Papers

Let $V$ be a compact connected manifold and $G$ a simply connected solvable Lie group. We study $G$-Lie foliations on $V$ from the point of view of $\mathcal{D}$-module theory, following the approach initiated by Dathe. To each singular foliation $\mathcal{I}$, we associate the $\mathcal{D}_X$-module $\mathcal{M}_{\mathcal{I}} = \mathcal{D}_X / \mathcal{D}_X \cdot \mathcal{I}$ and the derived ring $\mathcal{D}_{\mathcal{I}} := R\mathcal{H}om_{\mathcal{D}_X}(\mathcal{M}_{\mathcal{I}}, \mathcal{M}_{\mathcal{I}})$. We compute the D-irregularity $\mathrm{D\text{-}irr}(\mathcal{I})$ for three classes of solvable Lie foliations: regular homogeneous foliations, the Meigniez foliation with non-polycyclic holonomy group, and an explicit foliation on a compact 5-dimensional manifold. We show that the non-polycyclicity of the holonomy group $\Gamma$ is reflected in the non-vanishing of higher cohomology groups of $\mathcal{D}_{\mathcal{I}}$, establishing a new connection between the geometry of the holonomy group and $\mathcal{D}$-module invariants.

We present generic expressions for one-sided series of Bessel functions of the first kind: $\sum_{m=0}^{\infty} J_{Nm+p}(x)$ and $\sum_{m=0}^{\infty} (-1)^m J_{Nm+p}(x)$. The result is expressed in terms of incomplete Lipschitz-Hankel integrals of Bessel functions of the first kind, which can be represented as Kampé de Fériet functions.

In this paper, we prove Hare's $\epsilon$-unfair conjecture in [Computational progress on the unfair 0–1 polynomial conjecture, Experiment. Math. (2025), https://doi.org/10.1080/10586458.2025.2527786].

Let $p$ be a prime number and let $n\geq 1$ be an integer. We completely solve the exponential Diophantine equation $p^{x}+6^{y}=z^{2n}$ in positive integers $x,y,z$ for every prime $p$ such that the Legendre symbol $\leg{6}{p}$ equals $-1$, that is, for $p\equiv 7,11,13,17\pmod{24}$: the equation has no solution, with the single exception $(p,n,x,y,z)=(13,1,1,2,7)$. Since these residue classes contain half of all primes in the sense of Dirichlet density, this settles the equation for one prime out of two, uniformly in the exponent $n$. Two ingredients of the proof have independent interest. First, we show that the Ramanujan–Nagell type equation $p^{x}=2\cdot 6^{m}+1$ has no solution with $x\geq 2$, using Zsigmondy's theorem on primitive prime divisors; this removes the extra congruence hypotheses that appeared in earlier work of the author on the case $n=1$ and solves the open problems formulated there. Secondly, we prove that $13^{x}+6^{y}=z^{2}$ has $(x,y,z)=(1,2,7)$ as its unique solution, thereby completely resolving the exceptional prime. The Legendre condition is sharp: we exhibit solutions for primes with $\leg{6}{p}=+1$ realising each branch of the factorisation argument. All results are corroborated by extensive computational verification, and we state several open problems, including the complete classification of the solutions when $\leg{6}{p}=+1$.

This paper develops a spectral Koopman-Hopf framework for adversarial reachability analysis of nonlinear systems. By lifting the nonlinear drift dynamics into Koopman eigenfunction coordinates, the proposed approach transforms the original state-dependent Hamilton-Jacobi-Isaacs (HJI) equation into an approximate state-independent optimization problem in spectral coordinates. The transformed control and disturbance directions are approximated using least-squares spectral projections, and upper and lower bounds on the transformed Hamiltonian are used to derive corresponding bounds on the value function and backward reachable sets. Koopman eigenfunctions are computed using a path-integral-based Galerkin framework that avoids spatial discretization of the associated eigenfunction PDEs. Numerical examples demonstrate tractable approximation of backward reachable sets for nonlinear adversarial systems.

We prove Qian's conjecture on element orders and irreducible character codegrees.

Comparing nonlinear dynamical systems directly from trajectory data remains challenging because finite-horizon trajectory representations are generally coordinate dependent. Motivated by recent behavioral subspace approaches for linear systems, this paper introduces a Koopman-behavioral distance for nonlinear stochastic systems using multi-rollout trajectory data. State measurements are lifted through a common observable dictionary, and dominant lifted behavioral subspaces are compared using Grassmannian distances without explicit Koopman operator identification. We establish coordinate invariance, connection with recently proposed linear behavioral distances under exact lifted closure, and robustness to approximate closure. Numerical examples demonstrate coordinate invariance and parameter discrimination.

