Papers

We prove the Gromov-Hausdorff convergence of the state spaces, equipped with Connes' distance function, of the operator systems associated with spectral truncations for noncommutative $n$-tori, $n\ge 2$. The action of the $n$-torus on the noncommutative $n$-torus, together with an appropriate use of Bochner integrals, enables us to construct an $C^1$-approximate order isomorphism that establishes the convergence. The earlier convergence result for the classical case of tori [20] follows as a special case of our result. We conclude by studying the $C^*$-envelopes and propagation numbers of the truncated operator systems and show that the structural properties of the truncated operator systems for tori persist under deformation.

We equip Khovanov's colored $\mathfrak{gl}(2)$-homology groups with a degree-2 derivation. Over a field of characteristic $p>2$, the derivation specializes to a $p$-nilpotent differential, which leads to a categorification of colored $\mathfrak{gl}(2)$-homology at a $2p$th root of unity.

Gibbs samplers, and more generally component-wise Markov chain Monte Carlo algorithms such as Metropolis-within-Gibbs, can be implemented using either random-scan or deterministic-scan updates. How much convergence can depend on this choice of scanning rule has been a longstanding question. We study this problem through $L^2$ spectral gaps, measuring computational cost in units of component updates. For a $d$-block Gibbs sampler, the cost of random scan is at most twice that of any fixed deterministic scan, while the reverse cost ratio is at most of order $d^2$. We extend these comparisons to general reversible component-wise updates under the global block-wise contraction condition. If $K_j$ denotes the update of block $j$ and $P_j$ its Gibbs counterpart, and $\|K_j-P_j\|\leq \lambda_0<1$ for $j=1,\ldots,d$, then the cost of random scan is at most $2/(1-\lambda_0)$ times that of deterministic scan, while the reverse cost ratio is of order at most $d^2/(1-\lambda_0)$. Examples show that the cost bounds for random scan relative to deterministic scan are asymptotically sharp and that the joint dependence on $d$ and $(1-\lambda_0)^{-1}$ in the reverse comparison cannot be improved uniformly. This work was assisted by generative AI, including for formal verification of mathematical results in Lean. The human author reviewed and verified the mathematical content and takes full responsibility for the results.

We propose an antithetic coupling that reduces the computational cost of gradient random-walk methods for one-dimensional convection-diffusion equations. The methods represent the solution's spatial derivative by particles with positive or negative masses, whose cumulative sums reconstruct the solution, and model diffusion by Gaussian displacements. The particles reorder and, for a nonlinear flux, interact. Antithetic sampling negates the displacements of a second simulation, which positively correlates the contributions of particles matched by position when their masses have opposite signs. Our coupling instead gives matched particles of opposite sign identical displacements and retains opposite displacements for equal-sign pairs. We prove that each simulation keeps the distribution of an uncoupled run as the matching changes, so the averaged solution keeps the bias of one simulation. For fixed particle configurations and matching, the coupling minimizes each matched pair's conditional variance contribution after diffusion among all couplings with the same Gaussian marginal distributions. From the same configurations, its variance reduction over antithetic sampling at the final diffusion stage has a closed form for any flux. For the heat equation, a martingale decomposition extends both results to each step's contribution to the terminal variance. In a Burgers' equation test, the coupling meets a tolerance for the spatially integrated mean squared error with 4 simulations, against 34 for antithetic sampling and 65 for independent sampling. In a second test, it meets a common error tolerance in about 40% less time than pairing particles only within each sign.

We present a novel numerical integration method for non-periodic functions over the high-dimensional unit cube $[0,1]^s$ by a specially crafted (unequally-)weighted quadrature rule using transformed lattice points, with the additional option of subsampling. The method is designed for integrands whose mixed derivatives up to order $\alpha$ are square-integrable with respect to a product Chebyshev density. The ingredients are (i) periodizing the integrand, (ii) approximating the smooth periodic function by a kernel interpolant at lattice points, (iii) integrating exactly the product of the kernel interpolant and the nonsmooth density, together with subsampling alternatives for (ii) and~(iii). With $|J|$ corresponding to the number of function evaluations, we achieve the convergence rate close to the order $|J|^{-(\alpha-1/2)}$ and $|J|^{-\alpha}\,($1.110721$)^s$, where the implied constants are independent of $s$ under favorable conditions. We can interpolate these two results to trade between the convergence rate and the growth in $s$. We provide numerical experiments to demonstrate our theory.

