Papers

In many real-world domains, knowledge is inherently vague or imprecise - features that classical ontology languages, based on crisp Description Logics (DLs), are unable to capture. This shortcoming poses particular challenges for applications in the Semantic Web and Explainable Artificial Intelligence (XAI), where robust reasoning over graded information is essential. Fuzzy ontologies address this limitation by enriching DLs with fuzzy logic, enabling the expression of partial truth and supporting more nuanced modelling of real-world knowledge. We present fuzzy-dl-owl2, a complete re-engineering in Python of the fuzzyDL reasoner and the Fuzzy OWL 2 framework. The former is an expressive fuzzy DL reasoner, while the latter allows for defining fuzzy ontologies within OWL 2. Our contribution addresses several shortcomings of the original software, including semantic inconsistencies, rigid architectural design, and limited solver integration. The re-implementation features a modular class hierarchy tailored for extensibility, supports a broader range of Mixed-Integer Linear Programming (MILP) solvers (including open-source alternatives), and corrects IRI ambiguities arising from overlapping ontological elements. Furthermore, a dedicated Python library (pyowl2) has also been developed to handle OWL 2 annotations in a standards-compliant manner, improving interoperability with existing Semantic Web tooling and resolving IRI ambiguities. The resulting framework offers a portable, extensible, and theoretically grounded platform for reasoning with fuzzy ontologies, suitable for both research and deployment in vague-aware systems. Performance tests have also been conducted that show improved execution times w.r.t. the original Java implementation. The source code and full documentation are publicly available to facilitate community adoption and further development.

Streetlights shining across rippled water often produce tall, narrow reflections with strikingly parallel sides. These light pillars appear almost architectural, yet the water surface is neither vertical nor smooth. We develop a geometric optics model that explains the phenomenon using the specular reflection rule, projection geometry, and the physics and statistics of surface slopes. A rippled water surface can be viewed as an ensemble of small facets (tangent planes on waves) acting as tiny mirrors. As one looks farther across the water (coordinate x), the physical width of the region whose facets reflect the light into the eye (or camera/pinhole) increases, but the pinhole projection onto the image plane compresses this widening by a factor proportional to 1/x. The two effects cancel, producing a reflection whose image width remains constant. This paper appears to be the first to formally document this. (Deviations from the strict cancellation at both ends of the image are also explained.) The analysis clarifies how earlier qualitative treatments failed through a lack of identifying the role of projection geometry. The model also qualitatively explains the brightness variation along the pillar, and why extended objects such as buildings do not produce elongated reflections. The treatment is intended for scholars of introductory optics and for those interested in the physics and mathematics of everyday visual phenomena. We conclude by outlining how the same framework extends naturally to more complex viewing situations, such as the setting sun viewed from a cliff top over the sea.

The macroscopic instability of soft materials undergoing extreme deformations is traditionally viewed as a pure structural or mechanical failure. Driven by the quest to uncover universal principles across disparate physical systems, we bridge two vibrant yet seemingly disconnected research frontiers: macroscopic finite-strain solid mechanics and quantum-like topological physics. Here, we demonstrate that the classical elastic surface instability of a deformed hyperelastic manifold is not merely a mechanical bifurcation, but fundamentally a topological phase transition. By incorporating Lie group metric evolution into a generalized Stroh formalism, we map the highly nonlinear geometric frustration onto an algebraic surface impedance matrix $\mathbf{H}$. For a semi-infinite hyperelastic half-space under finite compression, we analytically map the system to a one-dimensional Dirac Hamiltonian, where the macroscopic mechanical stretch acts as a tunable knob for the Dirac mass. We reveal that the onset of surface wrinkles marks a topological transition from a trivial to a non-trivial phase characterized by a quantized step in the winding number, naturally giving rise to a robust, macroscopically localized zero-energy edge state. This fundamental linkage unifies macroscopic symmetry breaking with the topological paradigm, opening a new theoretical pathway for programmable smart soft matter.

