Papers

We maximize the average torsional rigidity among open subsets of $\mathbb{R}^2$ with fixed inradius whose complements consist of pairwise disjoint balls of fixed radius $\varepsilon$ with mutual distances bounded below. We prove that, provided $\varepsilon$ is sufficiently small, the optimal value is asymptotically attained by sequences of domains whose complements exhibit a regular honeycomb structure. The proof is reduced, via Delaunay triangulations, to the analysis of a mixed Dirichlet–Neumann problem on triangular domains with circular cutouts at their vertices. Our approach is quite general and extends to other variational energies, such as the Cheeger constant, for which we also derive an explicit bound of the ratio $\varepsilon/$ inradius.

Zilber's celebrated quasiminimality conjecture states that a subset of $\mathbb C$ defined by polynomials and exponentials is either countable, or has countable complement. Koiran has asked whether the same could be true replacing $\exp$ with any unary entire function. We give a negative answer to Koiran's question, using results from the theory of holomorphic approximation.

A mean shift between two data sources can be easy to detect but hard to remove without substantially changing their representations. We cast its removal as a statistical decision problem: from noisy differences between paired calibration measurements in $\mathbb{R}^d$, learn one linear map, applied to both sources under a hard distortion budget, that leaves as little of the shift as possible on fresh data. We derive the exact finite-sample minimax risk over all such maps, $(d-k) \mathbb{E}[1/(d+2J)]$ with $J\sim\mathrm{Pois}(\kappa/2)$, where the budget allows deleting $k$ directions and $\kappa$ is the calibration signal-to-noise ratio. Projecting out the mean calibration difference attains it without knowing $\kappa$ or the noise scale. This exposes a detection-repair gap: detecting the shift needs only $\kappa\gg\sqrt d$, whereas removing a fixed fraction of it at constant distortion needs $\kappa\asymp d$, as for estimating its direction. Standard linear concept erasers (MP, SAL, LEACE) remove the same calibration difference, so the formula gives, before fitting, exactly how much shift they leave on fresh data and how much calibration a target requires. The limit is robust: pairing keeps it exact for non-Gaussian shared content, the projection keeps its guarantee under anisotropic noise, and selective abstention cannot close the gap. On paired clinical and wearable sleep EEG, where differences between participants act as calibration noise, the formula predicts the device shift left in new participants, and more recordings per person soon stop helping. Together, these results tell whether a correction that falls short needs a better method, more recordings, or more participants.

We prove a local Hartman-Grobman theorem at zero for the unforced Navier-Stokes equation on the three-dimensional torus in the strong topology of $H^s$ for every $s>7/2$. More precisely, we construct a homeomorphism between open positively invariant neighborhoods of zero that conjugates the Navier-Stokes semiflow to the heat semigroup. We develop an infinite-dimensional boundary-time scheme built on a nested hierarchy of exact asymptotic dynamical factors. Their fiber geometry yields Sobolev block coordinates with autonomous finite-dimensional factor systems, on which boundary-time topological conjugacies can be constructed at every finite level and then assembled on a common neighborhood into the desired $H^s$ conjugacy. The infinite-dimensional assembly relies on new spectral estimates for the divergence-free transport structure, which control derivative loss and the blockwise tails of the finite-level conjugacies and their inverses without an additional spectral-gap assumption.

We consider the type AIII (quasi split) quantum symmetric pair subalgebras $U_q'$. We define a new Cartan subalgebra in one of the two subfamilies. Together with Letzter's Cartan subalgebra in the other, we construct a triangular decomposition of $U_q'$ via the Letzter map. Then we define Verma modules as induced modules and prove that finite dimensional simple $U_q'$-modules are quotients of our Verma modules. These treatments are uniform in both subfamilies. In the second part we define a diagrammatic category $Web^B$ that controls polynomial representations of $U_q'$. We prove a multiplicity-free decomposition of the quantum exterior powers $\bigwedge^k\mathbb{V}$ as a $U_q'$-module, using a explicit eigenspace decomposition of certain dot morphisms in $Web^B$. As applications, we deduce the multiplicity-freeness of the involved anti-spherical Hecke module. For $\bigwedge^k\mathbb{V}$, we explicitly determine the highest weight vectors and their weights, thereby determining the isomorphism classes of their irreducible summands. Finally, we determine the kernel of the $Web^B$-action on certain finite dimensional representation category of $U_q'$.

