Papers

A novel collocation scheme is presented for elliptic-type differential equations with degenerate coefficients and homogeneous Dirichlet boundary conditions. The use of weighted orthogonal Chebyshev polynomials for the basis functions leads to stiffness matrices with sparse structure, enabling efficient direct calculations. By an orthogonal projection, rigorous analyses are devoted to deriving a-priori error estimates of spectral accuracy in two norms. Furthermore, ample numerical experiments are conducted and compared with error data, convergence rates, condition numbers and $N$-$\log$ curves to confirm the theoretical analyses results. Our proposed method achieves spectral accuracy and handles boundary singularities efficiently, as demonstrated by theoretical analyses and numerical experiments.

We study matrix perpetuities, that is, solutions to affine fixed-point equations of the form \[ \mathbf{X} \stackrel{d}{=} \mathbf{A}\,\mathbf{X} \,\mathbf{A}^\top+\mathbf{B},\qquad (\mathbf{A},\mathbf{B})\mbox{ and }\mathbf{X} \mbox{ are independent}, \] with particular emphasis on the empirical spectral distribution of the solution. We first establish existence and uniqueness results by relating the problem to classical vector perpetuities. Under orthogonal invariance, we then prove a compression identity for symmetric multiplicative convolution and show that principal submatrices of a matrix perpetuity are themselves lower-dimensional matrix perpetuities. For positive semidefinite, orthogonally invariant models, we prove a finite-dimensional spectral Kesten theorem: we obtain precise power-law tail asymptotics for the expected empirical spectral distribution, show that its tail is governed by the largest eigenvalue, and relate it explicitly to the tail of any diagonal entry. We also prove that, in the subcritical regime, the expected empirical spectral distribution of matrix perpetuities converges weakly, as the dimension tends to infinity, to the distribution of the corresponding free perpetuity. Our results are illustrated by matrix Beta prime perpetuities, for which explicit limiting spectral distributions are available.

In this paper, we investigate the notion of the \textit{$\lambda$-chromatic polynomial} of a graph, which enumerates the number of distinct $L(2,1)$-colorings using colors from a prescribed finite set. We prove that the $\lambda$-chromatic polynomial of a graph with $n$ vertices is a monic polynomial of degree $n$ and provide a combinatorial interpretation via lattice point enumeration within the framework of inside-out polytopes. Moreover, we compute the $\lambda$-chromatic polynomial of complete graphs $K_n$ using lattice path enumeration, and we develop a block-gap technique to derive the $\lambda$-chromatic polynomials for complete bipartite and multipartite graphs. Our approach unifies geometric, combinatorial, and algebraic methods to provide a systematic treatment of $\lambda$-colorings across various families of graphs.

In this work, we study the application of a hybrid mixed domain decomposition(HMDD) method for the rotor-stator coupling of a permanent magnet synchronous machine. For this, we derive a variational formulation on the electric machine inspired by hybridized discontinuous Galerkin methods using a mixed magnetostatics problem, an affine material law and boundary conditions respecting the symmetry of the motor. We are then able to locate the resulting finite element method within the HMDD framework. This enables us naturally to transfer the well-posedness results and error estimates for the HMDD method to the finite element method considered in this work. Lastly, as a proof of concept, we consider an academic example and compare the resulting magnetic flux density and potential lines to their counterparts obtained by a well-established in-house code using iso-geometric analysis.

We introduce and study a one-parameter Hermitian deformation of the Yamabe problem on compact complex manifolds. The deformation is defined by adding natural torsion terms to the Riemannian scalar curvature, and includes both the classical Yamabe equation and a scalar-curvature equation arising in locally conformally Kähler geometry as a momentum map. We analyze criteria for the existence of solutions, and discuss several examples.

