Many geometry processing pipelines implicitly assume their input data is a manifold, or is sampled from one, with a unique tangent plane at every point. Geometric data, however, routinely contains sharp features like edges, corners, self-intersections, branching junctions, and other singularities, rendering standard methods ill-defined at these points. To bring geometry processing to these and other singular spaces, we introduce the ``tangent blow-up,'' a representation inspired by algebraic geometry that restores structure at singularities by lifting to the product of the ambient space and the Grassmannian of tangent planes. After iterating this construction, points that coincide in position but differ in tangent direction, curvature, or higher-order contact become well-separated. We equip the tangent blow-up with a product metric and define discretized differential operators, such as the gradient, divergence, and Laplacian, directly in the lifted domain. We demonstrate our framework across geodesic computation, segmentation, surface parameterization, and curvature estimation.

We study the sticky Cox--Ingersoll--Ross (CIR) process in one dimension, a diffusion on $[0,\infty)$ with a sticky boundary condition at the origin, arising as the marginal process in a sparse Bayesian inference framework based on Hadamard--Langevin dynamics. For the parameter range $δ\in(1,2)$, in which the origin is accessible but not absorbing, we prove well-posedness of the process and uniqueness of its invariant measure, which is a mixture of a point mass at zero and a weighted gamma-type density on the interior. We derive an explicit Green's function for the resolvent in terms of confluent hypergeometric functions, and use this to construct an exact sampler for the invariant measure in the zero-potential case. For a non-trivial potential $G$, we establish existence and uniqueness of the tilted invariant measure via a Girsanov change of measure, and develop two sampling algorithms: a Metropolis--Hastings corrected sampler that targets the invariant measure exactly, and a cheaper, biased unadjusted Langevin algorithm (ULA) for a boundary-clamped variant of which we prove a first-order expansion of the stationary bias with an explicit constant: the leading error is a rank-one transfer of mass $K_\star h|\log h| $ onto the atom, so the total-variation bias is of exact order $h|\log h | $ -- independent of $δ$ -- whenever the potential has nonzero boundary drift. Numerical experiments confirm the predicted behaviour: the Metropolis--Hastings sampler achieves the target invariant measure at all step sizes, while the ULA bias follows the proven first-order law, including its constant.

The calculation of inclusive processes that involve the production of many particles is a challenge for lattice QCD, a Euclidean-space method that is far removed from real-time, multiparticle production. A new approach to this problem based on Nevanlinna-Pick interpolation has been proposed by Bergamaschi et al. Here we extend their method by exploring the propagation of the statistical and systematic errors that accompany a lattice QCD calculation through this interpolation process. A simplified example of a multiparticle spectral function is studied with a focus on the possible applications of these methods to the calculation of inclusive heavy-particle decays.

The function of the zebra's striped coat has been debated since Darwin and Wallace. A growing body of comparative and experimental evidence supports the hypothesis that the stripes act primarily as a defence against visually orienting biting Diptera - in particular tabanids (horse flies), glossinids (tsetse flies) and culicids (mosquitoes). The mechanisms proposed for this protection range from polarotactic disruption and silhouette break-up to motion-based illusions arising in the Reichardt-type motion detectors of the insect visual system. In this work we focus on a complementary, purely optical mechanism: the Moiré interference that arises when a periodic striped stimulus is sampled by the periodic ommatidial lattice of an insect compound eye. We develop a linear, shift-invariant Fourier model of the diptera compound eye, parameterised from published optical data on diurnal Culicidae, and apply it to images of zebra coats observed at biologically relevant viewing. The model predicts that, in a band of approach distances of approximately 1-5 m, the interaction of the stripe pattern with ommatidial sampling generates parasitic spatial frequencies that are absent from the physical stimulus and that fall within the spatial-frequency window most relevant to host fixation and landing control. A post-retinal motion-detector stage demonstrates that these parasitic frequencies translate into spurious local motion vectors, consistent with the empirical observation that tabanid and glossinid flies fail to land cleanly on striped surfaces. Our results are therefore consistent with the biting-fly hypothesis of zebra striping.

