We investigate the probabilistic reasoning capabilities of large language models through a controlled benchmarking study on discrete probability problems. We constructed two datasets, respectively a set of standard exercises and a set of counterintuitive exercises, designed to trigger heuristic reasoning, and evaluated 8 state-of-the-art models, each tested with and without Chain-of-Thought prompting. Models achieve an average accuracy of 0.96 on standard problems but only 0.59 on counterintuitive ones. We further provide empirical evidence of token bias: performance drops by over 20% when canonical formulations are replaced by disguised variants. Embedding misleading suggestions in the prompt reduces performance by up to 34%, with no model proving immune. Taken together, the reported findings suggest that current LLMs are not yet genuine probabilistic reasoners, despite their success in advanced mathematical problems.

This manuscript contains a collection of counterintuitive problems in discrete probability, together with detailed solutions. The dataset was constructed as part of a broader research project investigating the capabilities of the latest-generation Large Language Models (LLMs) in solving discrete probability problems, in order to assess whether LLMs tend to make systematic reasoning errors associated with known cognitive biases. The problems collected here are specifically designed to challenge heuristic reasoning strategies that often lead to intuitively appealing but mathematically incorrect conclusions. The dataset combines several types of problems. Some are adapted from classical probabilistic paradoxes and cognitive-bias literature, while others originate from recreational mathematics sources or were developed by ourselves following similar principles. The primary purpose of this document is to provide a transparent and publicly accessible reference for the problems used in our experimental evaluation of language models, as well as providing detailed human-made solutions. At the same time, we believe that this collection may also prove useful for future research on probabilistic reasoning, cognitive biases, and the evaluation of reasoning capabilities in artificial intelligence systems.

We study two invariants, rate and slant, that describe the extremal behavior of the maximal shifts in the minimal free resolution of the residue field over a standard graded algebra. We provide bounds and computations of these invariants for a variety of rings, including compressed level algebras, rings defined by general forms and rings defined by monomials. We also prove an asymptotic additivity property of the maximal shifts for certain classes of rings, including Golod rings, which allows to interpret slant as a limit.

Let \(X\) be a smooth complex projective variety and let \(D=D^p+D^c\) be a simple normal crossing divisor, where \(D^p\) is a cusp divisor and \(D^c=\sum_{i\in I}D_i^c\) is a compact divisor whose components carry rational parabolic weights \(q_i/p_i\). We study the parabolic Higgs bundle \[ E_*=(\Omega_X^1(\log D^p)\oplus\mathcal O_X)_* \] whose only non-zero compact weights occur on the conormal lines of the components \(D_i^c\). The equality case of the parabolic Bogomolov--Gieseker inequality is formulated on the iterated root stack \(X[\sqrt[p_i]{D_i^c}]_{i\in I}\), obtained by taking the \(p_i\)-th root along every \(D_i^c\). We prove that equality produces a flat trace-free adjoint harmonic bundle, a principal \(\mathrm{PU}(n,1)\)-variation, and a period map to the unit ball. In root coordinates \(z_i=w_i^{p_i}\), its normal form is \[ \xi_i=u_i(w)w_i^{p_i-q_i}, \qquad u_i(0)\neq0. \] Thus the general equality case gives a branched complex-hyperbolic structure; it is an unramified orbifold ball uniformization exactly in the standard case \(q_i=p_i-1\) for every \(i\). Conversely, a branched complex-hyperbolic structure with this local normal form gives a mixed Poincaré--cone current representing \(c_1(K_X+\Delta)\); this class is big, nef, and \(\Delta\)-admissible. The induced Hodge metric gives parabolic polystability with respect to every admissible big and nef class, and the parabolic Chern equality.

