Papers

The vertex arboricity $\mathrm{va}(G)$ of a multigraph $G$ is the minimum number $k$ for which $V(G)$ can be partitioned into $k$ subsets, each of which induces an acyclic subgraph of $G$. By definition, if $\mathrm{va}(G)= k$, then the chromatic number, $\chi(G)$, satisfies $k\leq \chi(G)\leq 2k$. Fundamental results by Borodin from 1976 and Bollobás and Manvel from 1979 imply an analog of Gallai's lower bound on the number of edges in a $(2k-1)$-critical graph. We consider a slight generalization of vertex arboricity in the setting of DP-coloring. Using this framework, we derive lower bounds on the number of edges in graphs critical for vertex arboricity and for list arboricity that are better than Gallai's bound, along with similar bounds in our DP-setting.

Extreme Learning Machine (ELM) computes output weights analytically using the Moore-Penrose pseudoinverse. Although this leads to fast training, its numerical stability depends strongly on the conditioning of the hidden layer matrix. This paper studies pseudoinverse-based ELM from a spectral perspective. We show that the smallest singular value governs perturbation amplification in the output weights, while the condition number provides a quantitative measure of hidden-layer instability. We compare SVD-based pseudoinverse computation with iterative hyperpower methods and discuss width-dependent conditioning through a random feature interpretation. Experiments on synthetic matrices and ELM benchmarks show that SVD-based methods remain the most reliable under ill conditioning, while iterative methods are more sensitive to spectral properties. The results suggest that ELM stability is fundamentally governed by the singular value structure of the hidden layer matrix.

This paper proposes a robust optimization formulation to calculate dynamic operating envelopes (DOEs) to safely operate unbalanced three-phase distribution systems. Unlike conventional formulations that satisfy network constraints only at the envelope bound, the robust formulation covers the entire envelope range. We formulate a robust non-linear programming (NLP) problem with the full AC power flow equations, as well as an approximate linear programming (LP) model. Numerical simulations are run with real-world data from Belgium and two different distribution test feeders. The paper compares the conventional approaches with their robust counterparts and examines the trade-off between constraint violation and envelope size as well as accuracy and solve time aspects.

In this paper, we study the approximation of gradient flows of harmonic maps, which serve as model problems for applications in micromagnetics, liquid crystals, and nonlinear plate bending. Harmonic maps are vector fields that are critical points of the Dirichlet energy subject to the constraint that the vector field be unit length pointwise. Most existing time-stepping schemes for gradient flows deal with the constraint by linearizing the unit length constraint at every step, which involves solving for the solution increment in the tangent space of the constraint. These schemes lead to robust control over the violation of the constraint, but require solving degenerate saddle point systems at every step that may be difficult to precondition. In this paper, we propose a scheme that first computes the unconstrained increment and then projects this increment pointwise onto the tangent space. With an additional stabilization, this scheme is energy stable under mild step size restrictions and provides robust control of the unit length constraint violation. Our new scheme only requires the solution of decoupled symmetric positive definite systems at every step, which translates to a large increase in computational efficiency. We also propose a computable a posteriori criterion and a variable time-stepping procedure that guarantee the stability of the scheme. We conclude with computational examples demonstrating the efficacy of the scheme, and present a computational extension of the scheme to nonlinear plate bending.

Renormalization provides a framework for relating microscopic models of physical systems to effective descriptions at larger length scales. This procedure is studied for the boundary states of non-chiral two-dimensional topologically ordered models. The initial data consist of renormalization fixed points built from representations of finite-dimensional $C^*$-Hopf algebras, which are then perturbed by uniform on-site noise quantum channels and repeatedly coarse-grained. The resulting flows admit an intrinsic algebraic description in terms of completely positive maps on the $C^*$-Hopf algebra or, equivalently, positive linear functionals on its enveloping $C^*$-Hopf algebra. Their iteration is governed by convolution powers, and convergent trajectories yield new matrix product density operator fixed points, described by finite $*$-quantum hypergroups. This provides a concrete physical interpretation of such structures. For finite group algebras and their duals, we provide explicit classifications via Goursat's lemma for groups. Finally, we formulate and prove a quantum generalization of Goursat's lemma for finite-dimensional $C^*$-Hopf algebras, a result of independent interest, which gives an explicit structural description of all convergent renormalization trajectories.

