Papers

Let $(\mathcal{C}, J_{\mathcal{C}}, \omega_{\mathcal{C}}, g_{\mathcal{C}})$ be a Calabi-Yau cone and $(M, J, \omega, g)$ an open Ricci-flat Kähler manifold. We show that, if there exists a diffeomorphism $\Phi: \mathcal{C} \setminus \overline{B_1(o)} \rightarrow M \setminus K$, for some compact $K \subset M$, such that $\Phi^{*}J$ is asymptotic to $J_{\mathcal{C}}$ and $C^{-1} \omega_{\mathcal{C}} \leq \Phi^{*} \omega \leq C \omega_{\mathcal{C}}$ for some $C \geq 1$, then $(M, g)$ is asymptotically conical (AC) with tangent cone at infinity given by $(\mathcal{C}, d_{g_{\mathcal{C}}})$. As a consequence, we obtain that any Ricci-flat Kähler metric on $T^{*}S^n$ which is quasi-isometric to the Stenzel metric must be equal to the Stenzel metric up to scaling and diffeomorphism. Similarly, any Ricci-flat Kähler metric on $\mathcal{O}_{\mathbb{P}^1}(-1)^{\oplus2}$ which is quasi-isometric to the Candelas-De la Ossa metric must be equal to the Candelas-De la Ossa metric up to scaling and diffeomorphism. This provides new examples of complete Calabi-Yau manifolds for which a Liouville-type theroem holds.

We develop an energy-stable implicit convex-splitting BDF2 discretization (CS-BDF2) of the Cahn–Hilliard–Navier–Stokes equations. For the Cahn–Hilliard equation, BDF2 analyses can establish energy stability by testing the phase equation in the (H^{-1}) metric. For CHNS, this test is not compatible with the coupled energy estimate: the momentum equation is tested by (\bfu^{n+1}), while the transported phase equation is tested by (\mu^{n+1}) so that transport cancels capillary work. The chemical-potential relation must then be paired with the BDF2 phase increment ((3\phi^{n+1}-4\phi^n+\phi^{n-1})/2); its nonlinear part must produce a BDF2 bulk-energy difference, up to nonnegative higher-order history terms. To overcome this difficulty, we introduce a new BDF2-compatible convex-splitting approximation of the nonlinear bulk force that directly yields a discrete bulk-energy identity and enables a discrete energy analysis for the CHNS system. Specifically, we discretize the bulk force (f(\phi)=\phi^3-\phi) by (\chi(\phi^{\dagger,n+1},\phi^{\dagger,n})-\phi^{*,n+1}), where (\chi(a,b)=\tfrac14(a^2+b^2)(a+b)), (\phi^{\dagger,n+1}=\tfrac{3\phi^{n+1}-\phi^n}{2}), (\phi^{\dagger,n}=\tfrac{3\phi^n-\phi^{n-1}}{2}), and (\phi^{*,n+1}=2\phi^n-\phi^{n-1}). This discretization is based on the shifted BDF2 identity ((3\phi^{n+1}-4\phi^n+\phi^{n-1})/2=\phi^{\dagger,n+1}-\phi^{\dagger,n}). With a matching discretization of the reversible coupling terms in CHNS, the scheme is mass conservative, uniquely solvable, and unconditionally energy stable. We prove second-order convergence for the phase variable, chemical potential, velocity, and pressure.

This paper considers holomorphic modular forms for $\Gamma_1(N)$ of integral weight of the form $$f^{(N)}_{\mathbf a}(\tau) =q^{s} (q^{N};q^{N})_{\infty}^{a_0}\prod_{j=1}^{\lfloor N/2 \rfloor}(q^j,q^{N-j};q^N)_\infty^{a_j}, \quad \mathbf a = (a_1, \ldots, a_{\lfloor N/2 \rfloor}),$$ for fixed $a_0=2k \in 2 \Bbb Z_{\ge 0}$. We show that the number of relevant exponent vectors $\mathbf a$ is finite and characterize them in terms of the $\mathbb{Q}$-rational cuspidal divisor class group of $X_{1}(N)$. Effective procedures are given for counting the admissible exponents by enumerating the corresponding polytopes. This leads to formulas for the number of exponent vectors in terms of quasipolynomials in $k$.