This work presents a prototype implementation of a noncoherent multiple-input multiple-output (MIMO) system based on Grassmannian signalling, where neither the transmitter nor receiver requires channel state information (CSI), operating at a carrier frequency of 15 GHz in the frequency range 3 (FR3) upper mid-band envisioned for 6G. To the best of our knowledge, this is the first prototype implementation of Grassmannian signaling reported in the literature beyond sub-6 GHz bands. We identify the motivation of applying Grassmannian signaling, a form of unitary space-time modulation (USTM), in this band for wireless and mobile applications with low spectral-efficiency requirements but strict reliability and power-efficiency demands. With this idea in mind, we present a system design integrated with orthogonal frequency-division multiplexing (OFDM) that addresses the challenges associated with the higher carrier frequencies of FR3 while providing reliable operation across a broad range of scenarios, including urban and rural environments as well as vehicular communications. We then present the prototype implementation of the proposed design, based on software-defined radio (SDR) devices integrated with radio-frequency (RF) mixers, filters, and power amplifiers to ensure adequate operation in the targeted band. Over-the-air (OTA) experiments based on symbol error rate (SER) measurements under non-line-of-sight (NLoS) and line-of-sight (LoS) scenarios, and across different data rates, show good agreement with software simulations, thereby validating the prototype implementation.

This paper addresses an extension of the pickup and delivery problem with time windows and multiple stacks (PDPTWMS). We call the new extension the pickup and delivery problem with time windows and requests split across multiple stacks (PDPTWRSMS). Applications of the PDPTWRSMS arise in less-than-truckload transportation systems where loads are packed as pallets or other standardised containers. In the PDPTWRSMS, the vehicle's loading space is divided into compartments of finite capacity. Loading and unloading in each compartment must obey the last-in-first-out (LIFO) policy. Moreover, a request can be split among all compartments, and a single request can also exceed a single compartment's capacity. Thus, each request is only limited to the vehicle's capacity. Computational experiments show that applying this extension can lead to substantial routing cost reductions (the total distance travelled and/or the number of vehicles used), and, compared with versions of the PDPTWMS where compartments are not restricted to a LIFO policy, splitting is worth more once the vehicle has three or more compartments, though less when it has two deep ones. We apply a modified version of the recently proposed method of fragments. Results confirm that this approach can solve many PDPTWRSMS instances and significantly outperforms the current state-of-the-art method for the PDPTWMS.

We construct smooth Einstein metrics with positive scalar curvature on $\#_k({\mathbb S^2}\times {\mathbb S^2})$ and on $k\mathbb{CP}^2\#(2k+1)\overline{\mathbb{CP}}{}^{2}$, for $k\geqslant2$. In particular, we show that positive Einstein metrics exist on infinitely many homeomorphism types of closed $4$-manifolds. The first family shows that any simply connected spin $4$-manifold carrying metrics with positive scalar curvature is homeomorphic to an Einstein $4$-manifold. Wick rotation gives associated Lorentzian static axisymmetric Einstein metrics with positive cosmological constant and arbitrarily many horizons with the same surface gravity. The metrics are obtained by varying the cone angle $2\pi\beta$ of a prescribed singularity along a torus, starting from the product metric on $\mathbb S^2\times\mathbb S^2$ and from the symmetric Kähler-Einstein metric on $\mathbb{CP}^2\#3\overline{\mathbb{CP}}{}^2$. At $\beta=\frac{2}{k+1}$, changing the period of the angle around the torus removes the cone singularity and changes the topology. For both families, as $k\to+\infty$, the metrics collapse along the torus orbits to an explicit $2$-dimensional disk, whose boundary has Hausdorff dimension $\frac{4}{3}$. At the scale of curvature, they converge to complete periodic Ricci-flat metrics with Kasner asymptotics: the Euclidean Myers-Korotkin-Nicolai metric for the first family, and a new one for the second.