Functional dependence measures have become an important tool in the analysis of nonlinear time series and are typically formulated with respect to a given innovation representation of the process. This note points out that the probability space on which such representations yield the expected memory loss properties may not always coincide with the natural dynamical probability space of the model. We exhibit classes of uniformly ergodic autoregressive processes for which the behavior of the natural innovation coupling undergoes a qualitative transition as the model parameter varies. For this family of models, this transition coincides with a change in the sign of an associated Lyapunov exponent. In particular, a positive Lyapunov exponent may prevent the forgetting of initial perturbations along trajectories driven by the same innovations, despite uniform ergodicity of the associated Markov chain. These observations highlight the importance of carefully specifying the underlying probability space when interpreting or applying functional dependence measures.

This is the first paper of a project on new integral models $\mathcal{X}(N)$ of the modular curve $X(N)$. The final results for a general level $N$ will be obtained in the second paper, while this paper is devoted to giving all necessary background and definitions applicable to any $N$ and then working out the case of $\mathcal{X}(p)$ with all possible details. We define $\mathcal{X}(N)$ as the closure of $Y(N)$ in the space $\overline{\mathcal{M}}_{1,N^2}=\overline{\mathcal{M}}_{1,\Gamma}$, where $\Gamma=(\mathbb{Z}/N\mathbb{Z})^2$, and show that for $N=p$ it is the blowup of the Katz-Mazur model $\widetilde{\mathcal{X}}(p)$ at all supersingular points, and hence $(\mathcal{X}(p),Y(p))$ is the minimal toroidal resolution of $(\widetilde{\mathcal{X}}(p),Y(p))$. In fact, it is even log smooth over $(\mathbb{Z},\mathbb{Z}[1/p])$, but this is special for the case when $p=N$. One can tautologically view $\mathcal{X}(p)$ as the moduli space of $\Gamma$-stabilized genus-1 curves $(E,\Gamma)$ which can be smoothed to an elliptic curve labelled by its $N$-torsion, but our main results provide explicit criteria of the smoothability: $\mathcal{X}(p)$ parameterizes $\Gamma$-equivariant stable genus-1 curves $(E,\Gamma)$ such that the action satisfies two explicit conditions formulated in the paper.

Let $\alpha(n)$ denote the length of the shortest non-trivial two-variable law for the symmetric group $S_n$. Buskin's quantitative subgroup-separability argument gives the classical lower bound $\alpha(n)\geq 2n-O(1)$. In this short note we give an improvement by proving that $\alpha(n)\geq \frac52 n-O(1)$.

We study a weighted-threshold version of the coupon collector problem in continuous time. Each type $i$ is discovered at rate $\lambda p_i$ and, once discovered, contributes weight $w_i$, where $p$ and $w$ are probability vectors. The stopping time when the total weight of the discovered types first exceeds a fixed threshold $\theta\in (0,1)$ is called the quorum time. We first prove concentration estimates and compare the quorum time with the corresponding deterministic threshold time obtained from the mean discovered weight. When all discovery rates are equal and the largest individual weight tends to zero, the first-order asymptotics are universal and do not depend on the weight vector. We then analyze the aligned Zipf family $p_i = w_i \propto i^{-s}$. This model has three regimes: a deterministic linear scale for $0\le s < 1$, a critical scale $H_NN^\theta$ at $s=1$, with an explicit leading constant, and a non-degenerate random hitting-time limit for $s>1$. Finally, we show that the expected quorum time need not be monotone in the Zipf exponent.

