Papers

We introduce a natural Markov chain on chord diagrams, which, at every step, selects two random chords and swaps them if doing so preserves the diagram's genus. This generalizes the chord swap chain on the Catalan structure of non-intersecting chord diagrams. We show that for fixed genus, the chain mixes in polynomial time.

We consider $N$ independent random walks $X^1,\dots,X^N$ in the lattice $\mathbb{Z}^d$ and prove limit theorems for the competitive range $\mathcal{R}_n^k$ of the $k$-th random walk $X^k$, which corresponds to the number of distinct sites that it has discovered before any of the other $X^\ell$, $\ell\ne k$, up to time $n$. This is a natural object to study foraging mechanisms in population ecology, in which context it is also natural to ask how the effect of competition for the access to resources affects the number of resources consumed by each individual. We work with random walks in the domain of attraction of a $\beta$-stable law and focus on the regime $d/\beta\in[1,3/2)$, in which classical results for the range show that the fluctuations are described by the renormalized self-intersection local time of the limiting process. We establish a central limit theorem in which a competition term emerges, thus answering the two previous questions we asked. We end the paper with a brief discussion on the remaining regimes $d/\beta\ge3/2$, in which the fluctuations are Gaussian and are not affected by the competition, and $d/\beta<1$ in which no strong law of large numbers holds and we expect the effect of the competition to strongly affect the first-order asymptotics.

We design exact algorithms for the ham-sandwich and centerpoint theorems for polytopal measures. Our key observation is that the cap-volume function of such a measure, i.e., the volume cut off by a halfspace, is piecewise rational on a natural decomposition of the space of oriented hyperplanes. This lets us recast prescribed-proportion cutting problems as semialgebraic feasibility problems. For fixed ambient dimension, this yields polynomial-time algorithms to decide the existence of cuts, describe the full solution set, and sample or enumerate solutions. We extend this framework to the center transversal theorem, showing that spaces of deep affine flats are semialgebraic, which holds for centerpoints. We further show that the set of centerpoints of a convex polytope coincides with its floating body at level $1/(d+1)$, a useful semialgebraic description.

We study a Brown–York type mass for closed hypersurfaces in four-dimensional asymptotically flat manifolds. The reference mean curvature is defined intrinsically as the trace of the positive solution of the contracted Gauss equation. For large uniformly convex hypersurfaces with controlled scale, we derive an expansion consisting of a boundary term converging to the ADM mass and a shape-dependent correction. For the four-dimensional analogue of the nearly round surfaces of Shi–Wang–Wu, this correction vanishes under a natural decay compatibility condition.

This work develops an algebraic framework for merging two $3$-anchored bundles over the same base manifold, equipped with mutual actions and two twisted cocycle terms, so as to obtain a $3$-Lie algebroid structure on the corresponding Whitney sum. We also record the purely algebraic counterpart of this construction, namely the bicocycle double cross product $3$-Lie algebra, obtained by removing the anchor and Leibniz-type compatibility conditions. The resulting framework provides a unified setting for $3$-Lie algebroids and contains, as special cases, unified products, double cross products, semi-direct products, cocycle extensions, and direct products.

Hybrid beamforming is a promising solution for high-frequency multi-antenna wireless systems, but its implementation is constrained by the cost and complexity of analog phase-shifter (PS) networks. Although sub-connected architectures simplify the analog network, their conventional realization still requires a dedicated PS for each antenna, causing considerable layout area, wiring, calibration, and control overheads. To address this issue, this paper proposes a novel static-connection architecture with sparse PSs for ultra-low-cost sub-connected hybrid beamforming, where antennas within each sub-array share a PS through an optimized fixed PS-to-antenna connection matrix. The proposed architecture preserves static connections while enabling dynamic beam control via adaptive PS phase-shift adjustments and digital precoding. For the single-radio-frequency (RF)-chain scenario, the sparse-PS connection design is transformed into an antenna-grouping problem, with analytically characterized structural properties and an efficient algorithm. For the multi-RF-chain scenario, we develop a quality-of-service (QoS)-majorization-minimization (MM) algorithm to handle the mixed discrete-continuous optimization problem. Numerical results demonstrate that the proposed architecture reduces the PS count while preserving most beamforming capability of the traditional full-PS sub-connected architecture. In particular, the proposed design achieves PS-count reductions of 37.5% and 62.5% in single-RF-chain and multi-RF-chain systems, respectively, while avoiding deep-null and grating-lobe degradations associated with deterministic connection schemes. These results provide engineering insights into static sparse-PS sharing: the key to hardware-efficient hybrid beamforming is not merely reducing the PS count, but also preserving essential analog-domain degrees of freedom through optimized PS connection topologies.

