Papers

Jackson and Thomassen conjectured that every $2k$-strong digraph admits a spanning $k$-strong oriented subdigraph [Ann. N. Y. Acad. Sci. 555 (1989) 402-412]. The conjecture holds for $k=1$ but other than some partial results that have been obtained for general $k$ in some special families of digraphs, including symmetric digraphs, the conjecture remains wide open in general. Even the existence of an integer $K$ such that every $K$-strong digraph has a 2-strong spanning oriented subdigraph is open. As a natural optimization counterpart, the minimum spanning $k$-strong subdigraph (MSSS$_k$) problem, is to find the minimum number of arcs in a spanning $k$-strong subdigraph of a $k$-strong digraph. This problem is NP-hard already for $k=1$ as it generalizes the hamiltonian cycle problem. In this paper, we address both problems simultaneously for the class of split digraphs, by constructing sparse spanning $k$-strong oriented subdigraphs. Specifically, we prove that every $k$-strong split digraph $D = (V_1, V_2; A)$ with minimum semi-degree {$\delta^0(D)\geq 26k+15$} contains a spanning $k$-strong oriented subdigraph with no more than $kn+k|V_1|+98k^2+38k+3$ arcs, where $kn + k|V_1|$ is tight and the $k^{2}$ term is tight up to a constant factor. For the class of $k$-strong tournaments with minimum semi-degree at least $26k+15$ our results improve the bound obtained by Kang in [Combin. Probab. Comput., 27:892-907, 2018].

Let $s782$ be the 2-cusped hyperbolic 3-manifold in the SnapPy census. Its spherical CR uniformization was established in \cite{JWX2023} using the Ford domain of the complex hyperbolic triangle group $\Delta_{4,4,\infty;\infty}$. By comparing the combinatorial structures of the Ford domain of $\Delta_{4,4,\infty;\infty}$ and the Dirichlet domain of $\Delta_{4,4,n;\infty}$, we prove that for each $n \geq 5$, the Dehn filling of $s782$ along the slope $(n-1)\mathcal{m}_1 + \mathcal{l}_1$ on its second cusp admits a spherical CR uniformization, where $(\mathcal{m}_1, \mathcal{l}_1)$ denotes the meridian-longitude system of a cusp in SnapPy notation.

Let $X_N=\{x_1,\dots,x_N\}\subset \Sph^2$ be a spherical $t$-design of strength $t\ge2$, and let $\mu_X$ be its empirical measure. Minkowski's theorem associates with $X_N$ a convex polytope $P_X\subset\R^3$, unique up to translation, whose facet normals are the design nodes and whose facets all have area $4\pi/N$; equivalently, $S_{P_X}=4\pi\mu_X$. We show that this realization transfers polynomial exactness into exact convex geometry: the normalized surface tensors of $P_X$ agree with those of the unit ball through order $t$, and mixed volumes are exact against convex bodies whose support functions are spherical polynomials of degree at most $t$. We next derive quantitative shape information. Spherical Jackson approximation yields a $1$-Wasserstein discrepancy $W_1(\mu_X,\sigma)=O(t^{-1})$, while degree-two exactness gives a uniform nondegeneracy condition. Combined with quantitative inverse stability for Minkowski's problem, this implies, after Steiner normalization, \[ d_H(P_X,B)=O(t^{-1/2}),\qquad \alpha(P_X,B)=O(t^{-3/4}), \] for every spherical $t$-design, without assumptions on cardinality, separation, covering radius, or spectral conditioning. Projection bodies retain the full $O(t^{-1})$ scale, separating linear surface-area observables from nonlinear reconstruction of the body. In the critical regime $N=(t+1)^2$, a uniform spectral lower bound for the sampling Gram matrix further forces the facet normals to be separated at the wavelength scale $t^{-1}$. The construction extends to $\Sph^d$, with the universal Hausdorff rate $O(t^{-1/d})$.

