Papers

In this work, we present a computational model to investigate transdermal insulin delivery using coated microneedles. A detailed skin geometry incorporating a coated microneedles was developed to analyze insulin release through the different skin layers and to evaluate the influence of key transport parameters. The model represents the major skin layers: the stratum corneum, viable epidermis, and dermis. Unstructured grids were used to achieve a reliable resolution of the model. The simulations provide insights into the permeation of insulin from the coated microneedles and the transport and distribution across the different skin layers. Finally, the simulation results were compared with experimental data to evaluate the predictive capability of the model.

We study packings of annuli of a common width, allowing each ring to nest inside the hole of a larger one. The objectives of maximizing contact area and cardinality diverge: area is superadditive in the radius, cardinality is not. Under superincreasing radii, every descending greedy maximizes every positive, strictly increasing, superadditive objective. More strongly, any choice among feasible containers yields the lexicographically maximal feasible set, for containers of arbitrary shape in every dimension. This placement irrelevance holds unconditionally for at most three rings and fails at four in disks and squares; twin instances exclude every universal rule based only on the observable state. Write $\rho=\max_i(\sum_{j>i}r_j)/r_i$. The additive model has threshold exactly $1$. For disks we prove the exact global threshold $\tau=\varphi$, with no failure at $\rho\le\varphi$, for every finite inventory, even with independent hole radii. The key geometric theorem states that, under golden tail bounds, an entire disk list fits a circular container if and only if its three largest disks fit; this supplies the uniform exchange of parents that the threshold proof needs. The Tribonacci constant $T\approx1.83929$ remains the exact floor of a rigid subfamily. A dimension-reduction lemma transfers spherical sharpness results to all dimensions $d\ge2$, and a separate argument proves the golden threshold for at most five rings in those dimensions. For square pans, a Cartesian confinement criterion gives twins and a family proving $\tau_{\square}\le Y\approx1.6845$; its optimality is open. For independent holes, the exact universal area guarantee under $\rho\le\kappa<1$ is $\min(1,\kappa^{-2}-1)$, with threshold $1/\sqrt2$. The repository has 122 Lean theorems. Euclidean geometry, forest assembly and continuity remain written proofs; numerical checks do not substitute for them.

X-ray coronary angiography is the clinical gold standard for coronary artery disease during real-time cardiac interventions, but provides only 2D projections of inherently 3D vessels. Existing learning-based 2D-to-3D reconstruction methods typically require wide angular coverage or multiple views, assumptions that are rarely satisfied in routine practice where only two projections with narrow angular separation are available. To address these challenges, we propose NeCA++, a multi-stage self-supervised neural radiance field (NeRF) framework tailored to clinically realistic acquisition constraints. The framework decomposes reconstruction into two stages that progressively refine spatial support and representation capacity. In the first stage, a coarse 3D representation of the vasculature is reconstructed, restricting the subsequent optimisation to regions with a higher likelihood of vessel presence, termed an active region. Afterward reconstruction is restricted to this region while higher-resolution representations are progressively activated to recover fine vascular details. This multi-stage strategy focuses learning on anatomically plausible regions, mitigates gradient dilution under extreme sparsity, and stabilises global topology before recovering fine vascular branches. Furthermore, two vessel-specific regularisations are introduced: a ray-aligned constraint to reduce projection-induced ambiguity, and a bimodal density penalty to enable early vessel-background separation. Extensive experiments across three datasets (ImageCAS, ASOCA, and Synthetic RCA) and four angular configurations demonstrate consistent superiority over state-of-the-art baselines, particularly under clinically realistic narrow-angle settings, while achieving reconstruction within 58 seconds per case.

