Local regression is widely used to explore spatial heterogeneity, but anisotropic or effectively low-dimensional neighborhoods can produce ill-conditioned local solves, causing coefficient variation driven by numerical artifacts rather than substantive structure. Such instability is often hidden when estimation relies on implicit tuning or optimization without exposing local diagnostics. This paper proposes Gimbal Regression (GR), a deterministic, geometry-aware local regression framework for stable and auditable estimation. GR constructs directional weights from neighborhood geometry using explicit orientation objects and deterministic safeguards, and computes local coefficients by a closed-form solve. Theoretical results are stated conditional on the realized neighborhood configuration, under which the estimator is a deterministic linear operator with finite-perturbation stability bounds. Simulations and empirical examples demonstrate predictable computation, transparent diagnostics, and improved numerical stability relative to common local regression baselines.

The free string spectrum is highly degenerate, with a degeneracy that grows exponentially with the mass. Turning on a non-vanishing string coupling $g_s$ introduces interactions, rendering massive string states unstable and allowing them to decay into lower-mass states, with mixing constrained by Lorentz invariance. This behavior is expected already at one-loop level. The imaginary part of the one-loop mass correction is related to the width of the decay into two lower-mass states at tree level, whereas the real part is generally IR-divergent and needs regularization and renormalization. The analysis simplifies for states in the first Regge trajectory. In particular, we consider the one-loop mass corrections for these states in the NS-NS sector of Type-II string theories. We explicitly construct the related vertex operators and exploit the properties of elliptic functions in order to obtain a closed form expression for the integral over the insertion point. We further regularize the divergences of the integral over the modular parameter of the torus by means of the $i\varepsilon$-prescription in string theory. Finally we extract numerical results for the mass correction up to level $N=4$.

Large Language Models (LLMs) have revolutionized code generation, evolving from static tools into dynamic conversational interfaces that facilitate complex, multi-turn collaborative programming. While LLMs exhibit remarkable proficiency in generating standalone code snippets, they often struggle to maintain contextual consistency during extended interactions, creating significant obstacles in the collaboration process. Existing benchmarks primarily emphasize the functional correctness of the final output, overlooking latent quality issues within the interaction process itself, which we term Interaction Smells. In this paper, we conduct an empirical study on sampled real-word user-LLM interactions from WildChat and LMSYS-Chat-1M datasets to systematically investigate Interaction Smells in human-LLM code generation tasks from the perspectives of phenomena, distribution, and mitigation. First, we establish the first taxonomy of Interaction Smells by manually performing open card sorting on real-world interaction logs. This taxonomy categorizes Interaction Smells into three primary categories, i.e., User Intent Quality, Historical Instruction Compliance, and Historical Response Violation, comprising nine specific subcategories. Next, we quantitatively evaluate six mainstream LLMs (i.e., GPT-4o, DeepSeek-Chat, Gemini 2.5, Qwen2.5-32B, Qwen2.5-72B, and Qwen3-235B-a22b) to analyze the distribution of Interaction Smells across different models. Finally, we propose Invariant-aware Constraint Evolution (InCE), a multi-agent framework designed to improve multi-turn interaction quality through explicit extraction of global invariants and pre-generation quality audits. Experimental results on the extended WildBench benchmark demonstrate that this lightweight mitigation approach significantly improves the Task Success Rate and effectively suppresses the occurrence of Interaction Smells.

We study band modulations and topological transitions in a one-dimensional periodic bead-on-string chain. Using an exact transfer-matrix formulation of the wave equation with periodically modulated mass density, combined with numerical spectral searches and tabletop experiments, we characterize band gaps and localized midgap states. We interpret these states by mapping the system to the Su-Schrieffer-Heeger (SSH) model and its low-energy (1+1)-dimensional Dirac theory. This framework reveals that the robust states are topological solitons bound to boundaries or engineered domain walls in the Dirac mass. Through this mapping, we provide an intuitive account of how band structure controls topological phase changes in mechanically realizable lattices.

