Papers

Over any commutative ring containing $\tfrac16$, we study Lie algebras $L$ of type $\mathrm{E}_8$ that arise from the Tits–Kantor–Koecher (TKK) construction on a Brown algebra, and their twisted forms. We construct a smooth scheme $\mathbf{Y}$ of pairs of extremal elements in $L$. When $L$ arises from the TKK-construction, we express the automorphism group, of type $\mathrm{E}_8$, as an $\mathrm{E}_7$-torsor over $\mathbf{Y}$. We show that twisting by this torsor produces the graded isomorphism classes of those algebras isomorphic to $L$, and parametrize these classes by using $\mathbf{Y}$. We show that this torsor is non-trivial, yielding isomorphic Lie algebras of type $\mathrm{E}_8$ that are not graded isomorphic, as opposed to the behaviour over fields.

We study dual-cascade flux laws for the stochastic forced–dissipative surface quasi-geostrophic (SQG) equation on a large periodic box. For statistically stationary solutions, under a weak anomalous dissipation assumption, we derive rigorous mixed third-order structure-function laws for the dual cascade: a Yaglom-type law for the direct cascade of surface potential energy (SPE) and a mixed flux law for the inverse cascade of the Hamiltonian. We further establish complementary scaling laws: a cubic small-scale law for the Hamiltonian structure function and a large-scale vanishing law for the SPE structure function. In particular, both Hamiltonian laws appear to be new for SQG even as an explicit third-order structure-function relation. We also prove Onsager-type obstruction results showing that sufficiently regular stationary families cannot sustain the corresponding non-zero fluxes: $B^s_{3,\infty}$-regularity above the Onsager threshold $1/3$ rules out the direct SPE flux, while sufficient low-frequency Besov regularity rules out the inverse Hamiltonian flux. These results provide a rigorous formulation of the SQG dual-cascade phenomenology in a stochastic stationary setting.

We introduce avalanche homology as a new (di)graph homology theory, based on the dynamics of the sandpile model. Avalanche homology is the simplicial homology of the avalanche complex generated from the sets of unstable vertices at the time steps of the sandpile dynamics. In this work we focus on digraphs, and our main results give the homotopy types of the avalanche complex for directed paths and directed cycles for certain initial configurations of the sandpile dynamics. Even for such simple digraphs a wide range of topologies can arise, and we compare this to the directed flag complex and to the recently introduced burning homology. Furthermore, the dynamics yields very naturally a filtered simplicial complex, and hence persistent avalanche homology.

Computing matrix pseudospectra over a prescribed region requires evaluating the smallest singular value of $C-zI$ at a large number of grid points, which can be prohibitively expensive for large-scale matrices. We develop a recycling-based framework for accelerating such computations for both dense and sparse matrices. The main idea is to exploit the correlation between singular value problems at neighboring grid points by adaptively recycling singular subspaces computed at previously visited points by an iterative SVD solver. We develop fast Rayleigh-Ritz-SVD procedures for extracting Ritz singular pairs from the recycled singular subspaces, together with fast residual evaluation procedures, with an overall cost that scales linearly with the number of recycled samples. When the iterative SVD solver admits preconditioning, we propose using a two-level preconditioner whose projection subspaces are recycled. Numerical experiments demonstrate that the proposed recycling strategies yield substantial speedups over existing methods while maintaining the accuracy of the computed pseudospectra.

This text provides an introduction to multiple hypothesis testing. It covers various error criteria and testing procedures, and includes references to relevant R packages. An earlier version of this text served as the lecture notes for a PhD-level course on multiple testing.