The Art Gallery Problem (AGP) asks for the fewest guards that see all of a simple polygon. It is $\exists\mathbb{R}$-complete, hence NP-hard. We show that for a particular class of polygons, confining guards to a single edge makes AGP exactly and efficiently solvable. We call this the Strait Guarding Problem (SGP). Its input is a weak visibility polygon (WVP): a simple polygon where every point is seen from some point of one fixed edge, the base. SGP places the fewest guards on the base that jointly see the whole polygon. First, a structural fact: guards on the base edge that cover the boundary already cover the entire interior, turning a two-dimensional covering problem into a one-dimensional one. Our main result is the Witness-Guard Algorithm, which solves SGP exactly in $O((n + \mathrm{OPT} \cdot \rho)(\log n + \log \mathrm{OPT}))$ time, where $\rho$ is the number of reflex vertices in the WVP and OPT is the minimum number of guards. It is output-sensitive and certifies optimality by a witness set of size OPT derived from its output. We also study the guarding-the-vertex version and prove a tight $\Theta(n \log n)$ bound, with the lower bound following from Sorting. As a corollary of SGP, we obtain two results for altitude terrain guarding (ATG), a special case that SGP generalizes. We give a linear-time perfect-guarding algorithm, improving the previous $O(n^2 \log n)$ bound of Daescu, Friedrichs, Malik, Polishchuk and Schmidt. We also resolve their problem on the minimum guarding altitude, in $O(nk + k^2 \log k)$ time, improving on the $O(k^2 \lambda_{k-1}(n) \log n)$ bound of Kang, Kim and Ahn.

The Gaussian kernel is a widely used similarity measure underlying kernel methods such as kernel PCA and spectral clustering, but computing Gaussian kernel distances for many pairs of points can be expensive. Using Random Fourier Features (RFF), Chen and Phillips [ALT 2017] showed that for points in a $d$-dimensional Euclidean ball in ${\mathbb R}^N$, $t=\Omega((d/\varepsilon^2)\log(dR/\varepsilon))$ features suffice to preserve all pairwise Gaussian kernel distances within a $(1\pm\varepsilon)$ factor with high probability. We establish a uniform relative-error embedding theorem for the more general setting of an arbitrary positive-reach submanifold $\mathcal M\subset{\mathbb R}^N$ of intrinsic dimension $d$. We show that $t=O((d/\varepsilon^2)\log(\operatorname{vol}(\mathcal M)^2N^{2d}/(\operatorname{vol}(B_1^d(0))^2\operatorname{rch}(\mathcal M)^{2d}\varepsilon^{2d+1}\delta)))$, or approximately $O((d^2/\varepsilon^2)(\log N+\log(1/(\varepsilon\delta))))$, RFFs suffice, with probability $1-\delta$, to preserve the Gaussian kernel distance between every pair of manifold points up to relative error $\varepsilon$. Thus the bound depends only logarithmically on the ambient dimension and on manifold parameters such as volume and reach, while retaining the $1/\varepsilon^2$ Euclidean rate. We also prove a topological consequence: under the same RFF embedding, persistent homology is preserved in the sense that weighted Cech and Rips filtrations built from Gaussian kernel power distance are $(1\pm\varepsilon_\star)$-interleaved, where $\varepsilon_\star$ accounts for both distance distortion and kernel-weight approximation.

We prove that every configuration of four through eight points in three-dimensional space, with no four points coplanar, has a connected full geometric flip graph under $2 \leftrightarrow 3$ flips. Every point remains fixed and present throughout the sequence. The proof brings each tetrahedralization into placing form at a convex hull vertex: the tetrahedra not incident to that vertex fill the convex hull of the remaining points. This reduction allows us to establish connectivity by induction, using the connectivity of regular tetrahedralizations. The main geometric tool is radial projection, which turns the tetrahedra incident to a hull vertex into a planar triangulation. When this planar triangulation is regular, varying its lifting heights produces legal spatial flips that progressively shrink the region occupied by the incident tetrahedra and reach placing form. Planar lifting criteria and compatibility constraints between the projections at different hull vertices resolve the remaining small cases. For eight points, at most 35 flips are needed to reach placing form.

Automatically generating high-quality conforming hexahedral (hex) meshes from general input boundaries remains a challenging problem, despite the numerical advantages hex elements offer in simulation. Grid-based adaptive refinement, followed by hanging-node removal, is the most robust fully automatic choice. Among two-refinement methods, primal templates have better mesh quality than dual templates but depend on a strong balancing condition that over-refines the grid; neither family guarantees planar quadrilateral (quad) faces. This paper generalizes recent primal three-refinement templates to a two-refinement scheme under a moderate balancing condition, a relaxation of the strong one. 32 special and five fundamental patterns resolve all 144 symmetry-reduced configurations, followed by either a fast greedy algorithm or an integer linear programming (ILP) algorithm that trades extra runtime for a further reduction in element count. The scheme is the first two-refinement method in which all hexes have planar faces, and its meshes feed directly into a recent quality-guaranteed hex mesh reconstruction algorithm. At comparable element counts, it attains a similar Hausdorff ratio (HR) and a higher minimum scaled Jacobian (min SJ) than its three-refinement counterpart.

