the limiting distribution of the circular-triad count in the relevant asymptotic regime. Combining this with procedures for testing regularity and uniformity, we design an algorithm for constructing confidence intervals for the consistency coefficient that is asymptotically valid for all tournamentons. We also obtain structural characterizations of tournamentons for which the limiting distribution of the number of circular triads exhibits specific degeneracies. These results can also be viewed through the lens of tournament quasirandomness and may be of independent interest.

The Neural Spectral Element Method (NSEM) evaluates each network only at fixed Legendre-Gauss-Lobatto quadrature nodes and replaces all derivative calls with precomputed spectral differentiation matrices. The resulting deterministic loss enables limited-memory BFGS (L-BFGS) to reach residuals of 10^-9 to 10^-10. A Kosloff-Tal-Ezer coordinate map resolves electrochemical boundary layers, while a mesh-free neural mortar framework couples multi-element domains. On the four-example Poisson-Nernst-Planck (PNP) benchmark of Huang and co-workers, NSEM attains 10^-4 to 10^-7 relative pointwise error with two orders of magnitude fewer collocation points than the adaptive-resampling PINN baseline. Both a tanh multilayer perceptron (MLP) and a basis-aligned Legendre Kolmogorov-Arnold Network (KAN) backbone attain spectral accuracy within the same NSEM infrastructure, with the KAN requiring roughly half the Adam steps to enter the L-BFGS basin of attraction on the 1D PNP benchmark.

Necklace polynomials arise in various fields of mathematics, including combinatorics, Lie theory, and Galois theory over finite fields. Their arithmetic properties have been extensively studied in recent years. In this article, we introduce a new class of rational necklace functions that unifies two well-studied families of polynomials: necklace polynomials and Fekete polynomials. We describe several ways in which cyclotomic polynomials appear as factors of these rational necklace functions. Our results unify and generalize various earlier work on necklace polynomials and on Fekete polynomials. In particular, we describe a surprising phenomenon in which certain Galois groups play a hidden role in the appearance of new cyclotomic factors that are not covered by these previous works.

Continuous-variable quantum key distribution (CV-QKD) requires highly efficient reconciliation techniques to operate at low signal-to-noise ratios and long distances. Multidimensional reconciliation addresses this challenge by transforming the physical Gaussian quantum channel into a virtual binary-input additive white Gaussian noise (BIAWGN) channel, enabling the use of modern errorcorrecting codes. In this work, we review the principles of multidimensional reconciliation, with a particular focus on high-dimensional constructions beyond the algebraic dimensions 1, 2, 4, 8. We describe the construction of the virtual channel, discuss practical coding schemes for reverse reconciliation, and analyse their integration with linear error-correcting codes. We also present an opensource simulation framework, HDirac, implementing multidimensional reconciliation for arbitrary dimensions, and use it to evaluate state-of-the-art LDPC codes. The results highlight key trade-offs between dimension, reconciliation efficiency, and frame error rate, providing practical guidance for CV-QKD system design.

Let $\mathcal{R}$ be an expansion of the real field such that every subset of $\mathbb{R}$ definable in $\mathcal{R}$ either has interior or is a finite union of discrete sets. Answering a question by Chris Miller, we show that for every $n\in \mathbb{N}$ and every definable subset $A\subseteq \mathbb{R}^{n+1}$ there is $N\in \mathbb{N}$ such that for all $x\in \mathbb{R}^n$ either $A_x$ has interior or is the union of $N$ discrete sets.

Reduced-order models (ROMs) are widely employed to describe complex system dynamics when simulations with full-order models (FOMs) are computationally prohibitive. This study presents POD-AS-PRS, a novel model-reduction framework based on the active subspaces (AS) technique, which performs dimensionality reduction in both the state and parameter spaces, enabling efficient and high-fidelity approximations of quantities of interest (QoI). The approach employs proper orthogonal decomposition (POD) to extract low-dimensional coefficients from CFD snapshots, which are inputs to a residual neural network (ResNet) with linear layers to learn their nonlinear mapping to QoI. Reverse-mode automatic differentiation (AD) is utilized to compute gradients with respect to the coefficients, enabling AS analysis to identify influential modes by shifting the analysis to the POD coefficient space, thereby achieving a dual-stage dimensionality reduction driven by QoI sensitivity rather than modal energy. A surrogate model is subsequently constructed using a polynomial response surface (PRS) based on AS-derived active variables, retaining only the highly influential POD coefficients to ensure accurate and efficient QoI reconstruction. The framework is validated on periodic and chaotic bluff-body flows, demonstrating high accuracy with few influential parameters, while AD-based gradients achieve a two-order-of-magnitude speed-up over finite-difference approximations. Sensitivity analysis further reveals that the influential coefficients are not necessarily proportional to modal energy, highlighting the critical flow structures. Consequently, POD-AS-PRS identifies a low-dimensional manifold of sensitivity-dominant parameters that govern the QoI, elucidating the essential flow structures and their coupling with control parameters, thereby enabling efficient and accurate QoI reconstruction.

