Papers

Recently, the first and third authors initiated a study of arithmetical relationships between Ramanujan's theta functions $\varphi(-q)$ and $\psi(q)$. In this work, we prove eight families of internal congruences modulo arbitrary powers of $3$ and $5$ for infinite series related to the two theta functions. We also show that there exist isomorphisms between the congruence families for $\varphi(-q)$ and $\psi(q)$, which can be realized by the study of congruence multiplicities previously studied by Garvan, Sellers, and the second author. We believe that such equivalences are exclusive, at least on the congruence subgroups $\Gamma_0(6)$ and $\Gamma_0(10)$. In the end, we show how a simple manipulation of function field extensions allows us to predict whether additional isomorphisms to our congruences occur.

Fixing the quadratic proxy in a sub-gamma bound at the true variance leaves a scale to optimize, and a two-sided infinitely divisible law generally requires different scales in the two directions. For a centered law with finite nonzero variance, we normalize its Kolmogorov canonical measure and multiply the resulting variable by an independent Beta(1,2) variable. The signed remainder obtained in this way gives exact variational formulas for the right and left scales. We prove that a directional scale vanishes exactly when the Levy measure has no jumps in that direction, establish reflection, scaling, convolution, Levy-time, and opposite-jump perturbation rules, and recover the Levy triplet from the remainder law. The formulas give the two Gamma scales for bilateral Gamma laws and, for centered Skellam laws, the exact transition points p_+=(2+sqrt(3))/4 and p_-=(2-sqrt(3))/4 between local and interior control; when positive jumps are rare, the right scale is asymptotic to 1/log(1/p). The pair therefore records jump direction and the mechanism that controls the variance-exact sub-gamma pole.

A graph $G$ is minimal Ramsey for a graph $H$ if every $2$-colouring of the edges of $G$ contains a monochromatic copy of $H$, but for every proper subgraph of $G$, there is a $2$-colouring that does not contain such a monochromatic copy. Characterizing minimal Ramsey graphs is a widely studied problem. Recent research in this field includes the characterization of the size of the set $\mathcal{M}_2(H)$ of all minimal Ramsey graphs for $H$, or finding the smallest minimum degree among all graphs in $\mathcal{M}_2(H)$. In this paper, we introduce a game theoretic analogue of the above concept by considering the Maker-Breaker $H$-game on a graph $G$. In this game, two players, Maker and Breaker, alternately claim unclaimed edges of $G$, and Maker wins if in the end of the game the graph spanned by Maker's edges contains a copy of $H$. Otherwise, Breaker wins the game. We call a graph $G$ winnable for $H$ if Maker has a winning strategy for the Maker-Breaker $H$-game on $G$, and we call it minimal winnable if additionally Breaker wins the $H$-game on every proper subgraph of $G$. Along the lines of minimal-Ramsey theory, we characterize all graphs $H$ for which there exist infinitely many minimal winnable graphs, and we prove tight bounds for the smallest minimum degree among all these minimal winnable graphs. Amongst others, we obtain precise results for trees, cycles, cliques, and complete bipartite graphs. In general, we find many similarities between the Ramsey setting and the Maker-Breaker setting, but we also show substantial differences.

Although almost all graphs are asymmetric – having no nontrivial global automorphisms – they may still possess local symmetries in the form of isomorphisms between induced subgraphs, i.e., partial automorphisms. We study such local symmetries via asymmetric depth, defined in terms of the maximum rank of a nontrivial partial automorphism. We prove a tight upper bound on asymmetric depth in the class of planar graphs and identify the extremal graphs: duals of IPR fullerenes attain the maximum already on $47$ vertices. Our main structural result concerns the IPR fullerenes that are neither maximally asymmetric nor symmetric. In such a cage no purely local action realises a low asymmetric depth, and we show that the map which does realise it cannot be confined to a small part of the cage either: neither to a single face, nor behind an interface of at most $5-k$ edges, $k \le 3$ being the deficiency. A cage of asymmetric depth $2$ or $3$ is therefore not asymmetric in one place; it carries a broken symmetry invisible to its automorphism group. Such cages are rare – under $2\%$ of the asymmetric IPR fullerenes at $n = 118$. In all $727$ of them the largest partial automorphism is a near-mirror reflection, which we state as an explicit conjecture. We also extend the asymmetric depth bound to graphs of higher genus.