We study a simple 1-parameter perturbation of the regular holonomic Trace System satisfied by a complex power of the root of the universal polynomial of degree k as a holomorphic function of the coefficients. We prove that these systems have many analogous properties than the Trace System studied in [4] and we prove that they are, in general, minimal extensions of a simple pole meromorphic connection on a rank $k$ trivial bundle on $\mathbb{C}^k$. We also examine the structure of these $D$-modules for the special values of the parameters. This explicites many examples of perverse sheaves associated to representations of the $\pi_1$ of the complement of the hyper-surface $\{\sigma_k\Delta(\sigma) = 0\}$ in the affine space with coordinates $\sigma_1,\ldots,\sigma_k$, where $\Delta(\sigma)$ is the discriminant of the universal monic polynomial of degree $k$, $P_\sigma(z) := z^k + \sum_{h=1}^k (-1)^h \sigma_h z^{k-h}$.

This paper concerns a class of non-divergence nonlinear elliptic equations driven by the cone degenerate Laplacian, which is motivated by cone calculus. We establish the existence of viscosity solutions by proving the Alexandrov-Bakelman-Pucci and Hölder estimates. Furthermore, we obtain the existence of weak solutions by proving the equivalence between weak solutions and viscosity solutions.

We investigate the convergence of partial sum processes based on a strictly stationary $\beta$-mixing sequence of random variables. The convergence in the space of continuous function as well as in Hölder spaces is considered. The conditions are close to optimality.

The transition to electric heavy-duty freight is constrained by a strong interdependence between vehicle adoption and charging infrastructure deployment. While system dynamics models are well-suited to simulate these socio-technical feedback loops, their reliance on conditional logic and heuristic rules makes the underlying dynamics difficult to interpret mathematically. This limits the direct application of formal systems and control theory. This paper proposes a solution to that problem through a general hybrid system identification framework that extracts a continuous-time analytical surrogate from simulations of complex system dynamics. The approach combines the sparse identification of nonlinear dynamics algorithm with neural ordinary differential equations to form a grey-box model. The first component identifies the dominant interpretable dynamics under causal coupling constraints, while the neural residual captures unmodelled nonlinearities. The framework is trained using multiple shooting over a 40-year horizon. The resulting model reproduces the training trajectories with a normalized root-mean-square error below 4%, while maintaining reliable predictive accuracy when evaluated on unseen initial conditions.

Let $G$ be a connected, undirected simple graph. Denote by $L(G)$ the Laplacian matrix of $G$, and let $m_{G}(\lambda)$ be the multiplicity of an eigenvalue $\lambda$ of $L(G)$. When $G$ is a tree $T$ with $n \ge 6$ vertices, Tian et al. [Discrete Mathematics, 2026] proved that if $T$ is reduced and contains no pendant $P_3$, then \[ m_{T}(1) \le \frac{n-6}{4}, \] and they gave a complete characterization of the graphs for which equality holds. In this paper, we further investigate the above problem. Still assuming that $T$ is a tree with $n \ge 7$ vertices which is reduced and has no pendant $P_3$, we prove the following results. If $m_T(1) \neq \frac{n-6}{4}$, then \[ m_{T}(1) \le \frac{n-7}{4}, \] and we give a complete characterization of the graphs for which equality holds. If, moreover, $m_T(1) \neq \frac{n-6}{4}, \frac{n-7}{4}$, then \[ m_{T}(1) \le \frac{n-8}{4}, \] and we also give a complete characterization of the extremal graphs.

In this work we establish weighted Poincaré inequalities for multivariate Liouville distributions, which are a generalization of the Dirichlet distribution. We also consider continuous elliptically contoured distributions, whose density levels are unions of hyperellipsoids. Our approach is based on a transport argument which allows weighted Poincaré inequalities to be transferred between probability measures. We apply our results to global sensitivity analysis and illustrate their practical use in a flood model case study, where the structure of dependence of the input variables is encoded by classical copulas.