Suppose we observe two sets of $n$ Gaussian vectors in $\mathbb{R}^d$, with the promise that, after applying a permutation of $[n]$ and a rotation of $\mathbb{R}^d$, the two sets are $\rho$-correlated. The Procrustes matching problem asks us to recover the unknown permutation of $[n]$ that aligns the two sets. The problem is well-studied in the low-dimensional regime $d=O(\log n)$, but the high-dimensional regime $d\gg \log n$ has remained largely uncharted: prior matching guarantees require nearly perfect correlation $\rho=1-o(1)$, even for information-theoretic recovery. Our main result is a polynomial-time algorithm for exact recovery at constant correlation. The algorithm works by computing and comparing weighted counts of a specially chosen family of “wide” trees. So long as $d\ge \mathrm{polylog}(n)$, the algorithm succeeds with high probability for any $\rho^2>\sqrt{\alpha}$, where $\alpha\approx 0.338$ is Otter's tree-counting constant. We complement this algorithmic result with an improved information-theoretic guarantee, showing that exact recovery is possible when $\rho^2 \gtrsim \max\{\log n/d,\sqrt{\log n/n}\}$. We also carry out a low-degree advantage calculation, which suggests that the condition $\rho^2 > \sqrt{\alpha}$ is necessary for any tree-counting algorithm.

We show that the lower edge of the appropriately scaled size $n$ Laguerre beta-ensemble with parameter $a=a_n$ converges to the $\operatorname{Airy}_{\beta}$ process as $n\to \infty$ when $a_n\to \infty$ and $\tfrac{a_n}{n}\to 0$. This completes the picture of the possible edge scaling limits of the Laguerre beta-ensemble with a fixed $\beta>0$. When $a_n\gg (\log \log n)^3$ our proof establishes operator level convergence of the inverse of the scaled Dumitriu-Edelman tridiagonal matrix to the inverse of the stochastic Airy operator. Our methods allow us to prove similar operator level limits for the known soft edge scaling limits of the Laguerre and Gaussian beta-ensembles. For $a_n\le (\log n)^{1/2}$ we give a different argument that relies on coupling and a result of Dumaz-Li-Valko for the transition between the hard and soft edge limits of the Laguerre beta-ensemble.

Assuming that the asset price $X$ follows a constant elasticity of variance process, this paper studies the optimal prediction problem $\inf_{0\leq \tau\leq T}\mathbb{E}|X_\tau-\ell|$, where the infimum is taken over stopping times $\tau$ of $X$ and $\ell$ is a hidden aspiration level independent of $X$. Adopting the aspiration level hypothesis, we show that a class of admissible laws of $\ell$ leads to optimal trading boundaries which are located relative to the median interval of $\ell$ and serve as predictors of the resistance and support levels. The existence of these boundaries is proved and nonlinear integral equations are derived to characterise them uniquely. In the positive drift case the stopping set is bounded by two curves, while in the negative drift case the stopping set is described by a single boundary.

We study $\pi$-equigeodesic vectors associated with homogeneous fibrations, namely vectors that are geodesic with respect to every homogeneous metric making the projection a Riemannian submersion. We obtain an algebraic criterion characterizing such vectors and apply it to classical flag manifolds and Ledger-Obata spaces. As a framework for this study, given Lie groups $K\subseteq H\subseteq G$ with $H$ and $K$ closed in $G$, and a fixed $G$-invariant metric $g_b$ on $G/H$, we describe the family of $G$-invariant metrics $g$ on $G/K$ for which the natural projection $\pi:(G/K,g)\to(G/H,g_b)$ is a Riemannian submersion. We also give a criterion for the fibers of $\pi$ to be totally geodesic.

Belief propagation (BP) is widely used for data association (DA) in target tracking. Existing convergence analyses of BP for DA address only the two-way correspondence between targets and measurements, where each target generates at most one measurement per scan. Multipath DA (MPDA) allows a single target to produce multiple measurements via distinct propagation paths, creating a three-way correspondence among targets, paths, and measurements, for which a complete convergence proof has not yet been provided. We provide such a proof for the BP updates in MPDA, establishing convergence to a unique fixed point. Simulations illustrate the convergence behavior of BP in MPDA and demonstrate a favorable accuracy–efficiency trade-off relative to both single-scan and two-scan variants of the multiple-detection multiple-hypothesis tracker.