This paper studies the computational implementation of proper orthogonal decomposition reduced-order models (POD-ROMs) for evolutionary parametric time-dependent partial differential equations (PDEs). For a one-parameter model, many papers in the literature build the correlation matrix by projecting onto $L^2(\Omega)$ even though optimal pointwise error estimates are proved when projecting onto $H^1_0(\Omega)$. We compare both scenarios and observe the similar performance in practice. Additionally, to get the POD approximation it is necessary to compute the nonlinear term in the reduced equations. This requires a high computational cost that increases when the problem gets more complex. Numerical results in this paper show different approaches to address this issue. Discrete Empirical Interpolation Method (DEIM) is the most efficient approach. It reduces computational time by approximately half compared to computing the whole FEM formulation by a tensor construction while maintaining the same accuracy. For a multiparameter model, we analyze the new and standard method proposed in \cite{newmethod} using a two-dimensional Brusselator model to complement the results and support the error analysis with a more complex system.

We resolve the long-standing problem of establishing interior \(C^2\) estimates for admissible solutions of the graphical scalar curvature equation in every dimension \(n\ge 3\). More precisely, we prove interior curvature estimates for admissible solutions to the constant graphical scalar curvature equation. The proof combines Jacobi inequalities with a two-surface maximum principle and a two-surface Pogorelov estimate.

The Boltzmann equation plays an important role in modeling mesoscopic behavior in a wide range of scientific and engineering applications. However, its numerical solution is computationally expensive due to the high dimensionality of the model and the nonlinear nonlocal collision operator, especially for steady-state problems that require iterative solvers. This cost becomes prohibitive for inverse problems, where the induced optimization problem requires repeated forward solves. In this work, we propose a reduced-order model (ROM) for the parametric Boltzmann equation to address this computational challenge. The ROM constructs a low-dimensional approximation space for the parameter-induced solution manifold through a residual-based greedy strategy, and the reduced solution is then obtained via residual minimization over the reduced space, subject to mass conservation. The overall efficiency of the ROM is achieved by exploiting the quadratic structure of the collision operator and a precomputed separable approximation of the collision kernel. The resulting ROM is further applied to a thermally-driven inverse problem for reconstructing collision parameters from the observed macroscopic temperature data. This is accomplished either by directly replacing the PDE constraint with the ROM, leading to a bilevel optimization formulation, or by reformulating the task as a single-level optimization problem through the Karush–Kuhn–Tucker (KKT) conditions. Numerical experiments in both collision-dominated and transport-dominated cases are performed to demonstrate the efficiency and accuracy of the proposed ROM and its effectiveness in inverse problems. In particular, the resulting inverse problem is computationally much more tractable, achieving speedups of several orders of magnitude over that based on the full-order model while maintaining comparable accuracy.

We prove sharp both-sided estimates for the $W$-invariant trigonometric Dunkl heat kernel in rank one case, including root systems $A_1$ and $BC_1$. Consequently, we obtain sharp bounds for the non-compact Jacobi heat kernel in the full range of the type parameters.

This paper investigates probability-maximizing sequential change detection, a formulation in which performance is measured by the probability of stopping within an admissible interval of duration $\xi$ after a change rather than by the expected detection delay. Earlier work introduced this viewpoint in a Bayesian setting and subsequently formalized it under Lorden- and Pollak-type minimax criteria for the case in which successful detection must occur on the first post-change observation, i.e., $\xi=1$. This paper generalizes this framework in two directions. First, the decision maker is allowed to stop within a prescribed window of $\xi\in\mathbb{N}$ post-change observations. Second, the monitored process is allowed to experience multiple, non-overlapping transient change episodes with unknown onset times and durations, so that success consists of stopping within the admissible window associated with any one of these episodes. False alarms are controlled through an average run-length constraint. For exact finite-sample analysis, the paper introduces a survival-weighted average success criterion, which represents the probability of successfully detecting a randomly encountered change opportunity conditional on the detector being active at its onset. It is established that this criterion admits an exact representation as the expected truncated Shiryaev–Roberts (TSR) statistic at the stopping time normalized by the average run length, and it characterizes its exactly optimal stopping rule. The optimal procedure has finite memory and aggregates the likelihood-ratio evidence corresponding to all possible change onsets within the most recent window and compares the resulting TSR statistic with a state-dependent continuation boundary obtained from an optimal-stopping formulation.

