We construct for a Schur concave function $f$ on the set of quantum states a tight upper bound on the difference $f(ρ)-f(σ)$ for a quantum state $ρ$ with finite $f(ρ)$ and any quantum state $σ$ $m$-partially majorized by the state $ρ$ in the sense described in [1]. We also obtain a tight upper bound on this difference under the additional condition $\frac{1}{2}\|ρ-σ\|_1\leq\varepsilon$ and find simple sufficient conditions for vanishing this bound with $\,\min\{\varepsilon,1/m\}\to0\,$. The obtained results are applied to the von Neumann entropy. The concept of $\varepsilon$-sufficient majorization rank of a quantum state with finite entropy is introduced and a tight upper bound on this quantity is derived and applied to the Gibbs states of a quantum oscillator. We also show how the obtained results can be reformulated for Schur concave functions on the set of probability distributions with a finite or countable set of outcomes.

The family of $(k,\ell)$-sparse graphs, introduced by Lorea, plays a central role in combinatorial optimization and has a wide range of applications, particularly in rigidity theory. A key algorithmic problem is to decide whether a given graph is $(k,\ell)$-sparse and, if not, to produce a vertex set certifying the failure of sparsity. While pebble game algorithms have long yielded $O(n^2)$-time recognition throughout the classical range $0 \leq \ell < 2k$, and $O(n^3)$-time algorithms in the extended range $2k \leq \ell < 3k$, substantially faster bounds were previously known only in a few special cases. We present new recognition algorithms for the parameter ranges $0 \le \ell \le k$, $k < \ell < 2k$, and $2k \leq \ell < 3k$. Our approach combines bounded-indegree orientations, reductions to rooted arc-connectivity, augmenting-path techniques, and a divide-and-conquer method based on centroid decomposition. This yields the first subquadratic, and in fact near-linear-time, recognition algorithms throughout the classical range when instantiated with the fastest currently available subroutines. Under purely combinatorial implementations, the running times become $O(n\sqrt n)$ for $0 \leq \ell \leq k$ and $O(n\sqrt{n\log n})$ for $k< \ell <2k$. For $2k \leq \ell < 3k$, we obtain an $O(n^2)$-time algorithm when $\ell \leq 2k+1$ and an $O(n^2\log n)$-time algorithm otherwise. In each case, the algorithm can also return an explicit violating set certifying that the input graph is not $(k,\ell)$-sparse.

The Energy Conserving Descent (ECD) algorithm was recently proposed (De Luca & Silverstein, 2022) as a global non-convex optimization method. Unlike gradient descent, appropriately configured ECD dynamics escape strict local minima and converge to a global minimum, making it appealing for machine learning optimization. We present the first analytical study of ECD, focusing on the one-dimensional setting for this first installment. We formalize a stochastic ECD dynamics (sECD) with energy-preserving noise, as well as a quantum analog of the ECD Hamiltonian (qECD), providing the foundation for a quantum algorithm through Hamiltonian simulation. For positive double-well objectives, we compute the expected hitting time from a local to the global minimum. We prove that both sECD and qECD yield exponential speedup over respective gradient descent baselines--stochastic gradient descent and its quantization. For objectives with tall barriers, qECD achieves a further speedup over sECD.

We study a nonlocal diffusion equation of porous medium type featuring a generalised fractional pressure with spatial anisotropy. We construct a finite element method for the numerical solution of the equation on a bounded open Lipschitz polytopal domain $Ω\subset \mathbb{R}^{d}$, where $d = 2$ or $3$. The pressure in the model is defined as the solution of fractional elliptic problem involving the fractional power of a second order differential operator, in terms of its spectral definition. Under suitable assumptions on the fractional order and the coefficients of the operator, we rigorously prove convergence of the numerical scheme. The analysis is carried out in two stages: first passing to the limit in the spatial discretization, and then in the time step, ultimately showing that a subsequence of the sequence of finite element approximations defined by the proposed numerical method converges to a bounded and nonnegative weak solution of the initial-boundary-value problem under consideration. Finally, we present numerical experiments in two dimensions illustrating the computational aspects of the method and highlighting the interplay between nonlocal effects and spatial anisotropy under different configurations. We also show numerically the failure of the comparison principle and exponential decay of the numerical solution to a steady state.