Multi-cause observational studies contain information about unmeasured confounding through the dependence structure among causes. However, literal imputation of the unobserved confounder is often more complex than learning a lower-dimensional substitute score that preserves the shared assignment variation needed for stable causal adjustment. The deconfounder (Wang and Blei, 2019) and related substitute confounder methods exploit this idea, but flexible assignment models can fit the joint distribution of the causes while producing scores that over-encode the treatment vector, collapse overlap, or capture single-cause variation. We develop a Bayesian factor assignment framework for learning sparse substitute confounders that retain coarse multi-cause dependence with shrinkage priors. The theory is stated at the level of posterior concentration, factor score contraction, and overlap-preserving assignment geometry and therefore does not rely on a particular shrinkage prior. Under these conditions, the proposed regression-adjusted estimators are consistent for mean potential outcomes when the corresponding latent variable identification assumptions hold. Shrinkage priors provide a natural tool for latent structural learning: they favour low-dimensional factors supported by multiple causes, discourage effectively single-cause factors, and induce an ordering of the latent factors through progressive shrinkage. Synthetic experiments illustrate the roles of signal strength, outcome validity, and geometry-aware regularization. In an Alzheimer's Disease Neuroimaging Initiative (ADNI) baseline analysis, sparse substitute scores recover much of the adjustment obtained by directly conditioning on invasive cerebrospinal-fluid biomarkers, while collapse diagnostics identify when fitted factors reduce to individual observed measurements.

Different kinds of definable patterns in the models of a first-order theory, such as the order property, the tree property, or the ($n$-)strict order property, allow us to distinguish theories according to their logical complexity. The complexity distinctions given by these definable patterns play a central role in model theory. However, a rigorous definition of the notion of a model-theoretic patterning property has yet to be established. We start by discussing different proposals from the literature for making the notion of a model-theoretic patterning property rigorous. Some examples will include the straight definability of Shelah, which will describe properties definable by a pattern of consistency and inconsistency in a formula and its negation, and the poset definability of Garcia and Mennuni, covering properties definable by interpreting a partial order embedding a given poset. We will also introduce a higher-arity version of straight definability. In our first main result, we will answer open questions of Bailetti and Garcia-Mennuni, showing that the $n$-strict order property $\mathrm{SOP}_{n}$ is straightly definable and poset definable even for integers $n \geq 4$. This will complete the categorization of all of the classical classification-theoretic properties as straightly definable. Our other main result will concern properties that are straightly definable without negation: the positively straightly definable properties defined by Bailetti. We will show using Saracino's theorem and results of Bodirsky, Bodor and Marimon that, in any countably categorical theory, implications between positively straightly definable properties must be exhibited at the level of $\exists\forall$-formulas. This will have special consequences under the assumption that $\mathrm{SOP}_{2}$ is equal to $\mathrm{SOP}_{3}$.

The Mixed-Radix One-Time Pad (MR-OTP) extends the classical OTP to heterogeneous alphabets while preserving perfect secrecy. We provide a practical, bias-free method to convert raw binary entropy from a QKD source into uniform mixed-radix keys by identifying Horner's method and its inverse as the natural mapping between binary integers and mixed-radix tuples. We show that naive modular reduction induces bias and prove that rejection sampling restores uniformity with optimal expected cost. We establish end-to-end information-theoretic security for single and multi-session pipelines, quantify efficiency gains, present a batched extractor, and give unconditional and conditional results on the Base Recovery Problem.

We prove that there is no local quasiconformal map between the limit set of a convex-cocompact Kleinian group and the Julia set of a postcritically finite rational map, provided that both are Sierpiński carpets. This contrasts with the recent results by Y. Luo, D. Ntalampekos and Y. Luo, M. Mj, S. Mukherjee for gasket and tree-like spaces, respectively.

We present sufficient conditions for the semi-global exponential stability of nonlinear systems whose dynamics have both slow and fast time variations. Unlike most existing results, the fast variation is non-periodic, thereby allowing a wider class of systems, especially switched systems with fast (non-periodic) switching and those with quasi-periodic variations; we therefore rely on general averaging to construct an average system. It is assumed that the average system admits a time-invariant equilibrium that is globally exponentially stable when the slow variation is frozen, i.e., remaining at a fixed value. This slow variation is allowed to be discontinuous in time, provided its total variation (flows and jumps) is bounded. The main result is illustrated using a nonlinear switched system with slow-fast non-periodic switching.