We solve the open problem of constructing a Bayes-optimal iterative signal recovery algorithm for linear-Gaussian multiuser communication systems with random precoding at the transmitters. Specifically, we consider the received signal model $\mathbf{y} = \sum_{u} \mathbf{H}_u \mathbf{\Xi}_u \mathbf{s}_u + \mathbf{n}$, where $\mathbf{n}$ is white Gaussian noise, $\{\mathbf{H}_u \in \mathbb{C}^{L \times L}\}$ are discrete-time channel matrices – modeling a wide class of generally time-varying and dispersive linear channels with possibly multiple antennas – and the precoding matrices $\{\boldsymbol{\Xi}_u \in \mathbb{C}^{L \times N_u}\}$ are drawn independently from a right-unitarily invariant random matrix ensemble. We consider generic non-separable (coded) systems where the users' signals $\{\mathbf{s}_u\}$ follow general (non-factorizing) distributions. For this model, we introduce a novel orthogonal/vector approximate message passing (OAMP/VAMP)-type framework, including an algorithm and its high-dimensional (but finite-sample) analysis. From an algorithmic standpoint, the proposed method can be interpreted as an interpolation between Minka's expectation propagation (EP)–a widely used method in machine learning–and OAMP. Our main theoretical contribution is the explicit finite-sample analysis of the proposed algorithm. Furthermore, we analyze the associated inference problem via a replica-symmetric (RS) ansatz by using a novel disorder-averaging technique. Both the (rigorous) high-dimensional analysis of the algorithm and the RS ansatz reveal the same decoupling principle, establishing that the proposed algorithm is asymptotically Bayes-optimal under the validity of the RS ansatz.

Motivated by the twin homomorphism problem for Coxeter groups and the corresponding Hecke monoids, we find a large class of its solutions originating from standard homomorphisms of Artin monoids and their compositions. These homomorphisms are expected to be injective when they are optimal and injective on generators, which generalizes the homogeneous homomorphisms and the famous Tits conjecture settled by Crisp and Paris. We classify disjoint standard homomorphisms and conjecture the complete classification when the domain is of rank two.

These lecture notes contain the material presented at one of the mini-courses of the Workshop on Character Varieties and Higgs Bundles held in Liberia, Guanacaste, Costa Rica, in August 2025. They also contain some exercises for the students attending the conference. This manuscript contains the basic ideas and constructions about $e$-polynomials in character varieties and the state of the art of certain research in the field, plus some new further directions. We introduce mixed Hodge structures and $e$-polynomials, together with a series of arithmetic (counting points over finite fields) and geometric (stratification into parabolic types) techniques to compute them. We include a complete example of the calculation of the $e$-polynomial for the ${\rm GL}_3$-character variety of the free group. Finally, we extend the geometric stratification into parabolic types to a general reductive group $G$ to obtain explicit motivic expressions for the $G$-character varieties, and reduce certain topological mirror symmetry conjectures for these moduli spaces.

Community detection is a key task in network analysis, providing insight into the structural organization of complex systems. Effective resistance, a graph-theoretic metric derived from electrical network theory, has emerged as a powerful tool for evaluating connectivity and influence within networks. This paper proposes an effective resistance-based community detection algorithm that calculates the similarity between nodes using effective resistance values and produces a weighted graph. The sparse graph used in the algorithm is generated after computing the minimum spanning tree (MST) of the weighted graph and adopting a threshold sparsification strategy on non-MST edges. A maximum modularity approach is adopted using the Clauset-Newman-Moore algorithm on the resultant sparse graph. This algorithm is evaluated for both synthetic and real-world networks, demonstrating its effectiveness compared to popular existing methods. The result shows that the effective resistance-based approach accurately captures the structures of the community while maintaining computational efficiency.

Electric service vehicles (ESVs), such as mobile chargers and drone-based service units, are becoming an important operational resource for on-demand service systems. Unlike conventional spatial servers, ESV operations are shaped by battery limits and recharging needs, which affect dispatch feasibility and spatial deployment decisions. We develop an energy-constrained hypercube spatial queueing model that embeds battery-state dynamics into the classical hypercube framework and uses a semi-Markov representation to estimate steady-state performance. We then formulate a joint location–zoning problem for station placement and service zone design. The resulting large-scale mixed-integer nonlinear program admits a set partitioning reformulation whose column coefficients are not available in closed form. We therefore develop a Branch-Price-and-Evaluation framework for set partitioning problems with externally computable column coefficients: upper-bounding surrogates guide pricing, and iterative exact evaluation updates the coefficients of active columns. Computational results show that explicit energy modeling significantly reduces false service promises and yields more credible planning decisions. They also reveal a load-dependent reversal in zoning: pooling is preferable under light demand, whereas tighter zoning becomes more profitable as demand increases. Over the tested range, profitability is driven more by zoning than by battery improvement, suggesting that managers should get service zone design right before investing in battery upgrades; this caution is reinforced by the counterintuitive finding that larger batteries may delay replenishment and reduce fleet readiness under sparse demand. These findings show that energy feasibility is not merely a matter of battery-capacity expansion, but a design dimension that shapes service-zone configuration.

