Papers

For each $n \geq 2$, we construct an ascending chain of irreducible lattices in the product of $n$ homogeneous trees. Moreover, for each pair of integers $m_1, m_2 \geq 1$, we define explicitly a bi-reversible automaton $\mathcal B$ such that the group $G_{\mathcal B}$ defined by the automaton $\mathcal B$ has finiteness length $m_1$ (i.e. it is of type $\mathrm{F}_{m_1}$ but not of type $\mathrm{FP}_{m_1+1}$), and the group $G_{\mathcal B^*}$ defined by the dual automaton has finiteness length $m_2$. Both constructions rely on the consideration of $S$-arithmetic groups in the affine group of a global function field.

We present a machine learning approach for developing subgrid-scale (SGS) parametrizations in coarse simulations of partial differential equations. We utilize structure-preserving neural networks and entropy variables to learn subgrid fluxes in coarse simulations of the Burgers' equation. In particular, we employ a decoupled neural network architecture explicitly separating the subgrid corrections into two distinct components: a conservative Flux Potential network and an Eddy Viscosity network. We demonstrate that this reduced-order framework maintains high physical fidelity, accurately reproducing the energy spectrum, spatial and temporal correlation functions, and dynamical characteristics of the full-scale system. Furthermore, we show that our approach is robust and applicable to parameters outside the training regime.

We study matrix product factorizations (MPFs) in symmetric association schemes: identities $A_SA_T=A_U$ where $A_S,A_T,A_U$ are loopless unions of basic relations and the ordinary matrix product is again a $0$-$1$ adjacency matrix. We give equivalent structural and spectral criteria for MPFs, derive valency and rank restrictions, and analyze several standard families. For $2$-class schemes, the only nontrivial loopless MPF comes from the scheme of the $5$-cycle. For $P$-polynomial schemes, the distance-regular recurrence gives strong restrictions on products $A_1A_i$. We also prove a universal pentagon theorem for the case $A_SA_T=J-I$, and show that extremal rank forces all non-zero eigenvalues of $A_U$ to be $\pm k(U)$, hence gives bipartiteness. Finally, in Hamming schemes we obtain rank obstructions and classify MPFs of the form $A_1A_T=A_U$: in $H(d,2)$, for $d\ge2$, the only non-zero loopless example is $A_1A_d=A_{d-1}$, which is trivial since $A_d$ has valency $1$; for $q>2$, no non-zero example occurs.

An $(n,R)_q$-covering sequence is a cyclic sequence, over the finite field $\F_q$, whose consecutive $n$-tuples form a code of length $n$ and covering radius $R$. An $(n,m,R)_q$-covering-sequences code is a set of cyclic sequences of length $m$, over $\F_q$, whose consecutive $n$-tuples form a code of length $n$ and covering radius $R$. These codes are the best building blocks for $(n,R)_q$-covering sequences. We show, for small radii, how cyclic codes and constacyclic codes with small covering radius, can be used to construct such sequences of short length and such codes with a relatively small number of sequences and a total number of codewords in the associated covering code. Sequences with small radius whose length approaches asymptotically to optimality are constructed, especially for an alphabet of prime power size large enough. With the same construction, interesting codes are also constructed for larger radii.

This paper addresses stabilization of MIMO systems with uncertain time-varying diagonal input direction. We propose a modular switching sign-compensation layer acting as an outer wrapper around a nominal controller. Unlike Nussbaum-type gains, monitoring functions, or binary adaptive mechanisms, the method uses only bounded sign changes that preserve the nominal control magnitude and its properties. The compensation layer uses adaptive variables built from nominal Lyapunov quantities to search for the unknown input-sign configuration based on schedulers. Two schedulers are developed: a vector scheduler, where each input channel explores its own sign compensation and admits an online trapping certificate, and a scalar pattern scheduler, where one variable visits all diagonal sign matrices and gives a design-time recovery guarantee on sufficiently long constant-sign intervals. Once the correct sign configuration is set, the actual closed loop coincides with the nominal closed loop and the original nominal stability property is recovered. The approach is illustrated on a flight roll-reversal problem, a visual-servoing benchmark, and an underground-reservoir control example motivated by human induced-seismicity mitigation.