In this paper, by using the arithmetic properties of the Gauss periods and character sums over cyclic groups, we study the cyclotomic matrix $$A_k(\chi)=\left[G_N(\chi^{ki+ki})\right]_{0\le i,j\le \varphi(N)/k-1},$$ where $N=p^m$ is a prime power, $\varphi(\cdot)$ is the Euler totient function, $k$ is a divisor of $\varphi(N)$, $\chi$ is a generator of character group $\widehat{(\mathbb{Z}/N\mathbb{Z})^{\times}}$, and $$G_N(\chi^{ki+kj})=\sum_{x\in\mathbb{Z}/N\mathbb{Z}}\chi^{ki+kj}(x)e^{2\pi ix/N}$$ is the Gauss sum over $\mathbb{Z}/N\mathbb{Z}$.

This dissertation provides the first systematic study of symmetric consensus-breaking bifurcation to periodic multiconsensus in multi-agent systems. It analyzes this for three classes of multi-agent systems based on three different types of memory, whose closed-loop dynamics equations form delay differential equations of retarded type, neutral type, and pseudoneutral type - a subclassification of retarded type equations introduced in this dissertation which bridges retarded and neutral type delay equations. Equivariant twisted degree is used to analyze the symmetric global Hopf bifurcation problem in these systems, i.e. bifurcation from a stable consensus to periodic multiconsensus. This shows how the effects of memory allow self-organizing agents to move beyond mere stationary consensus. Theoretical results for the global Hopf bifurcation and symmetric classification of periodic multiconsensus solutions across all three systems are provided, and numerical results are conducted to both validate and enhance the theoretical predictions by providing stability information on the branches which is not obtainable by the degree alone. These principles are demonstrated in three real-world applications: one involving the control of formations of UAVs, allowing them to maintain their overall spatial relationships while dancing in complex selectable oscillations; and two more in networked asset markets featuring different traders with different memory-based strategies, showing how similar mechanisms can be responsible for economic cycles of bubbles and crashes. Finally, we also numerically investigate resonant double Hopf bifurcations in the neutral delay system, showing strong evidence of a breakdown to chaos via the Ruelle-Takens-Newhouse scenario and the existence of riddled basins.

In this paper, we study generalized rank weights (GRWs) and extended generalized poset weight (EGPWs) of codes over rings via a Galois connection approach. First, we show that various coding-theoretic properties related to generalized weights, including security drops of a code employed in wire-tap channel of type II, connections between generalized weights of a Gabidulin code and its associated Delsarte code, (generalized) Singleton bound, MDS discrepancy of a code, characterizations of MDS, near MDS, $i$-MDS, MRD, near MRD, $i$-MRD, (dually) quasi-MRD codes as well as evasive property of subspaces, can be reformulated in terms of Galois connections. Next, we study GRWs and rank profiles defined for modules over principal ideal rings, especially those over chain rings. Generalizing GRWs defined for vector spaces over fields, we establish a singleton bound and a Wei-type duality theorem, characterize MRD, near MRD and dually quasi-MRD codes and determine their GRWs; moreover, we characterize $i$-MRD codes and establish a scattered bound for $(h,h)$-evasive codes over chain rings, generalizing counterpart result established for vector space over finite fields. Finally, we propose and study EGPWs and extended poset profiles defined for modules with a composition series, which in fact form a Galois connection. Generalizing EGPWs defined for modules over finite Galois rings, we establish a Wei-type duality theorem for modules over arbitrary quasi-Frobenius rings, which unifies the two Wei-type duality theorems derived in both \cite{32} and \cite{33}.

