Papers

This study presents a statistical and machine learning framework for characterizing a hydrogen-based multi-energy system (H-MES) using one year of high-resolution operational data. Statistical analysis revealed a binary operation driven by renewable surplus, with solar irradiance explaining 45.7% of rank-based variance in hydrogen production, a large effect by conventional standards. Only high-irradiance periods triggered meaningful electrolyzer engagement, while electricity demand exerted a weaker inverse suppression effect ($\epsilon^2 = 0.126$). Multiple regression confirmed electrolyzer power as the dominant linear predictor, with a synergistic solar-wind interaction. Notably, Random Forest analysis ranked wind output first in predictive importance despite its weak bivariate correlation (r = 0.167), revealing non-linear dynamics invisible to parametric methods. A sequence model exploited strong 24-hour autocorrelation (r = 0.845) for operational forecasting, while a reinforcement learning agent optimized hydrogen revenue dispatch. The core contribution is demonstrating that statistical and machine learning approaches are complementary for H-MES modeling and control.

Problem C7 from the 2006 IMO Shortlist gives \(A(P)-B(P)=V(P)-1\) for a generic convex polyhedron, where \(A(P)\) counts antipodal vertex pairs and \(B(P)\) counts antipodal edge-midpoint pairs. For an arbitrary convex polyhedron \(P\subset\mathbb{R}^3\), define \(\delta(P)=V(P)-1-A(P)+B(P)\). We construct a two-dimensional antipodal square complex \(X(P)\) and prove \(H_0(X(P);\mathbb{Z})\cong\mathbb{Z}\), \(H_1(X(P);\mathbb{Z})\cong\mathbb{Z}/2\), and \(H_2(X(P);\mathbb{Z})\cong\mathbb{Z}^{\delta(P)}\). Consequently, \(\delta(P)\geq 0\), extending the generic identity to the inequality \(A(P)-B(P)\leq V(P)-1\). The proof uses a directed double cover, a polyhedral support blow-up over the normal sphere, and the Vietoris–Begle mapping theorem. Independently, Euler integration on the projective normal fan gives an exact local formula for the defect; in three dimensions, only exact edge–facet and facet–facet opposite pairs contribute. We also determine the integral image of the square-boundary map: it is the even-cycle lattice of the antipodal graph, with nonzero Smith invariant factors \(1,\ldots,1,2\). Applications include a zero-defect criterion, centrally symmetric and extremal formulas, and an explicit description of defects and primitive belts for pyramids over polygons.

We study ideals in a polynomial ring with partially linear (virtual) resolutions. We establish an effective bound beyond which their powers coincide with a power of the maximal ideal, proving a slightly weaker version of the Eisenbud-Huneke-Ulrich conjecture for a more general class of ideals. We also obtain an upper bound on their Castelnuovo-Mumford regularity, extending Macaulay's bound and a theorem of Eisenbud-Huneke-Ulrich. We introduce higher syzygy bundles, which generalize the classical Green-Lazarsfeld syzygy bundle. The key ingredient is the relationship between the syzygies of partially linear ideals and the sheaf cohomology of higher syzygy bundles.

We study convex bodies whose translates by a fixed lattice cover space and for which every proper convex subbody loses this property. In three dimensions, the convex hull of independent translates of the six Kuhn tetrahedra always gives a lattice covering. We give a geometric proof using auxiliary tetrahedra and the parity of the covering multiplicity. We also prove polytopality and boundary restrictions for minimal covering bodies, and show that every three-dimensional parallelohedron admits a Kuhn representation with respect to its face-to-face tiling lattice. Constructions from Reeve tetrahedra give non-symmetric minimal covering bodies with eight vertices in dimension three and centrally symmetric ones with sixteen vertices in dimension four, with unbounded volumes for the integer covering lattice. They have pairwise distinct arithmetic contact types, which record lattice contacts and the faces containing them. The covering theorem also gives a finite intersection criterion for a prescribed lattice basis, with three intersection tests in the centrally symmetric case.

In this paper, we introduce and study the notions of (n,Q)-ideals ans phi-(n,Q)-ideals of commutative rings.