While the positive Grassmannian is deeply understood through the rich combinatorics of plabic graphs and positroid cells, its tropical counterpart, the positive tropical Grassmannian Trop$_{>0}G(k,n)$, has lacked a comparable structural framework for general $k$. Both the global face structure of Trop$_{>0}G(k,n)$ and the internal metric geometry of the tropical linear spaces it parametrizes have remained largely uncharted. This paper develops a systematic algebraic and polyhedral foundation that resolves this gap. The engine of our framework is a fundamental tropical duality, analogous to the duality between cluster variables (or more precisely, their $u$-coordinates) and $\mathbf{g}$-vectors, pairing two families of objects introduced by the first author: the planar basis of tropical Plücker vectors and the planar cross-ratios on the positive configuration space. We prove that this duality links the fan structure of the positive tropical Grassmannian to the noncrossing fan of Santos, Stump, and Welker, yielding a global bijection between integer points of $Trop_{>0}G(k,n)$ and noncrossing tableaux. We then study how this discrete combinatorial data controls the continuous metric geometry of positive tropical linear spaces. We realize the bounded complex of an integer positive tropical linear space as the subdifferential of a central roof function on the hypersimplex, and use this realization to embed it into a dilate of the fundamental alcoved simplex. The dilation factor, and hence the geometric diameter of the complex, is governed by a single invariant, the planar kinematics ($K) weight, which we show equals the number of columns in the associated noncrossing tableau. The results of this work are applied in our parallel work on scaffolds for higher tropical Grassmannians.

Scaffolds are the one-dimensional skeleta of high-dimensional flag simplicial complexes of nonpositive curvature. They generalize the phylogenetic trees of Trop G(2,n) to arbitrary $k$, drawing together SL(k)-web bases, affine buildings, the combinatorics of the positive tropical Grassmannian and low-dimensional topology. We prove that scaffolds model points in all tropical Grassmannians via a $k$-point distance function. In this paper, we study in detail CAT(0) planar graphs, which are positive scaffolds for the tropical Grassmannian of three-planes. CAT(0) planar graphs are directed versions of the diskoids of Fontaine-Kamnitzer-Kuperberg, planar dual to SL(3)-webs. Our main result is the construction of a unique representation of any given integer positive tropical Plucker vector by a normal CAT(0) planar graph. We show that any normal CAT(0) planar graph embeds into the tropical linear space as a Lam-Postnikov membrane, and embeds into the Keel-Tevelev membrane within the affine building. We show that Early's planar basis expansion can be computed directly from the strand combinatorics of the dual web, and connect this expansion to Petersen-Pylyavskyy-Speyer's noncrossing tableaux, explored further in our companion paper.

The goal of the paper is to present two simple proofs of the Lagrange Inversion Formula for formal power series. Both proofs are non-external in the sense that they use concepts that do not go beyond the scope of basic tools of formal power series analysis.

We classify when a spin and a non-spin irreducible character of the alternating group have proportional 2-modular reductions. Equivalently, we classify when such a pair of characters are proportional on elements of odd order.

Let $\Gamma$ be an additive abelian group equipped with a commutative factor $\omega$. We describe the simple Lie colour algebras and the associated untwisted affine Lie colour algebras graded by $\Gamma$, which fulfil the Cartan-Weyl paradigm. The quantised universal enveloping algebras of these (affine) Lie colour algebras are constructed, which are colour analogues of the Drinfeld-Jimbo quantum groups including the latter as the special case of trivial $\Gamma$. We develop the quasi-triangular Hopf colour algebraic structure of these ``colour quantum groups'', which has immediate applications in areas such as knot theory and statistical mechanics.

We study two families of quandles arising from Coxeter quandles. One is the quandle defined by Andruskiewitsch-Graña, which is the set of roots with binary operation defined by using the negatives of reflections. We observe that this is realized as a conjugation quandle in a Pin group. The other, which we call a rotational $D_n$ quandle, is the set of some right angle rotations in the Coxeter group of type $D_n$ with binary operation given by conjugation. We determine their inner automorphism groups, and observe that they are quite similar.

We introduce a family of constants \[ C_m:= \lim_{n\to\infty} \left( \sum_{k=1}^n \operatorname{Li}_m\!\left(\frac1k\right) - \log n \right), \] which may be regarded as polylogarithmic analogues of Euler's constant. We study their basic properties and derive representations in terms of iterated logarithmic integral structures associated with the gamma function. We further introduce associated polylogarithmic zeta potentials and polylogarithmic gamma functions, establish differential relations and integral representations, and describe logarithmic branch asymptotics near the singular points. As an application, we relate the constants \(C_m\) to special values of certain Dirichlet series involving the Riemann zeta function.