Researchers are increasingly prompting language models with cultural personas so that they answer questionnaires as diverse human groups would. However, studies often report that these persona responses do not predict the models' behavior. We argue that whether the persona is Western or non-Western, the model's answers carry a cultural preference, and that this preference masks the variables that could predict its behavior. We examine the prompted preference at two levels: aggregate Big Five scores and response geometry, the correlation matrix between dimensions within each respondent, represented as a symmetric positive-definite (SPD) matrix. In DeepSeek-V3 and GPT-4o, when the item order is fixed, the prompted preference appears at both levels. Randomizing item order reduces it in the response geometry, while leaving it in the Big Five scores: the Big Five still separate the two personas at $95$–$97\%$, while response geometry falls to chance. In randomized administration, the response geometry still contains information about the respondent. In DeepSeek-V4-Flash, SPD features are associated with whether the same instance sorts Asian American public figures as “foreign” in a text IAT ($p = .047$), while the Big Five do not. In human studies conducted using randomized item order, the response geometry identifies people from test-retest sessions in $81$ – $85\%$, against chance $0.4\%$, and predicts sex and, weakly, age. In addition, SPD features also predict complex human behavior (e.g., cumulative GPA, $R^2 = 0.28$, permutation $p = .003$), where the Big Five do not.

In this paper we study a class of stochastic chemical systems that, in general, do not satisfy the property of detailed balance nor the property of complex balance. These systems are obtained by adding sources and sinks to conservative chemical systems. This procedure is a way to define rigorously stochastic chemical systems in contact with reservoirs. We prove that these systems are non-explosive Markov chains and we prove that they converge to a steady state as time tends to infinity. The stationary solution are out of equilibrium solutions. We conclude the paper by applying our results in order to describe fluxes of molecules through some membrane channels.

We compute the weight distribution of the third-order Reed–Muller code RM(3,11) of length 2048. The weight enumerator is assembled from the coset weight enumerators of f+RM(2,10), evaluated for representatives of all 3691560 nonzero GL(10,2)-orbits of Boolean cubic forms in ten variables. The computation rests on a structural theorem: a nondegenerate Boolean cubic form admits a nondegenerate hyperplane restriction, except for a single orbit in each odd dimension. The same pass determines the second-order nonlinearity of every cubic form: the relative covering radius of RM(2,10) in RM(3,10) is 408, attained on 179 orbits. This raises the best known lower bound on the covering radius of RM(2,10) from 400 to 408. A complementary heuristic search shows that the relative covering radius of RM(6,10) in RM(7,10) is at most 32, improving the previous bound of 50.

We study the problem of deterministically computing the exact root of a sparse polynomial in the multivariate setting. Let $f \in \F[x_1,\ldots,x_n]$ be a nonzero polynomial that is an exact $e$-th power, say $f = g^e$. Suppose $f$ is $s$-sparse, has an individual degree of at most $d$, and a total degree of $D = \tdeg(f)$. We prove a sparsity bound on the base polynomial $g$: \[ \|g\|_0 \le s^{D(2d+2)/e + 1}. \] Based on this bound, we develop a deterministic algorithm that computes the base $g$. % In contrast to the general deterministic factorization algorithm of Bhargava, Saraf, and Volkovich \cite{BhargavaSarafVolkovich2020}, which achieves only a quasi-polynomial dependence on the input parameters, our algorithm is polynomial-time in the setting where the total degree $D$ is bounded. Specifically, the overall complexity is \[ \mathrm{poly}\left(s^{O(Dd)}, n, d, D\right) + s\cdot R(e), \] % where $R(e)$ denotes the cost of constructing a single $e$-th root of a scalar in the base field $\F$, and, when $\operatorname{char}(\F)\mid e$, the cost of computing a single Frobenius root of a scalar. % This term is field-dependent, and over finite fields, $\mathbb{Q}$, or number fields with a suitable representation, it is absorbed into the polynomial complexity bound. % Within the bounded total-degree regime, this yields a deterministic polynomial-time algorithm for exact-root computation.