For each integer $N\geq 3$ we construct examples of $N$-component hyperbolic links $L\subset\mathbb{S}^3$ whose exterior contains an incompressible spanning planar surface $P\subset X_L$ with one boundary component on each boundary torus of $X_L$ of nonmeridional and nonintegral slope, thus providing counterexamples to a recent conjecture of M. Eudave-Muñoz and M. Ozawa. The case $N=3$ is the crucial one to consider: all such link pairs $(L,P)$ are classified and found to be generated by the structure of the exterior of hyperbolic Eudave-Muñoz knots. More generally, necessary and sufficient conditions on integers $p_1,p_2,p_3\geq 2$ are given for the existence of a 3-component link in $\mathbb{S}^3$ whose exterior contains a spanning pants with boundary slopes of the form $a_i/p_i$. A key role in the analysis of 3-component link pairs is played by the properties of the embeddings of three mutually disjoint and nonparallel primitive circles on the boundary of a genus two handlebody. These are classified in general and in the special case when the handlebody is part of a genus two Heegaard decomposition of $\mathbb{S}^3$ associated with a 3-component link pair. The hyperbolic links with $N\geq 4$ components whose exterior contains a spanning planar surface with nonmeridional and nonintegral boundary slopes are constructed via an inductive process that starts with any of the classified 3-component hyperbolic link pairs.

We study reaction-diffusion equations with finite signed measure delays and nonlinear, possibly nonlocal feedback. We treat abstract reactions that are Lipschitz on bounded $L^2$-balls with quadratic coercivity, and pointwise cubic polynomials with positive leading coefficient in spatial dimensions at most three. For affine-growth feedback and time-independent forcing in $H^{-1}(\Omega)$, both classes generate global weak solution semiflows on $X=C([-r,0];L^2(\Omega))$. Besides total-variation Lipschitz stability, we obtain quantitative weak-star stability with modulus $\varpi(d)=d\log(e+d^{-1})$ for the abstract class and $\varpi(d)^{1/2}$ for cubic reactions, where $d$ is the bounded-Lipschitz distance between delay measures. The estimates are uniform on bounded sets of continuous histories and yield concentration and atomic-quadrature rates. A damping condition gives common compact absorption for the abstract class. Cubic coercivity removes that smallness condition on every fixed total-variation-bounded kernel class. In both cases the global attractors are upper semicontinuous in $X$ and in $C([-r,0];H_0^1(\Omega))$.

We discuss optimal recovery for classes of multivariate convex functions from given point samples, as well as the sampling numbers of these classes, corresponding to optimal sample choices. Upper and lower bounds for either variant are established when the reconstruction error is measured in $L_p$ for $1\leq p\leq \infty$. These bounds match, sometimes up to logarithmic factors, and therefore characterize the respective optimal rate of decay. For classical smoothness classes such as Sobolev, Hölder or Besov spaces, it is well known that the optimal decay rate of sampling numbers can be achieved by sampling on uniform tensor product grids and using linear methods of reconstruction, such as piecewise polynomial interpolation. One of the main findings in this paper is that for classes of convex functions, these procedures generally produce suboptimal rates, except when $p=1$ and $p=\infty$, and are outperformed by nonlinear reconstruction methods that do not employ tensor product grids.

Let $A$ be either a classical generalized Weyl algebra (also known as a noncommutative deformation of type A Kleinian singularity) or the enveloping algebra $U(\mathfrak{sl}_{2})$ over $\mathbb{Z}/p^n\mathbb{Z}.$ In this paper we compute the center and derivations of $A.$ More specifically, we show that the center of $U(\mathfrak{sl}_2)$ is generated by the Casimir element over the ring of the Witt vectors (of length $n$) of its $p$-center. Our description of derivations of $A$ implies that if the ground ring is a field $k$ of characteristic $p>2,$ then the restriction homomorphism $HH^1_{k}(A)\to Der_{k}(Z(A), Z(A))$ from the first Hochschild cohomology of $A$ to $k$-derivations of the center is an isomorphism.

Motivated by Norine's conjecture, this paper investigates monochromatic antipodal paths in $2$-edge-coloured hypercubes and simplicial complexes. Our main method relies on a topological criterion applied to triangulated $2$-skeleta. We show that any antipodal colouring of a centrally symmetric, simply connected $2$-complex yields a monochromatic path linking an antipodal pair of vertices. By symmetrically triangulating opposite square faces in certain classes of colourings, we obtain a topological verification of Norine's conjecture for these classes. We also establish quantitative bounds for cases in which only a limited number of square faces present structural obstructions.