Fluorescent labeling is a cornerstone of DNA visualization and a key enabler of random access in DNA-based data storage. However, the stochastic nature of biochemical processes, including synthesis, hybridization, and optical readout, induces burst synchronization errors within the resulting labeling sequences. To address this critical challenge, we formally introduce \emph{burst $t$-deletion/insertion $\mathcal{A}$-labeling codes,} designed to correct a single burst of $t$ deletions or insertions in the label domain. Our contributions are threefold. \begin{itemize} \item Fundamental limit. We establish an information-theoretic lower bound of $\log_4 n + \mathcal{O}(1)$ on the redundancy of any such code for all $t \ge 1$ with $t \mid n$. To the best of our knowledge, this resolves the first information-theoretic lower bound even for the single-error case \(t=1\). \item Explicit construction. For $t \ge 2$, $t \mid n$, and $n \ge 7t + 3$, we propose explicit encoding and decoding algorithms, both running in $\mathcal{O}(n^2)$ time. A novel generalized Run-Length Limited (RLL) constraint is introduced to bridge the structural mismatch between the DNA encoding domain and the label error domain. \item Asymptotic optimality. The proposed scheme achieves redundancy $\log_4 n + (t-1)\log_4 \log_{8/3} n + \mathcal{O}(1)$, matching the dominant term of the lower bound up to a small $\mathcal{O}(\log\log n)$ overhead, rendering the construction asymptotically optimal for fixed $t$. \end{itemize}

We demonstrate that all anti-self-dual Einstein metrics with non–zero cosmological constant $\Lambda$ locally arise from solutions of a single second order PDE introduced by Lipstein and Nagy. We show how this equation fits into the heavenly formalism of Plebański, and establish a Lax pair. Finally we show how Plebański's second heavenly equation arises in the limit as $\Lambda\rightarrow 0$.

Let $G_{n,p}$ be the binomial random graph of density $p$ and let $X_H$ be the number of copies of a fixed graph $H$ in $G_{n,p}$. We prove asymptotically tight bounds on the logarithmic upper-tail probability of $X_H$ whenever $H$ is a connected, irregular graph with maximum degree $\Delta \ge 2$ and $p \ge n^{-1/\Delta - \varepsilon_H} (\log n)^{\omega(1)}$ for an explicit $\varepsilon_H >0$. These bounds are expressed in terms of a new variational problem that generalises the combinatorial optimisation problem arising from the naïve mean-field approximation. This new variational problem includes an entropy term that corresponds to the large number of embeddings of certain highly structured graphs in $K_n$. For a certain class of irregular graphs $H$ that we call stable, we show that this description of the upper-tail probability is valid in a range of densities that is optimal up to a poly($\log\log n$) factor. For a further subclass of stable graphs, which includes all irregular complete bipartite graphs, we show that this range of densities is optimal up to a multiplicative constant.

Non-Euclidean optimisation methods with matrix-valued updates, such as Muon and Scion, have recently shown strong empirical performance for training Transformer models, yet their theoretical advantages over Euclidean methods remain poorly understood. We address this gap in the heavy-tailed non-convex regime, where stochastic gradients have bounded $p$-th central moments, $p \in (1,2]$. We show that certain non-Euclidean methods achieve optimal sample complexity under stronger stationarity measures, while Euclidean methods incur additional dimension-dependent costs. As a consequence, for $m \times n$ matrices, Muon finds an $\varepsilon$-stationary point in nuclear norm within $\mathcal{O}\left(\min\{m, n\} \frac{\Delta_1 L}{\varepsilon^2} \left(\frac \sigma \varepsilon \right)^{\frac p {p-1}}\right)$ samples, absorbing heavy-tailed noise without extra dimension dependence, unlike Euclidean methods. We further prove this sample complexity, including its dimension dependence, is optimal for all first-order methods under nuclear-norm stationarity. Experiments on large language models support our theory. Surprisingly, our results suggest that other Schatten geometries beyond the spectral geometry of Muon can perform competitively in certain settings.

