Papers

Bluetooth Low Energy (BLE) employs MAC address randomisation – via Resolvable Private Address (RPA) – to mitigate long-term device tracking on public advertising channels. Existing research has shown that advertising packets contain metadata and structural features that allow re-identifying a target device via manually crafted rules. In this work, we investigate the feasibility of automating the process of tracking BLE devices despite MAC randomisation by leveraging machine learning algorithms for the signature creation. Based on the actual Bluetooth traffic from target devices, we characterise the persistence of advertising-layer features across RPA changes and formulate device linkage as a supervised classification problem. Using simple decision tree classifiers as a proof-of-feasibility approach, we evaluate the distinguishability of target and non-target devices under varying address rotation patterns. Our results reinforce prior work demonstrating that advertising-layer metadata can enable device re-identification under MAC randomisation, to the point where such linkage can be automated using standard supervised learning techniques, without any specific knowledge of the technology.

Electrospray ionic liquid ion sources operating in the pure ion regime emit metastable, low solvation number ion clusters whose post-emission dissociation modifies plume composition, energy distributions, and diagnostic observables. While field-free measurements support first-order, thermally activated decay on microsecond timescales, the acceleration region is characterized by rapidly varying electric fields and nanosecond residence times such that dissociation becomes strongly field-enhanced. In this work, microcanonical ensemble molecular dynamics (MD) trajectories are used to infer dissociation kinetics of EMI-BF4 ion clusters under controlled internal energy and applied electric field. Positive and negative dimers, trimers, and tetramers are simulated at temperatures between 600 to 1000 K and uniform fields of 10^6 to 10^9 V/m. Dissociation lifetimes are extracted using a connectivity-based fragmentation criterion, and product channels are classified to obtain pathway-resolved branching probabilities. Across all solvation numbers, lifetimes are field independent in the weak field limit, but collapse by orders of magnitude above 10^8 V/m. Cluster polarity dependence is most prominent in the temperature-controlled regime and diminish as electrostatic work dominates. The MD simulations further reveal that field-driven changes in dissociation topology, including a transition in trimer breakup from neutral pair evaporation to charged core ion ejection, and three competing tetramer pathways involving single neutral loss, double neutral emission, and core ion ejection. The MD dataset is reduced to compact parameterizations of {\tau_n}(E,T), yielding transferable kinetic inputs for multiscale plume transport models.

Gravitational evidence firmly establishes the existence of dark matter (DM), yet its non-gravitational interactions and fundamental nature remain unresolved. In this work, we demonstrate that a chirality boost-spin gauge field, which emerges naturally in the General Standard Model formulated within gravitational quantum field theory, offers a vector DM candidate. We construct the corresponding low-energy effective field theory for this DM with axial-vector couplings to Standard Model fermions and systematically investigate its signatures in direct detection, indirect detection, and collider experiments. Comparing our predictions against current experimental data places stringent constraints on the DM parameter space, with the viable regions within the further reach of future searches.

We demonstrate that forking in monadically NIP theories satisfies a very strong form of triviality over arbitrary sets, generalizing Baldwin-Shelah in monadically stable theories and analogous to Shelah's result for finite satisfiability over models in monadically NIP theories. This implies in particular that strict non-forking independence is particularly well behaved, and that the forking relation induces a canonical semilinear partial order on singletons. We also give an example of an $\omega$-categorical monadically NIP theory so that no finite set is an extension base.