Since Green (2005), the Fourier-analytic transference principle has dominated the landscape of combinatorial theorems relative to sparse arithmetic sets. We demonstrate a different approach using arithmetic regularity. This is more versatile and has the potential to succeed when no obvious `dense model' is forthcoming. Moreover, we contend that, just as the traditional circle method disassembles an arithmetic problem into real and $p$-adic parts which can be solved individually, the arithmetic regularity method generalises this to yield an additional `combinatorial' factor. This framework leads directly to a correct lower bound on the number of configurations in a dense set. We illustrate this using a system comprising a linear equation together with a higher-degree equation.

In a recent paper the $\gamma$-Boost scheduling policy was shown to minimize the tail of the response time distribution in a light-tailed M/G/1-queue. This policy schedules jobs using a boosted arrival time, defined as the arrival time of a job minus its boost, where the boost of a job depends on its exact job size. The $\gamma$-Boost policy can also be used when only partial job size information is available, such as the type of an incoming job. In such case the boost $b_i$ of a job depends solely on its type $i$ and $\gamma$-Boost was shown to optimize the tail among all boost policies, where a boost policy is fully determined by the $b_i$ values. In the partial information setting $\gamma$-Boost relies on two types of information: job types and arrival times. This paper focuses on the problem of minimizing the tail in a light-tailed M/G/1-queue in the partial job size information setting when the scheduler only makes use of the job types and does not exploit arrival times. Prior work showed that in case of $2$ job types the so-called Nudge-$M$ policy minimizes the tail in a large class of scheduling policies. In this paper we introduce the $\gamma$-CounterBoost policy in the partial information setting with $d \geq 2$ job types and prove that it minimizes the tail in an even broader class of scheduling policies called Contextual CounterBoost policies. The $\gamma$-CounterBoost policy reduces to the Nudge-$M$ policy in case of $d=2$ job types.

In this paper, we construct families of unimodal maps with wild Cantor attractors and study, via symbolic dynamics, the interplay between recurrence properties of the critical orbit. Using kneading and co-kneading techniques, we provide a symbolic characterization of persistent recurrence in terms of cutting and co-cutting times, and relate it to other recurrence conditions, including the Collet–Eckmann condition. As an application, we obtain uncountable families of non-renormalizable unimodal maps of sufficiently high critical order admitting wild attractors. We also analyze the structure of postcritical dynamics in recurrent regimes, constructing examples with embedded odometers and with minimal homeomorphic dynamics on the critical $\omega$-limit set, and showing that a minimal homeomorphism together with regular recurrence does not imply conjugacy to an adding machine. Finally, we prove that longbranched combinatorics is incompatible with persistent recurrence, thereby clarifying the boundary between different recurrence regimes.

We study sign uncertainty principles for trigonometric polynomials of prescribed degree $N$ with respect to a symmetric Borel measure $\mu$ on the unit circle $\mathbb{R}/\mathbb{Z}$. For each such measure, we determine the smallest radius of the last sign change for trigonometric polynomials with non-positive $\mu$-integral. We further extend these results to polar measures on higher-dimensional spheres $\mathbb{S}^d$, showing that the extremal problem reduces to the one-dimensional case via the polar part of the measure, and we establish a polynomial analogue on $[0,1]$ using orthogonal polynomials on the real line.

We investigate a three-dimensional ghostly Hamiltonian realisation of the fully degenerate resonant sixth-order Pais-Uhlenbeck oscillator. On the classical level, the phase-space flow is non-diagonalisable and decomposes into two complex-conjugate Jordan chains of length three, explaining the appearance of oscillatory solutions with secular terms. Upon quantisation, we construct intertwining operators whose quadratic combinations generate a hidden spectrum-generating $\mathfrak{u}(2,1)$-algebra. The associated descendant spaces are finite-dimensional invariant subspaces carrying non-trivial Jordan structure. Although these spaces admit a natural decomposition into irreducible modules of a distinguished $\mathfrak{sl}_2$-subalgebra, this decomposition does not in general coincide with the Jordan decomposition of the Hamiltonian. We further derive a tri-Hamiltonian formulation from Lie point symmetries of the classical flow and show that the corresponding Hamiltonians are naturally encoded by the same hidden algebra. Nevertheless, unlike in the non-resonant case, no positive-definite linear combination of them generates the same dynamics. Finally, we analyse the common centraliser of the tri-Hamiltonian family in $U(\mathfrak u(2,1))$, showing that the natural higher-order candidate $Q$ is reducible and yields no independent classical or quantum integral. The model thus provides a resonant higher-derivative system in which hidden $\mathfrak{u}(2,1)$ symmetry, classical and quantum Jordan structures, and multi-Hamiltonian geometry coexist.