This work presents an algorithm, together with its correctness proof, to determine the minimum perimeter polygon (MPP) for digital objects given as regular complexes in the triangular plane tiling. Such objects are edge-adjacency-connected sets of triangle tiles that have no end tiles, and the point set union of all their tiles forms a simple polygon. Nevertheless, the boundary paths of the objects are not assumed to be simple. Then the MPP is a weakly simple polygon that coincides with the relative convex hull (i.e., geodesic hull) of a set $A$ with respect to a simple polygon $B$, where $A\subset B$, but $A$ is not necessarily a polygon, in fact it is generally not connected. Our MPP algorithm relies on constructing and iteratively constraining cones of visibility through forthcoming boundary tiles, it uses the structure of the canonical boundary path, the MPP frontier is the shortest polygonal curve following this path. We also propose a boundary tracing algorithm to obtain such paths from the objects.

A contiguous cartogram is a map whose area is proportional to a quantity such as population. The defining condition, that the Jacobian determinant of the deformation equals the density, is one equation for two unknown functions, so every cartogram method adds a tie-breaker, usually implicitly. Optimal transport makes the tie-breaker explicit: among all density-equalising maps of the frame onto itself, take the one that moves the population least in the mean-square sense. By Brenier's theorem that map is the gradient of a convex potential, so it has no local rotation anywhere and cannot fold. We compute this map for the world population of 2025 on a 4096 by 4096 Mercator grid (10 km cells) from the GHS-POP raster, using the fixed-point Monge-Ampère iteration of Benamou, Froese and Oberman with spectral Poisson solves, continuation in the population share and an ocean-only buffer, on a laptop GPU. Against a Gastner-Newman diffusion cartogram of the same density we find the same transport cost to within 0.4 per cent and density errors of a few per cent for both, but a median local rotation of 8.8 degrees for diffusion against 0.01 for transport, and a median anisotropy of 6.30 against 3.93. The construction extends to local refinement, by transporting a city's 100 m population onto the area measure the global map assigns to it, and to a semi-discrete counterpart: 8,192 Laguerre cells of 1.00 million people each. Code, data provenance and every figure's inputs are public.

Let ${R^\ast}(n)$ be the least radius of a central circle to which nonoverlapping circles of radii $1,\ldots,n$ are externally tangent. We prove that ${R^\ast}(n)={C_\ast} n^2+o(n^2)$ and characterize ${C_\ast}$ by finite linear programs with an explicit error tending to zero. The reduction preserves arbitrary orders and all pairwise constraints: the limiting problem places marked points on a line at pairwise separation at least the geometric mean of their marks. Concatenation with a bounded boundary cost proves existence, and balanced finite-word programs supply matching effective upper and lower bounds. Their certified gap is $(1/k+1/r)/\pi$, before directed arithmetic error, for $k$ mark types and words of length $r$. A quantitative reflected-block recovery theorem supplies genuine permutations and full ring geometry, with a countable extension and a strict four-block improvement. The explicit interval is $C_{\mathrm{term}}+\eta_{\mathrm{width}}\le{C_\ast}\le U_4$; neither endpoint is asserted sharp. In particular, the coefficient $1/8$ proposed in the preceding finite study is false. An elementary expression for ${C_\ast}$, efficient high-precision evaluation and global floating-circle structure remain open.