Recent results established the maximum forest number $f(B)$ for balanced bipartite graphs under Ore-type degree sum conditions. In this paper, we extend these results by determining the exact value of the maximum forest number as a closed-form formula for general bipartite graphs under Ore-type conditions, answering an open question posed by Yu. We prove that the maximum forest number is bounded by a discrete optimization over at most six critical structural coordinate points dictated by hyperbolic density constraints. Furthermore, we establish that deciding whether a specific balanced bipartite graph on $2n$ vertices has a forest number of at least $n+2$ is NP-complete. This implies that while the maximum possible forest number can be exactly bounded, computing the exact forest number for a given graph remains computationally intractable, and a simple structural characterization via finite forbidden induced subgraphs cannot exist unless P = NP.

As is well known in the literature, the category Mon(V) of monoids in a monoidal category V satisfies various fundamental categorical properties, at least when the monoidal base V is correspondingly well-behaved. In particular, Mon(V) is monadic over V as soon as free monoids exist, while if V is cocomplete or locally presentable and its tensor b is sufficiently compatible with the appropriate colimits, then Mon(V) inherits the analogous property. In the present work, we extend such results to the context of double categories. More precisely, we identify conditions on a double category D under which one can show that the category Mnd(D) of monads in D is monadic over the category of endomorphisms End(D), is cocomplete or even locally presentable. We also tackle the issue of local presentability in the dual case Cmd(D) of comonads. In these results, our assumptions on the double category D revolve around notions of colimit, in particular those of parallel and stable local colimits, as well as a notion of local presentability of a double category which has been introduced in previous work.

In this paper, we first propose two different quantities on graphs, namely local entropy and cohesion, then design two corresponding flows for edge weights: the local entropy flow and the cohesion flow. We establish the global existence and uniqueness of solutions for both flows and investigate their asymptotic behaviors, including the case that the limit goes to positive infinity. Moreover, they can be applied to fundamental network analysis tasks, including community detection and node classification. Empirical evaluations demonstrate that our method achieves performance competitive with Ollivier Ricci flow and Lin-Lu-Yau Ricci flow on benchmark network analysis tasks. In experimental scenarios, we first apply the cohesion flow to evolve the edge weights of the graph, and then apply the local entropy flow to further update the resulting weighted graph. Both flows are computationally efficient, leading to a significant reduction in overall computational cost and improved scalability.

We introduce and analyze several grouped variants of the method of reverse importance sampling (RIS) for estimating a partition function from samples of the Boltzmann distribution $p(x)=e^{ \betaU(x)}/Z(\beta)$. Ordinary RIS weighs each sample separately. By contrast, our proposed grouped RIS (GRIS) methods are based on assigning the samples into groups (or batches) of size $k\ge 2$ and applying a joint weight function to each group. The focal point of the research is the quest for a tractable weight function that would yield the smallest possible mean squared error (MSE). A simple identity relates the normalized MSE to the chi-squared divergence between the joint-weight distribution and the distribution of the $k$-fold sum of independent energies. Our first theoretical finding is that any weight that improves on ordinary RIS ($k=1$) must couple the group components. In other words, it must not be a product-form function across those components, as product-form weight functions always worsen the MSE. Our second, and more important, finding is that, without loss of optimality, it is sufficient to seek weight functions that depend only on the total energy, $\sum_iU(x_i)$, of the group (group-energy weight functions); for the sliding-window variants, the analogous result is open. This finding simplifies both the theoretical analysis and the application of the method substantially. For $k=2$ and $k=3$, the MSE associated with non-overlapping (NOL) groups is reduced by $20$–$65\%$ across three examples. We then propose two additional variants of GRIS, both based on sliding-window grouping (as opposed to NOL grouping). The first applies a fixed weight sliding window (FSW) across all (cyclic) shifts of the sliding window, and the second allows a variable-weight sliding window (VSW). The FSW scheme improves on the NOL one, and the VSW improves even further, as will be demonstrated numerically.