We establish $L^p$- and $W^{1,p}$-stability of the $L^2$-projection onto mapped Lagrange finite elements on hybrid meshes consisting of triangles and convex quadrilaterals arising from adaptive mesh refinement. If $K$ is the (tensor product) degree of polynomials of the discretisation, then we show, in particular, $W^{1,2}$-stability for all $K\geq 2$ for the Q-RG and Q-RB refinements. This extends results by Ali, Funken, and Schmidt (2022) which hold for the range $2 \leq K \leq 9$ for initial meshes consisting of parallelograms. Our proof relies on an extension of the technique by Diening, Storn and Tscherpel (2021) to general convex quadrilaterals.

We study mesoscopic linear spectral statistics for two ensembles of random quantum graphs. These are defined by a discrete graph $G$ and a unitary-matrix-valued function $U(k)$ indexed by directed edges of $G$. The matrix function $U(k)$ is constructed from unitary matrices $U^{(v)}$ indexed by the neighbours of each vertex $v$. The first ensemble is obtained by sampling the underlying discrete graph uniformly from the set of $d$-regular graphs. The second ensemble is obtained by sampling $U^{(v)}$ uniformly from the Haar measure, independently for each vertex. We prove that the variance of a linear spectral statistic in the large graph limit on polynomial mesoscopic scales coincides with that of the Gaussian Orthogonal/Unitary Ensemble.

We introduce an alternative generative framework based on a nonlinear modification of the classical Ornstein–Uhlenbeck dynamics. The proposed dynamics admits both a microscopic description through an interacting particle system and, in the mean-field limit, a macroscopic formulation given by a nonlinear Fokker–Planck equation with a superlinear drift term. We show that, for suitable choices of the model parameters and sufficiently large initial mass, the forward dynamics exhibits condensation phenomena by proving the loss of $L^2$ regularity of the solution in finite time. Building upon this formulation, we derive a stabilized reverse-time partial differential equation that reconstructs the initial distribution from the asymptotic state of the forward dynamics, thereby extending the generative paradigm beyond the classical score-based framework. Furthermore, we introduce numerical discretizations of both the forward and reverse processes that accurately capture the asymptotic behavior of the continuous model while successfully reconstructing the initial distribution. Numerical experiments in one and two spatial dimensions validate the proposed methodology and illustrate its application to density filtering through successive iterations of the generative process.

We establish a new regularity phenomenon of continuous functions. Specifically, given any continuous function $f$ and arbitrary $\epsilon>0$, we construct a Lipschitz perturbation $g_\epsilon$ whose Lipschitz seminorm is less than $\epsilon$ such that every level set of $f+g_\epsilon$ has Lebesgue measure zero.

The Poisson compound decision problem is a long-standing problem in statistics, in which empirical Bayes methods are used to estimate Poisson means under a mixture model. We study this problem from the viewpoint of $g$-modeling, comparing two nonparametric strategies for estimating the unknown mixing distribution: a Bayesian empirical Bayes strategy, based on the Dirichlet process posterior, and a quasi-Bayesian empirical Bayes strategy, based on Newton's algorithm. The latter is computationally attractive, but its relationship with the Bayesian strategy requires theoretical justification. Under a Poisson mixture model with a “true”, or oracle, mixing distribution, we establish concentration rates for the marginal probability mass functions induced by the Bayesian and quasi-Bayesian estimates. These rates are then translated into rates of decay for the corresponding regrets, interpreted as excess Bayes risks, and used to prove a frequentist merging result between the Bayesian and quasi-Bayesian empirical Bayes strategies. We also extend the analysis to the multidimensional Poisson compound decision problem. Numerical experiments on synthetic data illustrate that the quasi-Bayesian strategy achieves accuracy comparable to the Bayesian strategy, while requiring substantially fewer computational resources, especially in the multidimensional setting.