Incentives in experimental design are often misaligned: experimenters design and finance experiments to seek regulatory approval, while regulators seek to maximize social-welfare. We propose a framework to resolve this conflict, wherein regulators set a minimum welfare threshold, and experimenters optimize designs subject to this constraint. It requires no knowledge of experimenters' private preferences or costs and mitigates strategic Bayesian persuasion. Under normal priors, Neyman-allocation is always the optimal-sampling strategy, regardless of specific objectives. We also characterize the optimal stopping-rule. A numerical study calibrated to clinical-trial data shows sample-size reductions of over 48% relative to classical designs attaining the same social-welfare.

One of the fundamental challenges of quantum gravity is to understand how the microscopic degrees of freedom of the cosmological horizon shape the evolution of the Universe. One possible approach to this problem is based on the Barrow--Tsallis entropy. This entropy accounts for both quantum gravitational effects and the nonextensive effects inherent in any long-range interaction. By employing an inverse cosmographic reconstruction of the model parameters, we derive a relation between the Barrow parameter, which encodes the microscopic deformation of the horizon geometry, and the Tsallis parameter, which characterizes macroscopic nonextensivity. Within the IR--UV correspondence, this relation determines the scaling of the microscopic length uncertainty in terms of the current cosmographic parameters and demonstrates how long-range nonextensive effects alter the standard Karolyhazy-type scaling. We also used our method for finding the parameters of cosmological models to evaluate the feasibility of using fractional derivatives to describe the late evolution of the Universe. The resulting relationships are exact. Therefore, the uncertainty in the relationship between the model parameters depends only on the current uncertainty in the values of the cosmographic parameters.

We introduce a de Rham-Hodge framework induced by a vector field on a compact, oriented smooth manifold. Using a vector field induced bundle isomorphism on differential forms, we define a vector field induced Hodge $L^2$-inner product, codifferential, and Hodge Laplacian. Unlike classical deformations, such as the drifting and Witten-type Hodge Laplacians, the induced Laplacian modifies the principal symbol and gives rise to an anisotropic Laplace-Beltrami type operator on functions. We establish the resulting de Rham-Hodge theory for closed manifolds, including the ellipticity of the induced Hodge Laplacian and the corresponding Hodge decomposition and isomorphism results. We further extend the framework to manifolds with boundary by imposing certain vector field induced boundary conditions, which are necessary to restore the adjointness between the differential and induced codifferential, and to obtain a well-posed boundary value problem. Under these boundary conditions, we establish analogues of the Hodge-Morrey and Friedrichs decompositions. We also discuss several structural properties of the framework, including its relation to anisotropic Laplace-Beltrami operators, its spectral behavior in several explicit examples, and its invariance under isometries.

Public access to pulse-level and control-electronics interfaces in commercial quantum computing has bifurcated. This paper proposes a six-axis rubric for measuring control-plane openness, the layer between gate-level circuit specification and physical control electronics, defined operationally so that the same evidence produces the same grade across vendors. The rubric is validated three ways: a blinded re-grading pass, thirty-nine days after the evidence cutoff, that tests whether the cited evidence and the level definitions alone reproduce the recorded grades; a boundary-case methodology that fixes where each level begins and ends; and a published grading protocol that lets others reproduce and contest any cell. We establish that the rubric measures change rather than describing a snapshot by comparing the catalog against the documented control plane before the February 2025 removal of pulse-level access from IBM hardware, and reporting the cells that moved. The rubric is applied to thirteen commercial vendors across superconducting, trapped-ion, neutral-atom, and photonic modalities as of May 1, 2026, as its first application, and one of the three harms the rubric is designed to detect is demonstrated through a reproduction-access audit of five pre-2025 IBM Qiskit Pulse experiments against the access available on current hardware, carried through to a client-side structural port of the audit's selected target to Rigetti Quil-T. The catalog ships as a separate machine-readable artifact under CC-BY-4.0 with per-cell source URLs (https://doi.org/10.5281/zenodo.20163276). The catalog readings will change as vendor policies shift; the rubric is the contribution that survives them.