In this paper, we analyze the structure of the dimension spectrum of continued fraction expansions with coefficients restricted to the generalized Fibonacci sequence. Let $F_{(a_1,a_2)}$ denote the generalized Fibonacci sequence starting with the positive integers $a_1<a_2$. We prove that the continued fractions whose digits lie in $F_{(a_1,a_2)}$ have full dimension spectrum for every $(a_1,a_2)$ such that $a_1 \geq2$, or $a_1=1$ and $a_2\geq3$. On the other hand, using the numerical tools developed by Falk and Nussbaum, we show that the dimension spectrum has a gap for continued fractions with digits restricted to each of the sets $F_{(1,2)}$ and $F_{(2,1)}$, where $F_{(2,1)}$ denotes the set of Lucas numbers. Moreover, for $F_{(1,2)}$ and $F_{(2,1)}$, we prove that the dimension spectrum always contains a non-trivial interval.

Erasure coding is a key technique for providing fault tolerance in modern distributed storage systems. In practice, as storage systems evolve, the parameters of the deployed erasure code may need to be adjusted to accommodate changes in storage scale, reliability requirements, and disk failure rates. Such adaptation is achieved through code conversion, which transforms data encoded by an initial code into data encoded by a final code. Convertible codes are designed to carry out this transformation efficiently while preserving desirable code properties. In this work, we study code conversion between systematic optimal-distance locally repairable codes (LRCs) in the global split regime, using read bandwidth as the conversion-efficiency metric. Specifically, we focus on the parameter range $g^I,g^F \leq r$, where the numbers of initial and final global parity nodes are at most the local information dimension $r$. Over this entire parameter range, we derive lower bounds on the read bandwidth of stable optimal-distance locally repairable convertible codes (LRCCs) via an information-theoretic approach, without imposing any linearity assumption on the initial codes, the final codes, or the conversion procedure. We then develop constructions based on MDS array codes with prescribed repair or alignment properties. Depending on the relative sizes of $g^I$ and $g^F$, we handle the construction separately in the three cases $g^F=g^I$, $g^F>g^I$, and $g^F<g^I$, and show that each attains the corresponding lower bound. This yields a complete characterization of the optimal read bandwidth for stable optimal-distance LRCCs over the entire parameter range $g^I,g^F\le r$.

We consider the problem of optimally distributing resources from a set of suppliers to a set of consumers in the presence of general transportation constraints, and including heterogeneous flexibility in the marginal constraints. Such problems frequently arise in a variety of practical settings; for example, in the context of sharing economy applications, where one is not only interested in the transportation plan, but also its realisation, and in other problems that involve the study of martingales. Our principal contribution in this paper is to consider a generalisation of the classic entropically regularised Optimal Transport formulation in which such problems can be solved with a Sinkhorn algorithm. In particular, we present provably convergent Sinkhorn-like algorithms for solving this class of problems, and provide examples to both illustrate the utility of our approach as well as its efficacy.

This paper develops a Bessel-chromatic expansion framework associated with fractional powers of the Bessel-Laplace operator. The construction combines methods of weighted polynomial approximation and of fractional differential operators. Using the spectral representation of $(-\Delta_a)^{\frac{1}{2}}$, we define Bessel-chromatic derivatives and apply them to weighted spherical means both at a general point and at the origin. Different classes of weights on finite and infinite intervals are considered, with particular attention to cases where the inverse Hankel transform is explicit. The convergence of the expansions is studied through Cesàro and de la Vallée Poussin means. In the bandlimited case, the method gives reconstruction formulas for weighted spherical means and, under suitable assumptions, recovery formulas for the original function. Numerical examples illustrate the decay of the Bessel-chromatic coefficients and the accuracy of the corresponding reconstructions.