Proximity gaps are a property of error correcting codes that arise in the study of Interactive Oracle Proofs (IOPs) and Succinct Non-interactive Arguments of Zero Knowledge (SNARKs). Recent work of Goyal and Guruswami has established near-optimal proximity gaps for many families of codes, including subspace design codes, as well as random ensembles like random linear codes, Reed-Solomon codes with random evaluation points, and Gallager's ensemble of LDPC codes (Goyal & Guruswami, 2025). However, the parameters for these latter randomized ensembles are worse than the parameters for subspace design codes, and degrade as the degree ell increases. In this work, we obtain improved proximity gaps for random ensembles of codes, including random linear codes, Reed-Solomon codes with random evaluation points, and Gallager's ensemble. Quantitatively, our results for these random ensembles match the results that Goyal and Guruswami attained for subspace design codes. In fact, our techniques are a black-box transference from subspace design codes: any progress on subspace design codes will automatically lead to analogous progress for these random ensembles. To obtain our results, we extend the Local Coordinate-wise Linear (LCL) property framework developed by Levi, Mosheiff, and Shagrithaya and by Brakensiek, Chen, Dhar, and Zhang to a row-span constrained version (Levi, Mosheiff & Shagrithaya, 2025; Brakensiek, Chen, Dhar & Zhang, 2025). This allows us to cast curve-decodability – a property that implies proximity gaps – directly as a row-span constrained LCL property, and make use of that machinery. In contrast, because curve-decodability is not obviously a vanilla LCL property, prior work had worked with a proxy property instead, leading to the aforementioned parameter losses.

This paper investigates the behaviour of a fluid characterized by a viscosity simultaneously depending on pressure and shear rate within a Hele-Shaw cell featuring a sharp corner geometry. The study extends previous analyses conducted on purely pressure-dependent (piezo-viscous) and yield-stress fluids, providing a new perspective on confined complex flows. Motivated by practical applications related to designing biomedical devices and flows of relevance to biomedicine area, thin film technologies, injection molding – to name only a few – the flow configuration considered here can highlight essential features of complex fluid behavior in narrow-gap geometries around a sharp edge. Starting from the governing equations for an incompressible generalized Newtonian fluid and employing an appropriate rheological model, we derive the modified flow equations adapted to the Hele-Shaw flow. A particular solution is obtained near the corner region. Numerical simulations complement the theoretical results, illustrating the influence of the rheological parameters on the flow behavior.

The theory of quantum filtering (of quantum continuous measurements) was developed by V.P. Belavkin about 40 years ago. Since then it attracted attention of numerous investigators including mathematicians, theoretical and experimental physicists. However, the rigorous mathematical theory of the filtering equations for mixed states in basic infinite-dimensional quantum systems remained an open problem, which was resolved by the author recently. This survey paper presents in full the mathematical theory of quantum filtering equations, their rigorous derivation from basic principles, the corresponding law of large number limits (propagation of chaos) and related topics. Applications to feedback control, quantum dynamic and mean-field games are discussed.

We investigate a special class of nonlinear infinite dimensional systems. These systems are obtained by modifying the second order differential equation that is part of the description of conservative linear systems out of thin air introduced by M. Tucsnak and G. Weiss in 2003. The modified differential equation contains a new nonlinear damping term, that is maximal monotone and possibly set-valued. We show that this new class of nonlinear infinite dimensional systems is incrementally scattering passive (hence well-posed). Our approach uses the theory of maximal monotone operators and the Crandall-Pazy theorem about nonlinear contraction semigroups, which we apply to a Lax-Phillips type nonlinear semigroup that represents the whole system. We illustrate our result on the n-dimensional wave equation.

The dynamics of a mathematical model for a chemostat-type reactor is investigated. The model describes the temporal evolution of suspended and wall-attached bacterial populations, with the latter represented as a one-dimensional biofilm, subject to a non-reproducing growth-limiting substrate and a reaction product formed through bacterial utilization of the substrate. In particular, it is shown that, in the regime where the trivial (washout) equilibrium is unstable, there exists a unique nontrivial equilibrium that is locally asymptotically stable. Under slightly stronger assumptions, uniform persistence and global asymptotic stability of the nontrivial equilibrium are established.

