Papers

We study new algorithms for Contextual Bandits with Knapsack. In these problems, there are finitely many types of customers, products, and resources. Each product is made from a fixed combination of resources, and resources have finite capacity. A decision maker must assign each arriving customer one out of a set of multiple possible products. Every assignment of a customer to a product will generate a random reward, which equals an unknown linear function of customer and product features, plus a noise term. The objective is to jointly learn the mean reward function, and to make online assignments to minimize the expected revenue loss relative to an optimal policy that knows the reward function. We propose a natural and simple extension of the Upper-Confidence-Bound (UCB) family of algorithms and apply re-optimization techniques. We show that by taking advantage of re-optimization, our algorithm achieves an average regret of $O(\frac{(\ln T)^3}{T})$ where $T$ is the horizon length. Our bound significantly reduces the $O(\frac{1}{\sqrt{T}})$ bound in the literature for closely related dynamic-pricing problems that are based on re-optimization.

Let $\Omega(n)$ denote the number of prime factors of $n$, counted with multiplicities. We study the set $Acc^\Omega(x)$ of weak-$^*$ limits of the sequence $\frac{1}{N}\sum_{n\leq N}\delta_{T^{\Omega(n)}x}$ in $\sigma$-compact dynamical systems $ (X,T)$, demonstrating that if $x \in X$ is quasi-generic for an ergodic measure $\mu$, then $\mu \in Acc^\Omega(x)$. This extends a result of Bergelson and Richter, who studied the problem in the setting of uniquely ergodic systems. We give a more precise description of the set $Acc^\Omega(x)$ in the case of the horocycle flow on non-compact quotients of $SL(2,\mathbb{R})$. We show that for every non-periodic $x\in X$, in addition to Haar measure, there exists sequences $(s_n), (c_n) \subseteq \mathbb{R}$ such that $$ \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-\frac{r^2}{2}}\nu^{i}_{s_n-2\log|1+c_nr|} dr\in Acc^{\Omega}(x), $$ where $\{ \nu^{i}_{s} \}_{i \leq k}$ denotes the one parameter family of periodic measures in each of the $k$ inequivalent cusps. Depending on Diophantine properties of the non-periodic point $x$, we show that $Acc^\Omega(x)$ contains a full two parameter family of such periodic measures, as well as the Dirac measure at each cusp. In particular, these results yield almost-everywhere divergence of pointwise averages along $\Omega(n)$ for the non-compact horocycle flow.

We consider the problem of identifying the paths most likely to occur as geodesics in last passage percolation. Heuristics suggest that these modal paths should be the extreme corner paths, going straight between the corners of the cube of accessible vertices. We identify three mechanisms which favour such corner paths, and show that they are more likely to appear than all but a vanishingly small portion of paths. We make more specific comparisons in exponential last passage percolation, where moderate deviation estimates may be used to show that corner paths are nearly modal in a precise sense. Finally, we show a form of monotonicity of geodesic probabilities in a special case and conjecture that this holds in general.

We show that for any closed atoroidal 3-manifold $M$, there are only finitely many isotopy classes of links that can arise as singular orbits of pseudo-Anosov flows on $M$.

One of the most effective approaches to computations in algebraic $K$-theory is trace methods, which compare algebraic $K$-theory with topological Hochschild homology and topological cyclic homology. In recent work, two of the authors, together with Gerhardt, construct an equivariant refinement of topological Hochschild homology ($\mathrm{ETHH}$) which receives a trace map from Merling's genuine equivariant algebraic $K$-theory. In this paper, we perform foundational computations of $\mathrm{ETHH}$ that can serve as input for future computations of $\mathrm{ETHH}$ and equivariant topological cyclic homology. Namely, we compute $\mathrm{ETHH}(H\underline{\mathbb{F}}_p)$ for odd primes, showcasing the complexity of Bökstedt periodicity in this setting. Furthermore, we give computations of $\mathrm{ETHH}$ for the equivariant complex cobordism spectra $MU_G$ and $MU_{\mathbb{R}}$.