We present a combinatorial optimization problem for the strategic deployment of UAVs equipped with 5G antennas to assist rescue operations in regions hit by natural disasters. Our goal is to optimize the placement of UAVs to provide coverage in flying ad-hoc networks among given candidate sites. Our formulation aims to maximize signal coverage and minimize interference while ensuring network connectivity. To mitigate interference effects, we incorporate the use of multiple frequencies. We formulate this problem as an integer quadratic program (IQP). We present numerical solutions obtained via the CPLEX solver and conduct a preliminary analysis of the problem's scalability in realistic network configurations. Our findings reveal a significant exponential increase in Time-to-Solution (TTS) as the number of sites grows, which poses a critical challenge in urgent, time-sensitive scenarios. To address this issue, approximate suboptimal solutions can be produced by enforcing a time limit on the solver. Although these solutions are not optimal, they preserve connectivity in most cases, providing a practical trade-off between solution quality and computational times that remain within feasible limits for real-time UAV redeployment. Recognizing the limitations of classical solvers in these contexts, we explore quantum computing as a promising alternative. Specifically, we reformulate the problem as a quadratic unconstrained binary optimization (QUBO) problem, suitable for most quantum algorithms. Through high-performance computing emulation, we show that the quantum adiabatic algorithm (QAA) can accurately solve small-scale instances, paving the way for future application of quantum computing to large-scale, time-critical optimization problems in disaster response.

We introduce relative versions of non-nef loci and restricted base loci, and investigate a relative version of Boucksom–Broustet–Pacienza's conjecture that these two loci coincide. We prove the conjecture holds for normal integral schemes projective and flat over a Dedekind domain of characteristic zero with mild singularities, such as BCM-regularity. This result is a mixed characteristic analogue of results of Cacciola–Di Biagio in characteristic zero and Sato in positive characteristic. To prove this result, we show that BCM-regularity is preserved under perturbations by divisors with small vanishing order.

By refining the approach developed in \cite{AGH-2015, GS2026a}, we establish resolvent estimates in $L^p_\sigma (\Omega)$ for the Stokes operator in a bounded Lipschitz domain $\Omega$ in $\R^3$ for any $(3/2)-\e< p\le \infty$, where $\e>0$ depends on $\Omega$. As a consequence, the Stokes operator generates a uniformly bounded analytic semigroup in $L^p_\sigma (\Omega)$. The results are particularly surprising, as it is long believed that the $L^p$ resolvent estimates in three-dimensional Lipschitz domains may fail for large $p$. In the Appendices we prove a theorem on interpolation between $L^2_\sigma(\Omega)$ and $L^\infty_\sigma(\Omega)$. Given $1< p< \infty$ and a bounded Lipschitz domain $\Omega$ in $\R^d, d\ge 2$, we show that the resolvent estimate cannot hold in $L^p(\Omega; \C^d)$ unless the Helmholtz projection is bounded on $L^p(\Omega; \C^d)$. This explains the counter-example constructed in \cite{Deuring-2001}.