We propose and study a compartmental epidemiological model that incorporates both waning immunity and mutation-driven immune escape into a single-strain framework. The model, termed SIRCm, extends the SIRC (Susceptible-Infected-Recovered-Cross-immune) model, in which recovered individuals pass through an intermediate cross-immune class before returning to full susceptibility. In SIRCm, both the rate of immune waning and the rate of immune escape are amplified by a feedback function tied to the current level of infection. We study two formulations of this feedback, a transmission-driven model, in which mutation opportunities arise at the point of new infection, and a prevalence-driven model, in which they arise in proportion to the infected population alone. For both models and any feedback strength, we prove existence of an endemic equilibrium using persistence theory, and characterize its behavior analytically in the weak and strong-feedback limits, recovering the SIRC and SIS models, respectively, as limiting cases. Unlike the SIRC model, we show that the SIRCm endemic equilibrium is unstable in certain parameter regimes. This leads to novel behavior, even in the absence of seasonal forcing.

We study the emergence of cuspons through wave breaking, with the Camassa–Holm equation as a principal model. For a broad open class of smooth initial data, the first singularity occurs at a unique point and has the sharp local Hölder regularity \(C^{3/5}\). We construct a continuation of the Lagrangian flow through this singular time and reconstruct from it a conservative Eulerian weak solution. We show that the first singularity does not simply persist after breaking; instead, it bifurcates immediately into an anticuspon–cuspon pair, each having the sharp local Hölder regularity \(C^{2/3}\), with opposite one-sided slope orientations. The two singular branches move apart from the original breaking point, and the portion of the profile between them reverses its local monotonicity. The proof is based on an extended inverse-slope variable that remains regular through wave breaking. At the first singular time, this variable has a nondegenerate quadratic zero, which splits immediately afterward into two simple zeros. The corresponding degeneracy of the Lagrangian flow map converts the quadratic first-breaking geometry into the \(C^{3/5}\) profile and the post-breaking simple-zero geometry into the \(C^{2/3}\) cuspon profiles. The generalized Hunter–Saxton family provides an explicit realization of the same Lagrangian mechanism and also allows us to track the subsequent interaction and annihilation of the singular branches. Thus an anticuspon–cuspon pair emerges dynamically from smooth initial data through wave breaking, rather than being imposed through a singular traveling-wave ansatz.

Spacecraft rendezvous and proximity operations (RPO) are currently planned through an expertise-intensive process in which engineers translate high-level operational intent into safe, dynamically feasible trajectories, creating a bottleneck to scalable operations. Large language model (LLM)-based agents could offer an intuitive interface for this process, although their outputs are not inherently grounded in orbital dynamics, operational constraints, or the structure of admissible spacecraft maneuvers. To exploit their semantic reasoning while ensuring the generated plan's physical validity, this paper presents a hierarchical framework for spacecraft task-and-motion planning (TAMP) that grounds LLM reasoning in a graph of reusable behaviors and domain-specific planning modules. Within this framework, a pretrained LLM maps a natural-language command to a partial mission specification. The associated planners then resolve unspecified decisions within the admissible operational space. Finally, trajectory optimization converts the completed mission specification into a dynamically feasible trajectory. Numerical experiments demonstrate that this architecture substantially improves intent recovery over direct LLM generation, achieving 98% exact recovery of partial mission specifications across all evaluated splits when backed by frontier LLMs. Additional test-time-compute experiments show that, for a compact 9B model, verifier-guided revision increases exact recovery from 75% to 88%, while broader behavior-plan search independently improves selection among admissible trajectory realizations. Overall, these results establish a scalable and auditable foundation for language-driven agentic planning of spacecraft RPO.