This paper tackles the Electric Capacitated Vehicle Routing Problem (E-CVRP) through a bilevel optimization framework that handles routing and charging decisions separately or jointly depending on the search stage. By analyzing their interaction, we introduce a surrogate objective at the upper level to guide the search and accelerate convergence. A bilevel Late Acceptance Hill Climbing algorithm (b-LAHC) is introduced that operates through three phases: greedy descent, neighborhood exploration, and final solution refinement. b-LAHC operates with fixed parameters, eliminating the need for complex adaptation while remaining lightweight and effective. Extensive experiments on the IEEE WCCI-2020 benchmark show that b-LAHC achieves superior or competitive performance against eight state-of-the-art algorithms. Under a fixed evaluation budget, it attains near-optimal solutions on small-scale instances and sets 9/10 new best-known results on large-scale benchmarks, improving existing records by an average of 1.07%. Moreover, the strong correlation (though not universal) observed between the surrogate objective and the complete cost justifies the use of the surrogate objective while still necessitating a joint solution of both levels, thereby validating the effectiveness of the proposed bilevel framework and highlighting its potential for efficiently solving large-scale routing problems with a hierarchical structure.

The Bell colouring graph $\mathcal{B}(G)$ of a graph $G$ is the graph whose vertices are the partitions of the vertex set of $G$ into independent sets, with an edge between two partitions if and only if one can be obtained from the other by changing the part of a single vertex of $G$. Given a natural number $k$, the Bell $k$-colouring graph $\mathcal{B}_k(G)$ and the upper-Bell $k$-colouring graph $\mathcal{B}_{\geq k}(G)$ are the induced subgraphs of $\mathcal{B}(G)$ consisting of all partitions with at most $k$ parts and at least $k$ parts, respectively. We determine precisely when two finite graphs have isomorphic Bell colouring graphs. In particular, we show that every $n$-vertex graph $G$ with no vertices of degree $n-1$ is uniquely determined by its Bell colouring graph $\mathcal{B}(G)$, and by its upper-Bell colouring graph $\mathcal{B}_{\geq k}(G)$ if $k\leq n-2$. We also show that every $n$-vertex graph with maximum degree $Δ(G)< \frac{1}{9}n-\frac{1}{3}$ is uniquely determined by its Bell $k$-colouring graph $\mathcal{B}_k(G)$ if $k>χ(G)$. By taking graph complements, each of these results can be restated in terms of partitions into cliques.

We establish sufficient conditions on discrete subsets of $\mathbb{R}^d$ for them to form a uniqueness or a non-uniqueness pair for the fractional Laplacian. Specifically, assuming that $f=0$ on $Λ$ and that $(-Δ)^sf=0$ on $M$, where $Λ, M \subset \mathbb{R}^d$ are discrete, we find sufficient conditions on these sets that force $f$ to vanish identically, and we provide examples in which non-uniqueness occurs. Some of the ideas used in the proofs also extend to a broader class of multiplier operators.

This work establishes that an optimal transport~(OT) problem regularized by a given $f$-divergence admits the same solution as another OT problem regularized by a different $g$-divergence, under an appropriate transformation of the cost function. This structural equivalence between OT problems regularized by distinct divergences, in the sense of sharing the same unique minimizer, is demonstrated within the framework of Polish spaces with bounded cost functions.

We develop a variationally consistent mesoscopic extension of Cosserat elasticity motivated by the breakdown of compatibility in classical formulations. By admitting compatibility-breaking perturbations, the classical theory ceases to remain closed under admissible variations, necessitating an enlargement of the constitutive framework. This leads naturally to a formulation in which torsion and curvature are treated as independent distributed measures of defects. The theory is constructed using a Palatini-type variational approach, with the coframe and connection as independent fields. The resulting Euler--Lagrange equations yield both the standard balance laws and defect-related excitation fields. Material invariance gives rise to configurational forces and moments, which emerge as Noether currents and are directly linked to defect transport governed by the Bianchi identities. The framework provides a unified description of defect kinematics, configurational mechanics, and microstructural evolution. Illustrative examples and numerical evaluations demonstrate how defect transport generates configurational forces and highlight the underlying Maxwell-type structure of the theory. The proposed formulation offers a consistent geometric foundation for the analysis of structured solids with evolving internal geometry and provides a basis for future developments in defect dynamics and dissipative processes.

