Papers

We study the membership of commutators $[b,T]$ of Calderón–Zygmund operators in the Schatten–Lorentz classes in the matrix-weighted setting. Working on a space of homogeneous type $(X,\rho,\mu)$ with a matrix $A_2$ weight $W$, we characterise, for $p\in(1,\infty)$ and $q\in[1,\infty)$, the Schatten–Lorentz norm $\|[b,T]\|_{S^{p,q}(L^2(W))}$ in terms of an oscillation (Besov-type) norm $\|b\|_{\mathrm{Osc}^{p,q}}$ of the symbol: for all $p\in(1,\infty)$, finiteness of $\|b\|_{\mathrm{Osc}^{p,q}}$ is necessary; and for $p>p(\eta)$, where $p(\eta)$ depends on the regularity of the kernel of $T$ and a dimensional parameter of the space, the two-sided estimate $\|[b,T]\|_{S^{p,q}(L^2(W))}\approx\|b\|_{\mathrm{Osc}^{p,q}}$ holds. Extending a recent unified framework for unweighted and scalar-weighted versions of such results by one of us to the matrix-weighted setting, our approach combines the dyadic representation of singular integral operators with the nearly weakly orthogonal (NWO) sequences of Rochberg and Semmes.

Recently, two exponentiations of ultrafilters have been introduced in [7]. We rephrase them in nonstandard terms, and use the nonstandard formulations to characterize some of their properties. We also study their relations with divisibility by introducing factor-picking functions and highly exponential ultrafilters.

Fix a finite alphabet $\Sigma$ having at least two letters, and natural numbers $n,k\geq 1$. We denote by $D_n(\mathbf{\Pi}^0_k)$ the class of differences of $n$ subsets of the Cantor space $\Sigma^\omega$ in $\mathbf{\Pi}^0_k$, and by $\checked{D}_n(\mathbf{\Pi}^0_k)$ the class of complements of sets in $D_n(\mathbf{\Pi}^0_k)$. We prove, for any natural numbers $n,k\geq 1$, the existence of a finite alphabet $\Sigma$ having at least two letters and a finitary language $U_{n,k}$ over $\Sigma$ accepted by a one-counter automaton such that the $\omega$-power $$U_{n,k}^\infty:=\{ w_0w_1\cdots\in\Sigma^\omega\mid\forall i\in\omega ~~w_i\in U_{n,k}\}$$ is $D_{2n-1}(\mathbf{\Pi}^0_k)$-complete. We also prove a conditional result for the dual classes $\checked{D}\_{2n}(\mathbf{\Pi}^0\_k)$: under the assumption that there exists a finitary language $V_{n,3}$ accepted by a one-counter automaton, such that its $\omega$-power $V_{n,3}^\infty$ is $\checked{D}_{2n}(\mathbf{\Pi}^0_3)$-complete, there would also exist, for any integer $k>3$, a finitary language $V_{n,k}$ accepted by a one-counter automaton, such that its $\omega$-power $V_{n,k}^\infty$ is $\checked{D}_{2n}(\mathbf{\Pi}^0_k)$-complete. Finally we add another conditionnal result about Wadge degrees of rational $\mathbf{\Delta}^0_2$ omega powers, which completes the recent results of [FGL26].

We study the asymptotic behaviour of the nonlocal perimeter of tubular neighbourhoods. For a closed $C^{1,1}$ submanifold of Euclidean space, we show that a suitable rescaling of the nonlocal perimeter of its tube converges to an explicit constant times the Hausdorff measure of the submanifold. The constant is the fractional perimeter of the unit ball in the codimension of the submanifold. For finite point sets, we obtain an exact tube formula in terms of a generalised hypergeometric function. In contrast with the classical Weyl formula, this expansion has infinitely many terms, contains both noninteger and integer powers of the tubular radius, and has coefficients that depend on the relative positions of the points rather than only on their intrinsic geometry. We also establish a point-set expansion for general analytic kernels.