In this paper, we introduce maximal green sequences of infinite length in abelian length categories and establish natural bijections among infinite maximal green sequences, finest stability data, and complete backward $\operatorname{Hom}$-orthogonal (briefly, CBHO) sequences of bricks. For cluster quiver patterns (equivalently, cluster algebras) of infinite rank, we define maximal green sequences of infinite length via $c$-vectors. Moreover, we prove that under suitable conditions such a maximal green sequence in a cluster quiver pattern of infinite rank can be categorified as a maximal green sequence of torsion classes in the corresponding representation category via a specific construction, thereby providing an example of a maximal green sequence in an abelian length category. This framework establishes a connection between infinite maximal green sequences in representation theory and maximal green sequences for cluster algebras of infinite rank.

The Laplacian spectral radius of a graph is the largest eigenvalue of its Laplacian matrix. Previously, upper bounds for the Laplacian spectral radius were proposed using a backward-reconstruction procedure starting from expressions equal to $2x$ and substituting local degree data. A numbered list of 68 such candidate bounds was subsequently investigated, resulting in the refutation of 30 of these bounds; two additional bounds were later refuted in a separate study. This paper updates the status of the remaining 36 candidate bounds. Of these remaining bounds, we confirm 22 and refute 12, leaving only two upper bounds open. The valid bounds follow primarily from classical Laplacian spectral radius bounds and the Collatz–Wielandt comparison; the refutations are carried out through explicit counterexamples relying on equitable partitions.

The predictive approach to Bayesian inference accesses the posterior distribution via a sequence of one-step-ahead predictives, enabling inference via predictive resampling without Markov chain Monte Carlo. In the random-design regression setting, an explicit specification of the predictive design distribution is required, yet the impact of this choice has received little formal attention. We study the role of this predictive design distribution in parametric martingale posteriors for regression, and identify predictive notions of identifiability and design invariance that are essential for valid inference, particularly in the high-dimensional regression setting. Building on these foundations, we introduce a novel class of parametric martingale posteriors for regression that satisfies a weak form of these desiderata, and naturally accommodates the high-dimensional setting through regularization. We then illustrate our method through a simulation.

A central problem in statistical physics is to extend the Boltzmann distribution to nonequilibrium steady states (NESS). We prove that NESS on large dense disordered networks have Boltzmann-like occupation despite extensive entropy production. One consequence is that such NESS can conceal their nonequilibrium nature from measurements of static response to energy perturbations. We further prove that the active-matter heuristic of "low rattling" is asymptotically exact for these NESS. Intuitively, they spend a greater fraction of their time in states they leave more slowly. We show that rattling effectively extends the Boltzmann distribution to such NESS, as well as the broader class of "equiaccessible" steady states, which play a role in our analysis akin to that of equilibrium in linear response.

This paper resolves several open questions regarding the algebraic and metric inheritance of distance-based regularities under graph products, with a particular focus on Commutative Distance Degree-Regular (CDDR) graphs. We characterize the behavior of CDDR, distance mean-regular (DMR), and distance degree-regular (DDR) graphs under the strong and direct (tensor) products. To address the non-closure of classical regularities under the direct product, we introduce and study a novel, broader family of graphs: Distance Walk Regular (DWR) graphs. Furthermore, we relax the global commutativity condition by defining and structuralizing the class of $i$-CDDR graphs, providing deep characterizations within the realm of Cayley graphs. By developing systematic lifting operations, we construct infinite families of graphs with arbitrarily large order and diameter within specific structural intersections, effectively settling topological gaps in the current regularity hierarchy.

In this article, we study polynomials over $\mathbb{F}_{q^2}$ that do not permute $\mathbb{F}_{q^2}$. More precisely, we characterize polynomials of the forms $x^q + b x^2 + c x + d$ and $x^{q+1} + b x^q + c x + d$ over $\mathbb{F}_{q^2}$ according to whether they are permutation or non-permutation polynomials. To this end, we determine the exact number of zeros of these polynomials using existing results on certain special Weil sums.