Given integers $c\geq 2$ and $s\geq 0$, let $\mathsf{M}_3(c,s)$ denote the least integer such that every set of at least $\mathsf{M}_3(c,s)$ points in the plane, no three on a line, colored with $c$ colors, contains a monochromatic triangle with at most $s$ interior points. Further, let $\lambda_3(c)$ be the least integer such that $\mathsf{M}_3(c,\lambda_3(c))<\infty$. \citet{colorempty} proved that, for every $c\geq 2$, $$\left\lfloor\frac{c-1}{2}\right\rfloor \leq \lambda_3(c)\leq c-2.$$ Later, \citet{cravioto2019almost} improved the upper bound to $c-3$, for $c\geq 4$. In this paper, we refine their argument to obtain the following asymptotic improvement: $$\lambda_3(c) \leq c-\sqrt{c\log c}+o (\sqrt{c\log c} ),$$ for all sufficiently large $c$. We also show that every $c$-coloring of a sufficiently large Horton set contains a monochromatic triangle with at most $\lfloor \frac{c-1}{2} \rfloor$ interior points. This shows that the aforementioned lower bound on $\lambda_3(c)$ is sharp within the class of Horton sets. We conclude with a conjecture on the large-color asymptotics of $\lambda_3(c)$.

We present a framework for tracking topological features directly within continuous implicit models. Such models, including implicit neural representations (INRs) and multivariate functional approximations (MFAs), are increasingly adopted to represent scientific data without the resolution constraints of discrete grids. They offer compact, smooth, and differentiable representations of complex fields, enabling new opportunities for high-performance data storage, reconstruction, and analysis. Given a continuous implicit model, our method tracks the evolution of critical points by querying the model and its derivatives, thereby eliminating the need to resample onto a grid. This approach enables faithful feature tracking while avoiding discretization-induced artifacts such as aliasing. We demonstrate the generality of our framework across a range of implicit representations, including analytic functions, MFAs, and INRs, and show that it produces smooth, coherent critical point trajectories. By enabling feature tracking directly on continuous representations, our method supports a new class of feature-driven visualization workflows centered on implicit models.

A rotation system on $n$ elements assigns to each element a cyclic order of the other $n-1$ elements. A four-element subset is non-planar if its induced rotation system cannot be realized by a crossing-free drawing of $K_4$. As a combinatorial strengthening of Hill's conjecture on the crossing number of the complete graph, Archdeacon conjectured that every rotation system on $n$ elements has at least $H(n)=\frac{1}{4} \lfloor\frac {n}{2}\rfloor \lfloor\frac{n-1}{2}\rfloor \lfloor\frac{n-2}{2}\rfloor \lfloor\frac{n-3}{2}\rfloor$ non-planar four-element subsets. We computationally verify Archdeacon's conjecture for $n\leq 10$ and show that every extremal rotation system in these orders is realizable by a simple drawing. With computer assistance, we prove that every rotation system on $n$ elements has at least $(8/9 - o(1)) H(n)$ non-planar four-element subsets. We also present a proof by hand for a weaker lower bound of $(2/3-o(1)) H(n)$. Finally, extending recent work of Felsner on antipodal pairs in drawings, we show that Archdeacon's conjecture holds for antipodally shellable rotation systems.

Boundary representation (B-rep) is the standard format used by modern CAD systems for parametric 3D models. It turns out, the exact same solid can be represented by different B-reps: for example, two engineers using different operations, a geometry kernel rebuilding the file, and an export setting repartitioning faces will lead to different B-reps even though the underlying solid remains the same. We show that existing B-rep encoders are not robust to variation in the B-rep with the same solid on perturbations applied to standard benchmarks, naturally occurring variations inherent to CAD software, and differences in how designers model the same part via a human dataset we created in FreeCAD. The performance of popular B-rep encoders often collapses catastrophically. We propose the canonical region graph, an input representation whose nodes, features and coordinate frame are derived from the solid itself and show theoretical invariance guarantees on repartitioning and rigid motions. It matches the strongest baseline on standard benchmarks, and is stable under every perturbation we test.

A point set is universal for planar graphs on $n$ vertices if every such graph has a straight-line drawing without crossings whose vertices belong to the set. We construct universal point sets of size $n^{1+o(1)}$, improving the previous quadratic upper bound. Our construction uses the reduction of Bannister, Cheng, Devanny, and Eppstein from universal point sets to superpatterns for $213$-avoiding permutations. We represent these permutations by ordered rooted forests and construct a small family of intervals containing every such forest. The result follows from a straightforward bound on the size of the family of intervals. GPT-6 Astra assisted in developing the construction and proof.

