Papers

A resolving set in a graph is a set of landmarks whose distance vectors distinguish all vertices. We use information theory to prove lower bounds for metric dimension and class dimension in distance-regular graphs and association schemes arising from finite geometry. The core idea is that, for a fixed landmark, a random object usually lies in one overwhelmingly likely distance or relation class. For classical dual polar graphs, with rank and type fixed and $q\to\infty$ through the admissible field orders, we prove $\mu(\Gamma(q,d,e))=\Theta_{d,e}(q^e)$ for $d\geq 2$ and $e>0$. The lower bound uses opposition as the typical distance. For the upper bound, we take, for each of a constant number of $(d-1)$-dimensional singular subspaces, all generators containing it. For Grassmann graphs, bilinear forms graphs, and attenuated-space schemes, we obtain lower bounds of the same exponential order as the known incidence constructions.

We study a distinguished one dimensional restriction associated with planar vector fields whose linearization is a nonzero nilpotent matrix of index two. This restriction admits an intrinsic interpretation on the nilpotent eigendirection, while its scalar representation in Jordan coordinates satisfies a simple transformation law under admissible changes of Jordan chain. Its quadratic Taylor coefficient coincides with the intrinsic quadratic Bogdanov–Takens coefficient. % For rational restrictions, we show that the Taylor coefficient sequence satisfies a finite linear recurrence determined by the minimal denominator. Its degree defines a recurrence degree that is invariant under admissible changes of Jordan chain and gives the minimal order of an eventual homogeneous recurrence. Exact cancellations may therefore be detected directly from the restriction before nonlinear normal form transformations are performed. Applications to four planar models exhibit different recurrence degree and cancellation mechanisms, illustrating the complementarity between the distinguished restriction and smooth nilpotent normal form theory.

We study the edge-power cost of the labelled tree generated by the $\gamma$-strategy, a constant-gain rule for sequential Euclidean connections. Starting with $x_0=p_0$, each input point $p_i$ is attached to $x_{i-1}$, and the state is updated by $x_i=\gamma x_{i-1}+(1-\gamma)p_i$. Retaining $x_i$ subdivides the insertion segment into a spine edge and a leaf edge. The memory parameter $\gamma$ controls how long earlier points influence subsequent attachment points. We minimize the sum of the $\alpha$-powers of the edge lengths under independent uniform input and arbitrary input sequences. For uniform points in the unit ball, the stationary problem has a transition at $\alpha=1$. Its continuous extension is minimized at the boundary for $0<\alpha\leq1$, while every global minimizer is interior for $\alpha>1$. We determine the finite optimizer in the joint window $\alpha_N=1+\varepsilon_N$, $\varepsilon_N\log N\to\lambda$. Below an explicit threshold it lies on the $N^{-1/2}$ scale, at the threshold its scale is $\sqrt{\log N/(N\log\log N)}$, and above the threshold it approaches an explicit stationary root with two computable corrections. A second threshold identifies the governing correction, and differentiated estimates prove eventual uniqueness. At $\alpha=3d+8$, the linear coefficient at the stationary endpoint changes sign and a branch of strict local maxima enters the parameter interval. For arbitrary input sequences, the optimal parameter and asymptotic worst-case edge-power cost per point are explicit for $0<\alpha\leq3$. At high powers, periodic antipodal block inputs give explicit lower bounds which, with a separation argument, show that the optimized cost is asymptotic to $2\log2/\log\alpha$. Exact results for powers two and four, a rational recursion for every even power, and a high-dimensional expansion complete the analysis.

We study a PDE-constrained optimization problem for an elliptic equation with the spectral fractional Laplacian and a linear combination of Dirac measures as the forcing term; the controls are the amplitudes of these singular sources. We prove existence and uniqueness of an optimal solution and derive first-order optimality conditions. We then propose a discretization based on finite elements. Since the set of admissible controls is finite dimensional, the control variable itself does not require discretization. We conclude by deriving a priori error bounds