Since the introduction of network psychometrics, several connections to statistical models in "classical" psychometrics (i.e., IRT, SEM, GLM) as well as to approaches from other research fields have been established. In this paper, these developments have been reviewed and synthesized and, based on an exploratory literature search, further advanced and presented in an accessible visual format. This perspective opens up promising opportunities to extend the psychometric-toolbox by incorporating and learning from statistical methodologies developed in other research domains, which often address similar or even identical problems. Highlighting these methodological commonalities may also foster collaboration across research fields that have traditionally remained largely independent. Moreover, awareness of these connections may render methodological development more systematic and goal-directed and may enable a meaningful division of labor, for example between the development of statistical methodology and its practical implementation for empirical research through software tools. Finally, these methodological advances provide new opportunities for empirical research and may contribute to a reconciliation with longstanding conceptual issues concerning psychometric constructs and, more broadly, psychological phenomena.

With the next generation of both electromagnetic and gravitational wave observatories beginning to come online, rapid analysis methods for kilonova data are becoming increasingly important in astronomy. Traditional Bayesian parameter estimation using Markov chain Monte Carlo (MCMC) is time-consuming and relies on explicit likelihood approximations that can break down when modeling uncertainties are significant. We develop a simulation-based inference (SBI) framework for kilonova parameter estimation using density-estimation likelihood-free inference. The framework uses a Gaussian process emulator trained on $\sim 1300$ POSSIS simulations. We demonstrate that SBI provides a rapid alternative to MCMC that is robust to likelihood misspecification. The standard Gaussian likelihood approximation fails to capture the non-Gaussian, correlated structure of emulator uncertainty; SBI learns this structure directly from forward simulations. Simulation studies show that the SBI method accurately recovers injected parameters, while the MCMC suffers from systematic bias caused by likelihood misspecification. This problem persists when analyzing AT2017gfo, where a subset of the MCMC posteriors pile up at prior boundaries and the SBI posteriors do not. The SBI framework infers a total ejecta mass of $\sim 0.087 M_{\odot}$ dominated by lanthanide-poor ejecta and excludes toroidal and peanut ejecta geometries at the 99th percentile for both components. The SBI framework generates $\sim 2 \times 10^{4}$ posterior samples in seconds.

We exhibit an equigenerated monomial ideal $I\subseteq K[x,y,z,w]$ with $\operatorname{reg}(\overline{I})>\operatorname{reg}(I)$. The ideal $I$ is generated in degree 4 and satisfies $\operatorname{reg}(I)=4$, while its integral closure $\overline{I}$ has a minimal generator of degree 5 and satisfies $\operatorname{reg}(\overline{I})=5$. This gives a counterexample to the polynomial-ring formulation of the Küronya--Pintye conjecture.

Soft, slender structures are ubiquitous in natural and engineered systems, with broad application potential from biomimetic materials to soft robotics. However, there is a notable lack of computational tools that simultaneously preserve high-fidelity continuum rod mechanics, scale to large interacting ensembles, and remain flexible across diverse biophysical settings. Here we introduce Elastica++, an open-source, high-performance implementation of the Cosserat-rod model for large-scale simulations of slender-body dynamics. Elastica++ combines performance-oriented kernels with shared-memory parallelism to sustain teraflop-scale throughput despite complex discretization domains and physical interactions. The framework further interoperates with external numerical solvers, supporting efficient multiphysics workflows. We demonstrate robustness and breadth through case studies spanning passive nest-like metamaterials, collective active-matter dynamics, cilia carpets, soft magnetic microrobots, and schooling swimmers. Elastica++ thus provides a missing foundation for high-throughput studies of emergent behavior in interacting assemblies of elastic slender structures.

Two constructions of the Darboux-Halphen system are discussed. In the Jacobi construction we start with transcendental functions which are fixed as the first integrals. In the Lie construction we use a single-valued representation of the simple Lie algebra $sl(2,\mathbb R)$ which is non-integrable in Lie's terminology.

Unconstrained binary integer programming (UBIP) is a challenging optimization problem due to the presence of binary variables. To address the challenge, we introduce a novel class of functions named sharp-peak functions (SPFs), which equivalently reformulate the binary constraints as equality constraints, giving rise to an SPF-constrained optimization. Rather than solving this constrained reformulation directly, we focus on its associated penalty model. The established exact penalty theory shows that the global minimizers of UBIP and the penalty model coincide when the penalty parameter exceeds a threshold, a constant independent of the solution set of UBIP. To analyze the penalty model, we introduce Karush-Kuhn-Tucker (KKT) points and a new type of stationarity, referred to as P-stationarity, and provide a comprehensive characterization of its optimality conditions. We then develop an efficient algorithm called Sha-Peak based on the inexact alternating direction method of multipliers. It converges toa P-stationary point at a linear rate or terminates at such a point within finitely many steps. These results are established under appropriate parameter choices and a single mild assumption, namely, the local Lipschitz continuity of the gradient over a bounded box. Finally, numerical experiments demonstrate its nice performance in comparison to several established solvers.