We construct a smooth embedded open annulus in Euclidean three-space with strictly negative Gaussian curvature and injective Gauss map that contains a simple closed asymptotic curve of zero normal self-linking. This gives an affirmative answer to Problem 1.1 of Ghomi and Raffaelli. The construction starts with two simple planar curves whose tangent vectors are positive multiples of one another. Their mixed integral is made zero by rounding a pair of polygons with one free length parameter.

We propose a new formula relating the rationalized Drinfeld associators obtained from the Knizhnik–Zamolodchikov equation to the quantum $6j$-symbols of the algebra $\mathcal{U}_q(\mathfrak{g})$: $ U_{q} = U \cdot \Phi^{KZ}_{\mathbb{Q}} $. We prove this relation in the case of the $\mathfrak{sl}_N$ Lie algebra for the tensor cube of the fundamental and the first symmetric representation. We also check it perturbatively for the $\mathfrak{sl}_N$ algebra for representations $[2]\otimes [1] \otimes [1]$, $[2]\otimes [2] \otimes [1]$, the non-trivial cases with multiplicities — $[2,1]^{\otimes 3}$, $[2,2]^{\otimes 3}$, and in the case of the $\mathfrak{so}_5$ Lie algebra for the tensor cube of the fundamental representation.

In this paper, we study the existence problem of completely independent spanning trees (CIST) in chordal graphs through appropriate hypergraph representations and their panchromatic and bipanchromatic colorings. First, we disprove a conjecture stating an exact relationship between the panchromatic number, the bipanchromatic number, and the minimum number of unique colors in an optimal panchromatic coloring of a hypergraph. Then, by relating CIST to panchromatic and bipanchromatic colorings of the associated hypergraphs, we derive structural conditions for their existence in chordal graphs and specifically strictly chordal graphs.

Let $\Omega=G/K$ be an irreducible bounded symmetric domain other than the unit disk, and let $S$ be its Shilov boundary. We study the two-parameter family of Shilov–Poisson transforms between homogeneous line bundles over $S$ and $\Omega$. For a transform $F$ of $L^2$ boundary data, we determine what finite Euclidean regularity up to the full boundary $\partial\Omega$ forces in the interior. Outside five explicit exceptional parameter families, sufficiently high finite regularity forces $F=0$. In each exceptional family, the conclusion is an interior covariant differential equation. In the scalar Poisson–Szegő case, it yields a finite-order Graham type theorem: the stated boundary regularity forces a Bergman-harmonic function to be pluriharmonic.

We extend the generalised Calderon Transfer Principle as presented in \cite{dBL} to the setting of trace preserving group actions of $\sigma$-compact locally compact Hausdorff groups on semifinite von Neumann algebras $\M$. At its essence the transfer principle consists of showing that a large class of regular operators in the group context may be translated to operators in the algebra context by means of this group action. The end result is a protocol for proving ergodic convergence results for group actions on noncommutative Orlicz space of semifinite von Neumann algebras. We start with proving the existence of the transferred operator in the noncommutative context. Two tools are developed for the purpose of actually proving convergence results: (1) a theory of what may be called 2-variable decreasing rearrangements for the algebra $L^\infty \overline{\otimes} \M$ (where ($\Gamma,\nu$) is a Radon measure space) and (2) a concepts of maximal operators suited to the present context. These tools are then used to lift the ergodic convergence results presented in \cite{dBL} to the noncommutative setting. In closing we present examples illustrating the application of the tools

We study bounded, simply connected, convex, and C2-smooth domains in the space R^n for finite n greater than 1. For such domains we use the diameter to find balls with the property that they are included in this ball, while the diameter of the ball is chosen to be minimal. With a simple transformation we are able to map our domain onto a domain inside the open unit ball. We derive estimates for the Poincare constants of scalar and vector functions on those domains. The main ingredient is the first eigenvalue of the Laplacian or the Stokes operator with vanishing Dirichlet traces, respectively (we use results of [21]). Additionally we apply the relation of the first eigenvalues and one eigenfunction of the (scalar) Laplace and the Stokes operator cf. [21]. Our estimates for the Poincare constants are new. The dimension n of the space has significant effect on the quality of our estimates. This is illustrated with the help of an example.