Efficiently and robustly analyzing shape data is critical across many scientific disciplines. While chirality is a fundamental property in numerous applications - most notably in molecular science - existing shape analysis metrics fail to distinguish between a shape and its mirror image. To address this gap, we introduce a multilinear generalization of the Gromov-Wasserstein objective. Under mild assumptions, this objective yields a distance between shapes, represented as probability distributions quotiented by a symmetry group $G$. In particular, for $G = SO(d)$, we introduce the Chiral Gromov-Wasserstein ($\mathrm{CGW}$) distance, sensitive to chirality. We establish robustness properties for the multilinear Gromov-Wasserstein distances and develop efficient algorithms to compute them, reformulating the underlying optimization problem by projecting couplings onto a low-dimensional space. We derive algorithms for both local and approximate global solutions, yielding a fully polynomial-time approximation scheme for these problems. We validate the framework through numerical experiments that demonstrate the effectiveness of $\mathrm{CGW}$ as a shape metric for chiral objects.

The category TFS of thin flat surfaces, introduced by Khovanov, Qi, and Rozansky, is a strict symmetric monoidal category whose morphisms are compact oriented surfaces with corners arising as neighbourhoods of immersed graphs in a strip. We prove two main results. First, a normal form theorem: every connected viewable tf-cobordism is equal, in TFS, to a canonical composite determined by its topological type. Second, a sufficiency theorem: we establish a list of relations in the TFS category which are sufficient, meaning that any two composites of generators representing the same tf-cobordism are related by a finite sequence of the listed relations. Together, these results give a complete generators-and-relations presentation of TFS.

We study the Segre invariant on rank two $H$-stable vector bundles with fixed Chern classes on Hirzebruch surfaces $\Sigma_{n}$. For each vector bundle on these surfaces, we determine the possible values of the Segre invariant and compute the number of global sections of these bundles and present applications to Brill-Noether Theory.

We provide a rigorous Laughlin-pump argument that establishes the fractional quantization of the Hall conductance directly from the Laughlin state as a topological index. Our proof uses a recently defined index of a pair of pure states together with an infinite matrix product state analysis that is rigorous on a sufficiently thin cylinder. By virtue of the connection to an index, our result implies, in particular, that Laughlin states are representatives of topologically stable fractional quantum Hall phases.

We study the Coxeter-Dickson $E_8$ order as a common integral support for the octonionic, para-octonionic and compact real Okubo products. The three algebra structures have the same additive lattice, the same positive composition norm and hence the same $240$ norm-one vectors, namely the roots of $E_8$ and the vertices of the Gosset polytope $4_{21}$. Their multiplicative structures are nevertheless different and this difference is already visible in the idempotents: among the same $240$ roots one finds respectively $1$, $57$ and $12$ nonzero integral idempotents. We show that these three counts organize the common root system in three increasingly polarized ways. The octonionic product leaves the full $E_8$ geometry unseparated; the $57$ para-octonionic idempotents reproduce the $E_7$ contact decomposition $1+56+126+56+1$; the $12$ Okubo idempotents split into four mutually orthogonal oriented $A_2$ triangles and hence determine canonically an $A_2^4$ subsystem. Choosing one triangle as the external $A_2$ then yields the $A_2+E_6$ Magic Star, while the remaining nine idempotents determine the trinification subsystem $A_2^3\subset E_6$. This provides an arithmetic interpretation of the $E_8\supset E_7$ and $E_8\supset E_6+A_2$ decompositions directly from integral idempotents.

Recent work by Wanchun Shen shows that $K_2$-regular affine local complete intersections of any dimension in characteristic zero are smooth. In this note, we show that the same holds for affine toric varieties.

We study path-width and the closely related parameter line-width in countably infinite graphs. Our first result characterises the graphs of finite path-width: they are the graphs that do not have infinitely many vertices of infinite degree, do not have infinitely many pairwise disjoint infinite paths, and contain no subdivision of some finite tree of maximum degree 3. We then investigate universality under the subgraph relation for graphs of bounded path-width or line-width. In particular, we prove that there exists a universal graph with line-width $\mathcal{O}(k^2)$ for the class of graphs with line-width at most $k$. In contrast, we show that no graph of finite path-width is universal for the class of locally finite graphs with path-width $1$. Finally, we show that for each $k\geq 2$, every universal graph for the class of graphs with path-width at most $k$ has line-width at least $k + 1$.