Three-dimensional bioheat simulation aims to predict transient temperature distributions in biological tissue and is commonly modeled using the Pennes bioheat equation, which combines thermal diffusion, perfusion-mediated heat loss, and external heat generation. In this study, we consider a controlled 3D Pennes bioheat simulation under a localized heat-source condition inspired by microwave ablation (MWA). To evaluate neural approximation performance, we compare three neural partial differential equation (PDE) solvers under the same controlled simulation: a spatial Fourier-feature physics-informed neural network (PINN), a generic PINNMamba temporal subsequence model, and Pennes Physics-Informed Mamba (PPIM). PPIM builds on the temporal subsequence model by incorporating conditioned heat-source input and Pennes-aware state-space model (SSM) decay initialization. All three neural models are trained under the same conditions with the same Pennes residual, and an explicit finite-difference method (FDM) solution is used only as the numerical reference. In a representative 600~s run, PPIM achieved the lowest MAE, relative $L_1$ error, and relative $L_2$ error among the evaluated neural solvers. Error maps further showed that the remaining PPIM errors were more concentrated near the heat-source region than across the rest of the domain. These results indicate that PPIM is effective for approximating the FDM reference final temperature field in this controlled simulation. The source code is available at https://github.com/muvYun/PPIM.

Hadwiger conjectured in 1943 that every graph with no $K_t$ minor has chromatic number at most $t-1$. Delcourt and Postle proved that every graph with no $K_t$ minor has chromatic number $O(t\log\log t)$. We build on their result to improve this bound to $O(t\log\log\log t)$.

We study bridge indices of graphs, virtual graphs, and their handlebody classes. We construct virtual ravels for which the difference between bridge index and weighted overpass bridge index is unbounded. The lower bound uses exponentially many colorings for one fixed flow in a $\Z_2$-family of biquandles. For each negative Euler characteristic we construct classical almost unknotted graphs of one fixed trivalent graph with arbitrarily large bridge index. We also obtain unbounded even bridge indices for ravels of each fixed bouquet rank. A separate almost unknotted family has an unbounded difference between its spatial-graph bridge index and the bridge index of its regular neighborhood. Finally, edge connected sums of Suzuki curves give exact constrained and unconstrained bridge indices whose difference is unbounded.

This is an advanced introduction to the various types of normal random variables, with most of the needed preliminaries included. We first discuss the probability basics, standard central limits, and the theory of the usual, real normal variables, with examples, illustrations and numerous formulae. Then we go on a similar discussion regarding the complex normal variables, and the Rayleigh variables too. We then move to arbitrary dimensions, with a discussion regarding the Gaussian vectors, and related probability laws, featuring some functional analysis, and geometry and physics too. Finally, we provide an introduction to the quantum versions of the normal variables, and notably to those coming from free probability and random matrices.

We give a detailed proof that every set-sized symmetric extension of a model of $\mathsf{ZF}+\mathsf{VP}$ again satisfies $\mathsf{ZF}+\mathsf{VP}$, where $\mathsf{VP}$ is formulated for arbitrary set-sized languages. For each standard $n\geq 1$, the same preservation theorem holds for $\mathsf{VP}(\Pi_n)$, with set parameters and arbitrary set-sized languages. We also prove that $\mathsf{ZF}+\mathsf{VP}$ and $\mathsf{ZFC}+\mathsf{VP}$ are equiconsistent, and that $\mathsf{ZF}+\mathsf{VP}$ proves the existence of a proper class of Löwenheim-Skolem cardinals. Relative to $\operatorname{Con}(\mathsf{ZF}+\mathsf{VP})$, the axiom $\mathsf{DC}$ is independent of $\mathsf{ZF}+\mathsf{VP}+\neg\mathsf{AC}$. The preservation theorem also yields Feferman-Lévy and full Solovay models satisfying $\mathsf{VP}$. We consider whether every countable model of $\mathsf{ZF}+\mathsf{VP}$ has a class-generic extension satisfying $\mathsf{ZFC}+\mathsf{VP}$.

