Papers

We study travelling-rotating solutions of the Schrödinger map equation into the sphere, viewed as tangent profiles of rigid vortex filaments. Two first integrals reduce the profile equation to a scalar cubic equation for the vertical component, giving an elliptic-function description and explicit closure conditions. We prove bifurcation from the equatorial branch at $\lambda_k=R\sqrt{k^2-1}$, $k\ge2$, and establish a global continuation alternative inside the regular non-polar class. The possible boundary mechanisms are pole contact, vertical collapse, and double-root degeneration. Numerical continuation of the equatorial branches suggests convergence to the north-pole boundary. Up to gauge, the reconstructed vortex filaments are of Kida type.

Trust-region methods differ chiefly in how they update the radius, and each rule has traditionally been analysed on its own terms. We show that three structural conditions on the radius sequence suffice: a lower bound relative to the gradient norm, a contraction on unsuccessful iterations, and a controlled expansion on successful ones. A mechanism is weakly admissible if it satisfies the first two, and strongly admissible if it satisfies the first and third. Strong admissibility yields the optimal worst-case complexity $\mathcal{O}(\varepsilon^{-2})$ for first-order stationarity, and extends global convergence to model Hessians growing linearly with the iteration; weak admissibility gives $\lim_{k\to\infty}\|\nabla f(x_k)\| = 0$ under uniformly bounded model Hessians. Both hold for every rule in the class, so a new mechanism is covered by verifying two inequalities rather than by a fresh convergence proof. We verify admissibility for five mechanism classes (fixed-factor, step-driven, retrospective, criticality-anchored and gradient-scaled), prove convergence of the retrospective update under linearly growing model Hessians, and extend the Lyapunov framework of Curtis and Scheinberg (2020), Wang and Yuan (2022) to three distinct scaling factors with decoupled step acceptance, covering $\eta = 0$; its stochastic version specialises to the deterministic gradient-scaled rule. Part II treats the asymptotic behaviour of the radius and convergence to second-order critical points.

We extend the construction of Steinberg algebras of ample groupoids to étale semicategories. We also relate ample semicategories to Boolean restriction semigroups via a representation result extending previously known results for categories. Furthermore, we prove a reconstruction result which characterises an abstract algebra $A$ with a certain Cartan-like restriction subsemigroup $B$ (subject to conditions resembling those defining quasi-Cartan pairs) as the Steinberg algebra of the ultrafilter groupoid of $B$. In this way we obtain a twist-free extension of previous Steinberg algebra reconstruction results.

Let $G$ be a locally compact, second countable, amenable group acting on a finite von Neumann algebra $(\mathcal{M},\tau)$ by trace-preserving automorphisms. In this article, we establish a Jacobs-de Leeuw-Glicksberg decomposition for this action, yielding a decomposition of $\mathcal{M}$ into its almost periodic and weakly mixing components. We also prove a noncommutative version of the van der Corput lemma. As an application, we establish a noncommutative Wiener-Wintner theorem for amenable group actions on finite von Neumann algebras.

In 1934, Romanoff proved that the set of positive integers representable as the sum of a prime and a power of two has positive lower density. Erdős later constructed an infinite arithmetic progression of odd integers none of which admits such a representation. Let \(F_n\) be the Fibonacci sequence. In this paper, we prove that the set of integers of the form \(p+F_{2^k}+F_q\), where \(p,q\) are primes and \(k\ge0\), has positive lower asymptotic density. The same holds for the set of integers not of this form.

