We introduce a new dynamical system: the wind-tree tiling billiards. This system studies trajectories of a ray in Euclidean space which has a negative refractive index when encountering rectangular obstacles located at lattice points. We show that for almost every configuration of the system, trajectories with initial vertical direction are trapped in an infinite strip of the plane. This result is reminiscent of the propagation of light rays in Eaton lenses, as shown by Frączek and Schmoll.

The teapot effect refers to the tendency of a poured liquid to cling to the lip of a container and run down the outside. It is a familiar but physically rich example of flow separation. We present a low-cost experiment for introductory physics laboratories that uses 3D-printed cups, a simple flow regulator, and basic surface treatments to explore this phenomenon in a classroom setting. Students measure the run-off length along the outer wall as an accessible indicator of sticking versus separation and use it to compare the effects of flow velocity and surface wettability. Rather than attempting a full quantitative test of research-level models, the activity is designed to illustrate the inertial-capillary picture of the teapot effect in a form that is experimentally straightforward and pedagogically effective. The experiment connects a familiar everyday observation to fluid inertia, wetting, and interfacial forces in a form that is well suited to introductory instruction.

Controlling polytype selection in polymer-derived silicon carbide (SiC) remains challenging since stacking sequences are determined locally at the nucleation front. Here, we demonstrate an interface-driven strategy to bias SiC polytype evolution by introducing compositionally complex TiVCrMoC3 MXene nanosheets at the preceramic stage. Under spark plasma sintering (1900 C, 70 MPa), which typically stabilizes cubic beta-SiC, the MXene partially transforms into multicomponent carbide structures and generates two distinct heterogeneous interfacial states: reconstructed carbide/SiC interfaces that locally disrupt stacking sequences and promote hexagonal ordering, driving the emergence of alpha-SiC; and coherent MXene/SiC interfaces that preserve cubic stacking. Mechanical testing further reveals peak performance at an optimal MXene loading where interfacial reconstruction is most pronounced, with an around 82% increase in Young's modulus and 42% improvement in fracture toughness. These findings highlight interfacial polytype engineering via two-dimensional carbide templates as a promising route for directing crystal structure evolution in polymer-derived ceramics.

In this work, we obtain exact solutions to the $(n+2)$-dimensional Einstein Field Equations with a non-zero cosmological constant for $n > 1$. These solutions depend on a set $\{ A_a, a=1,2,\ldots , m \}$ of pairwise commuting constant matrices in $\mathfrak{sl} ( n, \mathbb{R} )$ and on a constant matrix $g_0$ in $\mathcal{I} (\{ A_a, a=1,\ldots , m \})$, determined in previous work. Different choices of $\{ A_a, a=1,\ldots , m \}$ and $g_0$ correspond to different solutions. As examples, we show how to obtain the de Sitter metric, the Anti-de Sitter metric, the Birmingham metric, the Nariai metric and the Anti-Nariai metric in higher dimensions. The generalized Nariai and Anti-Nariai solutions are direct topological products of $AdS_{\frac{n}{2} + 1} \times H^{\frac{n}{2} + 1}$, $dS_{\frac{n}{2} + 1} \times S^{\frac{n}{2} + 1}$, $AdS_2 \times H^n$, $AdS_n \times H^2$, $dS_2 \times S^n$ and $dS_n \times S^2$. In addition, we study a solution in the context of cosmology.