Let $\mathbf{G}=\{G_1,\dots,G_{2n}\}$ be a collection of not necessarily distinct balanced bipartite graphs of order $2n$ with a common bipartition $V=(X,Y)$. A cycle $C$ with $V(C)\subseteq V$ and $|E(C)|\leq 2n$ is called rainbow in $\mathbf{G}$, if there exists an injection $\phi\colon E(C)\to [2n]$ such that $e\in E(G_{\phi(e)})$ for each $e\in E(C)$. The graph collection $\mathbf{G}$ is said to be rainbow bipancyclic if it contains a rainbow cycle of every even length from 4 to $2n$. In this paper, we show that if $e(G_i)\ge n^2-n+1$ for each $i\in[2n]$, then $\mathbf{G}$ is rainbow bipancyclic unless $G_1=G_2=\dots=G_{2n}\cong B_{n}^1$, where $B_{n}^1$ is the graph obtained from $K_{n,n}$ by deleting all edges in its one subgraph $K_{n-1,1}$. As a corollary, we obtain that if $\rho(G_i)\ge \rho(B_{n}^1)>\sqrt{n(n-1)}$ for each $i\in[2n]$, then $\mathbf{G}$ is rainbow bipancyclic unless $G_1=G_2=\dots=G_{2n}\cong B_{n}^1$, which strengthens Theorem $5$ of [Electron. J. Combin., 33(3)(2026), \#P3.29] and Theorem $1.6$ of [Linear Algebra Appl., 743(2026), 250–270] from rainbow Hamiltonicity to rainbow bipancyclicity. We also characterize all bipartite graph collections that are not rainbow bipancyclic under the condition $\rho(G_i)\ge \sqrt{n(n-1)}$ for each $i\in[2n]$

Spectral duality structures (SDS) explain why a distinguished family of initial distributions makes the absorption probabilities of resetting processes insensitive to the resetting mechanism. Here we study their axioms. We prove that the six defining conditions are mutually independent and determine what each one contributes. The key tool is a structure theorem valid for an arbitrary permutation and an arbitrary positive weight: every insensitive distribution is supported on a union of cycles balanced at a common level, and the level of a cycle is fixed by the geometric mean of the weight on it. From it, positivity selects the admissible root, the constant orbit product is equivalent to the existence of an insensitive distribution of full support, and the involution is equivalent to the response rank being equal to the number of free weight parameters. Without the duality relation, insensitivity of a single distribution is cheap and generically isolated; duality turns it into structural neutrality of the separatrix. The full-rank condition splits into two layers: a weaker condition is equivalent to maximal response rank, while full rank also excludes insensitivity at any other level. We then show that a single SDS is internally rigid: compatible semigroup structures force a constant weight, compatible abelian group structures obey a precise arithmetic condition, congruences form a simple lattice, and the automorphism group is computed explicitly. Finally, the axioms do not determine the orientation of the response; in the two-site case we give the exact condition under which the geometric and analytic sign criteria agree.

An edge of a multigraph is rich with respect to a proper edge-coloring if all the edges adjacent to it receive mutually distinct colors, and a proper edge-coloring is a rich-neighbor coloring if every non-isolated edge has a rich edge adjacent to it. The least number of colors in such a coloring is the rich-neighbor chromatic index chi'_rn(G). So far it was studied for subcubic graphs only, and multigraphs of larger maximum degree are considered here for the first time. The only upper bound valid in general was the quadratic one inherited from the strong chromatic index, while a single rich edge already requires 2D-1 colors in a D-regular graph, where D denotes the maximum degree. We give an upper linear bound chi'_rn(G) <= 6D-8 which holds for every loopless multigraph G with D >= 3, and we show that a coloring realizing this bound can be found in polynomial time. Consequently chi'_rn(G) = Theta(D). For D-regular graphs with D >= 3 the lower bound is 2D-1, and the upper bound is 5D-8. The bound 5D-8 holds more generally for every multigraph with D >= 3 which has a dominating P_{>=3}-system, and we characterize these multigraphs: such a system exists if and only if k_2(G-T) <= 2|T| holds for every set T of vertices, where k_2 counts the components on two vertices. In particular the bound holds for every graph with 3 <= D <= 4d-4, where d denotes the minimum degree. The paper is concluded with a conjecture on the tight upper bound on chi'_rn(G).