Kaplansky's zero-divisor conjecture asserts that the group ring of a torsion-free group over a field has no zero divisors. It holds for every group with the unique-product property, so a counterexample can only come from a torsion-free group without unique products. In lectures in 2021, Gardam announced that the torsion-free $\tilde{A}_2$ lattice $\Gamma = \langle a, b \mid a b a^2 b^{-1} a^2 b^{-2}, a b^3 a b^4 a^{-1} b \rangle$ does not have unique products and presented it as a new candidate group: it has property (T), and the known methods for proving the conjecture do not apply to it. To our knowledge, no proof of the announcement has been published. We give a proof checked by the Lean 4 kernel and stated against Mathlib's UniqueProds class. The witness is an explicit pair of finite subsets with $|A| = 32$ and $|B| = 28$ in which each of the 896 products coincides with another product. For 658 products the certificate is an identity in the free group; the other 238 certificates are explicit products of conjugated relators, 970 conjugates in all, checked by free reduction. A homomorphism onto $\mathbb{Z}/42$ shows that each pair $(u,v)$ differs from its partner $(u',v')$ as a pair of group elements, which is all the theorem requires. Together with a homomorphism onto the alternating group $A_4$ it also shows that the listed words are pairwise distinct, so the sets have exactly 32 and 28 elements. The witness and certificates come from an untrusted search program and are re-checked by Lean. The development uses only the axioms propext, Classical.choice and Quot.sound, with no sorry and no native_decide. The mathematical statement is Gardam's. To our knowledge this is the first verification in a proof assistant of a unique-product failure in a torsion-free group; torsion-freeness of $\Gamma$ is taken from Gardam and is not formalized here.

High-resolution orbital imagery offers a rich record of the Martian surface, but sparse geological labels limit supervised representation learning. We present MarsRecon, a geospatially aware pipeline for learning visual and multimodal representations from HiRISE observations of Olympus Mons. The pipeline calibrates NASA Planetary Data System products, extracts valid georeferenced patches, and trains a masked autoencoder on unlabeled imagery. Increasing input resolution and filtering invalid tokens reduced held-out reconstruction loss from 0.1751 to 0.1342 in the principal Stage A model series. We then freeze the visual encoder and align its features with observation text, coordinates, and local–global image context. The strongest current local-primary model achieves image-to-text recall@10 of 0.3787, text-to-image recall@10 of 0.9161, and local-to-global recall@10 of 0.4350 on the held-out test split. These results establish a working Mars-specific pretraining and retrieval pipeline; further crop-overlap controls and downstream geological evaluations are needed to assess the broader utility of its embeddings.

We prove that the Conway knot has infinite order in the concordance group, and in particular that it is not slice. Ours is an alternative, independent proof with respect to those of Piccirillo and of Donatone, Kegel, Lewark, and Truöl.

The brain fluid clearance system can be studied with intrathecal contrast-enhanced magnetic resonance imaging (MRI). By injecting a tracer into the spinal canal and performing MRI over 24–48\,h, one can measure how a tracer injected into the cerebrospinal fluid spreads through the brain. As this is an invasive procedure requiring several MRI scans, we ask how much of the later tracer distribution can be predicted from the pre-contrast scan and elapsed time alone. Using 104 training patients (878 MRI volumes), we compared a direct image-to-image regression model with a more complex generative model based on conditional flow matching, and tested both on 23 held-out patients and on 51 patients from a separate cohort with idiopathic normal pressure hydrocephalus (iNPH). Relative to simply copying the pre-contrast scan, the direct model reduced the mean absolute error (intensities normalized to $[0,1]$) from 0.089 to 0.036, about 60%, in held-out patients and from 0.075 to 0.056, about 25%, in the iNPH cohort. The reduction held in every sampled elapsed-time interval and brain compartment, and errors remained largest at 20–30\,h and in the ventricles, where enhancement is strongest. Its predicted change maps agreed with the observed tracer enhancement rather than only reproducing anatomy. The generative model was less accurate in both cohorts. A substantial part of the later tracer distribution is therefore predictable from the patient's brain anatomy as seen on pre-contrast MRI, and the added model complexity gave no benefit at this data scale. Whether this accuracy suffices for a given clinical use remains to be established; with prospective validation, such forecasts could help simplify the MRI protocol used to assess brain clearance.