The supersaturation problem asks, for a fixed $r$-graph $\mathcal F$, for the minimum number of copies of $\mathcal F$ in an $n$-vertex $r$-graph with $\ex(n,\mathcal F)+q$ edges. Mubayi conjectured a local form of supersaturation under a stability hypothesis: if $\mathcal F$ is non-$r$-partite and stable, meaning roughly that the extremal $\mathcal F$-free construction is unique and all near-extremal $\mathcal F$-free $r$-graphs are close to it, then this minimum should be at least $q c(n,\mathcal F)$, where $c(n,\mathcal F)$ is the minimum number of copies created by adding one edge to the extremal $\mathcal F$-free $r$-graph. We disprove this conjectured local lower bound in the strongest possible quantitative sense that the ratio to the conjectured lower bound can be made arbitrarily small. More precisely, for every $r\ge2$ and every $K>1$, we construct a stable non-$r$-partite $r$-graph $\mathcal F$ such that, for all sufficiently large $n$ and every $1\le q\le\delta n$, there is an $n$-vertex $r$-graph with $\ex(n,\mathcal F)+q$ edges containing at most $K^{-1}q c(n,\mathcal F)$ copies of $\mathcal F$. This arbitrarily large failure already occurs when $q=1$.

For a bounded linear operator acting between Banach spaces, its metric generalized inverse is the analog to the prominent Moore-Penrose inverse for operators acting between Hilbert spaces. This generalized inverse is well-defined for Banach spaces that are strictly convex and reflexive. Previous studies had been restricted to closed range situations, where the metric generalized inverse constitutes a continuous homogeneous mapping. The focus of the present study is on the ill-posed situation in the sense of Nashed, when the governing operator has a non-closed range. We define and analyze iterative schemes as the Landweber and the Schulz-Newton method, as well as parametric schemes with focus on a specific Tikhonov method. Both types are called regularizations aimed at approximating the metric generalized inverse. As a fundamental feature of such schemes we observe a dichotomy. This emphasizes that these schemes, when applied to elements of the domain of the metric generalized inverse, approximate the corresponding best approximate solutions well, whereas the resulting approximations will be asymptotically unbounded if they are applied to elements that do not belong to the domain of the metric generalized inverse.

We introduce isoregular theories, in which it is possible to express existential quantification up to unique isomorphism, as typically used to characterise category-theoretic universal constructions, such as limits. We then develop a functorial semantics for isoregular theories and prove that their 2-categories of models are accessible with flexible limits. We apply these results by showing that a number of 2-categories of interest in general category theory, categorical algebra, and categorical logic are models of isoregular theories, thereby establishing that they are accessible 2-categories with flexible limits and obtaining a number of new free constructions.

Balancing exploration and exploitation remains a central challenge in metaheuristic optimization. To address this issue, this paper proposes Bézier Walk Evolution (BWE), a geometry-driven optimization framework that reformulates evolutionary search as adaptive trajectory construction in the decision space. BWE integrates Bézier curve modeling with a distance-aware random walk mechanism to generate topology-guided search trajectories. By adaptively varying the curve order during evolution, the proposed method enables a smooth transition from diversified global exploration to refined local exploitation. Higher-order Bézier curves leverage multiple population-derived control points to enhance search diversity, while lower-order curves generate near-linear trajectories to improve convergence efficiency. This adaptive geometric search mechanism provides an interpretable alternative to conventional nature-inspired designs. Extensive experiments on 41 benchmark functions from the CEC2017 and CEC2022 suites, spanning dimensions from 10 to 100, show that BWE achieves strong overall performance and favorable scalability compared with 7 classical and 6 state-of-the-art optimizers, including L-SHADE and CMA-ES. Additional evaluations on five constrained engineering design problems further demonstrate the practical applicability and robustness of BWE.