The deterministic incompressible Navier-Stokes equations are physically incomplete: any viscous fluid at finite temperature must exhibit thermal fluctuations whose form is dictated by the fluctuation-dissipation relation. We formulate the stochastic Navier-Stokes equations with the kinematically selected deformation Laplacian on compact Riemannian manifolds with strictly negative Ricci curvature. The fluctuation-dissipation relation, derived from a topological (Poincaré lemma) argument, uniquely determines the noise from the viscous operator. For the spectrally truncated system, we prove that the unique stationary distribution is the Gibbs measure (Gaussian in the mode amplitudes, because the nonlinear convective terms preserve energy), and that convergence to equilibrium is exponentially fast with rate at least $2\nu\lambda_\Def$, where $\nu$ is the kinematic viscosity and $\lambda_\Def$ is the spectral gap of the deformation Laplacian. The spectral gap satisfies $\lambda_\Def \geq \kappa^2$ when $\Ric \leq -\kappa^2 g$, and is independent of the volume of the domain. On flat space, the analogous thermalisation rate vanishes in the infinite-volume limit. The equilibrium velocity-velocity correlation function decays exponentially in geodesic distance, in contrast to the algebraic decay on flat space. These results provide a rigorous statistical-mechanical foundation for viscous fluids on negatively curved manifolds and illustrate how the geometry of the domain controls not only the deterministic dynamics but also the approach to thermal equilibrium.

We show that virtually nilpotent groups are coarse median if and only if they are virtually abelian. The main idea is that the sub-Riemannian geometry of the asymptotic cone obstructs the existence of a locally convex Lipschitz median of finite rank. As an application, we deduce that non-compact lattices in the isometry group of a rank 1 symmetric space of non-compact type other than real hyperbolic space are not coarse median. This establishes the remaining case in the classification of lattices with the coarse median property initiated by Haettel. The same approach applies more generally to complete finite-volume non-compact Riemannian manifolds $M$ of pinched negative sectional curvature: if at least one cusp cross-section does not admit a flat metric, then $\pi_1(M)$ is not coarse median.

For graphs $G$ and $F$, we say that $G$ is $F$-free if $F$ does not occur as an induced subgraph of $G$. This paper is concerned with the following question: Given an $F$-free graph $G$ having two vertices between which there exists at least one Hamiltonian path, how many Hamiltonian paths between these endpoints must exist (in terms of the order of $G$)? Our main result shows that there exists a sharp dichotomy. More precisely, we show that if $F$ is not an induced subgraph of $P_3+sP_1$ for any integer $s \geq 0$, then there exists an infinite family of $F$-free graphs having two vertices between which there exists a unique Hamiltonian path. On the other hand, we prove that if $F$ is an induced subgraph of $P_3+sP_1$ for some integer $s \geq 0$, then any $F$-free graph having two vertices between which there exists a Hamiltonian path contains exponentially many such paths between these two vertices. Our proofs use Ramsey-theoretic methods, a result on the existence of two vertices with low degree in graphs containing a unique Hamiltonian cycle, a path variant of Thomassen's red-independent weakly green-dominating sets, and a structural analysis of Hamiltonian paths in $P_3+sP_1$-free graphs. As an algorithmic consequence we obtain that for every fixed $s \geq 1$, given a Hamiltonian $sP_1$-free graph together with a Hamiltonian cycle, one can decide in linear time whether a second Hamiltonian cycle exists and construct one if it does.

Exact \(k\)-leaf powers are graphs whose edges are exactly the pairs of leaves at distance \(k\) in a tree. We prove explicit structure theorems for exact leaf powers on several representative graph families that test different exact-distance phenomena. Our most detailed root-classification theorem concerns chordless cycles: all exact \(5\)-leaf roots of \(C_\ell\), \(\ell\ge 8\), are described by a complete terminal block language. We also prove that the \(t\)-square ladder \(L_t\) is an exact \(5\)-leaf power if and only if \(t\le 2\). In contrast, dense bipartite square structures are often representable: among block-complete multipartite graphs, the exact \(5\)-leaf powers are precisely the bipartite members, and every bipartite co-cluster graph, including every crown \(K_{n,n}-M\), is an exact \(k\)-leaf power for every \(k\ge 5\). Finally, we give parity classifications for complete multipartite graphs and multipartite block graphs at larger exact distances, and isolate a sharp fan boundary at exact distance six.