Vision-language foundation models such as CLIP and SigLIP provide widely used representations for multimodal learning systems. While these models are typically compared through downstream performance, such evaluations often do not explain how their representations differ structurally. In this work, we study this problem through the task of Contrastive Embedding Clustering: identifying sample subsets that are weakly clustered under one representation but strongly clustered under another. We propose Kernel Optimization for Discrepancy Analysis (KODA), a kernel-based framework for contrastive representation comparison and alignment. KODA constructs unified multimodal kernels through modality-wise kernel composition and formulates discrepancy discovery as a constrained optimization problem that searches for coherent structures in one representation while suppressing coherence in a reference representation. This yields interpretable discrepancy directions associated with specific sample subsets and modality interactions. To scale KODA to large vision-language datasets, we develop randomized low-dimensional approximations of joint kernels using random projections, including Random Fourier Features for shift-invariant kernels. Empirically, KODA identifies consistent and interpretable discrepancy structures across vision-language representations and provides sample subsets for representation alignment. The code is available at https://github.com/yokiwuuu/KODA.

We prove functional scaling limits for interpolated random walks whose increments are functions of a stationary Gaussian sequence. In this setting, the classical Dobrushin-Major-Taqqu theorem describes the scaling limit when the covariance has a regularly varying, non-summable tail, while the Breuer-Major theorem describes the limit in the summable regime. We strengthen these convergence results to functional convergence in Hölder topology and, in the summable regime, in rough Hölder topology. These stronger topologies are useful because many operations on paths, such as Young integration and solution maps of differential equations, are continuous in (rough) Hölder topology, but not in Skorokhod topology.

Graph burning is a deterministic discrete-time process that models the propagation of information within a network as a set of fires that spread in a graph. The associated graph parameter, the burning number, measures the speed of that spread. The smaller the burning number of a graph is, the faster information spreads in the corresponding network. In this paper, we mainly study the burning number from a structural point of view. In particular, we structurally characterize all graphs with burning number 2. We then proceed to compute the burning number of split graphs and $\{ K_2+2K_1, P_4, C_4\}$-free graphs. In particular, we show that connected split graphs and connected $\{ K_2+2K_1, P_4, C_4\}$-free graphs are well-burnable. Lastly, we provide a blueprint on how to investigate the burning number of a given social network.

We study a distributional generalization of the matrix completion problem in which each entry of the target matrix is a probability distribution rather than a scalar. In this setting, only a subset of matrix entries is observed, and even for observed entries, the underlying distributions are not directly accessible; instead, we observe finitely many samples drawn from them. To represent distributional entries, we employ kernel mean embeddings and introduce a notion of Tucker rank for distribution-valued matrices to capture their low-rank structure. The infinite-dimensional nature of kernel embeddings poses significant methodological challenges. To address this, we introduce functional unfolding operators that link the proposed distributional low-rank structure to the classical Tucker rank for finite-dimensional tensors. Based on this framework, we propose a novel estimator for distributional matrix completion. We establish non-asymptotic error bounds that characterize the statistical performance of the estimator. Extensive experiments on synthetic data and a real-world application demonstrate the effectiveness of the proposed method.

It is well-known that any class of simple graphs, that is characterized by finitely many forbidden minors, also admits a characterization by finitely many forbidden topological minors; furthermore, the list of forbidden topological minors may be derived from the list of forbidden minors. We prove a similar result in Matching Theory. Our Main Theorem states that any class of matching covered graphs, that is characterized by finitely many forbidden $S$-minors that are cubic, also admits a characterization by finitely many forbidden conformal minors that are cubic as well; once again, the list of forbidden conformal minors may be derived from the list of forbidden $S$-minors. In order to establish the above, we first prove that every matching covered graph has one of two graphs as a conformal minor – either $K_4$, or the $\Theta$ graph (that is, two vertices joined by three edges). (In fact, we need and prove a much stronger statement.) This is reminiscent of a theorem due to Lovász: every nonbipartite matching covered graph has one of two graphs as a conformal minor – either $K_4$, or the triangular prism $\overline{C_6}$. As applications of our Main Theorem, we deduce known 'forbidden conformal minor characterizations' of pfaffian near-bipartite graphs, and of pfaffian solid graphs, using their respective known 'forbidden $S$-minor characterizations'.

