Papers

Gaussian-process Bayesian optimization (GP-BO) excels at black-box optimization of costly functions, e.g., hyperparameter optimization (HPO) and multi-agent system (MAS) design. Convergence-rate guarantees exist for select methods, notably GP upper confidence bound (GP-UCB), but require a fixed kernel. Critically, the kernel encodes how input proximity affects objective value similarity. When raw coordinates poorly match this geometry - as with log-scaled hyperparameters or localized peaks - input warping can greatly improve sample efficiency, yet known GP-UCB proofs require a fixed kernel. We propose Finite-Library Input-Warped Bayesian Optimization (FLIWBO), which selects warps from a finite library of smooth input maps by any history-dependent rule. It adapts the input geometry to accelerate learning while retaining high-probability convergence guarantees under mild hypotheses, with an explicit $\sqrt(N_\varepsilon)$ library-size cost. Controlled diagnostics show that finite-library warping repairs planted geometry mismatches and identify FLIWBO failure cases. Across four repeated benchmarks - warped synthetic objectives, a confidence-fence trap, and Fashion-MNIST HPO - FLIWBO-UCB beats raw-coordinate GP-UCB under misspecified geometry, escapes traps that defeat even oracle-warp expected improvement, and recovers much of the gain from manual log scaling, while leading the tested methods that admit a matching regret guarantee. A 20-dimensional MAS design study further shows feasibility under costly noisy evaluations. Code for experiments is available: https://github.com/edvin-ketabati/bogp-paper-experiments.

Learned graph simulators provide an efficient alternative to high-fidelity solvers for granular dynamics. However, granular motion depends strongly on inter-granular contact history, which is difficult to preserve when particle contacts form, break, and rearrange. Existing simulators mainly store temporal information in node features or node-level memory. Here we introduce TRACE, a graph-network simulator that stores interaction history directly on contact edges. Each edge maintains a persistent memory updated by attention-based message passing and a gated recurrent unit, while an edge-identity dictionary preserves this memory as the contact graph changes. A physics-structured decoder predicts inter-granular normal and tangential contact forces, enforces the Coulomb friction limit, and applies equal-and-opposite internal forces. The model is trained with single-step pretraining followed by autoregressive rollout fine-tuning. We evaluate TRACE on 2D and 3D granular column-collapse benchmarks. In both cases, TRACE produces stable, physically consistent long-horizon rollouts, closely reproducing the final deposit geometry and the kinetic energy released during collapse. Compared with graph network simulator (GNS) and node-memory graph neural simulator (NMGNS), TRACE reduces long-rollout position error by 31-62% and final-deposit error by 58-89% across the two benchmarks, while using fewer parameters and maintaining near-zero particle interpenetration. TRACE also achieves 12.2$\times$ and 8.9$\times$ speedups over the material point method (MPM) reference solver in 2D and 3D, respectively. Our code is available at https://github.com/Data-Driven-Computational-Geotechnics/TRACE.

Homology nilpotency of a simply connected minimal Sullivan algebra $M$ is the least integer $n$ for which $(M^+)^{n+1}$ is contained in an acyclic differential ideal. We develop lower- and upper-bound criteria, including methods for rational cell attachments, and construct an explicit finite-type minimal Sullivan algebra $M$ satisfying $nil_h(M)=3<Hnil(M)=4$. Complete strictly coordinated retractive towers identify the obstruction: they characterize homotopical nil-length while exposing the additional quotient defect measured by homology nilpotency. We then show that this rigid defect disappears after stabilization by degreewise finite wedges of simply connected rational spheres: $$ Hnil_s(X)=cat_o(X) $$ for every simply connected rational space $X$ of finite type.

We prove that fully independent SparseStack achieves the oblivious subspace embedding parameters conjectured by Nelson and Nguyen (FOCS 2013): $m=O((d+\log(1/\delta))/\varepsilon^2)$ rows and $s=O(\log(d/\delta)/\varepsilon)$ nonzero entries per column for distortion $\varepsilon$ and failure probability $\delta$ on any fixed $d$-dimensional subspace, with explicit constants. The proof turns random-matrix concentration into a problem in finite-dimensional linear algebra. A coupling first reduces the moment estimates to a model with independent finite-valued entries. We represent these variables by multiplication operators, so their matrix moments become exact matrix elements of a deterministic operator on a finite tensor product. The central estimate bounds the contribution of $\ell\ge1$ occupied sites sharing the external factor $\mathbb{R}^d$ by $d+\ell-1$ rather than $d\ell$, yielding additive dependence on the dimension and the moment order. This approach controls both spectral edges without Gaussian comparison. The main theorem has been formally verified in Lean 4.