Recently, the authors of this paper introduced logarithmic Hochschild (co)homology of logarithmic spaces in a geometric way using formality of derived intersections. In this paper, the authors extend the decomposition theorem for the logarithmic Hochschild (co)homology of firm orbifolds to general logarithmic orbifolds and consider two applications of the decomposition theorem. First, we consider two versions of a symmetric product and compute the logarithmic Hochschild homology of them. Second, we show that logarithmic Hochschild homology is invariant under root stack operations.

Inspired by the structure of $srs$ Laves networks in $\mathbb{R}^3$ that underpin the celebrated gyroid surface, we construct a Laves network of identical three-coordinated vertices on $S^3$ with double-twist. This network is a subset of the vertices and edges of the 600-cell, and can be viewed as a bipartite graph of disjoint 24-cell vertices inscribed in the 600-cell. We describe mutually entangled realizations of this network on $S^3$, and describe their relation to the well-known $srs$ Laves network structure in $\mathbb{R}^3$.

For stochastic Hamilton-Jacobi (SHJ) equations, instability points are the space-time locations where two eternal solutions with the same asymptotic velocity differ. Another fundamental structure in such equations is shocks, which are the space-time locations where the velocity field is discontinuous. In this work, we study the KPZ fixed point, the central object of the KPZ universality class, which can be viewed as a prototype--albeit degenerate--of an inviscid SHJ equation in one spatial dimension. We describe the geometric structure of the instability region and give a detailed and precise analysis of its interplay with the shock structures of the two eternal solutions. We show that these shock structures allow one to reconstruct the instability region. Along the way, we obtain a complete classification of all possible configurations of semi-infinite geodesics emanating from arbitrary space-time points, in the directed landscape--the random environment in which the KPZ fixed point evolves.

This paper investigates the robust stability problem of a feedback system in the presence of uncertainties induced by graphical regions in the plane where the scaled relative graphs (SRGs) reside. Our main results are developed using a novel and intuitive concept, the Davis-Wielandt shell, together with its connection to SRGs and related variants. We first study a matrix robust nonsingularity (MRN) problem for two types of graphically induced uncertainty sets: one with prior information on $θ$ and one without. In the former case, we show that, whenever the uncertainty-inducing region is mirror symmetric about the $θ$-axis, the separation between a specific variant of the SRG and the region provides a necessary and sufficient condition for MRN. When the region is asymmetric, the necessity generally fails. This recovers the necessity of the small gain condition, and reveals the necessity of small angle conditions and sectored-disc conditions at the matrix level. In the latter case, we show that an additional $θ$-circular connectivity property is required to obtain necessary and sufficient conditions. Building on these MRN results, we then derive sufficient conditions for robust stability of multi-input multi-output (MIMO) linear time-invariant (LTI) systems under frequencywise symmetric uncertainties. In addition, connections with existing system characteristics such as disc-boundedness are discussed and exploited to obtain state-space characterisations for angle-bounded and mixed gain-angle-bounded systems. Based on these results, we construct a $θ$-angle-gain profile of a system that provides an intuitive visualisation of its feedback robustness against conic and sectorial uncertainties.