Upcoming radio surveys will probe the sky with unprecedented depth and sky coverage, enabling a broad range of cosmological and astrophysical applications, as well as powerful synergies with experiments at other wavelengths. The preparation and scientific exploitation of these surveys require realistic mock catalogues that capture the complexity of the radio sky and the interplay of its emitting components. We present a modular and extensible algorithm for generating empirical simulations over the full radio sky, i.e. a solid angle of $4π$ steradians ($f_{\rm sky}=1$), down to redshift $z=5$, comprising both radio continuum and line emission. The framework combines a simulated dark-matter light-cone with empirically sampled galaxy populations and a probabilistic galaxy-halo assignment scheme, producing self-consistent mock catalogues including multiple radio populations on the same light-cone. We release two public catalogues: a shallow catalogue, fully constrained by existing observational data and limited to flux thresholds of $S_\text{1.4 GHz}^\text{lim} \sim 8\times10^{-5}\ \text{Jy}$ at $1.4\ \text{GHz}$ and $S_\text{21}^\text{lim} \sim 2\ \text{Jy}\cdot\text{Hz}$ for the HI 21 cm line; and a deep catalogue extending the calibrated empirical model to better sensitivities, broadly matching future SKAO surveys, with flux limits of $S_\text{1.4 GHz}^\text{lim} \sim 4\times10^{-5}\ \text{Jy}$ and $S_\text{21}^\text{lim} \sim 0.3\ \text{Jy}\cdot\text{Hz}$. The catalogues include radio continuum active galactic nuclei and star-forming galaxies, together with HI-emitting galaxies, for a total of more than 260 million sources in the shallow catalogue and more than 1 billion in the deep catalogue. We validate the simulations by analysing their statistical properties: the mocks reproduce the targeted clustering and population statistics while retaining minimal physical assumptions.

A central challenge in nanoscale superconductivity is to understand and exploit the combined action of quantum confinement and proximity effects in experimentally realistic metallic heterostructures. We theoretically investigate superconducting bilayer heterostructures in which these two effects coexist. Using a generalized Eliashberg framework that incorporates both quantum confinement and proximity coupling, we show that their interplay can substantially enhance the superconducting critical temperature. In particular, the theory predicts superconductivity in selected bilayers whose constituent materials are nonsuperconducting or only weakly superconducting in the bulk. These results identify quantum-confined bilayers as a promising route to engineering emergent superconductivity in metallic heterostructures.

We propose Majorana-assisted nonlocal spin correlation as a manifestation of Majorana nonlocality in quasi-one-dimensional (1D) Kitaev spin liquids. Focusing on the flux-free sector of the Kitaev honeycomb model in a quasi-1D geometry, we uncover its topological nature and show that it hosts Majorana zero modes localized at both ends, which are stabilized by finite-size-induced topology. We further show that the nonlocal Majorana fermion parity operator, $P_{\text{MF}}=iγ_{\text{L}}γ_{\text{R}}$, is mapped to a nonlocal spin-string operator, producing an end-to-end spin correlation proportional to the product of $P_{\text{MF}}$ and total fermion parity operators when local perturbations remove redundant ground-state degeneracies while preserving the Majorana and total fermion parities in the flux-free sector. Numerical calculations confirm a finite nonlocal spin correlation generated by these Majorana zero modes without any local magnetization. Our results establish a concrete signature of intrinsic Majorana nonlocality in quantum spin liquids.

Given a sequence of uniformly convex norms $ ϕ_h $ on $ \mathbf{R}^{n+1} $ converging to an arbitrary norm $ ϕ$, we prove rigidity of $ L^1 $-accumulation points of sequences of sets $ E_h \subseteq \mathbf{R}^{n+1} $ of finite perimeter, that are volume-constrained almost-critical points of the anisotropic surface energy functionals associated with $ ϕ_h $. Here, almost criticality is measured in terms of the $ L^n $-deviation from being constant of the distributional anisotropic mean $ ϕ_h $-curvature of (the varifold associated to) of the reduced boundaries of $ E_h $. We prove that such limits are finite union of disjoint, but possibly mutually tangent, $ ϕ$-Wulff shapes.