Dynamical low-rank approximation has become a widely used numerical method in diverse disciplines. Its main idea is to represent the matrix or tensor-valued solution to a time-dependent differential equation as a low-rank factorization. The evolution of the factorization leads to highly stiff dynamics and requires the derivation of novel time integration methods that are not prone to this stiffness. A promising family of integrators are basis-update & Galerkin (BUG) integrators as they enable implicit time integration and structure–preservation. However, current BUG integrators are limited to second–order accuracy while general-order BUG integrators are designed as projections of explicit time integration methods, thus severely limiting their use. In this work, we propose general-order BUG integrators that do not rely on an explicit time integration scheme while requiring a smaller number of basis functions to achieve high-order accuracy. We prove a general order error bound for the proposed augmented and parallel BUG integrators and demonstrate their behaviour for a series of stiff and non-stiff numerical benchmarks in which they significantly outperform previous BUG versions.

In the framework of network dynamics, learning models, and neural tangent kernels (NTK), we show that the corresponding linearized dynamics leads naturally to a semigroup formulation. More precisely, in our analysis of input/output models, the time-dynamics is presented via special semigroups of linear operators on Hilbert spaces, together with an associated class of semigroup perturbations. In this context, we then present new and explicit a priori perturbation-bound results: for the fixed-kernel linearization constructions arising in the NTK setting, we prove norm-bounds on the corresponding semigroup perturbations, in the form of explicit finite-time perturbation estimates. We further present refinements on prescribed task spaces, Cesàro-averaged (ergodic) comparisons estimates, and versions in which the lower spectral edge assumption is replaced by a spectral-distribution condition. We also extend the comparison to nonautonomous NTK evolutions through piecewise-frozen approximations, record a corresponding discrete Euler specialization, and offer worked examples in order to illustrate our perturbation-bound estimates.

Let $a,b$ be integers with $1\le b\le a$, and let \[ G_0=1,\qquad G_1=a+1,\qquad G_{n+2}=aG_{n+1}+bG_n. \] For real-valued functions which are additive in the greedy $G$-digits, we prove a necessary-and-sufficient criterion for the existence of a limiting distribution. The criterion consists of a first-order drift series and a quadratic digit-energy series, and it recovers the Zeckendorf theorem when $a=b=1$. The main difficulty is necessity: a one-step transfer matrix detects all non-maximal digit values, while the maximal digit becomes visible only in a two-step product. A weighted Euclidean norm symmetrizes the untwisted companion matrix and makes both contractions occur at the true Perron scale. Sufficiency follows from a two-dimensional Perron product lemma with square-summable transverse perturbations and a convergent, not necessarily absolutely convergent, Perron drift. The limiting characteristic function admits a scalar infinite-product representation in a neighbourhood of the origin and a global matrix-product representation at arbitrary frequencies.

We study a nonautonomous compartmental model on the probability simplex for radical vote-share dynamics under delayed policy implementation. The model distinguishes a target regime from an effective regime that adjusts with finite speed and includes a persistent alienation reservoir that can later be mobilised by radical actors. In a symmetric benchmark, persistent alienation destroys the global one-threshold picture: below the local instability threshold, a radical-free centrist–alienated equilibrium may coexist with a stable positive equilibrium, separated by a saddle branch created in a saddle-node bifurcation. By hyperbolic persistence, this coexistence survives near the symmetric baseline, while numerical exploration identifies a wider bistable region under asymmetric perturbations. Delayed implementation further separates target and effective threshold passage, producing a parameter-space lag and an additional state-space lag before a visible response in radical support. The analysis combines a Perron–Frobenius threshold for the frozen radical-free equilibrium, a geometric characterisation of positive equilibria, local adiabatic tracking under slow uniformly subcritical drift, and explicit delay bounds for transversal threshold passage. Under initial matching, the parameter-space delay scales as \(O(\kappa_\theta^{-1})\); in a monotone ramp example the bound is nearly attained, whereas the state-space lag is substantially larger and less sensitive to implementation speed. Thus threshold restoration is preventive rather than curative: once the trajectory enters the basin of the radicalised attractor, returning below the local threshold need not restore the radical-free regime.