In the design of computational methods for Bayesian inverse problems, costly forward model evaluations make it difficult to sample from or compute the posterior. This motivates the need for approximate forward models that are cheaper to evaluate. We consider reduced-order forward models which exploit the lower-dimensional structure in the Bayesian inverse problem by projecting to the "likelihood-informed subspace" of the parameter space where the prior-to-posterior update is significant. However, the theoretical properties of these reduced-order forward models and their impact on the solution of the Baysian inverse problem are not always well-understood. In this work we consider linear Gaussian inverse problems with a possibly singular prior covariance matrix. We analyse a recently proposed reduced-order model which uses a Petrov-Galerkin projection to likelihood-informed subspaces that arise in optimal low-rank approximations of the posterior covariance matrix. We bound the error in the resulting approximation of the root prior-preconditioned Hessian of the data misfit. Based on this we also bound the errors of the approximate posterior covariance and mean. Our analysis shows that this reduced-order model recovers the exact posterior when the rank of the reduced-order model is equal to the "intrinsic dimension" of the inverse problem, i.e. the rank of the prior-preconditioned Hessian. Two numerical experiments from structural engineering illustrate the performance of our bounds.

We construct a new family of $K_s$-free graphs that leads to improved lower bounds for Ramsey numbers across a wide range of parameters. For any fixed $s \ge 4$, we show that the off-diagonal Ramsey numbers satisfy $r(s, k) \ge k^{s-2 + o(1)}.$ For $s \ge 6,$ this improves the best known lower bound of the form $r(s, k) \ge k^{\frac{s+1}{2} + o(1)}$ which was first established by Spencer in 1977 and has since only seen logarithmic improvements. This nearly matches the best known upper bound which is of the form $r(s, k) \le k^{s-1 + o(1)}$ and which is widely believed to give the correct exponent. More generally, we show that if $s, k/s \rightarrow \infty$, then $r(s, k) = \left(\frac{k}{s}\right)^{(1+o(1)) s},$ where the upper follows from the seminal work of Erdős and Szekeres in 1935. We also obtain improved lower bounds for Ramsey numbers extremely close to the diagonal as well as for diagonal multicolor Ramsey numbers.

Many problems in computational science and engineering become one-to-many after coarse graining, partial observation, or inverse reconstruction: a resolved state may not determine a unique subgrid forcing, a structural descriptor may not determine a unique effective response, and a low-resolution observation may correspond to many plausible high-resolution fields. In such settings, deterministic surrogates may learn a well-defined mathematical object while still missing application-relevant uncertainty. This tutorial develops a self-contained module centered on the conditional-mean barrier: the point at which a squared-loss predictor has reached the conditional mean and the remaining error is irreducible aleatoric variance. We give two diagnostics for locating this barrier, residual-feature orthogonality and the coefficient of determination against its explained-variance ceiling, and prove that adding latent randomness to a squared-loss predictor collapses it back to the conditional mean. Crossing the barrier therefore requires a loss that scores distributions rather than point predictions. We briefly organize common distributional objectives, including negative log-likelihood, moment and observable matching, variational objectives, adversarial divergences, and score matching, by the feature of the conditional law each targets. The emphasis is the boundary itself and a finite-data procedure for recognizing it, rather than a survey of methods beyond it. CPU-based demonstrations on a two-branch law and a two-scale Lorenz-96 closure problem show how the diagnostics distinguish deterministic underfitting from residual distributional variability.

We show that every stable UCT Kirchberg algebra has a principal étale groupoid model, and thus contains a C$^*$-diagonal. Every unital UCT Kirchberg algebra $A$ for which $[1_A]_0$ has infinite order in $K_0(A)$ is also covered by our methods. In particular, we obtain a principal étale groupoid model for the Cuntz algebra $\mathcal{O}_\infty$.

Simulations of chaotic systems can only produce high-fidelity trajectories if the initial and boundary conditions are well specified. When these conditions are unknown but measurements are available, variational state estimation can reconstruct a trajectory that is consistent with both the data and the governing equations. A key open question is how many measurements are required for accurate reconstruction, making the full system trajectory observable from sparse data. We establish observability criteria for variational state estimation applied to the Kuramoto-Sivashinsky equation by linking its observability to embedding theory for dissipative dynamical systems. For a system whose attractor lies on an inertial manifold of dimension $d_M$, we show that $m \geq d_M$ measurements ensures local observability from an arbitrarily good initial guess, and $m \geq 2d_M + 1$ implies global observability using a gradient-based smoother since the only critical point on $M$ is the global minimum. We also analyze optimization-dynamic limitations that persist even when these topological conditions are met, including drift off the manifold, degeneracy of the Hessian, negative curvature, and vanishing gradients. To address these issues, we introduce a robust reconstruction strategy that combines non-convex Newton updates with a novel pseudo-projection step. Numerical simulations of the Kuramoto-Sivashinsky equation validate our analysis and show practical limits of observability for chaotic systems with low-dimensional inertial manifolds.