Feedforward neural network (NN) expressivity is typically studied by emulating optimal basis-expansion schemes. While powerful, this perspective is incomplete: it primarily captures complexity through regularity, and therefore does not distinguish intuitively simple and complicated objects with comparable regularity, such as the square-root function and a typical Brownian path. The guiding message is that neural networks should be viewed not only as flexible basis functions, but also as models of computation. If a function is computable by a real-valued circuit over a prescribed elementary gate language, then it can be computed to comparable accuracy by an NN with explicit depth, width, and non-zero-parameter bounds controlled by the depth, width, gate count, and gate structure. Thus, neural-network complexity is not governed by regularity alone, but also by algorithmic complexity. We then show that any definable NN model satisfying a natural parallelization condition, allowing possibly multivariate non-linearities such as attention or layer normalization, is a universal approximator if and only if it contains a non-affine nonlinearity. The scope of our theory is illustrated by deducing universal approximation guarantees for continuous functions, minimax-optimal approximation guarantees for Besov classes, logarithmic-error complexity for holomorphic functions, and by showing that NNs can emulate numerical algorithms such as Newton-Raphson root finding and power iteration without architecture-specific arguments. Its precision is illustrated by shortest-path computation on $k$-vertex graphs: compiling the tropical dynamic-programming circuit yields NNs with O(log(1/{\epsilon})) non-zero parameters, exponentially improving in 1/{\epsilon} over the generic $O({\epsilon}^{-c k^2})$ Lipschitz-approximation scale, for a constant c>0.

The convergence of the self-consistent field iterations in Kohn-Sham density functional theory can be significantly hindered by the presence of small eigenvalues in the dielectric matrix, which are often associated with electronic phase transitions in magnetic systems. In this work, we study this type of convergence issues and propose a new preconditioning scheme to mitigate them. Our preconditioning scheme is inspired by the Stoner model and based on a non-interacting susceptibility that neglects orbital variations. We demonstrate the effectiveness of our approach on a range of ferromagnetic systems, showing that it can significantly reduce the number of iterations required to achieve convergence in the vicinity of magnetic phase transitions.

Total variation diminishing (TVD) and total variation bounded (TVB) properties are crucial for controlling spurious oscillations in numerical solutions of conservation laws. In the classical Runge–Kutta (RK) discontinuous Galerkin (DG) framework, enforcing these properties is intrinsically tied to TVD time integrators, more commonly known today as strong-stability-preserving (SSP) methods. This reliance imposes severe structural restrictions, including order barriers and incompatibility with various fully discrete DG formulations, ranging from the recent RKDG method with compact stencils (cRKDG) to the widely established Arbitrary DERivative (ADER) DG method. To bypass these constraints, we propose a novel trace-limited corrector framework that preserves the TVD/TVB-in-the-means properties using generic, non-SSP time stepping. Based on Harten's lemma, our key insight is that total variation stability is dictated solely by the cell-average update in the final corrector stage. Consequently, we modify the traces in the numerical fluxes exclusively in the final stage, leaving the intermediate predictor stages unconstrained. This strategy decouples oscillation control from the SSP restriction, accommodates standard RKDG, cRKDG, and ADER-DG predictors, and retains the compactness of the cRKDG framework. We also prove that the limiter does not activate in smooth regions, thereby preserving the underlying accuracy. Finally, numerical experiments are presented to demonstrate the capabilities and robustness of the method.

Let $G$ and $H$ be finite simple graphs and assume that either both are undirected or both are directed. We introduce and study the ideal of weak graph homomorphisms $I_{G\to H}$. We characterize all graphs $G$ and $H$ for which every (equivalently, some) power of $I_{G\to H}$ has a linear resolution. Moreover, unmixedness, Cohen-Macaulayness, projective dimension and Castelnuovo-Mumford regularity of these ideals are studied.

This paper studies extensions of adaptive randomized pivoting (ARP), recently introduced for matrix column subset selection, to tensors in the t-product framework. We propose two constructions. ARP-T-CUR applies matrix ARP-cross at the nonredundant Fourier frequencies and uses conjugate symmetry to preserve real-valued tensors. Assuming that the selected intersections are nonsingular almost surely, the matrix theory gives a direct expected-error bound for this Fourier-slicewise approximation. T-ARP instead selects common lateral and horizontal slices, with the same indices used at every Fourier frequency. This common-index constraint requires a new analysis. Under frequency-alignment and frequency-wise rank-growth assumptions, we prove an expected-error bound and recover the matrix ARP factor $r+1$ when the frequency-wise sampling distributions are aligned. We also derive bounds for tensor cross approximation and t-DEIM. The numerical experiments show that both proposed methods give accurate tensor approximations and perform favorably against the considered slice-selection baselines. A separate full-enumeration experiment verifies the assumptions and the expected-error bound for common-index T-ARP.