We study the complementary version of Hedetniemi's problem for infinite graphs. We prove that if a graph $G$ and its complement $\overline{G}$ are both uncountably chromatic while their categorical product is countably chromatic, then $|V(G)|=\omega_1$. Assuming $\diamondsuit$, we construct a graph $G$ on $\omega_1$ such that $\chi(G)=\chi(\overline{G})=\omega_1$ and $\chi(G\times\overline{G})=\omega$; the construction uses two suitably chosen minimal Countryman lines. We also define a c.c.c. forcing of cardinality $\omega_1$ that adds a graph with the same properties. It remains open whether ZFC alone proves the existence of such a graph.

We study a heterogeneous diffusion process with a position-dependent diffusion coefficient in the presence of independent recruitment and mortality. This framework generalises heterogeneous diffusion under stochastic resetting, which is recovered as the special case of equal recruitment and mortality rates. At long times, the system approaches a non-equilibrium stationary state with a mean squared displacement that saturates to a value independent of the recruitment rate. Crucially, this saturation limit directly encodes the underlying transport regime via an effective exponent applied to the diffusion coefficient, and we characterise the non-trivial transition to this stationary regime. We further derive the mean first-passage time to a target and show that, for a fixed recruitment-to-mortality ratio, an optimal mortality rate minimises the search time. This "mortality-assisted search" effect is confirmed by numerical evaluation. Reinterpreting recruitment and mortality as infection and removal processes, we apply the model to epidemic spreading and show that a moderate removal rate can counter-intuitively accelerate, rather than delay, the arrival of infection at a new spatial region. Beyond epidemiology, our framework offers broad relevance to systems featuring concurrent particle creation and destruction, such as bacterial growth, tracer diffusion in reactive fluids, or cellular population dynamics in crowded environments.

We propose a one-step implicit-explicit (IMEX) Runge-Kutta framework for high-order time integration of geometric flows. Starting from the Dziuk and Deckelnick-Dziuk parametric finite element formulations of curve shortening flow, we recast the spatially semidiscrete equations as systems of ordinary differential equations and construct second- and third-order schemes. Each stage requires only a linear solve. The key ingredient is a reduced-coefficient structure that allows the variational problem at each stage to be posed on the geometry generated by the preceding stage. The Runge-Kutta stages thus supply the required prediction geometry without a separate prediction procedure. The resulting schemes require no additional starting values. The framework also extends directly to the Barrett-Garcke-Nürnberg formulations of mean curvature flow and surface diffusion for curves and surfaces. Numerical experiments support the designed orders of temporal convergence and indicate that suitable coefficient choices yield mesh quality comparable to that of the corresponding first-order methods.