We consider a class of strongly coupled nonsmooth systems consisting of a semilinear evolution inclusion and a differential inclusion governed by state-dependent maximal monotone operators. Our main contributions are fourfold. First, we collect the well-posedness, compactness, and Painlevé–Kuratowski continuity properties of the parameterized solution map required for the subsequent optimization analysis. Second, for Bolza-type optimization over the solution set, we prove the existence of optimal pairs, establish continuity properties of the value function, and derive upper semicontinuity of the optimal-solution map. Third, we study fixed-parameter optimal control, simultaneous control-parameter design, min–max robust control, and Hurwicz-type compromise control under parameter uncertainty, and we establish existence results for each formulation. Fourth, we report numerical experiments for sweeping-type systems that illustrate the sensitivity and robustness phenomena predicted by the theory.

The Shifted Boundary Method (SBM) sidesteps body-fitted meshing by shifting boundary conditions onto a surrogate boundary and correcting for the displacement through Taylor expansions. Despite its broad analysis and application, scalable iterative solvers for the incompressible Stokes equations remain underdeveloped. We present a block preconditioner for SBM–Stokes discretisations that uses the velocity block together with a pressure mass matrix as a Schur complement approximation. Because the SBM system is non-symmetric, classical operator preconditioning does not apply directly; a field-of-values analysis instead shows that the non-symmetric SBM contributions act as asymptotically small perturbations of a standard saddle-point operator, yielding mesh-independent GMRES convergence on sufficiently fine meshes. Numerical experiments demonstrate iteration counts under refinement across geometries of increasing complexity. We expose a coarse-mesh regime in which an under-resolved grid produces elevated iteration counts, an artefact of insufficient resolution that vanishes once the mesh captures the geometry.

We present a GPU-oriented formulation of continuous high-order finite elements in which the redundant, cell-wise (element-local) vector is the persistent primary representation of all field data, rather than a transient stage of matrix-free operator evaluation. We prove that, given a preconditioner whose image is continuous, the entire flexible conjugate gradient iteration can be carried out exactly on this unassembled representation: a simple primal-dual pairing identity shows that all Krylov scalars computed from local data coincide with those of the assembled solve, so inter-element communication is confined entirely to the preconditioner. The required direct stiffness summation (DSS) is then realized without indirect gather-scatter, atomics, or coloring, by a dimensionally-split cascade of one-to-one face exchanges that provably accumulates edge and vertex contributions as a byproduct of sequential axis passes; unstructured macro-block interfaces and $h$-adaptive hanging nodes are handled by disjoint topological kernels and a shadow-cell wrapper that leaves the high-throughput sweeps untouched. The cell-wise storage decouples the memory layout from the mesh topology, and we exploit this freedom to benchmark blocked layouts that trade memory coalescing against element contiguity. Numerical experiments on modern GPUs demonstrate that the resulting operator evaluation and solver outperform state-of-the-art matrix-free implementations, signifficantly exceeding throughput of existing implementations.

We present a fully variational, model-independent formulation of the Cut Finite Element Method (CutFEM) for finite-strain elasticity. The discrete problem is derived from a single augmented energy functional consisting of the bulk hyperelastic energy, the Nitsche terms that impose the boundary conditions weakly, and the ghost-penalty stabilisation. At each nonlinear iterate, the residual is the exact first variation of this functional with the adaptive Nitsche weight frozen, while the correction uses a symmetric, coercive approximation of its Hessian. Automatic differentiation (AD) generates the first Piola–Kirchhoff stress tensor and the elasticity tensor directly from the scalar energy density, avoiding manual re-derivation when exchanging hyperelastic models. To our knowledge, this is the first unfitted finite-strain scheme combining an energy-only, model-independent construction with AD and an accuracy analysis at unfitted boundaries. Analysis of the linearised problem solved at each quasi-Newton step establishes cut-independent coercivity, continuity, and an $O(h^{-2})$ condition number bound, yielding a quasi-optimal convergence theorem for regular solutions through the Brezzi–Rappaz–Raviart framework. Numerically, the method attains optimal $h$-convergence for quadratic and cubic elements on a smooth test case. Furthermore, we quantify the method's accuracy limit at mixed Dirichlet–Neumann junctions using the Kolosov–Muskhelishvili characteristic equation. The exact solution's corner singularity caps the convergence rate identically for fitted and unfitted methods. Local mesh refinement removes this bound: we verify the recovery of optimal rates numerically for first-order elements, and prove that the unfitted discretisation inherits the rate the underlying refinement attains, with cut-independent constants.