A proper edge-coloring of a graph $G$ is a D-coloring if every subgraph isomorphic to $K_4-e$ is rainbow. The minimum number of colors in such a coloring is the D-chromatic index $\chi'_D(G)$. Wang conjectured that every planar graph of maximum degree $\Delta \ge 4$ satisfies $\chi'_D(G) \le 9$ for $\Delta = 4$, $\chi'_D(G) \le 10$ for $\Delta = 5$, and $\chi'_D(G) \le 2\Delta - 1$ for $\Delta \ge 6$. We prove that every planar graph $G$ satisfies \[ \chi_D'(G) \leq \begin{cases} 9, & \Delta(G) \leq 4, \\ 10, & \Delta(G) = 5, \\ 2\Delta(G) - 1, & \Delta(G) \geq 33. \end{cases} \] Each bound is best possible in its stated range. Consequently, Wang's conjecture remains open only for $6 \le \Delta \le 32$.

In this paper, we propose a compression framework for weighted graphs in which the graph topology is transmitted losslessly and edge weights are compressed lossily. A challenge in the lossy compression of edge weights is that the underlying relationships between edges are ambiguous. To address this issue, we first transform the unweighted graph into the corresponding line graph, whose nodes represent the edges of the original graph and whose edges encode the relationships between them. The line graph transform allows us to regard edge weights as a graph signal defined on the line graph. Instead of transmitting the edge-weight vector, we first transform it with a graph filter bank on the line graph. Then, quantization and entropy coding are performed on the transformed coefficients of the edge weight vector. In addition to the lossy compression method, we formalize edge smoothness on the line graph and show that it serves as a measure of the difficulty of compression. The proposed smoothness measure can be easily calculated without converting to a line graph. This provides insight into the expected compression performance of a given weighted graph. Experiments on synthetic and real-world data validate the effectiveness of the proposed method by comparing it with existing matrix preprocessing methods.

We construct Kakeya sets in arbitrary dimension $d\geq 2$, generalizing the classical Perron tree construction beyond dimension $2$. The Kakeya sets we construct have $\delta$-neighbourhood of volume at most $C|\log\delta|^{-(d-1)}$, which improves on previously known constructions, and which is conjecturally optimal. We further derive consequences for the $L^p$-boundedness of radial Fourier multipliers in terms of Besov spaces with logarithmic smoothness.

This survey offers a geometric and topological perspective on translation surfaces, with a focus on aspects that are often overshadowed by the dynamical viewpoint. After presenting several equivalent definitions of translation surfaces, particularly those based on polygonal models and fundamental domains, we explore the structure of their moduli spaces in the spirit of Thurston, including natural stratifications. The final part is devoted to the study of the period map, the realisation of representations as period characters and the fibres of the holonomy map. This approach highlights a foundational, yet sometimes underrated, facet of the theory.

We consider adaptive finite element methods for symmetric second-order linear elliptic PDEs, where the adaptive algorithm steers the local mesh refinement as well as an iterative algebraic solver. Under abstract assumptions on the underlying a-posteriori error estimator and the solver, we prove that the usual adaptive algorithm leads to unconditional full R-linear convergence, independently of the user-chosen adaptivity parameters. For sufficiently small parameters, this guarantees optimal complexity in the sense that the decay rate of an appropriate quasi-error is optimal with respect to the overall computation cost (and hence time) measured in terms of the usual nonlinear approximation classes. Unlike available results in the literature, the main focus is on the analytical understanding of non-residual estimators like averaging-based estimators as proposed by Zienkiewicz and Zhu or estimators based on equilibrated fluxes.