Let f be a labeling of the edges of a finite graph G by positive integers, and let the weight of a path be the sum of the labels of its edges. The labeling is a geodesic Leech labeling if the weights of the geodesics are exactly 1, 2, ..., t_gp(G), each occurring once, where t_gp(G) is the geodesic path number of G. Let W_n be the wheel on n vertices, a hub joined to an (n-1)-cycle. Our main result is an upper bound: if n >= 5 and W_n is geodesic Leech, then n <= 40. The proof quantifies, via a finite Fourier kernel, the Sidon-type structure of the spoke labels, in which only the cyclically adjacent pairs are allowed as defects, and closes the last three cases with a six-variable Parseval argument. In the other direction, explicit labelings of W_7, ..., W_13, found by a computer search, answer in the negative a problem of Lakshmanan S. and Manattu, who had found labelings of W_5 and W_6 and expected every W_n with n >= 7 to be a non-geodesic Leech graph. Writing E for the set of n >= 5 for which W_n is geodesic Leech, we obtain {5, 6, ..., 13} is contained in E, which is contained in {5, 6, ..., 40}.

We improve the currently best known bound for the norm of the embedding of $PW^2$ into $PW^q$ for all $q>2$. We show that our bound is asymptotically optimal as $q\to \infty$. Our approach combines the recently established theorem of optimal Paley-Wiener concentration with a Hardy-Littlewood-Pólya majorization argument.

This paper presents the closed-form Eshelby's tensor for an arbitrarily oriented polyhedral inclusion in a bimaterial domain under general uniform eigenstrain. Existing bimaterial solutions are mainly restricted to special inclusion shapes or dilatational eigenstrains, because the bimaterial Green's function contains two Boussinesq's displacement potentials in addition to the harmonic and biharmonic potentials in Kelvin's solution. This paper derives the missing domain integrals of the two Boussinesq's potentials by reducing the volume integrals to surface and elementary line integrals. The formulae provide the complete elastic and thermoelastic bimaterial Eshelby's tensors, which are verified against analytical solutions for spherical and cuboidal inclusions parallel to the bimaterial interface, and finite element results of an inclined cuboid. Singularity analysis demonstrates that the interface-related contribution remains regular when the inclusion is separated from the bimaterial interface, while additional logarithmic singularities arise when an edge or vertex touches the interface without increasing the dominant singularity order.

We study the Hunter–Saxton equation on the real line and its global dissipative solution in the energy space $ L^\infty(\mathbb R)\cap \dot H^1(\mathbb R). $ We prove strong ill-posedness through instantaneous failure of Sobolev regularity at and below the Lipschitz threshold. More precisely, for every $s\in(1,\nicefrac32]$, we construct $ u_0\in L^\infty(\mathbb R)\cap\dot H^1(\mathbb R)\cap\dot H^s(\mathbb R) $ whose unique global dissipative solution satisfies $u\notin C([0,T]; \dot H^s (\mathbb R))$, for every $T>0$. The constructions differ substantially in the subcritical and critical regimes. For $1<s<\nicefrac32$, we superpose rescaled, localized bubbles with increasingly negative slopes; subcritical scaling preserves $\dot H^s$-summability, while the explicit characteristic formula produces norm inflation. At $s=\nicefrac32$, where scaling yields no smallness, we use logarithmically distributed compactly supported multiscale profiles whose breaking times converge to zero. A one-sided localization principle and an almost-orthogonality estimate then transfer the inflation of individual profiles to the full dissipative solution.

We investigate the optimization problem of minimizing a nonsmooth function that satisfies a nonsmooth version of the descent lemma over a nonempty and closed but not necessarily convex set. The objective function belongs to the class of upper-$\mathcal{C}^2$ functions, whereas the constraints may promote a sparse or low-rank structure. We propose a projected subgradient method with two different globalization strategies: (a) a nonmonotone linesearch and, under additional assumptions, (b) an auto-conditioned method, where the stepsize is given by a formula depending on data from past iterations. We show that both methods converge to solutions that satisfy a stronger stationarity concept than one would expect from the subdifferential sum-rule, which is particularly important since the optimization problems of interest are inherently nonconvex. Finally, we present promising numerical results when applying the algorithm to an MPEC-style problem as well as the matrix optimization problems MAXCUT and Robust PCA.

