Papers

Different representations of nonlinear dissipative Hamiltonian and port-Hamiltonian (pH) differential-algebraic equations (DAE) systems are derived and compared. Using global geometric and algebraic points of view, including singular Lagrange and Dirac structures, as well as Morse families and their skew counterparts, constraints are incorporated in the models. The results are illustrated specifically with constrained mechanical systems using kinematic and geometric constraints. The relation between the different representations is studied and in particular the conversion of Lagrange into Dirac algebraic constraints and vice versa is discussed. To incorporate dissipation also maximally monotone structures are studied.

We prove the sign-pattern and limiting-transition assertions of the Berkovich–Dhar conjecture modulo five, covering both the square and the cube of the finite Borwein product. In each power, the coefficients in residue class zero are strictly positive, while the sequences in residue classes three and four have exactly one positive-to-negative sign change after zero terms are omitted. We determine all four transition constants, correct the preliminary numerical estimates in the conjecture, and give their linear corrections with explicit bounded errors. A common four-arc expansion governs both powers. Near each vanishing amplitude, strict negativity of a weighted adjacent difference determines the transition. The cubic case also has two identically vanishing components in the infinite product; a shifted saddle expansion resolves the resulting finite-product boundary terms. Positive series identities, uniform remainder estimates, and an exact integer recurrence complete the proof for every positive integer.

Let $(a_n)$ be a positive null sequence with $\liminf_{n\to\infty}a_{n+1}/a_n>0$. For every $\eta>0$, we construct a compact set $E\subset[0,1]$ with Lebesgue measure greater than $1-\eta$ containing no nontrivial affine copy of $\{a_n:n\ge1\}$. No monotonicity is assumed. The proof adapts a finite-tree construction for geometric sequences to grids chosen from a subsequence with two-sided ratio bounds. We also obtain one such set for any prescribed countable family of sequences, with infinitely many distinct points omitted from every affine copy.

Many convex optimization methods use affine lower models to certify that a candidate solution is near optimal. Keeping full gradients can be costly when the informative directions occupy a small, initially unknown subspace. We develop accuracy certificates that store projected affine minorants together with bounds on the discarded components. The certificates remain valid as the subspace, queries, and certificate weights are chosen adaptively. For quadratic regularization, we derive an exact correction that accounts for when each discarded component was recorded, and a compact problem for selecting certificate weights. We also characterize the strongest lower bound supported by the compressed records when an ambient direction remains unused. The certificates can be used as stopping criteria for methods that provide valid affine minorants. We also analyze an adaptive cutting-plane construction whose oracle bound depends on the discovered dimension. Experiments on kernel SVM and $\nu$-SVR problems show reduced certificate communication and stopping times under an emulated bandwidth constraint.

The (global) average order of dominating sets of a graph is the average number of vertices of its dominating sets. Analogously, the local average order of dominating sets is the average number of vertices of its dominating sets containing a fixed vertex. In this paper, we show that the local average order of dominating sets of a graph with $n$ vertices is at least $\frac{n+1}{2}$, with equality if and only if the degree of the fixed vertex is $n-1$. Furthermore, for a graph on $n$ vertices without isolated vertices, we show that $\frac{5n-1}{6}$ is an upper bound for the local average order of dominating sets. Additionally, we give a proof of an exact formula for the local average order of dominating sets when the degree of the fixed vertex is $n-2$, and determine an upper bound for the local average order of dominating sets when the fixed vertex is an $l$-stem ($l \geq 2$).

Exponential ergodicity of stochastic reaction networks has attracted considerable attention in recent years [SIAM J. Appl. Dyn. Syst. 24, 1668-1710 (2025)]. Here, we provide a structural classification of the long-time behavior of reducible stochastic linear reaction networks under the $L^1$-Wasserstein and total variation distances. The classification is determined by the maximal eigenvalue $\lambda_{\max}$ of the first-order influx matrix $A$, the position of the zero-order influx vector $b$ relative to the left nullspace of $A$, and the conservation-law structure of the network restricted to the persistent species. We first prove that every stochastic linear reaction network is non-explosive. In the stable regime $\lambda_{\max}<0$, the process converges exponentially fast to a unique stationary distribution. In the critical regime $\lambda_{\max}=0$, when the zero eigenvalue of $A$ is semisimple and $b$ is orthogonal to the left nullspace of $A$, exponential convergence occurs if and only if a regularity condition on the restricted network is satisfied. If these two spectral conditions hold but the regularity condition fails, convergence occurs only in total variation and is non-exponential. In the divergent regimes, no closed irreducible positive recurrent class can contain an interior state, and any stationary distribution, if it exists, must be supported on the boundary. Moreover, by employing a coupling method, we obtain the optimal convergence rate in the exponentially convergent cases.