In this paper, we study finite dimensional Lie algebras with respect to the length of their chief series. We examine separately the semisimple, solvable, and mixed cases over fields of characteristic zero, and we establish several structural results describing Lie algebras of prescribed chief length. In the solvable and mixed settings, the problem is reduced to the study of irreducible modules arising from suitable quotient actions. In positive characteristic, we investigate semisimple Lie algebras by means of Block s structural theorem and obtain several results concerning Lie algebras of chief length two.

In this note, we provide several equivalent descriptions of the tropicalization of definable sets in a polynomially bounded o-minimal expansion of a real closed field. We show that the tropicalization can be described in multiple ways: as the image under the valuation map, as the image under the tropicalization map of the archimedean points in the o-minimal spectrum, in terms of initial degenerations of the definable set and initial nonnegativity cones, or via the real tropicalization of polynomials belonging to the corresponding nonnegativity cone. Moreover, we prove analogous results for the fine tropicalization of definable sets, where the valuation map is replaced by the RV sort map, which records not only the valuation of a point but also its angular component.

We propose a kernel-based estimator of the velocity field governing the transport and diffusion of $d$-dimensional interacting particles. Assuming the initial positions are i.i.d. with law $\mu_0$, we establish consistency of the estimator with an explicit mean-squared error rate of order $h^2 + N^(-2/(d+2))$, where $h$ denotes the time-discretization step and N the number of particles. The analysis covers two distinct settings: the Fokker-Planck equation, where we recover the underlying potential function, and the McKean-Vlasov equation, where we deconvolve the interaction kernel driving the mean-field dynamics.

We prove rigidity and classification results for the quasilinear Liouville equation associated with the $n$-Laplacian on complete noncompact Riemannian manifolds with nonnegative Ricci curvature. Our first result shows that, under a sharp logarithmic lower bound, the ambient manifold must be isometric to the Euclidean space and the solution must be one of the standard bubbles. We also prove a finite-mass rigidity theorem under the corresponding sharp asymptotic lower bound. We show that any logarithmic lower bound forces positive asymptotic volume ratio and one-endedness of the manifold. Finally, we construct solutions on nonflat manifolds with nonnegative Ricci curvature showing the sharpness of our hypotheses.

We establish boundary Hölder regularity for weak solutions to a class of nonlinear elliptic Dirichlet problems with nonstandard growth posed on Reifenberg flat domains. More precisely, for any prescribed exponent \(\alpha\in(0,1)\), we show that weak solutions are \(\alpha\)-Hölder continuous up to the boundary provided that the Reifenberg flatness parameter is sufficiently small. The proof combines an iterative boundary decay argument with an ABP-type maximum principle in the Orlicz–Sobolev setting.

We introduce a Grid-Based Multinode Shepard Collocation Method (GBMSC) for two-dimensional elliptic boundary value problems on rectangular domains. The method combines Shepard-type partition functions with tensor-product Lagrange interpolation on overlapping local Cartesian subgrids. The Cartesian structure avoids the local unisolvency search required by multinode Shepard constructions on scattered data, while the local character of the approximation yields sparse collocation matrices and moderate conditioning. We establish the main structural properties of the method, including the cardinal property, tensor-product polynomial reproduction, and local interpolation estimates. Numerical experiments for Poisson problems with Dirichlet and mixed boundary conditions confirm the polynomial reproduction property up to round-off accuracy and show regular convergence for smooth non-polynomial solutions. Comparisons with Multinode Shepard Collocation Method (MSC) and Kansa's collocation methods indicate that the proposed discretization preserves high accuracy with significantly better conditioning than the RBF-based schemes considered.

In 1986, Sussmann proposed an expansion of the Chen series as an infinite product of exponentials of Hall basis elements multiplied by coefficients defined through a simple induction. He proved that this product converges locally in time for bilinear control systems associated with matrices or bounded operators. In this note, we prove that Sussmann's product also converges locally in time for control-affine systems driven by analytic vector fields. The difficulty is that one must keep track of the structure of the involved Lie brackets, as the natural decay of the explicit coefficients is not enough to counterbalance the natural growth of the iterated Lie brackets.