We consider the Dirac equation in a real setting to show its close relationship to Maxwell's equation. In addition to analysing the classical Dirac mass term, we also inspect the alternative Majorana mass term and discuss their quaternionic structure and its consequences.

This work provides a counterexample to the previously asserted non–existence of future null lines in strongly causal spacetimes $(M, \mathbf{g})$ foliated by a past–omniscient timelike foliation $\mathcal{F}$ with a compact leaf space $Q$. We introduce a new hypothesis requiring the timelike vector field that defines $\mathcal{F}$ to be causally decreasing. Under this condition, the causality requirement on $M$ can be relaxed to the distinguishing level. In the absence of this hypothesis, we demonstrate that more restrictive conditions, such as global hyperbolicity, are necessary to ensure the absence of future null lines. Finally, we derive sufficient conditions on the scale function of Generalized Robertson–Walker (GRW) spacetimes to ensure that the natural timelike vector field is causally decreasing, thereby guaranteeing the non–existence of future null lines.

A point source of light is placed inside a billiard with a smooth, convex, closed boundary. For any integer $n$, the $n$-th caustic by reflection, denoted by $\Gamma_n$, is the envelope of light rays that have undergone $n$ reflections in such a billiard after emanating from the source. It has been conjectured by Gil Bor and Serge Tabachnikov that for an elliptical billiard, $\Gamma_n$ has exactly four ordinary cusps; this problem is a billiard variation of Jacobi's Last Geometric Statement, which concerns the number of cusps in the conjugate locus of a point on an ellipsoid. Gil Bor, Mark Spivakovsky, and Serge Tabachnikov have proven that $\Gamma_n$ has at least four ordinary cusps. In this paper, we present a proof that $\Gamma_1$ has exactly four ordinary cusps, using billiards in complex spaces.

We are interested in the so-called multiple-node immunization problem for complex networks under attack by a viral agent. It consists in identifying and removing a set of nodes of size $k$ in a graph to maximize the impeding of virus spread. A few approaches have been proposed in the literature based on numerical and theoretical insights on how classical models for virus spread evolve on graphs. Based on the analysis of these models, the maximal eigenvalue of the adjacency matrix of the graph has become a classical measure of how resilient the network is. Thus, a clear, well-explored approach for multiple-node immunization consists of identifying a set of $k$ nodes in such a way that the reduced network, obtained by removing these nodes, has a minimal largest eigenvalue. This spectral optimization problem turns out to be a computationally hard problem for which only greedy algorithms offer good solutions at efficient computational time. Among those, the so-called Netshield algorithm represents one of the reference choices. The latter is, in fact, a clearly defined algorithm aiming at optimizing a certain sub-modular functional, called shield-value, which approximates the original optimization problem. We propose here a novel procedure, based on random walk kernels and related random spanning forests, to build a new algorithm, referred to as K-shield, which enhances Netshield searching performance at the same computational complexity. We give theoretical insights behind this novel method, which could also be used for other optimization problems, and then test it via numerical showcase experiments on various benchmarks.

We develop a duality-based formulation of the dense graph limit for a variational model of biological transportation networks, where edge conductivities balance pumping power against metabolic cost. In contrast to the pressure-based approach of our previous work, which required conductivities to be uniformly positive, the present formulation allows general nonnegative conductivity kernels. The kinetic energy is defined through a dual variational principle, which remains meaningful for degenerate integrable kernels and assigns infinite energy when the associated nonlocal Poisson problem is not solvable. Using this formulation, we prove $\Gamma$-convergence in the sense of Mosco of the semidiscrete network energies to a continuum energy on symmetric nonnegative kernels. The convergence is obtained in the natural $L^\gamma$ topology dictated by the metabolic term. The $\Gamma$-$\liminf$ inequality follows directly from the dual formulation, while $\Gamma$-$\limsup$ recovery sequences are constructed by positive regularization of the conductivity kernels.

The Poisson Nernst Planck PNP system constitutes a canonical stiff coupled PDE problem where the charge density prefactor produces extreme coefficient ratios and the electric double layer imposes sharp boundary layers. Physics informed neural networks PINNs are appealing here because they require no mesh and differentiate through the physics automatically. Spectral bias and multi task loss imbalance however have limited their accuracy on stiff PNP systems. We present the first systematic data free benchmark of eleven PINN configurations organised into four strategy groups on a physically parametrised one dimensional PNP model for a lithium symmetric cell implemented within NVIDIA PhysicsNeMo Sym and validated against a finite volume method FVM reference. Root mean square errors RMSE span across architectures. The balanced residual decay rate BRDR scheme matches Neural Tangent Kernel NTK performance for concentration fields while reducing mean wall clock time making it the preferable strategy under compute constraints. Loss landscape geometry corroborates the RMSE ranking. We release an open source PhysicsNeMo Sym implementation for reuse on stiff coupled PDE problems in computational mechanics.