We consider dimensionality reduction for high-dimensional observations accompanied by a supplied partition into two or more clusters. The objective is not to construct a low-rank projection, but to retain an interpretable subset of the original coordinates that preserves the distributional information distinguishing the clusters. For each coordinate, the proposed procedure compares the cluster-specific empirical distribution functions through a several-sample Kolmogorov-Smirnov separation statistic. We formalize the resulting marginal cluster support and establish simultaneous finite-sample concentration over all coordinates, explicit bounds for false inclusions and omissions, and exact support recovery when the minimum distributional separation dominates the high-dimensional stochastic error. We also quantify the dimension inflation induced by using an unadjusted testing level and give a familywise-error-controlled version. Under a conditional sufficiency condition, sure screening preserves the full-data posterior cluster probabilities, mutual information, and Bayes risk; an additional result characterizes robustness to imperfectly estimated cluster labels. The procedure is invariant to strictly increasing coordinate transformations and can retain low-variance cluster signals that principal components may discard. We further develop average dual information, a criterion combining partition agreement after transformation with structural coverage of cluster-relevant coordinates, and derive its basic properties and consistency. Simulations illustrate the theory, the interpretability of the selected coordinates, and the distinction between cluster-directed screening and variance-directed projection.

We solve four questions raised by Bayart concerning coefficient summability for multilinear forms with prescribed supports. For every $m\ge 2$ and $d\in[1,m]$, we determine the product-summability exponent and the multilinear summability invariant: \[ \mathrm{prod}(m,d) =\min\left\{\frac{m}{d},\,m-\lceil d\rceil+1\right\}, \qquad \gamma_{\mathrm{mult}}(m,d) =\min\left\{m-\lceil d\rceil+1,\frac{2m}{d+1}\right\}. \] In particular, $\gamma_{\mathrm{mult}}(4,2)=8/3$, showing that the multilinear invariant need not be an integer. We also prove that, for every $d\in[1,m]$, there is a single infinite support of exact combinatorial dimension $d$ on which the dimensional Hardy–Littlewood bound is attained over both scalar fields for every anisotropic parameter $\mathbf p=(p_1,\ldots,p_m)$ with $\sum_j 1/p_j<1$, simultaneously across the two regimes separated by $\sum_j 1/p_j=1/2$.

Reactive sedimentation of mixtures composed by biological and inert solid matter including liquid substrates is modelled by a system of convection-diffusion-reaction equations. In this process, the small solid particles dispersed in the viscous fluid settle due to gravity while simultaneous chemical reactions take place between solid and liquid components (substrates). Among the applications of reactive sedimentation processes, the principal use is in the simulation and control of secondary settling tanks (SSTs) in water resource recovery facilities (WRRFs). The governing equations include a nonlinear strongly degenerate parabolic equation for the total concentration of solids, and transport equations including reaction terms for the solid components and substrates. To accurately approximate the model equations, a high-order finite volume scheme is developed using polynomial reconstructions. Two reconstruction methods are used, a second-order monotonic upstream-centered scheme (MUSCL), and a third-order central weighted essentially non-oscillatory (CWENO3) method. In addition, for the time discretization, we employ strong stability preserving Runge-Kutta (SSPRK) schemes of second- and third-order. The CWENO3 scheme is complemented with limiters, and in the case of zero diffusion, the designed high-order schemes are shown to preserve an invariant-region property. This property is particularly relevant as it ensures that physically relevant numerical solutions are obtained at each time iteration. In addition, high-order approximations for the diffusion terms are proposed to approximate the complete model equations. Simulations of denitrification in SSTs under a continuously operated regime in WRRFs demonstrate the performance of the model and its discretization. Finally, convergence tests show the order of accuracy of the developed scheme and the invariant-region preservation.

We prove an explicit lower bound of optimal quadratic order for the anticanonical volume of every complex $\epsilon$-log canonical weak del Pezzo surface, with no restriction on its Picard number or its singularities. More precisely, the volume is at least $\min\{c_{exc},\epsilon^2/3032064\}$, where $c_{exc}>0$ is an explicit absolute constant. Quotients of the projective plane show that the exponent two is optimal. We also prove cubic upper bounds for the cyclic determinant mass and for the Picard number of a smooth rational log Calabi–Yau pair. We give geometric examples showing that these two auxiliary exponents are optimal in their respective settings.