Let $Q$ be a translation invariant indefinite quadratic form in $8$ variables. Let $\mathcal{P}$ be the set of all primes. Let $\mathcal{A}\subset\mathcal{P}\cap\{1,\dots,X\}$. By developing the recent work of Green, we prove that subject to a mild condition on the form $Q$ there exist pairwise distinct primes $p_{1},\dots,p_{8}\in \mathcal{A}$ satisfying $Q(p_{1},\dots,p_{8})=0$ provided that \[\frac{|\mathcal{A}|}{X/\log X}\gg_{Q}(\log\log X)^{-c}\] for some absolute constant $c>0$.

The nonparametric maximum likelihood estimator (NPMLE) of a Gaussian location mixture maximizes the likelihood over the infinite-dimensional space of mixing distributions. The maximizing mixing distribution can be nonunique, and the classical bound on its number of atoms grows linearly with the sample size $n$. We show that a vanishingly small random perturbation of the likelihood yields exact polylogarithmic sparsity. The resulting randomly reweighted NPMLE maximizes a weighted likelihood whose independent weights, taken to be Gamma in our analysis, concentrate around one as $n$ grows. With high probability, it is unique, has $O\{(\log n/\log\log n)^d+\log n\}$ atoms in dimension $d$, nearly maximizes the ordinary likelihood, and estimates the mixture density at a Hellinger rate that is parametric up to logarithmic factors. This sparsity holds for the estimator itself, not for an approximation of it, and requires no support penalty. The proof rests on an effective-dimension principle for positive kernel mixtures: low-dimensional variation of the fitted values controls the support of every extreme point of the set of maximizers. Numerical illustrations verify that the reweighted NPMLE has Hellinger risk and support size comparable to those of the ordinary NPMLE.

We characterize admissible BLD remetrizations of branched self-covers of spheres by linear local contractibility of their canonical pullback length metrics. A relative homology obstruction involving the total degree of an inverse image component yields uniform porosity of branch values. Explicit degree-two cusp covers of $\mathbb S^n$, $n\ge3$, have uniformly porous branch values of Hausdorff dimension $n-2$ but admit no compatible Ahlfors $n$-regular, linearly locally contractible BLD source metric. Short homologically essential loops provide the obstruction, showing that porosity is necessary but insufficient for admissibility. We also prove quasisymmetric invariance and sharp degree dependence of porosity for sphere-to-sphere covers in dimension two.

Let $\PG(2,q)$ denote the projective plane over the finite field $\mathbb{F}_q$ where $q\geq2$ is a prime power. For a positive integer $t$, let $B_{t}[\PG(2,q)]$ denote the $t$-page book obtained from $t-$many copies of the $(q+1)-$graph $\PG(2,q)$, sharing a common edge. Employing the $\mathsf{L}^p-$method and using the incidence structure of the projective plane, we establish that the upper bound of Turán density of $B_{t}[\PG(2,q)]$ is $(q(q+1)^{-1})^{\frac{1}{qt}}$. For the $(q+1)-$graph $\PG(2,q)$, the previously known upper bound on its Turán density was $1-\binom{q^2}{q}^{-1}$. As a consequence of our estimate, this upper bound is improved to $(q(q+1)^{-1})^{\frac{1}{q}}$.

We prove the 2143-3421 case of Burstein's conjecture on the distribution of descent-top sets by constructing an explicit bijection between the two avoidance classes. The bijection also preserves the pinnacle set and reverses the left-to-right order of the pinnacle values. We combine an involution on maximum Cartesian trees with exchanges of the final entries of maximal decreasing runs. On an intermediate avoidance class, we prove that the exchange procedures terminate and that the output in each direction is independent of the choices made. This gives an explicit inverse. Restricting the bijection to Dumont permutations of the first kind proves the conjecture of Burstein and Jones that 2143 and 3421 are Wilf-equivalent on this class