We prove real-rootedness for the Poincaré polynomial \[ P_n(t)=\sum_{i=0}^{n-3} \dim H^{2i}(\overline{\mathcal M}_{0,n};\mathbb{Q})t^i \] of the Deligne--Mumford moduli space $\overline{\mathcal M}_{0,n}$ of stable $n$-pointed rational curves, proving a conjecture of Aluffi--Chen--Marcolli. The proof starts from the Keel--Manin--Getzler recurrence, but its main new idea is a bivariate deformation $F_m(y,t)$ of the Poincaré polynomial. This deformation reveals a hidden interlacing structure not visible in the one-variable recurrence. For fixed $t<0$, the zero set of $F_m$ in the $y$-direction is controlled by a Sturm--Rolle argument on the interval $0<y<1-t$. The original polynomial is recovered on the slice $y=1$, and the ordered crossings of the moving roots through this slice give both real-rootedness and strict interlacing. Consequently, the Betti numbers of $\overline{\mathcal M}_{0,n}$ form an ultra-log-concave sequence. We further prove real-rootedness and ultra-log-concavity for the Poincaré polynomial of the Fulton--MacPherson space $\mathbb{P}^1[n]$ of $n$ ordered points in degenerations of the complex projective line. The proof for $\overline{\mathcal M}_{0,n}$ was obtained through an iterative AI-assisted workflow with Co-Mathematician, an agentic frontier-model system developed by Google DeepMind. Our role was to formulate the problem, evaluate the proposed proof attempts, identify gaps and request corrections, compare the developing argument with the literature, and refine the presentation of the final proof. Our additional human contribution was to observe that a similar residual deformation strategy applies to the Fulton--MacPherson spaces $\mathbb P^1[n]$, yielding the corresponding real-rootedness theorem.

Let X be an affine variety. The local finiteness of a Lie subalgebra h of Lie(Aut(X)) is equivalent to the existence of an algebraic subgroup G of Aut(X) such that h is contained in Lie(G). Let h be a solvable Lie subalgebra of Lie(Aut(X)) generated by a finite collection of locally finite Lie subalgebras. The authors of [ arXiv:2507.09679 ] wondered whether h is itself locally finite. After presenting some criteria for the local finiteness of h, we answer this question in the affirmative in the particular case where X is the affine plane.

The geometrically defined wide class of time-like surfaces in $\mathbb R^3$, admitting real asymptotic lines is considered. A fundamental theorem of Bonnet-type is obtained for these surfaces. It states that a surface in this class is determined (up to a motion) by four invariant functions, satisfying some natural PDEs. Then canonical parameters are defined for these surfaces and it is proved that such a surface is determined (up to a motion) in canonical parameters with only two invariant functions (which in particular can be the Gauss and the mean curvature), satisfying a partial differential equation, equivalent to the Gauss equation.

This paper develops a unified explicit solution theory for optimal execution through sequential limit-order placement in a limit order book. Rather than controlling only the trading speed of a metaorder, we determine how individual limit orders should be quoted over time. The model incorporates signal-dependent drift, price impact, inventory risk, and execution risk, with fills modeled by point processes whose intensities depend on the submitted quotes. We formulate four execution criteria: expected terminal wealth, expected terminal wealth with running inventory penalty, CARA utility of terminal wealth, and CARA utility with running inventory penalty. For general price-impact and inventory-penalty functions, we derive the corresponding HJB equations and show that all four problems reduce to a triangular finite-dimensional structure which can be solved explicitly, leading to fully explicit value functions and optimal quotes across all cases. We also prove well-posedness, admissibility, and verification results. The explicit formulas reveal connections between quoting strategies under different criteria, support long-horizon asymptotic analysis, and show numerically that signal-dependent drift can substantially affect optimal execution.