Assuming a smoothness property for mixed-characteristic shtuka spaces and hyperbolic localization for twisted sheaves, we determine the inertial restriction of the Fargues-Scholze $L$-parameter for irreducible depth-zero representations. Let $E$ be a non-archimedean local field of residue characteristic $p$, let $G$ be an unramified connected reductive group over $E$, and let $\pi$ be an irreducible smooth depth-zero representation of $G(E)$ over $\overline{\mathbb{F}}_{\ell}$ where $\ell$ is a prime number which does not divide $p|\pi_0Z(G)|$. Let $\varphi_{\pi}^{\operatorname{FS}}$ be the Fargues-Scholze $L$-parameter associated to $\pi$. We describe the restriction of $\varphi_{\pi}^{\operatorname{FS}}$ to the inertia subgroup in terms of Deligne-Lusztig theory. We prove this by computing the inertia action on the cohomology of shtuka spaces with depth-zero level structure. As an application, we prove that the Fargues–Scholze $L$-parameters agree with DeBacker–Reeder $L$-parameters over $\overline{\mathbb{Q}}_{\ell}$.

We study the boundary regularity of solutions for the tempered fractional Laplacian \[ (-\Delta)^s_\lambda u=f \quad \text{in }\Omega, \qquad u=0 \quad \text{in }\mathbb{R}^n\setminus\Omega, \] where \(0<s<1\), \(\lambda>0\), and \(\Omega\subset\mathbb{R}^n\) is a bounded Lipschitz domain satisfying the exterior ball condition. The tempered operator is defined by \[ (-\Delta)^s_\lambda u(x) = c_{n,s}\operatorname{P.V.} \int_{\mathbb{R}^n} \frac{u(x)-u(y)}{|x-y|^{n+2s}} e^{-\lambda|x-y|}\,dy, \] which was originally proposed in [Koponen, Phys. Rev. E (1995), 1197-1199]. % Exponential tempering preserves the leading singularity of % the kernel but destroys its homogeneity, preventing a direct % application of classical boundary regularity estimates. % Extending these estimates [Ros-Oton and Serra, J. Math. Pures Appl. (2014), 275–302] to the tempered setting therefore % requires overcoming the difficulties caused by the loss % of scale invariance. By decomposing the operator into the classical fractional Laplacian and the correction operator \(\mathcal Q_\lambda^s\), and combining approximation, localization, and iterative arguments with interpolation inequalities, we establish the global estimate \[ \|u\|_{C^s(\mathbb{R}^n)} \le C\|f\|_{L^\infty(\Omega)}. \] For more regular data, we derive interior and boundary-weighted Schauder estimates. In particular, under suitable regularity assumptions on \(f\), these estimates yield \[ |D^\nu u(x)| \le C\,\delta(x)^{s-|\nu|}, \quad |\nu| > 0, \] where \(\nu\) is a multi-index and \(\delta(x)=\operatorname{dist}(x,\partial\Omega)\). These bounds quantify the possible growth of higher-order derivatives near the boundary and provide a regularity foundation for numerical error analysis of the tempered problem.