The anti-Ramsey number $AR(n,F)$ is the maximum number of colors in an edge-coloring of the complete graph $K_n$ containing no rainbow copy of $F$. I prove that every spanning linear forest on an even number $n\ge6$ of vertices whose components have even orders and which has an isolated edge has the same anti-Ramsey number as a perfect matching on $n$ vertices. The orders of the longer path components may be different.

We propose two new Douglas-Rachford type algorithms for solving generalized DC programming problems. More precisely, we deal with the minimization of the sum of two possibly nonsmooth convex functions from which a Fréchet differentiable convex function with a Lipschitz continuous gradient is subtracted, in a Hilbertian framework. We introduce an inertial Douglas-Rachford DC algorithm and a parametrized inertial Douglas-Rachford DC algorithm for solving optimization problems of this type, and study their convergence behavior. Numerical experiments illustrate the behavior of the proposed algorithms when solving sparse recovery and robust classification problems, highlighting their superior performance in comparison with their state-of-the-art counterparts.

We study the quantized $K$-theoretic Coulomb branches of framed star-shaped quivers with central gauge dimension $n\geq2$, $r-1$ one-dimensional outer gauge vertices, and a one-dimensional framing at the central vertex. For $n=2$ and arbitrary $r\geq1$, and for every $n\geq2$ with $1\leq r\leq5$, we prove that the loop-graded Koszul-perverse heart is a monoidal categorification of the associated quantum cluster algebra over $\mathbb{Z}[v^{\pm1}]$. This quantum cluster algebra coincides with its upper quantum cluster algebra. For all $n\geq2$ and $r\geq1$, we construct an explicit initial seed and identify the associated upper quantum cluster algebra with the Grothendieck ring of this heart. Every mutation is realized by a short exact sequence, and every quantum cluster monomial is represented by a simple object.

Addario-Berry, Reed, Scott, and Wood (JoCG 2026) proved that the $n$-dimensional associahedron has chromatic number at most $22500\cdot\log_3 n+O(1)$. We present a much simpler proof that the chromatic number of the $n-1$-dimensional associahedron is at most $6 \log_2 n + 18$. Since this paper's appearance as a preliminary abstract in the Japan Conference on Discrete and Computational Geometry, Graphs, and Games held from September 7-10, 2026, the bound has been improved to $O(\log \log n)$ by Oum and Wood (arXiv September 30, 2026). It is hoped that the simple method here could give insights to further improvements.

This paper studies time-varying composite optimization problems with convex but not necessarily strongly convex objective functions, on compact domains and with bounded temporal variability. As a baseline, we first analyze the time-varying proximal gradient (TV-PG) method. We show that its asymptotic function value gap is bounded by $O(\delta^{1/2})$ as $\delta\to0$, where $\delta$ measures the temporal variability of the objective function. This dependence on $\delta$ is tight in the worst case for TV-PG. Our main contribution is a time-varying accelerated proximal gradient method that improves the asymptotic upper bound on the function value gap to $O(\delta^{2/3})$. The method builds on a novel acceleration scheme whose momentum parameter is determined by the current iterates, with no explicit dependence on the iteration index. We establish the bounds for both the baseline and accelerated methods through a unified analysis based on potential functions. Numerical experiments on a synthetic problem and autoregression problems with real-world datasets demonstrate the tracking performance of the proposed algorithm.

In this paper, we first construct a universal quasi-Polish space Y with the property that every quasi-Polish space is homeomorphic to a closed subspace of Y. This gives a hyperspace F(Y) of all quasi-Polish spaces which is itself a quasi-Polish space. We then prove that the collection of all Polish closed subspaces of Y forms a coanalytic, non-Borel subset of F(Y). In particular, this collection is {\Pi}11-complete.