We obtain necessary and sufficient conditions for a block of an Ariki-Koike algebra to have the property that all its associated multipartitions have no addable node with a given residue. This leads to a classification of Scopes equivalence classes for Ariki-Koike algebras in terms of their pyramid numbers and Scopes vectors, generalising that for level 1 and for core blocks. Our proof is independent of the results of Richards for level 1.

Pointwise error analysis of discontinuous Galerkin (DG) methods for the Poisson equation has received considerable attention during the past two decades. However, on convex polyhedral domains, existing analyses can only establish optimal error estimates in the broken $W^{1,\infty}$ seminorm for several DG methods. Since the broken $W^{1,\infty}$ seminorm does not control discontinuities across mesh interfaces, a complete discrete $W^{1,\infty}$ theory for discontinuous approximations on convex polyhedral domains has remained unavailable. In this paper, we develop an interface Green's function framework for the complete discrete $W^{1,\infty}$ analysis of DG methods on convex polyhedral domains. The proposed framework introduces new interface Green's functions that represent jumps of discontinuous approximations across mesh interfaces. Its central analytical ingredient is a new local energy estimate for these Green's functions, obtained by exploiting a cancellation between neighboring discrete delta functions. This estimate differs fundamentally from existing Green's function estimates and enables us to derive maximum-norm estimate for interface jumps without introducing additional logarithmic factors.

For fixed integers $s\ge t\ge2$, let $\operatorname{ex}(n,n,n,K_{s,t})$ denote the maximum number of edges in a tripartite $K_{s,t}$-free graph with $n$ vertices in each part. When $s\ge(t-1)!+1$, let $r$ be the largest integer satisfying $s\ge(t-1)!r^{t-1}+1$. Using the quotient norm graphs of Alon, Rónyai and Szabó, we prove that \[ \operatorname{ex}(n,n,n,K_{s,t}) \ge \left(\frac{3}{2^{1/t}}r^{1-1/t}+o(1)\right)n^{2-1/t}. \] Improving an upper bound of Tait and Timmons, we prove that, for all $s\ge t\ge 2$, \[ \operatorname{ex}(n,n,n,K_{s,t})\le \left(\frac{3}{2^{1/t}}(s-t+1)^{1/t}+o(1)\right)n^{2-1/t}. \] Together, these bounds recover the results for $t=2$, and give the new asymptotic formula \[ \operatorname{ex}(n,n,n,K_{3,3}) =\left(\frac{3}{\sqrt[3]{2}}+o(1)\right)n^{5/3}. \] Analogous results extend to $k$-partite graphs containing no $K_{s, t}$ whose $s$-vertex or $t$-vertex side lies in a single part. As an application of our tripartite construction, we determine the tripartite multicolor Ramsey number of $K_{3,3}$ asymptotically.

We construct an explicit pure-swirl solution of the forced three-dimensional Navier–Stokes equations in a circular cylinder. For every $T>0$ and $1\le p,q<\infty$ with $1/p+1/q>1$, the force belongs to $L^q(0,T;L^p(D))$ and is smooth for $t<T$. Moreover, the same construction always yields the additional energy-class property $f\in L^1(0,T;L^2(D))$. The solution is classical on $[0,T)$, extends strongly in $L^2(D)$ to the unique Leray–Hopf solution at time $T$, and satisfies the energy equality, while $\|v(t)\|_{L^\infty(D)}\to\infty$ as $t\uparrow T$. We determine the exact mixed-norm ranges of both the force and the velocity and obtain two-sided rates for the force, the velocity supremum norm and the enstrophy. The construction refines Zhang's profile by an annular cancellation that preserves smoothness at the symmetry axis. It is a direct, self-contained sharpening of the $k=1$ part of our previous weighted construction and supersedes its endpoint discussion. Since the convection term is absorbed by the pressure, the same example applies to the forced Stokes system.