We study absorption at the origin in radial pressureless Euler flow in $\mathbb{R}^n$, $n\geq2$. For rotation-invariant nonnegative locally finite initial mass and bounded radial initial velocity vanishing at the origin, we prove global existence and uniqueness in a full-space radial entropy class with a one-sided Lipschitz bound and an initial kinetic-energy trace. Absorption is permanent, including retention of any initial origin atom. For finite positive mass with finite second moment, a Lagrangian projection representation characterizes the absorbed mass and its left limits at positive times through extreme minimizers of a free-position primitive. We derive a radial scalar momentum balance with a nonnegative Stieltjes boundary impulse measure. The minimum value of the same primitive yields a weighted Stieltjes identity equating accumulated time-weighted impulse with the integral of initial radii over absorbed labels, valid also for jumps and singular continuous absorption; the boundary impulse measure and the absorption measure are mutually absolutely continuous on $(0,\infty)$. For such finite-mass data with common total mass, strong $L^2$ convergence of initial quantiles and Lagrangian velocities, together with convergence of initial origin masses, yields convergence of absorbed mass at continuity times of the limiting absorption and local weak convergence of absorption measures on $(0,\infty)$. Finally, even with a fixed smooth initial radial probability density, every probability measure compactly supported in $(0,\infty)$ can be realized as the absorption measure by choosing a bounded radial initial velocity; in particular, purely singular continuous absorption occurs.

An isometric immersion $X: \Sigma^n \longrightarrow \mathbb{E}^{n+1}$ is biharmonic (BHH) if $\Delta^2 X = 0$, i.e., if $\Delta H =0$, where $\Delta$ and $H$ are the metric Laplacian and the mean curvature vector field of $\Sigma^n$ respectively. More generally, biconservative hypersurfaces (BCHs) are isometric immersions for which only the tangential part of the biharmonic equation vanishes. BCHs are first described and then constructed.

In this article, we demonstrate that there exists an infinite family of lens spaces L(p,q) that cannot be obtained from the 3-sphere by two integral surgeries. The first such lens space in our family is L(177,29).

We consider inverse problems regularized by a trained neural network with convex activations and unrestricted weights. The resulting optimization problem is generally nonconvex, and gradient-based methods may converge to poor local minima. We introduce a penalized epigraph lifting of the network and show that the lifted problem admits a difference-of-convex (DC) decomposition. For Lipschitz continuous activations, we establish finite exactness: the original objective is recovered exactly at a finite and explicitly computable penalty threshold $\bar\rho$. Building on this structure, we propose H-DC (Homotopic DC), a homotopic scheme that solves a sequence of DC subproblems with increasing $\rho$, warm-starting each level from the previous solution. Numerical experiments on synthetic problems, a manifold denoiser regularizer, and image inpainting with a learned Mumford–Shah regularizer show that H-DC consistently escapes poor local minima and reaches solutions with lower objective values than gradient-based methods from the same initialization.

We study the inviscid limit of the one-dimensional viscous Burgers equation from smooth initial data through nondegenerate shock formation and subsequent shock development. Chaturvedi and Graham \cite{CG} obtained sharp estimates of order $\nu^{1/4}$ up to the shock-formation time. We extend their analysis beyond this time, comparing the viscous solution with the inviscid entropy solution evolving from the same initial data. With the shock-formation time normalized to $t=0$, we prove a pointwise error bound on a fixed interval $0\leq t \leq \bar{\varepsilon}$, where $\bar{\varepsilon}>0$ is independent of the viscosity $\nu$. The bound consists of a term of order $\nu^{1/4}$ and an exponentially localized contribution originating from the viscous shock layer. The proof is based on matching an outer expansion, an inner expansion near the formation point, and a viscous shock-profile expansion. Combined with the results of Chaturvedi and Graham, this yields a quantitative description of the inviscid limit from the initial time through the development of the young shock.

Extending the methods previously established for plane sextics and quartic surfaces, we develop a unified and systematic framework for the equisingular deformation classification of projective models of $K3$-surfaces of arbitrary polarization. We treat all polarizations of degree $2n$ with $n\le 4$, including the higher-degree cases of sextic surfaces in $\mathbb{P}^{4}$ and octic surfaces in $\mathbb{P}^{5}$. The computational machinery developed here is degree-independent and provides a complete tool for enumerating and distinguishing the deformation classes arising from singular projective models of $K3$-surfaces. This framework lays the foundation for future investigations in higher degrees, particularly the geometry of equisingular strata, extremal configurations, real structures.