We give a presentation of a groupoid of transformations of quilt decompositions of a surface, which are pants decompositions of the surface with the addition of "seams". We consider colored quilt decompositions of a surface, where one side of a quilt decomposition is colored. We also give a presentation of a groupoid of transformations of colored quilt decompositions of a surface. These presentations of such groupoids are realized by constructing connected and simply-connected two-dimensional cell complexes whose 0-cell are these up-to-isotopy structures on the surface. In the latter part of this paper, we show that Torelli group (or more precisely, a torsion-free subgroup of a mapping class group of a surface) acts on these complexes freely.

Let \(G\) be a simple and simply connected algebraic group of simply-laced type. For each positive braid word \(\beta\), and for the complete strong duality datum attached to a \(Q\)-datum, we construct an explicit based monoidal categorification of the quantum cluster algebra of the braid variety \(X(\beta)\). We identify this algebra with both a localized quantum Grothendieck ring and a localized level-\(\geq1\) subalgebra of the bosonic extension algebra. Under these identifications, quantum cluster monomials correspond simultaneously to real simple modules and normalized global basis elements. We construct the specialization homomorphism at \(q^{1/2}=1\) and prove that it recovers \(\CC[X(\beta)]\); in particular, the resulting integral form is a flat quantum deformation. We also give intrinsic Lusztig parameters for cluster variables attached to double strings, identify the corresponding quantum grid minors, and establish generalized quantum \(T\)-systems.

Physical decoherence can preserve the microscopic strategic neutrality (i.e. zero effective field and equal potential of the two homogeneous strategic orientations) of a quantum game while changing the thermodynamic regime of the corresponding interacting population. We demonstrate this for an Eisert–Wilkens–Lewenstein (EWL) Stag Hunt embedded as independent nearest neighbor encounters on a square lattice. For the restricted strategies $\mathsf{Q}=i\mathbb{Z}$ and $\mathsf{D}=i\mathbb{Y}$, noisy two-player payoff matrices are determined for phase damping, depolarization, and amplitude damping and mapped exactly to channel dependent Ising parameters $\mathfrak{J}(\Gamma,p)$ and $\mathfrak{H}(\Gamma,p)$. Phase damping and depolarization show the clearest contrast: they share the same microscopic neutrality branch $\mathfrak{H}=0$, while only depolarization suppresses the interaction as $(1-p)^2$. At $\beta=1$, this produces an exact depolarization-driven square-lattice critical point at $p_{*}\approx 0.233460\ldots$, whereas phase damping remains in the ordered coexistence regime along the same neutrality branch. Monte Carlo finite size scaling is consistent with two-dimensional Ising criticality and distinguishes field driven coexistence below $p_{*}$ from a smooth crossover above it. Amplitude damping additionally reveals a strong dependence on channel placement: the post-strategy neutrality branch reaches $\Gamma=0$ at $p=1/3$ and then disappears. Resource negativity further shows that microscopic two-qubit entanglement and collective interaction strength are distinct quantities. The resulting extended lattice remains an ordinary classical Ising system.

We develop an efficient and accurate multiscale computational framework for predicting the buckling and post-buckling behaviour of elastic beams containing periodically distributed soft inclusions. The framework extends our previous multiscale, patch, computational homogenisation for linear elasticity by incorporating nonlinearity, and thus enables accurate prediction of both the buckling onset and the subsequent post-buckling response. Microscale computations are performed only within a sparse set of small subdomains (patches), while the macroscale behaviour is recovered through a proven patch-coupling algorithm. The scheme is assessed through quarter- and half-domain patch computations for beams with inclusion-to-matrix Young's modulus ratios ranging from 0.001 to 1. The results show that reducing the inclusion stiffness lowers both the critical buckling strain and the critical buckling stress, indicating an increased susceptibility to instability, while producing a milder post-buckling response with smaller transverse deflections and stress drops. Eigenvalue analysis of the Jacobian matrix accurately predicts the onset of instability and the corresponding critical strain and stress. Bifurcation diagrams of the nonlinear buckled configurations under compressive loading, and a quantitative analysis of the effect of the interpolation order on the predicted buckling and post-buckling responses, are also presented. Comparisons with full-domain simulations demonstrate that the proposed framework accurately predicts both the buckling threshold and the post-buckling behaviour while substantially reducing the computational cost. The methodology is readily extendable to heterogeneous beams, plates, shells, and other engineering structures.