For a graph $F$ with at least one edge, a $k$-uniform hypergraph is Berge-$F$-saturated if it contains no Berge-$F$, but adding any missing hyperedge creates a Berge-$F$. The saturation number $\text{sat}_k(n,\text{Berge-}F)$ is the minimum number of edges in such a hypergraph on $n$ vertices. English, Gordon, Graber, Methuku and Sullivan conjectured a linear upper bound for every fixed finite family of forbidden graphs. We prove the single-graph case: $\text{sat}_k(n,\text{Berge-}F)=O_{F,k}(n)$ for every fixed integer $k\ge2$ and every fixed finite simple graph $F$ with at least one edge. The proof combines a sparse construction with degree and matching arguments.

We construct a finite-dimensional Auslander-regular algebra $R$ with global dimension three, which is right diagonal but not left diagonal; moreover, any simple right $R$-module with grade $i$ has projective dimension $i$, but there is a simple left $R$-module with grade one and projective dimension two. It answers a question raised by Iyama negatively.

In this paper, we study the global regularity of oblique boundary value problems for n-1 type augmented Hessian equations on a bounded domain, without imposing any convexity condition either on the domain or on the matrix-valued function in the n-1 type augmented Hessian equations. We establish a global existence and uniqueness theory for classical elliptic solutions by deriving global a priori estimates up to second-order derivatives. Besides the known applications for n-1 type Monge-Ampere operators in optimal transportation and geometric optics, the general theory here embraces equations raised by Li-Sheng, Some Dirichlet problems arising from conformal geometry. Pac. J. Math. 251, 2011, 337-359 and Guan-Qiu-Yuan, Fully nonlinear elliptic equations for conformal deformations of Chern-Ricci forms. Adv. Math. 343, 2019, 538-566.

This letter investigates age-of-information (AoI) minimization in multi-user wireless networks operating in the finite-blocklength (FBL) regime, which is critical for low-latency transmission of short state-update packets. While rate-splitting multiple access (RSMA) provides a powerful and flexible framework for interference management in multi-user FBL systems, the joint optimization of its parameters, such as precoding vectors, power allocation, and rate-splitting ratios, to guarantee information freshness results in analytically intractable complexity. To address this challenge, we propose an actor–critic deep reinforcement learning (DRL) framework to learn dynamic resource-allocation policies in multi-user multiple-input single-output (MU-MISO) broadcast channels. Simulation results show that the proposed RSMA-RL framework achieves consistently lower AoI than the state-of-the-art benchmarks, with substantial gains observed at low signal-to-noise ratio (SNR) and short blocklengths, while matching benchmark performance at high SNR with significantly lower online complexity via a single neural-network forward pass at execution.

For a graph $G$ and a set $B\subseteq V(G)$, the spread $\mathrm{sp}(B)$ of $B$ is the difference between the largest and the smallest degree in $G$ of a vertex of $B$, and for an integer $k\geq 0$ the parameter $\mathrm{sp}(G,k)$ is the largest cardinality of a set $B$ with $\mathrm{sp}(B)\leq k$. Caro, Lauri and Zarb asked for the minimum of $\mathrm{sp}(G,k)$ over the maximal planar graphs of order $n$; we write $\mathrm{MP}(n,\delta ,k)$ for this minimum over the maximal planar graphs of order $n$ and minimum degree $\delta \in \{3,4,5\}$. This manuscript is intended as an exposition of the subject of spread in the degree sequence of a graph $G$, with emphasis on maximal planar graphs. We apply the general lower bound of the companion paper \cite{P1} to this class, together with further ideas, some based on Caro–West and Caro–Lauri–Zarb and some new. In particular we asymptotically determine the values of $\mathrm{MP}(n,\delta ,k)$ for every pair $(\delta ,k)$ with $\delta \in \{3,4,5\}$ and $k\geq 0$. This solution recovers the case $\mathrm{MP}(n,\delta ,0)$, which is the repetition number $\mathrm{rep}(G)$ introduced by Caro and West.