In this paper, we study the torsion problem on bounded, smooth, strictly horo-convex domains in hyperbolic space. We prove that if the diameter of the domain is sufficiently small, then $-\sqrt{u}$ is strictly convex, where $u$ denotes the torsion function. The main ingredients of our proof are the constant rank theorem, a boundary convexity estimate, and a deformation argument for the domain. We also show that the exponent $1/2$ is optimal, even among strictly horo-convex domains of arbitrarily small diameter.

Let $\omega_{\mathrm{FS}}$ be the normalized Fubini–Study form on $\mathbf P^n$, with $n\geq2$. We prove that the one-pole Lelong-number range in $DMA(\mathbf P^n,\omega_{\mathrm{FS}})$ is exactly $[0,1]$: for every $\lambda$ in this interval there is a Green function with a single pole, Monge–Ampère measure equal to the Dirac mass at that pole, and Lelong number $\lambda$. This answers Question~9 in the survey of Dinew–Guedj–Zeriahi, where the problem is attributed to Coman and Guedj. The construction adapts Li and Xia's local zero-Lelong-number scheme through variable degrees and homogeneous finite stages, while also establishing the exact Lelong number and membership in the global $DMA$ class. We then study the relation between the one-pole range $\mathcal R_\alpha(x)$ and the Seshadri interval $[0,\varepsilon(\alpha,x)]$. An application of Koike's equivalence theorem gives a point on a degree-one del Pezzo surface where the Seshadri endpoint is not attained. Conversely, a finite-pullback criterion and an explicit finite morphism show that every ample rational class on a product of projective spaces realizes its full Seshadri interval at every point.

Let $G$ be a connected, simply connected complex simple algebraic group and $\mathscr{A}_G=G/U^+$ its base affine space, whose elements are called decorated flags. We introduce the quantum configuration space of decorated flags $\mathcal{O}_q(\mathrm{Conf}_K \mathscr{A}_G)$ and initiate its algebraic study, based on the representation theory of quantized enveloping algebras. Our algebra $\mathcal{O}_q(\mathrm{Conf}_K \mathscr{A}_G)$ gives a quantum analogue of the configuration space $\mathrm{Conf}_K \mathscr{A}_G$ of $K$ decorated flags, which provides local building blocks for the Fock–Goncharov moduli space $\mathscr{A}_{G,\Sigma}$ of decorated twisted $G$-local systems on a marked surface $\Sigma$. We establish basic algebraic properties of $\mathcal{O}_q(\mathrm{Conf}_K \mathscr{A}_G)$ such as the domain property, quantum normalization of representatives, the quantum cyclic shifts, the quantum Wilson lines, whose classical counterparts have been fundamental in the study of $\mathscr{A}_{G,\Sigma}$. Moreover, we construct quantum seeds for $\mathcal{O}_q(\mathrm{Conf}_4 \mathscr{A}_G)$ by transporting the Berenstein–Zelevinsky quantum cluster structure on $\mathcal{O}_q(G)$ via quantum Wilson lines, and prove that $\mathcal{O}_q(\mathrm{Conf}_4 \mathscr{A}_G)$ coincides with the corresponding quantum cluster algebra and its upper counterpart after the localization at frozen variables. The exchange matrices for our quantum seeds agree with the Goncharov–Shen exchange matrices. We also show that quantum seeds for $\mathcal{O}_q(\mathrm{Conf}_4 \mathscr{A}_G)$ restrict to those for $\mathcal{O}_q(\mathrm{Conf}_3 \mathscr{A}_G)$. Finally, we construct quantum seeds for $\mathcal{O}_q(\mathrm{Conf}_K \mathscr{A}_G)$, $K\geq 5$, and prove that the corresponding quantum cluster algebras contain the quantum configuration space $\mathcal{O}_q(\mathrm{Conf}_K \mathscr{A}_G)$.

An action of a group is binary if it induces the group of automorphisms of some homogeneous edge-coloured directed graph. In this paper we study the quasisimple groups $G$ with $G/Z(G)$ a sporadic simple group—we produce a complete classification of their binary actions.