Direct satellite-to-device (DS2D) communication is emerging as a transformative paradigm for extending ubiquitous connectivity and edge computing capabilities to remote and underserved regions within 6G non-terrestrial networks. However, practical deployment faces dual critical challenges: i) dynamic satellite channel conditions (e.g., severe Doppler shifts, fast fading) and constrained satellite computing resources in cooperative scenarios; and ii) unauthorized satellite access introduces significant spectrum security threats in non-cooperative scenarios. To address these challenges, we propose a versatile DS2D system that supports cooperative task offloading and non-cooperative access monitoring. For cooperative DS2D communications, we integrate a channel estimation module with a dueling double deep Q-network (D3QN) to dynamically optimize task offloading strategy. For non-cooperative DS2D communications, we propose Transformer-based models to enable blind signal detection and automatic modulation classification (AMC). Simulation results show that: 1) The D3QN algorithm reduces average latency by up to 225% compared to static association policies. 2) Our signal detection model achieves an average presence detection probability of 90.5% for DS2D signals. 3) The proposed AMC algorithm achieves superior performance across different signal-to-noise ratios (SNRs), with a 9.4% accuracy gain in low-SNR regimes compared to existing methods.

For a finite graph $H$, a connectivity graph-code is a family $\mathcal C\subseteq 2^{E(H)}$ such that $A\triangle B$ is a connected spanning subgraph of $H$ whenever $A$ and $B$ are distinct members of $\mathcal C$. Let $m(H)$ denote the maximum size of such a family, and let $f(d)$ be the largest integer $q$ for which $m(H)=q$ for infinitely many pairwise nonisomorphic $d$-regular graphs $H$. Restricting codewords to the edges incident with a vertex gives $f(d)\leq 2^d$. Alon proved equality for all sufficiently large $d$ and asked whether it holds for every $d\geq 4$. We answer this question affirmatively. More precisely, for every $d\geq 4$ we construct infinitely many finite simple $d$-regular bipartite graphs carrying a linear connectivity graph-code of dimension $d$. The construction begins with a vector-labelled copy of $K_{d,d}$. For $d\geq 7$, the required labelling follows from a probabilistic count over an irreducible conjugacy class in $\mathrm{GL}_d(2)$; explicit matrices, verified by a short exact exhaustive program, cover $d=4,5,6$. Cyclic voltage lifts then produce the required infinite families.

We investigate the dynamics of Chebyshev's method applied to the polynomial family $p_n(z)=z(z^n-1)$ for $n>1$. The resulting map is denoted by $C_n$. It is proved that the immediate basins corresponding to the non-zero roots are unbounded and simply connected. We also show that the Julia set of $C_n$ is connected. It is proved that the immediate basin of the root at the origin exhibits a different behavior: it is unbounded for $n\leq 16$ and bounded for $n\geq 17$. We establish that $C_n$ is convergent whenever $n\leq 16$ or $n$ is odd. Finally, we determine the symmetry group of $C_n$ and prove that it coincides with the symmetry group of the polynomial $p_n$, thereby confirming, for this family, a conjecture proposed by Nayak and Pal.

We obtain a new, conceptually simpler, unconditional proof that more than $67.25\%$ of the non-trivial zeros of the Riemann zeta function are simple and on the critical line, and that at least $83.62\%$ of the non-trivial zeros are distinct. Our approach also yields two new unconditional estimates on simple zeros and zeros on the critical line. More precisely, we prove that the proportion of zeros that are simple or lie on the critical line (or both) is at least $88.76\%$, and that the average of the proportions of simple zeros and of zeros on the critical line is at least $83.62\%$. A proof of the bounds for simple zeros on the critical line and for distinct zeros was very recently produced by an internal research version of Claude developed by Anthropic and subsequently verified by two mathematicians at Anthropic, Levent Alpöge and Ralph Furman, whereas our two additional estimates are neither stated nor proved in the Claude paper. The argument produced by Claude is technically intricate, and its main mechanism is not immediately transparent. It combines several ingredients from linear algebra, including a finite-dimensional matrix representation of Weil's Hermitian form and a rank–trace inequality for Hermitian matrices, with a second moment calculation over the zeros using the explicit formula. Our new approach proceeds by replacing the entire finite-dimensional matrix framework by a single Hilbert space inequality, which allows for a direct application of Montgomery's theorem on the pair correlation of zeros of the zeta function, in the unconditional form obtained by Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh.