We study the $\rm{PGL}_2$ skein algebras of small surfaces, focusing on the closed torus and the once-punctured torus. We provide explicit finite presentations by generators and relations for the $\rm{PGL}_2$ skein algebra of the torus and once-punctured torus, analogous to those of Bullock and Przytycki for the Kauffman bracket skein algebras. We give explicit bases of the $\rm{PGL}_2$ skein algebras of the torus and once-punctured torus and describe the multiplicative structure of the basis for the torus, analogous to the product-to-sum formula of Frohman and Gelca. We twist the basis for the torus to obtain one with structure constants in $\mathbb{N}[q,q^{-1}].$ We compute the kernels of natural maps between the $\rm{PGL}_2$ and Kauffman bracket skein algebras associated to the once-punctured torus and closed torus. We compute the center of the $\rm{PGL}_2$ skein algebra of the torus, which depends on the specialization of the quantum parameter $q.$

We give a randomized polynomial-time algorithm that colors any promised $3$-colorable graph on $n$ vertices with $\smash{O(n^{4/23}) = O(n^{0.17391\ldots})}$ colors, improving on the recent bounds of $O(n^{0.19539})$ by Bansal, Huang, and Lee and Narang and Tang who obtained $O(n^{(13-\sqrt{97})/18+\epsilon})=O(n^{0.17506\dots + \epsilon})$ colors for every fixed $\smash{\epsilon>0}$. To prove our result, we start from a fixed-level semidefinite relaxation, where we use a finite-depth recursion on Gaussian covers. Fixing a root vertex, we group vertices by correlation with the root vector. Here, each step extends a cover of directions by one edge and transfers it to a successor group. Our key analytic ingredient is a variance bound for Gaussian maxima: for a maximum of $m\geq 2$ centered linear forms with coefficient norms at most $r$, mean $\mu$, and variance $v$, we prove $v\leq r^2-\mu^2/(2\log m)$ using Chen's Gaussian convexity theorem. Together with a variance-scale lower-tail estimate, this controls the threshold loss at each extension, which shows that root-conditioned vector colorings can either extract a large independent set from a group or bound its size, forcing a contradiction after constantly many steps. The resulting sparse-case guarantee combines with the dense progress bound of Kawarabayashi, Thorup, and Yoneda, and the recursion's numerical inequalities are verified via rational interval arithmetic.

Quantum low-density parity-check (QLDPC) codes are promising candidates for low-overhead fault-tolerant quantum computation, but their practical use requires fast, low-complexity, and reliable decoders. Belief propagation (BP) is attractive because of its local message-passing structure, yet standard flooding BP often suffers from convergence failures on QLDPC codes due to short cycles, degeneracy, and symmetric decoding trajectories. Learned sequential BP improves convergence by using a reinforcement-learning policy to choose variable-node update orders, but a single learned trajectory can still be sensitive to unfavorable local Pauli decisions. We propose a learned parallel bit-flipping sequential BP decoder with a new quaternary score combining syndrome gain, a quantized Pauli log-likelihood penalty, and Q-table lookahead at hypothetical post-flip states. Our decoder first runs learned sequential BP for a fixed number of iterations. If the syndrome is not satisfied, it constructs quaternary bit-flipping candidates, each corresponding to changing the current Pauli decision of one qubit. The selected candidates initialize independent learned sequential BP continuations from the same decoder state. Since these continuations are independent, they can be executed in parallel, so testing several candidates mainly increases parallel hardware resources rather than sequential decoding latency. Simulations on representative QLDPC codes over the depolarizing channel show that the proposed decoder improves the reliability of learned sequential BP while having a parallel low-latency structure.

We give an explicit three-dimensional Hadamard matrix of order $22$. The example is presented by listing its twenty-two $22\times22$ two-dimensional slices.