We study stochastic linear bandits under a natural combination of batching and communication constraints: the time horizon is partitioned into batches of equal size $B$, and during each batch the learner sends $B$ requested arm pulls to an agent, who then observes the corresponding $B$ rewards and responds with a single bit of feedback to the learner. For each batch, the learner specifies the 1-bit quantization rule the agent uses, which may depend on all previously received bits but not on any past rewards directly. This setting addresses a significant yet unexplored “middle ground” between previous models having per-round quantization only or total bit budgets only. We establish a minimax lower bound showing that $\Omega(B\min\{d,\log\lvert \mathcal{A} \rvert\})$ regret is unavoidable due to the 1-bit communication bottleneck, even in the absence of noise. Combined with standard statistical limits, this yields a general lower bound of $\widetilde{\Omega}(B\min\{d,\log\lvert \mathcal{A} \rvert\} + \sqrt{dT \min\{d,\log\lvert \mathcal{A} \rvert\}})$. We develop two phased-elimination algorithms based on $G$-optimal designs and 1-bit mean estimation. The first achieves $\widetilde{O}(dB + d\sqrt{T})$ regret, matching the lower bound up to logarithmic factors when $\lvert \mathcal{A} \rvert = \exp(\Omega(d))$, and the second incorporates a safe-arm identification and warm-start procedure to obtain $\widetilde{O}(B\log\lvert \mathcal{A} \rvert + d^{3/2}\sqrt{B} + \sqrt{dT\log\lvert \mathcal{A} \rvert})$ regret, which is near-optimal in broad scaling regimes of $(\lvert \mathcal{A} \rvert, B, d, T)$. Together, our results demonstrate that a single bit of feedback per batch suffices to nearly match the minimax regret of unconstrained linear bandits in broad scaling regimes, even for batch sizes as large as $\Theta(\sqrt{T})$.

We adapt the notion of mixing of all orders to random dynamical systems and use it to establish quenched limit theorems. Assuming quenched exponential mixing of all orders, we prove both the central limit theorem and the Poisson limit theorem. Compared to existing approaches, this framework offers two main advantages. First, it allows for substantially weaker assumptions on the random constants appearing in mixing estimates: in the exponential mixing regime, logarithmic integrability suffices for the central limit theorem, while in the Poisson case, no integrability condition is required. Second, it applies naturally to invertible systems, where standard methods based on $L^{\infty}$-type estimates or spectral decay are less well suited.

Let $(x_n)_{n\geq 1}$ be a sequence of distinct points on the unit circle. After the first $n$ points are inserted, the circle is divided into $n$ intervals. For a fixed integer $r\geq 1$, let $M_n^{(r)}$ and $m_n^{(r)}$ denote respectively the largest and smallest total lengths of $r$ consecutive intervals. A theorem of de Bruijn and Erdős gives \[ \limsup_{n\to\infty}\frac{M_n^{(r)}}{m_n^{(r)}}\geq 1+\frac1r . \] The case $r=1$ is sharp and gives the classical factor $2$. The cases $r\geq 2$ remain much less understood. We prove the improved lower bound \[ \limsup_{n\to\infty}\frac{M_n^{(r)}}{m_n^{(r)}} \geq 1+\frac{r}{r^2-1} \qquad (r\geq 2). \] In particular, for two consecutive intervals the lower bound becomes $5/3$, improving the de Bruijn–Erdős bound $3/2$.

Contemporary data-driven and technology-integrated era, various matrix-valued integer-valued time series, such as cross-regional crime statistics, multi-category sales records, and network traffic matrices, exhibit high dimensionality, complex structures, and strong row-column intertwined dependencies. Although the existing matrix integer-valued autoregressive (MINAR) model provides a framework that directly handles matrix data and captures bidirectional row-column dependencies, it suffers from limited interpretability and inflexible structural representation, as its parameters often lack clear empirical meaning and the model cannot separately distinguish the effects arising from rows, columns, and lagged dynamics. To overcome these drawbacks, this paper proposes the additive matrix integer-valued autoregressive (Add-MINAR) model. By introducing an additive structure that explicitly decomposes the matrix response into row effects, column effects, and lagged effects, the proposed model not only preserves the matrix-valued nature but also significantly enhances parameter interpretability and structural flexibility. Two estimation methods, namely projection estimation and iterative conditional least squares estimation, are developed for parameter identification and inference, and their asymptotic properties, including consistency and asymptotic normality, are rigorously established. Simulation results show that the iterative conditional least squares estimator generally outperforms the projection estimator in most scenarios. Empirical analysis of Chicago crime data further demonstrates that the Add-MINAR model achieves superior in-sample fitting and out-of-sample forecasting performance compared to benchmark models such as MINAR, making it particularly suitable for practical applications with explicit row-column interaction features.