In this article, we present a unified algebraic-combinatorial framework for computing explicit, piecewise rational, and combinatorially indexed parametric formulas for volumes and higher moments of slices and slabs of polyhedral norm balls. Our main method builds on prior work concerning a combinatorial decomposition of the parameter space of all slices of a polytope. We extend this framework to slabs, and find a polynomial-time algorithm in fixed dimension. We also exhibit computational methods to obtain moments of arbitrary order for all slices or slabs of any polyhedral norm ball, and an algebraic framework for analyzing their critical points. In addition, we present an experimental study of the $d$-dimensional unit cube. Our analysis recovers and reinterprets the known volume formulas for slabs and slices of the two- and three-dimensional cubes, first obtained by König and Koldobsky. Moreover, our method identifies a new complete family of fourteen rational functions giving the volumes of slices and slabs of the four-dimensional cube. We further compute explicit higher moments of slices and slabs in dimensions two and three, and derive explicit formulas for moments of arbitrary order for slices of the two-dimensional cube, describing their critical points.

In the NP-hard \textsc{Group Closeness Centrality Maximization} problem, the input is a graph $G = (V,E)$ and a positive integer $k$, and the task is to find a set $S \subseteq V$ of size $k$ that maximizes the reciprocal of group farness $f(S) = \sum_{v \in V} \min_{s \in S} \text{dist}(v,s)$. A widely used greedy algorithm with previously unknown approximation guarantee may produce arbitrarily poor approximations. To efficiently obtain solutions with quality guarantees, known exact and approximation algorithms are revised. The state-of-the-art exact algorithm iteratively solves ILPs of increasing size until the ILP at hand can represent an optimal solution. In this work, we propose two new techniques to further improve the algorithm. The first technique reduces the size of the ILPs while the second technique aims to minimize the number of needed iterations. Our improvements yield a speedup by a factor of $3.6$ over the next best exact algorithm and can achieve speedups by up to a factor of $22.3$. Furthermore, we add reduction techniques to a $1/5$-approximation algorithm, and show that these adaptations do not compromise its approximation guarantee. The improved algorithm achieves mean speedups of $1.4$ and a maximum speedup of up to $2.9$ times.

This paper studies how to estimate an individual's taste for forming a connection with another individual in a network. It compares the difficulty of estimation with and without the assumption that utility is transferable between individuals, and with and without the assumption that regressors are symmetric across individuals in the pair. I show that when pair-specific regressors are symmetric, the sufficient conditions for consistency and asymptotic normality of the maximum likelihood estimator that assumes transferable utility (TU-MLE) are also sufficient for the maximum likelihood estimator that does not assume transferable utility (NTU-MLE). When regressors are asymmetric, I provide sufficient conditions for the consistency and asymptotic normality of the NTU-MLE. I also provide a specification test to assess the validity of the transferable utility assumption. Two applications from different fields of economics demonstrate the value of my results. I find evidence of researchers using the TU-MLE when the transferable utility assumption is violated, and evidence of researchers using NTU-model-based estimators when the validity of the transferable utility assumption cannot be rejected.

In this methods paper, we show how to tridia\-go\-nalize two families of bosonic multimode systems: optomechanical and Bose-Hubbard hamiltonians. Using tools from number theory, we devise a rendering of these systems in the form of exact $D \times D$ tridiagonal symmetric matrices with real-valued entries. Such matrices can subsequently be exactly diagonalized using specialized sparse-matrix algorithms that need on the order of $D \ln(D)$ steps. This makes it possible to describe systems with much larger numbers of basis states than available to date. It also allows for efficient diagonal representation of large, accurate, symplectic split-operator propagators for which we moreover show that the required basis changes can be implemented by simple re-indexing, at marginal computational cost.

This paper introduces Conchordal, a bio-acoustic instrument for generative composition whose sonic agents are governed by artificial life dynamics within a psychoacoustic fitness landscape. The system is built on Direct Cognitive Coupling (DCC), a design principle requiring that generative dynamics operate directly within a landscape derived from psychoacoustic observables and read from that landscape without symbolic harmonic rules. The environment integrates roughness and harmonicity into a continuous consonance field without presupposing discrete scales or explicit harmonic rules. Agents adjust pitch through local proposal-and-accept dynamics under a crowding penalty, regulate survival via consonance-dependent metabolism, and entrain temporally through Kuramoto-style phase coupling. Four experiments are reported: (1) consonance search produces structured polyphony with enriched consonant intervals; (2) consonance-dependent metabolism yields survival differentials that vanish when recharge is disabled; (3) a minimal hereditary adaptation assay shows that parent-guided respawn plus metabolic selection can accumulate more structured polyphony without adult hill-climbing; and (4) a shared oscillatory scaffold organizes rhythmic timing under external forcing. A supplementary mechanism check reports one possible composer-configurable bridge by which spectral state can modulate temporal coupling. These findings show that a psychoacoustically derived landscape serves as an effective artificial-life terrain, yielding self-organization, selection, synchronization, and lineage-level accumulation in a non-traditional computational medium. At the level of the model, the same landscape therefore functions both as ecological terrain and as an internal proxy for musical coherence.