The scalar auxiliary variable (SAV) method replaces the nonlinear part of the free energy by a positive scalar $r(t)=\sqrt{\mathcal{E}_{_\mathcal{N}}[\phi]+C}>0$, thereby yielding linear, unconditionally energy-stable schemes for gradient flows. At the discrete level, however, the standard backward Euler and Crank–Nicolson discretizations provide no guarantee that the computed $r^{n+1}$ remains positive, an inconsistency with the continuous definition that contradicts the square-root ansatz and may compromise long-time robustness. Although the SAV method has been widely applied, this subtle but consequential issue has received little attention. We first characterize this failure quantitatively by deriving a sharp criterion and a sufficient condition on the time step size, and construct an explicit counterexample showing that sign loss occurs for parameters of practical relevance. Rather than modifying the definition of $r$ as in existing positivity-preserving variants, we retain the square-root form and reformulate the discrete evolution from $r_t$ to $(r^{2})_t$, which converts the scalar equation into a convex quadratic with a strictly negative constant term, always yielding a unique positive root. For the Crank–Nicolson scheme, the product-form discretization $r^{n+1}r^{n}$ preserves this quadratic structure, while conventional alternatives do not. The resulting schemes incur the same computational cost as the original SAV method and are proved unconditionally energy-stable. Numerical experiments for the Cahn–Hilliard equation confirm the predicted positivity, energy stability, and convergence rates.

Density peak clustering (DPC) connects each observation to its nearest neighbor of higher density and identifies cluster centers as high-density observations with unusually large nearest neighbor uphill shifts. The resulting uphill paths from observations to cluster centers, however, can be irregular and unstable in low-density regions, making the clustering assignments sensitive to local perturbations and obscuring the population geometry of the DPC graph. In this paper, we introduce gradient-guided density peak clustering (GGDPC), which performs a gradient ascent step before each nearest neighbor uphill search. We develop a stability theory that relates the GGDPC graph to the gradient ascent flow of the population density. In particular, we establish consistency of GGDPC under five complementary criteria: recovery of local modes, adjusted Rand index, dendrogram (cluster tree), path length, and waterfall measure. Together, these results provide new statistical, geometric, and topological interpretations of DPC-type clustering algorithms.

We prove that at every infinite cardinality, every group which is neither abelian of exponent greater than two nor generalized dicyclic admits a graphical regular representation, settling the infinite-group part of Watkins's conjecture. We also determine the Cayley index of every infinite group: it is $1$, $2$, or $8$, according to its algebraic type, and in every case the index is attained by a connected Cayley graph. For every infinite group $G$ of cardinality $\kappa$, we construct $2^\kappa$ pairwise nonisomorphic Cayley graphs with exactly the unavoidable inverse-pair symmetries, diameter two, and $\kappa$ common neighbors at every distinct pair. The principal tool recovers a continuous ordinal hierarchy from an alternating adjacency baseline with bounded-degree errors: robust finite patterns identify the initial classes, successive twin quotients recover the layers, and their finite exception packets determine the translation action. The reconstruction applies without a group action and is stable under additional layerwise bounded-degree edits. Further results give closed Cantor-cube families with prescribed finite data in the regular cases, sharp cofinality-dependent graph properties, and optimal three-valued shortest-path metrics.

For a fixed digraph $F$, let $\operatorname{ex}_2^+(n,F)$ be the maximum of $\sum_{v\in V(D)}d_D^+(v)^2$ over all $n$-vertex $F$-free digraphs. Ai et al. [arXiv:2606.03520, 2026] asked for which self-converse tournament $F$ one can determine $\operatorname{ex}_2^+(n,F)$. Let $TT_r$ denote a transitive tournament on $r$ vertices and $\text{RT}_5$ denote a regular tournament on 5 vertices. Note that both $TT_r$ and $\text{RT}_5$ are self-converse. Iľkovič [arXiv:2609.05042, 2026] determined $\operatorname{ex}_2^+(n,F)$ for $F=TT_4$. In this paper, we determine $\operatorname{ex}_2^+(n,F)$ for $F$ being $TT_r$ or $\text{RT}_5$, and the latter confirms a conjecture due to Iľkovič.

We characterize extreme points of the (positive) unit balls of Marcinkiewicz spaces with respect to their natural quasi-norms. Having these results at hand, we show that every positive surjective isometry on a noncommutative Marcinkiewicz space is of elementary form. In particular, a linear mapping on a noncommutative Marcinkiewicz space over an atomic von Neumann algebra with all atoms having the same trace is a positive surjective isometry if and only if it is the restriction of a Jordan $*$-isomorphism which preserves the singular value functions. We also study (not necessarily positive) surjective isometries on weak $L_p$-spaces affiliated a $\sigma$-finite non-atomic von Neumann algebra for $1<p<\infty$, Marcinkiewicz sequence spaces, and weak $\ell_p$ operator ideals, $p>0$, as well as one-parameter groups of isometries on the latter two.