Classical channel decoding typically treats all information sequences as equally likely and relies primarily on the channel observations and code structure, without exploiting statistical or semantic structure in the source data. Although source compression is designed to remove redundancy, practical source coding can leave substantial residual structure that conventional channel decoders do not exploit. Modern multimodal data sources, including text, speech, and images, exhibit rich statistical and semantic dependencies that foundation models can learn and exploit to improve channel decoding. This article introduces semantic error control coding (SECC), which seamlessly integrates the semantic structure of the source into encoding and decoding through a foundation model. The semantic source prior, represented by the model's a priori probability of the source content, directs code redundancy toward semantically important content at the encoder, and improves reliability estimation, candidate search, and error detection/correction at the decoder. The channel code keeps its algebraic structure, and its constraints ensure that the semantic suggestions from the foundation model comply with this structure. We describe the SECC framework, classify its design methods into three approaches, and demonstrate representative schemes on text sources. The demonstrated schemes show several decibels of coding gain over conventional decoding on AWGN channels, and reach error rates below the normal approximation bound. Finally, we identify several open challenges.

The work of Baverez, Guillarmou, Kupiainen, and Rhodes [BGKR24] is the starting point of this paper. It shows that analytic changes of boundary parametrizations act differentiably on Liouville amplitudes, with derivative given by Virasoro operators and a scalar anomaly term. We compute this scalar term. Its holomorphic part is a Schwarzian boundary integral, which gives a geometric explanation of the Virasoro central term. We then derive local Ward identities on disks, annuli, and pairs of pants. They give finite recursions for descendant matrix coefficients. From these recursions we recover the polynomial factorization of normalized pair-of-pants coefficients. In the annular zero-weight limit, we recover the Shapovalov form. For a pair of pants with two incoming boundaries, we recover the formal chiral vertex-operator coefficients. We also give a geometric proof of smoothness in the bulk insertion points and derive the genus-zero arbitrary level BPZ equations for degenerate bulk insertions.

In 1965, V. I. Arnold established a structure theorem guaranteeing the existence of a foliation by invariant surfaces for general three-dimensional steady Euler flows with non-constant pressure. In this paper, we investigate what foliation structures can arise in steady Euler flows. We consider a toroidal domain foliated by the level sets of a flux function $\Psi$, and prove that every $C^{1}$ steady Euler flow $(\boldsymbol{u},p)$ satisfying the assumptions $\iota_{\boldsymbol{u}}d\Psi=0$ and $p=p(\Psi)$ admits the tangential flow representation \[\boldsymbol{u}=c_1(\Psi)\boldsymbol{\xi}^{1}+c_2(\Psi)\boldsymbol{\xi}^{2},\] for some lifted solenoidal vector fields $\boldsymbol{\xi}^{1}$ and $\boldsymbol{\xi}^{2}$ associated with a natural basis of weighted harmonic one-forms on the toroidal leaves. Moreover, the flux function $\Psi$ satisfies a single scalar equation, referred to as the normal flux equation. These characterizations reveal the general foliation structure of steady Euler flows, with the Clebsch representation and the Grad–Shafranov equation recovered as the axisymmetric special case.

Let $\lambda, \mu, \nu$ be distinct partitions such that $\lambda, \mu\subset \nu$ and $|\nu/\lambda|=|\nu/\mu|=1$. We prove Butler's positivity conjecture posed in 1994: the expansion of the Macdonald intersection polynomial \[ \frac{T_\lambda\widetilde{H}_\mu(X;q,t)-T_\mu\widetilde{H}_\lambda(X;q,t)}{T_\lambda-T_\mu} \] in terms of the Schur function basis has coefficients in $\mathbb{Z}_{\geq 0}[q,t]$.