For a semiconcave function with linear modulus, we introduce the cut locus through a touching approach and prove that it coincides with the variational cut locus defined via the Lax–Oleinik semigroup, independently of the Hamiltonian. Cut points are characterized by the emptiness of the proximal subdifferential of $\phi$, which we use as the analytic criterion behind the touching definition. We introduce the degree of regularity $R_\phi$, which measures the $C^{1,1}$ deviation of $\phi$, and prove the quantitative relation $R_\phi(x)\sim 1/\tau_{\phi,H}(x)$ with the cut time function, valid pointwise up to explicit truncation constants. From this relation we derive a separation estimate for calibrated curves, showing that no conjugate points occur before the cut locus. This estimate supplies the tools for the propagation results. Cut points propagate globally along generalized characteristics for the evolutionary Hamilton–Jacobi equation, and Alexandrov points propagate forward along calibrated curves, with the second derivative satisfying a matrix Riccati equation. We also show that the cut locus is a Lebesgue null set and give a streamlined proof of Alexandrov's theorem. Finally, we give necessary and sufficient conditions for the cut locus of a weak KAM solution to be closed, in terms of the cut time function, the $C^{1,1}$ support, and the degree of regularity.

We study collapsing finite-time singularities of the unnormalized Kähler–Ricci flow on Fano bundles arising in the analytic minimal model program. For a Fano bundle $X^n\rightarrow Y^m$, we prove a maximal splitting theorem by the Kähler-Ricci flow that every tangent space at a fixed limiting point splits globally as \((\C^m,g_{\rm E},J_0)\times(Z',d',J')\). If the fibre has complex dimension one, we prove a global Type-I bound for the full curvature tensor and show that the ambient and intrinsic diameters of every fibre are uniformly comparable to \(\sqrt{T-t}\). Furthermore, every tangent flow at any fixed limiting point is the round shrinking cylinder \(\C^m\times\PP^1\).

Ordinary differential equations (ODEs) underlie models in science and engineering, and many applications need derivatives of their solutions with respect to parameters. Ensembles of independent trajectories suit graphics processing units (GPUs), but current GPU software forces a trade-off: the fastest ensemble solvers cannot be differentiated in reverse mode at the speed they solve, and the solvers built for differentiation solve more slowly. No single tool has yet offered a reverse-mode gradient at the speed of a fused-kernel solve. We present GRADSOLVE, an open-source JAX library for solving and reverse-mode differentiating low-dimensional ODE ensembles on NVIDIA GPUs. It records the steps an adaptive solver accepts and differentiates a fixed-step replay of them; the returned gradient is the exact discrete adjoint of those steps, the same derivative Diffrax returns by default, obtained more cheaply from a fixed-length chain than from an adaptive loop. It targets ensembles differentiated many times against one recorded mesh, keeps Diffrax as a fallback, and supports explicit and Rosenbrock integrators. Used as a solver, GRADSOLVE's forward-only kernel ran 2.8x faster than DiffEqGPU.jl; used for gradients, once a record exists, it computed them 5.6-14.1x faster than Diffrax's checkpointed adjoint at matched forward-state accuracy across three GPU generations, the advantage narrowing on large ensembles and, on stiff systems, down to parity at tight accuracy. GRADSOLVE is released at https://github.com/ECLIPSE-AI4Science/gradsolve.