We extend the two-dimensional singular Zienkiewicz element to three dimensions, leading to a novel $H^2$-conforming finite element on tetrahedral meshes based on rational shape functions. Besides the finite element construction and its conformity analysis, we develop an exact iterative integration procedure for the associated class of rational functions. The resulting formulae allow for the exact integration of the basis functions, their derivatives, and the products occurring in finite element assembly.

We study the singularity categories of simple singularities of the same dimension over an algebraically closed field of positive characteristic, and show that, as in characteristic zero, these categories are not equivalent as triangulated categories unless the underlying singularities are analytically isomorphic. In contrast to the characteristic zero case, simple singularities in positive characteristic cannot be distinguished solely from the Auslander-Reiten quivers of their singularity categories. To address this, we extend to positive characteristics a theorem by Hua and Keller, which asserts that the 0th Hochschild cohomology of the dg singularity category of an isolated hypersurface singularity in characteristic zero is isomorphic to the Tyurina algebra of the defining polynomial. Furthermore as an application, we determine the condition for the singularity category of a rational double point (i.e., a simple singularity of dimension two) to be standard. We prove that such a category is standard if and only if the defining polynomial is weighted homogeneous.

We study the similarity problem for representations of compact and discrete quantum groups. We first prove that every non-degenerate contractive representation of a compact or discrete quantum group is automatically unitary. Then, for compact quantum groups, we develop an interpolation method for constructing explicit non-unitarizable non-degenerate representations with norms arbitrarily close to $1$. In particular, this applies to all non-Kac compact quantum groups whose dual has subexponential growth, including the Drinfeld–Jimbo quantum groups $G_q$, as well as to the non-Kac free unitary quantum groups $U_F^+$ with $F\in\operatorname{GL}_2(\mathbb{C})$. On the discrete quantum group side, we establish a quantum analogue of the classical lifting principle for non-unitarizable uniformly bounded representations. Combining this lifting principle with recent results on maximal Kac quantum subgroups, we construct explicit non-unitarizable non-degenerate representations with norms arbitrarily close to $1$ for all unitary free quantum groups $\mathbb FU_F$ and all non-amenable orthogonal free quantum groups $\mathbb FO_F$.

This paper broadens the range on which Fouvry and Radziwiłł's results on nearly balanced convolutions apply. In particular, let $\alpha_m$ and $\beta_n$ be sequences supported on $m\sim M$ and $n\sim N$ where $\beta_n$ is equidistributed for small moduli, and let $Q=X^{\frac 12+\varepsilon}$. We find that \begin{gather*}\sum_{q\sim Q}\left|\mathop{\sum\sum}_{\substack{n\sim N,m\sim M \\ mn\equiv a\pmod q}}\alpha_m\beta_n-\frac{1}{\phi(q)}\mathop{\sum\sum}_{\substack{n\sim N,m\sim M \\ (mn,q)=1}}\alpha_m\beta_n\right|\ll \frac{X}{\log^A X} \end{gather*} if $N=X^{\frac 12+\delta}$ and $M=X^{\frac 12-\delta}$ with $0<\delta<\frac 1{68}$, which improves Fouvry and Radziwiłł's $0<\delta<\frac 1{112}$. To prove this, we sharpen Bettin and Chandee's famous result on trilinear forms with Kloosterman fractions in the case where some of the sums are over subdyadic intervals.