Generalized Reed-Solomon (GRS) and Gabidulin codes have been proposed for various code-based cryptosystems, though most such schemes without elaborate disguising techniques have been successfully attacked. Both code classes are prominent examples of the isometric families of (generalized) skew and linearized Reed-Solomon ((G)SRS and (G)LRS) codes which are obtained as evaluation codes from skew polynomials. Both GSRS and GLRS codes share the advantage of achieving the maximum possible error-decoding radius and thus promise smaller key sizes than e.g. Classic McEliece. We investigate whether these generalizations can avoid the known structural attacks on GRS and Gabidulin codes. In particular, we prove that both GSRS and GLRS codes decompose into GRS subcodes and are thus efficiently distinguishable from random codes with a square code method. This applies to all parameters for which the code length $n$ and its dimension $k$ over the field $\mathbb{F}_{q^m}$ satisfy $m + 1 < k < n - \tfrac{1}{2} (m^2 + 3m)$. The distinguishability extends to GSRS and GLRS codes with Hamming-isometric disguising. We further relate these findings to existing distinguishers for GRS, Gabidulin, and LRS codes, and extend known results on duals of SRS and LRS codes to the generalized setting allowing nonzero column multipliers. Finally, we provide explicit transformations between GSRS and GLRS codes, clarifying the algebraic relationship between the skew and linearized frameworks.

The Sauer-Shelah-Perles Lemma is a cornerstone of combinatorics and learning theory, bounding the size of a binary hypothesis class in terms of its Vapnik-Chervonenkis (VC) dimension. For classes of functions over a $k$-ary alphabet, namely the multiclass setting, the Natarajan dimension has long served as an analogue of VC dimension, yet the corresponding Sauer-type bounds are suboptimal for alphabet sizes $k>2$. In this work, we establish a sharp Sauer inequality for multiclass and list prediction. Our bound is expressed in terms of the Daniely--Shalev-Shwartz (DS) dimension, and more generally with its extension, the list-DS dimension -- the combinatorial parameters that characterize multiclass and list PAC learnability. Our bound is tight for every alphabet size $k$, list size $\ell$, and dimension value, replacing the exponential dependence on $\ell$ in the Natarajan-based bound by the optimal polynomial dependence, and improving the dependence on $k$ as well. Our proof uses the polynomial method. In contrast to the classical VC case, where several direct combinatorial proofs are known, we are not aware of any purely combinatorial proof in the DS setting. This motivates several directions for future research, which are discussed in the paper. As consequences, we obtain improved sample complexity upper bounds for list PAC learning and for uniform convergence of list predictors, sharpening the recent results of Charikar et al.~(STOC~2023), Hanneke et al.~(COLT~2024), and Brukhim et al.~(NeurIPS~2024).

Recently, the statistical properties of empirical Entropic Optimal Transport (EOT) have attracted great interest, as this quantity has been shown to be useful for complex data analysis, among other reasons due to its computational efficiency. In several applications, it has been observed that the EOT plan provides valuable information beyond just the optimal value. For example, in cell biology, colocalization analysis based on the EOT plan has been introduced as a measure for quantification of spatial proximity of different protein assemblies. Despite recent progress in the analysis of its risk properties, a precise understanding of its statistical fluctuations to make it accessible for inference remains elusive to a large extent. In this paper, we derive asymptotic weak convergence result for a large class of functionals of the EOT plan, in which the colocalization process is included. The proof is based on Hadamard differentiability and the extended delta method. As an application, we obtain uniform confidence bands for colocalization curves and bootstrap consistency. Our theory is supported by simulation studies and is illustrated by real world data analysis from mitochondrial protein colocalization.

Let $V$ be a grading-restricted vertex algebra and let $A^\infty(V)=U^\infty(V)/Q^\infty(V)$ be the associative algebra constructed by Huang, where $U^\infty(V)$ is the space of column-finite infinite matrices with entries in V and $Q^\infty(V)$ is an ideal of a (nonassociative) algebra structure on $U^\infty(V)$ defined by Huang. Huang introduced families of elements in $Q^\infty(V)$ and conjectured that these elements generate $Q^\infty(V)$. We discover and prove that Huang's elements all satisfy what we call ``the diagonal shift property". On the other hand, in the case that $V$ is the rank one Heisenberg vertex operator algebra, we construct infinitely many linearly independent elements in $Q^\infty(V)$ that do not satisfy the diagonal shift property. As a corollary, we disprove Huang's conjecture.