A classical extremal problem on progression free sets is to determine the maximum size of a $3$-term arithmetic progression free set in algebraic structures, for instance in intervals of integers or in finite vector spaces. To determine the minimum size of a complete $3$-term arithmetic progression free set is a lower-end analogue of this problem. It is also closely related to complete caps and saturating sets in finite geometry. A simple counting argument shows that the order of magnitude of the minimum size is at least the square root of the cardinality of the structure. Addressing two open problems, we show that this lower bound is essentially tight. First, for every cyclic group $\mathbb{Z}_m$, we give explicit constructions of complete $3$-AP-free sets whose size is less than $2\sqrt m$. For $m\ge81$ the constructed sets satisfy the stronger, so-called complete $(2,-1)$-avoiding property; the remaining cases $m<81$ are covered by a finite verification. Second, we resolve the vector space variant in a weaker sense by showing that for every fixed odd prime $p$ and $\varepsilon>0$, there is a constant $C_{p, \varepsilon}$ such that \[ a(3\text{-}\mathrm{AP},\mathbb{F}_p^n)\le C_{p, \varepsilon}\,n^{1+\varepsilon}\,p^{n/2} =p^{n/2+o(n)} \] holds for the minimum size $a(3\text{-}\mathrm{AP},\mathbb{F}_p^n)$ of a complete 3-AP-free subset of $\mathbb{F}_p^n$, for all $n\ge1$.

A multiset of $n$ unordered points in $\mathbb{R}^r$ – a point cloud, or, for $r=2$, a persistence barcode of birth-death pairs – is a point of the orbit space $\mathbb{R}^{nr}/S_n$ for the symmetric group $S_n$ permuting the rows of an $n \times r$ matrix; a separating family of invariants on this space is exactly a complete set of permutation-independent coordinates. We provide one that is explicit, small, and stable, in the max-plus (tropical) setting: for all $n \geq 1$ and $r \geq 1$, the $\binom{n+r}{r}$ basic $r$-symmetric tropical polynomials, of degree at most $n$, separate the orbits of $S_n$ on $\mathbb{R}^{nr}$. This settles in full a problem left open in [Kubo, J. Pure Appl. Algebra 223 (2019) 72-85], where separation was known only for $r=2$ and special cases of $r \geq 3$, and yields a family far smaller and of lower degree than the general separating sets from Derksen's recent theory of tropical invariants for permutation actions ($nr + (nr)!/n!$ invariants of degree $O(n^2 r^2)$). The proof is elementary and constructive: the basic values are identified with a transportation problem, and the multiset is recovered from the dual by an explicit algorithm. We further show the coordinate map is a bi-Lipschitz embedding for all $n$ and $r$, being an injective max filter bank (via the bi-Lipschitz theory of max filtering), with an explicit Lipschitz constant for the forward bound and a fully explicit, dimension-free distortion when $r=1$. Finally we determine when the pairwise values suffice (exactly $n \leq 3$) and show that invariants on at least three columns and of degree less than $n$ are necessary in general, the obstruction being a standard non-uniqueness configuration from discrete tomography.

Can a finite set of lattice points determine many rectangles and few isosceles triangles? This turns out to be a surprisingly interesting question in combinatorial geometry that we answer using basic analytic number theory combined with a finite-field construction. The result is useful because it gives obstructions to Mizohata–Takeuchi-type estimates in the setting of the paraboloid. Specifically, we establish transference between Euclidean and periodic weighted $\mathrm{L}^2$ estimates for solutions to the Schrödinger equation, and then relate the failure of the latter to quantities tied to combinatorial problems, such as the one above. By completing this programme we give new explicit combinatorial counterexamples to the paraboloid case of the Mizohata–Takeuchi conjecture, which was recently shown to be false by Cairo for curved hypersurfaces.

Owing to the effectiveness of Tensor Train (TT) decomposition in managing high-order tensors, low-rank tensor completion within the TT-format has emerged as a prominent research focus. In this paper, we leverage the left-orthogonal property of the TT-decomposition to construct a novel quotient manifold and introduce a family of admissible Riemannian metrics. Within this geometric framework, we propose a new approach to constructing retractions compatible with the quotient structure, realized via two novel retractions based on recursive polar and QR decompositions that respect the recursive orthogonalization structure of the TT format. We then derive Riemannian gradient descent and conjugate gradient methods to solve the tensor completion problem. Theoretically, our approach streamlines the horizontal projection by reducing the number of unknowns per block from a quadratic dependence on the TT-ranks to a near-half scaling, thereby enhancing computational efficiency over conventional quotient-based methods. Numerical experiments demonstrate that the proposed algorithms achieve reconstruction accuracy comparable to state-of-the-art TT-based geometric methods.