For a graph $G$, let $\cpn(G)$ and $\ccn(G)$ denote the minimum numbers of cliques whose edge sets partition and cover $E(G)$, respectively, and put $f(n)=\max_{|V(G)|=n}\bigl(\cpn(G)-\ccn(G)\bigr).$ In 1983, Erdős, Faudree, and Ordman asked whether there is a sequence of graphs $G_n$ such that $|V(G_n)|=n$ and $\cpn(G_n)-\ccn(G_n)=n^2/4+O(n)$. The question appears as Problem 66 in Chung's survey \cite{ChungProblems} and is also listed on the UCSD Erdős Problems website. Caccetta, Erdős, Ordman, and Pullman proved that $f(n)=n^2/4-o(n^2)$. We prove that $f(n)=\left\lfloor\frac{n^2}{4}\right\rfloor-\Theta(n^{4/3}),$ and hence answer the question in the negative.

In this paper, we introduce a pseudo-differential operator associated with the linear canonical Dunkl transform. For a particular class of symbols, we prove that this operator defines a continuous linear mapping from the Schwartz space into itself. We further establish an amplitude integral and kernel representation of the operator and investigate the smoothness and decay properties of the associated kernel. Subsequently, we establish an $L^1$-norm inequality for the pseudo-differential operator on the linear canonical Dunkl Sobolev spaces. Moreover, we extend the pseudo-differential operator to tempered distributions, proving its continuity. We then investigate the $L^2$-boundedness of the operator for the symbol class $S^m_0$. Finally, as an application, we employ the pseudo-differential operator to study non-homogeneous and nonlinear parabolic partial differential equations.

Seymour's second neighborhood conjecture asserts that every finite oriented graph has a vertex with at least as many exact second outneighbors as outneighbors. Established cases include tournaments, proved by Fisher (1996), and oriented graphs of minimum outdegree at most six, proved by Kaneko and Locke (2001); a recent preprint of Sadhukhan, Sandeep, and Sen (2026) treats minimum outdegree seven. For dense incomplete graphs, Fidler and Yuster (2007) proved the conjecture when the missing edges form a matching, a star, or a clique, Ghazal (2012) extended this direction to generalized stars, and Dara, Francis, Jacob, and Narayanan (2022) proved it when the missing edges can be partitioned into a matching and a star. We give a short counting proof of the conjecture for every oriented graph of order $n=2\delta+2$, where $\delta$ is the minimum outdegree, with no prescribed structure on the missing edges. Together with Fisher's tournament theorem, this implies the conjecture for every oriented graph satisfying $n\le2\delta+2$. Combined with the known minimum-outdegree results, this raises the best lower bound known to us on the order of a counterexample from $16$ to $17$ and, conditional on the preprint of Sadhukhan, Sandeep, and Sen (2026), from $18$ to $19$.

The dual Stiefel-Whitney classes wbar_j(M) of a manifold M are elements of H^j(M;Z/2) which give information about embedding and immersing M in Euclidean space. We consider the question of finding, for each n, the largest j such that there is an n-dimensional Spin manifold with wbar_j nonzero. We obtain upper and lower bounds. Since one theorem proves existence of manifolds without giving explicit examples, we consider a parallel question of finding explicit manifolds with wbar_j nonzero for j as large as possible.

For a finite group $H$, let $\nu(H)$ denote the maximum order of a nilpotent subgroup of $H$. We prove that every finite solvable transitive permutation group $P$ on a finite set $\Omega$ has a subset $\Delta\subseteq\Omega$ such that $|P:P_\Delta|\ge\nu(P_\Delta)$. We also prove that if a finite solvable group $H$ acts faithfully and completely reducibly on a finite module $V$, then some $x\in V$ satisfies $|H:H_x|\ge\nu(H_x)$. Consequently, we settle Gluck's well-known conjecture, open since 1985: every finite solvable group $G$ satisfies $|G:\mathbf{F}(G)|\le b(G)^2$, where $\mathbf{F}(G)$ is the largest normal nilpotent subgroup of $G$ and $b(G)$ is the largest degree of an irreducible complex character of $G$.