We prove new large value estimates for the Riemann zeta function on the critical line. For instance, we improve the upper bound on the measure of $t\in [T, 2T]$ with $|\zeta(1/2 + it)| \geq T^{1/8}$ for the first time since Hardy-Littlewood (1923). Our results imply improved upper bounds on the $k$-th moment of zeta for every $4 < k \leq 12$. We show in particular that \begin{equation*} \int_{0}^{T} |\zeta(\tfrac{1}{2} + it)|^6 \,d t \ll_\varepsilon T^{\frac{5}{4} - \frac{1}{60} + \varepsilon}. \end{equation*} The main ingredient is a new large value estimate for exponential sums with square-root phases, and certain perturbations. Those arise in the expansion of the short second moment of zeta.

We prove that the weak Pareto solution set of a multi-objective optimization problem with convex polynomial objectives and constraints is connected, assuming neither boundedness of the constraint set nor a constraint qualification. Since this set is semi-algebraic, its connectedness implies that it is path-connected whenever it is nonempty. In particular, convex quadratic multi-objective optimization problems have connected weak Pareto solution sets; this answers the weak Pareto part of an open question posed by N.~D.~Yen in Recent Developments in Vector Optimization (Springer, 2012).

Photoacoustic tomography (PAT) is a hybrid imaging modality that combines high optical contrast with high ultrasonic resolution for biomedical imaging applications. In this work, we investigate the inverse problem of recovering the initial pressure distribution from boundary measurements in the presence of nonlinear acoustic propagation and viscous attenuation effects. To model these phenomena more accurately, we consider a nonlinear damped viscoelastic wave equation incorporating spatially varying sound speed, temporal attenuation, and nonlinear propagation mechanisms. We first establish the well-posedness of the corresponding forward problem using a Galerkin approximation combined with energy estimates and a fixed-point argument. For the inverse problem, we derive existence, uniqueness, and local uniqueness results under suitable assumptions through a harmonic extension reduction, spectral Laplace transform techniques, and observability estimates. To numerically reconstruct the initial pressure field, we develop a hybrid reconstruction framework that combines a convolutional neural network (CNN) with a gradient-free optimization strategy based on the sequential quadratic Hamiltonian (SQH) method derived from Pontryagin's maximum principle. The CNN is used to generate an informative initial guess, while the SQH framework enforces the governing PDE dynamics during the reconstruction process. Numerical experiments demonstrate that the proposed hybrid strategy significantly improves reconstruction quality, contrast, and robustness compared to standalone time-reversal and CNN-based approaches.

We construct indefinite theta representations for a family of signed generalized Nahm sums of arbitrary rank \(r\geq2\). After explicit normalization and boundary correction, one signed branch gives the even components of a \((2r-1)\)-component holomorphic theta vector associated with a lattice of signature \((1,1)\). Euler-type multiple sums provide the odd components, yielding formulas uniform in both the rank and the component index. Using Zwegers' theory, we determine the weight-one modular transformation laws of the real-analytic completion, with an explicit finite sine matrix for the \(S\)-transformation. We decompose the nonholomorphic correction into a finite sum of products of unary theta functions and one-dimensional error-function series, and compute the antiholomorphic derivative of the completion. Together with the completed transformation laws, these formulas give the inhomogeneous transformation law of the holomorphic vector. We also evaluate a linear combination of two signed branches as a difference of Jacobi product quotients. This identity and further evaluations for two families of symmetrizable matrices and their duals, combined with known results, give multiple-sum representations for every component of four associated product vectors. These representations are finite combinations of generalized Nahm sums with the prescribed matrices and symmetrizers; in the dual families, half-period translates are absorbed into signs and constant phase factors.