We establish lower bounds on the orbit cardinalities of $\text{SL}(2,\mathbb{Z})$ on the $\mathbb{F}_p$-points of the generalized Markoff surfaces given by $x^2 + y^2 + z^2 - xyz = \tau+2$. We show that outside of the reductions of a certain exceptional finite set of $\overline{\mathbb{Q}}$-points found by Dubrovin and Mazzocco, every orbit has cardinality $\ge p$. Together with a recent result of Nathaniel Kingsbury-Neuschotz, this establishes, for all but finitely many primes $p$, the classification conjectures of McCullough-Wanderley (2013), and gives a complete classification of the connected components of the Hurwitz stack of $\text{SL}(2,\mathbb{F}_p)$-covers of elliptic curves, only branched above the origin. This extends earlier work of Bourgain-Gamburd-Sarnak and the author in the Markoff case $\tau = -2$. Our method studies certain noncongruence modular curves attached to a $\text{SL}(2,\mathbb{Z})$-orbit. These modular curves parametrize elliptic curves with a $\text{PSL}(2,\mathbb{F}_p)$-cover. We use this moduli interpretation to show that the noncongruence modular curves involved have bad reduction, as a cover of the $j$-line, at some place above $p$. This implies that the degree of the cover is $\ge p$, which translates into the desired bound on orbits. Key inputs include a result of Bouw-Wewers on bad reduction of three-point $\text{PSL}(2,\mathbb{F}_p)$-covers of the projective line, and the general stable reduction theory of Henrio, Obus, Raynaud, and Wewers.

Opinion dynamics models relate interpersonal influence to consensus or persistent disagreement. We study a discrete-time model combining weighted-average and weighted-median updates on a fixed directed network. Individuals update synchronously with possibly different adjustment coefficients. We characterize equilibria through local balance conditions and opinion level sets, and establish convergence to a Lyapunov-stable equilibrium using nonexpansiveness and Krasnosel'skii–Mann theory. We define type-dependent cohesive sets: each weighted-average member assigns all its weight to the set, whereas each weighted-median member assigns at least one half. Consensus is reached from every initial condition if and only if no two such sets are disjoint. Otherwise, an open set of initial conditions leads to persistent disagreement. Replacing weighted-median individuals by weighted-average individuals preserves consensus whenever it is already guaranteed for all initial conditions. Four numerical studies examine the number and locations of weighted-average individuals. The frequency of reaching a prescribed consensus tolerance is nondecreasing with their fraction in the sampled ensembles, but individual realizations need not follow this trend.

Classical harmonic analysis is built upon Lebesgue (or more general) measure. Such a framework is remarkably successful for sets of full ambient dimension, but becomes intrinsically insensitive to lower-dimensional geometry: curves, surfaces, and more generally fractal subsets of \(\mathbb R^n\) with zero \(n\)-dimensional Lebesgue measure. In contrast, Hausdorff content provides a dimension-sensitive notion of size for lower-dimensional sets or fractal sets, but harmonic analysis in this setting is still largely limited to maximal operators. In this paper, we establish a weighted two-parameter good-\(\lambda\) inequality under Hausdorff content setting, providing a route toward other general operators from maximal operators over Hausdorff content. To do this, we firstly identify an appropriate reverse Hölder class $\mathcal R\mathcal H_{q,\delta}$ associated with Hausdorff content, and then prove the self-improving property of $\mathcal R\mathcal H_{q,\delta}$ by establishing the fundamental equivalence between reverse Hölder weights and capacitary Muckenhoupt weights. As direct applications, we obtain weighted Fefferman–Stein inequality and some boundedness on general operators over Hausdorff content. The methodology developed in this paper, even when reduced to the Lebesgue measure case, also provide novel insights, which shows that the good-\(\lambda\) principle is not intrinsically tied to an underlying measure, but can extended to a non-additive framework.