Using the strong discrete Morse theory developed by Fernández, we define the complex $\mathcal{SM}(K)$ of strong discrete Morse matchings on a simplicial complex $K$, as well as the pure subcomplex $\mathcal{SM}_{pure}(K)$ generated by the facets in $\mathcal{SM}(K)$ of maximal dimension. For most complexes $K$ these objects are proper subcomplexes of the complexes ${\mathfrak M}(K)$ and ${\mathfrak M}_{\textrm{pure}}(K)$ defined by Chari–Joswig using all Morse matchings on $K$. The homotopy types of the latter are not well-understood in general, but they are known when $K$ is the path $P_n$ with $n$ edges, the cycle $C_n$ with $n$ edges, the star $S_n$ with $n$ leaves, the $n$-simplex $\Delta^n$ ($n\le 3$), and the boundary $\partial\Delta^n$ ($n\le 3$). in this paper we compute the homotopy types of $\mathcal{SM}_{pure}(C_n)$ and $\mathcal{SM}(K)$ for $K=P_n,S_n,\Delta^n,\partial\Delta^n$ for all $n$. We also compute the homology of $\mathcal{SM}(C_n)$ for $n\le 21$.

We investigate weak solutions to the two-dimensional quasi-geostrophic equations without dissipation. We establish global existence of weak solutions for temperature bounded and lacking spatial decay and velocity in the space $L^2_{ul}(\mathbb{R}^2)$. Our methods rely on a spectral Serfati identity, which we use to establish uniform $L^2_{ul}$ bounds on a sequence of velocities satisfying the dissipative equations. These bounds, combined with a maximum principle on the scalar temperature, allow us to pass to the zero-dissipation limit, giving global-in-time weak solutions.

We prove that, for all $\beta>0$, the Dobrushin-Lanford-Ruelle (DLR) equations for the unit-intensity $\mathsf{Sine}_\beta$ point process have a unique stationary solution under a particular finite electric energy condition. Uniqueness fails if this condition is dropped. This answers a question of Dereudre-Hardy-Leblé-Maïda and gives a canonical statistical physics characterisation of $\mathsf{Sine}_\beta$. The main technical ingredient is a relative entropy estimate which allows us to compare the conditional distribution in a finite box of a DLR solution to a corresponding circular $\beta$ ensemble.

Recently, Huang and Satriano determined the sharp dimension threshold for Batyrev's non-negativity conjecture on stringy Hodge numbers: the conjecture holds in dimensions at most four and fails in every dimension at least five. We show that their counterexamples fit into a single broad three-parameter family $Y_{m,d}:=V(\sum_{i=0}^m s_i g_i(x)+h(x))\subset\mathbb{P}^{m+d+1}$, where each $g_i$ is homogeneous of degree $d-1$ and $h$ is homogeneous of degree $d$, whose singular locus is a linear space $\Lambda\cong\mathbb{P}^m$. Blowing up $\Lambda$ gives a log resolution with one smooth exceptional divisor of discrepancy one. We compute the Hodge-Deligne polynomials of both the resolution and the exceptional divisor from their projective-space fibrations over $\mathbb{P}^d$, derive a master formula for $E_{\mathrm{st}}(Y_{m,d}\times(\mathbb{P}^1)^n)$, and prove the structure formula $h_{\mathrm{st}}^{p,q}(Y_{m,d}\times(\mathbb{P}^1)^n)=A_{p,q}+B_{p,q}-C_{p,q}$, where the three terms are non-negative. Thus the only possible source of negativity is $-C_{p,q}$, which is governed by the primitive middle cohomology of $Z_{m,d}=V(g_0,\ldots,g_m)\subset\mathbb{P}^d$. Off the diagonal, every negative entry equals $-C_{p,q}$, a binomially weighted sum of shifted primitive Hodge numbers of $Z_{m,d}$. On the diagonal, the behaviour is governed by the parity of $m+d$. We also obtain a complete classification for $0\le m\le d-1$, $d\ge3$, and $n\ge1$. The only non-counterexamples are $(m,d)=(0,3)$ for every $n\ge1$, and $(m,d)=(2,3)$ for $n\ge4$.

The mean squared displacement (MSD) is a cornerstone in the analysis of diffusion processes in complex media. When the system is heterogeneous and, in particular, when single-particle trajectories are short, it is essential to extract maximal information from each measured trajectory. This is typically done by time-averaging squared increments and examining the scaling of the time-averaged MSD in log-log space. However, classical regression methods perform poorly in this setting because time-averaging introduces correlations aggravated by those inherent to anomalous diffusion. We tackle these limitations by applying a generalized least squares framework, which substantially reduces variance and bias in diffusion parameter estimates, especially for short (around 100 points) and ultra-short (around 10 points) trajectories. The method is fully automated and requires no supervision. Furthermore, it enables prediction of estimation error probability density, which is asymptotically Gaussian, for both classical and enhanced approaches. Leveraging this prediction, we introduce a specialized deconvolution algorithm that reconstructs the underlying particle ensemble structure from experimental data.