We consider the post-detection analysis of change-points for multivariate time series, with the goal of identifying which coordinates are responsible for a detected change. After a change-point has been located by an offline detection algorithm, we propose post hoc statistical procedures to determine whether the change occurs in either of two predefined blocks of coordinates or in both. Our methods rely on two-sample testing procedures with a particular focus on nonparametric tests; we provide theoretical guarantees for Type I error control. Simulations and a real-data experiment demonstrate the strong performance of the proposed procedures.

We study strong temporal approximation of periodic stochastic Korteweg–de Vries equation driven by small additive \(Q\)-Wiener noise of amplitude \(\mathcal O(\varepsilon)\), \(0<\varepsilon\ll1\) under $H^s$-regularity, $s=1,2$. This is a challenging problem, due to the additional derivative term in the nonlinearity and the lack of suitable exponential moment bounds for the exact solutions. Exploiting the small-noise regime, we first decompose the solution into a deterministic KdV flow and a stochastic component; then we linearize the obtained stochastic equation and approximate the resulting equation by means of Fourier analytic techniques. Combining the small-noise linearization error, the discretization error of the linearized equation, and the deterministic temporal approximation error, and assuming suitable conditions on the deterministic solver, we get strong errors of size $\mathcal{O}(\tau+\varepsilon^2)$ under \(H^1\)-regularity and of size $\mathcal{O}(\tau^2+\varepsilon^2)$ under $H^2$-regularity. To the best of our knowledge, these are the first strong error bounds achieved for numerical time approximations of the stochastic KdV at low regularity.

We study the worst-case false discovery rate (FDR) of the Benjamini-Hochberg procedure for both one- and two-sided Gaussian tests with general correlation matrices. For each setting and given target FDR level $q$, we construct a family of finite Gaussian models whose FDR divided by $q$ diverges as $q\downarrow0$. This disproves any universal multiplicative FDR bound. Moreover, we establish new FDR lower bounds that are independent of the number of hypotheses. For two-sided tests, the worst-case FDR over the number of hypotheses, mean vector, and correlation matrix is at least an explicit ${\ell_{=}}(q)>q$: \[ \ell_{=}(q)=\frac{q\sqrt{\log(1/q)}}{2\sqrt{\pi}}+c_\ell q+o(q), \qquad c_\ell=0.6492828\ldots. \] For one-sided tests, we show a lower bound $\ell_{\le}(q)>q$: \[ \ell_{\le}(q)=\frac{q\sqrt{\log(1/q)}}{\sqrt\pi} +\frac q2+o(q). \] Finally, we prove the leading constants are sharp for a class of correlation matrices for both one- and two-sided tests.

We study the existence of standing wave solutions for the following quasilinear Schrödinger equations with critical growth on the Heisenberg group $$ -\Delta_{\mathbb{H}} u +V(\xi)u-\Delta_{\mathbb{H}} (\left|u\right|^{2\alpha})\left|u\right|^{2\alpha-2} u= \lambda \left|u\right|^{q-2}u + \left|u\right|^{p-2}u \text{ in }\mathbb{H}^N $$ where $\mathbb{H}^N$ is Heisenberg group, $\Delta_{\mathbb{H}}$ is Kohn Laplacian operator, $4\alpha <q<p \leq 2\alpha Q^{*}$, \(Q^{*}= \frac{2Q}{Q-2}\) is the critical Folland–Stein exponent, $\lambda$ and $\alpha $ are positive parameters, $\alpha > \frac{1}{2}.$ By a suitable nonlinear change of variables, the quasilinear equation is transformed into a semilinear one, allowing the use of variational methods in the Folland–Stein Sobolev space $S^{1,2}(\mathbb{H}^N)$. Applying the mountain pass theorem together with a concentration–compactness argument adapted to the sub-Riemannian framework, we establish the existence of a nontrivial solution.