We define a homological Conley index for a class of hybrid dynamical systems. This is achieved by factoring through the hybrid suspension semiflow, which views a class of hybrid dynamics as a continuous semiflow. We establish that this invariant is well-defined, invariant under continuation, and satisfies the Ważewski property and other reconstruction results. We compute the index for several examples, showcasing how classical reconstruction theorems apply in this setting to characterize hybrid dynamics.

In this paper, we show that there are $57$ nonnegative integer solutions $(a,b,x)$ to the inequality $1\le \lvert x^{2} - 2^{a}\cdot 3^{b}\rvert < 3\max\{a, b\}$ and we list them explicitly. The inequality is converted into a statement about how closely $x/q$ approximates irrational number $\sqrt{d}$ for $d\in\{2,3,6\}$, where $q$ is an integer which is $3$-smooth, after which Worley's theorem on rational approximations via continued fractions is applied to parametrise the solutions and a $p$-adic lower bound for a linear form in logarithms due to Bugeaud and Laurent is applied to find a rather large bound on $\max\{a,b\}$. We finish with an application of the LLL algorithm to reduce this bound.

We show that finite morphisms of smooth algebraic varieties naturally give rise to Cartan–Coxeter type structures. Starting from the self-fiber product $X\times_YX$ of a finite morphism $Q:X\to Y$, we construct local symmetry operators and intrinsic Cartan-type invariants from the geometry of its non-diagonal irreducible components. This provides a mechanism for reconstructing root-theoretic structures directly from algebraic correspondences rather than from reflection groups. A central part of the theory is a rank-two geometry associated with pairs of non-diagonal components. We establish a rank-two reduction theorem, derive explicit trace and determinant formulas for the corresponding operators, and obtain a classification into elliptic, parabolic, and hyperbolic transport types. These results yield intrinsic analogues of Cartan matrices, Coxeter transformations, exponents, and Dynkin diagrams associated with finite morphisms. We further prove rigidity theorems showing that the structures arising from a single finite morphism are highly constrained. To obtain richer geometries, we introduce transport atlases of compatible local finite covers equipped with connection data, leading to nonlinear Cartan fields with variable local geometry. This places classical Weyl and complex reflection geometries within a broader correspondence-based root theory extending beyond finite reflection groups.

A convex scene communicates more than shape: the pattern of which groups of objects are mutually apart and which cross is a discrete relational payload. We treat the apartness table of a finite family of convex bodies, a separoid, as a source signal; a renderable convex scene as its encoder; and the rendered image as a noisy visual channel from which the apartness structure is decoded. For disjoint index sets $A,B$, the source bit records whether $\operatorname{conv}(\bigcup_{a\in A} C_a)$ and $\operatorname{conv}(\bigcup_{b\in B} C_b)$ are disjoint. Within this view, apartness-preserving rendering becomes a rate–distortion problem: the rate is a differentiable geometric code length for the carrier scene, while the distortion is closure-aware and weights maximal separations and minimal Radon partitions by the number of consequences they control. A differentiable support-function realization turns separability into a soft directional margin and represents each separation by a distribution over witnessing directions, yielding a variational lower bound on apartness mutual information $I(\Sigma;Y)$ and an information-theoretic account of view selection. Experiments on planar convex scenes show that scenes are recovered from the apartness table alone at 99.9% bit accuracy, with the certificate skeleton already determining the full table; coordinate quantization gives a clean operational rate–distortion frontier where certificate distortion is more stringent than Hamming error; and rendered $48\times48$ images transmit about 0.72 of the apartness-graph entropy under mild noise. Increasing the viewpoint-robustness term widens separating cones with only a modest geometry-rate surcharge. The result is a certificate-aware rendering objective for scenes whose purpose is to make relational convex structure recoverable rather than merely pixel-faithful.