For training-data-based model risk prediction, $K$-fold cross-validation~(CV) is widely used to mitigate the well-known over-optimism of the empirical risk and is often regarded as reliable. However, for binary classification via empirical risk minimization, our numerical studies reveal a surprising phenomenon: $K$-fold CV may perform poorly in estimating class-specific risks, even worse than the empirical estimator. We perform a higher-order asymptotic analysis showing that $K$-fold CV may converge at a slower rate, whereas the empirical estimator exhibits a second-order asymptotic bias that explains its over-optimism. These findings motivate a novel two-step procedure for model risk prediction, termed cross-audit projection (CAP). The cross-audit step adopts the same resampling scheme as $K$-fold CV to estimate over-optimism in subsamples, while the asymptotic-theory-informed projection step adjusts for the reduced sample size in bias correction of the empirical risk. The resulting CAP estimator is first-order asymptotically equivalent to the empirical risk while achieving second-order asymptotic unbiasedness. An accompanying inference procedure is also developed. Simulation studies support theoretical advantages of CAP and demonstrate favorable finite-sample performance. An application to breast cancer detection further illustrates the proposed method.

Stochastic resources such as wind farms, electric vehicle aggregators, and demand-side assets are increasingly participating as reserve providers in ancillary service markets. To manage delivery uncertainty, system operators impose minimum reliability thresholds on such providers. Energinet, the Danish transmission system operator (TSO), has pioneered this approach through the P90 requirement, requiring stochastic providers to make accepted reserve capacity bids available with at least 90% probability. Yet this threshold is set by regulatory convention, not optimization: no existing framework treats it as a design variable or characterizes the cost-reliability trade-off it governs. This paper closes that gap. We develop a bilevel optimization framework in which the TSO in the upper level sets the reliability threshold endogenously while providers in the lower levels respond through reliability-constrained bidding, with chance constraints reformulated analytically using a Weibull tail distribution. Applied to the Nordic frequency containment reserve for disturbances (FCR-D) market, the cost-optimal threshold lies below P90 in the studied cases, with cost reductions by up to 14.5% relative to the fixed standard. Dynamic hourly thresholds yield a further reduction of up to 2.4%, suggesting efficiency gains may increase in larger and more diverse reserve markets.

The solution space of the homogeneous Dirichlet problem for the fractional Laplacian $(-\Delta )^{a}$ ($0<a<1$) or a pseudodifferential generalization $P$, on a bounded open set $\Omega \subset R^n$ with $C^{1+\tau }$-boundary, $$ Pu=f \text{ on }\Omega ,\quad u=0 \text{ on }R^n\setminus \Omega , $$ is analysed in detail. It is shown, both for solutions in Sobolev spaces of Bessel-potential type $H_q^t$ and in Hölder-Zygmund spaces $C_*^t$, that the solution space for $f$ of regularity $s\in [0,\tau -2a)$ is the direct sum of a component $\dot H_q^{2a+s}(\bar\Omega)$ resp. $\dot C_*^{2a+s}(\bar\Omega)$ with full regularity $2a+s$ and a component of the form $d^a$ times a lifting of boundary values by Poisson operators. Here $d(x)=dist(x,\partial\Omega )$. This extends to non-smooth problems results known in the $C^\infty $ setting. The knowledge is used to establish gradient estimates for $a>1/2$, e.g. estimating $d^{1-a+s}\nabla (u/d^a)$ in terms of norms of $f$ and $u$, both in $H_q^t$-spaces and $C_*^t$-spaces. This is entirely new in the case of Bessel-potential spaces; it extends previous results by Fall and Jarohs in Hölder spaces. A new tool is introduced: $\dot H^{s+t}_q(\bar\Omega)\subset d^s\dot H^{t}_q(\bar\Omega)$ holds for $s,t\ge 0$ with $s+t<\tau $.