Repeated elliptic solves on domains with evolving boundaries arise in moving-interface simulation, design, and reactive navigation. Even when a fixed Cartesian grid avoids remeshing, rebuilding all boundary interactions for every configuration can limit the efficiency of repeated solves. We develop a static–dynamic boundary reduction for an unfitted lattice Green's function method on prescribed moving planar domains. Like boundary integral and boundary element methods, the formulation reduces the problem to boundary-supported unknowns through a Green representation. Its construction, however, reverses the usual order: the Cartesian operator is discretized before the Green representation is formed, rather than representing the continuous problem first and then discretizing the boundary. This discretize-then-represent viewpoint avoids boundary meshes and singular quadrature. The method also separates interactions associated with stationary geometry from those affected by motion, reuses the invariant part throughout a simulation, and updates only couplings involving the changing boundary. Boundary conditions are imposed at true interface intersections, lattice-kernel data are reused, and the interior field is reconstructed by a fast sine-transform solver. The principal contribution is an implemented and validated update strategy for translating, deforming, appearing, and topology-changing obstacles.

In this article, we investigate the following modified quasilinear equation with parameter driven by the $Q$-subLaplacian: \begin{align*} \begin{cases} -\Delta_Q u - \Delta_Q\bigl(|u|^{2\alpha}\bigr)\,|u|^{2\alpha-2} u = \lambda f(\xi,u) & \text{in } \Omega, \\[2mm] u = 0 & \text{on } \partial\Omega, \end{cases} \end{align*} where $\Delta_Q(\cdot):= \mathrm{div}_{\mathbb{H}}\bigl(|\nabla_{\mathbb{H}}(\cdot)|^{Q-2}\nabla_{\mathbb{H}}(\cdot)\bigr)$ denotes the $Q$-subLaplacian on the Heisenberg group $\mathbb{H}^N$, $Q=2N+2$ is the homogeneous dimension, $\Omega \subset \mathbb{H}^N$ is a smooth bounded domain, $\lambda>0$, $\alpha > \frac{1}{2}$, and $f$ has critical or subcritical exponential growth of order $\exp\bigl(\beta|t|^{2\alpha Q/(Q-1)}\bigr)$. We prove three results: the existence of a nontrivial positive weak solution in the critical case for all large $\lambda$, and the existence of a least-energy nodal solution with exactly two nodal domains, under subcritical and critical exponential growth. A change of variables $u=g(v)$ reduces the problem to a quasilinear problem whose energy functional is of class $C^1$; the exponent $2\alpha Q/(Q-1)$ arises from the growth of $g$. We handled the exponential growth using the sharp Moser-Trudinger inequality of Cohn and Lu. The positive solution is obtained by the mountain pass theorem and the nodal solutions by minimization on a nodal Nehari set.

We study the effect of magnetic vector potentials on periodic Grover walks on finite graphs. The magnetic vector potential is introduced through the framework of quantum graphs, which induces the Grover walk as a special case. We regard the magnetic vector potential as a perturbation of a periodic Grover walk and investigate the robustness of its periodicity. Our analysis reveals that the response to such perturbations depends on the spectral structure of the underlying graph. In particular, when the graph possesses at least one non-simple eigenvalue, we derive a Hermitian matrix that characterizes the robustness of its periodicity. As a consequence, for initial states orthogonal to the eigenspaces of the unperturbed Grover walk corresponding to the eigenvalues $\pm1$, we show that the perturbed dynamics is asymptotically described by a continuous-time quantum walk generated by this Hermitian matrix.