The Casten-Holland and Matano theorem for interior reactions states that no nonconstant stable solutions exist in convex domains $Ω$ of $\mathbb{R}^n$ under zero Neumann boundary conditions. In this paper we establish that the analogous statement fails for boundary reactions when $n=2$ (that is, for harmonic functions in $Ω$ with a Neumann reaction term on its boundary $\partialΩ$). For instance, nonconstant stable solutions exist when $Ω$ is a square, or a smooth strictly convex approximation of it. In regular polygons of many sides, which approach the circle, we can prove the existence of as many nonconstant stable solutions as wished. Instead, in the circle such stable solutions do not exist. More importantly, we can predict the existence or not of nonconstant stable solutions, as well as the location of its boundary "vortices" $(p,q)$, through the properties of a real function defined on $\partialΩ\times\partialΩ$ (the renormalized energy) which depends only on the conformal structure of the domain $Ω$. This requires the development of a new Ginzburg-Landau theory for real-valued functions and the analysis of the half-Laplacian on the real line.

Precise sensing and control of spatial mode content is essential for the performance of precision optical systems, particularly interferometric gravitational-wave detectors, where misalignment and mode mismatch can lead to significant optical losses and degraded quantum noise suppression. Conventional approaches, including heterodyne wavefront sensing and phase camera techniques, are effective but can be limited by hardware complexity and systematic uncertainties arising from restricted reference-beam overlap. This paper presents a novel two-step deep learning pipeline for robust beam diagnostics based solely on beam intensity images. In the first stage, a multi-intensity-image convolutional neural network (CNN) performs accurate mode decomposition, recovering the complex modal content of distorted beams. In the second stage, the predicted mode coefficients are fed into a downstream regression network that simultaneously estimates all eight degrees of freedom (DoFs) associated with misalignment and mode mismatch, including beam tilt, lateral offset, and waist size and position mismatches in both transverse directions. The proposed CNN-based framework achieves a mean absolute error (MAE) of 0.0034 in the mode decomposition stage, which propagates to a total MAE of 0.0062 in the recovered beam imperfection parameters at the final stage. This corresponds to an average residual optical loss of 39 ppm per DoF (310 ppm total). This approach relies only on standard CCD imaging and is robust to random intensity noise, eliminating the need for complex interferometric hardware. The results demonstrate that the proposed deep learning pipeline enables real-time, high-accuracy wavefront sensing and mode-mismatch diagnostics, providing a scalable and hardware-efficient tool for improving the stability and sensitivity of precision optical systems.

We study dispersive properties of the one-dimensional Schr{ö}dinger equation with a short-range array of delta interactions. More precisely, we consider the self-adjoint operator obtained by perturbing the free Laplacian on the line with a real-valued sequence of Dirac delta potentials and belonging to weighted ${\ell}$^1(Z) spaces. Under suitable decay assumptions on the coupling constants and in the absence of a zero-energy resonance, we establish the L^1 (R) $\rightarrow$ L^$\infty$ (R) dispersive estimate with decay rate |t|^{-1/2} for the associated Schr{ö}dinger group. The proof relies on a limiting absorption principle in weighted spaces, explicit representation of the resolvent kernel in terms of Jost solutions and Born series expansion of the Friedrichs extension of the perturbed operator.

Let $Z\subset\mathbb{C}^N$ be an $n$-pseudoconcave subset, for $1\leq n<N$, which is locally the graph of a continuous function over a closed subset of $\mathbb{C}^n\times\mathbb{R}$. We show that $Z$ can be realised as the disjoint union of $n$-dimensional complex manifolds. In particular, the same conclusion can be made for any $n$-pseudoconcave subset $Z$ of the graph $Γ(g)$ of a continuous function $g:D\subset\mathbb{C}^n\times\mathbb{R}\to\mathbb{R}\times\mathbb{C}^p$, for $n\geq 1$ and $p\geq 0$.

We prove a mathematical theorem that solution for all $t > 0$ of the two-dimensional (2D) Kolmogorov flow governed by Navier-Stokes (NS) equations with periodic boundary condition keeps the same spatial symmetry as its smooth initial condition. The proof of a similar theorem for the three-dimensional NS equations is given in the appendix. These mathematical theorems can be used to check the correctness and reliability of numerical simulations of NS turbulence. For example, they support the corresponding CNS (clean numerical simulation) results of the 2D and 3D turbulent Kolmogorov flows [1-3] that remain the same spatial symmetry in the whole time interval of simulation, but do not support the corresponding DNS (direct numerical simulation) results that lose the spatial symmetry quickly. In other words, these DNS results violate these mathematical theorems. Thus, these mathematical theorems rigorously confirm that the spatiotemporal trajectories of NS turbulence given by DNS are indeed quickly polluted by numerical noises badly. All of these indicate that CNS can indeed provide helpful enlightenments to deepen our understanding about turbulence and besides approach some mathematical truths about NS equations.