We study eigenfunctions of quantum star graphs in the large edge number limit through the edge-mass distributions associated with their semiclassical measures. For generic edge lengths, we show that these distributions can realize every admissible multifractal scaling law along suitable subsequences of eigenvalues. We also prove a constructive result for quasi-equilateral star graphs. Starting from prescribed probability measures, we construct graphs and locate eigenvalues inside spectral clusters whose eigenfunctions reproduce the same scaling behavior. These results show that quantum star graphs form an explicit model realizing the full range of admissible multifractal behavior between localization and equidistribution.

We associate the Orlik–Solomon sheaf with a finite geometric lattice and compute its sheaf homology. We show that this homology concentrates in top degree, admitting a convolution-type decomposition into a principal ideal OS piece tensoring with a principal filter complement poset homology. Applications to uniform matroids provide interesting representations of symmetric groups.

We prove expression rate bounds of finite-parametric, spectral and neural surrogates for holomorphic maps between separable Hilbert spaces. The surrogates have an encoder-approximator-decoder architecture, with Karhunen-Loéve encoders and frame decoders. We prove expression rate bounds for two classes of finite-parametric surrogates: i) spectral surrogates obtained by N-term truncations of Wiener polynomial chaos expansions and ii) neural surrogates obtained by approximation of parametric maps with deep feedforward neural networks, ReLU and RePU activation functions and uniformly bounded weights. We work under an algebraic decay assumption on the eigenvalues of the covariance of the Gaussian measure on the input space. We obtain convergence rates for mean-square errors, and additionally in first-order Gaussian Sobolev spaces, to account for errors in the approximation of gradients.

A limiting case is worked out in which the causal action principle for causal fermion systems describing Minkowski space gives rise to the linear Fock space dynamics of perturbative quantum field theory, including non-abelian gauge fields and Dirac fields.

In recent work, Harman and the second author introduced some new infinite dimensional algebraic groups that generalize the classical groups. In this paper, we establish versions of the first and second fundamental theorems of invariant theory for them.

We consider the design of low-density parity-check (LDPC) codes for a given iterative decoder. While LDPC performance can be evaluated using simulation, density evolution (DE), or EXIT-chart analysis, selecting a parity-check matrix (PCM) remains a difficult combinatorial optimization problem. Existing approaches often rely on population-based search, random mutations, or genetic algorithms, which require careful tuning and incur high computational cost. Recent gradient descent (GD)-based methods optimize relaxed PCMs by differentiating through decoder simulations, but rely on noisy Monte Carlo estimates, line searches over soft matrix representations, and remain costly for long codes. Moreover, the loss is typically evaluated only at integer-valued PCMs. We focus on long protograph-based LDPC codes and propose a deterministic GD-based framework operating directly on a relaxed protograph representation. The loss is based on DE bit error rate (BER) and can be evaluated directly for relaxed protographs. To justify this relaxation, we associate the relaxed representation with an ensemble of binary PCMs and show that the proposed relaxed DE yields the ensemble-averaged DE performance. The resulting procedure supports standard GD optimization and achieves fast, reliable convergence through deterministic DE evaluation and informative gradients. Numerical results show that the optimized protographs outperform 5G LDPC codes with matching dimensions.