We study Bienaymé-Galton-Watson trees conditioned to have $n$ vertices and $k_n$ leaves, where $k_n\to\infty$ while remaining negligible compared to $n$, i.e. $k_n=o(n)$. More precisely, we determine the asymptotic distribution of the outdegrees of these trees. We first show that the tree is asymptotically binary: the number of vertices with two children is asymptotically equal to the number of leaves, while almost all remaining vertices have exactly one child. Then, we identify the scales at which vertices with larger outdegrees emerge. For every $d\ge2$, the critical scale $k_n \sim cn^{(d-1)/d}$, $c>0$, is the threshold for the appearance of vertices with outdegree $d+1$. Below this scale, such vertices are absent with high probability; at the critical scale, their number converges to a Poisson distribution; above it, they satisfy a law of large numbers. Our proofs rely on the coding of Bienaymé-Galton-Watson trees by their Łukasiewicz paths and asymptotic estimates for associated random walks.

We propose and analyze a class of evolving surface finite element methods for mean curvature flow of closed surfaces using Lagrange multipliers for preserving the energy-decreasing structure. The algorithm is based on the solution-driven formulation using Huisken's evolution equations for the normal vector and the mean curvature. The time discretizations use linearly implicit backward difference formulas (BDF) for the parabolic geometric variables and implicit Adams updates for the nodal positions. The approach also accommodates artificial tangential velocities of minimal-deformation-rate type. The resulting fully discrete algorithms are area-decreasing at every time step, with a prescribed decay rate determined by the computed mean curvature. We prove local existence and uniqueness of the discrete Lagrange multiplier and convergence of a simplified Newton iteration for its computation under weak regularity assumptions. Under stronger regularity assumptions, as used in the convergence theory for the underlying evolving surface finite element method, we derive optimal-order error bounds of order $h^k+\tau^q$ in the $H^1$-norm for finite elements of polynomial degree $k\ge 2$ and $q$-step BDF and $q$-step implicit Adams methods with $2\le q\le 5$, both without and with minimal-deformation-rate tangential motion. Numerical experiments for mean curvature flow of a sphere confirm the predicted convergence rates and show that the Lagrange-multiplier correction entails only a small computational overhead that is essentially independent of the mesh size and the time step size.

We prove two results on the fine structure of the binary digits of $3^{m}$. First, for every fixed period $p$, the number of positions at which the binary expansion of $3^{m}$ breaks $p$-periodicity grows in order like $\log m/\log\log m$; equivalently, no window of the expansion deeper than a fixed power of $\log m$ is $p$-periodic. Second, the finite binary word formed by the low-order digits of $3^{m}$ has full low-order subword complexity: its complexity function satisfies $p_{3^{m}}(n)\ge n+1$ for every length $n$, once $m$ is large enough.

We develop a systematic theory of $(n-1)$-truncated maps, called $n$-covering maps, in a fixed $\infty$-topos $\mathscr{E}$, guided by the analogy with classical covering spaces. We prove an equivalence of $n$-categories between $n$-coverings over a pointed connected object $(X,x)$ and $\infty$-actions of the fundamental $n$-group $\Pi_n(X,x)$ on $(n-1)$-truncated objects, which restricts to a classification of pointed connected $n$-coverings in terms of sub-$n$-groups of $\Pi_n(X,x)$. We study the $n$-group of deck transformations $\mathscr{D}\mathrm{eck}(p)$, identifying it with $\Pi_n(X,x)$-equivariant autoequivalences of the fiber $F$. For normal $n$-coverings, it is further described as a quotient of $\Pi_n(X,x)$, yielding a classification of such coverings in terms of normal subgroups of $\pi_n(X,x)$. For an arbitrary $n$-covering, the deck $n$-group arises as a quotient of a suitable normalizer. Our approach relies on a careful study of $n$-groups and their $\infty$-actions, on the use of univalent universes, and on an internal Yoneda embedding. When $n=1$ and $\mathscr{E}$ is the $\infty$-category of homotopy types, our results recover the classical theory of covering spaces. We further illustrate the theory in sheaf and étale $\infty$-topoi, where the external deck group recovers cohomology of the base, and in cohesive $\infty$-topoi, where it recovers the $1$-covering theory of manifolds.