In this thesis, we study the spectral properties of dynamically defined aperiodic metric and discrete graphs. Our goal is to determine to what extent spectral properties of discrete one-dimensional ergodic Schrödinger operators persist when the aperiodicity is manifested through the geometry rather than through a potential. The graphs considered here are inspired by one-dimensional aperiodic tilings, and are called tiling graphs and decorated $\mathbb{Z}$-graphs. For a large family of metric tiling graphs equipped with the standard Laplacian, we show that the spectrum is of zero Lebesgue measure, and is a generalized Cantor set up to a possible discrete set of energies. For decorated $\mathbb{Z}$-graphs, we further show that for a Baire-generic and Lebesgue almost-sure choice of the decoration edge lengths, the spectrum is a generalized Cantor set. We then study the integrated density of states (IDS) for metric and discrete decorated $\mathbb{Z}$-graphs. We prove a gap labelling theorem, which characterizes the set of possible values taken by the IDS inside spectral gaps. We show that the gap labels are contained in the Schwartzman group associated with the dynamical system generating the graph, up to a geometric scaling factor. Lastly, we consider the Dry Ten Martini Problem for discrete Sturmian decorated $\mathbb{Z}$-graphs, asking whether all possible values predicted by the gap labelling theorem are indeed attained by the IDS inside spectral gaps. We answer this question negatively, by identifying a large set of gap labels which are not attained due to jump discontinuities of the IDS. We then show that away from these jump discontinuities, the periodic approximants for Sturmian graphs display the same combinatorial structure as the standard Sturmian Hamiltonians, and use this to obtain an explicit characterization of the realized gap labels for Sturmian comb graphs.

We study contraction properties of non-stationary continuous-time mean-field games (MFGs) under discounting and entropy regularization. The state of the representative agent evolves according to a controlled continuous-time Markov chain, and both the state and action spaces are finite. In contrast to the undiscounted case, we show that, under a sufficiently large discount rate, finite-horizon MFGs admit a horizon-independent contraction condition, which also coincides with the corresponding infinite-horizon non-stationary contraction condition. As a byproduct, we obtain an explicit convergence rate between finite- and infinite-horizon mean-field equilibria. For each finite horizon, we further derive a refined contraction criterion from the spectral radius of a positive operator that majorizes the propagation of policy errors, and show that its large-horizon limit agrees with the horizon-independent contraction factor. Finally, we provide an explicit error bound between discounted and undiscounted finite-horizon regularized equilibria.

The rows and columns of the character table of the symmetric group $S_n$ are both naturally indexed by partitions of $n$. Let $D(n)$ denote the number of conjugacy classes of $S_n$ whose column contains no zero entry. The identity column is always zero-free, so $D(n)\geq 1$. It is known that $D(n)\ll n^2$. We prove that $D(n)\ll n^{3/4}$. Second, we prove for almost all positive integers $n$ that $D(n)\ll_B n^{1/2}(\log n)^B$ for every $B>5/6$, with a quantitative bound for the exceptional set, using work of Matomäki and Radziwill. Finally, we offer a heuristic supporting our conjecture that $D(n)\ll_{\varepsilon} n^{\varepsilon}$. AxiomProver formalized the results in this paper in Lean assuming preexisting literature.

We develop a Dirichlet-beta counterpart and a structural extension of the coefficient-extraction framework for hyperbolic and logarithmic tangent integrals. For integers $m\geq n\geq1$ with $m+n$ even, the zeta-type formula \[ \int_0^\infty\frac{\tanh^{m+1}x}{x^{n+1}}\,dx = (-1)^{(m-n)/2} \sum_{p=\lceil n/2\rceil}^{(m+n)/2} \binom{2p}{n}(2^{2p+1}-1) \frac{\zeta(2p+1)}{\pi^{2p}} [u^{m+n-2p}](u\cot u)^{m+1} \] admits the beta-type analogue \[ \int_0^\infty \frac{\tanh^m x}{x^n\cosh x}\,dx = (-1)^{(m-n)/2} \sum_{p=\lceil n/2\rceil}^{(m+n)/2} 2^{2p}\binom{2p-1}{n-1} \frac{\beta(2p)}{\pi^{2p-1}} [u^{m+n-2p}] \frac{u}{\sin u}(u\cot u)^m. \] The proof uses derivative polynomials, a beta kernel, and a coefficient-collapse argument. We then establish strict sign alternation for several coefficient families. In the boundary shifted-hyperbolic case the sign is controlled by generalized Bernoulli polynomials, while the arctanh and logarithmic tangent families are identified with continuous dual Hahn and continuous Hahn polynomials. Their zero distributions turn the observed alternation into structural statements valid for all admissible indices. Finally, with $\lambda(s)=(1-2^{-s})\zeta(s)$, we show that the parallel zeta–lambda and beta identities are the two binary specializations of a single Lerch-transcendent scheme with $\varepsilon\in\{0,1\}$. The same convention $\varepsilon=0$ for the beta case and $\varepsilon=1$ for the zeta–lambda case is used throughout. Besides the shifted hyperbolic, odd-$\sinh$, logarithmic tangent, and $\tanh$ families, this framework yields a further unified pair of reciprocal-arctanh integral formulae on $(0,1)$.