We prove a fixed finite-window structural theorem for sharp localized Navier-Stokes packages, formulated as both a local-to-clean detection theorem and an anti-phantom principle. The result addresses whether a defect visible in the baseline quotient geometry can disappear after pressure-tail enrichment, residual transfer, quotienting, and clean-to-local detector comparison. Under synchronized representatives, baseline-to-tail visibility, component comparison, residual-ledger closure, detector comparison, chart visibility, and a clean quotient gap, any baseline-visible defect is either detected by the localized detector or charged to an explicit quotient-residual ledger. Quantitatively, M_Lambda^loc(D-zeta_*) >= c_{Lambda,0} Dist_{loc,intg,0}(D,Gamma^{intg}*{Lambda,adm}) - E^{quot}*{Lambda,0}(D). The proof assembles three modules: pressure-tail visibility, componentwise residual-ledger closure, and detector comparison. The anti-phantom interpretation is that a baseline-visible defect cannot be simultaneously detector-silent and residual-cheap. We also record provenance for the imported quotient interface, finite-dimensional pressure-tail models, explicit matrix realizations of the structural inputs, NS-generated coordinate realizability, compactness criteria for clean pressure images, and reduced pressure/tax kernel-free criteria.

In bilevel optimization, optimistic and pessimistic follower behaviors are the most commonly used forms to define how the follower ties-breaks among multiple optimal solutions. In this work, we go beyond these extreme tie-breaking behaviors and investigate the intermediate bilevel optimization program (I-BO), where the follower's selected optimal response is a decision-dependent random event, with a probability measure influenced by the leader's decision. We formally introduce a class of such endogenous measures, including the special case of strong-weak decision-dependent I-BO. We reformulate the I-BO as a Transformed I-BO (T-I-BO) with exogenous uncertainty by defining inverse and Markov-chain transformations, which represent the follower's response as a function of the leader's decision and exogenous randomness. We handle the T-I-BO's uncertainty via sample-average approximation (SAA), and we propose tailored approaches for its SAA program according to the chosen transformation. Computationally, our methods solve reasonable-sized instances efficiently and outperform the deterministic equivalent when available. Furthermore, experiments stress the critical need to accurately model follower tie-breaking behavior, particularly depending on its alignment with the leader's objective, as misspecification leads to suboptimal leader decisions.

In 1921, Kuratowski gave the now-standard definition of ordered pair in the context of set theory. This paper studies the problem of defining ordered tuples as sets in the hierarchy of simple types, of which Kuratowski's construction is a special case.

We construct exact, positive-energy, normalizable wave-packet solutions of the Dirac equation in the axisymmetric potential $V=-\,v_0/\rho$ – to our knowledge, the first such solutions in any external potential. Remarkably, one family comprises only elementary functions whose longitudinal profiles reproduce the free-Schrödinger Hermite–Gauss wave packets in the nonrelativistic limit. All packets share two striking features: (i) a probability density that is pointwise decoupled from spin orientation – despite the inherent spin-orbit coupling of the Dirac equation – and (ii) a complete freezing of their time evolution at the critical coupling $v_0\to\hbar c/2$. We also present a simple scheme that maps solutions of the 2D Helmholtz equation to further exact Dirac wave packets.

Distributed stochastic gradient descent (SGD) is limited by communication rather than computation, since each iteration requires an AllReduce across processes. Communication-avoiding SGD (CA-SGD) amortizes communication over $s$ iterations by replacing $s$ consecutive AllReduces with a single AllReduce of an $sb\times sb$ Gram matrix, trading more computation and bandwidth for fewer synchronization points. Modern GPUs with matrix hardware and reduced-precision formats offset this by accelerating the Gram GEMM and shrinking BF16 traffic. We study mixed-precision CA-SGD for generalized linear models on NVIDIA GPUs. Our finite-precision analysis decomposes the local rounding error of one CA-SGD outer iteration into nine independent precision choices, depending on the hardware only through its low-precision unit roundoffs, so the resulting recipes transfer in principle across GPU generations. The recipe stores the input matrix and margin vector in low precision, computes the Gram matrix from low-precision inputs with high-precision accumulation, communicates it in high precision, and performs the inner recurrence and weight updates in high precision. On NERSC Perlmutter A100 GPUs, mixed-precision CA-SGD matches FP32 SGD loss within $0.5\%$ on logistic, linear, and Poisson problems and reaches $5.1$–$6.8\times$ speedup over FP32 SGD on epsilon, SUSY, HIGGS, synth, and Poisson-synth. Our software is available at https://doi.org/10.5281/zenodo.20448273