We study the $R$-sub-bimodule structure of differential polynomial rings $R[x;\delta]$ by introducing the notion of strong simplicity, requiring each nonzero $R$-sub-bimodule of $R[x;\delta]$ to be either $R[x;\delta]$ or the truncation $\sum_{i=0}^n R x^i$ for some $n \in \mathbb{Z}_{\geq 0}$. Our main result gives a complete characterization: $R[x;\delta]$ is strongly simple if and only if $R$ is simple, ${\rm char}(R)=0$, and the derivation $\delta$ is outer. We provide examples illustrating both when strong simplicity fails and when it holds.

In this paper we investigate a stable space-time formulation for long-time industrial sound emission problems. To this end, we use a well-posed Galerkin formulation in space and time of the acoustic wave equation in $\mathbb{R}^3$, involving a hypersingular boundary integral operator. Our numerical experiments confirm that the resulting time stepping scheme is stable and accurate for complex acoustic problems in industrial geometries, in contrast to alternative well-known schemes. The proposed method is shown to be efficient for real-world problems, and we obtain very good agreement with physical acoustic measurements.

Giampietro and Darmon conjectured a formula for the norm of various algebraic numbers, obtained as infinite products of $p$-adic cross-ratios of CM points. These quantities arose from the $p$-adic uniformisation of Shimura curves and displayed strong parallels with the Gross–Zagier factorisation for the norms of the differences between two singular moduli. The conjectured formula was conditional on the genus of the Shimura curve being zero, and in earlier work, this formula was proved in most cases. In this work, we extend the validity of the factorisation formula beyond what was conjectured by Giampietro and Darmon to many more cases, by relating this to the genus of an Atkin–Lehner quotient of the Shimura curve being zero instead. To this end, we solve a $p$-inverted version of a counting problem that was previously considered in work of Howard and Yang.

We prove an extremal theorem for positive Ollivier/Lin–Lu–Yau curvature: every graph of order \(n\geq 8\) with more than \[ T(n)=\frac{n^2-3n}{2}-\left\lceil\frac{n}{2}\right\rceil+2 \] edges has positive Ollivier/Lin–Lu–Yau curvature, and this threshold is optimal. Moreover, for even $n\geq 12$, there exists a unique graph with $T(n)$ edges that has an edge with non-positive curvature. For $n=8,10$ and odd $n\geq 9$, the extremal graphs are not unique. This suggests a new class of extremal graph-theoretic problems arising from discrete curvature notions.

We study the critical coagulation–fragmentation equation with multiplicative coagulation kernel $a(s,\hat s)=s\hat s$ and constant fragmentation kernel $b(s,\hat s)=1$. Under the Bernstein transform, mass-conserving solutions correspond to solutions of a singular Hamilton–Jacobi equation studied by Tran and Van (Comm. Pure Appl. Math. 75 (2022), no. 6, 1292–1331). Through this correspondence they proved that mass-conserving solutions are unique on the full critical range $0<m\le1$, but could establish their existence only for $0<m<\tfrac12$. We identify a one-sided, convexity-type invariant that holds for Bernstein-transform data and is propagated by their viscous scheme as a genuine maximum-principle bound. We call it the half-slope invariant. It sharpens the curvature barrier and thereby extends mass-conserving existence to the entire critical range $0<m\le1$. Hence $m=1$ is the critical mass, confirming the threshold predicted by Vigil and Ziff (J. Colloid Interface Sci. 133 (1989), no. 1, 257–264). The same invariant appears in the radial partial-mass formulation of the two-dimensional Keller–Segel equation, whose critical mass is $8\pi$.