Geodesic polar coordinates (GPCs) provide an intrinsic parameterization over curved surfaces, but their accurate estimation remains challenging, particularly in the presence of anisotropic metrics, high curvature and complex topology. We introduce iLogMap, a method for computing GPCs in curved domains that recasts the angular component of the logarithmic map to a ground-state magnetic eigenproblem over the circumferential direction field of geodesic distance. Our method effortlessly extends to anisotropic metric tensors and solid volumes, enabling cylindrical and spherical parameterizations in tetrahedral meshes. Experiments on diverse shapes with varying genus confirm competitive angular accuracy and reduced metric distortion relative to heat-based methods, with improved performance on surfaces with boundary and domains with anisotropy. We demonstrate the utility of iLogMap in computational cardiology applications, where we use it to initialize spiral phases on atrial surfaces and estimate local activation patterns in ventricular models.

Computing the closest point on a clothoid is a recurring task in geometric design, road and railway alignment, and path planning. The efficient algorithm of Frego and Bertolazzi addresses this problem, but its candidate-selection analysis assumes at most one local minimum per search interval. We exhibit admissible configurations with two local minima, raising the question of whether the existing strategy accounts for every possible minimum. Using the geometry of the clothoid evolute, we prove that, for any query point and any proper no-inflection planar clothoid segment with tangent-angle variation at most $2\pi$, the squared-distance function has at most three stationary points; if all three are local extrema, their order is min-max-min. This establishes the completeness of the original candidate-selection logic beyond the one-minimum premise. It also shows that no interior search is needed when neither endpoint derivative test is active, allowing unnecessary midpoint searches to be omitted while retaining numerical fallback. Numerical experiments demonstrate reductions in iteration count and evaluation time.

We consider online packing of axis-parallel squares into a unit-width strip under the Tetris and gravity constraints: An incoming square must be lowered from above along a monotonic downwards path until it reaches support from below. Fekete, Kamphans, and Schweer [Algorithmica, 2014] gave an algorithm with asymptotic competitive ratio $34/13\approx2.6154$ in this model. We present $\mathrm{AsymmetricSlots}$, a recursive algorithm based on splitting each slot into a wide and narrow subslot. The proof uses a local charging argument: squares that are large relative to its associated slot pay for the height they create with their own area, while smaller squares are balanced between the two subslots and may use a bounded temporary credit. For a suitable split parameter $p^\star$, we prove $\mathrm{AsymmetricSlots}{p^\star}(\sigma)\le 2.37332 \operatorname{OPT} (\sigma)+O(1)$ for every input sequence $\sigma$. Additionally, we show that the same framework gives an algorithm with asymptotic competitive ratio $O(\kappa)$ for rectangles of aspect ratio at most $\kappa$, and a matching $ \Omega(\kappa)$ lower bound shows that the dependence on $\kappa$ is asymptotically optimal. For the square algorithm, we give a lower bound of $2$ on its asymptotic competitive ratio.

In this paper we introduce a generalization of Helly's theorem closely connected to Bárány-Gromov overlap theorems (also called selection lemmas). Our main result implies both the topological colorful Helly of Kalai and Meschulam and Karasev's topological centerpoint theorem. We further investigate the topological fractional Helly theorem from this overlap perspective, and show an overlap theorem for dense complexes (a continuous second selection lemma for tame maps).

Despite the prominence of hyperbolic surfaces in mathematics, basic algorithmic questions about them, even computing the distance between two points, have remained open, leaving many features of these surfaces inaccessible. The classical machinery assumes a polyhedral structure absent on a smooth surface. We remove these obstacles. We begin with an efficient $O(g^2)$ algorithm for the distance between two points, where $g$ is the genus of the surface. Building on it, we obtain an $O(g^2 \log g)$ method for answering distance queries from a fixed source and, as a consequence, for recentering a Dirichlet domain around an arbitrary point. This understanding of distances on the surface then lets us approximate the diameter to within any $\eps$ in time $O(g^3 \log g / \eps^2)$. We further show that the diameter, a single real number encoding a great deal about the surface, is exactly computable. Its hyperbolic cosine is an algebraic number over the field encoding the coefficients of the hyperbolic isometries defining the surface.