Quantitative analysis of experimental powder X-ray diffraction data remains challenging when evaluating complex multiphase materials and noisy measurements. We introduce a hybrid quantum neural network framework designed to extract quantitative parameters, such as phase weight fractions and scale factors, directly from one-dimensional powder diffraction patterns without iterative refinement. The model combines noise-aware classical simulator pre-training with fast downstream fine-tuning on quantum processing unit features, ensuring stability against hardware decoherence. We demonstrate the practical utility of this approach by deploying the trained network onto an IBM quantum computer to analyse experimental X-ray diffraction computed tomography datasets from a three-phase solid oxide fuel cell containing ca. 10,000 patterns and a four-phase lithium-ion battery containing ca. 20,000 patterns). The network successfully reconstructs quantitative spatial phase maps in strong agreement with classical Rietveld refinement, paving the way for using quantum computing hardware to analyse real-world materials characterisation data.

Introduction. Open collaborative knowledge graphs such as Wikidata increasingly ground agentic artificial intelligence, information retrieval, and language modeling systems, making systematic auditing of their demographic representation and overall equity a research imperative. Methods. Herein, we present an open-source auditing platform that ingests over 10 million statement bindings representing over 6 million humans on Wikidata via QLever, and evaluates representation of gender, sexual orientation, geography, birthplace urbanicity, ethnicity, multilingual coverage of labels, descriptions, and aliases, occupation, and select intersectional pairs of these entities. It does so by making use of Chi-square goodness-of-fit tests, 95% Wilson-score confidence intervals, and disparity ratios, in light of Rubin's missingness taxonomy. Results. Women accounted for 28.71% (CI +/-0.04) of all humans in Wikidata with a stated gender. 38.26% of humans had a citizenship statement, with Western Europe and North America (WENA) representing approximately 53% of such statements. Among 1.8 million birthplaces that could be classified, rural birthplaces were observed in 2.48% of cases (in comparison to 27.4% global baseline). Fewer than 1.2% of entities carried an ethnicity statement, and non-English Wikidata descriptions covered 18.2% of items. Discussion. Our findings reveal significant missingness across the evaluated axes. Ethnicity and sexual orientation were the most critically under-documented (missing statements) while rural birthplaces and non-WENA citizenship were the most underrepresented.

For a monic irreducible $f \in \mathbb{Z}[x]$ of degree $n$ and an integer $c$, we study the palindromic transform $F(x) = x^n f(x + x^{-1} + c)$, which produces a polynomial of degree $2n$. We give a discriminant formula $\operatorname{disc}(F) = f(c+2)f(c-2)\operatorname{disc}(f)^2$, sufficient conditions for the irreducibility of $F$, and a criterion showing that $F$ is monogenic when $f$ is monogenic and $f(c+2)f(c-2)$ is squarefree, together with a matching non-monogenicity criterion. Iterating the transform yields monogenic number fields of degree $2^k n$ from a fixed base polynomial. As an explicit example, we construct a monogenic field of degree 512 from $x^4 + x + 1$.

A Stieltjes clock allows effective time to advance continuously, remain unchanged over intervals, or jump. We study whether a compact-valued set-valued map evolving relative to a Stieltjes clock admits a single-valued selection that passes through a prescribed graph point while retaining the regularity and variation of the multifunction. For g-Hölder exponents $\alpha \geq 1$, we prove that regularity can be preserved without increasing the g-Hölder seminorm, with no monotonicity assumption on g. If g is additionally nondecreasing, one prescribed-point selection preserves both this regularity and the Hausdorff variation on every subinterval. The same selection consequently preserves all finite Riesz p-variations associated with nondecreasing external clocks. For $\alpha>1$, the structure becomes jump-driven. For left-continuous nondecreasing Stieltjes clocks, continuous clock evolution cannot generate variation: all variation is carried by jumps. We obtain exact jump decompositions for both the set-valued map and its selection, together with explicit atomic formulas for Riesz p-variation. Examples show that the principal regularity, variation, and jump bounds are attained. For compact-convex Euclidean-valued maps, we also examine the case $0<\alpha<1$. In this setting, the Hölder exponent can still be preserved, but preservation of the same constant through a prescribed point holds in one dimension and can fail in higher dimensions. Finally, without a quantitative Hölder bound, we characterize exactly when every compact-valued Hausdorff g-continuous map admits a prescribed-point g-continuous selection: precisely when the clock image is zero-dimensional.