In the excursion-set formalism, the mass distribution of primordial black holes (PBHs) is derived from the first-passage time of a random walk describing the density contrast as the coarse-graining scale varies. We address two recent concerns that have been raised about this approach. First, it was argued that the random walks are subject to colored (i.e. correlated over time) noise, making the first-passage-time problem cumbersome. We show that this arises from an incorrect separation of drift and noise when sampling on the Hubble-crossing surface: if Fourier modes are uncorrelated, the noise is strictly white. Moreover, sampling along the Hubble-crossing surface precludes using the density dispersion as a time variable, explaining some pathologies. Sampling instead on a synchronous surface removes both issues. This requires solving a first-passage-time problem with a moving barrier, for which we provide an efficient numerical framework. Second, it was suggested that cloud-in-cloud (i.e. that large black holes may engulf smaller ones) is irrelevant for PBHs and that the excursion set is therefore not needed. While valid for widely separated scales, this statement fails for broad power spectra with enhanced continua of modes. We further show that Press-Schechter estimates neglecting boundary evolution can break down even without cloud-in-cloud effects. Our results establish the robustness and necessity of the excursion-set formalism in realistic PBH formation scenarios.

We report a complete computation of next-to-next-to-leading order (NNLO) helicity and transversity Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) splitting functions, in both space-like and time-like kinematics. These results are obtained from the next-to-next-to-next-to-leading order (N$^3$LO) twist-2 matching of polarized transverse-momentum-dependent (TMD) parton distribution and fragmentation functions, including helicity, quark transversity, and linearly polarized gluons. We compare our results with existing calculations in the literature and discuss both agreements and discrepancies. Our results provide all perturbative ingredients required for the computation of N$^3$LO differential cross sections below the resolution scale $q_{T\mathrm{cut}}$ in transverse-momentum subtraction and enable next-to-next-to-next-to-next-to-leading logarithmic (N$^4$LL) resummation of $q_T$ observables in the Sudakov region. We further determine the small-$x$ structure of the polarized matching coefficients through N$^3$LO. These fixed-order results furnish the data for future small-$x$ resummation in polarized TMD factorization, where high-energy logarithms and Sudakov logarithms become simultaneously relevant. Establishing a consistent joint treatment of polarized small-$x$ evolution and transverse-momentum resummation remains an important open direction toward uniform precision in spin-dependent phenomenology. Our results provide essential theoretical input for precision spin physics at the forthcoming Electron-Ion Collider.

Growth and decay are system-level properties of chemical reaction networks (CRNs) relevant from prebiotic chemistry to cellular metabolism. Their properties are typically analyzed through the kinetics of particular models, which requires specification of the full set of kinetic laws and parameters. In this work, we derive stoichiometry-based constraints on the growth (or shrinkage) rate, in the balanced-growth regime of scalable CRNs. The resulting bounds are controlled by a topological quantity, the maximum amplification factor, defined via a von Neumann max-min problem over feasible fluxes as illustrated by numerical tests on random-network ensembles of CRNs. We argue for the relevance of our results in the context of origin of life studies but also for designing synthetic chemical reaction networks.

We study the topological phase transition in a two-leg Su-Schrieffer-Heeger (SSH) ladder by redefining the unit-cell structure. For both identical hopping dimerization pattern (uniform) and alternate hopping dimerization pattern (staggered) along the legs of the ladder, we demonstrate that different unit-cell choices generate different topological phases and phase transitions. In the uniformly dimerized ladder, variation of the inter-leg coupling induces a transition from topological phase to another topological phase through a gapless region. In contrast, the staggered dimerization configuration exhibits a richer phase structure, supporting both topological-topological and trivial-topological transitions occurring through a single gap-closing point, depending on the unit-cell definition. The phases are characterized through bulk-boundary correspondence, edge-state analysis, and bulk topological invariants. Interestingly, we obtain that while all the topological phases host two zero energy edge states each, the topological phase for the staggerred dimerization case at small inter-leg coupling hosts four edge states. We then perform topoelectric circuit simulation and experiments to observe the signatures of the topological phases. By using circuit impedance and voltage responses we establish the emergence of distinct edge modes in the circuit. Our analysis provides a route to engineer topological edge modes in a two-leg ladder SSH set-up.