We study the Brill–Noether theory of higher rank sheaves on surfaces $X$ with $H^1(X,\mathcal{O}_X) = 0$ and its interactions with the classical Brill–Noether theory of line bundles on curves lying on $X$. Denoting by $\mathrm{Spl}(v)$ the moduli space of simple sheaves on $X$ with Chern character $v = (r,c_1,\mathrm{ch}_2)$, we introduce the Brill–Noether loci $BN^k(v)^\circ \subseteq \mathrm{Spl}(v)$ of \emph{generically $k$-generated} sheaves and show, under assumptions on $c_1$ and $K_X$, certain well-behavedness results. In particular, if $c_1$ is indecomposable and $X$ is a K3 surface, we generalize a Brill–Noether theorem for sheaves on K3s due to Yoshioka. This is achieved by employing He's deformation theory of the moduli space of coherent systems and a globalization of the Lazarsfeld–Mukai construction, providing a correspondence between the Brill–Noether theories of generically $r$-generated sheaves on $X$ and that of line bundles on curves $C \in |c_1|$. The same techniques yield upper bounds on the dimension of $\mathcal{W}^{r,\mathrm{bpf}}_d(|c_1|) \setminus \mathcal{W}^{r+1}_d(|c_1|)$ consisting of basepoint-free line bundles in terms of the number of endomorphisms of associated Lazarsfeld–Mukai bundles, for arbitrary $c_1$. This generalizes a result of Aprodu and Farkas in the context of Green's conjecture beyond K3 surfaces. In the case $r = 1$ of pencils, we obtain smoothness and expected-dimension results under an effectivity condition on $-K_X|_C$ for $C \in |c_1|$. As an application, we prove Green's conjecture for the general curve on a rational surface with a nef and effective anticanonical divisor.

Let $F$ be a function field over an algebraically closed field $K$ and $S$ a finite dimensional $K$-subspace of $F$. The square of $S$ is spanned by all products of pairs of elements in $S$. We conjecture that if $\dim S^2 \leq 3 \dim S - 4$, then $S^2$ must contain a Riemann–Roch space of dimension at least $2 \dim S - 1 + g$, where $g$ is the genus of $F$. This generalizes a theorem of Freiman from additive combinatorics, stating that small sumsets must contain long arithmetic progressions. We prove our conjecture in the case that $S$ is contained in a Riemann–Roch space of dimension at most $3/2\dim S + 1$. For the proof we study the annihilator of $S$ and introduce the notion of weight for linear forms.

Given a convex lattice polygon $\Delta\subset \mathbb R^2$, let $N_\Delta$ be the count of rational, nodal curves in the linear system of the ample line bundle $L_\Delta$ on the toric variety $\mathbb P_\Delta$ defined by $\Delta$, having fixed intersection with the toric boundary. We show that the relative Jacobian of the linear system defines an integrable system that has a mirror dual in the language of constructible sheaves on a two-torus microsupported on a Legendrian link. In this setting, we define the dual counting problem using an analogue of rulings of Legendrian links in three-space, then prove equivalence with $N_\Delta$. We perform calculations of $N_\Delta$ in several examples using constructible methods, tropical curve counting, and localization in logarithmic Gromov-Witten theory to demonstrate the equality of the different approaches. Moreover, the rulings give rise to a stratification of the moduli of constructible sheaves. We conjecture that this ruling decomposition recovers the refined tropical invariants of Block-Göttsche, and hence encodes higher-genus logarithmic Gromov–Witten invariants, by a theorem of Bousseau.