Exact periodic solutions of the incompressible Navier-Stokes equations arise when the convective field is a pure gradient and can be absorbed into the pressure. We construct solutions outside this class, in arbitrary spatial dimension, by an iteration in which each step solves a linear diffusion problem forced by the transverse part of the preceding convective field. The construction is dimension-independent, and for a cyclic family of initial fields the first two iterates are available in closed form. Carried out numerically in three dimensions, the iteration converges: at Re = 10 successive iterates contract with an observed factor 0.43 and the limit is a fixed point of the discretised map to a relative defect of 8.2 x 10^-5. The observed factor exceeds unity by Re = 30, which identifies the practical range of the iteration rather than a proved convergence boundary. We also show what goes wrong when the projection is omitted, as it was in an earlier iteration of the author's: the iterates cease to be divergence-free and an apparent growth appears that belongs to the expansion and not to the flow. For an n-dimensional generalisation of the initial field the fraction of the convective field not absorbed by pressure is exactly 2*sqrt(2)/3 in every dimension n >= 3, and the longitudinal fraction exactly 1/3. The divergence of each returned source is a sharp diagnostic, failing by sixteen orders of magnitude at the step where the projection is omitted; the cell kinetic energy is a weaker one, detecting gross failure only.

We construct commutative rings whose derived categories have non-spatial frames of smashing ideals, thereby answering in the negative questions of Balchin–Stevenson and Aoki.

For a $k$-uniform hypergraph $\mathcal H$, the shadow of $\mathcal H$ is the graph whose edges are the pairs covered by a hyperedge. In this paper, for all sufficiently large $n$, we determine the $n$-vertex $k$-uniform hypergraphs of maximum adjacency-tensor spectral radius whose shadow has no $K_t$ minor, for every $t\ge k+1$, and those whose shadow has no $K_{s,t}$ minor, for every $2\le s\le t$ with $s+t\ge k+1$ and every residue of $n-s+1$ modulo $t$; outside these ranges the problems are trivial. This extends to uniform hypergraphs the theorem of Tait on graphs with no $K_r$ or $K_{s,t}$ minor, whose remaining residues were settled by Zhai and Lin. In each case the extremal hypergraph is unique, and it is the $k$-clique hypergraph of the join of a clique with a graph that we call the light part. For $K_{s,t}$ the answer depends on $j=k-s+1$. When $j\le1$, the maximum has order $n^{(k-1)/k}$, and the light part is the one found by Zhai and Lin for the adjacency matrix, including its exceptional components. When $j\ge2$, a regime that does not occur for graphs, the maximum has order $n^{(s-1)/k}$ and $t$ enters its leading constant. The light part then consists of copies of $K_t$ and one smaller clique, with a single exception: for $(k,s,t)=(9,8,8)$ and $n-s+1\equiv2\pmod 8$, the complement of the Petersen graph appears. When the smaller clique has between $1$ and $j-1$ vertices, the extremal graph is not unique. In particular, for $t=8$, $4\le s\le7$, $k=s+1$ and $n-s+1\equiv2\pmod 8$, the clique hypergraph of the extremal graph of Zhai and Lin is not extremal. For $j\ge2$ the light part is determined by a weighted clique inequality, which for $j\ge3$ follows from a weighted form of the closed-neighborhood counting of Chao and Dong.