We show that for tame valued fields of equal characteristic with divisible value group, the $C_i$ property lifts from the residue field to the valued field under suitable hypotheses on the residue field. We apply this transfer principle to prove Kato-Kuzumaki's conjecture in full generality for several arithmetically significant fields, for instance the field $\mathbf{C}(x_1,\dots,x_m)(\!(t_1)\!)\dots(\!(t_n)\!)$, and the perfections of both $\overline{\mathbf{F}}_p(x_1,\dots,x_m)(\!(t_1)\!)\dots(\!(t_n)\!)$ and $\mathbf{F}_p(\!(t_1)\!)\dots(\!(t_n)\!)$. Finally, we prove that $\mathbf{Q}_p$ satisfies the strong $C_1^1$ property, thereby answering a question of Wittenberg.

In traditional banking, repeated deposit-and-lend cycles let a single dollar of reserves support multiple dollars of claims. Decentralized finance produces an analogous structure with tokens. Constructing a Token Graph of 10,200 tokens across 200 blockchains, this paper maps the resulting hierarchy and shows that, by late 2025, each dollar of base assets supports $4.7 of total claims. An embedded yield correction disentangles two channels that raw data conflates: a compositional channel, where lending protocols concentrate in deeper tiers and mechanically raise average yields; and a liquidity channel, where each derivation step reduces secondary-market depth and depresses yields in liquidity-sensitive pools. The liquidity channel concentrates in DEX pools and vanishes in lending pools. A yield decomposition shows that the tier gradient operates entirely through fundamental protocol yields, not incentive-token emissions; quantile regressions reveal that the structural associations concentrate in the upper tail of the yield distribution, with near-zero effects at the median. These findings reframe DeFi's "double counting" as a structural risk question and identify liquidity fragmentation as the primary mechanism associated with yield variation across the token hierarchy.

We study the Higgs branch and associated vertex operator algebra (VOA) of 4d $\mathcal{N}=2$ superconformal field theories (SCFTs) from the geometric engineering of IIB superstring on canonical threefold singularities. For terminal singularities, we explain how to derive the 4d Higgs branch from their small resolution. We also investigate singularities with compact 4-cycles in their crepant resolution, and discuss different ways to compute their Higgs branch. Using a symplectic duality argument, we propose the first examples of 4d $\mathcal{N}=2$ SCFTs with the E-type Kleinian singularities as their Higgs branches, and conjecture their associated VOA to be affine E-type W-algebra. Many new VOAs with no known W-algebra descriptions are found, with conjectured associated varieties. We investigate the singularities associated with lisse VOAs and propose predictions for the BPS quivers of $D_N^N[k]$ and $E_7^{14}[k]$ from the perspective of deformed singularities. We further analyze the structure of the Schur index using the Coulomb branch IR formula, derive the expressions for the Schur index corresponding to these two classes of singularities, and illustrate, in a general setting, how the Schur index is determined by the BPS quiver.

Quantifying the natal kick distribution of pulsars is essential for understanding supernova physics and binary evolution, yet measurements are historically limited by the lack of radial velocity data. Most previous studies rely on transverse velocities under the assumption of spatial isotropy. In this work, we reconstruct the intrinsic three-dimensional (3D) velocity distribution for a curated sample of 18 pulsars by explicitly incorporating the observational constraint of spin-velocity alignment. Using a hierarchical Bayesian framework that accounts for measurement uncertainties, we compare nine candidate velocity distribution models. We find that a Gamma distribution provides an adequate description of the inferred 3D velocities; however, the modest Bayes factor (1.65 relative to a single Maxwellian) indicates that the current data lack sufficient resolving power to discriminate decisively among the models considered. The Gamma model is characterized by a peak velocity of $237^{+67}_{-84}$ km s$^{-1}$. The reconstructed 3D velocities under alignment are systematically lower than those inferred under isotropy, indicating that projection effects can bias individual kick estimates high, while leaving the overall population scale largely unchanged within uncertainties. A complementary analysis of 465 pulsars with transverse velocity estimates favors a Log-Normal distribution for the full sample, while isolated young pulsars remain consistent with a Gamma-like profile. Our results underscore the importance of geometric assumptions in population inference and highlight the need for larger samples with improved distance and spin-axis measurements to place tighter constraints on natal kick physics.