We study the landscape function on a discretized interval. The discrete landscape has an explicit closed form, and its spectral coefficients can be computed exactly. We show that these coefficients lie in explicit abelian extensions of $\mathbb{Q}$ and obtain the bound $$[\mathbb{Q}(c_k(N)):\mathbb{Q}]\leq\varphi(N)$$ for every odd $k$. The Chandra–Jain conjecture, asserting that $[\mathbb{Q}(c_1(N)):\mathbb{Q}]=\varphi(N)$ for all $N\geq3$, has since been proved by Q. Zhou (Zenodo, doi:10.5281/zenodo.21935814). We then reduce higher modes to the coprime case. Exact computations for all 624 pairs with $3\leq N\leq50$ reveal only the degrees $\varphi(N/d)$ and $2\varphi(N/d)$ for non-coprime modes, where $d=\gcd(N,k)$. This motivates the Second Chandra–Jain Conjecture that this degree dichotomy holds generally. We also discuss connections with parity, the Arnold cat map, and Lefschetz numbers.

We investigate the collision dynamics of two bright sphalerons in a (1+1)-dimensional deformed $\phi^6$ scalar field theory with a symmetric potential possessing false vacua. Two one-parameter realizations of the model, referred to as the barrier and well models, are considered and their static and linear instability properties are first reviewed. We then study head-on collisions of boosted sphalerons over a broad range of initial velocities and deformation parameters. The scattering dynamics exhibit a rich variety of final states, including the production of kink-antikink pairs, long-lived oscillons in true and false vacuum, multiple oscillons propagating in false-vacuum regions, and radiative decay. A particularly remarkable outcome is the emergence of a long-lived bubble of the false broken vacuum bounded by a kink-antikink pair, which repeatedly collapses and re-expands before eventually decaying into an oscillon. These results demonstrate that deformed $\phi^6$ theories with false vacua exhibit considerably richer sphaleron dynamics than previously known and provide new insight into the role of unstable localized configurations in nonlinear field theories.

The classical multinomial, negative multinomial, hypergeometric, and negative hypergeometric distributions are naturally organized by two features of the sampling scheme: sampling with or without replacement and stopping after a fixed number of draws or a fixed number of failures. We complete the graphical analogue of this scheme for decomposable graphs by adding graphical hypergeometric and graphical negative hypergeometric distributions to the previously introduced graphical multinomial and graphical negative multinomial models in Danielewska et al. (2025). The resulting four families provide a unified parametric framework for graphical modeling of multivariate count data, in which dependence and admissible configurations are encoded by a graph. They interpolate between products of univariate distributions for the empty graph and the corresponding classical multivariate distributions for the complete graph, while retaining explicit Markov factorizations and tractable sampling representations. We further develop a unified Bayesian hierarchy based on graphical Dirichlet-type distributions, obtaining explicit posterior and predictive laws. The framework is particularly natural for count data arising under exclusion or incompatibility constraints. We discuss several such applications and illustrate its practical potential using Rydberg-atom excitation data.

We develop a recursive algebraic framework for the compatibility matrices arising in moving-boundary problems for the heat equation. Such matrices naturally appear in the analysis of boundary trace operators, including Dirichlet-to-Neumann maps and first-passage problems for Brownian motion. The construction is based on two recursive hierarchies: transported differential operators in the interior and boundary operators generated by successive differentiation of the boundary condition. Their interaction produces a recursive family of compatibility matrices whose size grows linearly with the order of the transported operator. The main contribution of the paper is a decomposition of every compatibility matrix into a universal reference matrix and a sequence of transport perturbations. This representation isolates the boundary geometry from the recursive transport corrections and shows that only the transport rows change at each recursive level. Consequently, the determinant of the compatibility matrix may be interpreted as the determinant of a reference matrix together with successive low-rank perturbations. The recursive construction is illustrated through explicit examples, which suggest additional algebraic structure in the reference determinants. The resulting recursive representation provides a systematic organization of compatibility matrices and establishes an algebraic foundation for future investigations of recursive determinant formulas, boundary trace operators, and Dirichlet-to-Neumann maps for moving-boundary problems.