Dirac asked in 1970 whether for every k >= 4 there is a k-vertex-critical graph without critical edges; Jensen settled all k >= 5, and only k=4 remains open. Following Skottova and Steiner, call a graph G a (4,1)-graph if chi(G)=4, chi(G-v)=3 for every vertex v, and chi(G-e)=4 for every edge e; they proved delta(G) >= 6 and lambda(G) >= 6 for every (4,1)-graph and asked whether a 6-regular (4,1)-graph exists. We prove three results about this 6-regular case. Theorem A (computational): there is no 6-regular 4-vertex-critical graph on n <= 15 vertices, except for a unique graph (up to isomorphism) on n=13, whose 13 critical edges form a Hamilton cycle; hence any 6-regular (4,1)-graph has at least 16 vertices. Theorem B: in a 6-regular (4,1)-graph every 6-edge-cut is either the edge star of a vertex or has both shores of size at least 15; consequently every 6-regular (4,1)-graph on at most 29 vertices is super-6-edge-connected. Theorem C (all sizes): no shore of a nontrivial 6-edge-cut in a 6-regular (4,1)-graph induces a bipartite graph; more generally, a shore whose deficiency is concentrated on two vertices forces them to receive equal colours in every proper 3-colouring. The proof of Theorem B rests on an exact classification of the 3x3 cut matrices of 6-edge-cuts in (4,1)-graphs (exactly 21 matrices, five types up to row/column permutations) together with a boundary-shortfall lemma; the unique near-miss is K_{3,3,3} minus a rainbow 3-matching. Several supporting lemmas are machine-checked in Lean 4/Mathlib.

Let $X_r\sim N_r(0,1)$ be the centered unit-scale generalized Gaussian random variable with density proportional to $\exp(-|x|^r/2)$. We prove that, for $p,q>0$, there exists a strictly positive random variable $V$, independent of $X_q$, such that $X_p\stackrel{d}{=}VX_q$ if and only if $p\le q$. Moreover, the law of $V$ is unique. For $p<q$, put $a=1/p$, $b=1/q$, and $\alpha=b/a=p/q$. If $S_\alpha$ is a positive $\alpha$-stable random variable with Laplace transform $\mathbb{E}\exp(-uS_\alpha)=\exp(-u^\alpha)$, set $W_0=S_\alpha^{-b}$, let $W$ be the $W_0$-size-biased version of $W_0$, and define $V_{p,q}=2^{a-b}W$. Then $X_p\stackrel{d}{=}V_{p,q}X_q$. For $p>q$, the required Mellin quotient, viewed as the candidate characteristic function of $\log V$, is unbounded by Stirling's formula, and hence cannot be a characteristic function. The factor laws form a multiplicative cocycle, $V_{p,r}\stackrel{d}{=}V_{p,q}V_{q,r}$, for $p\le q\le r$, where the factors on the right-hand side are independent copies. Thus the Mellin quotient isolated by Dytso, Bustin, Poor and Shamai is realized constructively throughout the $p<q$ branch. In particular, $\Phi_{p,q}$ is positive definite exactly in the range $p\le q$, and the inverse Fourier–Mellin candidate density in the remaining $p<q$ branch is a genuine nonnegative probability density. The known Gaussian-base and bounded-parameter product cases are recovered as parts of a single positive scale-mixture classification.