Given two continuity equations with density-dependent velocities, we provide a new formula for the Wasserstein distance between the solutions in terms of the difference of velocities evaluated at the same density. The formula is particularly attractive to deduce quantitative estimates and rates of convergence for singular limits. We illustrate it using several examples. For the porous medium equation with exponent $m$, we prove that solutions are Lipschitz continuous with respect to $m$, providing a quantitative version of the result of Bénilan and Crandall. This result can be extended to a general aggregation-diffusion equation. We also study the limit $m \to \infty$ (the so-called mesa problem or the incompressible limit) and we recover the rate of convergence $1/{\sqrt{m}}$. Last but not least, we improve the rate of nonlocal-to-local convergence for the quadratic porous medium equation from recently obtained $\sqrt{\varepsilon}$ to numerically conjectured $\varepsilon$.

We show that the hyperkahler geometry of $T^*\mathbb{CP}^{n-1}$ can be described algebraically by the affine scheme of rank-1 projections, and that this description simultaneously yields explicit $SU(n)$-equivariant isometric embeddings \[ T^*\mathbb{CP}^{n-1} \hookrightarrow \mathbb{R}^{(n^2+1)^2}, \] as well as a generalization of the hyperkahler geometry of $T^*\mathbb{CP}^{n-1}$ to arbitrary commutative rings with involutions (and some noncommutative ones). In particular, we obtain para-hyperkahler and complex hyperkahler manifolds by taking the rings to be the split-complex numbers and bicomplex numbers, respectively. The functor of points of the scheme of rank-1 projections is the functor that maps a commutative ring $\mathcal{R}$ to the space of idempotents in $M_n(\mathcal{R})$ whose images are rank-1 projective modules. In particular, its space of $\mathbb{C}$-points is identified with $T^*\mathbb{CP}^{n-1}$.

This study explored healthcare professionals' perspectives on the management of Type 1 Diabetes Mellitus (T1DM) through a two-part questionnaire. The first part examined how clinicians prioritise and apply current clinical guidelines, including the relative importance assigned to different aspects of T1DM management. The second part investigated clinicians' perceptions of patients' ability to interpret data from the glucose monitoring devices and to make appropriate treatment decisions. An online questionnaire was completed by 19 healthcare professionals working in diabetes-related roles in the United Kingdom. The findings revealed that blood glucose management is prioritised within clinical guidance and that advice is frequently tailored to individual patient needs. Additionally, clinicians generally perceive that data presented in glucose monitoring devices is easy for patients to interpret and based on these data, they believe that patients occasionally make correct treatment decisions.

In this article, we study the repetitive Penrose process for the Kerr-Taub-NUT black hole (BH). First of all, we briefly review the spacetime of the Kerr-Taub-NUT BH, including horizon and ergosphere structures. The results indicate that the event horizon and ergosphere radii increase under the influence of the gravitomagnetic charge $l$. Subsequently, we find by using the irreducible mass of the BH that the extractable energy decreases with the rise of the gravitomagnetic charge. We then turn to the repetitive Penrose process by writing the conservation laws and setting the corresponding iterative stopping conditions. Furthermore, we numerically calculate the change in the BH's parameters, along with the corresponding quantities of the repetitive Penrose process, for each iteration.