We construct a nested version of the non-commutative Hilbert scheme and embed the nested Hilbert scheme of points on $\mathbb{C}^n$ as the commutativity locus. In the $\mathbb{C}^2$-case, we exhibit this locus as the zero locus of two different sections of bundles and use this description to equip the nested Hilbert scheme of points with a perfect obstruction theory equivalent to that of Gholampour, Sheshmani and Yau. We study the torus equivariant pushforward of the virtual structure sheaf under the map of nested Hilbert schemes forgetting the largest subscheme of the nesting. Using a map of the bundles on the non-commutative Hilbert scheme, we prove that this pushforward is a twist of the virtual structure sheaf on the lower level. Using localization, we show that the twist is by a constant class with values corresponding to the equivariant Euler characteristic of a tautological class of the Hilbert scheme of points. From this, we derive a closed formula for the multivariate generating series of the equivariant virtual Euler characteristic of the nested Hilbert scheme of points.

The log canonical threshold (LCT) is a fundamental invariant in birational geometry and singularity theory, measuring the complexity of an analytic singularity through discrepancy and valuation data on a log resolution. In this work we investigate the asymptotic behaviour of sublevel-set volumes associated with principal analytic ideals, equivalently with holomorphic function germs. Building on the classical theory of local zeta functions and Mellin asymptotics, we introduce the visible spectrum, the set of actual poles of the local zeta function, and show that it is determined by the asymptotic expansion of the volume. Conversely, we prove that this spectrum, together with its multiplicities and coefficients, can be recovered recursively from the volume asymptotics by an explicit reconstruction procedure. We also give complementary interpretations in terms of arc spaces, where the divisorial exponents appear both as ratios of vanishing orders along generic divisorial arcs and as normalized codimension growth rates of divisorial cylinders. Taken together, these results establish an explicit correspondence between the visible spectrum and the asymptotic expansion of sublevel-set volumes, providing an intrinsic metric characterization of the visible spectrum itself.

We introduce and develop the theory of braided cogroupoids, a class of algebraic structures generalizing cogroupoids in a braided setting. We show that braided cogroupoids induce monoidal equivalences between the associated comodule categories, and we generalize Majid's transmutation and bosonization of braided Hopf algebras to the cogroupoid setting. Several examples are studied in detail, including the braided $SL_{n}$ cogroupoid and the braided bilinear cogroupoid.

We establish a criterion for Property $N_p$ for line bundles on a class of smooth projective toric varieties. More precisely, we prove that if a smooth projective toric variety $X$ of dimension $n\ge2$ satisfies the uniform unimodularity condition and the Thomsen stratification intersection-number condition, then any line bundle $L$ on $X$ with $L\cdot C\ge n-1+p$ for every $T$-invariant curve $C$ satisfies Property $N_p$. We also show that these two conditions hold for several families of toric varieties and are preserved under finite products.

To our knowledge, there is no rigorous mathematical justification of the Reynolds equation for a spherical bearing. In this article, we demonstrate that the solution of the Stokes problem in a domain between two closely spaced spheres converges, as the distance between the spheres approaches zero, to the solution of a "Reynolds equation".

In this paper, we investigate the discrete SIS (Susceptible-Infected-Susceptible) models. We focus on minimizing epidemic spreading over networks by extending an existing edge deletion algorithm to the SIS model. To achieve this, we employ the mean-field approximation to linearize the network dynamics into a deterministic SIS model. We analytically demonstrate that the total number of infections is upper-bounded by a super-modular function, thereby ensuring the efficiency of the edge-deletion approach. To evaluate the proposed method, we conduct experiments on synthetic Erdos-Renyi networks and the real-world dataset collected from BBC Pandemic Haslemere app. Numerical simulations validate our theoretical results, confirming that both configurations converge to the stable, disease-free equilibrium.