We introduce a general methodology to construct globally unstable trajectories near the normally hyperbolic invariant manifold (NHIM) of a saddle-center-center Hamiltonian equilibrium. The proposed approach builds upon the classical geometric mechanism of searching for trajectories near heteroclinic connections between invariant tori of the NHIM. The main ingredient is the Birkhoff normal form near the NHIM, which allows us to control the dynamics of the hyperbolic and center components. We construct transition maps on the NHIM that encode the dynamics along heteroclinic connections. By combining the transition maps with the inner dynamics, given by the restriction of the Hamiltonian flow to the NHIM, we show that these connections can be efficiently (in terms of the travel time) shadowed by true trajectories of the full Hamiltonian system. Finally, the efficiency of this methodology is illustrated in the case of the NHIM associated to the libration point $L_1$ of the Spatial Restricted 3-Body Problem (SR3BP) for the Sun-Earth system.

In this paper, we extend the notion of a removable subcategory to $n$-exact categories, and prove that an ideal quotient by a subcategory of projective-injectives inherits an $n$-exact structure from the original category if and only if the subcategory is removable. In the Frobenius case, the quotient remains Frobenius, and the stable $(n+2)$-angulated categories are equivalent. We give a sufficient condition for a subcategory of projective-injectives to be removable. We also show that the ideal quotient construction is compatible with cluster tilting. Finally, we apply the construction to $2$-cluster tilting subcategories of the category of ACM bundles on a Geigle–Lenzing projective plane, obtaining quotients for both natural and relative $2$-exact structures.

We study the no-flux problem for a taxis-type migration-consumption system with logistic damping in a bounded planar domain, where the motility function \(\phi\) vanishes at zero and degenerates algebraically of order \(\alpha\ge1\). We prove uniform boundedness in \(L^\infty(\Omega)\times W^{1,\infty}(\Omega)\) for every fixed damping coefficient \(b>0\), without any smallness assumption on the initial data, thus improving the previous \(L^2\)-boundedness results. For \(\alpha>1\), we further prove that the solution converges exponentially to the spatially homogeneous state \((a/b,0)\), with an explicit admissible rate. The proof combines a weighted Gagliardo–Nirenberg type inequality adapted to the degeneracy with the dissipative structure of the signal equation. Therefore logistic damping drives the system toward homogeneity, despite the loss of uniform elliptic regularization.

In 1965, Chowla conjectured that the values of the Liouville function become asymptotically uncorrelated at consecutive natural numbers. Recently, the two-point case is resolved by OpenAI. In 2026, Charamaras and Richter conjectured a generalization of the Chowla conjecture to the asymptotic independence of the consecutive prime Omega functions. In this article, we prove the two-point case of their conjecture. As a consequence, we obtain a dynamical generalization of the two-point Chowla conjecture.

We present a self-contained, algorithmic construction of almost invariant subspaces [Panati et al. (2003), De Nittis and Panati (2010), Freund and Teufel (2016)] for Hofstadter-like Hamiltonians $H_b $ acting on $\ell^2(\mathbb{Z}^2,\mathbb{C}^n)$. More precisely, for rational values of $b_0/(2\pi)$ we describe an iterative procedure that takes a “locally gapped" projection $P_{0}$ (possibly topologically non-trivial) which commutes with $H_{b_0}$, into a projection $P_{\varepsilon}$ that commutes with $H_{b_0+\varepsilon}$ up to an error of order $\varepsilon^\infty$. Our construction insures that the Chern number of $P_{\varepsilon}$ equals the Chern number of $P_0$ as long as $P_\varepsilon$ exists. Moreover, its integrated density of states varies linearly with $\varepsilon$ (the “Středa formula"). We also show that any spatially localized Parseval frame of $P_{0}$ can be mapped into to a localized Parseval frame of $P_{\varepsilon}$. We choose to work with a discrete model in order to keep the technicalities to a minimum level, with no need for advanced pseudo-differential calculus. Nevertheless, we are confident that most of our results can be extended to (unbounded) magnetic (pseudo)differential operators, but the technical burden would be much heavier. This is left for future work.