We obtain upper bounds for the number of real zeros of functions of the form $$ f(x) = \sum_{k=1}^{n} c_k \bigl(P_k(x)\bigr)^{\alpha_k}, $$ where $c_k, \alpha_k \in \mathbb{R}$ and each $P_k$ is a real polynomial of degree at most $d$ that is non-negative on an interval $I\subset \mathbb{R}$. We improve previously known exponential upper bounds for the number of roots on $I$ to bounds that are polynomial in $n$, linear in $d$, and independent of the exponents $\alpha_k$. For linear combinations of square roots of positive quadratic polynomials on $\mathbb{R}$ we prove the linear bound $2n$, answering a question of N.~Alon. A modification of the argument yields a linear bound for a question of A.~Gabrielov, D.~Novikov, and B.~Shapiro related to Maxwell's conjecture. The article describes two independent approaches: an elementary ODE method in the general case, which also gives a polynomial bound for the number of critical points of one dimensional Gaussian mixtures, and a PDE method for the case of positive quadratic polynomials, which connects the problem to the number of nodal domains of solutions to $\Delta u + \lambda u = 0$ on the punctured hyperbolic plane. As a byproduct of the second approach, we describe a curious relation between axially symmetric harmonic functions on $\mathbb{R}^3\setminus\{(x,0,0)\}$ and Laplace-Beltrami eigenfunctions on the hyperbolic plane with eigenvalue $1/4$.

We study Gaussian-restricted barycenters for quadratic two-sided Kullback–Leibler unbalanced optimal transport with independent marginal penalties and no coupling entropy. Exact profiling of the barycenter mass reduces the problem to a smooth Gaussian shape functional with endogenous Gibbs weights. We establish global attainment, derive the stationary moment equations, and construct a reverse-KL majorization–minimization (MM) iteration whose full sequence converges from every nondegenerate Gaussian initialization to a stationary fixed point. The diagonal second variation induces a parallel-sum tensor coupling the Bures–Wasserstein and Fisher–Rao metrics; its finite-mass extension admits a radial cone representation. Under common penalty scaling, global minimizers converge to a Gaussian Wasserstein barycenter with effective weights; for sufficiently large penalties, the minimizer is unique and admits a first-order analytic expansion. In the small-penalty regime, the distance of every global minimizer to the compact maximizer set of a weighted Chernoff affinity functional vanishes with respect to the mean–covariance parameter distance. Numerical experiments illustrate MM descent, local contraction, and the two penalty limits.

A probability coordinate chart is a continuous strictly increasing bijection that transports observations to the open unit interval, where averaging is always well defined. For barycentric inference, the first coordinate moment is returned through the inverse chart. Higher initial coordinate moments similarly generate initial Kolmogorov moments by pullback, while centred coordinate moments remain on the probability scale. We develop inference for these quantities under fixed, intrinsic, and estimated charts. For a fixed benchmark chart, the coordinate mean is the primary inferential object. Confidence intervals are constructed on the probability scale and then transported through the inverse chart, preserving the geometry and allowing asymmetry on the observation scale. Hoeffding's inequality also provides finite-sample distribution-free intervals. The intrinsic case requires different treatment. The empirical cumulative distribution function is not an admissible chart, and its natural generalised plug-in construction collapses exactly to a middle order statistic. Feasible intrinsic inference therefore reduces to median inference. This self-induced functional is distinguished from the frozen oracle construction, which has a different asymptotic variance. For estimated location-scale charts, the asymptotic expansion contains a calibration correction for the first-order effect of learning the chart from the same sample. This yields an influence-function variance estimator and a bootstrap procedure that recalibrates the chart in every resample. Joint covariance theory is developed for vectors of initial and centred coordinate moments, with initial Kolmogorov moments obtained by inverse-chart pullback. The framework separates variation on the probability scale, amplification by the inverse chart, and uncertainty from chart calibration.

Recent advances in equation discovery methods such as SINDy have highlighted the growing interest in identifying governing parameters and models directly from data. In this work, we take a complementary approach grounded in analysis and numerical PDE methods: we recover an unknown viscosity in steady Navier-Stokes equations (NSE) from partial incompressible flow observations using continuous data assimilation (CDA). We propose a simple and efficient parameter recovery algorithm and also a nonlinear solver for CDA-NSE. Together, this creates a highly efficient technique for recovering an unknown viscosity from partial solution data. Our analysis establishes the well-posedness of steady CDA-NSE, quadratic convergence of the parameter recovery algorithm, and quadratic convergence of a CDA-Picard + CDA-Newton nonlinear solver. Numerical experiments illustrate that the methods are very effective in restoring parameters quickly, even with poor initial guesses.