In the previous papers in the series, we characterized partial cross-sections for general flows, in the spirit of Fried's work on global cross-sections. In this paper, we deduce several consequences for Anosov flows. We provide a homology criterion for the existence of a partial cross-section in a given cohomology class. Additionally, there are at most finitely many partial cross-sections in that cohomology class. We deduce that on a 3-dimensional hyperbolic manifold, any Anosov flow is homologically full.

We present a non-intrusive version of the index-aware learning framework introduced in arXiv:2309.00958. Index-aware learning itself is an approach for learning the time and parameter dependent solutions of differential-algebraic equations (DAEs), in particular those describing electric circuits. A central feature of the approach is that it ensures the learned solutions to fulfill the inherent constraints of the DAE, such as e.g. Kirchhoff's laws in the case of electric circuits. This is achieved by leveraging a decoupling of the DAE into its differential and algebraic parts, with the non-intrusive version of the approach additionally relying on results from arXiv:2604.20475 and arXiv:2107.07755. We illustrate the approach using a filtered buck converter as an example and compare both the intrusive and non-intrusive versions. The code for the example is openly available.

This paper develops, analyzes, and validates a fast algorithm for computing guided modes within bent interfaces and non-periodic defects in high-contrast resonator crystals, where the Floquet–Bloch theory is not applicable. We first establish the resolvent convergence of the governing continuous operator to the discrete capacitance operator. This result rigorously justifies the reduction of the continuous spectral problem to a discrete eigenvalue problem. Then, we develop a truncation scheme of the discrete operator, named the patch approximation, and derive a rigorous error estimate for the patch approximation. Finally, we validate the accuracy and efficiency of our scheme through various examples. Our framework provides a general, computationally efficient, and rigorously justified approach to simulate guided modes in non-periodic systems of high-contrast resonators.

The Ricci matrix $R_T$ of a finite tree encodes its discrete Einstein metrics via the Perron eigenvector, with Lin-Lu-Yau's Ollivier Ricci curvature: $\kappa = -\lambda_{\max}(R_T)$. We show that edge subdivision, the natural operation of lengthening a tree, can decrease, preserve, or increase $\lambda_{\max}$. Compressing each branch into a scalar feedback function via the Schur complement reduces the spectral problem to a one-dimensional Chebyshev equation. We obtain an exact one-step trichotomy, a scalar transmission equation for arbitrary length, and the long-chain limit. Examples on double stars, including an asymmetric case where subdivision strictly increases $\lambda_{\max}$, illustrate the theory.

We study the global structure of bifurcation diagrams for semilinear elliptic Dirichlet problems with supercritical nonlinearities in the unit ball. In particular, we focus on the borderline dimension $N = 10$, where the qualitative behavior of the bifurcation diagram is not determined solely by the growth rate of the nonlinearity. We show that the bifurcation curve is monotone, yielding uniqueness of classical solutions, for a class of nonlinearities including $f(u) = \exp((u+1)^p)$ with $p > 1$ and iterated exponential functions. Our approach is based on the construction of suitable singular subsolutions that satisfy a Hardy-type stability condition, avoiding the need for explicit representations of singular solutions. As a consequence, we show that, in dimension $N = 10$, these nonlinearities exhibit the same qualitative bifurcation diagram as the classical Gel'fand problem. We also characterize the monotonicity of the bifurcation curve in terms of the existence of global-in-time unbounded solutions to the associated parabolic problem.

We compare notions of admissible surfaces for Cauchy fluxes, formulated as understanding when the normal trace of the underlying stress field can be represented by a measure. If this field is unbounded, the problem of admissibility is necessitated by the fact that the normal trace need not admit a measure representation on every regular surface, but only on “almost all” such surfaces. We compare an approach based on a precise majorant introduced by Šilhavý (Arch. Ration. Mech. Anal. 116.3 (1991)) with a Minkowski-type condition introduced by Chen, Torres and the first author (Arch. Ration. Mech. Anal. 249.6 (2025)) by showing that, under mild geometric conditions, the former condition implies the latter. We also show, by means of an explicit construction, that the latter admissibility condition can allow for arbitrary measure concentrations as a normal trace.