Short tandem repeats (STRs) are low-entropy regions in the genome, consisting of a short (1-6 bp) unit that is consecutively repeated multiple times. They are known for high mutational instability, due to so-called stutter-mutations, in which the number of units in the run increases or descreases. In particular, STRs with repeat unit length of 1-2 bp are prone to mutate even within several cell divisions. The extremely rapid accumulation of variation makes them interesting phylogenetic markers for retrospective single-cell lineage reconstruction. Here we model their mutational dynamics at the level of individual repeat unit type and then aggregate length variations over many STR loci with the aim of obtaining a very fast ``molecular clock''. We calibrate our model based on several datasets with known lineage structure prepared from cultured cells. We find that the mutational dynamics of STRs are reasonably consistent for a given cell line, but vary among different ones. This suggests that the dynamics are not entirely explained by mutations in caretaker genes, rather, various other factors play a role -- possibly tissue origin and differentiation state. Further data and research is necessary to asses their relative effects.

In this work, we investigate the existence of multiple positive solutions for a weakly coupled system of nonlinear elliptic equations governed by Pucci extremal operators. Specifically, we consider the system: \[ \begin{cases} -{M}_{λ_1,Λ_1}^+(D^2u_1) = μf_1(u_1, u_2, \dots, u_n), & \text{in } Ω, \\ -{M}_{λ_2,Λ_2}^+(D^2u_2) = μf_2(u_1, u_2, \dots, u_n), & \text{in } Ω, \vdots \\ -{M}_{λ_n,Λ_n}^+(D^2u_n) = μf_n(u_1, u_2, \dots, u_n), & \text{in } Ω, \\ u_1 = u_2 = \dots = u_n = 0, & \text{on } \partialΩ, \end{cases} \] where $ {M}_{λ,Λ}^+ $ represents the Pucci extremal operator, $ Ω$ is a bounded domain in $ \mathbb{R}^N $ with smooth boundary, and the nonlinear functions $ f_i: [0, \infty)^n \to [0, \infty) $ belong to the $ C^{1,α} $ class. Our main results establish the existence and multiplicity of solutions for sufficiently large values of the parameter $ μ> 0 $. The analysis relies on the method of sub and supersolutions, in conjunction with fixed-point arguments and bifurcation techniques.

The rapid advancement of high-performance computing infrastructure and its extended application produce an increasing amount of waste heat. This heat constitutes an unsustainable loss of energy as well as requires cooling solutions that transcend conventional thermal management. Here, we demonstrate a novel mechanism that converts vertical waste heat supply directly into horizontal fluid motion, enabling autonomous, self-powered pumping in microenvironments. Our approach is based on a concept that combines geometric symmetry breaking with heterogeneous thermal conductivities to induce local thermocapillary Marangoni flows. We provide an implementation of the concept as well as an experimental and numerical proof-of-concept, showing good agreement between the respective flow fields. The approach is scalable and operates under realistic areal heating conditions. It enables versatile pumping designs for microtechnological applications, lab-on-a-chip architectures, passive thermal management and heat-driven microfluidic systems.

Counterdiabatic (CD) driving enables efficient quantum state preparation, but it requires implementing highly nonlocal adiabatic gauge potentials (AGP) that are impractical to compute and realize in large many-body systems. We introduce a \textit{weighted nested-commutator} (WNC) ansatz to approximate AGP using local operators. The WNC ansatz generalizes the standard nested-commutator ansatz by assigning independent variational weights to commutators of local Hamiltonian terms, thereby enlarging the variational space while preserving a fixed operator range. We show that the WNC ansatz can be efficiently optimized using a local optimization scheme. Moreover, it systematically outperforms the nested-commutator ansatz in preparing one-dimensional matrix product states (MPS) and the ground state of a nonintegrable quantum Ising model. We then numerically demonstrate that CD driving based on the WNC ansatz significantly accelerates the preparation of 1D MPS for system sizes up to $N = 1000$ qubits, as well as the two-dimensional Affleck-Kennedy-Lieb-Tasaki state on a hexagonal lattice with up to $N = 3 \times 10$ sites.