This paper characterizes random spherical codebooks over the real additive white Gaussian noise channel in the high signal-to-noise ratio (SNR) regime in which the blocklength is fixed, the SNR per real channel use tends to infinity, and the codebook size grows with SNR. The ensemble exhibits a sharp error-probability transition governed by the intrinsic dimension of the sphere and the codebook-growth order. Below the critical codebook-growth order, the ensemble-average error probability vanishes; at the critical order, it converges to a nontrivial limit; and above that order, it approaches one. For a fixed target average error probability, this transition yields the high-SNR expansion of the ensemble-achievable data rate. The achievable rate and the corresponding converse rate bound have the same high-SNR prelog, establishing first-order optimality within the deterministic equal-energy, average-error class. Their ratio tends to one for every fixed blocklength and target error probability, but their additive difference approaches a strictly positive limit that depends on both. We characterize this rate-bound gap jointly in blocklength and error probability. At fixed error probability, it vanishes with increasing blocklength, with a universal leading $1/n$ behavior and reliability dependence first appearing at the next order. For error probabilities that decrease exponentially with blocklength, we identify the threshold between vanishing and nonvanishing limiting gaps. We also derive blocklength laws for meeting a prescribed gap as the target error probability becomes more stringent.

This paper investigates a class of nonlinear elliptic obstacle problems involving measure data, where the nonlinearity depends on the variable of the possibly unbounded solution itself. We establish pointwise gradient estimates for the solutions in terms of Riesz potentials.

This paper determines the structural and proof-theoretic consequences of the forcing condition in relational quantum modal logic, under which every modal transition available at a world is also available at every world compatible with it. We prove that modal successor sets are constant on compatibility components, so boxed truth sets belong to the Boolean algebra of unions of these components. The relation holding exactly between worlds in the same component assigns to each stable proposition its greatest lower and least upper approximations by unions of components, and in hard superselection models these are exactly the approximations by central propositions. We adopt local validity for sequents with multiple conclusions to give a semantics for modal excluded middle on frames with several components. A connectedization obtained by adding one point then shows that component frames, equivalence frames satisfying the forcing condition, and connected compatibility frames with universal modal accessibility have the same logic for sequents with one conclusion. Finally, maximal consistent pairs yield a canonical model, and the calculus obtained by adding T, 4, and B is proved sound and complete for the three frame classes. These results characterize the logical scope of the forcing condition and establish a complete proof theory for component modalities.

We express in polar form the Dirac operator as left-multiplication by an assigned matrix: we use this technique to find its right-resolvent for low-dimensional spaces.

In this work, we present an intrusive stochastic Galerkin formulation of the one-dimensional shallow water linearised moment equations expressed in conservative variables, using the pseudospectral product for generalised polynomial chaos expansions. The shallow water linearised moment equations constitute a hyperbolic system of partial differential equations with an arbitrary number of equations that enhance the accuracy of the standard shallow water equations. Without loss of generality, we assume for both the theoretical analysis and the simulations that the uncertain parameter is the friction coefficient. For the new stochastic Galerkin shallow water linearised moment equations, we derive an energy equation, analyse the hyperbolicity - since this property is not preserved by the stochastic Galerkin projection - and introduce a regularisation to ensure hyperbolicity for the linear case of the shallow water linearised moment equations. Through numerical tests, we demonstrate the accuracy of the new stochastic Galerkin formulation in comparison with a non-intrusive Monte Carlo method, showing that the stochastic Galerkin approach achieves comparable accuracy with significantly faster run times.