Barycentric algebras are an abstraction of the notion of convex sets, defined by a set of equations. We study semitopological and topological barycentric algebras, in the spirit of a previous study by Klaus Keimel on semitopological and topological cones (2008), which are special cases of semitopological and topological barycentric algebras. For example, the space of all continuous valuations (a very close cousin of measures) over a topological space is a topological cone, while probability valuations form a topological barycentric algebra, and subprobability valuations form a pointed topological barycentric algebra. Among other results, we show the existence of free semitopological cones over semitopological barycentric algebras and over pointed semitopological algebras, we investigate which semitopological barycentric algebras embed into semitopological cones and which pointed semitopological barycentric algebras embed strictly into semitopological cones. We study notions of local convexity, which split into weak local convexity, local convexity, local affineness and local linearity. We show that the weakly locally convex topological barycentric algebras are exactly the affine retracts of locally affine topological barycentric algebras. On locally convex barycentric algebras, we show sandwich theorems, extending theorems by Roth and Keimel on cones. A running theme of this paper is the notion of barycenters, which we progressively generalize until we reach a general notion of barycenters of continuous (resp., subprobability, probability) valuations, inspired by a definition of Choquet. We conclude with a general barycenter existence theorem, whose proof relies on the study of the Smyth poweralgebra, namely the topological barycentric algebra of all non-empty convex compact saturated subsets of a topological barycentric algebra.

In this paper, we present a novel numerical framework for solving a specific biological reaction-diffusion-advection system of cancer growth in three dimensions (3D) using particles of variable mass. We adopt empirical particle measures to represent cell density and dynamically construct the concentration fields of multiple related chemical species throughout the 3D domain. Efficient interaction between the particles and the spatial grid is achieved through a Particle-in-Cell (PIC) algorithm, while diffusion in space is solved rapidly using a spectral method. We demonstrate that for this particular system, the rate of change of particle mass remains bounded over finite time intervals. Furthermore, in addition to the inherent positivity preservation of cell density guaranteed by the empirical particle measures, the concentrations constructed by the algorithm are also unconditionally positivity-preserving on the spatial grid. Moreover, we present a rigorous error analysis for the proposed method, and numerical experiments confirm the theoretical convergence rates. To the best of our knowledge, this is the first numerical work to solve this system in three dimensions, wherein a rapid spread of cells driven by haptotactic flux is observed, similar to the behavior documented in the two-dimensional case.

We study the sticky Cox--Ingersoll--Ross (CIR) process in one dimension, a diffusion on $[0,\infty)$ with a sticky boundary condition at the origin, arising as the marginal process in a sparse Bayesian inference framework based on Hadamard--Langevin dynamics. For the parameter range $\delta\in(1,2)$, in which the origin is accessible but not absorbing, we prove well-posedness of the process and uniqueness of its invariant measure, which is a mixture of a point mass at zero and a weighted gamma-type density on the interior. We derive an explicit Green's function for the resolvent in terms of confluent hypergeometric functions, and use this to construct an exact sampler for the invariant measure in the zero-potential case. For a non-trivial potential $G$, we establish existence and uniqueness of the tilted invariant measure via a Girsanov change of measure, and develop two sampling algorithms: a Metropolis--Hastings corrected sampler that targets the invariant measure exactly, and a cheaper, biased unadjusted Langevin algorithm (ULA) for a boundary-clamped variant of which we prove a first-order expansion of the stationary bias with an explicit constant: the leading error is a rank-one transfer of mass $K_\star h|\log h| $ onto the atom, so the total-variation bias is of exact order $h|\log h | $ -- independent of $\delta$ -- whenever the potential has nonzero boundary drift. Numerical experiments confirm the predicted behaviour: the Metropolis--Hastings sampler achieves the target invariant measure at all step sizes, while the ULA bias follows the proven first-order law, including its constant.

We introduce a de Rham-Hodge framework induced by a vector field on a compact, oriented smooth manifold. Using a vector field induced bundle isomorphism on differential forms, we define a vector field induced Hodge $L^2$-inner product, codifferential, and Hodge Laplacian. Unlike classical deformations, such as the drifting and Witten-type Hodge Laplacians, the induced Laplacian modifies the principal symbol and gives rise to an anisotropic Laplace-Beltrami type operator on functions. We establish the resulting de Rham-Hodge theory for closed manifolds, including the ellipticity of the induced Hodge Laplacian and the corresponding Hodge decomposition and isomorphism results. We further extend the framework to manifolds with boundary by imposing certain vector field induced boundary conditions, which are necessary to restore the adjointness between the differential and induced codifferential, and to obtain a well-posed boundary value problem. Under these boundary conditions, we establish analogues of the Hodge-Morrey and Friedrichs decompositions. We also discuss several structural properties of the framework, including its relation to anisotropic Laplace-Beltrami operators, its spectral behavior in several explicit examples, and its invariance under isometries.