Trajectory optimization is a critical component for enabling safe and reliable autonomous operations in space exploration. As space missions increase in frequency, complexity, and scope, there is a growing need to rapidly formulate mathematically sound trajectory optimization problems that accurately reflect mission objectives and operational constraints. However, translating mission intent into tractable analytical formulations for trajectory optimization requires substantial domain expertise. This paper presents a framework that leverages large language models (LLMs) to translate natural language descriptions of mission requirements and constraints into executable trajectory optimization code and corresponding mathematical formulations. Experiments in spacecraft rendezvous scenarios demonstrate a high success rate in reconditioning a convex trajectory optimization problem from semantic mission requirements. Ultimately, this work highlights the potential of LLMs to bridge high-level intent and formal optimization models, enabling more flexible and efficient trajectory design of spacecraft.

We study almost everywhere divergence of rectangular partial sums of double trigonometric Fourier series along variable regions concentrated near the diagonal. Fefferman's theorem shows that, in two variables, unrestricted rectangular summation is radically different from the one-dimensional Carleson-Hunt theory: there are continuous functions whose rectangular Fourier sums diverge everywhere. Bakhvalov proved that this phenomenon persists even when the indices are restricted to a fixed cone-shaped neighbourhood of the diagonal. On the other hand, the diagonal summation results of Tevzadze and Fefferman, and the later theorem of Antonov for shrinking cones, show that convergence is restored when the aperture is of order $1/n$. We prove that Antonov's aperture condition is sharp in the $L^{2}$-scale. For every positive nonincreasing sequence $\{\lambda _{n}\}$ with $% \sup_{n}n\lambda _{n}=\infty $, we construct a function $f\in L^{2}(\mathbb{T% }^{2})$ whose symmetric rectangular Fourier partial sums fail to converge almost everywhere along the corresponding variable cone. The construction combines a Fefferman-type analytic block with frequency separation and independent random translations. We also observe that the sufficiency part of Antonov's theorem does not require monotonicity of $\{\lambda _{n}\}$.

Carbery posed a question of sharpened triangle inequalities for families of operators in the Schatten–von Neumann classes $S_p$, $p\geq 2$. He established a weaker form of the desired estimate for even integer values of $p$. In the commutative setting the corresponding sharp inequality (with optimal exponent $p'=\frac{p}{p-1}$) was recently obtained for all integer $p\geq 2$. In the present work we resolve Carbery's question completely in the non-commutative setting: we prove the sharp inequality $$ \Big\|\sum_{j} T_{j}\Big\|_{S_{p}}\leq \|(\alpha_{ij}^{p'})\|^{1/p'}_{\ell_{2}\to \ell_{2}} \Big( \sum_{j} \|T_{j}\|^{p}_{S_{p}}\Big)^{1/p} $$ for all $p\geq 2$ and all countable sequences of operators $(T_{j}) \subset S_{p}$, where $\alpha_{ij}$ are almost orthogonality coefficients. The proof is based on a block-operator reduction and a complex interpolation of the polar parts of the blocks.

We give a sharp lower bound for the total mean curvature of a convex hypersurface in Euclidean space in terms of the length of a shortest nontrivial closed geodesic, generalizing a result of Álvarez Paiva for convex surfaces. This result is based on a sharp lower bound for the mean width of a convex hypersurface in terms of its Birkhoff invariant, which gives sharp lower bounds for a broader array of total curvature functionals. We also characterize spheres as the unique convex hypersurfaces whose planar sections containing chords of maximal length are all as long as possible.