We develop a moving-contour Hankel framework for encoding local zero configurations of the Riemann $\Xi$-function. Weighted contour integrals of the logarithmic derivative $\Xi'/\Xi$, expressed in a holomorphic coordinate associated with the contour, are identified with the power moments of a finite atomic measure supported at the coordinate images of the enclosed zeros. This representation yields exact zero-free and rank criteria and, for conjugation-compatible contour-coordinate pairs, an inertia formula: once the matrix order is at least the number of distinct coordinate nodes, the negative index equals the number of distinct nonreal conjugate pairs. Consequently, the Riemann hypothesis admits a local finite-dimensional Hankel-positivity formulation, although establishing this positivity independently of the zero set remains unresolved. As the contour moves, the Hankel matrix evolves by a continuous congruence flow between zero crossings and undergoes finite-rank jumps at crossing events. An isolated zero produces a signed rank-one jump, whereas a nonreal conjugate pair produces a rank-two indefinite event in a real-axis circular scan that meets the pair. Removing the continuous coordinate drift yields a piecewise-constant matrix process from which crossing coordinates, multiplicities, and zero locations can be recovered. Numerical experiments validate the contour quadrature, indicator fields, continuous flow, crossing signatures, and recovery procedure.

The paper considers the construction of two new orthogonal multiwavelets with supercompact support by using the Fast Bauer's method for matrix spectral factorization on the matrix product filter of the orthogonal CL multiwavelet filter. The new multiwavelets possess orthogonality, symmetry/antisymmetry, and one of them provides better coding and smoothness than other supercompact multiwavelets. The performance of the new multiwavelet filters in subband-based edge detection, grayscale and color image compression and 1D and 2D signal denoising is compared with the GHM, SA4, CL, Integer Haar and Alpert multifilters. The comparative analysis shows that new multiwavelets can provides better human visual measures, SSIM and MS-SSIM in image compression and denoising applications.

We characterize minimizers in Gamow's liquid drop problem.

We prove that the centroid Banach–Mazur diameter of planar convex bodies is strictly less than $4$, (slightly) improving the prior upper bound $69/17\approx 4.05882$ due to Lassak. The proof combines a sharp covariance-ellipse sandwich, whose equality cases in any single direction occur only for triangles, with a compactness argument: an extremal pair at distance $4$ would consist of two triangles, which are linearly equivalent and hence must in fact be at distance $1$. The resulting gap below $4$ is uniform but nonquantitative.

Machine Learning and more specifically Deep Learning involves solving large scale nonconvex optimization problems. Several algorithms have been proposed in the literature, that seem to achieve satisfactory practical efficiency for difficult instances, the Stochastic Gradient Method being the most rudimentary, while still outperforming more recent algorithms at a number of learning tasks. A major open question about the current methods used in deep learning is to understand their convergence properties. Following a line of previous works about the long time behavior of gradient-type algorithms, %and in particular the recent contributions from Azizian et al., we present a new approach for studying the asymptotic properties of a wide family of methods from an ergodic theoretical viewpoint. Our main results include a study of the expected time for a stochastic optimisation algorithm to reach a certain small neighborhood of a minimizer and show that this reaching time distributes exponentially around its average, which is given by the inverse of the stationary measure of the target. The assumptions on the Stochastic Gradient noise include the Gaussian and the Sub-Exponential assumptions.

In correspondence beginning on August 8, 2026, V. S. Monakhov and I. L. Sokhor communicated the following question to the author: does every finite group have an unrefinable chain of subgroups whose successive indices are nondecreasing? We answer this question in the negative, even among soluble groups. For every odd prime p, we construct a soluble matrix group of degree five over the field with p elements. The group has order 64 times the eighth power of p and has no such chain. These counterexamples form an infinite family whose smallest member, obtained when p equals three, has order 419904. The construction is based on an example due to Kohler, and the obstruction follows from three elementary calculations of maximal subgroup indices.