Movement disorders (MDs) frequently co-occur, yet phenomenological and severity assessment shows substantial inter-rater variability. Markerless video could improve reproducibility, but prior work is largely single-symptom, depends on standardized acquisition, and lacks validation and transfer across ages and sites. We combined two foundation models into one frozen backbone: Segment Anything Model 3 (SAM 3) for dense, per-frame markerless segmentation summarized into geometric, contour and grid kinematic signals, and TabICLv2, a tabular foundation model, for in-context multi-label classification of eight hyperkinetic MD phenomenologies. Trained on standardized recordings of 21 adults and 4 controls, it transferred unchanged to two independent datasets, pediatric (n=12) and tremor-dominant adult (n=20), assessed with the CODY-SAMP scale; only the patient-level decision step was recalibrated per site. Under clinician consensus labels, false positives fell to zero in both datasets. Dystonia recovered perfectly (7/7 pediatric; 15/15 adult held-out), chorea fully in children (3/3), and tremor was recovered in adults (11/15) once a tremor-rich cohort made it evaluable, through recalibration alone. Per-region effect-size analysis gave clinically coherent, phenomenology-specific signals and identified myoclonus as the principal failure. Against YOLOv8 sparse keypoints, the dense representation matched under clinician permissive labels (Jaccard 0.63 vs 0.63) and was markedly more robust under clinician-label consensus (0.93 vs 0.76). This frozen foundation-model backbone with light per-site calibration yields transferable, interpretable, conservative multi-label phenotyping of co-occurring hyperkinetic MDs across ages and from standardized to routine video, adding robustness on high-confidence, clinician-agreed labels. Prospective multi-centre validation is required before clinical use.

Understanding eco-evolutionary dynamics presents significant mathematical challenges, particularly regarding predictability, equilibrium, and the mathematical representation of fitness. Using Open System Dynamics and Stochastic Analysis, we develop a theoretical framework to characterize fitness change and its relationship with predictability. We explore the limiting behavior of biomass abundances modeled by stochastic processes, deriving continuous-time diffusion counterparts from discrete-time models. Focusing on neutrality and the search for equilibrium, we analyze diffusions in two distinct state spaces. For relative abundances on a d-dimensional simplex, we demonstrate the central role of orthogonal polynomials in model construction. For non-proportional abundances in the positive orthant of d-dimensional space, we detail two specific families of models. Finally, we address statistical inference by presenting numerical examples for relative abundances, illustrating how orthogonal polynomials facilitate maximum likelihood parameter estimation.

I believe that the development of AI tools will lead to a transformation of mathematics, both professionally and culturally. However, it will be a tragic missed opportunity if the massive shifts in culture and practice that are underway are motivated and influenced solely by AI. We must aspire to more than this. We can and should collaboratively strive for a future where our cultures, practices, and values are grounded first and foremost in empathy, compassion, justice, and humanity. In this article, I call to action every member of the mathematical community, whether you view yourself as a researcher, teacher, scholar, student, professional, hobbyist, or a combination of these and more.

We analyze the formation of domain structures in multiaxial ferroelectrics with an open, electrochemically compensated surface. The model combines vector polarization, surface electrochemistry, electrostatics, compatible strain, and competitive adsorption on common sites. We construct homogeneous phase maps and analyze polarization instabilities, first assuming perturbations that are uniform through the film thickness and then allowing depth-dependent polarization profiles. Nonlinear continuation follows the growth of these modulations into developed domains. Optimization of their period and tests of their stability establish how the initial instability wavelength relates to the preferred spacing of an established domain structure. We also consider surface ordering above a stable, weakly anisotropic bulk as a constitutive limit relevant to ferroelectric relaxors, providing insight into possible origins of their periodic surface modulations. Surface energies with the same quadratic terms can produce the same initial instability but favor different developed states, including periodic domains and uniform in-plane polarization. This analysis provides a procedure for connecting the initial development of ferroelectric modulations to amplitude selection, wavelength adjustment, and the subsequent evolution of the resulting domain structures.