We introduce weak $A_n$-formality for Galois cochain algebras and, for profinite groups whose mod-$p$ cohomology is concentrated in degrees at most two, characterise weak and genuine $A_n$-formality through finite embedding problems and the projective system they form. We then study $A_n$-formality for number fields. Let $K$ be a number field and let $p$ be an odd prime with $\mu_p\not\subset K$. For $p\geq 5$, and for $p=3$ under a natural local cyclotomic condition, we prove that every associated finite $A_3$-embedding problem is solvable. In contrast, for every odd $p$ with $\mu_p\not\subset K$, we show that no compatible family of finite-stage solutions exists, and hence $C^\bullet(G_K,\mathbb F_p)$ is not genuinely $A_3$-formal. The proof uses Chebotarev density, governing fields, and the Gras-Munnier criterion to obstruct the compatibility of finite-stage solutions, and it builds on previous work of Maire-Mináč-Ramakrishna-Tân. This shows that genuine $A_3$-formality detects a global compatibility obstruction for number fields invisible to every individual finite embedding problem and invisible to obstructions arising from individual strong Massey vanishing problems.

Conditional independence testing is a ubiquitous problem in scientific discovery. The widely employed model-X assumption shifts the modelling burden from the dependence of the output on the inputs to the dependencies within the inputs. Log-optimal e-variables have been studied in this setting, but it remains unclear how to incorporate machine learning models into their design. Other approaches test exchangeability directly, yielding an e-variable with lower power in theory but, surprisingly, higher power in practice. We explain this phenomenon by decomposing the error into null enlargement, approximation, and estimation error. The decomposition shows that GRO e-variable estimates can be beaten because of their worse approximation and estimation errors, and we explore intermediate null hypotheses between model-X conditional independence and exchangeability to reduce these errors. Moreover, the model-X assumption often only holds up to an estimation error, invalidating exact type-I error guarantees. We provide estimation error bounds that accommodate triple robustness results, achieving fast convergence rates.

This paper addresses the design of flatness-based tracking controllers for backward-flat nonlinear discrete-time systems. First, we show that backward-flat discrete-time systems can be constructively obtained from differentially flat continuous-time systems by applying an implicit Euler discretization to a structurally flat triangular representation. Subsequently, two approaches for the exact linearization of backward-flat systems are considered. Besides a dynamic feedback based on prelongations, we show that a linearizing feedback involving lower-order backward-shifts of the flat output can be employed without explicitly implementing the corresponding dynamic extension. This allows the order of the resulting tracking error dynamics to be reduced while requiring only stored values of past system trajectories. Based on the resulting linear input-output representation, flatness-based tracking control laws are derived. The proposed discretization and controller design are illustrated for a 2D gantry crane.

We show that a free-by-cyclic group with infinite order monodromy is finite index rigid, that is, any two isomorphic finite index subgroups have the same index.

We study a strong form of genuine multipartite nonlocality (GMNL), where an $n$-partite scenario admits a quantum strategy generating outcomes belonging to a restricted set with certainty, whereas every strategy built from nonclassical resources shared among at most $k<n$ parties fails this criterion with positive probability. This generalizes what is referred to as strong contextuality/nonlocality, or pseudotelepathy, beyond ruling out local models. For the simplest five-party Bell scenario of two binary-outcome measurement settings per party, we use the inflation technique to demonstrate that a game with a perfect quantum strategy witnesses a robust separation from models built from bipartite resources under local operations and shared randomness (LOSR), establishing the behavior is LOSR-GMNL. We provide broader context for this result by investigating its simulability with resources shared among three or four parties under different resource models and alternative definitions of GMNL. A central open question is whether $n$-party quantum resources can achieve such a separation from models built from $(n-1)$-party resources, for $n\ge 3$. We make progress here by constructing a family of robustly LOSR-GMNL games with $\epsilon$-optimal quantum strategies experimentally indistinguishable in the limit from true probability-one-winning strategies for any desired level of precision.

We consider entire solutions $u$: $\mathbb{R}^n \rightarrow \mathbb{R}$, $n \ge 2$, of the equation $\operatorname{div}(\nabla f(\nabla u)) = 0$ with convex density $f$: $\mathbb{R}^n \rightarrow \mathbb{R}$ given for example by $f(\nabla u) = \sum_{i=1}^n |\partial_i u|^{p_i}$ with exponents $p_i \ge 2$ including Giaquinta's example for the choice $p_1 = \dots = p_{n-1} = 2$, $p_n = 4$. If $n \le 2 + \min\{p_i\}$ holds, then we have the Liouville property: the boundedness of $u$ implies its constancy.