We introduce the class of relatively weakly convex functions, which extends the classical notion of weak convexity by measuring nonconvexity relative to a distance-generating function. We investigate the fundamental properties of this function class, establishing characterization results, calculus rules, and illustrative examples. We further analyze the associated optimization landscape and identify a neighborhood of the set of global minimizers that is free of saddle points. Motivated by this geometric structure, we propose the Projected SubGradient Algorithm (PSGA) with several step-size strategies. Under a sharpness error bound, we prove that, when initialized within this saddle-point-free neighborhood, the iterates generated by PSGA converge to a global minimizer for each of the proposed step-size strategies. Furthermore, linear convergence is established for the geometrically decaying step-size strategy.

Let $k\geq 2$ be a positive integer. It is known that the set of visible lattice points from the origin in $\mathbb{Z}^k$ has a translation bounded pure point diffraction spectrum. We investigate these properties for sets of points simultaneously visible from a finite set of lattice points $ \{\mathbf{x}_1,\dots,\mathbf{x}_n\} \subseteq \mathbb{Z}^k$. We provide explicit formulas for the coefficients of the diffraction spectrum. Additionally, we generalize our procedure to show that the set of visible points from the origin in certain classes of cut-and-project sets has a translation bounded pure point diffraction spectrum.

In this article, we present a self-contained proof of Gromov's dihedral rigidity conjecture on scalar curvature in the three-dimensional case. The proof avoids many of the technical complications that arise in higher dimensions, while still illustrating the essential ideas of the general approach developed in arXiv:2112.01510 (version 6) and arXiv:2203.09511. It is significantly shorter than the proof of the general case and is intended to be more accessible.

In the study of energy-preserving methods for Hamiltonian systems, polynomial continuous-stage Runge–Kutta methods play an important role. Necessary and sufficient conditions for such methods to be energy-preserving have already been established. They are energy-preserving if the matrix $M\in \mathbb{R}^{s\times s}$ defining the method is symmetric, and the converse holds under the assumption that a certain $s\times \infty$ matrix $\Phi^\mathrm{CSRK}$ has full row rank. It was conjectured in Remark 3 in Miyatake and Butcher (SIAM J. Numer. Anal., 2016) that the full-rank assumption should always hold for every consistent polynomial continuous-stage Runge–Kutta method. In this paper, we prove the conjecture by showing that the matrix $\Phi^\mathrm{CSRK}$ has full row rank under the standard consistency condition. The proof is a direct application of the polynomial moment problem solved by Pakovich and Muzychuk (Proc. Lond. Math. Soc., 2009).

Let $E$ be a stable holomorphic vector bundle over a compact Kähler (or Gauduchon) manifold $(M,\omega_g)$. We show that for any real number $\mu>0$ and any initial Hermitian metric $h_0$ on $E$, there exists a unique iteration sequence $\{h_m\}$ satisfying $$ \Lambda_{\omega_g}\left(\sqrt{-1}R^{h_{m+1}}\right) =(\lambda_E-\mu)h_{m+1}+\mu h_m, $$ and $\{h_m\}$ converges smoothly to a Hermitian-Einstein metric $h_\infty$ on $E$ satisfying $$ \Lambda_{\omega_g}\left(\sqrt{-1}R^{h_{\infty}}\right) =\lambda_Eh_\infty, $$ where $\lambda_E\in \mathbb R$ is the stability constant. A key feature of this proof is that it is independent of Donaldson's variational framework and applies to non-Kähler manifolds.

Let $\Omega$ be a bounded domain in $R^d$. Denote by $\lambda_k$ (resp. $\mu_k$) the eigenvalues of the Laplace operator in $\Omega$ with Dirichlet (resp. Neumann) boundary conditions. Denote by $\Psi = \Psi (d,k,\Omega)$ the shift of indices in the inequality $\mu_{k+\Psi} \le \lambda_k$. We are interested to describe the behaviour of $\Psi$ for large $d$. We prove that a) $\Psi (d,1,\Omega) \ge C (e/2)^d$ for all domains $\Omega$; and b) $\Psi (d,k,\Omega) \ge C (e/2)^d$ for all $k$ and all convex domains $\Omega$.