We study the problem of learning Gaussian mixture models under overparameterization. Prior work has shown that while overparameterization is essential for avoiding spurious local optima and enables global recovery of the ground-truth model using the gradient-EM (expectation-maximization) algorithm, it can dramatically slow down the local rate of convergence. Under certain assumptions on the mixture weights, we show that a standard divergence measure minimized by statistical learning procedures possesses a manifold of slow growth on which the well-known Polyak stepsize reduces the loss geometrically, and design a gradient-based method that converges to minimizers at a locally linear rate. Additionally, we show that our method converges to nearly optimal solutions – up to a natural misspecification threshold – for mixtures with arbitrary weights. At a high level, the method alternates between several "short" gradient descent steps that approach the manifold and "long" Polyak steps that contract the distance to minimizers. Our results suggest that slow convergence is not an intrinsic challenge of overparameterization, but can be overcome by exploiting the favorable structure of the loss landscape.

Let $F$ be a non-Archimedean local field, let $L=\breve F$, and let $G=\mathrm{GL}_n$. Let $M\subset G$ be a standard Levi subgroup and let $b\in M(L)$ be basic in $M$, but not necessarily basic in $G$. For a dominant cocharacter $\mu$, we study the reduction-to-Levi morphism $\beta:X^G_\mu(b)\to \bigsqcup_{\mu_M\in S_M(\mu,\nu_b)}X^M_{\mu_M}(b)$ for affine Deligne–Lusztig varieties in the affine Grassmannian. Using an Iwasawa factorization relative to $P=MN$, we reduce the fiber condition to explicit Frobenius-twisted lattice equations in the off-block coordinates. In the Drinfeld case, where the base $X^M_{\mu_M}(b)$ is zero-dimensional, we prove that $\beta$ is globally trivial with constant affine-space fiber in the non-basic cases considered. More generally, in the minuscule case we develop a nonzero-slope lattice-theoretic criterion which shows that the fibers are affine spaces and that $\beta$ is Zariski locally a trivial affine-space bundle in the non-basic cases considered. We also give examples in the non-minuscule setting where the fibers need not be affine spaces.

We introduce the space of continuous and translation invariant area measures, which are measure-valued functionals on the space of convex bodies satisfying a certain locality condition. Our main result shows that the space of $\mathrm{GL}(n,\mathbb{R})$-smooth area measures coincides with the space of area measures obtained by integration with respect to the normal cycle. We show how this result yields Hadwiger-type classification results for continuous area measures that are equivariant with respect to compact groups acting transitively on the unit sphere. In addition, we establish a general density criterion for invariant submodules and show that mixed area measures generate dense submodules with respect to suitable topologies on the space of continuous area measures. As a byproduct, we discuss how McMullen's Conjecture can be obtained directly from the representation of $\mathrm{GL}(n,\mathbb{R})$-smooth translation invariant valuations on convex bodies in terms of integration with respect to the normal cycle.

We show that the Hutchinson operator of an Iterated Function System with an attractor is equicontinuous if and only if it is continuous; in particular, every finite Iterated Function System with an attractor has equicontinuous Hutchinson operator. We also characterize equicontinuity of Hutchinson operators generated by a natural class of continuous multifunctions in terms of continuity of their limit maps.

Let $A\subset N$, and define its ordered representation function $r(n)=\#\{(a,b)\in A\times A:a+b=n\}.$ The Erdos–Turan conjecture asserts that, if $r(n) > 0$ for all sufficiently large $n$, then $r(n)$ is unbounded. Konstantoulas proved a density-theoretic version: if the upper density of $E=N\setminus(A+A)$ is less than $1/10$, then $\limsup_{n\to\infty} r(n)> 5$. In this paper, we improve Konstantoulas' constant to $7/32$. We also prove that $D(E)< 1/2$ implies $\limsup_{n\to\infty} r(n) > 3$, and give a conditional criterion forcing $\limsup_{n\to\infty} r(n)>7$.