As a promising solution for extremely large-scale arrays, reconfigurable holographic surfaces (RHS) can be integrated with integrated sensing and communication (ISAC) to form the holographic ISAC paradigm, where significant beamforming gains provided by RHS can improve both communication and sensing performance. However, most existing works on holographic ISAC assume that RHS elements can control the amplitudes of radiated ISAC signals in a continuous manner, which is hard to implement in practice. In this paper, we investigate how the limited radiation amplitudes of the RHS influence ISAC system performance. Specifically, we first derive closed-form lower bounds for both communication rate and sensing SINR to establish a tight upper bound on the minimum radiation amplitude quantization bits. Based on this, we further explore how the communication-sensing performance tradeoff impacts the quantization bits. Numerical results validate our theoretical analysis.

We study the semiclassical limit of SL_2-opers on the four-punctured sphere in Nekrasov-Rosly-Shatashvili Darboux coordinates. Using exact Wentzel-Kramers-Brillouin (WKB) connection formulae, we express the trace coordinates of the corresponding SL_2(C) character variety as finite Laurent sums of Voros exponentials. Tropicalizing these formulae and the NRS relations gives a chamberwise integer affine linear system for the leading logarithms of the NRS coordinates. In chambers where this system is unimodular and the selected cycles form a primitive symplectic pair, the leading asymptotics agree, up to flavor-period shifts, with Seiberg-Witten periods of the N=2 SU(2) theory with N_f=4 fundamental hypermultiplets. We verify this mechanism in a sample chamber and in the weak-coupling degeneration. No global coordinate-independent recovery theorem is claimed; non-unimodular or degenerate chambers are treated as limitations of the chosen NRS chart. In the weak-coupling degeneration, we show that the NRS chart can be chosen compatibly with the plumbing limit so that the resulting chamber is unimodular and non-degenerate away from tropical walls.

We study the propagation of coupled surface and internal equatorial internal waves. A model of two vertically stratified fluid layers with different constant densities is employed. Taking Coriolis forces into account, we derive explicit solutions to the linearized governing equations which assumes irrotational fluid motion in both layers separately, and further obtain the dispersion relation which determines the phase speeds of propagating surface and internal waves. We prove a result on solutions to the dispersion relations which greatly simplifies our subsequent analysis of the nonlinear dynamical systems which describe the motion of the fluid in the upper layer. Phase portraits for all possible streamlines in both fluid layers are presented, while furthermore a Lagrangian description of the fluid flow is obtained, and the particle trajectories of the fluid particles are determined.

We develop a spectral based coefficient of determination to measure how well the spectral density of a stationary linear process is represented by the class of MA($q$) models. Using periodogram-based estimators, we establish asymptotic normality, derive tests for the MA($q$) hypothesis, and construct procedures for determining the smallest order $q$ achieving a prescribed approximation quality.

The singularities of a neurovariety have so far always turned out to lie at subnetworks: functions already computed by a smaller architecture of the same kind. This is what makes them readable as a bias of training towards simpler models, and whether they exhaust the singular locus is left open in the literature. We show that they do not. There are singular points at which every neuron is active, no two neurons are redundant, and the parametrisation is unramified, so that nothing visible at a single parameter signals the singularity. We call these exceptional, and we show that, after quotienting by the group of architectural symmetries, which acts freely on the full parameters, they can arise in only three ways: a rank drop of the differential of the parametrisation, a disconnected fibre, or a failure of properness. A small dimensional example is a fully connected network with two hidden layers of width two and quadratic activations. All the examples we produce are of the second kind, so that what fails at an exceptional singularity is not the simplicity of the model but the possibility of recovering the parameter, up to symmetry, from the function it computes.

In this paper, we study a sequential workforce management problem in a contingent labor setting with uncertainty in both worker production and labor supply. A firm seeks to maximize cumulative profit by maintaining an active team of fixed size while learning worker productivity over time. We emphasize two critical operational frictions in this problem: replacing workers is costly, and workers may not be available immediately for hiring because of, for example, prior job commitments, scheduling constraints, or onboarding procedures. Thus, hiring decisions take effect only after a random delay. We formulate this problem as a stochastic multi-play bandit with costly switching and delayed actions, and develop a learning-based hiring policy, DR-UCB (DelayedReplacement-UCB), that makes replacement and hiring decisions sequentially through learning cycles. In each cycle, the policy uses real-time production data to determine when to initiate workforce changes and which workers to replace and hire. We show that the leading-order regret of the proposed policy matches its lower bound in its dependence on the time horizon. Our numerical experiments show that DR-UCB outperforms benchmark policies.