The computability of real numbers and functions using Turing Machines has been a central area of theoretical computer science since the mid-20th century. In the late 20th century, it was shown that chemical reactions can serve as a basis for computation using the Chemical Reaction Network (CRN) model. Recent advances in computing real numbers using Deterministic Chemical Reaction Networks (DCRNs) have identified numerous classes of DCRN-computable real numbers. In parallel, the works of R. Landauer and C. H. Bennett, spanning the 1960s to the early 2000s, showed that reversible computing offers significant advantages over irreversible methods, particularly in energy efficiency, motivating extensive research on reversible computation. In this work, we investigate the computability of real numbers using Reversible Chemical Reaction Networks (RevCRNs). The paper has two primary contributions: (1) establishing relationships among CRN-computable real number classes including Lyapunov CRN ($\mathbb{R}_{LCRN}$), Real-Time CRN ($\mathbb{R}_{RTCRN}$), rational numbers ($\mathbb{Q}$), and RevCRNs ($\mathbb{R}_{RevCRN}$), with key results: (i) $\mathbb{Q}$ is a strict subset of $\mathbb{R}_{RevCRN}$; (ii) the set of positive algebraic numbers ($ALG$), $\mathbb{R}_{LCRN}$, and real numbers computable by 1-species RevCRN ($\mathbb{R}_{RevCRN}^{1s}$) are equal; (iii) $\mathbb{R}_{RTCRN}$ and $\mathbb{R}_{RevCRN}$ exhibit non-empty overlap; and (iv) the set of real numbers computable by detailed-balanced RevCRNs ($\mathbb{R}^{DetBal}_{RevCRN}$) is a subset of $ALG$; and (2) exploring the existence of a hierarchy within $\mathbb{R}_{RevCRN}$. Finally, we leave open the exact relationship between $\mathbb{R}_{RevCRN}$ and $\mathbb{R}_{RTCRN}$ while conjecturing a general hierarchy of RevCRN-computable reals.

The time-optimal solution for a double integrator is a foundational result: its time-displacement law $T_r = 2\sqrt{L}$ for a normalized step $L$ is structure that numerical methods cannot provide. We solve the time-optimal control for rest-to-rest maneuvers of a double integrator coupled to a harmonic oscillator, deriving the first analytical solution and closed-form time-displacement law. Synthesis reduces to a single scalar inversion. This is the model to which the two-mass-spring system, the one-bending-mode flexible structure, and the linearized overhead crane reduce. A Pythagorean identity $T^2 = T_r^2 + 2T_s^2$ decomposes the optimal maneuver time into the rigid-body minimum and a synchronization time $T_s \leq \pi$, the cost of flexibility, giving the sharp envelope $2\sqrt{L} \leq T \leq 2\sqrt{L + \pi^2/2}$. The penalty is scale-dependent. For small maneuvers $T \propto L^{1/4}$: one oscillator mode costs as much as two additional integrators. For large ones, flexibility is asymptotically free. It vanishes at the natural motions, where the oscillator completes whole cycles. A closed-form sensitivity formula proves the maneuver time non-monotone in stiffness. Adding flexibility to a rigid body never shortens a maneuver, yet softening an already flexible structure can: stiffer is not always faster. The natural motions are robust design targets since first-order sensitivity to stiffness vanishes there.

In our previous article [arxiv:2410.05120], we introduced the notion of a finite dimensional 3-Hilbert space, categorifying Baez's 2-Hilbert spaces. In this article, by further categorifying Baez's higher linear algebra, we provide useful tools for working with 3-Hilbert spaces, including, generalized scalar multiplication, orthonormal bases, and unitary adjoints for operators. We use these tools to endow the $\mathrm{C}^*$-3-category of 3-Hilbert spaces with a self-enrichment. We prove a Unitary Yoneda Lemma/Riesz Representation Theorem for 3-Hilbert spaces: the Yoneda embedding is an isometric equivalence. Finally, we define a unitary version of the Deligne product on 3-Hilbert spaces and prove that it satisfies an isometric version of the folding trick.