Strategic classification studies learning settings in which individuals can modify their features, at a cost, in order to influence the classifier's decision. A central question is how the sample complexity of the induced (strategic) hypothesis class depends on the complexities of the underlying hypothesis class and the cost structure governing feasible manipulations. Prior work has shown that in several natural settings, such as linear classifiers with norm costs, the induced complexity can be controlled. We begin by showing that such guarantees fail in general - even in simple cases: there exist hypothesis classes of VC dimension $1$ on the real line such that, even under the simplest interval neighborhoods, the induced class has infinite VC dimension. Thus, strategic behavior can turn an easy learning problem into a non-learnable one. To overcome this, we introduce structure via a geometric definability assumption: both the hypothesis class and the cost-induced neighborhood relation can be defined by first-order formulas over $\mathbb{R}_{\mathtt{exp}}$. Intuitively, this means that hypotheses and costs can be described using arithmetic operations, exponentiation, logarithms, and comparisons. This captures a broad range of natural classes and cost functions, including $\ell_p$ distances, Wasserstein distance, and information-theoretic divergences. Under this assumption, we prove that learnability is preserved, with sample complexity controlled by the complexity of the defining formulas.

We exhibit an equigenerated monomial ideal $I\subseteq K[x,y,z,w]$ with $\operatorname{reg}(\overline{I})>\operatorname{reg}(I)$. The ideal $I$ is generated in degree 4 and satisfies $\operatorname{reg}(I)=4$, while its integral closure $\overline{I}$ has a minimal generator of degree 5 and satisfies $\operatorname{reg}(\overline{I})=5$. This gives a counterexample to the polynomial-ring formulation of the Küronya--Pintye conjecture.

Soft, slender structures are ubiquitous in natural and engineered systems, with broad application potential from biomimetic materials to soft robotics. However, there is a notable lack of computational tools that simultaneously preserve high-fidelity continuum rod mechanics, scale to large interacting ensembles, and remain flexible across diverse biophysical settings. Here we introduce Elastica++, an open-source, high-performance implementation of the Cosserat-rod model for large-scale simulations of slender-body dynamics. Elastica++ combines performance-oriented kernels with shared-memory parallelism to sustain teraflop-scale throughput despite complex discretization domains and physical interactions. The framework further interoperates with external numerical solvers, supporting efficient multiphysics workflows. We demonstrate robustness and breadth through case studies spanning passive nest-like metamaterials, collective active-matter dynamics, cilia carpets, soft magnetic microrobots, and schooling swimmers. Elastica++ thus provides a missing foundation for high-throughput studies of emergent behavior in interacting assemblies of elastic slender structures.

Two constructions of the Darboux-Halphen system are discussed. In the Jacobi construction we start with transcendental functions which are fixed as the first integrals. In the Lie construction we use a single-valued representation of the simple Lie algebra $sl(2,\mathbb R)$ which is non-integrable in Lie's terminology.

Unconstrained binary integer programming (UBIP) is a challenging optimization problem due to the presence of binary variables. To address the challenge, we introduce a novel class of functions named sharp-peak functions (SPFs), which equivalently reformulate the binary constraints as equality constraints, giving rise to an SPF-constrained optimization. Rather than solving this constrained reformulation directly, we focus on its associated penalty model. The established exact penalty theory shows that the global minimizers of UBIP and the penalty model coincide when the penalty parameter exceeds a threshold, a constant independent of the solution set of UBIP. To analyze the penalty model, we introduce Karush-Kuhn-Tucker (KKT) points and a new type of stationarity, referred to as P-stationarity, and provide a comprehensive characterization of its optimality conditions. We then develop an efficient algorithm called Sha-Peak based on the inexact alternating direction method of multipliers. It converges toa P-stationary point at a linear rate or terminates at such a point within finitely many steps. These results are established under appropriate parameter choices and a single mild assumption, namely, the local Lipschitz continuity of the gradient over a bounded box. Finally, numerical experiments demonstrate its nice performance in comparison to several established solvers.

It is proven that every cohomology class of the moduli space $M_{g,m}$ for any $2g+m\geq 3$, $m\geq 1$ can be represented combinatorially by a ribbon quiver with at most four-valent vertices. The "at most four"-valency condition is sharp.