The LogQ algorithm encodes Quadratic Unconstrained Binary Optimization (QUBO) problems, which are often encountered in the industry (portfolio optimization, fleet optimization, charging stations, etc.). It was developed within the framework of quantum computing, designed as a pragmatic approach to quantum combinatorial optimization that drastically reduces the number of required qubits and quantum circuit depth. While LogQ has recently been made compliant with gradient-inspired methods, greatly improving parameter optimization efficiency, it still faced hurdles regarding Pauli decomposition and measurement overhead. We here demonstrate that LogQ can be fully reformulated within a classical framework, which effectively eliminates the need for Pauli decomposition and bypasses the measurement challenges altogether. This finally leads to a classical heuristic based on a non-linear continuous relaxation of variables and is, to the best of our knowledge, novel. The LogQ story illustrates how quantum computing can inspire classical algorithms, leading to so-called "quantum-inspired" methods.

We prove that a Le Page-type inequality is also valid for metrically characterizing those JB$^*$-triples that are commutative. More precisely, we establish that the following statements are equivalent for any JB$^*$-triple $E$: $(a)$ $E$ is commutative. $(b)$ There exists $γ>0$ satisfying $$\big\|\{a,b,\{x,y,z\}\}\big\|\leq γ\ \! \big\|\{x,y,\{a,b,z\}\}\big\|, \hbox{ for all } a,b,x,y,z\in E.$$

We consider nonlinear GMRES (NGMRES) as an acceleration technique for the Navier-Stokes Picard iteration, a direction that has not previously been explored. We identify the optimal norm for the least squares optimization problem arising in the NGMRES algorithm, and establish a convergence analysis for NGMRES with general depth that proves NGMRES scales the Picard Lipschitz constant by the gain of the optimization problem. To our knowledge, this is the first convergence proof for NGMRES that identifies the mechanism responsible for convergence acceleration. Numerical experiments demonstrate that the convergence estimates are remarkably sharp. In addition, NGMRES greatly improves the performance of the Picard iteration, even in cases where the unaccelerated iteration diverges.

In 2005, Parreau proved that if a measure preserving system is not strongly mixing then it contains a non-trivial factor that is disjoint from every strongly mixing system. Taking this construction as the starting point, we develop the complementary notions of $\mathcal U$-generated and $\mathcal U$-mixing systems, for a set $\mathcal U$ of ultrafilters, and use them to recover several classical results in ergodic theory as special cases of a unified framework. We prove that a system is $\mathcal U$-mixing if and only if it is disjoint from all $\mathcal U$-generated systems. In fact, we show that if $\mathcal Y$ is a $\mathcal U$-generated system and $\mathcal Z$ is disjoint from every $\mathcal U$-mixing system, then any joining of $\mathcal Y$ and $\mathcal Z$ remains disjoint from all $\mathcal U$-mixing systems. We also show that every partially rigid system is a finite extension of some $\mathcal{U}$-generated system.

The doubling conjecture predicts that a manifold admits positive scalar curvature with mean convex boundary if and only if its double admits positive scalar curvature. We show that it holds true for manifolds where the inclusion of the boundary satisfies a certain split-condition on fundamental groups. Our proof is based on surgery-techniques for positive scalar and mean curvature. If the boundary is non-connected, we use existence of area-minimizing hypersurfaces and the monotonicity-formula. Furthermore, we investigate if a psc-metric on a closed manifold can be adjusted so that a given embedded hypersurface is minimal, stable minimal or totally geodesic. While not true in general, such an adjustment is possible in many cases.

Large Language Model-based Hyper Heuristic (LHH) has recently emerged as an efficient way for automatic heuristic design. However, most existing LHHs just perform well in optimizing a single function within a pre-defined solver. Their single-layer evolution makes them not effective enough to write a competent complete solver. While some variants incorporate hyperparameter tuning or attempt to generate complex code through iterative local modifications, they still lack a high-level algorithmic modeling, leading to limited exploration efficiency. To address this, we reformulate heuristic design as a Bi-level Optimization problem and propose \textbf{BEAM} (Bi-level Memory-adaptive Algorithmic Evolution). BEAM's exterior layer evolves high-level algorithmic structures with function placeholders through genetic algorithm (GA), while the interior layer realizes these placeholders via Monte Carlo Tree Search (MCTS). We further introduce an Adaptive Memory module to facilitate complex code generation. To support the evaluation for complex code generation, we point out the limitations of starting LHHs from scratch or from code templates and introduce a Knowledge Augmentation (KA) Pipeline. Experimental results on several optimization problems demonstrate that BEAM significantly outperforms existing LHHs, notably reducing the optimality gap by 37.84\% on aggregate in CVRP hybrid algorithm design. BEAM also designs a heuristic that outperforms SOTA Maximum Independent Set (MIS) solver KaMIS.