We prove the real-rootedness conjecture of Lin, Ma, Ma, and Zhou (2021) for multiset Eulerian-Narayana polynomials, including simplicity and negativity of all zeros after removing the factor t. Using a forest decomposition and Lagrange inversion, we derive a differential formula and a first-order factorization of the differential operator. We use these factors to show that the nonconstant gamma-coefficients and ordered gamma-zeros are nondecreasing, with corresponding bounds given by the Narayana and Eulerian polynomials. The same factorization gives totally nonnegative coefficient transition matrices. We also prove strict interlacing of the zeros of generating polynomials arising from adjacent columns of the transition matrices.

For the $N$-body problem in dimension $d \ge 3$, we give an alternative proof of a theorem of da Luz and Maderna on the nonexistence of free-time minimizers defined on the entire real line, and extend it from the Newtonian potential to a larger class of singular subharmonic pairwise interaction potentials. The proof is based on the following generalization of the Cheeger-Gromoll splitting theorem: let $(M,g)$ be a connected Riemannian manifold and let $U$ be a smooth positive function such that $\mathrm{Ric}_g \ge \tfrac{\Delta U}{2U}g$, and the metrics $Ug$ and $U^{-1}g$ are both complete; if the metric $Ug$ admits a line, then the function $U$ is constant and $(M,g)$ splits off that line. We apply this splitting theorem to the Jacobi-Maupertuis metric associated with the $N$-body problem; a quantitative version of Marchal's theorem enables us to reduce to the case of smooth potentials.

This paper studies the focusing Calogero–Moser derivative nonlinear Schrödinger (CMdNLS) equation under the nonzero boundary condition \(|u(t,x)|\to 1\) as \(|x|\to+\infty\). In sharp contrast to the classical paradigm for the cubic nonlinear Schrödinger equation, namely that focusing nonlinearities support only bright solitons on a zero background, we show that the focusing CMdNLS equation admits a rich family of dark multi-soliton solutions on a nonzero background, whose existence is driven by a spectral gap \([-1,0]\) in the point spectrum of the Lax operator, together with the Hardy space structure. The main results are threefold. First, we establish an explicit formula for general solutions in the Hardy–Zhidkov space \(\mathcal Z^2_+\), via a semigroup method based on new commutator estimates between the Lax–Beurling shift semigroup and Toeplitz operators. This approach circumvents the non-commutativity of the unbounded generator with the time derivative and requires no weighted hypothesis on solutions or on the domain of validity of commutator formulas. Second, the algebraic definition of dark multi-soliton potentials is shown to be equivalent to two spectral conditions on the Lax operator. Together with the conservation of a generating functional, this yields the invariance of the dark \(N\)-soliton set \(\mathcal U_N\) under the CMdNLS flow. Third, the action-angle variables on \(\mathcal U_N\) are constructed, which establishes the complete integrability of the focusing CMdNLS equation on \(\mathcal U_N\). Combining these results, we obtain an inverse spectral formula, from which it follows that the poles of dark multi-soliton solutions satisfy a complexified Calogero–Moser system. Long time asymptotics and uniform Sobolev estimates are also derived.

In this paper, we construct an explicit correspondence between Engel and Pierce expansion digit sequences, obtained by adding $n-2$ to the $n$th Engel expansion digit, and show that it induces a homeomorphism from the set of irrationals in the unit interval onto the complement of a countable dense subset of this set. This yields a transfer principle for Baire category between the two expansions, and we determine exactly which Borel classes are preserved. We then identify this homeomorphism with the composition of two maps considered by Moroz (2027), and study the continuity of the second map, which sends modified Engel expansions to Pierce expansions. We determine its set of discontinuities, show that every discontinuity is a jump, and prove that the map is nowhere monotone. In particular, this confirms, for this map, the nowhere monotonicity and the continuity outside the exceptional set conjectured by Moroz (2027). As applications, we characterize the non-increasing position-dependent weights for which the associated series of Pierce expansion digits diverges on a comeager set, and we relate the convergence exponents of the Engel and Pierce expansion digit sequences.