We prove that, for almost every uniformly random configuration of $N$ particles in a two-dimensional periodic square domain, there exists a unique nearest stealthy hyperuniform configuration, up to floppy-mode displacements, when a finite set of $m$ low-$k$ Fourier modes is constrained to vanish exactly. Under an explicit full-rank Jacobian assumption, the displacement component orthogonal to the tangent space of the finite-mode hyperuniform manifold is uniquely determined, while residual tangent-space degrees of freedom correspond to first-order floppy modes that preserve the constrained low-$k$ density fluctuations. We restrict attention to the regime of constrained-mode fraction $\chi=m/(dN)$ in which such configurations are known to remain disordered rather than crystallize. To corroborate this framework, we implement a gradient-based generator network that iteratively displaces particles to minimize a combined loss functional of hyperuniformity, short-range repulsion, and smoothness, and whose dynamics is expected to approximate the theory's minimal orthogonal projection onto the hyperuniform manifold. The network drives generic random configurations toward disordered hyperuniform representatives, progressively suppressing long-wavelength density fluctuations as measured by the structure factor and number-variance exponent\textcolor{black}{, over a reciprocal-space window that extends well beyond the explicitly constrained modes and is bounded, as we show, by the structure-factor sum rule. Local bond-orientational analysis and the absence of Bragg peaks confirm that the resulting states remain disordered}. This integrated analytical and computational approach provides a rigorous geometric foundation for understanding stealthy hyperuniformity and for generating its disordered representatives from generic random configurations.

In this paper, we study the $p$-adic zeros of the Tribonacci sequence $T_n$, for a prime $p$. In particular, we complete the analysis that was done in a 2024 paper of Yuri Bilu by investigating the case of $p=11$. In addition, we investigate the primes $p$ for which at the first two values $T_n$ and $T_{n-1}$ which are both divisible by $p$, both of these are actually divisible by $p^2$, proving under the assumption of a generalisation of the $abc$ conjecture that there are infinitely many $p$ which are not such.

Once diagnosed, cancer requires a fast, reliable and preferably cost-efficient assessment of the current state and potential progression of the disease. A new method for estimating tumor cell diffusivity $D$ and proliferation rate $\gamma$ in the context of the mechanistic reaction-diffusion equation from single-point-in-time routine biopsies aims to deliver just that, and quantities computed from the parameter estimates have recently been tested as a new biomarkers for risk-stratification in radiotherapy. Here, we extend the findings of this previous work by providing a first theoretical validation. The method is applied to in-silico biopsies which are generated by solving the two-dimensional reaction-diffusion equation for different growth terms (exponential and logistic) with a Dirac-Delta initial condition, and transforming the continuous results into spatial point patterns via a form of reverse coarse-graining. If no information about tumor age is used, in short-term experiments the original dispersion length $\sqrt{D/\gamma}$ could be retrieved with a relative root mean squared error (RRMSE) of around 8% and an $\text{R}^2$-value of 0.97. In long-term experiments, the RRMSEs ranged from 8 to 14% and the $\text{R}^2$-values from 0.75 to 0.98. The scaled front velocity $\sqrt{D \cdot\gamma}$, which can only be estimated if information about tumor age is available, was retrieved with an RRMSE of 7% and an $\text{R}^2$ of 0.98 in both, the short-term and the long-term experiments.

Methods for water demand forecasting and leak detection are based on public datasets, and for water those are scarce, short, or released only after an undocumented cleaning process, hiding defects of the deployment they came from. We present Tethys: 91 months of hourly water consumption data from 24 buildings of a municipal water network, published with a quantitative account of its quality, rather than in place of one. Raw availability is 59.4%, while loss is not random: 4 fleet-wide outages totalling 595 days interrupt the entire estate at once. We show that the aggregation producing the released files silently introduced 205,200 impossible decreases in a cumulative index, and that correcting it is a one-line change. Because the meters are cumulative, the readings bracketing a short gap fix the volume that passed through it, so 75.9% of hours rest on a measurement, while 24.1% are reported as unknown. We release the dataset, its per-hour provenance, and the pipeline that produces it.