We construct neural approximation classes native to the graph spaces H(div) and H(curl), in two and three dimensions and, for H(div), in any dimension. Every realization lies in the space for all parameter values, and with kinked potentials, such as ReLU networks, the admissible jumps appear at finite width. The classes are images of scalar and componentwise networks under fixed operators of the de Rham complex, and do not involve a mesh or finite element emulation. For H(div) in R^n two native classes are given on an equal footing, with a skew-symmetric potential $A$: $\mathrm{Div}\,A+R_nq+\mathbf{h}$, with the divergence $q$ as an explicit unknown, and $\mathrm{Div}\,A+\mathbf{z}$ with an $H^1$ field $\mathbf{z}$; for H(curl) the analogous classes are $\mathrm{grad}\,\phi+Sr+\mathbf{h}$ in two dimensions and $\mathrm{grad}\,\phi+\mathbf{z}$ in two and three dimensions. In all of them every interface jump of the field is carried by the potential term, $\mathrm{Div}\,A$ or $\mathrm{grad}\,\phi$, while the remaining part has no interface jump (it is an $H^1$ field in the regular-decomposition classes); the classes with $\mathbf{z}$ are the componentwise approach enriched by this term. Known or learned interface geometry enters the potential through factors with trainable amplitudes, and the remaining part if the divergence jumps. The classes lead to the FOSLS-deRhaNN method, first-order system least squares with de Rham neural networks, whose loss is the least-squares functional posed in the natural spaces of the weak formulation; for elliptic equations this includes $H^{-1}$ right-hand sides and $H^{1/2}$ Dirichlet data. Elliptic equations with discontinuous coefficients and curl-curl problems are treated as instances, with the functional equivalent to the error; linear transport with discontinuous solutions and conservation laws with shocks use the same flux classes.

The 150 years old Sylvester's conjecture asserts that every prime $p\equiv 4,7,8\mod 9$ is the sum of two rational cubes. Recently, a proof of this old problem is obtained by the work of mine for the case 4,7 \cite{Yin} and the work of Burungale-Tian for the case 8 \cite{BT}. In this paper we give a shorter proof by deep mining my result in \cite{Yin} combining with the ideas of my previous work on case 8 in \cite{Yin8}. In particular, we use the ordinary trace and the Kronecker congruence of modular functions, but do not use the $\sqrt{-3}$-division boundary and Fermat isogeny as in \cite{BT}.

In this paper we develop a parabolic version of the method of moving planes on systems with a mixed local and nonlocal diffusion operator, that being $\mathcal{L} = -(-\Delta)^s + \Delta$, where $(-\Delta)^s$ is the integral fractional laplacian. We then use this result to study the quenching phenomena occurring in said systems with a singular absorption term of the type $(1-u)^{-p}$. We prove that there exists a range of multiplicative parameters for which every solution presents quenching, and that for any initial datum $u_0$, there exists a small enough multiplicative parameter such as the solution is globally defined. We also prove that the quenching phenomena cannot happen arbitrarily close to the boundary, but they are contained in a compact subset of the domain.

We study cyclic triorthogonal codes in prime dimension $p$, on which a transversal third-level diagonal gate acts as a logical non-Clifford gate. For odd $p$, we use a definition of triorthogonality that does not require self-orthogonality. For cyclic codes of length coprime to $p$, triorthogonality reduces to a sumset condition on the spectral support, which makes exhaustive search feasible: for two natural constructions, we enumerate all maximal codes of this kind for $p=3$ up to length $121$ and $p=5,7$ up to length $96$. For qubits, we prove that a cyclic code of odd length is triply even if and only if it is classically triorthogonal, giving a complete description of triply-even cyclic codes, and we determine all such codes up to length $763$. For qudits, we find codes such as $[[19,1,3]]_3$ and $[[7,1,3]]_7$, which improve on the shortest Reed-Muller and Reed-Solomon codes in both overhead exponent and distillation threshold. The latter is the first member of an infinite family of $[[p,1,2\lfloor p/6\rfloor+1]]_p$ codes.

We study an interior inverse scattering problem for the two-dimensional Helmholtz equation with backscattering data, where the source and receiver coincide. Such data are common in applications, yet existing uniqueness results are almost exclusively established for full multistatic data with separated sources and receivers. We consider two main measurement regimes. In the first regime, data are collected on a closed curve inside the unknown domain. Exploiting the meromorphic dependence of the backscattering field on the wavenumber, we show that the residues at the poles determine the diagonal of the spectral projection kernel on the curve, and hence the visible part of the Dirichlet spectrum. This yields an unconditional spectral comparison: the symmetric difference of the two Dirichlet spectra is contained in the spectrum of the interior domain enclosed by the curve. If, in addition, the common visible eigenvalues have either matching multiplicities or matching restriction ranks, we obtain an area estimate bounding the discrepancy of the two domain areas by the area of this interior domain. In the second regime, data are given on a nonempty open subset of the domain, and a single non-resonant frequency already uniquely determines a bounded connected Lipschitz domain. The proof combines interior real-analyticity of the regular part of the Helmholtz Green function with its boundary blow-up.