We study $n$-dimensional contests between two players with heterogeneous effort costs, where each dimension (battle) is modeled as a Tullock contest. Prize-allocation rules are identity-independent, budget-balanced, and weakly increasing in the number of victories. Players' costs can be separable across battles or exhibit cross-battle externalities. We identify a tight sufficient condition under which a unique equilibrium exists and is in pure strategies, for all admissible prize-allocation rules and all degrees of player asymmetry. Under this condition, we characterize the effort-maximizing prize-allocation rule: the entire prize goes to the player who wins more battles than the opponent by at least a prespecified margin, and is split equally if neither player meets this threshold. In the symmetric-player case, the majority rule is optimal if $n$ is odd. Interestingly, cross-battle cost externalities do not change the optimal prize allocation rule in our setting.

We study Host-Kra factors and the spectral type of nilsystems, without assuming ergodicity. In the ergodic case, it is known that the spectral type splits into a discrete component and a Lebesgue component of infinite multiplicity. Our main result extends this decomposition to nonergodic nilsystems.

An $SO(3,3)$ BF-type gauge theory is formulated on a six-dimensional spacetime of split signature $(3,3)$, interpreted as the pre-electroweak-symmetry-breaking phase. A MacDowell--Mansouri-type symmetry breaking to $SU(2)\times SU(2)$ is implemented, and the corresponding stabilizer and coset structures are computed. The curvature decomposes into chiral sectors, and effective tetrads are introduced using components of the higher-dimensional connection. The resulting left and right sectors are formulated as constrained BF/Plebanski-like theories with appropriate simplicity and reality conditions. The six-dimensional theory yields two overlapping four-dimensional Lorentzian sectors of opposite signature, related via gluing constraints across their intersection. In the first sector, the selfdual two-forms ($Σ^{(+)}$) satisfy simplicity constraints that select the non-degenerate branch and reproduce Einstein gravity. Subsequently, the $SU(2)_R\times U(1)_{Y{\rm dem}}\to U(1)_{\rm dem}$ breaking pattern is outlined which admits an ultra-soft regime consistent with current phenomenological bounds under sufficiently suppressed couplings. In the second sector, the antiself dual two-forms ($Σ^{(-)}$) satisfy analogous simplicity constraints, realizing weak gauge dynamics as gravity on the opposite-signature sector. Subsequently, the $SU(2)_L\otimes U(1)_Y$ electroweak symmetry is realized within the Yang--Mills branch of the BF theory which incorporates the standard Higgs mechanism $SU(2)_L\otimes U(1)_Y \to U(1)_{\mathrm{EM}}$, recovering the conventional electroweak $W^\pm$, $Z$, and photon spectrum.

It was recently demonstrated that imposing the condition $P_{r} = -ρ$ on the radial pressure of a galactic halo can lead to regular black-hole solutions for certain density profiles, such as the Dehnen and Einasto models. In the present work, we show that some of the most commonly used halo profiles, including the Hernquist model, do not yield regular geometries under the same condition, but instead support black-hole solutions that retain a central singularity

We propose a multi-scale lung model to investigate spatio-temporal distributions of ventilation variables. Lung envelope and large airway geometries are derived from CT scans; smaller airways are generated using a physiologically consistent algorithm. Tissue mechanics is modeled using nonlinear elasticity under small deformations, coupled with local air pressure from fluid dynamics within the bronchial tree. Airflow accounts for inertia and static airway compliance. Simulations employ finite elements. Using this model, we explore spatio-temporal airflows and shear stresses distributions.