We introduce a notion of labelled cycle for normal labelled spaces and prove a reduction theorem for Leavitt labelled path algebras. We show that every nonzero element can be reduced, by suitable left and right multiplication, either to a nonzero scalar multiple of a projection or to a polynomial supported on a labelled cycle without exits. This extends the classical reduction theorem for Leavitt path algebras of directed graphs and its analogues for ultragraph Leavitt path algebras and subshift algebras. As applications, we prove the graded uniqueness theorem and the Cuntz–Krieger uniqueness theorem for Leavitt labelled path algebras, and show that these algebras are semiprime and semiprimitive over fields.

We study logarithmic topological Hochschild homology (log THH) of the truncated Brown–Peterson spectra $\mathrm{BP}\langle n\rangle$. We first construct compatible prelog structures on $\mathrm{BP}\langle n\rangle$ generated by the classes $v_i$. Then we compute the $V(n)$-homotopy groups of log THH of $\mathrm{BP}\langle n\rangle$ for the prelog structures generated by $v_n$ and $(v_1,\cdots,v_n)$, under the assumption that the Smith–Toda complex $V(n)$ exists as a ring spectrum. At height $2$ and for $p\geq7$, we also compute the $V(2)$-homotopy groups of log THH of $\mathrm{BP}\langle 2\rangle/v_1$ relative to $v_2$.

Recent finite-time analyses of nonlinear two-time-scale stochastic approximation show that under contractive assumptions the slow iterate $Y_k$ with stepsizes $\beta_k=\Theta(k^{-1})$ and $\alpha_k=\Theta(k^{-a})$, $a\in(1/2,1)$, generally satisfies a mean-square rate of order $k^{-a}$; decoupled $k^{-1}$ rates require strong local linearity. We identify a sharp regularity-dependent boundary. In a rate-determining normal form where the slow drift contains a locally linear leakage and a nonlinear remainder of order $1+\rho$ ($\rho\in[0,1]$), the uncorrected recursion satisfies \[ \mathbb{E}\|Y_k\|^2 \le C\bigl(k^{-1}+k^{-a(1+\rho)}\bigr), \] and a matching scalar Gaussian lower bound shows that the slower term is unavoidable without modifying the update. Thus the decoupled $k^{-1}$ rate is guaranteed for the uncorrected recursion exactly when $a(1+\rho)\ge 1$. This lower bound concerns only the naive update; it is not an information-theoretic obstruction. We demonstrate this by equipping the normal-form recursion with an auxiliary online bias estimator \[ M_{k+1}=M_k+\gamma_k(R(X_k)-M_k),\qquad \beta_k\ll\gamma_k\ll\alpha_k, \] and subtracting $M_k$ from the slow update. Under the same stability, moment, and remainder assumptions, the corrected recursion achieves $\mathbb{E}\|\widetilde Y_k\|^2=O(k^{-1})$ for every $\rho\in[0,1]$, including regimes where the uncorrected update provably suffers the slower rate. Finally, we prove localized transfer theorems that extend the phase-transition mechanism to general nonlinear TTSA in fast-manifold coordinates. The proofs are non-asymptotic and rely on two Abel-transform cancellations: one for the locally linear fast-error leakage, and one for the tracked nonlinear bias.

We extend the construction of the nonabelian Fourier transform proposed in our previous paper from the space of unipotent elliptic characters to the space of unipotent compact characters of a reductive $p$-adic group, and we investigate its relation with the branching to maximal compact open subgroups. We prove that this Fourier transform is compatible with parahoric restriction for certain groups, including $\mathrm{SL}_n, \mathrm{GL}_n$, and $G_2$. We also relate the compact setting to the elliptic setting using parabolic induction.