In this paper, we study Lipschitz spaces adapted to Dunkl–Schrödinger operators $\mathcal{L} = -\Delta_k + V$, where $\Delta_k$ is the Dunkl Laplacian and $V \ge 0$ belongs to the reverse Hölder class $\mathrm{RH}^q(dw)$ with $q > \max(1, \mathbf{N}/2)$. For $0 < \beta < 2$, we establish the full equivalence between the heat semigroup-based Lipschitz spaces $\widetilde{\Lambda}_{\mathcal{L},k}^{\beta/2}(\mathbb{R}^N)$ and the pointwise weighted Zygmund classes $\Lambda_{\mathcal{L},k}^{\beta}(\mathbb{R}^N)$ defined via the critical radius function $m(\mathbf{x})$. Remarkably, this characterization shows that the spatial smoothness component decouples completely from the underlying reflection group symmetries and root systems. As an intermediate result, we establish new equivalences for inhomogeneous weighted Lipschitz spaces associated with the unperturbed Dunkl Laplacian $\Delta_k$. A central difficulty in this context — arising from the necessity to disentangle the Euclidean distance from the orbit metric $d(\mathbf{x},\mathbf{y})$ — is resolved through a detailed geometric analysis. Finally, we provide pointwise Gaussian upper bounds for the time derivatives of the Dunkl–Schrödinger heat kernel, which are of independent interest for future work in harmonic analysis on Dunkl structures. Crucially, we do not assume the potential $V$ to be $G$-invariant under the action of the Weyl group, which introduces major geometric difficulties resolved here via a local analysis of the orbit metric.

Quantum time-marching algorithms for transport PDEs often represent variable coefficients and forcing through register-expanding dilations, block-encoding oracles, or repeated postselection. We present an alternative algorithm for a forced variable-coefficient advection-diffusion equation in flow-inspired skew-symmetric form that incorporates spatially varying velocity, viscous dissipation, and persistent source injection with a peak logical requirement of $n_q+1$ qubits. A centered skew-symmetric discretization makes the advection operator strictly skew-Hermitian for arbitrary velocity profiles, enabling an ancilla-free unitary realization using a Gray-code Trotter sequence of controlled-$R_y$ rotations. Diffusion is applied in the Fourier basis through a uniformly controlled rotation on one postselected ancilla, which is measured, reset, and reused between the two diffusion half-steps, while the source is incorporated classically through second-order Strang splitting. Statevector simulations for $N=16$ and $32$ recover second-order temporal convergence against high-accuracy classical solutions, while Richardson extrapolation gives fourth-order accuracy and reduces kernel calls by factors of four to fourteen. Independent tests through $N=256$ confirm second-order spatial consistency. We further show that the per-step ancilla failure probability is proportional to the instantaneous viscous dissipation rate, making postselection cost self-regulating over a fifty-fold viscosity range. Stable evolution is demonstrated for $5\times10^4$ time steps without observable secular error growth, while Gray-code advection accounts for $71$–$95\%$ of transpiled controlled-NOT gates. The fixed-width kernel provides a qubit-efficient building block for near-term hardware studies, although classical readout and state re-preparation remain the main obstacles to coherent multistep evolution.

A new family of Darboux integrable partial differential equations is constructed. The minimal orders of integrals in both characteristics can simultaneously be arbitrarily high for equations within this family. This suggests that the complete list of Darboux integrable equations may contain substantially more equations than are currently known.