We study prophet inequalities for $n$ i.i.d. base rewards subject to multiplicative discounts across $S$ phases. For equal-length phases and a common discount factor $\widetilde\gamma$, we characterize the optimal competitive ratio of single-quantile policies as $n\to\infty$ with $S$ and $\widetilde\gamma$ fixed. The resulting sharp guarantee describes the dependence on the phase parameters: it recovers $1-1/e$ without discounting and approaches $1/2$ in the worst-case regime. We also establish a distribution-free comparison that bounds the discounted prophet by a fractional i.i.d. maximum with effective horizon $B$, the sum of the discount weights across all observations. The comparison yields acceptance probability $1/B$, determined only by the phase profile. This calibration attains the canonical optimum even among single-quantile policies that may choose their acceptance probability using the known base distribution. The comparison also gives guarantees for heterogeneous discounts and unequal phase lengths. A complementary optimal-stopping upper bound establishes the same limiting optimum of $1/2$ for arbitrary stopping rules in the canonical model. Finally, the same effective-horizon calibration sets acceptance probability $1-\widetilde\gamma$ under infinite-horizon geometric discounting. This explicit quantile rule attains Boshuizen's (1994) known optimal worst-case ratio $1/(1+\widetilde\gamma)$ without solving a Bellman equation.

We study a Mean Field Control system arising in the management of fisheries with a special emphasis on non-uniqueness issues. Namely, we focus on a situation where a group of players coordinate in order to harvest a fishery in the most efficient way possible. A major challenge in such modelling is the coupling between the dynamics of fish population, which we model through a reaction-diffusion equation, and that of the players, which is seen through the lens of Mean Field Control. The resulting evolution system consists of four coupled equations. A central issue, both in the analysis and from the modelling perspective, is the uniqueness of solutions of this system. By focusing on the ergodic (or static) counterpart of the evolution equation, we show that one should in general expect the emergence of multiple solutions. Our approach relies on the theory of bifurcation, and the bifurcation parameter we take is the (biologically relevant) total amount of food available to the population. We also give refined uniqueness criteria that allow to bypass several limitations of previous works on this type of system [39]. This fits within two growing research lines: one on the optimal harvesting of fisheries [40, 39], one on questions of non-uniqueness in Mean Field Games and Mean Field Control [3, 27, 36].

Quantitative verification of neural networks requires reasoning about probabilities under substantial uncertainty in both input distributions and their dependence structure. In realistic settings, this information is often only partially specified, and assuming precise probabilistic models can lead to unreliable results. We propose a sound framework for quantitative verification under imprecise probabilistic information, combining interval belief structures to represent marginal uncertainty with imprecise copulas to model uncertain dependence. We develop a propagation method for imprecisely coupled interval belief structures through feed-forward neural networks. Using mixed imprecise copula volumes, we derive sound push-forward constructions through affine transformations and activation functions. The resulting output can provide guaranteed lower and upper bounds on probabilistic safety properties, valid for all probability models compatible with the specified imprecise inputs.

This contribution studies the Boltzmann scheme on a “D2T4”grid constructed on meshes using equilateral triangles. The center of each triangle is connected to itself and to three other triangles via the edges of the mesh. We adopt the multiple relaxation time approach. Applications for diffusion and acoustics problems are considered. Consistency analysis is particularly delicate. We propose an approach based on taking bipoints into account. We derive equivalent partial differential equations for diffusion and acoustics. These systems of equations are then approximated numerically using the D2T4 lattice Boltzmann method. A comparison with an analytical calculation in the case of periodic boundary conditions shows the convergence of the D2T4 lattice Boltzmann scheme.