We establish a strong comparison principle for viscosity solutions of fully nonlinear elliptic equations driven by second-order Hamiltonians that may be discontinuous with respect to all variables. The analysis relies only on a controlled superlinear growth in the gradient variable together with a structural monotonicity in the unknown, and requires neither ellipticity nor continuity of the coefficients. The proof is based on the construction of tailored viscosity test functions obtained as solutions of auxiliary eikonal-type equations. These functions compensate for the lack of regularity and allow for a fully local comparison argument despite the complete discontinuity of the Hamiltonian. This yields a robust comparison principle in open subsets of $\mathbb{R}^N$. As a consequence, we prove the existence of continuous viscosity solutions via a Perron's method adapted to discontinuous frameworks, combined with Ishii's semicontinuous envelope technique. The class of Hamiltonians covered includes linear and quasilinear equations with merely Borel measurable coefficients, as well as Hamilton-Jacobi-Bellman equations arising in stochastic control and differential games.

Feedback optimization is a control approach for driving a dynamical system to the solution of an optimization problem by interconnecting the plant with an algorithm. Existing stability guarantees typically rely on timescale separation, enforced by conservative gain bounds that limit transient performance and require a pre-stabilized plant. This paper revisits the robust control perspective on feedback optimization. We formulate the plant-optimizer interconnection as a generalized plant, where the cost gradients are characterized by Zames–Falb Integral Quadratic Constraints. Classical timescale-separation bounds are recovered as a special case of static multipliers, with dynamic multipliers yielding substantially tighter stability margins. The formulation also enables IQC based synthesis of dynamic output feedback controllers that jointly stabilize the plant and optimize transient performance, with possible model uncertainty absorbed into an uncertainty channel. For constrained problems, the framework extends to dynamic controllers that generalize projected gradient flows. Numerical examples illustrate the benefits and flexibility of the proposed approach.

In this follow-up article of a multi-part series on (strictly) preperiodic point-counting, we inspect an astonishing relationship between the set of (strictly) $1_{m}$-preperiodic points of a polynomial map $\varphi_{d, c}$ defined by $\varphi_{d, c}(z) = z^d + c$ for all $c, z \in \mathcal{O}_{K}$ and the coefficient $c$, where $K$ is any number field of degree $n\geq 1$, $d>2$ is an integer and $m\in \mathbb{Z}_{\geq 1}$ is any fixed (eventual period). As before, we study counting problems that are inspired by torsion point-counting in arithmetic statistics and (strictly) preperiodic point-counting in arithmetic dynamics. In doing so, we then first prove that for any prime $p\geq 3$ and for any fixed $\ell \in \mathbb{Z}_{\geq 1}$ and fixed (eventual period) $m\in \mathbb{Z}_{\geq 1}$, the average number of distinct $1_{m}$-preperiodic integral points of any $\varphi_{p^{\ell}, c}$ modulo prime $p\mathcal{O}_{K}$ is unbounded or zero as $c\to \infty$. Inspired further by work of Doyle-Poonen along with conjectural work of Hutz and $\textit{abc}(\textit{d})$-conditional work of Panraksa on $K$-rational preperiodic points of any $\varphi_{(p-1)^{\ell}, c}$ for any prime $p\geq 5$ in arithmetic dynamics, we then also prove that for any fixed (eventual period) $m \in \mathbb{Z}_{ \geq 1}$, the average number of distinct $1_{m}$-preperiodic integral points of any $\varphi_{(p-1)^{\ell}, c}$ modulo prime $p\mathcal{O}_{K}$ is unbounded or zero as $c\to \infty$. Finally, we then apply counting and distribution results from arithmetic statistics, and then get counting and statistical results on arithmetic objects arising naturally in our polynomial discrete dynamical settings. Together with the previous articles in the series, it might then be fair to view this article as an attempt to contribute to "arithmetic statistical dynamics".

In 2002, V. N. Temlyakov established a criterion for the convergence of the weak greedy algorithm in a Hilbert space for a given weakness sequence $ \tau = \{t_1,t_2,\ldots\} $. The criterion requires verifying a certain limiting relation for every nonnegative square-summable sequence. We give an equivalent closed-form criterion: the weak greedy algorithm converges if and only if $ \sum_{n=1}^{\infty}(1+ n\sum_{k=1}^{n}t_k^2 )^{-1/2}t_n^2=+\infty $.