In this paper, we explore a variable-coefficient Fokas-Lenells system with gain/loss. Applying a nonlocal symmetry reduction, the system leads to a PT-invariant variable-coefficient Fokas-Lenells equation (vcFLE) with specific constraints on dispersion and nonlinearity coefficients. We prove the Lax integrability of the given system and construct one- and two-soliton solutions using a nonstandard Hirota bilinearization method. We show that, depending on the soliton parameters, the solutions can retain or violate the underlying symmetry. With appropriate choice of dispersion and nonlinearity, the system shows periodic-wave and soliton structures.

Every smooth closed Riemannian surface of genus zero is isometric to the unit sphere carrying a conformal metric, the round metric multiplied by a positive factor. This paper solves the Poisson equation and the eigenvalue problem of the Laplace–Beltrami operator of such a metric with smooth spherical splines, taking the conformal factor as the only description of the geometry. Because the Dirichlet energy is conformally invariant in two dimensions, the stiffness matrix is the round-sphere matrix for every conformal metric, and the factor enters only the load vector, the mean-value constraint and the weighted mass matrix. A surface embedded in space with a conformal parametrization, a surface of revolution, and a surface given only as a triangle mesh are instances of the same problem that differ in how the factor is obtained; the first two are treated here, and the factor is checked against an area identity before any equation is discretized. The unknown is a spline on a fixed spherical triangulation in broken Bernstein–Bézier form, with $C^r$ smoothness imposed algebraically through edge functionals and the conforming space realized by a null-space matrix computed once and reused for every metric. Each step of the computation is stated with the formulas it needs. Experiments on the round metric, a prolate spheroid, a dumbbell of revolution, an Evans–Fung red-blood-cell profile and a prescribed factor with no embedding show the rates $d+1$ in $L^2$ and $d$ in energy for spline degree $d$, the rate $2d$ for eigenvalues with reproduction of the even-order eigenvalues of the round metric up to degree $d$ to roundoff, and, against parametric surface finite elements on the same triangulations, smaller errors at comparable numbers of unknowns for degrees four and six.

We introduce Bernstein constraint complexes, a representation of finite element differential complexes in which every triangle of a triangulation keeps its own Bernstein–Bézier coefficients and global continuity is imposed only through smoothness functionals attached to edges and vertices. Continuity of scalar splines, tangential continuity of spline vector fields, normal continuity of spline vector fields, and higher smoothness differ only in the edge functionals. No global basis is constructed. The structural result is a coefficient-level commuting identity: on every edge the smoothness functionals of a Bernstein derivative are explicit combinations of the smoothness functionals of its argument, through univariate Bernstein difference matrices. This gives a local proof that the kernels of the smoothness matrices form a subcomplex, for the de Rham traces and for componentwise $C^r$ spline profiles, and a computable row-space test for candidate smooth profiles. We realize the lowest-order Powell–Sabin exact sequence in this form and confirm its exactness by ranks. The constrained Galerkin problem is solved in the broken coefficients with exact smoothness equations; we prove equivalence with the conforming method, treat semidefinite element operators through a pseudoinverse interface system, and give a singular pencil whose finite spectrum is the constrained Maxwell spectrum. Under the covariant and contravariant surface Piola maps the reference functionals of the de Rham sequence transfer unchanged to conforming curved triangulations. Experiments compare the null-space method, the augmented Lagrangian iteration of Awanou, Lai and Wenston, and direct solution of the saddle system on the same problems, and cover planar source and eigenvalue problems, a $C^1$ biharmonic solve, the exact sphere, and an embedded hyperboloid.

It is proved that the affine theory of the full Riemann tensor constitutes the extended theory of gravity. The fundamental object here is a spacetime symmetric connection. Its physical interpretation is that of a field of local inertial frames. In this approach, gravity arises, in a natural way, as a local version of Newton's First Law. A variational formulation of the theory is presented, based on the results introduced in the article co-authored by the present author and one of the supervisors. The general framework is supported by several examples. A central result of the dissertation is the transition from the affine picture to the metric picture. A simplified version of this procedure – valid for a specific class of Lagrangians. It was proved there that the non-metricity in the affine picture can be interpreted as a matter field in the metric picture. This is precisely Hermann Weyl's interpretation of electromagnetism. Here, the transformation from the full Riemann tensor theory to the conventional metric theory is examined in full generality for the first time and illustrated using the previously introduced examples.