We prove that for every even integer $N\geq 4$ or $N\in\{3,5,7\}$, every axially symmetric solution to the $Q$-curvature-type problem $$ \alpha P_N u + (N-1)!(1-\frac{e^{Nu}}{\int_{\mathbb{S}^N} e^{Nu}dw})=0 \ \ \ \ \ \mbox{on} \ \mathbb{S}^N $$ is constant, provided that $ \alpha\ge\frac{1}{2}$ and $\alpha \not =1$. The proof starts from a Gegenbauer expansion and its weighted $\ell ^2$ estimate on coefficients. Then one key estimate is the refined estimate of the seminorm $\lfloor G\rfloor^2$ using a specially designed integration-by-parts identity for general even integer $N\geq 4$. For $N\in\{3,5,7\}$, the seminorm estimates are established via a descent to a three–dimensional nonlocal difference quotient identity.

We prove a general upper bound for scaling windows of sparse monotone covering problems. From that, we deduce that for every fixed value $k\geq 3$, the window of random $k$-SAT is $O(n/\log n)$, improving the Friedgut-Bourgain bound of $O(n/\log\log n)$. We also show that random signed Not-All-Equal-$k$-SAT and hypergraph non-two-colourability (Property B) have windows of the same order. Our upper bound for scaling windows of sparse monotone covering problems follows from a general strategy combining the strengthening of Bourgain's sharp threshold theorem by Keevash, Lifshitz, Long, and Minzer with a local-to-random replacement principle. This provides a systematic method for proving scaling-window bounds, addressing a question of Perkins.

Ordered-statistics decoding (OSD) post-processing substantially improves the performance of belief propagation (BP) ensembles, but it also removes their natural syndrome-based stopping criterion. The high computational complexity of ensemble decoding and OSD can be reduced by sequential activation of ensemble members paired with an early termination. Instead of a syndrome-based termination we therefore apply dynamic ensemble termination (DET) to sequential ensembles: members are evaluated one at a time and decoding stops when the estimated risk that an unseen member would correct the current decision falls below an offline-fitted threshold. With hardware-oriented design in mind, we instantiate the framework with layered normalized min-sum component decoders, a row-boosted ensemble, conditional low-order OSD on the dual code, and a conventional low-cost decoder gate. A decision tree estimates the residual risk from statistics of the current candidate list. Member-dependent thresholds then target a prescribed frame error rate (FER) loss. We validate this framework on mutliple IEEE 802.11 low-density parity-check (LDPC) codes with blocklengths \(648,1296\), and \(1944\) at rates \(1/2\) and \(5/6\). At each code's design point, where the full \(64\)-member list reaches an FER of \(10^{-3}\), the DET decoder processes \(1.04\) to \(1.15\) members on average. On the \((648,540)\) code, it reduces BP work by \(39\times\) and OSD activations by \(18\times\) against the full list while retaining an signal-to-noise ratio (SNR) gain of about \(0.4\,\mathrm{dB}\).

Collections of diverse observations often share one acquisition, processing, geometric, or channel corruption, while only an unpaired clean reference set is available. For a prescribed low-dimensional correction shared across observations, the observed and clean reference sets support inference only through the response of fixed moments. We formulate this problem as two-sample moment calibration and report a rank-aware information state combining local rank, scaled moment sensitivity, source-separated covariance, and a moment compatibility residual. Full rank gives local moment identifiability, whereas kernel directions remain unresolved to first order. A unified linearization separates observed-set and reference-set uncertainty. Under covariance weighting, the weakest scaled singular value determines worst-direction asymptotic amplification. For an orientation-preserving planar-similarity correction shared across observations, ensemble centroids and a nonzero third-order complex moment yield closed-form global population identification of translation, rotation, and isotropic scale under matched-population and no-clipping assumptions. Controlled validation tests the predicted $N^{-1}$ and $\sigma_{\min}^{-2}$ laws, Gaussian efficiency, and interval coverage. Bounded applications report color corrected-output quality, channel magnitude-response calibration, and a separate paired geometric de-beautification result. The framework therefore reports missing or weak information instead of treating every fitted correction as identified.