We prove that every convex body $K\subset\mathbb{R}^3$ of constant width $w>0$ satisfies \[ \operatorname{Vol}(K)\ge \frac{1651}{4000}w^3=0.41275\,w^3. \] This improves the certified lower bound $0.411040473721188\ldots\,w^3$ announced by Hyra and the earlier spectral bound $4\pi w^3/33$ of Nishioka. The main new ingredient is a contact-tetrahedron estimate near equality in Jung's circumradius inequality. Four balanced contacts of a minimum circumscribed ball form a near-regular tetrahedron; constant width then forces, beyond the tetrahedron itself, spherical regions over its faces, spindle sectors along its edges, and additional face-edge sectors. These regions can be selected with pairwise disjoint interiors and quantified by the total squared-edge deficit. A sharp convexity argument relates that deficit to the circumradius, and a symmetrization argument controls the four face contributions with only quadratic loss. This gives the analytic bound $\operatorname{Vol}(K)>0.41275318$ near the Jung endpoint. For the complementary circumradius range we use Hyra's radial transport/tangent-majorant mechanism together with closed-form contact and antipodal profiles. The antipodal contribution has an elementary primitive. For the contact contribution, a second-order concavity estimate converts the angular Jacobian into an explicit rational lower density; that density is monotone in the circumradius. The whole intermediate range therefore reduces to a single endpoint calculation, giving $\operatorname{Vol}(K)>0.41275168$. For smaller circumradius the tangent-majorant inequality with its nonnegative gap discarded already gives $\operatorname{Vol}(K)>0.41308109$. The proof is entirely analytic and requires no computer-assisted certificate.

The dilation of a geometric graph measures how much longer the path between pairs of points becomes when restricted to graph edges, rather than following the direct path through the ambient space. The minimum dilation tree of a point set is the spanning tree with minimum dilation, where edge lengths in the tree are given by distances in the ambient space. In the Euclidean plane, computing the minimum dilation tree is NP-hard, but no hardness of approximation result is known. On the other hand, the minimum spanning tree is an $(n-1)$-approximation to the minimum dilation tree, but no asymptotically-better approximation algorithm is known for general point sets in the Euclidean plane. We give the first sublinear approximation algorithm for the minimum dilation tree in the Euclidean plane. Our approximation ratio is $\tilde{O}(n^{14/15})$ and our algorithm runs in polynomial time. This resolves an open problem proposed by Eppstein in 1996.

We study the Longest Edge Bisection (LEB) process as a dynamical system on the projective shape space of simplices. A long-standing conjecture going back to Adler and Rivara-Levin and motivated by finite-element mesh refinement, often taken as a standing assumption, is that this procedure is non-degenerate and, in fact, in a certain way periodic. We prove: \begin{itemize} \item There are 3-dimensional simplices such that the longest edge-bisection algorithm degenerates. \item There is an open set of 4-dimensional simplices on which the longest edge-bisection algorithm degenerates. \item If parametrizing the space of $d$-dimensional simplices by independent standard Gaussian vectors, then as $d$ increases, a random simplex degenerates asymptotically almost surely. \end{itemize} This is realized through exhibiting hyperbolic behaviour of the LEB process. We also exhibit elliptic behaviour that is nonperiodic.

Delaunay triangulations are a fundamental class of plane spanners, and determining their worst-case stretch factor has been a longstanding problem in computational geometry. We prove an upper bound of \(1.65\), improving the bound of \(1.998\) due to Xia (2011) and reducing the gap to the known lower bound of \(1.5932\) by a factor of more than seven. Our proof works with the chains of circumdisks introduced by Xia, along which a path between two sites is assembled disk by disk. Xia measures such a path against a quantity attached to the whole chain, and because that quantity is not additive, his induction has to be carried alongside a separate global estimate. Our main idea is to measure the path against the progress it makes along the segment joining the two sites. This quantity is additive, so the bound becomes a Bellman recursion that forgets all but one number about the disks already passed, and we show that the bound holds if and only if a potential on the current state satisfies three local inequalities. The smallest feasible potential is the value function of that recursion, so searching for a potential becomes the problem of fitting this value function from above. The geometry of the disks reduces the fit to a linear program over functions of one variable, in which a GPT-based multi-agent system that we developed found a feasible point, certified in exact arithmetic.