One of the challenges of the unitary dual problem is the daunting number of possible representations to consider. This article describes an approach to narrowing the search space for minimal principal series representations, in terms of the “fundamental parallelepiped” (or “FPP”): the set of linear combinations of fundamental weights with coefficients in the interval $[0,1]$. An earlier conjecture of the author, which was rooted in work of Barbasch and then subsequently vastly generalized by Vogan (and recently proven by Davis and Mason-Brown), asserts that the FPP houses all dominant infinitesimal characters for which minimal principal series have a unitarizable quotient. We present a technique to prove this “FPP inequality” in specific examples, demonstrated here for the split real form $E_{8(8)}$, by introducing a limiting theory of intertwining operators as $\nu$ approaches $\infty$ in directions of fundamental weights. This limiting theory is inspired by Wilfried Schmid's work on variation of Hodge structure. As an application, we show that the unitary set for a particular minimal principal series consists of the closure of a single open alcove.

Maximum-angle conditions are standard finite-element mesh hypotheses that permit anisotropic triangles excluded by minimum-angle or shape-regularity assumptions. For exact spherical triangles, however, the angles of the chordal affine core and the intrinsic spherical angles need not coincide, while radial geometry introduces curvature-scale distortion. We give an algebraic characterisation of the intrinsic spherical maximum-angle condition through a dimensionless quantity computed from the three vertex vectors. A uniform positive lower bound on this quantity is equivalent to a uniform spherical maximum-angle bound and requires neither spherical-angle nor spherical-area evaluation. We identify the support-plane geometry linking the chordal circumradius to radial distortion and derive sharp comparisons between the intrinsic spherical and chordal semi-regularity parameters. In particular, a locality-independent comparison holds with sharp constant $2/\sqrt3$ and a characterised equality case. An area-based parameter is shown to be smaller than the spherical semi-regularity parameter, with sharp constant one in the local flat limit. For spherical finite-element meshes, the vertex criterion implies uniform chordal semi-regularity with an explicit constant, while relative refinement makes the radial-distortion factors converge uniformly to one. Thus, the intrinsic criterion, together with relative refinement, provides the geometric controls used in anisotropic finite-element analysis without imposing a minimum-angle or shape-regularity condition.

The odd Vassiliev conjecture asserts that all Jacobi diagrams with an odd number of legs vanish modulo the AS and IHX relations. Equivalently, the conjecture asserts that Vassiliev invariants cannot distinguish a knot from its inverse. The conjecture is elementary for one 1 or 2 loops, and was proven by Moskovich and Ohtsuki for 3 loops. We establish the conjecture for 4 loops. Our proof uses Jacobi diagrams and weight systems with polynomial coefficients, commuting colored legs, and several vanishing criteria.

Answering a question of Todorcevic, we prove higher-dimensional canonization theorems for the topological Ramsey space $\mathrm{FIN}^{[\infty]}_2$. Our results build upon the work of Lopez-Abad and continue the line of research initiated by Erdős and Rado, and further developed by Pudlák and Rödl, Prömel and Voigt, Taylor, and Klein and Spinas. We identify the canonical functions on fronts of $\mathrm{FIN}^{[\infty]}_2$ and develop an extension of the separating-mixing technique of Prömel and Voigt to canonize arbitrary functions $g:\mathcal{F}\to\omega$, where $\mathcal{F}$ is a front of $\mathrm{FIN}^{[\infty]}_2$. We further extend our canonization theorem to arbitrary Borel maps $g:\mathrm{FIN}^{[\infty]}_2\to\mathbb{R}$, establishing new canonical Ramsey theorems for Polish spaces. In particular, we provide a complete classification of the canonical functions on $\mathrm{FIN}^{[n]}_2$, together with an explicit formula for their number as a function of $n$.