We develop a boundary-centric semantic framework for higher-order quantum computation, building on the Kelly-Laplaza description of compact closure and Abramsky's execution account. In the semantic carrier Perm(C), morphisms are complex-linear combinations of polarized boundary linkings, composed by execution. Finite-family addresses provide coherent control over finite-level quantum registers (qudits) while retaining the multiplicative boundary structure. We identify essential unitarity, a boundary condition extending ordinary unitarity to higher-order interfaces. On positive qudit registers it coincides with ordinary matrix unitarity; at higher order it expresses preservation of information across the full polarized boundary. We define a unit-free coherent quantum core generated by multiplicative wiring, unitary gates on positive qudit registers, and contextual coherent control, and prove that every one of its morphisms is essentially unitary. The framework realizes an applied coherent quantum switch and the unitary stages of equal-ratio one-slot supermap dilations with explicit memory. An extended abstract of this work was accepted for QPL 2026 and is forthcoming in its proceedings.

Adaptive patching has been proposed for time-series forecasting: allocate finer patches where the sequence looks locally informative. We compare three adaptive patching methods with tuned uniform counterparts that use one fixed patch size throughout the input, while preserving each method's backbone family and shared training protocol. Across the evaluated long-horizon benchmarks, we observe mostly small accuracy differences and no consistent advantage for adaptive patching; uniform variants generally train faster and avoid adaptive routing at inference. Motivated by these observations, we develop a theoretical framework for budgeted patch allocation through a common resolution-density representation of uniform, nonuniform, and overlapping layouts, independently of the patching rule. Within a separable shared-loss model, we derive an exact criterion for when nonuniform allocation improves on uniform allocation at the same budget. We characterize modeled gains locally through a quadratic surrogate and bound them globally under strong convexity and a bounded loss derivative. The analysis distinguishes where resolution is placed from how much it is redistributed: even an allocation that correctly ranks regions by their weights can redistribute too much resolution and increase modeled loss. Without coupling across time, positive regional weights preserve the minimizers of the shared loss. When the shared loss has a common minimizing rate, the remaining absolute headroom for improvement tends to zero as the uniform rate approaches it. These results characterize the roles of regional weighting, placement, and redistribution strength under the stated model. The empirical findings support evaluating adaptive patching against a tuned uniform baseline and weighing any accuracy gain against the additional training and inference cost of adaptive routing.

We study simultaneous alternating power iteration for fixed-order asymmetric rank-one spiked tensor models. Our main contribution is a finite-iteration local theory that is independent of any particular initialization. Once the iterates enter a sufficiently small neighborhood of the planted rank-one direction, their error decomposes into a geometrically decaying transient and an intrinsic noise floor caused by fixed orthogonal noise contractions at the planted point. The deterministic finite-sample conditions are stated explicitly, but under a coarse fixed-order multilinear noise event they reduce to a conservative high-signal regime for fixed or slowly expanding local radii. We then separate the warm-start mechanism from any specific spectral construction. A generic one-sweep principle shows that, if a sign-compatible initializer has correlation \(\gamma_N\), first-sweep noise level \(a_N\), and \(a_N/(\gamma_N^{d-1}\omega_{N,d})\to0\), then one can choose an expanding radius \(r_N=o(\omega_{N,d})\) for which the first sweep enters the local basin. After entry, the local affine contraction yields convergence to the unique informative local fixed point in that basin. For centered-Gram initialization, we verify the required correlation and same-sample first-sweep noise bound under i.i.d. finite-fourth-moment noise by a signal-preserving noise-only leave-one comparison and an averaged leave-one slice-contraction estimate, which we call a pressed-back estimate. The leave-one comparison keeps the spike fixed and averages over the deleted coordinate, so planted coordinates enter through \(\ell_2\)-weighted sums rather than worst-case incoherence bounds.

We introduce noncommutative geometry techniques in order to reinterpret the Penrose–Sparling non-Hausdorff twistor space of the anti-self-dual Coulomb field by means of an explicit etale gluing groupoid and its convolution algebra. This algebraic model keeps track of both the identified open part and the two non-separated copies of the source quadric. We compute two kinds of Chern–Connes pairings. The strict tangent-module analogue, obtained from \([T_{\mathbb R}\CP^3\otimes\C]\), vanishes because \[ \operatorname{ch}_3(T^{1,0}\CP^3)+ \operatorname{ch}_3(T^{0,1}\CP^3)=0. \] By contrast, the Penrose–Sparling Coulomb line bundle \(\calC_n\) defines a \(K_0(A_Q)\)-class, and the relative cyclic cocycle supported on the two non-separated copies of a ruling line \(L\subset Q\) gives \[ \mathcal Q(\calC_n)= \frac12\left\langle \varphi_L^+-\varphi_L^-,[\calC_n]\right\rangle=n. \] Thus the source-adapted cyclic pairing recovers the Coulomb charge. We also formulate the non-abelian version in principal-bundle language. For a connected complex reductive group \(G\), a maximal torus \(T\subset G\), and a cocharacter \(\lambda:\C^*\to T\), the source is a principal \(G\)-bundle modification of type \(\lambda\) along \(Q\).