Let $S_m^n$ be the surface obtained by blowing up $\mathbb{P}(1,1,m)$ at $n$ general smooth points, where $m\geq2$. For every very ample Cartier divisor $H$ on $S_m^n$, we prove that $\mathrm{Affcone}_H(S_m^n)$ is flexible when $n\leq m+1$. For $n=m+2$ and $n=m+3$, we prove that $\mathrm{Affcone}_H(S_m^n)$ is generically flexible.

We study the problem of learning from data a parametric Nash equilibrium (NE) problem that is monotone (or strongly monotone) for all parameter values. In the presence of local and shared convex constraints, monotonicity enables efficient computation of generalized Nash equilibria of the learned game. We consider two learning scenarios: (i) direct learning of the NE problem from samples of the agents' costs, and (ii) inverse learning of surrogate agents' costs from samples of their best responses. We propose two methods to solve these tasks. The first is a penalty-based approach that promotes monotonicity of the learned game during training. The second, based on a representation theorem we introduce for a broad class of monotone games, parameterizes the agents' costs so that monotonicity of the NE problem is guaranteed by construction, for all parameter values, regardless of the training data used. We illustrate the applicability of the proposed methods on several numerical examples. A Python library and the examples reported in the paper are available at https://github.com/bemporad/learn_monotone_games.

In this article, we study a generalisation of Rényi's car-parking problem in which isosceles trapeziums are sequentially deposited on a strip. We derive an explicit formula for the parking constant in terms of the lengths of the two bases, recovering the classical rectangular and triangular models as special cases. We further obtain quantitative finite-size asymptotics for both the expected number and the variance of deposited trapeziums, with convergence rates that depend explicitly on the geometry of the deposited particle. In particular, although the recursive construction involves two different substrate geometries, their variances have the same leading asymptotic density. Finally, we show that the parking constant depends non-monotonically on the ratio of the two base lengths and possesses a unique minimiser, so that the least efficient shape is a genuine trapezium rather than a triangle.

In many systems, the interaction between a deformable surface or interface and the surrounding bulk fluid is coupled with intrinsic spatiotemporal dynamics within the moving surface. Examples include tumor growth, biological tissue morphogenesis, cardiac mechanics, multi-phase surfactant chemistry, additive manufacturing, clothing wear-and-tear, and reactive combustion flows. Solving such problems requires both geometric computing algorithms to track and resolve the surface and numerical methods to solve the coupled governing equations in the surface and the surrounding bulk phase. Here, we present a fully meshfree numerical solver for such coupled bulk-surface problems with deformable interfaces. The presented solver tracks the surface implicitly, solving for the dynamic surface geometry based on stress balance coupled to surrounding fluid phases. We show convergence for a mass-conserving case on a growing sphere and solve bulk-surface problems with incompressible Navier-Stokes fluids coupled to in-surface nonlinear reaction-diffusion dynamics. Finally, we show a model of biological morphogenesis, solving simultaneously for the dynamic surface shape and the fields on the curved surface with two-way coupling.

We study three identical bosons with contact attraction of strength mu on the square lattice at total quasimomentum K = 0 and K = pi. First, at K = 0, for all sufficiently large mu there are exactly two eigenvalues below the essential spectrum. The trimer branch follows from Kato perturbation theory at the simple isolated eigenvalue 3 of the pair-contact operator, uniformly in K; the atom-dimer branch follows from a Feshbach-Schur reduction whose limiting equation coincides with the second Birman-Schwinger (Faddeev) eigenvalue of the finite-rank principal part. The spectral gap between the two branches grows linearly in mu. Second, at K = pi the trimer branch is unchanged up to O(mu^-2), the two-particle threshold equals -mu + 4 exactly, and the gap to it is 2mu - 2 plus lower order corrections. The second bound state disappears because the exchange operator changes sign, S_pi = -S_0: the s-channel eigenvalue that exceeds 1 at K = 0 becomes negative, while the only positive atom-dimer channel at K = pi is the odd p-channel, bounded by 2 - 4/pi < 1. Consequently the operator has exactly one eigenvalue below the threshold minus 36/(mu - 10). Third, the ground state is exponentially localized in the relative coordinates. The decay exponent is positive unconditionally; under a non-cancellation hypothesis it is bounded above by log(3mu) plus lower order terms, while finite-volume computations indicate the slowest tail is the atom-dimer dissociation channel with decay rate one half log(2mu). A dictionary between the principal-part method, Kato theory, and the atom-dimer threshold matrix is given, together with the three-dimensional constants for comparison.