Morphological calculus represents decompositions of geometric objects by polynomial-like expressions in a symbol for the real line, but scalar operations suppress the incidence and attachment maps needed to reconstruct the space. We formulate a finite semantics using script chain complexes. The cell-count polynomial satisfies $$ \mathcal M_S(t)=\mathcal P_{S,\mathbb K}(t)+(1+t)\mathcal B_{S,\mathbb K}(t), $$ where $\mathcal P$ is the Poincar'e polynomial and $\mathcal B$ records boundary ranks. Over $\mathbb Z$, Smith labels separate unit pairs, representing refinement overhead, from non-unit pairs carrying torsion. We distinguish Cartesian products from bundles, derive the monodromy defect for mapping tori, and replace scalar gluing by a finite bar construction. Mapping cones, joins, and double mapping cylinders arise as reduced models. We apply the construction to bivector discrepancies in Sommen's calculations. In dimension four, Hodge decomposition identifies the unit bivector sphere with $S^2*S^2$; the angular excess is a contractible relative complex from inserting the rank-two midpoint. In dimension five, the $SO(5)$-action on $S(\Lambda^2\mathbb R^5)=S^9$ has principal orbit $SO(5)/T^2$ and singular orbits $\widetilde G_2(\mathbb R^5)$ and $\mathbb{CP}^3$. For standard Bruhat cell structures, the reduced double-mapping-cylinder inventory differs from sphere homology by seven unit-labelled pairs. We also treat Borel–Moore realizations of selected noncompact symbols, define division as an action-certified partial operation, and describe the orbit-type face diagram for unit bivectors in dimension six. There the correction polynomial has nonnegative coefficients, while compatibility of the face attachments remains open. These examples separate subdivision and attachment data from quotient actions and support conditions suppressed by scalar notation.

This paper proposes a contextual quality-diversity evolutionary reinforcement-learning controller, CQD-ERL, for the supervisory control of a tropical, water-cooled chiller plant and its associated air side. Rather than converging to a single scalarised policy, the controller maintains a product archive of specialised policies indexed jointly by a data- driven operating context, a cluster of daily weather and load regime, and a context-invariant behaviour descriptor, filled by a gradient-free evolutionary operator and a soft-actor-critic policy-gradient operator that share one replay buffer. Every action is filtered through a deterministic safety shield before execution. The controller is trained on a two-tier reduced-order environment representing the latent load, cooling-tower approach and humidity constraints of a Singapore commercial building, and is evaluated over a full annual backtest against an ASHRAE Guideline 36 baseline.

We study spectral clustering in the presence of a confounding latent geometry. The leading eigenvectors may then be dominated by the latent geometry rather than by the communities. Nevertheless, we show in a block latent-space model that communities can be recovered from eigenvectors deeper in the spectrum. We analyze the spectral properties of the adjacency matrix through a limiting integral operator and use its structure to develop DBSPEC, a density-based spectral clustering algorithm that requires only approximate localization of the informative eigenvalue and is robust to poor eigenvalue separation. Crucially, this approach handles general latent geometries, overcoming restrictions to homogeneous toroidal models in prior works. Our theoretical predictions for the location of the informative eigenvalue notably align with observations in real-world experiments.

A classical theorem due to Friedgut, Kalai and Naor asserts that if a function $f\colon \{0,1\}^n\to\{-1,1\}$ close to a degree $1$ function, then either $f$ or $-f$ is close to either the all $1$ function, or to $(-1)^{x_i}$ for some $i\in [n]$. We prove a version of their theorem for the Grassmann scheme over $\mathbb{F}_2$. More precisely, we prove if a function $f\colon \genfrac{[}{]}{0pt}{}{\mathbb{F}_2^n}{\ell}\to\{0,1\}$ is close to a degree $1$ function, then either $f$ or $1-f$ must be close to a function of the form $g(L) = \sum_{x\in\mathcal{X}}1_{x\in L}+\sum_{W\in\mathcal{W}}1_{L\subseteq W}$, where $\mathcal{X}\subseteq\mathbb{F}_2^n$ is a set of points and $\mathcal{W}$ is a set of hyperplanes in $\mathbb{F}_2^n$.