Binary compressive sensing (BCS) seeks to recover a $k$-sparse binary vector of length $n$ from $m$ linear measurements. Classical CS guarantees break down for $m < k$ and convex/greedy BCS algorithms with random Gaussian sensing matrices perform poorly. We introduce ISDP-MVSIC, which combines randomized semidefinite programming (SDP) sampling, majority voting (MV) and successive interference cancellation (SIC) across $L \ll n$ stages, wrapped in a residual-cost driven retry loop. The method exposes a tunable complexity–performance trade-off: for $n=100, 144$, raising the worst-case complexity $\mathcal{C}_{max}$ from $7.9 \times 10^9$ to $2.0 \times 10^{10}$ enables empirical exact recovery over $m/k \in [0.4,5.0]$ as the sparsity ratio $s=k/n$ decreases from $0.5$ to $0.1$, by practically targeting the undersampled regime.

We establish an anisotropic analogue of the celebrated theorem of Meeks-Simon-Yau: every minimizing sequence of surfaces within a fixed isotopy class converges to a smooth stable anisotropic minimal surface, with genus lower semicontinuity. This result also strengthens White's foundational existence theory for anisotropic minimal disks. As an application, we develop an anisotropic Simon-Smith min-max theory. In every closed $3$-manifold, we construct anisotropic min-max sequences within fixed isotopy classes whose limits are stable anisotropic minimal surfaces that are smooth except possibly at a single point. If the integrand satisfies either an ellipticity bound or a $C^3$-pinching condition, we remove the singular point by proving two independent removable singularity theorems for anisotropic minimal surfaces that are smooth and stable away from finitely many points. These removable singularity results also allow to remove the singularities arising in the anisotropic Almgren-Pitts min-max construction in $3$-manifolds of De Philippis-De Rosa and in its multiparameter variants.

Quantifying how meaning propagates through communicative exchanges remains underdeveloped in computational linguistics. Here we introduce an information-theoretic framework that quantifies the directed flow of semantic content between interlocutors and decomposes multi-source contributions into redundant, unique, and synergistic components. Our approach leverages large language models as probabilistic estimators of natural language to compute two measures: semantic transfer entropy (STE), which captures directed predictive influence between speakers, and semantic partial information decomposition (SPID), which resolves how multiple sources jointly shape a target's language. Across four experiments we show that the framework detects reduced information flow in cognitively rigid dialogue, captures the dominant role of persuaders in shaping discourse, distinguishes high- from low-quality psychotherapy by the directionality of therapist-client information exchange, and reveals synergistic premise contributions in argumentative essays. This framework opens new avenues for studying information dynamics in digital discourse, pedagogical interactions, clinical dialogues, and any domain in which the structure of linguistic exchange is of research relevance.

Recent advances in biomedicine generate high-dimensional single-cell data that describe cellular heterogeneity with unprecedented detail, but their geometric complexity and non-linear structure often limit the effectiveness of conventional statistical tools. Topological Data Analysis (TDA) provides a mathematical framework for characterizing the shape of data through persistent homology, which extracts structural features such as connected components and cycles across multiple scales. In this work, we propose a systematic two-level framework for evaluating topological representations in high-dimensional single-cell classification. The first level (\(R_1\)) performs statistical screening of topological descriptors based on separability between clinical groups, whereas the second level (\(R_2\)) evaluates their predictive utility in supervised classification models. This design makes it possible to compare representations not only in terms of discriminative performance, but also in terms of robustness to analytical choices. We illustrate the framework using bone marrow flow cytometry data from pediatric acute lymphoblastic leukemia, with a particular focus on relapse stratification. The results show that different topological representations vary substantially in both statistical separability and predictive stability, with Betti Curves and Persistence Silhouettes showing more robust behavior than Persistence Images in this cohort. Overall, the study provides a reproducible methodological framework for the systematic comparison of topological descriptors in complex biomedical point clouds.

Let $R$ be a commutative Noetherian ring in which $2$ is invertible. We prove that a conjugate of Petrov's odd elementary unitary group is contained in the DSER elementary orthogonal group defined over projective modules. We also show a sufficient condition regarding the Witt index of the quadratic module with a hyperbolic summand $\mathbb{H}(P)$ which implies the surjective stability of DSER orthogonal ${\rm K}_1$