Lin, Frankl, and Wu proved that the product of the sizes of two cross-intersecting 3-uniform hypergraphs with covering number three is at most 121. They conjectured that requiring the families to be disjoint lowers the sharp bound to 100. We prove this conjecture. The smaller family has at most eleven edges; after fixing it, the other family may be enlarged to its external family of 3-transversals. Splitting by matching number leaves an elementary intersecting case and two finite kernels. We prove the correctness of both encodings and every pruning rule, and exhaustive C++20 computations give the external-blocker maxima 21, 19, 16, 14, 12, 11, 10, 9 for family sizes 4 through 11. Concrete witnesses are checked independently in Python. A pair of ten-edge families on six vertices attains product 100.

We study maximal (proper) filters in the complete lattice of partitions of an infinite set~$X$. In the language of uniform spaces, these are precisely the atoms of the lattice of zero-dimensional uniformities on $X$, introduced by Pelant and Reiterman and studied further by Pelant, Reiterman, Rödl and Simon. The first half of this paper recovers, sharpens, and extends their classification in purely partition-theoretic terms. {Call a maximal filter of partitions type I if it does not contain all finite partitions, and type II if it does.} Type I filters are induced, in an essentially unique way, by ultrafilters on families of pairwise disjoint doubletons. {A type II filter} determines a non-principal ultrafilter on $X$, the heart; the heart determines the filter precisely when {the former} is minimal in the Rudin–Keisler order. Each member $F$ of {a type II filter} gives rise to a closed fiber in $X^{*} = \beta X \setminus X$, {consisting of} the set of ultrafilters agreeing with the heart on $F$. We prove a trichotomy describing the topology of arbitrary fibers. We then show that the fibers do not encode the filter: fibers do not form a semilattice under intersection, the closure of an infinite discrete set of ultrafilters of a single Rudin–Keisler type (a sparse set, in our terminology) need not be a fiber, and a partition incompatible with a member of the filter may have a {fiber strictly larger than that member.} A representation that does succeed is nevertheless available in another category: the {type-II filters} with heart $\mathfrak{u}$ correspond to the maximal proper substructures of the ultrapower $X^X\!/\mathfrak{u}$ of the full structure on $X$. The topological representation problem remains open.

Nonlinear stabilization is indispensable for discontinuous Galerkin (DG) discretizations of hyperbolic conservation laws, yet it typically disrupts the delicate error structure required for superconvergence analysis. Consequently, existing theory has largely been restricted to linear or semi-discrete schemes lacking oscillation control. This paper bridges this gap by proposing a Runge–Kutta (RK) aligned oscillation-eliminating (OE) DG framework that restores the superconvergence properties. By synchronizing the pseudo-time step in the OE procedure with the cumulative RK stage coefficients, we unlock a cancellation mechanism for low-order interface errors that is inaccessible to standard OEDG formulations. We rigorously prove that this fully discrete scheme achieves $(k+2)$-th order superconvergence to a tailored projection of the exact solution for linear conservation laws in both one and two dimensions, while maintaining the non-oscillatory shock-capturing capabilities of the original method. Moreover, we establish a general guiding principle for designing a class of OE-type DG schemes that exhibit such superconvergence. Key theoretical innovations include the construction of stage-aligned correction functions to compensate for nonlinear OE sources and the discovery of a two-dimensional projection operator that preserves outflow-edge averages, a property essential to close the discrete shift estimates. Numerical experiments confirm the predicted superconvergence rates and demonstrate that RK alignment preserves the parameter-free robustness of OEDG for problems with strong discontinuities.

We give negative answers to two questions of X.Xu concerning the occurrence of the unit interval in nonquasialgebraic quasicontinuous domains. We construct, directly from the binary tree and the binary-value map, a well-founded quasicontinuous dcpo \[ P=2^{<\omega}\mathbin{\dot\cup}[0,1] \] which is not quasialgebraic. Moreover, $P$ contains no sub-dcpo isomorphic to $[0,1]$, and $[0,1]$ is not a Scott-continuous retract of $P$.