Coherent states bridge the gap between quantum and classical physics, but their overcomplete and nonorthogonal nature makes it difficult to identify the minimal discrete set needed to reconstruct quantum information. Finite spin-coherent tomography and discrete coherent-state operator bases are known, but here we address the more specific rank-resolved problem of preserving the canonical contravariant-symbol representation. We show that the canonical finite coherent-state formula reconstructs every operator in the rank-$S$ sector exactly if and only if the sampling points form a spherical $(2J+S)$-design. We call the associated configurations spin-coherent quantum designs. We further give a fully explicit positive-weight Gauss-Legendre construction that avoids the need for an equal-weight spherical design. Together, these results establish a unified framework for reading out physical observables from a handful of measurement samples, playing for spin systems the role that the so-called von Neumann lattice plays for canonical coherent states. Finally, we derive practical protocols for estimating moments of spin operators from these constructions, with direct applications to polarimetry, magnetometry, and quantum state tomography.

Given a Lie group $G$, a level $k$, and a Lie subgroup $H$ one can construct 2d conformal field theories by either 1.) gauging a nonanomalous $H$ symmetry of the WZW model constructed from $(G,k)$ or 2.) using an algebraic procedure known as the GKO coset construction. The two models are closely related, but not precisely the same: The gauged WZW model is identified with the corresponding GKO model coupled to a 2d topological field theory. The topological theory is characterized by a commutative Frobenius algebra derived from the endomorphisms of an algebra object in a modular tensor category constructed from $(G,H,k)$. The partition function on the torus of the two models differ by a factor of the dimension of this algebra of endomorphisms. Concrete examples are constructed and some applications to string theory and 2d Yang-Mills coupled to nonanomalous matter are briefly discussed. This paper is a summary of a longer companion paper.

We explain how the 'bar-cobar' construction for homotopy algebras reformulates the equations of motion of arbitrary gauge theories as gauge-covariant quadratic equations for an extended set of fields. The linear term of the new equations encodes the interactions of the original theory, while the quadratic term is universal. The extended fields include type-I multilocal fields, which depend on a set of coordinates and type-II multilocal fields, which depend on several sets of coordinates. The new equations of motion are the Maurer-Cartan equations of a differential graded associative algebra or Lie algebra. We show that every solution of the new equations is gauge equivalent to a solution with only type-I fields, that represents a solution of the original equations of motion. In string theory type-I fields are entangled states of the associated CFT, to be inserted across multiple punctures of a Riemann surface. General type-II states can also represent disconnected Riemann surfaces.

We study smooth projective varieties whose Frobenius-trace kernel, also known as the Tschirnhausen bundle associated with the Frobenius morphism, is ample. We show that any non-constant morphism out of such a variety must be finite onto its image, providing strong evidence that these varieties must be Fano varieties of Picard rank 1. Using infinitesimal representation theory and by constructing special Frobenius splittings, we show that generalized Grassmannian of classical type and type $\mathrm{G}_2$ have ample Frobenius-trace kernel, with the exception of certain low characteristic examples which are explained by the existence of exotic isogenies of the associated algebraic groups.

The continuum limit is a widely used theoretical construct for obtaining continuum field-theories of crystalline systems. We study the formulation of a continuum limit for gapped bosonic phases and ask whether their topological properties survive passage to the continuum. Using methods in algebraic topology and category theory, we give a rigorous formulation of the continuum limit and construct a surjective global map relating crystalline topological phases across all finite point-group symmetries to continuum invertible topological phases. Consequently, we find that some lattice phases admit no continuum limit, while distinct lattice phases that do admit a continuum limit can share the same continuum image, hence implying that some lattice topological data can collapse. Conversely, we find that every continuum invertible phase admits a faithful crystalline realization. In addition to the general framework, we apply it to several examples, including studies of a 2D rotation-symmetric phase, higher-order topological phases, and the mixed spin-lattice anomaly of the deconfined quantum critical point between an antiferromagnet and valence bond solid.