Kubo-Ando means are a rich class of important binary operations on the set of positive elements of an operator algebra. The aim of this article is to understand the algebraic structure of these operations. We generalize a recent result of L. Molnár for $C^*$-algebras to the class of split $C^*$-by-nilpotent algebras by using the method of analytic local expansion of binary operations. Eventually, we prove that in any $C^*$-algebra $\mathcal{A}$, the identity $[x,[y,z]]=0$ implies commutativity.

We consider a generalization of the Gaussian information bottleneck problem to jointly stable variables $X$ and $Y$. Specifically, for $X = Y + A$, where $A$ is stable and independent of $Y$, we find optimal stable linear encoders of the form $T = kX + N$, with $N$ stable and independent of $X$. We characterize the linear stable information bottleneck in closed-form and its critical value. Furthermore, we show that, except for the Gaussian case, such stochastic linear encoders are strictly suboptimal, while becoming asymptotically optimal at large compression rates. Our linear solution recovers the Gaussian information bottleneck for jointly Gaussian variables as a special case.

We construct an explicit rational \(\mathrm{SL}_2(\mathbb C)\)-action on the underlying vector space of the rank-two \(\beta\gamma\)-system. The construction is motivated by Creutzig's branching decomposition with respect to \(L_{-1}(\mathfrak{sl}_2)\) and a rank-one Heisenberg vertex algebra. We realize the raising and lowering operators as residues of lattice-type fields, prove that they preserve the original \(\beta\gamma\)-system despite being defined on localizations, and establish their commutation relations and integrability. The resulting action commutes with the affine \(L_{-1}(\mathfrak{sl}_2)\)-action and all nonzero Heisenberg modes, while the Heisenberg zero mode serves as its Cartan operator. Each affine multiplicity space is thereby identified with \(V_n\otimes\mathcal F_{\mathrm{osc}}\), where \(V_n\) is the irreducible \(\mathrm{SL}_2(\mathbb C)\)-module of highest weight \(n\) and \(\mathcal F_{\mathrm{osc}}\) is the Heisenberg oscillator Fock space. Thus the Heisenberg charge decomposition is realized as the weight decomposition of \(V_n\). The infinitesimal action is not by vertex algebra derivations and does not preserve the symmetric conformal grading.

We study the integer sequence that enumerates cokernel-closed, idempotent split, full, additive subcategories of the category of finite dimensional complex representations of a uniformly oriented $A_n$ quiver. Using a combinatorial model, we describe a non-obvious connection of this sequence to Catalan numbers and derive an implicit recurrence. We also describe some basic properties of the lattice underlying the sequences, in particular, we give an explicit description of the meet irreducible elements of that lattice.

We study sparse locally private discrete mechanisms generated by a nonnegative kernel and an input-dependent admissible output support. The support set is intentionally small relative to the ambient alphabet and acts as a mechanism-design parameter. This formulation covers metric-ball supports, sparse exponential mechanisms, sparse staircase mechanisms, sparse randomized response, sparse additive kernels such as Skellam perturbations, and Poisson-binomial-inspired bounded-support channels. We give a general exact characterization of pure and approximate local differential privacy for this sparse-kernel class. Pure local differential privacy forces all supports to coincide, so genuinely input-dependent sparse supports are incompatible with pure privacy. In the approximate regime, the privacy defect decomposes exactly into support leakage and overlap excess loss. We instantiate the formula for several discrete mechanism families. For sparse staircase mechanisms, a shell-ratio condition eliminates overlap excess loss. For graph-metric supports, overlap of metric balls is necessary for nontrivial privacy. For sparse randomized response, the support-leakage term has a closed form in terms of support mismatch. For generic radius-truncated additive kernels and, in particular, sparse Skellam mechanisms, we obtain exact finite-support formulas in terms of the Skellam CDF and an exact Bessel-ratio overlap condition. Numerical evaluations illustrate how support radius and kernel shape separately affect support leakage and overlap excess loss.