This paper deals with the pullback measure attractor for a non-autonomous distribution-dependent stochastic lattice wave equation with nonlinear noise on $\mathbb{Z}$. We establish well-posedness in the energy space $\ell^{2}\times\ell^{2}$ and show that the solution generates a continuous non-autonomous dynamical system on $\mathcal{P}_{4}(\ell^{2}\times\ell^{2})$, which possesses a unique pullback measure attractor $\mathcal{A}$. Furthermore, we prove the convergence of the attractor $\mathcal{A}^{\varepsilon}$ as $\varepsilon\to 0^{+}$ to the attractor of the corresponding distribution-independent limit system.

In this paper, we apply persistent homology to study finite topological spaces by associating persistence diagrams with their elements through the furtherness function. This approach enables us to classify elements according to their topological evolution within the space. We further establish a bottleneck stability theorem showing that the bottleneck distance between the persistence diagrams associated with two elements is bounded above by their mutual furtherness.

This paper investigates the large deviation principle for 2D stochastic $g$-Navier–Stokes equations driven by multiplicative noise and defined on an unbounded channel-like domain. We first prove the existence and uniqueness of probabilistically strong solutions for the stochastic equation under small noise perturbations by means of the local monotonicity method. Subsequently, we show the well-posedness of solutions to the corresponding control equation and the convergence in distribution of the solutions for the stochastic equation as the noise intensity tends to zero. Finally, we establish the Freidlin-Wentzell type large deviation principle for the solutions of the stochastic equation by using the weak convergence method. To overcome the lack of compactness of Sobolev embedding on unbounded domains, we employ a scalar stream function together with a truncation technique to derive uniform tail estimates for the solutions of the control equation.

The KP-II equation was introduced by Kadomtsev and Petviashvili to explain the transverse stability of line solitary waves in shallow water. Using Darboux transformations, we prove the spectral stability of Y-shaped line solitons in an exponentially weighted space that captures the amount of waves receding behind the line solitons. We show that the spectrum of the linearized operator consists of the continuous spectrum of the zero background together with three residual spectral domains whose boundaries contain continuous spectra associated with the constituent line solitons. Exploiting the analytic dependence of the pairings with adjoint eigenfunctions on the spectral parameters, we prove that the orthogonality conditions associated with all adjoint eigenfunctions can be reduced to orthogonality conditions against three families of adjoint continuous eigenfunctions. Moreover, we investigate linear stability of Y-solitons in a subspace satisfying orthogonality conditions against adjoint continuous eigenfunctions associated with outgoing modulations along each line soliton constituting the Y-soliton.

This paper considers a four-dimensional Riemannian manifold with an additional circulant tensor structure $Q$. We prove that such a manifold is almost-Einstein if the curvature tensor preserves the action of the structure on the last two arguments. Additionally, we examine an associated Riemannian product manifold $(M, g, Q)$ where $P=Q^{2}$, deriving explicit expressions for the curvature tensor in terms of the metric and the product structure. The sectional and Ricci curvatures of these manifolds are determined using special bases generated by the images of a given vector under the structure $Q$. Finally, we construct a non-trivial example of a four-dimensional locally product manifold by embedding a four-dimensional hypersurface into a ten-dimensional Euclidean space.

Using the formalism of presheaf automata, we introduce new classes of automata for modelling concurrency that extend higher-dimensional automata and symmetric higher-dimensional automata. We study their languages and characterise these languages in terms of posetal cospans.

We propose Pathfinder-BP, a recursive decoder for topological quantum codes that combines belief propagation with controlled bit-flipping. An initial BP pass identifies and permanently fixes high-confidence errors. Upon convergence failure, the decoder recursively explores alternative error patterns by tentatively flipping low-confidence variables. On surface codes, Pathfinder-BP achieves a noise threshold and logical error rate comparable to minimum-weight perfect matching (MWPM). A crucial advantage of Pathfinder-BP is its highly parallelizable structure, which yields a worst-case decoding latency quasi-linear in the code distance, making it a suitable candidate for low-latency hardware implementation.