Mobile river monitoring robots must interpret obstacles and water boundaries that geographic waypoints alone cannot describe. On resource constrained platforms, converting imperfect visual predictions into timely and inspectable guidance is a distinct challenge. An object label or steering command does not explain which evidence supports a decision or when that evidence is unreliable. We present PAANI, an on-device perception to guidance architecture that combines a project trained YOLO11n detector and a custom MobileNetV3 Small semantic segmenter with timestamp aligned evidence fusion on Arduino UNO Q. Bounded tracking supplies object persistence, while an explicit corridor policy combines surface labels, accepted detections, urgency and mask uncertainty. Each final advisory exposes its contributing evidence and policy reasons. ROS 2 interfaces connect the local AI pipeline to a separate Gazebo vessel, localization and control testbed. Training uses 10,000 WaterScenes images for four-class detection and 1,127 MaSTr1325 images for segmentation, including 198 segmentation validation images. The selected FP32 ONNX models occupy 14.817 MB. Detector checkpoint test mAP at 0.5 IoU is 0.7388, while the separately evaluated rectangular ONNX export achieves validation mAP at 0.5 IoU of 0.7367. Segmentation ONNX validation mIoU is 0.9750. A five-minute UNO Q recording produced median and 95th percentile pipeline latencies of 467.8 ms and 580.3 ms at a configured 0.5 Hz cadence. The evaluation also identifies black input misclassification and a sampling rate mismatch that prevents the diagnostic apparent motion estimator from collecting sufficient evidence. These results support an inspectable and reusable edge robotics foundation while clearly distinguishing model accuracy and on-board execution from validated on-water collision avoidance.

We study the numerical evaluation of contour integral representations for solutions of the linearized second and fourth Painlevé equations. The construction combines a nonlinear background solution, continuation of canonical Lax-pair columns, normalization matching, and quadrature over cycles with decaying branches in distinct sectors. Accuracy is assessed using a reference fundamental matrix, differential-equation residuals, and variations of monodromy data computed independently from the spectral problem. An exact identity describes the propagation of initial-basis errors. A stagewise a posteriori refinement criterion uses increments in monodromy variations to allocate numerical accuracy. The contour algorithm is compared with the Dormand–Prince method on a regular background, in a region of rapid solution growth, and through hard loss of stability followed by approximately regular oscillations. Accumulated offsets in monodromy variations and subsequent drift are measured separately. The experiments show how drift depends on spectral conditioning, quadrature accuracy, and error allocation between initial and subsequent parts of the computation. Five additional initial-data and parameter cases demonstrate the dependence of comparative performance on the nonlinear background.

Open-vocabulary 3D Scene Graphs (3DSGs) ground each object node in a vision-language embedding, yet they record every entry as equally certain, so a robot querying the map cannot tell which of its entries are unreliable. Estimators of semantic uncertainty could supply that distinction, but they require repeated sampling of a model, training, or held-out labels, none of which are available to a deployed system at query time. We present a framework that exploits the detector confidence and the embeddings a 3DSG already stores, converts them into a probability that an entry is correct, and propagates that probability through the containment hierarchy into a belief that a room contains a queried class. Four signals, each paired with the object-level error it indicates, are converted to probabilities at the logit scale learned by the vision-language model and combined in closed form with no additional perception or training. Objects sharing a detector and a vocabulary fail together, so the framework aggregates them in the fully correlated limit, where an aggregation under independence would treat one repeated error as repeated evidence. Evaluated on HM3DSem against a state-of-the-art 3DSG system, the framework improves object retrieval and lowers the error of the room-level assertions of the graph it reads.