We develop the Gram/Wishart/Stiefel formulation of the \(N=2\), large–\(d\) planar endpoint theory of lattice BFSS/BMN matrix quantum mechanics on the lattice. The endpoint variables become rank–two Wishart eigenvalues and relative Stiefel angles, expressing the holonomy invariants \(A\), \(B\), and \(R^2=A^2+B^2\) in radial and angular Gram data. The large-\(R\) aligned contribution \(-A\) is absorbed into the Gaussian sector, producing the shifted mass \((\alpha_\Lambda)_{\rm eff}=\alpha_\Lambda-1/2\). The exact \(O(2)\) angular integral encodes this sector in a rank–two Bessel kernel. The pure \(-A\) theory, exactly solvable in Cartesian variables, fixes the leading Bessel/HCIZ structure: its exponential selects alignment, while its prefactor removes the spurious doubled Wishart entropy. We then apply this structure to the transverse \(B\)-type expansion and its non-polynomial toy completion. Finite polynomial truncations lead to an apparent large–\(d\) perturbativity bound incompatible with the continuum limit, but this bound is shown to be an artifact of truncation. After summing the local transverse completion and balancing the compensating \(+A\) term, the Wishart saddle is recovered with the physical shifted mass. The resulting continuum behavior reproduces the universal \(-2d\) contribution of the \(D_\Lambda\)-channel, while the genuinely anisotropic \(\beta_\Lambda\)-channel lies outside the scope of a pure transverse \(B\)-type description. Finally, we outline the emergent Grassmannian geometry encoded by the two endpoint planes. Their relative angular fluctuations form a \(2(d-2)\)-dimensional sector, whose entropy interpretation is developed in the sequel. For \(d\ge4\), the two planes generically span a relational four-dimensional subspace of \(\mathbb R^d\).

We initiate the study of the multiplicity of negative one of independence polynomials of graphs. In this article, we simply refer to this as the multiplicity of a graph. As applications, we provide a graph-theoretic description of trees whose independence complexes are contractible, give a new sufficient condition for independence polynomials of graphs to be log-concave, and finally, determine possible pairs $(\operatorname{mult}_{-1}P_G, \alpha(G))$, where $P_G$ denotes the independence polynomial of $G$, and $\alpha(G)$ the independence number. The study of the pairs $(\operatorname{mult}_{-1}P_G, \alpha(G))$ is equivalent to finding all pairs of the numerator degree and denominator degree of the Hilbert series of the edge ideal of $G$. We also use spectral graph theory to obtain results on the multiplicity of line graphs of forests. Finally, we give some translations and applications in combinatorial commutative algebra.

Using a result of Choi, we provide the first examples of pairs consisting of a closed hyperbolic three-manifold and a simple closed geodesic, such that there is no positive spun ideal triangulation for the manifold, spun about the chosen geodesic. In our first two examples, the closed manifold is the third manifold in the SnapPy census, also known as Vol3, and the geodesics are its systole and second systole. This provides evidence for the conjecture that Vol3 has no positive spun ideal triangulation for any choice of geodesic.

Hansen and Pitts (2006) introduced the problem of nonparametric estimation of the service-time distribution of an M/G/1 queue observed through its workload process at discrete times $t=0,1,\ldots,n$. Despite its seemingly simple formulation, obtaining an estimator with sharp risk guarantees for this observation model has remained an open challenge for nearly two decades. In this paper, we construct such an estimator and prove that its $L^1$-risk is $\mathcal{O}\left(\frac{\log n}{\sqrt{n}}\right)$ as $n\to\infty$. Remarkably, this nearly parametric convergence rate is achieved without assuming stationarity, stability, or knowledge of the arrival rate. Our approach is based on a two-stage screening procedure that uncovers a hidden conditionally independent compound Poisson structure within the dependent workload observations. This probabilistic reduction transforms the original estimation problem into a classical decompounding problem, making it possible to leverage existing nonparametric estimation techniques despite the complex dependence induced by the reflected workload process. More broadly, we hope that the proposed screening methodology will provide a useful framework for statistical inference from dependent stochastic systems beyond the classical assumption of stability.

Let $M_k$ denote a matching of size $k$. The classical Erdős matching conjecture asks for the maximum number of edges of an intersecting $r$-graph without $M_k$. The csae for $k=2$, which is known as intersecting $r$-graph, is established by Erdős, Ko and Rado. Hilton and Milner further determine the maximum number of edges of a non-trivial intersecting $r$-graph, where the intersecting $r$-graph $H$ is called non-trivial if $\cap_{e\in E(H)}e=\emptyset$. In this paper, we investigate the spectral analogues of the hpergraph matching problems and intersecting family problems. More precisely, for sufficiently large $n$, we determine respectively the maximum spectral radius of $M_{k+1}$-free and non-trivial intersecting $3$-graphs on $n$ vertices, and characterize the extremal hypergraphs.