The article studies query evaluation in parallel constant time in the CRCW PRAM model. While it is well-known that all relational algebra queries can be evaluated in constant time on an appropriate CRCW PRAM model, this article is interested in the efficiency of evaluation algorithms, that is, in the number of processors or, asymptotically equivalent, in the work. Naive evaluation in the parallel setting results in huge (polynomial) bounds on the work of such algorithms and in presentations of the result sets that can be extremely scattered in memory. The article discusses some obstacles for constant-time PRAM query evaluation. It presents algorithms for relational operators and explores three settings, in which efficient sequential query evaluation algorithms exist: acyclic queries, semijoin algebra queries, and join queries -- the latter in the worst-case optimal framework. Under mild assumptions -- that data values are numbers of polynomial size in the size of the database or that the relations of the database are suitably sorted -- constant-time algorithms are presented that are weakly work-efficient in the sense that work $\mathcal{O}(T^{1+\varepsilon})$ can be achieved, for every $\varepsilon>0$, compared to the time $T$ of an optimal sequential algorithm. Important tools are the algorithms for approximate prefix sums and compaction from Goldberg and Zwick (1995).

Developing non-perturbative methods to reveal exotic properties of strongly correlated fermionic systems remains one of the most essential tasks of theoretical physics. Tensor network methods with Grassmann algebra offer powerful numerical tools for fermionic many-body systems in the coherent-state path-integral representation. Despite their vast potential for both condensed-matter and particle-physics communities, Grassmann tensor network methods are somewhat underexploited in practical simulations. In this work, we provide a detailed, self-contained introduction to Grassmann tensor network methods, from the basics of the Grassmann tensor operations to the Grassmannization of typical tensor network algorithms. Furthermore, the resulting Grassmann tensor network methods are validated in several interesting models in both particle physics and condensed matter physics.

It is proven that every cohomology class of the moduli space $M_{g,m}$ for any $2g+m\geq 3$, $m\geq 1$ can be represented combinatorially by a ribbon quiver with at most four-valent vertices. The "at most four"-valency condition is sharp.

Existing Audio Deepfake Detection (ADD) systems often struggle to generalise effectively due to the significantly degraded audio quality caused by audio codec compression and channel transmission effects in real-world communication scenarios. To address this challenge, we developed a rigorous benchmark to evaluate the performance of the ADD system under such scenarios. We introduced ADD-C, a new test dataset to evaluate the robustness of ADD systems under diverse communication conditions, including different combinations of audio codecs for compression and packet loss rates. Benchmarking three baseline ADD models on the ADD-C dataset demonstrated a significant decline in robustness under such conditions. A novel Data Augmentation (DA) strategy was proposed to improve the robustness of ADD systems. Experimental results demonstrated that the proposed approach significantly enhances the performance of ADD systems on the proposed ADD-C dataset. Our benchmark can assist future efforts towards building practical and robustly generalisable ADD systems.

For $κ$ a regular uncountable cardinal, the higher Baire and Cantor spaces ${}^κκ$ and ${}^\kappa2$ (endowed with the ${<}κ$-box topology) have been relatively well-studied, but less is known about the case where $κ$ is singular. We will consider several spaces of functions and box topologies that could serve as higher Baire and Cantor spaces for singular cardinals. The ultimate focus of the article lies in studying cardinal characteristics of the ideal of $κ$-meagre subsets of these spaces.