The attitude dynamics of aircraft and spacecraft exhibit significantly nonlinear behaviour. In spacecraft, torque is generated through reaction wheels and control moment gyros. In aircraft, torque is generated using lift on control surfaces. In both cases, complex geometries, unique configurations, and internal/environmental changes imply that models must be identified, verified, and updated using in-flight experimental data. However, this data is often noisy, sparsely sampled, and partial in that modeled states may not be directly measurable. In this paper, we propose a method for estimating key parameters in realistic Ordinary Differential Equation (ODE) models of both spacecraft and aircraft dynamics. This method avoids the need to directly measure state derivatives by coupling sampled outputs using the Picard mapping -- an integral constraint on the solution of the parameterized ODE. This constraint is then enforced, and optimal parameter estimates are found using a gradient contraction algorithm. This algorithm is applied to well-studied models of spacecraft and aircraft motion. First, the algorithm is used to estimate the inertia tensor in a 4 control-moment gyro (CMG) model of spacecraft motion. Second, we estimate the 28 higher-order control surface coefficients in a model of the F/A-18 aircraft.

Let \(H(n)\) denote the Hilbert number, i.e.\ the maximal number of limit cycles of planar polynomial vector fields of degree \(\le n\). A classical lower-bound mechanism for \(H(n)\) is \emph{replication}: one pulls back a vector field by a polynomial map and lifts each existing limit cycle to several disjoint copies while controlling the resulting degree. In this paper we give a fully self-contained replication theorem based on the separable Chebyshev covering \[ Φ(u,v)=(T_m(u),T_m(v)). \] Using the \(m\) monotone full branches of \(T_m\) on \((-1,1)\), we prove that every degree-\(\le n\) polynomial vector field with \(k\) limit cycles gives rise to a degree-\(\le nm+m-1\) polynomial vector field with at least \(m^2k\) limit cycles. Consequently, \[ H(nm+m-1)\ge m^2H(n)\qquad (m\ge 2). \] We then extend the construction to general separable pullbacks \((u,v)\mapsto (p(u),p(v))\), show that Chebyshev attains the maximal possible branch count among degree-\(m\) separable pullbacks, and prove a quadratic ceiling for replication-only schemes: if one iterates separable pullbacks and no additional limit cycles are created beyond those forced by lifting, then the number of resulting limit cycles is at most quadratic in the final degree. This shows that superquadratic lower bounds, such as the known \(n^2\log n\)-type bounds, necessarily require mechanisms beyond pure separable replication. Finally, combining our replication theorem with the strongest currently published seed bounds, we obtain new explicit lower estimates in several degrees, including \begin{gather*} H(14)\ge 252,\qquad H(29)\ge 1080,\\ H(31)\ge 1380,\qquad H(39)\ge 2012. \end{gather*}

Hurwitz theory provides a large variety of enumerative problems related to algebraic geometry, mathematical physics, and combinatorics. We give a general framework to approach the large genus asymptotics of Hurwitz theory using only elementary methods and apply it to several types of Hurwitz numbers: single, double, or with an arbitrary numbers of fixed ramifications; simple and / or including completed cycles type of ramification and / or finitely many blocks of weakly monotone and / or strictly monotone types of ramifications. These, to the best of our knowledge, cover most of the Hurwitz numbers studied, and include for instance correlators of the HCIZ matrix model, Grothendieck dessins d'enfant, weighted Hurwitz numbers, and Gromov-Witten invariants of the Riemann sphere. We also apply our method to b-content Hurwitz numbers. As a specialisation, we recover some previously known about the large genus asymptotics of Hurwitz theory, namely classical results by Hurwitz and recent results of Do-He-Robertson, C. Yang, and results connected to recent work of X. Li.