For every \(0<\kappa<\sqrt{5}-2\), we prove global existence for the two-dimensional generalized parabolic Anderson model on the whole plane $\mathbb R^2$ with nonlinearity $F\in C_b^2(\mathbb R)$, driven by an enhanced noise $(\eta,\Psi)$. The noise $\eta$ has polynomially weighted spatial Besov–Hölder regularity $-1-\kappa$, and $\Psi$ is the corresponding renormalized second-order object. If $F''$ is globally Lipschitz, the solution is unique. The proof combines a weight-compatible annular high–low decomposition with a paracontrolled transport representation. The final remainder is estimated simultaneously in a weighted $L^\infty$ norm and in a higher-order weighted parabolic Hölder norm, using two strictly different polynomial weights. This weight gap absorbs the polynomial losses generated by the enhanced noise, the localization procedure, and the transport coefficient. Several refinements of earlier work allow the maximum-principle and Schauder estimates to yield a global a priori bound for a larger range of $\kappa$. Uniqueness is proved in a time-dependent exponentially weighted topology.

In this paper, we study analytic nonlinear partial differential equations for $\mathbf y=\mathbf y(x,\theta)\in \mathbb C^n$ of the form $x^2 \mathbf y'_x(1+\mathcal O(x)) + \mathbf y'_\theta+x \mathbf A \mathbf y = \mathcal O(x^2)$, $()'_z=\frac{\partial}{\partial z}$, $z=x,\theta$, with $x\in \mathbb C$ and $\theta\in \mathbb C /(2\pi \mathbb Z)$ denoting the independent variables. We show that solutions are $1$-sums (with respect to $x$) of Fourier series with coefficients of type Gevrey-$1$. The motivation for studying these equations is that so-called inner problems, associated with two-dimensional formal connections in unfoldings of local bifurcations, can be brought into this form (by looking for invariant manifolds in blowup coordinates). In the present paper, we give two examples: The zero-Hopf bifurcation and the resonant Hopf-Hopf bifurcation. Importantly, these inner problems are given by the unperturbed problem (i.e. at the bifurcation) and the invariant manifolds are expressed directly in phase space (through blowup coordinates associated with the local bifurcation). This contrasts the Lazutkin-based formulation of inner problems which is based upon a blowup of singularities (with respect to complex time) of approximate solutions; for local bifurcations this requires (artificial) coordinate transformations. We solve the PDE by extending the Banach-convolution-algebra-approach to Borel-Laplace by Bonckaert and De Maesschalck (2008) to account for analytic Fourier series. In further details, we apply the Borel transform with respect to $x$ (keeping $\theta$ fixed) and solve the resulting equation in an appropriate Banach space of Fourier series with coefficients that have exponential growth in the Borel plane. The solutions of the PDE are then obtained through application of the Laplace transformation.

Graph Neural Differential Equations (GNDEs) model continuous-time graph dynamics by parameterizing Neural ODE velocity fields with Graph Neural Networks. Their local, size-independent filters suggest a zero-shot size-transfer principle: train on a small graph and deploy on larger, similar graphs without retraining. We develop a quantitative theory for this principle on sparse random graphs sampled from graphons. We consider Graphon Neural Differential Equations (Graphon-NDEs) and adjoint Graphon-NDEs as the infinite-node limits of the forward and adjoint GNDE systems, and establish well-posedness. For an $n$-node random graph with sparsity parameter $\alpha_n$, we prove trajectory-wise convergence of GNDE solutions to Graphon-NDE solutions at rate $O((\alpha_n n)^{-1/2})$, up to logarithmic factors, with high probability. We also establish uniform-in-time convergence bounds for adjoint systems governing hidden-state and parameter gradients. We further study discretize-then-optimize (DTO) and optimize-then-discretize (OTD) training. Under explicit Euler discretization with $M$ steps, we show that DTO and OTD are asymptotically consistent, with hidden-state and local parameter-gradient discrepancies of orders $O(1/M)$ and $O(1/M^2)$, respectively, up to sparsity and logarithmic factors. Experiments on HSBM and tent graphons support the theoretical rates, while zero-shot transfer experiments across four graphon classes demonstrate accurate deployment of learned GNDEs on larger independently sampled graphs.