In this paper, we propose a relaxed CQ algorithm with dynamic stepsizes for the split feasibility problem (SFP), where the relaxed parameters are chosen adaptively. We establish the weak convergence of the proposed algorithm under standard assumptions and further derive its linear convergence under the bounded linear regularity property of the SFP. Numerical results on LASSO problem demonstrate that the adaptively over-relaxed CQ algorithm with dynamic stepsizes outperforms the dynamic CQ algorithm proposed in [10].

We define and study six ray class monoids associated to a level datum $(\mathcal{O}; \mathfrak{m}, \Sigma)$, where $\mathcal{O}$ is an order in a number field $K$, $\mathfrak{m}$ is an $\mathcal{O}$-ideal, and $\Sigma$ is a set of real places of $K$. The monoids extend the ray class group (and the ring class group) by including non-invertible classes consisting of non-invertible ideals and ideals not coprime to the modulus. We prove fundamental results on the structure of these ray class monoids.

For an almost simple complex algebraic group $G$ of semisimple rank $m$, we prove that a finite family of parabolic subgroups contains a common maximal torus whenever every subfamily of at most $m+2$ members does. We determine the resulting Helly number $(h(G))$ for almost simple groups of types $A$, $B$, $C$, $G_2$, $F_4$, $E_7$, and $E_8$, and obtain bounds differing by one in types $D$ and $E_6$. For general reductive groups of positive semisimple rank, the Helly number is the maximum of those of the almost simple factors. We apply these results to establish a fan-theoretic criterion characterizing smooth toric varieties on which every toric principal $G$-bundle splits equivariantly. In particular, for $d \ge 2$, this splitting property holds on $\mathbb{P}^d$ if and only if $d \ge h(G)$

Global existence of Leray-Hopf (type) weak solutions to the viscous primitive equations in two or three dimensions was established by Lions–Temam–Wang \cite{Lions1,Lions2,Lions3,Lions4} in the 1990s; however, the corresponding uniqueness, no matter in two or three dimensions, has been a long standing open problem since then. In this paper, we solve this open problem in the two-dimensional case. Precisely, we prove the uniqueness of Leray-Hopf weak solutions, with arbitrary $L^2$ initial data, to the viscous primitive equations in a periodic channel, subject to impermeability and stress-free boundary conditions on the solid boundaries.

We prove that for every $N\geq 3$, every solution to the $Q$-curvature-type problem $$ \alpha P_N u + (N-1)!\left(1-\frac{e^{Nu}}{\int_{\mathbb{S}^N} e^{Nu}dw}\right)=0 \ \ \ \ \ \mbox{on} \ \mathbb{S}^N $$ is constant, provided that $ \alpha\ge\frac{1}{2}$ and $\alpha \not =1$. The proof consists of a spectral gap estimate and pointwise estimates of the normalized density $\left(\int_{\mathbb{S}^N} e^{Nu}dw\right)^{-1}e^{Nu}$.

In this article, we derive various weighted p-numerical radius inequalities using the Schatten p-norm with the generalized Aluthge transform for operators in Hilbert spaces. In addition, the article explores classical numerical radius and norm inequalities for operators. The bounds obtained in this work improve upon several well-known previous results. In addition, we provide examples where we calculate and analyze upper bounds for the weighted numerical radius of different types of matrices using generalized Aluthge transformation.

We provide the first generic exact sampling algorithms with finite expected total cost for uniformly elliptic multivariate diffusions and the corresponding diffusion bridges with bounded coefficients and bounded continuous first derivatives. We formulate a computational cost model that counts work in terms of generation of Gaussian and uniform random variables to execute acceptance–rejection sampling, as well as function evaluations of the drift and diffusion coefficients, together with elementary operations such as sums and multiplications. Our constructions combine randomized parametrix expansions with a finite weighted graph that incorporates both bridge endpoints into the proposal. The exact sampling of the diffusion is minimax optimal jointly in the horizon and number of requested observations, while bridge sampling is minimax optimal in the number of observations for fixed horizon and endpoints.