Fairness in clustering has attracted sustained research interest, motivated by the need to ensure equitable representation of protected groups in machine learning applications. We study fair $k$-means clustering in Euclidean space, where the proportion of each protected group in every cluster must lie within specified lower and upper bounds. These constraints make it challenging to determine both cluster centers and point assignments. We propose an approximation algorithm that combines a linear programming relaxation with geometric transformations of the input to construct candidate center sets. Given a $\rho$-approximate algorithm for weighted $k$-means and any $\epsilon>0$, our algorithm returns a fractional solution whose cost is at most $1+(3-1/\Gamma)\rho+O(\epsilon)$ times the optimal integral fair cost, where $\Gamma\approx6.357$ is an upper bound on the integrality gap of the standard Euclidean $k$-means LP. With a PTAS as the subroutine, the approximation ratio becomes $3.8427+O(\epsilon)$, improving the previous factor of $5+O(\epsilon)$ to below $4$. The solution satisfies all fairness constraints exactly and can be rounded to an integral assignment with a bounded additive violation of fairness and no increase in cost. The same approximation guarantee extends to the $k$-sparse Wasserstein barycenter problem.

A convex body is mono-monostatic if, resting under gravity on a horizontal plane, it has exactly one stable and one unstable equilibrium position. Smooth mono-monostatic homogeneous bodies exist (the Gömböc of Domokos and Várkonyi), and Lángi proved that (homogeneous) mono-monostatic polyhedra exist; although no explicit example appears to have been published, to the author's knowledge. We construct explicitly two such mono-monostatic polytopes, the smaller one having $56946$ faces; importantly, we certify them: the polytope is presented as an intersection of half-spaces with rational data, and a certifying verification establishes, using exact rational arithmetic for every decisive comparison, that the body has equilibrium signature $(S,H,U)=(1,0,1)$ with respect to its own exact centroid, with explicit nondegeneracy margins. We describe the geometric obstructions that make naive discretisations of smooth mono-monostatic bodies fail, the adaptive construction that overcomes them, and the certification strategy. While the result itself is not strikingly novel, we emphasise the non-standard (but increasingly more common) methodology: the entire programme, i.e. experiments, constructions and the verifier itself, was implemented by AI agents under human mathematical direction. We argue that exact certification of numerically discovered objects is the natural contract between such AI-assisted workflows and mathematical standards of rigour.

Hilbert-space embeddings enable inference for populations of persistence diagrams, but separation between individual diagrams need not survive population averaging. We develop a framework for inference on population mean embeddings, with particular attention to the additive landmark representations PLACE and PALACE. Treating each diagram as one independent observation, we apply Hilbert-space limit theory to obtain covariance estimators, two-sample tests, and confidence balls under suitable moment conditions, without requiring a lower-distortion bound. For additive embeddings, we identify the population mean as an embedding of the mean counting measure and show that geometric separation of these measures alone cannot guarantee uniform testing power. We then introduce a model with latent template diagrams, missing features, and location perturbations. Under common or feature-specific prevalence conditions, a diagram-level lower-distortion certificate yields explicit lower bounds on population mean separation. These margins provide finite-sample uniform power guarantees, and an additional information-divergence comparison gives matching sample-complexity bounds over restricted scale ranges. Confidence sets yield lower bounds on transport separation of population mean measures and exclusion guarantees for specified structured alternatives. We also quantify how orthogonal truncation changes the certified signal and the approximation allowance needed for confidence sets targeting the full embedding, relating sample size, retained coordinates, and template separation. Simulations examine calibration, power, and coverage, and an analysis of resting-state connectivity from the Autism Brain Imaging Data Exchange illustrates the procedures.

In a weighted graph $G = (V, E)$, the maximum weight matching problem (MWM) asks for a matching (i.e. pairing) of its vertices, such that each vertex is paired with at most one other vertex and the total sum of weights of all edges connecting paired vertices is maximised. If the vertices of the graph correspond to points in the Euclidean plane and the weights to their pairwise Euclidean distances, we get the Euclidean maximum weight matching problem (Euclidean MWM). The best optimum-solution algorithm for this problem runs in $O(n^{2.5})$. Furthermore, there exists an FPTAS guaranteeing a $(1 - \epsilon)$-approximation ratio, which runs in $O(m \epsilon^{-1} \log \epsilon^{-1})$ time. Heuristics with a subquadratic running time (with respect to the number of vertices $|V|$) are known, but often yield solutions of a modest quality. In this paper, we present a novel algorithm for solving the Euclidean MWM running in $O(n \log n)$ time and providing excellent solution quality, especially for larger instances. In particular, in our computational tests, the algorithm yielded optimum or near-optimum solutions for all test instances; the worst observed optimality gap was less than $1.38\%$. This makes the algorithm highly attractive for practical applications, especially when exact methods become computationally prohibitive due to the size of the instance.