In earlier work with Cheng and Hua we showed that on a finite tree the Lin–Lu–Yau discrete Einstein equation is a single linear eigenvalue problem: the metric is the Perron eigenvector of an edge-indexed Ricci matrix. We extend this theory to unicyclic graphs. For geodesic cycle weightings of girth at least four, we characterize exactly when the same local matrix computes the true curvature; this is the balanced regime. Schur complementation of the pendant trees then reduces the eigenvalue problem to a periodic transfer equation on the cycle. We obtain closed formulas for bare cycles and regular suns, prove a sharp maximum-degree bound, and show that the spectral top of a fixed localized defect on a growing cycle converges to the top bound state of the corresponding infinite-line operator. This yields an explicit golden-ratio threshold for balance. Outside the balanced regime the active shortest paths and optimal transport constraints must also be tracked, and phenomena impossible on trees occur: the bare four-cycle has a one-parameter family of Einstein metrics, whereas a triangle with one pendant leaf has none. For every regular sun we construct the symmetric Einstein metric and prove that it is the unique Einstein metric, up to scale, among all positive weightings. As long as a normalized Ricci-flow trajectory remains in the balanced cone, its projective dynamics is generated by the fixed Ricci matrix.

We propose a Newton-based mixed precision iterative refinement framework for solving large-scale sparse continuous-time algebraic Riccati equations (CAREs). The framework computes the initial approximation and the inner Lyapunov correction equations in lower precision, while evaluating residuals and updating the solution in higher precision. To handle indefinite residuals and Newton correction terms in low-rank form, we introduce factor decomposition procedures with truncation strategies that preserve positive semidefiniteness and control rank growth. A first-order rounding error analysis derives a residual recurrence for the refinement process and relates stable mixed precision refinement to a Lyapunov operator conditioning threshold governed by the unit roundoff of the lower precision inner solves. We then present a concrete ADI-based realization, using NLR-ADI for the initial CARE approximation and LR-ADI for the inner Lyapunov correction equations. Compared with dense Lyapunov correction implementations, this realization reduces the main computations to shifted linear solves and low-rank factor operations, and we provide a solver-dependent complexity analysis. Numerical experiments on dense CARE over a range of condition numbers illustrate the conditioning effect described by the error analysis, and experiments on large-scale sparse CAREs show that the mixed precision framework is faster than the full double precision implementation while maintaining the same level of accuracy.

We establish a local obstruction to continuous Lorentzian extensions at a class of warped spacelike singularities. The criterion is expressed through two integrability conditions, a monotonicity condition on the relative warp factors, and divergence of the longitudinal factor; it does not require any symmetry of the closed fiber. The main geometric step is a radial compression of terminal causal traces, which replaces the rotational deformation available in spherical symmetry. The compression yields a compact chronological separator in an adapted boundary chart. Longitudinal translations then produce radial-slice distances that diverge intrinsically while remaining uniformly controlled in the extension chart. Next, we construct the canonical one-horizon Birmingham-Kottler spacetime in global Kruskal coordinates and classify all finite proper-time ends of its timelike geodesics. Every boundary-approaching finite maximizer supplied by a putative extension is thereby forced to the singular end. The local obstruction and the geodesic classification imply $C^0$-inextendibility for the one-horizon Birmingham-Kottler family with nonpositive cosmological constant and every closed connected fiber satisfying $\text{Ric}_{\gamma_\Sigma}=(n-2)k\gamma_\Sigma$, without assumptions of homogeneity, orientability, or simple connectivity.

We prove an all-directions Marstrand-Mattila projection theorem for self-affine measures and sets in $\mathbb{R}^d$. Under exponential separation, together with proximality and strong irreducibility assumptions on the linear parts, the projection of a self-affine measure onto every line has the expected Hausdorff dimension. If the proximality assumption is strengthened to strong pinching, then the same conclusion holds for the self-affine set $X$ itself, without any separation assumption. In the plane, strong irreducibility of the linear parts alone suffices, and this is sharp. As a corollary, if $X$ additionally has upper Minkowski dimension at most one, then its Minkowski dimension exists and equals its Hausdorff dimension, giving a partial affirmative answer to the folklore question of whether the Minkowski dimension exists for every self-affine set.