We establish sharp quadratic bounds on the relaxation time of simple random walk on connected regular graphs, with leading constants $3/(2\pi^2)$ and $1/\pi^2$ for even and odd orders, respectively. This resolves a longstanding conjecture known as Aldous–Fill spectral gap conjecture (2002). We also prove uniqueness and stability theorems for the corresponding cubic and quartic chains, and settle the quartic uniqueness conjecture posed by Abdi, Ghorbani, and Imrich (2021) and by Abdi and Ghorbani (2023). We establish sharp bounds on algebraic connectivity under minimum-degree and regularity constraints and prove the conjectured chain structure of the minimisers, identifying their repeating blocks. This resolves a longstanding conjecture of Guiduli and Mohar (1996) and a structural conjecture of Abdi and Ghorbani (2024). Our quantitative stability theorem shows that nearly minimum algebraic connectivity forces nearly maximum diameter, proving their diameter conjecture. We further obtain sharp relaxation-time bounds in terms of edge-connectivity. For nonregular graphs, we establish sharp bounds for the gap between maximum degree and adjacency spectral radius, confirming a conjecture of Liu (2024). As applications, we prove the sharp bounds on hitting and commute times conjectured by Aldous and Fill (2002) for regular graphs.

We settle the planar sharp Hessian-integrability conjecture of Armstrong, Silvestre, and Smart [Comm. Pure Appl. Math. 65 (2012), 1169–1184] for viscosity supersolutions of fully nonlinear uniformly elliptic equations. If $\kappa=\Lambda/\lambda$, then the optimal exponent in the $W^{2,\epsilon}$ regularity theory in the plane is exactly $$ \epsilon_2(\kappa)=\frac{2}{\kappa+1}. $$ The same mechanism reaches the known upper obstruction in dimension three throughout the full range $1\le \kappa\le 4$ and therefore gives $$ \epsilon_3(\kappa)=\frac{3}{2\kappa+1} \qquad(1\le\kappa\le4). $$ Beyond this threshold, it yields a closed algebraic lower bound and identifies the precise spectral configuration responsible for the remaining gap. The proof introduces a spectrally resolved continuum-in-opening mechanism for contact geometry. Throughout the contact evolution, the vertex Jacobian retains the negative index of the Hessian and its mean negative curvature—data erased by the classical pointwise reduction—and these variables are optimized only after integration over all openings. This reduces the analysis to a finite family of explicit one-dimensional kernels whose critical exponents recover the sharp planar threshold, identify the exact three-dimensional regime above, and expose a broader principle for extracting second-order regularity from the spectral geometry of contact sets.

We investigate secure communication over a classical multiple-access wiretap channel (MAC-WTC), specifically exploring the benefit of shared entanglement between the transmitters. Under the strict semantic security criterion, we derive an achievable rate region and a regularized expression for the secrecy capacity. We further establish a single-letter upper bound on the secure capacity of the MAC-WTC with entangled transmitters. Our results demonstrate that entanglement strictly enlarges the secrecy capacity of MAC-WTC compared to sharing only classical correlated randomness, a finding we illustrate using a pseudo-telepathy game example. Finally, we establish new strong soft-covering lemmas for the output statistics of multipleaccess channels (MACs) with entangled transmitters. Our results generalize existing results for non-entangled systems.

We prove an explicit Fricke transformation for the two-dimensional space generated by normalized reciprocal products on the cubic-residue cosets modulo 13. The transformation matrix is a scalar multiple of a matrix of differences of cubic Gaussian periods. Its projective action is defined over the cyclic cubic field, whereas the normalized matrix is defined over the real cyclotomic field. The proof uses generalized eta functions, a modular unit of degree two on $X_1(13)$, and a five-coefficient identity. A separate divisor argument lifts the projective transformation to the asserted linear transformation. We also record the Fricke images at an arbitrary prime $p\equiv1\pmod3$ and prove that their normalized sine constants belong to the associated cyclic cubic subfield.

We identify a class of simple yet nontrivial forcing notions that preserve the Ultrapower Axiom (UA) over canonical inner models, reaching the level of a strong cardinal. The forcings we consider are discrete product forcings whose index sets are chosen and spaced according to the structure of the underlying nonoverlapping extender sequence. As an application, we show that UA is consistent with the existence of a strong cardinal $\kappa$ such that $V\neq \text{HOD}_X$ for every $X\in V_\kappa$, answering a question of Goldberg.

In her graduate thesis, Gwyneth Harrison-Shermoen developed an abstract framework for approximating structures of a given theory, which generalized the notion of smoothly approximable structures. At the time it was unclear how broadly the framework applied due to a lack of examples. In this paper we will show that with slight modification to the framework, several well-understood $\mathrm{NSOP}_1$ theories are in fact approximable by simple structures. We discuss certain structural properties of the limit structure that can be obtained from the approximation and introduce a notion of limit dimension as a candidate for recovering Kim-independence in the limit structure. We also provide some examples of structures that are not approximable by structures of finite Morley rank.