We develop a localized persistent theory of commutative algebra for Stanley-Reisner rings, based on local cohomology supported at a coordinate prime rather than at the maximal ideal. The construction is modeled on the persistent Stanley-Reisner theory of Suwayyid and Wei (arXiv:2503.23482) and its functorial development for graphs and hypergraphs (arXiv:2512.17619), in which invariants of the face ring such as graded Betti numbers and f- and h-vectors are persisted across a filtration. That framework is built from the minimal free resolution and is thus Tor-theoretic; we work instead on the injective side, and the resulting modules record information localized at a single vertex, complementing the global picture given by maximal-support local cohomology. For a vertex prime $p_i = (x_j : j \neq i)$ we prove an exact $\mathbb{Z}^n$-graded decomposition of $H^q_{p_i}(k[\Delta])$ into the maximal-support local cohomology of the deletion and of the link of the vertex $i$, the first in $x_i$-degree zero and the second repeated in every positive $x_i$-degree; at the level of graded dimensions this recovers the vertex-prime case of Rahimi's bigraded formula. With Hochster's formula this yields a closed combinatorial description of every multigraded piece. Building on this structure we introduce per-vertex persistent local cohomology numbers, prove a persistent links-Hochster formula, obtain interval decompositions of the resulting reversed-arrow persistence modules and a bottleneck stability theorem, retain multiplication by the uninverted variable as a morphism of persistence modules that the two barcodes alone do not determine, and extend the theory to an arbitrary coordinate prime, where the multiplication maps of the uninverted variables assemble into a commuting Boolean diagram of persistence modules.

We study how far gradient descent (GD) can be accelerated by predetermined stepsizes in smooth convex optimization. Going beyond the classical $\Omega(n^{-2})$ first-order oracle lower bound of Nemirovsky and Yudin (1983), we prove an $\Omega(n^{-1.6342})$ non-anytime lower bound and an $\Omega(n^{-1.2408})$ anytime lower bound. These improve the recent $\Omega(n^{-1.932})$ non-anytime lower bound of Ma and Chen (2026) and the $\Omega(n^{-4/3})$ anytime lower bound of Tsai et al. (2026), respectively. Both results continue to hold when the stepsizes may be negative. Our anytime lower bound also shows that the $O(n^{-\log_2(1+\sqrt{2})})$ rate of non-anytime silver schedules (Altschuler and Parrilo, 2025; Grimmer et al., 2025) is unattainable in the anytime setting. This establishes a strict separation between the two settings.

Temporal graphs model dynamic networks by assigning positive integer time labels to the edges, while information propagates along temporal paths, whose edge labels are traversed in nondecreasing order. A temporal $\alpha$-spanner of a temporal graph with $n$ vertices is a temporal subgraph that approximates the minimum-hop temporal distance between every pair of vertices within a factor of $\alpha$. While general temporal graphs may not admit sparse temporal $\alpha$-spanners for any value of $\alpha$, temporal cliques are known to admit temporal $(2k-1)$-spanners of size $\widetilde{\mathcal{O}}(kn^{1+1/k})$ for every positive integer $k$. We present a simple recursive algorithm that computes, for every temporal clique on $n$ vertices, a temporal $3$-spanner of size $n^{1+2/\sqrt{\ln n}}=n^{1+o(1)}$, thereby improving the previous best upper bound of $\widetilde{\mathcal{O}}(n^{3/2})$. We also show that a modified version of our algorithm computes temporal $3$-spanners of size $\mathcal{O}(nL)$ when the lifetime is bounded by $L$, i.e., all time labels are in $\{1,\ldots,L\}$, thus improving the previous bound of $\mathcal{O}(2^Ln\log n)$. Both results are particularly striking in light of the known lower bound of $\Omega(n^2)$ on the size of temporal $2$-spanners, which already holds for temporal cliques of lifetime $L\geq 3$. Both algorithms rely on a new simple recursive decomposition that certifies temporal connectivity for a large collection of source-target pairs using only $\mathcal{O}(n)$ carefully selected edges and recursively processes only the remaining pairs. Besides yielding substantially improved upper bounds, this approach is significantly simpler than previous constructions.