For clique complexes $X_1\subseteq X_2$, normalized persistence in degree $d$ is $\operatorname{rank}[H_d(X_1)\to H_d(X_2)]/\dim H_d(X_1)$. Estimating it requires exact endpoint homology and the inclusion-induced map, even with inverse-polynomial endpoint Laplacian gaps. We prove that additive-error $1/24$ estimation is hard for $\mathsf{BQP}_{1}^{G_2}$, the perfect-completeness class over the exact gate set $G_2=\{X,\mathsf{CX},\mathsf{CCX},H\otimes H\}$, and hence for $\mathsf{BQP}_{1}$ over every finite gate set with entries in a cyclotomic field $\mathbb{Q}(\zeta_{2^k})$, even for unweighted clique complexes. The main tool is a finite-certificate kernel-transfer theorem. For a fixed palette of weighted clique gadgets satisfying finitely many exactly checkable local conditions, every unit chain $x$ of the full geometric complex satisfies $\operatorname{dist}(x,K)^2\le C(t\lambda^2+\langle x,\Delta x\rangle/(g\lambda^{26}))$, where $K$ is the embedded kernel of the simulated projector Hamiltonian, $g$ its gap, $t$ the number of gadgets, and $\lambda$ the private vertex weight. Since $\lambda$ is chosen independently of $g$, the geometric Laplacian has exactly $\dim K$ zero modes and a gap linear in $g$ above them. Exact fillings identify the endpoint homology with a quotient $V/W_A$ of the register cycle space, and nested term sets induce the natural quotient epimorphisms, so the persistent rank equals the later kernel dimension without any choice of compatible harmonic representatives. A fixed eight-dimensional label register turns a $\mathsf{BQP}_{1}^{G_2}$ verifier into instances with $\beta_d(X_1)=8$ and normalized persistence $3/4$ or $1/8$, and an established common-copy blow-up transfers everything to unweighted graphs.

We study the query complexity of estimating scalar summaries of zigzag bar lifetimes from snapshot-adjacency bits. For graphs $G_1,\ldots,G_m$ on $n$ labeled vertices, let $\ell_b$ be the snapshot lifetime of a degree-one bar $b$ of the intersection zigzag. For a probability generating function $\phi(x)=\mathbb{E}[x^R]$, the statistic $F_\phi=\sum_b\phi(\ell_b/m)$ includes normalized degree-$r$ total persistence and the mean generalized rank over a uniform time window. An exact identity underlies our algorithm: sample $R$ uniform times; the expected generalized rank between their minimum and maximum equals $F_\phi$. For graphs that rank is the circuit rank of an intersection graph, so a nonlinear barcode functional becomes an average of edge and component counts, and no barcode is computed. Without spectral-gap, homology-state, or QRAM assumptions, this gives a quantum estimator with additive error $\varepsilon n$ and $\widetilde O(\sqrt{m(K+n)}/\varepsilon)$ queries when a bound $K\ge F_\phi$ is supplied, against $\widetilde O(m\min\{n^2,(K+n)/\varepsilon^2\})$ classically, and an adaptive quantum variant with the same instance dependence. These estimators are optimal in two regimes. For every fixed power weight $x^r$, $r\ge2$, and for the uniform-window mean, the worst-case complexities are $\widetilde\Theta(n\sqrt m/\varepsilon)$ quantum and $\Theta(n^2m)$ classical. On sparse instances, under an explicit split-leakage promise met by power and binomial weights of logarithmic degree and the promise $F_\phi\le K$, they are $\widetilde\Theta(\sqrt{mK}/\varepsilon)$ and $\widetilde\Theta(m\min\{n^2,K/\varepsilon^2\})$. The classical lower bounds hold against fully adaptive algorithms, and fewer than $m$ such statistics cannot determine the positive-lifetime histogram. All bounds concern snapshot access; with an explicit update stream, near-linear full-barcode algorithms are known.