We prove a general Fueter Theorem over real alternative *-algebras. We show that a suitable power of the Laplacian maps Dunkl-regular functions to Dunkl monogenic functions with axial symmetries. Using the embedding of hypercomplex function theories in the class of Dunkl monogenic functions, we subsume several Fueter-type results known in the literature and obtain the most general form for the action of the Laplacian on function spaces over hypercomplex subspaces. We show that Fueter Theorems are in a one-to-one correspondence with a class of graphs, the Fueter trees, that describe the interactions between Dunkl-regular function spaces and the relation with the iterated Laplacian. We obtain that the number of distinct Fueter trees on a hypercomplex space of dimension $n+1$ is equal to the number of partitions in odd parts of the integer $n$.

We present an integral formulation of classical Yang-Mills theory coupled to fermionic and scalar matter fields in (1+1)-dimensional Minkowski spacetime. By reformulating the local dynamics in terms of loop-space holonomies, we demonstrate that the path independence of the holonomy eigenvalues constitutes a conservation law, yielding an infinite hierarchy of gauge-invariant, dynamically conserved charges. While a zero-curvature equation is associated with a necessary condition for this path invariance, we note that it is not strictly sufficient on its own. Employing a first-order symplectic formalism, we show that these non-abelian charges generate global symmetry transformations on the fundamental phase-space variables. We rigorously prove that these transformations preserve the physical dynamics, leaving the total Hamiltonian invariant up to first-class constraints. Furthermore, an analysis of the Poisson algebra reveals that these conserved charges are in involution, provided the boundary integration constant lies within the center of the gauge group. This exact, lower-dimensional framework provides a highly tractable setting to investigate the algebraic structures of these hidden symmetries and the meaning of the conserved charges as physical observables, establishing a classical foundation for exploring their role in the quantum regime, such as in strongly coupled lattice gauge theories.

We consider the problem of reconstructing missing data on a smooth manifold from incomplete and nonuniform samples. While classical methods for manifold approximation typically assume quasi-uniform data, their performance deteriorates significantly in the presence of large gaps or holes. We propose a unified framework for manifold data imputation that reduces the problem to function reconstruction on locally defined tangent spaces. The approach combines two complementary strategies. The first is a Fourier-based method that determines missing values by prescribing a decay rate of the discrete Fourier coefficients, thereby enforcing high-order smoothness through a global spectral criterion. The second is a local variational method based on minimizing high-order central differences, leading to sparse least-squares systems with favorable stability and conditioning properties. We establish a discrete inverse estimate linking decay of Fourier coefficients to uniform bounds on high-order divided differences, providing a theoretical foundation for the spectral approach. For the variational method, we analyze existence, uniqueness, and scaling behavior, showing that conditioning depends primarily on the geometry of the missing region. These functional reconstruction techniques are integrated with a moving least-squares projection framework to yield a practical algorithm for manifold completion. Numerical experiments, including reconstruction on surfaces with significant missing regions, demonstrate accurate and stable recovery without requiring a global parameterization. The proposed framework provides a flexible and effective approach to manifold data imputation in challenging settings with incomplete data.

We propose a theory based on dynamical systems to explain and predict the occurrence of extreme events, of which critical transitions form a subset. In fast-slow nonlinear systems, we identify a cascade of events preceding extreme events: (i) a slow regime, in which the fast covariant Lyapunov vectors (CLVs) are both tangent to the fast eigenvectors and remain transversal to the slow subspace; (ii) a transition regime, in which the fast eigenvalues become neutrally stable while the fast CLVs are no longer tangent to the fast eigenvectors; and (iii) a critical regime, in which a strong spectral gap in the eigenvalues causes both fast and slow CLVs to become tangent along the dominant fast direction, breaking the transversality between fast and slow subspaces. Building on this cascade, we propose two precursors to forewarn the occurrence of extreme events. We numerically test the theory and precursors on low- and higher-dimensional systems. The proposed precursors predict extreme events and critical transitions with 100% precision and recall. This work opens opportunities for time-forecasting extreme events using theoretically grounded precursors.