We construct a cohesive $\infty$-topos $\mathbf{H}_{\mathbb{Q}}$ equipped with a quantum modality – an idempotent product-preserving comonad $Q^{\diamond}$ with right adjoint $Q_{\bullet}$ satisfying the Beck–Chevalley compatibility conditions with the cohesive structure $(\Pi,\flat,\sharp)$. The model is the functor $\infty$-topos $\operatorname{Fun}(\mathbf{C}^{*}\mathbf{Alg}_{\mathrm{fd}},\; \mathbf{H}_{\mathrm{sm}})$, where $\mathbf{H}_{\mathrm{sm}}$ is the smooth cohesive $\infty$-topos and $\mathbf{C}^{*}\mathbf{Alg}_{\mathrm{fd}}$ is the category of finite-dimensional $C^{*}$-algebras with centre-preserving $*$-homomorphisms. Cohesion is lifted pointwise from $\mathbf{H}_{\mathrm{sm}}$; the quantum comonad is precomposition with the centre functor. We endow the topos with the Day convolution monoidal structure $\otimes_{\mathrm{Day}}$ induced by the tensor product of $C^{*}$-algebras and prove that $Q^{\diamond}$ is a strong monoidal comonad. The category of $Q^{\diamond}$-coalgebras is equivalent, via Gelfand duality, to the topos $\operatorname{Fun}(\mathbf{FinSet}^{\mathrm{op}},\mathbf{H}_{\mathrm{sm}})$ of discrete classical field theories. The comonad is interpreted as decoherence. This yields a cohesive linear $\infty$-topos in which the cartesian linear-logic structure degenerates, while the Day convolution provides a non-degenerate affine model of multiplicative intuitionistic linear logic. We also prove a synthetic no-cloning theorem and discuss the limits of the centre modality for representing quantum channels. This work provides the first rigorous instance of the cohesive linear framework and settles the open problem of finding a concrete model for cohesive linear homotopy type theory.

We study exact uniform sampling of permutations of length $n$ whose longest increasing subsequence (LIS) has prescribed length $k$. Our main result is, to the best of our knowledge, the first polynomial-time exact sampler for this problem, valid for every $1\le k\le n$. Via the Robinson–Schensted correspondence the problem reduces to sampling a Young diagram with first row of length $k$ from the hook-length-squared (conditioned Plancherel) law. We sample this shape one coordinate at a time, where each conditional weight, a sum over exponentially many completions, collapses, through the Cauchy–Binet formula, to a single coefficient of the determinant of a small polynomial matrix. A direct implementation runs in expected $\tilde O(n^4k^5)$ time in the word-RAM model. Aggregating the scores and exploiting the Hankel structure of the evaluated matrices reduces this to $\tilde O(n^2k^4)$. In the linear regime $k\in\Theta(n)$ we give a direct rejection sampler running in expected $O(n\log\log n)$ time, matching, up to constants, the cost of computing the LIS of one permutation. For the relaxed constraint $LIS(\pi)\le k$, plain rejection sampling gives expected $O(n\log\log n)$ time for every $k\ge4\sqrt n$, and so the worst case over all $k$ improves to $\tilde O(n^4)$.

In spite of the intensive study of cyclic codes and the recent construction of an infinite family of self-dual binary cyclic codes whose minimum distances have the square-root bound in IEEE Trans. IT, vol. 71, no. 4, 2025, it is still a 70-year-old open problem whether there is an infinite family of self-dual binary cyclic codes whose minimum distances have a lower bound better than the square-root bound. This paper settles this long-standing open problem in coding theory by presenting infinite families of such self-dual binary cyclic codes. As by-products, several families of cyclic codes with better parameters than those in some references are also constructed in this paper.

Ross and Yong conjectured a $K$-theoretic Kohnert rule for Grothendieck polynomials. Robichaux exhibited a counterexample to the Ross–Yong rule and proposed a revised ghost $K$-Kohnert rule, proving both rules hold for 321-avoiding permutations. We provide counterexamples to Robichaux's rule and give an explicit bijection showing that both the Ross–Yong and Robichaux rules hold for 1432-avoiding permutations. As an application, we provide a Kohnert-theoretic characterization of 1432-avoidance.