We construct realizable-volume models over every field for the factorial normalizations of homogeneous Lascoux, Lascoux-atom, and positive Grothendieck packets, including their minimal homogenizations and layers. In particular, the construction realizes the factorial normalizations of all Schubert and key polynomials and of the minimal sign-corrected homogeneous Grothendieck polynomials. The normalized polynomials are Lorentzian, and the ordinary supports are the lattice points of integral generalized polymatroids. On a Bott–Samelson tower, row and co-row filtrations assemble the local factors into globally generated bundles; a creation-state graph absorbs the remaining kernel factors by Chern flow. We also construct intrinsic algebras of joint Chern moments. Positive inverse-Chern presentations give these algebras Hard Lefschetz and Hodge–Riemann relations, and supply source-level Hodge completions of the packets. For globally generated tropical toric bundles in the sense of Kaveh–Manon, finite generating witnesses and matroid duality provide the presentations required by Larson–Partida's theorem, without representability. These constructions yield joint Chern-number inequalities, nonvanishing polymatroids, and equality criteria.

This article equips the universal moduli spaces of non-degenerate $\mathbb{Z}/2\mathbb{Z}$ harmonic spinors and eigenspinors with tame Fréchet manifold structures and proves that their natural projections to the corresponding parameter spaces are uniformly Fredholm. A unified construction yields the deformation theories of harmonic spinors and eigenspinors in dimension three and of chiral harmonic spinors in dimension four, recovering earlier work of Donaldson, Parker, and Takahashi. As an application, a deformation theory for non-degenerate $\mathbb{Z}/2\mathbb{Z}$ harmonic self-dual $2$-forms in dimension four is established.

A $B$-coloring of a graph is a proper edge-coloring in which every $4$-cycle is rainbow, and $q_B(G)$ denotes the minimum number of colors in such a coloring. Let $\Delta_2(G)$ denote the maximum number of common neighbors of two distinct vertices of $G$. We prove that, for integers $1\le d\le\Delta$, every finite simple $d$-degenerate graph $G$ with $\Delta(G)\le\Delta$ satisfies $$q_B(G)\le \Delta+(d-1)\Delta_2(G)\le d\Delta.$$ Consequently, $d\Delta$ is the exact maximum, with equality precisely for graphs containing $K_{d,\Delta}$. More generally, if $q_B(G)\ge d\Delta-s$, where $0\le s<\Delta$, then $G$ contains $K_{d,\Delta-s}$; if also $s<d$, then $G$ has at least $d-s$ vertices of degree $\Delta$ with the same open neighborhood. For $\Delta\ge3$, we further show that every $K_{3,\Delta}$-free 3-degenerate graph satisfies $q_B(G)\le3\Delta-2$; the example $K_{3,\Delta-1}$ shows that this bound is best possible up to one. For loopless multigraphs, we establish a sharp gap in the possible values of $q_B(G)$. For every integer $\Delta\ge3$, every finite loopless multigraph $G$ with $\Delta(G)\le\Delta$ satisfies $$q_B(G)\le\Delta(\Delta-1)$$ unless $G$ has a component isomorphic to $K_{\Delta,\Delta}$, in which case $q_B(G)=\Delta^2$. The bound $\Delta(\Delta-1)$ is attained by both $K_{\Delta,\Delta-1}$ and $K_{\Delta,\Delta}-e$. Consequently, among finite loopless multigraphs with maximum degree at most $\Delta$, no value of $q_B(G)$ lies strictly between $\Delta^2-\Delta$ and $\Delta^2$.

The exact N-soliton solutions are derived for the coupled negative first order Klein-Gordon (CNKG) equation subject to vanishing boundary conditions by using the inverse scattering method via Gelfand-Levitan-Marchenko equation. Based on the zero-curvature representation, the conservation laws, integrals of motion, and Hamiltonian structure of the aforementioned coupled nonlinear equations are constructed. The Jost functions and their analyticity properties for the Manakov spectral problem are recalled. The integral equations for the eigenfunctions are then used to formulate the Gel'fand-Levitan-Marchenko equations. Solving these equations establishes a direct correspondence between the kernel functions and the potential, yielding the general N-soliton expressions.