The evolution toward 6G communication systems is expected to rely on integrated three-dimensional network architectures where terrestrial infrastructures coexist with non-terrestrial stations such as satellites, enabling ubiquitous connectivity and service continuity. In this context, accurate channel models for satellite-to-ground propagation in urban environments are essential, particularly for user equipment located at street level where obstruction and multipath effects are significant. This work investigates satellite-to-urban propagation through deterministic ray-tracing simulations. Three representative urban layouts are considered, namely dense urban, urban, and suburban. Multiple use cases are investigated, including handheld devices, vehicular terminals, and fixed rooftop receivers operating across several frequency bands. The analysis focuses on the relative importance of competing propagation mechanisms and on two key channel parameters, namely the Rician K-factor and the delay spread, which are relevant for the calibration of channel models to be used in link- and system-level simulations. Results highlight the strong - and in some cases unconventional - dependence of channel dispersion and fading characteristics on satellite elevation, antenna placement, and urban morphology.

As the (JWST) pursues its observing journey, several thousands of icy-grain spectra are expected to be measured and analysed. The inventory of ices in particular, via the observations of background sources, is accessible for hundreds of lines of sight (LOSs) per molecular-cloud region, opening the possibility to add strong constraints on the solid phase chemistry in a vast domain of cloud densities. SynthIceSpec is a synthetic infrared (IR) spectrum generator that has been designed as a tool to support observing proposals and to test the outcome of chemical models. It is based on laboratory measurements of pure and mixed ices, where each vibrational component is fitted by a sum of Gaussian profiles. Given an initial ice chemical composition (either set by the user or the outputs of a chemical model), a full JWST spectrum is generated, to which the contribution of silicates; continuum, stellar photospheric absorption bands; and extinction law can be added. For the continuum, stellar photospheric models for a wide range of spectral types can be selected by the program, or, Spectral Energy Distribution (SEDs). We present a few use cases of SynthIceSpec: we probed the impact of dust temperature on CO_2 ice formation using IR data and gas-grain modelling. Next, we used SynthIceSpec to explore the detectability of the main feature of CH_3CN at 4.45 um in a cold core environment with the JWST, which was previously tentatively detected in YSOs. The detection thresholds we derive are reasonably low and observable, but identification is directly impacted by the photosphere absorptions that can greatly hinder identification. For some photostellar types, it could remain feasible. Coupled with the Estimated Time Calculator of the Space Telescope Science Institute, SynthIceSpec can be used to find the optimum observational setup for new observations.

We report on the epitaxial growth of antiferromagnetic Mn2Au on a Nb(001) substrate capped with a pseudomorphic layer of gold. We observe a layer-by-layer growth by means of medium-energy electron diffraction and confirm stoichiometry and surface structure by Auger electron spectroscopy and low-energy electron diffraction. Evaporation of 15 ML of ferromagnetic Fe on 12--17 ML of Mn2Au results in an exchange-coupled bilayer system with an exchange-bias shift that can be set by field-cooling from 400 K. Areas with and without exchange bias, with domain sizes in the range of tens of μm, are identified by Kerr microscopy. Postannealing the sample at or above 450 K after Mn2Au layer growth decreases the amount of areas where Fe magnetically couples to Mn2Au. We conclude that exchange coupling to an interfacial Fe layer depends on the interface termination of Mn2Au. Our findings provide insight into the growth process of Mn2Au and the coupling to an Fe layer. Our results point out the importance of growth, interface quality and termination on the magnetic properties of a Mn2Au/Fe bilayer which may help to improve material properties for spintronic applications.