For every integer $N\ge4$, we construct a fixed smooth, non-locally-conformally-flat metric on the $n$-dimensional unit sphere $\mathbb{S}^n$ for which the constant $Q_{2N}$-curvature equation admits an $L^\infty$-unbounded sequence of positive solutions. The construction applies for $n\ge2N+20$ when $N=4,5$, $n\ge2N+19$ when $6\le N\le8$, $n\ge2N+18$ when $9\le N\le17$, and $n\ge2N+17$ when $N\ge18$. For each $L \in \mathbb{N} \cup \{0\}$, the metric can be chosen arbitrarily close to the round metric in the $C^L$ norm. Together with the known cases $N=1,2,3$, this establishes noncompactness at every order, with dimension bounds that we expect to be optimal. We derive an explicit formula for the fixed-volume Hessian of total $Q_{2N}$-curvature in transverse-traceless (TT) directions at closed Einstein metrics satisfying $\operatorname{Ric}=(n-1)g$. Building on Juhl's formulas, we identify this Hessian, for every $N\in\mathbb{N}$ and $n>2N$, as a degree-$N$ polynomial in the Lichnerowicz Laplacian. For the metric perturbations generated by algebraic Weyl tensors in the construction, the quadratic reduced energy with fixed bubble center is a positive multiple of this Hessian restricted to a four-dimensional TT space. Differentiation with respect to the bubble scale then yields a finite matrix whose exact sign analysis establishes the required indefiniteness in the stated dimension ranges. After continuation to real $n$, this matrix changes from negative definite to indefinite at $n_N=2N+a_*+c_*N^{-1}+O(N^{-2})$ as $N\to\infty$, with $a_*\approx16.201871$ and $c_*\approx13.512880$, explaining the eventual bound $n\ge2N+17$.

Splitting the exponential and exponential-like functions can significantly reduce the overall computational cost of exponential integrators, providing a strong motivation for their rigorous analysis. In this paper, we analyze splitting errors for two noncommuting unbounded operators and derive new representations that include the popular Lie–Trotter and Strang cases. Moreover, within the abstract framework of strongly continuous semigroups, we prove the convergence of a split version of the exponential Euler method and a second-order exponential Runge–Kutta method for semilinear problems. Numerical experiments on a nonlinear dispersive equation and a diffusion–reaction equation demonstrate the theoretical findings, stability properties, and efficiency of the proposed split exponential integrators compared to the state of the art.

The stylic monoid $\mathrm{Styl}_n$, introduced by Abram and Reutenauer, is the quotient of the plactic monoid by the relations $x^2=x$, and its elements are represented by $N$-tableaux. Volkov showed that the Catalan monoid $\mathrm{Cat}_n$ of order-preserving, order-decreasing self-maps of $\{0,1,\ldots,n\}$ is a quotient of $\mathrm{Styl}_n$. However, the quotient map is defined on generators, and it is not apparent how to see, from an $N$-tableau, the map in $\mathrm{Cat}_n$ it corresponds to. In this paper we give a simple visual way to read off this map, and some of its main properties, from the $N$-tableau. The new ingredient is that we do not insist on drawing an $N$-tableau as a classical Young tableau: we allow the entries of each row to be shifted relative to the row below, as long as each entry stays above a smaller one. We call this a positioning, and prove that the column word read from any positioning is plactically equivalent to the usual column word; so every positioning can be used to compute the quotient map. We work with the tight positioning, in which each entry is pushed as far right as possible, and define the full core of an $N$-tableau: the part of each column that climbs by consecutive values from the bottom row. We call an $N$-tableau full if it equals its full core. We prove that passing to the full core does not change the image in $\mathrm{Cat}_n$, that full $N$-tableaux are in bijection with $\mathrm{Cat}_n$, and we show how to read the corresponding map directly off a full $N$-tableau.