The first paper worked out one family of four lattices in full – $|\det| = 12, 24, 36$ and $72$. This paper generalizes it. We enumerate the 44 symmetrizable rank-three hyperbolic Cartan matrices and their 56 depth-one edits, 100 root bases in all, and compute the obstruction space $S_{5/2}(\rho_L)$ of the 98 within our weight-$5/2$ budget: 10 vacuous, 34 unobstructed, 54 obstructed. The vacuous ten are the lattices Bruinier, Ehlen and Freitag call simple, read in signature $(2,3)$. Of their fifteen, our ten realize five; of the rest, five need more than three generators and cannot be the discriminant form of a rank-three lattice at all, three fail $|\det L| = 2k^2$, and two are simply not root bases. Within the rank-three hyperbolic world the simple lattices are exactly the Feingold-Frenkel neighbours of index $k \le 4$. The discriminant group $L'/L$ carries a finite quadratic form, and the finite group of its isometries acts on the weight-$3/2$ cusp forms for $\rho_L$, the bottom antisymmetric rung. We survey the lattices on which that action is absolutely irreducible of dimension at least two, so that the Petersson pairing there is pinned down up to a single scalar. The condition alone cuts the census to three: $L_4$ of the first paper, and two new ones at $|\det| = 40$ and $88$. The quaternion discriminants that occur are 6, 10 and 22, the three for which the Shimura curve $X^D$ has genus zero. Rigidity uses no quaternion input, so we record it as an observation, not a characterisation. Both invariants of the first paper – the shadow norm $\|\Xi\|^2$ and the Petersson scalar $t$ – were single points there. Three rigid lattices instead of one are where their special faces come off: $\|\Xi\|^2$ generalizes against $L(f,1)$ where the first paper read $L(f,2)$, and $t$ against an elliptic curve's imaginary period where it read $\Gamma(1/3)$. Both numerically, to 26-58 digits.

We prove the global existence and uniqueness of strong solutions to a constrained version of the damped nonlinear wave equation $$ \vartheta_{tt}+\gamma \vartheta_t-\Delta\vartheta+|\vartheta|^{p-2}\vartheta=0 $$ on a smooth bounded domain $\varOmega\subset\mathbb{R}^d$, where the evolution is projected onto the tangent space of the Hilbert manifold $$\mathcal{M} = \left\{ \vartheta\in L^2(\varOmega):\|\vartheta\|_{L^2(\varOmega)}=1 \right\}, $$ which is the unit sphere in $L^2(\varOmega)$. We assume that $$p\in[2,\infty)\ \text{ for }\ d=1,2, \ \text{ while }\ 2\leq p\leq \frac{2(d-1)}{d-2} \ \text{ for }\ d\geq3.$$ By employing the Faedo–Galerkin approximation method, together with suitable a priori estimates and compactness arguments, we establish the global well-posedness of the problem. In particular, we show that the Hilbert manifold $\mathcal{M}$ is invariant under the flow, and hence the $L^2$-constraint is preserved throughout the evolution. Using the Lusternik–Schnirelmann theory, we show that the corresponding stationary problem possesses at least countably many stationary solutions. We further investigate the long-time behaviour of solutions and prove that, along a subsequence, every solution of the constrained problem converges to a stationary solution by invoking Webb's theorem and Barbalat's lemma. When the initial data are sufficiently close to the first eigenfunction of the associated stationary problem, we show that the unique strong solution converges in $H_0^1(\varOmega)$ to the unique positive ground-state solution.

This survey paper provides a comprehensive study of unitarily invariant norms on the algebra of $n \times n$ matrices. This investigation leads naturally to the theory of symmetric gauge functions, a class of norms on $\mathbb{R}^n$ characterized by invariance and monotonicity properties. We develop the necessary framework by examining absolute and monotone norms and establishing their equivalence, thereby offering additional insight into the classical Hardy-Littlewood-Pólya theorem on majorization. The theory of majorization is further explored through its connections with doubly stochastic matrices, convexity, and fundamental results such as the Birkhoff and Radó theorems, as well as König's theorem on term rank and line rank. We also study weak majorization and derive a characterization that plays a crucial role in proving the monotonicity of symmetric gauge functions. On the spectral side, we review key results in matrix analysis, including the Courant-Fischer min-max theorem, the Cauchy interlacing theorem, and Ky Fan's majorization theorem, along with a weak subadditivity result for singular values of arbitrary matrices. These ingredients culminate in a detailed proof of von Neumann's characterization of unitarily invariant norms, which provides a complete and elegant description of this class of norms. Some illustrative examples, as well as the Ky Fan domination principle as an application, are also presented.

We give an example showing that the dual of an algebraic matroid need not be algebraic. Our example is a matroid of rank 6 on 10 elements which is algebraic in characteristic 2. It is obtained by gluing an algebraic realization of the non-Fano matroid to a realization of the Fano matroid. Its dual is shown not to be algebraic using the Ingleton-Main lemma.