Let $D_n$ be the dihedral group of order $2n$. Consider a continuous-time random walk on $D_n$ driven by arbitrary symmetric rates whose support generates $D_n$. For $p\in[1,\infty]$, we say the pair $(D_n,p)$ is rate-monotonic if for each fixed time $t$, the $\ell^p$-distance between the random walk's distribution at time $t$ and the uniform distribution is monotonically decreasing as a function of the rates. Lyons and White proved that $(D_n,2)$ and $(D_n,\infty)$ are rate-monotonic. Somewhat counterintuitively, they found several pairs $(D_n,p)$ with ${p\in[1,1.997]\cup[2.001,3.999]\cup[4.001,5.995]}$ that are not rate-monotonic, and they asked whether any such pairs exist with $p=4$ or $p=6$. We resolve their question, proving that $(D_n,2m)$ is rate-monotonic for all positive integers $m$ and $n$. In fact, we prove a generalization of this result to a broader family of groups that includes generalized dihedral groups, dicyclic groups, and generalized quaternion groups. In the other direction, we prove that for every real $p\geq 1$ that is not an even integer, there exists a positive integer $n$ such that $(D_n,p)$ is not rate-monotonic. The results of this paper were formally verified in Lean by AxiomProver assuming standard literature.

Generalized disjunctive programming (GDP) expresses mixed discrete-continuous decisions through Boolean indicators and disjunctions, and can be systematically solved via a library of methods proposed in the literature. The recent InfiniteGDP abstraction lifts this modeling layer to continuous domains such as time, space, and uncertainty, but only the big-M and hull reformulations, the two endpoints of the relaxation spectrum, have been generalized to the infinite setting. This work closes this gap by generalizing four other GDP solution methods to infinite-dimensional optimization: the multiple big-M reformulation, P-split reformulation, cutting plane reformulation, and the logic-based outer approximation algorithm. It further proposes MBM-GP, a novel Gaussian-process variant of multiple big-M that learns the big-M function over the infinite domain from a small subset of the subproblem solves. Moreover, these approaches are implemented in the Julia package InfiniteDisjunctiveProgramming.jl. The methods are benchmarked on case studies arising in dynamic and stochastic optimization. The results demonstrate how the generalized solution methods can outperform big-M and hull, with MBM-GP retaining the tightness of multiple big-M at a fraction of its reformulation cost.

Anosov flows have a long and rich history, firstly motivated by the study of geodesic flows in negative curvature surface by Anosov and Sinai. Not every closed manifold admits an Anosov flow for well-known reasons: the fundamental group of a 3-manifold \(M\) admitting an Anosov flow must have exponential growth, and \(M\) must be universally covered by \(\mathbb{R}^{3}\). Nevertheless, there are sufficient mechanisms for constructing distinct Anosov flows on admissible 3-manifolds, such as Dehn-Goodman-Fried surgery or playing with hyperbolic building blocks. A central problem in the field has been to determine the number of Anosov flows that can be supported by a single manifold. The question of whether there exists an infinite set of pairwise non-equivalent Anosov flows on a 3-manifold remains open to this day. However, there are several papers proving the existence of a manifold $M_n$ that admits $n$ pairwise inequivalent Anosov flows for any natural number $n$. In all known examples, the manifolds $M_n$ are composed of several geometric pieces. In the present paper, we prove the existence of a countable number of graph manifolds $M_{k,n}$, $k \in \mathbb{N}$ with a single geometric piece, each of which admits $n$ pairwise non-equivalent transitive Anosov flows. All previously known constructions of different flows on the same graph manifold were based on gluing geodesic flows. The nature of the flows constructed in this paper is completely different; they are constructed from a single hyperbolic plug, which is a suspension over a Morse-Smale diffeomorphism on a surface.

The Maximum Entropy on the Mean (MEM) method provides a flexible computational framework for solving inverse problems by combining data fidelity with entropy-based regularization. In practice, however, the prior distribution is typically unknown but can be estimated from data, giving rise to the empirical MEM method. We establish a parametric convergence rate of $O(n^{-1/2})$ in expectation for empirical MEM, improving upon the previously established $O(n^{-1/4})$ guarantee by King-Roskamp et al. (2026). Our proof is based on a novel stability analysis of the primal and dual optimization problems under perturbations of the underlying probability measure, relying only on foundational tools from convex analysis and probability. We further show that the MEM dual problem admits a reformulation as an expected risk minimization problem, thereby placing MEM within the modern framework of stochastic optimization and enabling scalable stochastic gradient algorithms for large-scale inverse problems. Together, these results place empirical MEM as a statistically and computationally efficient methodology for data-driven inverse problems.