We develop a general operator-theoretic route that turns Kato-type quasilinear evolution systems on a Banach scale $(Z,X)$ into finite-dimensional interacting approximations. The construction proceeds in two steps. First, one introduces a regularized family $(A_\varepsilon,f_\varepsilon)$ indexed by a scale parameter $\varepsilon>0$, for which the drift $A_\varepsilon[t,z]z+f_\varepsilon[t,z]$ takes values in an output space $Y$ suitable for discretization. Second, one discretizes this regularized dynamics by a sampling-reconstruction pair $(P_N,R_N)$ and obtains an interacting ODE on a finite-dimensional state space $V_N\simeq\R^{dN}$. Our main abstract theorem provides a quantitative estimate of the discrepancy $y_\varepsilon^N-y$ between the lifted discrete solution and the exact one, separating the regularization error $\chi(\varepsilon)$ from the discretization error $(1+L_\varepsilon)N^{-\gamma}$, where $L_\varepsilon$ measures the size of the regularized drift in the output norm. This makes explicit the trade-off between the regularization scale $\varepsilon$, the discretization scale $N$, and the possible deterioration of $L_\varepsilon$ as $\varepsilon\to 0$. As a running example, we focus on quasilinear PDEs on bounded Lipschitz domains with boundary conditions. We show that Burenkov's variable-step mollifiers provide a boundary-compatible kernelization: they regularize differential operators into explicit integral-interaction operators supported inside the domain and preserve boundary traces of sufficiently regular fields. In this setting one can choose an output space $Y$ for which $L_\varepsilon$ remains uniformly bounded, leading to algebraic convergence rates in $N$ for quasi-uniform discretizations.

Inspired by the problem of constructing ($\omega$-)pre-Lie algebra structures on the dual space of a pre-Lie algebra, we introduce the \(A\)-generalized Yang–Baxter equation as a generalization of the Yang–Baxter equation of pre-Lie algebras. We study its symmetric solutions through \(A\)-generalized Hessian pre-Lie algebras and split these solutions into two types. We further consider factorizable solutions of this equation and establish a one-to-one correspondence between them and generalized quadratic Rota–Baxter pre-Lie algebras of nonzero weight. By studying the structure of these algebras, we find all factorizable solutions. Finally, we study the structure of \(A\)-generalized Hessian pre-Lie algebras. In particular, we obtain a structural description via central and double extensions and classify low-dimensional non-trivial \(A\)-generalized Hessian pre-Lie algebras.

In 1983, Kusner conjectured that the largest equilateral set in $\mathbb{R}^{n}$ with metric $\ell_{p}$ has cardinality $n+1$ when $1<p<\infty$ and $2n$ when $p=1.$ This conjecture was proved only in the isolated cases $p=2$ and $p=4$, and was disproved when $1<p<2$. The best general upper bound $O_p(n^{\frac{2p+2}{2p-1}})$ is due to the celebrated work of Alon and Pudlák~[GAFA, 2003]. Our main contributions include: (1) We prove Kusner's conjecture for every dimension $n\ge 1$ when $2\le p\le 4$. More generally, for every integer $k\ge 0$ and every $p\in[4k+2,4k+4]$, every equilateral set in \(\mathbb{R}^{n}\) with metric $\ell_p$ has cardinality at most $(2k+1)n+1$. On the complementary intervals $p\in(4k,4k+2)$ with $p\geq 1$, we obtain the almost linear bound $O_p(n\log n)$. (2) We also consider the analogous problem on the torus $\mathbb{T}^n$, recently initiated by Alon, where the cyclic distance makes the problem substantially more delicate than in $\mathbb R^n$. We prove the almost linear bound $O_p(n\log n)$ for $1\le p\le 2$ and $O_p(n^{\frac{3}{2}-\frac{1}{p}})$ for every fixed real $p>2$, improving Alon's bounds $O_p(n^{2+\frac{2}{\lfloor p\rfloor}})$ for all finite $p\ge 1$.