We introduce a method for error-corrected quantum metrology where only partial quantum error correction (QEC) is needed to suppress local noise and maintain the probe states' super-standard-quantum-limit (super-SQL) sensing performance. This stands in contrast to the existing QEC-assisted sensing schemes in Phys. Rev. Lett. 112, 080801 (2014) and Phys. Rev. Lett. 112, 150802 (2014), where a probe state is encoded into the logical subspace of a quantum code and error correction involves measurements on all checks of the code. Here, we encode the probe states into superpositions of energetically different states of the underlying quantum code. For our probe states, error correction using a subset of checks is enough to suppress noise both before and after phase imprinting. We analyze the tradeoff in noise suppression. For noise parallel to our phase imprinter of weight $l$, we achieve a suppression of $p^δ$ where $p$ is the noise strength and $δ= \lfloor (l+1)/2 \rfloor$. We propose an adaptive imprinter weight increasing strategy to maintain super-SQL performance as we scale up the system. In all our examples, checks and phase imprinters are chosen to be local operators avoiding non-local connectivity.

This note proposes a simple polynomial-time method for constructing an ex ante stable school-choice lottery satisfying equal treatment of equals (ETE). We show that the ETE reassignment of any constrained efficient stable matching is ex ante stable, satisfies ETE, and is not ordinally dominated by any other ex ante stable lottery. We further show that there exists a constrained efficient stable matching whose ETE reassignment is not ordinally dominated by any ex post stable lottery.

We investigate the spectrum and polarization of radiation emerging from a subcritical X-ray pulsar using self-consistent radiation-hydrodynamic simulations of an accretion channel in a strong magnetic field. The polarized radiative transfer in the channel above the hot spot is simulated for the two normal modes, taking into account resonant Compton scattering in a strongly magnetized plasma and the effects of vacuum polarization. We show that the deceleration of the accreting matter in the subcritical regime is mainly governed by resonant scattering. Our simulations provide the velocity profiles of the plasma flow and demonstrate that vacuum polarization dominates over plasma birefringence, enhancing both the cyclotron spectral feature and the radiative deceleration of the plasma. We also find that the energy of the cyclotron feature increases with accretion luminosity, indicating a positive correlation consistent with previous observational results and theoretical interpretation.

The no-boundary wavefunction for slow-roll inflation on the 3-sphere exponentially favors a small universe, in sharp disagreement with observations. We show that this problem is resolved by changing the spatial topology to the 3-torus. The sum over the ${\rm SL}(3,\mathbb{Z})$ family of geometries, computed using the theory of automorphic forms for ${\rm GL}(3)$, produces a wavefunction favoring a large inflating universe with ${\cal N} \gtrsim 250$ e-folds. We also compute corrections to the CMB power spectrum induced by torus moduli fluctuations.

Let $Γ$ be a countable discrete amenable group, and let $A=l^\infty(Γ) \rtimes Γ$. It is shown that if $a, b \in A \otimes \mathcal K$ are positive elements such that $$\mathrm{d}_τ(a) < \mathrm{d}_τ(b),\quad τ\in \mathrm{T}(A),$$ then $a$ is Cuntz subequivalent to $b$. Moreover, consider the universal minimal set $(M, Γ)$. The simple C*-algebra $\mathrm{C}(M)\rtimesΓ$ is shown to be AH in the strong sense that there is an increasing net of unital sub-C*-algebras $A_λ\subseteq A$, $λ\in Λ$, such that each $A_λ$ is a simple (separable) $\mathcal Z$-absorbing approximately homogeneous C*-algebra with real rank zero and $A = \bigcup_{λ\in Λ} A_λ$. In particular, $\mathrm{C}(M)\rtimesΓ$ is approximately divisible.

Measuring the chemical and isotopic compositions of gas giants and brown dwarfs provides insights into their formation pathways and birth environments. 2MASS J0249-0557 c is an L2-type planetary mass companion ($\sim 12 M_{\mathrm{Jup}}$) orbiting a pair of brown dwarfs in the $β$ Pic young moving group and vicinity. Its mass places it at the intersection of planets and brown dwarfs, making it an interesting target for constraining formation pathways at the planet-brown-dwarf boundary. Using high-resolution spectroscopic data of the planet acquired with CRIRES+ mounted on VLT, we conduct atmospheric retrieval with the radiative transfer code \texttt{petitRADTRANS} and the nested sampling tool PyMultiNest. We retrieve a C/O ratio of $0.57\pm0.01$, a metallicity of [M/H] = $0.18\pm0.05$, and a $^{12}$CO/$^{13}$CO ratio of $95^{+23}_{-17}$. We also retrieve atmospheric compositions for two benchmark brown dwarfs in the $β$ Pic YMG, 2MASSI J0443+0002 and SIPS J2000-7523, using CRIRES+ data and find consistent compositions. Together with 2MASS J0249-0557 c's wide separation from its host, its compositional consistency with benchmark brown dwarfs supports gravitational collapse in a star-like manner as its most likely formation mechanism. These results deliver a homogeneous comparison of three substellar members in the $β$ Pic YMG and vicinity. Their solar-like abundances provide a baseline for exoplanet members in the same moving group, such as $β$ Pic b, 51 Eri b, and AF Lep b, whose host stellar compositions are difficult to measure. Future comparisons of atmospheric compositions among this moving group offer the potential to distinguish between formation mechanisms for its planetary members.