We study unitary quantum dynamics in noisy Brownian models with global continuous symmetries, such as $U(1)$ and $SU(2)$, focusing on Rényi entanglement entropies and hydrodynamic and non-hydrodynamic correlators. By mapping the averaged late-time dynamics to the low-energy physics of effective replica Hamiltonians, we find that the evolution is controlled by the quantum geometry of their ground-state manifolds, which is directly related to the geometry of $k$-commutants—the symmetry algebra of $k$ replicas of the system. In interacting systems, these $k$-commutants are generically determined solely by the symmetries of the system, independent of microscopic details of the noisy evolution. This allows us to use the time-dependent variational principle (TDVP) to provide simple geometric explanations for the sub-ballistic Rényi entanglement growth and the anomalous decay of non-hydrodynamic correlators in interacting systems with continuous symmetries. We find this behavior to be intimately connected to singularities within the $k$-commutant manifolds, arising from frozen “void” states in the Hilbert space that exist due to continuous on-site symmetries. This also demystifies the important role of voids in the dynamics of these observables, previously identified in $U(1)$ symmetric systems. We compare these behaviors in interacting systems with Abelian and non-Abelian continuous symmetries and in free-fermion systems, which differ in the geometry of their $k$-commutants. Ultimately, this work provides a general geometric framework for systematically studying observables in noisy systems with continuous symmetries, including Haar-random circuits.

We report solutions to five challenge problems posed by Doron Zeilberger and his collaborators, together with substantial partial progress on two further problems, and we describe the method by which they were obtained. The solved problems are: the Second Computational Chomp Challenge of Ekhad and Zeilberger, for which we exhibit a bar with three winning opening moves; the third challenge of Spahn and Zeilberger, asking whether the restricted permutation counts a_{r,s} and b_{r,s} are holonomic for all r,s>1, answered affirmatively; the First Rigorous Solid Standard Young Tableaux Challenge, for which we prove the conjectured second-order recurrence; the five-dimensional Geode Challenge of Amdeberhan, Kauers and Zeilberger; and Conjectures 2a and 2b of Kauers and Zeilberger, which we obtain from a local limit theorem for excursions of Markov-modulated random walks in cones. Several results of independent interest arise along the way: a staircase theorem constraining the winning opening moves of any Chomp bar, together with a parity theorem for square bars; an explicit algebraic generating function for reverse-Kreweras diagonal walks and a closed form for their diagonal-endpoint counts; a one-dimensional integral representation for diagonal Geode coefficients; and the identity that each Kauers-Zeilberger constant is a universal factor times the square of the apex value of a discrete cone-harmonic function. All of the work reported here was carried out in collaboration with a large language model. The paper sets out the division of labour, records the verification protocol this mode of work required, and documents the failures, which we regard as an essential part of the report.

A graph $G$ is called $CIS$ if each maximal clique intersects each maximal stable set of $G$, with maximality taken with respect to set inclusion. CIS graphs resemble perfect graphs in several respects and have interesting applications in game theory. The complexity of recognizing CIS graphs was posed as an open problem by Chvátal in the 1990s and has since led to conflicting conjectures. We settle the problem by showing that recognizing CIS graphs is $\mathsf{coNP}\text{-complete}$.

Basin is a numerical optimization library for the Rust programming language. Numerical optimization is the task of finding the inputs that minimize a function, and it is a fundamental element across the sciences: fitting a model to data, calibrating a simulation, training a machine learning model, or choosing engineering parameters that minimize cost. Basin gives users a single, consistent way to both state and solve such problems, with a broad catalog of solvers and first-class support for constraints.

The heavy ball GMRES (HBGMRES) method is one of Krylov subspace methods for linear systems combined with the restarted GMRES and the heavy ball method which is applied in optimization. HBGMRES not only keeps benefit of the restarted GMRES in limiting memory usage and controlling orthogonalization cost, but also is able to cover up the slow convergence problem in the restarted GMRES. Another type of Krylov subspace methods is GMRESR which consists of GCR as the outer algorithm and certain steps of GMRES as an inner method. Compared with HBGMRES, GMRESR gives the approximately optimal solution over the new search vector gained from GMRES and all previously kept search vectors in GCR. Even though GMRESR performs better than HBGMRES, it has slow convergence which is similar to GMRES and the restarted GMRES. Inspired by HBGMRES, we present the heavy ball GMRESR method (HBGMRESR) by using HBGMRES as the inner loop instead of using GMRES to salvage the lost convergence speed while still keeping the benefit of GMRESR that in the sense of a global minimization over some specific part of the Krylov subspace is done. Numerical tests on real data are presented to demonstratee the superiority of the new methods over GMRESR and HBGMRES.