Quark portal interactions $qqqN$ with a light singlet fermion $N$ make baryon number testable through missing transverse energy (MET). We find that present LHC data constrain scales up to 10 TeV (MET plus jet), 8 TeV (MET plus top) and 11 TeV (MET plus bjet). With increasing mass, or larger portal couplings, $N$ becomes less long-lived, and gives clean displaced vertex signatures, which encourage dedicated searches. We also narrow down viable mesogenesis models with color triplet scalars to a mass range $\sim y \,\cdot 3 \, \text{TeV}$ (with charm) and $\sim y \, \cdot 5 \, \text{TeV}$ (charmless) couplings $y$ to $b$ and lighter quarks, a window that can be scrutinized by HL-LHC. Interactions also induce rare decays of type baryon (meson) to meson (baryon) plus invisible, which complement high-$p_T$ searches and can prove baryon number violation. We explore charm decays $Λ_c \to (π,K) + \mathrm{invisible}$. Their branching ratios are subject to sizable hadronic uncertainties and require high luminosity flavor facilities such as a Tera-Z facility (FCC-ee, CEPC). Branching ratios of top quarks into one or two $b$-jets plus $N$ can reach few$\times 10^{-6}$.

We present a rotationally invariant viscous vertex model that accounts for both cortical and bulk dissipation of cells. The vanishing substrate-friction limit is enforced via Lagrange multipliers, which also provides a framework for implementing various boundary conditions, such as fixed boundaries and prescribed tractions. Building on this formulation, we introduce a slab-shear rheology protocol to extract an effective, coarse-grained tissue shear viscosity. Under polar or nematic activity, viscosity regulates the formation of elongated, spatially correlated cell-shape textures and stabilizes well-defined topological defects. Because the model remains well-posed at zero substrate friction, it is naturally suited to describing free-floating epithelia and organoids.

\textbf{Background:} The $^{3}$He($α,γ$)$^{7}$Be reaction plays a crucial role in the proton-proton chain and Big Bang nucleosynthesis, affecting solar neutrino fluxes and primordial element abundances. Experimental data at astrophysical energies remain uncertain due to the extremely low cross sections. \\ \textbf{Purpose:} This work uses a microscopic potential-model approach to construct the $^{3}$He+$α$ potential from the nucleon+$α$ interaction, aiming to describe low-energy elastic scattering and to calculate the astrophysical $S$ factor of the $^{3}$He($α,γ$)$^{7}$Be reaction. \\ \textbf{Method:} The nucleon-nucleus potential is derived from self-consistent Skyrme Hartree-Fock (HF) calculations extended to the continuum. The $^{3}$He+$α$ potential is then obtained by folding the HF potential with the $^{3}$He density. A small number of scaling parameters is constrained by elastic-scattering data.\\ \textbf{Result:} The scaled Skyrme HF potential and folded potential simultaneously reproduce the low-energy $p$+$α$ and $^{3}$He+$α$ $s$-wave phase shifts, respectively. The calculated astrophysical $S$ factor of $^{3}$He($α,γ$)$^{7}$Be shows good agreement with experimental data, yielding the recommended value $S_{34}(0) = 0.610 \pm 0.024$~keV~b. A moderate sensitivity of $S_{34}(0)$ to the choice of projectile density is also observed in the folding procedure. \\ \textbf{Conclusion:} The Skyrme HF-based potential provides a unified and predictive microscopic framework for describing both elastic scattering and radiative capture in light nuclei.

Price controls kill the incentive for arbitrage. We prove a Chaos Theorem: under a binding price ceiling, suppliers are indifferent across destinations, so arbitrarily small cost differences can determine the entire allocation. The economy tips to corner outcomes in which some markets are fully served while others are starved; small parameter changes flip the identity of the corners, generating discontinuous welfare jumps. These corner allocations create a distinct source of cross-market misallocation, separate from the aggregate quantity loss (the Harberger triangle) and from within-market misallocation emphasized in prior work. They also create an identification problem: welfare depends on demand far from the observed equilibrium. We derive sharp bounds on misallocation that require no parametric assumptions. In an efficient allocation, shadow prices are equalized across markets; combined with the adding-up constraint, this collapses the infinite-dimensional welfare problem to a one-dimensional search over a common shadow price, with extremal losses achieved by piecewise-linear demand schedules. Calibrating the bounds to station-level AAA survey data from the 1973--74 U.S. gasoline crisis, misallocation losses range from roughly 1 to 9 times the Harberger triangle.