We show that the Hurwitz problem for sums of squares can depend on the base field. More precisely, we construct an explicit formula of type $[12,12,18]$ over every field of characteristic different from $2$ in which $-1$ is a square, whereas no such formula exists over any formally real field. In particular, a formula of this type exists over $\mathbb Q(i)$ and over $\mathbb C$, but not over $\mathbb Q$ or over $\mathbb R$. This settles, in the negative, a longstanding conjecture of Shapiro from 1984, a conjecture of Adem from 1975, and answers a signed-formula problem raised by Shapiro in 2000.

The Lovász Local Lemma (LLL) is a central tool in probabilistic combinatorics, providing a sufficient condition under which a finite collection of undesirable events with limited dependencies can be simultaneously avoided with positive probability. This paper offers a self-contained expository treatment of the lemma and its strengthened versions, emphasizing mathematical foundations, conceptual clarity, and applications. We begin with a pedagogically motivated proof of the LLL based entirely on unconditional probability inequalities. Particular attention is given to the symmetric form of the lemma and several subsequent strengthenings. The paper also discusses a variety of classical applications of both the symmetric and asymmetric forms of the LLL in combinatorics and graph theory, including bounds for the edge-disjoint paths problem, satisfiability of Boolean formulas in conjunctive normal form, lower bounds on diagonal and off-diagonal Ramsey numbers, hypergraph coloring results, structural properties of directed graphs, and acyclic graph colorings. Additional observations and refinements are provided throughout. We also introduce the algorithmic framework of Moser and Tardos, highlighting its constructive counterpart to the LLL, together with an introduction to the entropy-compression principle. The lopsided LLL, a refinement of the LLL, is presented along with an application to the Latin transversal problem. We further discuss the cluster-expansion lemma and its relation to the LLL, and present an alternative treatment of the Latin transversal problem from the cluster-expansion perspective that yields an improved result. The paper concludes with a high-level overview of the iterated LLL, also known as the semi-random method.

We describe the design of a quantifier elimination framework for the complex numbers in the language of ordered rings supplemented with symbols for the imaginary unit, real parts, imaginary parts, and conjugates. Technically, we use a reduction to real quantifier elimination followed by a heuristic reinterpretation of the results within our complex framework. We present computational examples using a prototypical implementation of our approach in our Python-based open-source system Logic1.

This short note establishes an abstract Hales--Jewett theorem for semigroups equipped with a finite family of retractions. The proof relies on the interplay between retractions and tensor products of ultrafilters.

Evaluating the causal health effects of multivariate, continuous exposures, such as air pollution mixtures, is a critical public health challenge. A primary obstacle is the frequent violation of the positivity assumption, which renders the effects of standard deterministic interventions unidentified or heavily reliant on unreliable model extrapolation. In this paper, we develop a novel causal inference framework to address this challenge. We extend exponential tilting to multivariate exposures and address the critical question of how to compare different intervention directions fairly. This establishes a systematic framework for defining and evaluating various policy-relevant causal estimands, allowing researchers to address diverse scientific questions. We develop numerous methodological advancements, including efficient one-step estimation strategies, a Riemannian BFGS algorithm to solve a constrained manifold optimization problem, semiparametric efficiency bounds for causal estimands, minimax rates for estimators, and establishing asymptotic normality. We demonstrate our framework's utility by applying it to a nationwide environmental health dataset to identify the optimal strategy for reducing adverse health outcomes associated with a PM$_{2.5}$ chemical mixture.