CrCoNi medium-entropy alloys exhibit exceptional mechanical properties arising from pronounced chemical complexity, including short-range order (SRO), and low stacking fault energy, posing challenges for large-scale atomistic simulations. While most models focus on equimolar compositions, deviations from equimolarity provide an effective route to tuning properties, requiring transferable interatomic potentials that capture composition-dependent behavior. Here we develop a general-purpose machine-learned interatomic potential for the CrCoNi system within the neuroevolution potential (NEP) framework, achieving near first-principles accuracy with high computational efficiency. Trained on a comprehensive dataset spanning pure elements, binary and ternary alloys across a wide compositional range, diverse crystal structures and thermodynamic conditions, and based on spin-polarized \textit{ab initio} data, the model accurately reproduces equations of state, phonons, elastic constants, dislocation dissociation, surface and defect energies, melting temperatures and strain-induced phase transformations. It further captures SRO and its effect on stacking fault energies across both equimolar and non-equimolar compositions, in agreement with first-principles and experiments. In contrast to existing potentials, typically limited to equimolar alloys and less accurate for pure elements, the present model delivers consistent accuracy across the full compositional space while retaining superior efficiency. These results enable reliable atomistic simulations of composition-dependent behaviour and provide a framework for the design of non-equimolar CrCoNi alloys.

We consider multifractal Mandelbrot cascades supported on planar $C^2$ curves with nonvanishing curvature and show that their Fourier dimension is as large as possible, i.e., equal to the infimum of the lower pointwise dimension of the measure.

Assuming the generalized Lindelöf hypothesis for Dirichlet $L$-functions, we establish that the least prime $p\equiv a\pmod{q}$ satisfies $p\ll_{\varepsilon} q^{2+\varepsilon}$. This achieves a bound that nearly matches the classical estimate implied by the generalized Riemann hypothesis.

This paper develops an information-theoretic framework for algorithmic complexity under regular identifiable fibering. The central question is: when a decoder is given information about the fiber label in a fibered geometric set, how much can the residual description length be reduced, and when does this reduction fail to bring dimension below the ambient rate? We formulate a directional compression principle, proposing that sets admitting regular, identifiable fiber decompositions should remain informationally incompressible at ambient dimension, unless the fiber structure is degenerate or adaptively chosen. The principle is phrased in the language of algorithmic dimension and the point-to-set principle of Lutz and Lutz, which translates pointwise Kolmogorov complexity into Hausdorff dimension. We prove an exact analytical result: under effectively bi-Lipschitz, identifiable, and computable fibering, the complexity of a point splits additively as the sum of fiber-label complexity and along-fiber residual complexity, up to logarithmic overhead, via the chain rule for Kolmogorov complexity. The Kakeya conjecture (asserting that sets containing a unit segment in every direction have Hausdorff dimension n) motivates the framework. The conjecture was recently resolved in R^3 by Wang and Zahl; it remains open in dimension n >= 4, precisely because adaptive fiber selection undermines the naive conditional split in the general case. We isolate this adaptive-fibering obstruction as the key difficulty and propose a formal research program connecting geometric measure theory, algorithmic complexity, and information-theoretic compression.

Encoding continuous-variable quantum information in the optical domain has recently enabled the generation of large entangled states, yet robust implementation remains a challenge. Here, we present a straightforward protocol for generating multipartite entanglement based on spontaneous parametric down-conversion in a circular array of quadratic nonlinear waveguides. We provide a rigorous theoretical framework, including comprehensive derivations of the propagation equations and the identification of regimes where analytical solutions are possible. Crucially, our approach identifies the pump and detection configurations required to sustain and measure multipartite full inseparability across arbitrary propagation distances and for any number of waveguides $N=4 n$. This regime, elusive to standard numerical methods, represents a key requirement for scalable quantum protocols. Our scheme is inherently robust as it relies on phase-matched propagation eigenmodes, making it resilient against variations in sample length, coupling, and nonlinearity.

We extend results of Boros and Menzer on the alternative equation $f(x)f(y)=0$ for generalized polynomials $f$, and their theorems on the conditional inequality $f(x)f(y)\ge 0$ for generalized monomials $f$ of even degree. We use similar methods and ideas. We replace the largeness, of the respective Borel plane set $D$, in the measure or in the Baire category sense, by its largeness in the mixed measure-category sense.