We consider an incompressible Navier–Stokes system nonlinearly coupled with a conserved Allen–Cahn equation with a singular potential (e.g., of Flory–Huggins type). This model describes a mass-conserving two-phase flow with constant density and non-constant viscosity. First, in three spatial dimensions, we prove the existence and uniqueness of local-in-time strong solutions to the associated initial–boundary value problem, subject to no-slip boundary conditions for the velocity field and homogeneous boundary conditions for the phase field. Next, by means of a relative energy approach, we establish a conditional weak–strong uniqueness principle in three dimensions, as well as unconditional uniqueness of weak solutions in two dimensions. Finally, building on recent seminal results by the first and third authors, we prove for the first time that, in both two and three dimensions and for general singular potentials, every global-in-time weak solution asymptotically separates from the pure phases and converges to a unique equilibrium. This result is obtained under minimal assumptions on the viscosity coefficient. Moreover, under additional regularity assumptions on the viscosity, we combine the asymptotic strict separation property with the conditional weak–strong uniqueness principle to show that weak solutions undergo asymptotic regularization. As a consequence, convergence to equilibrium also holds in higher-order norms.

We derive theoretical bounds on the Kronecker rank of block-structured matrices that possess both inner and outer structure. Building on the matrix-to-tensor and tensor-to-matrix framework of Kilmer and Saibaba ($\textit{SIAM J. Matrix Anal. Appl.}$, 2022), we show that the Kronecker rank of a matrix $\mathbf{A}$ equals the dimension of the span of its distinct blocks (the $\textit{inner blockspan}$), and equals as well the dimension of the corresponding span for a permuted matrix $\mathbf{B}$ whose inner and outer structures are interchanged. We give two proofs of this equality: one via a direct dimension argument, and one via an explicit isomorphism between the outer blockspan of $\mathbf{A}$ and the column space of the second-mode unfolding of the associated tensor. These results yield nested containments that translate structural information, such as Toeplitz, Hankel, banded, or sparse patterns, into computable upper bounds on the Kronecker rank. We also establish a duality between element-level sparsity in $\mathbf{A}$ and block-level sparsity in $\mathbf{B}$. Numerical experiments confirm the theory across several classes of structured matrices and, for two sparse matrices drawn from the SuiteSparse collection, provide a structural explanation for observed singular value decay.

Anchored fixed point and monotone equation methods, including Halpern iteration, extra anchored gradient, and their relatives, add a vanishing pull toward a reference point to obtain last-iterate guarantees. Existing anchored variants often achieve sharp last-iterate guarantees, but from the update-level perspective the placement of the anchor can be algorithm-specific and conceptually opaque. We show that anchoring admits a single operator-side construction: regularize the operator queried by the base method with a vanishing Tikhonov term, then run the unmodified base method. Applied to the Picard iteration, this recipe reproduces the Halpern iteration; applied to the forward step, extragradient (EG), and past extragradient (PEG, also known as Popov's method), it yields three variants whose anchor placements inherit the base method's query pattern. The forward-step instantiation gives a new residual convergence guarantee, while the EG and PEG instantiations give new regularized variants. The four analyses share a residual recurrence, recovering the $O(1/k)$ Halpern residual-norm convergence rate, giving $O(1/\sqrt{k})$ for the regularized forward step, and giving $O(1/k)$ for the regularized EG and PEG variants in the unconstrained monotone Lipschitz setting.

We develop a systematic framework for constructing parameter-free algorithms for smooth nonconvex optimization. The framework is based on higher-order regularization: each step is computed from a regularized local model whose regularization exponent exceeds the order of the model error. This design makes the resulting method robust to misspecification of the regularization parameter and yields complexity bounds without backtracking or other acceptance tests. We apply the framework to gradient descent, Newton's method, the Gauss–Newton method, stochastic gradient descent, and PAGE. Without prior knowledge of problem-dependent parameters, the resulting algorithms achieve complexity bounds with optimal or best-known dependence on the target accuracy. When the problem-dependent parameters are known up to constant factors, suitable tuning also recovers the optimal or best-known dependence on those parameters.