Fan, Ma, and Xing constructed oriented-tangle invariants from finite-type two-parameter quantum algebras. In this paper, we construct an explicit parameter-transport comparison between two-parameter modules and their corresponding one-parameter modules, and we verify this comparison on every elementary oriented-tangle operator. After extending scalars to a common coefficient field, we prove that if $M_1$ is any finite-dimensional integrable type-1 simple highest-weight $U_{v,1}$-module whose weights lie in an admissible lattice and $M_t=\Phi_t(M_1)$ is its transported module, then for every oriented link $L$, the corresponding normalized invariants satisfy $I_{v,t}^{M_t}(L)=I_{v,1}^{M_1}(L)$. Consequently, the two invariants assign equal values to exactly the same pairs of oriented links and therefore have the same distinguishing power; this includes the vector representations of the finite classical types A, B, C, and D. Sean Clark's comparison of ordinary and super quantum knot invariants is obtained as a specialization of the same transport principle in which explicit scalar factors are allowed.

This paper introduces a novel second-order dynamical system driven by a forward-backward splitting operator for solving the structured inclusion $0\in(A+B)(x)$ in a real Hilbert space, where $A$ is a maximal $\rho$-comonotone operator and $B$ is a $\nu$-cocoercive operator. The well-posedness of the system is established, and Lyapunov analysis yields accelerated convergence rates of order $o(\frac{1}{t})$ for the velocity and $o(\frac{1}{t^2})$ for the forward-backward residual, together with weak convergence of the trajectories to $\operatorname{zer}(A+B)$. Temporal discretization further leads to a class of double inertial Halpern forward-backward splitting algorithms that encompasses the classical forward-backward splitting method and its inertial variants as special cases. Numerical experiments on split feasibility, sparse signal recovery, and image deblurring illustrate the effectiveness of the proposed algorithm.

For an integer $d\ge 3$, put $\Delta_d=\min{2^{d-1},d(d-1)}$. Let $a/q$ be reduced, let $P(X)=\frac{a}{q}X^d+\alpha_{d-1}X^{d-1}+\cdots+\alpha_0$, and let $\mathcal{I}$ be an interval of $H\le q$ consecutive integers. We prove $\left|\sum_{n\in\mathcal{I}}e(P(n))\right|\ll_{d,\varepsilon}q^{1/d}H^\varepsilon+H^{1-1/\Delta_d+\varepsilon}$. Consequently, for every prime $p>d$, every degree-$d$ polynomial $P\in\mathbb{F}p[X]$, and every interval $\mathcal{I}$ of $H$ consecutive integers with $p^{1/d}<H<p^{1/(d-1)}$, writing $H^d/p=H^u$, one has $\left|\sum{n\in\mathcal{I}}e_p(P(n))\right|\ll_{d,\varepsilon}H^{1-\min{u/d,1/\Delta_d}+\varepsilon}$. This strictly improves throughout the full natural short-interval window the best generic estimate obtained by combining classical Weyl differencing with the optimal Vinogradov mean value theorem.

For a graph G, its bondage number b(G) is the minimum number of edges whose deletion increases its domination number. Dunbar, Haynes, Teschner, and Volkmann conjectured in 1998 that every nontrivial planar graph satisfies b(G) <= Delta(G) + 1. We show that the truncated octahedral graph T has gamma(T) = 8 and b(T) = 5. Since T is planar and cubic, this gives b(T) = 5 > 4 = Delta(T) + 1 and disproves the conjecture. The finite parts of the verification are exhaustive: the direct verifier checks candidate dominating sets of sizes six, seven, and eight and all 58,905 four-edge sets. The targeted discovery search and an independently written verifier are described, and the complete C++20 verifier is included in the source archive.