In this paper, we use the recently introduced notation and terminology for magnetic systems, which provide a natural framework for extending techniques and results from the theory of geodesic flows to magnetic flows. In this context, we prove a magnetic analogue of the classical Freire–Mañé theorem and obtain some applications. As a consequence, we show that for an energy level of a magnetic system without focal points, either the metric entropy is positive or the magnetic curvature is identically zero.

This paper investigates a class of Reflected Mean-Field Stochastic Differential Equations when the noise is driven by a Brownian motion and an independent Poisson measure. We prove the existence and uniqueness of solutions and provide moments estimates for the state processes. We apply our result to derive a Feynman-Kac formula for the solution of an Integral-Partial Differential Equation with Neumann boundary conditions.

This paper develops a variational framework for regulated language generation. Starting from autoregressive token sampling, we derive the induced distribution over complete messages and relate it to an entropy-regularized Gibbs law. Regulation is modeled as an optimal discriminator whose convex-dual value is an f-divergence, and the generator-regulator interaction is formulated as a saddle-point problem. The framework applies to moderation, censorship, AI deception detection, compliance auditing, phishing defense, and manipulation control, where regulation concerns a distribution over possible messages rather than a single output. The equilibrium clarifies the tradeoff among utility, entropy, regulatory alignment, and finite-length detectability. Two finite-vocabulary case studies, censorship filtering and phishing defense, illustrate how the theory can be evaluated through utility, entropy, divergence, receiver-side scores, and detection probability.

We establish an unconditional analogue of Sylow's first theorem for finite skew braces, and deduce an analogue of Cauchy's theorem. We also prove an analogue of the existence part of Hall's theorem for finite skew braces with soluble additive and multiplicative groups. We make some observations regarding the number of Sylow subskew braces of a skew brace in various cases. By applying these results we streamline the classification of skew braces of order $ pq $, where $ p,q $ are distinct prime numbers.

Cholesky-QR is among the fastest algorithms for computing the thin QR factorization of tall-and-skinny matrices on GPUs, relying entirely on BLAS-3 operations. However, it is numerically unstable: forming the Gram matrix squares the condition number, causing breakdown when $\kappa_2(\boldsymbol{A}) \gtrsim 10^8$. We present MRCQR (Mixed-Precision Randomized Cholesky-QR), a stable GPU algorithm that addresses this limitation. MRCQR uses a subsampled randomized trigonometric transform to construct a preconditioner $\boldsymbol{R}_s$ that reduces $\kappa_2(\boldsymbol{A}\boldsymbol{R}_s^{-1})$ to near unity with high probability, then applies Cholesky-QR in double precision to the preconditioned matrix. The key insight – supported by perturbation analysis – is that the preconditioner requires far less accuracy than the final result: single (FP32) precision suffices when $\kappa_2(\boldsymbol{A}) \lesssim 10^8$, and half (FP16) when $\kappa_2(\boldsymbol{A}) \lesssim 10^4$. MRCQR produces an explicit orthogonal factor $\widehat{\boldsymbol{Q}}$ satisfying $\|\boldsymbol{I} - \widehat{\boldsymbol{Q}}^\top\widehat{\boldsymbol{Q}}\|_2 = \cal O(\mathbf{u})$ ($\mathbf{u} \approx 10^{-16}$, double-precision unit roundoff) for condition numbers up to $10^{16}$, far beyond the $10^8$ limit of CholQR2. Experiments on an NVIDIA H100 GPU show that MRCQR (FP16) outperforms rand-cholQR by $1.4$–$1.8\times$ across all tested column counts and is $1.8$–$13.5\times$ faster than cuSOLVER geqrf, while the FP16 sketch (used when $\kappa_2(\boldsymbol{A}) \lesssim 10^4$) is $2\times$ cheaper than FP64 at no accuracy cost.