The classical isocapacitary inequality states that, among all sets of fixed volume, the ball uniquely minimizes the capacity. While this result holds in the continuum, it fails in the discrete setting, where the isocapacitary problem may admit multiple minimizers. In this paper we establish quantitative fluctuation estimates for the discrete isocapacitary problem on subsets of $\mathbb{Z}^d$ as their cardinality diverges. Our approach relies on a careful extension of the associated variational problem from the discrete to the continuum setting, combined with sharp (continuum) quantitative isocapacitary inequalities.

Mobile gaming applications (apps) have become increasingly pervasive, including a growing number of games designed for children. Despite their popularity, these apps often integrate complex analytics, advertising, and attribution infrastructures that may introduce privacy and security risks. Existing research has primarily focused on tracking behaviors or monetization models, leaving configuration-level privacy exposure and children-oriented apps underexplored. In this study, we conducted a comparative static analysis of Android mobile games to investigate privacy and security risks beyond permission usage. The analysis follows a three-phase methodology comprising (i) designing study protocol, (ii) Android Package Kit (APK) collection and static inspection, and (iii) data analysis. We examined permissions, manifest-level configuration properties (e.g., backup settings, cleartext network traffic, and exported components), and embedded third-party Software Development Kit (SDK) ecosystems across children-oriented and general-audience mobile games. The extracted indicators are synthesized into qualitative privacy-risk categories to support comparative reporting. The results showed that while children-oriented games often request fewer permissions, they frequently exhibit configuration-level risks and embed third-party tracking SDKs similar to general-audience games. Architectural and configuration decisions play a critical role in shaping privacy risks, particularly for apps targeting children. This study contributes a holistic static assessment of privacy exposure in mobile games and provides actionable insights for developers, platform providers, and researchers seeking to improve privacy-by-design practices in mobile applications.

Recent DESI 2025 results have renewed interest in dynamical dark energy models with phantom crossing. In this work, we extend the running curvaton scenario by introducing a non-minimal gravitational coupling, $ξχ^2 R$. We show that this coupling can drive the late-time equation of state into the DESI-preferred region in the $(w_0,w_a)$ plane. We further find that, at leading order, the early-universe predictions of the original running curvaton model can be preserved through a parameter redefinition, while the scalar sector remains free of gradient instability in the regime considered. A parameter scan identifies viable regions and a benchmark solution that also shifts the background-inferred value of $H_0$ upward. These results suggest that the non-minimally coupled running curvaton provides a viable framework for DESI-motivated dark energy evolution, with possible implications for the Hubble tension.

We review the memoir \emph{heorie der Parallellinien} by Johann Heinrich Lambert, written in 1766. Lambert, a victim of the prejudices of his time, conceived this memoir as an attempt to prove the so-called parallel postulate of Euclid's \emph{Elements}, and consequently, the non-existence of the geometry that we now call hyperbolic geometry. In fact, by developing the foundations of a geometry obtained by replacing the parallel postulate with its negation while keeping Euclid's other postulates unchanged, Lambert was hoping to arrive at a contradiction. Of course, he failed in his endeavor, but these attempts at proving the parallel postulate implicitly contain, without Lambert having foreseen it, fundamental results of hyperbolic geometry, the discovery of which, by Lobachevsky, Bolyai and Gauss, was not to take place until the following century. Thus, Lambert's memoir (which he did not intend to publish but which was eventually published in 1895) constitutes one of the founding texts of non-Euclidean geometry. Spherical geometry is one of the three geometries of constant curvature, the other two being Euclidean geometry and hyperbolic geometry. In this sense, along with hyperbolic geometry, spherical geometry constitutes one of the two non-Euclidean geometries. In fact, Lambert, like Lobachevsky and others after him, understood the deep relationships between the three geometries: Euclidean, spherical, and hyperbolic, in particular the formal and the more profound analogies between the trigonometric formulae, the properties of birectangular isosceles quadrilaterals and of trirectangular quadrilaterals, the monotonicity properties (which can be formulated in terms of convexity properties) which hold in opposite senses in spherical and hyperbolic geometry which at some points he calls a sphere of imaginary radius. It is for these reasons that we decided to include in this volume, dedicated to spherical geometry, a chapter on this important memoir by Lambert, trying to highlight its most important ideas. This paper will appear as a chapter in the book ``Spherical Geometry in the Eighteenth Century I: Euler, Lagrange and Lambert'', ed. R. Caddeo and A. Papadopoulos, Springer Nature Switzerland, 2026.