In this paper we are studying the meromorphic integrability of a two-dimensional Hamiltonian system with a homogeneous potential of degree 6. The approach used in this work is the theory of the Ziglin-Moralez-Ruiz-Ramis-Simo. Within the scope of this theory, the study of such systems is reduced to determining the differential Galois group of a linear differential equation, obtained as a projection onto the tangent bundle of the phase curve of its non-equilibrium solution - Variation Equations (VE). In the case of Hamiltonian systems with homogeneous potentials, the variation equations are hypergeometric. If a standard approach is used to study such a system, it is necessary to calculate a Darboux point, which is not always easy. In this paper we can skip this difficulty by reducing VE to a Legendre equation. We use the results for commutativity of the Galois group of the associated Legendre equation for study a Hamiltonian system with a homogeneous polynomial potential of degree 6. The approach is different and answers are sought as to what exactly is happening in the gray areas of the classical results. For the full study, the second variations are used and conditions for a non-zero logarithmic term in their solutions are found. This is exactly the case when VE is solvable, but the unit component of the Galois group is not commutative.

When a statistical model $\{P_θ : θ\in Θ\}$ lacks analytically tractable likelihoods, parametric statistical inference based on data generated from an unknown underlying distribution $P$ can still be performed as long as simulations from the model are possible. This approach is called Simulation Based Inference (SBI). Statistical models are rarely exactly correct (that is, $P \notin \{P_θ: θ\in Θ\}$), and Robust SBI focuses on inferring a reasonable parameter even under model mis-specification. We focus on the setting where $P$ possesses potentially both geometric and Total Variation type discrepancies from $P_{θ^*}$. For this problem, we use a Kullback-Liebler informed robust Optimal Transport divergence, motivated by Empirical Likelihood considerations. We introduce a stochastic sub-gradient ascent algorithm with a convergence guarantee for estimating the semi-discrete version of this robust Optimal Transport divergence, and design a parallelized SBI algorithm which employs the regular bootstrap on top of minimum semi-discrete robust Optimal Transport for parameter uncertainty quantification. We demonstrate mathematically why the divergence is robust under a joint geometric plus Total Variation type contamination and then illustrate the robustness of inferences on a complex benchmark SBI task.

The microscopic mechanism by which doped holes disrupt the antiferromagnetic order is one of the fundamental questions in cuprates. In this work, we propose a unified microscopic theory in which doped holes form spatially localized Zhang-Rice singlets which actively mediate emergent spin exchange. Rather than acting as simple non-magnetic vacancies, these localized states introduce emergent next-nearest $J_2$ and third-nearest $J_3$ neighbor superexchanges. This dopant-induced exchange pathway generates significant magnetic frustration, naturally explaining the rapid collapse of the Néel AFM order and the emergence of a spin-glass phase on the hole-doped side. Our findings provide a comprehensive framework for understanding the complex doping-driven magnetic phase transitions and magnetic electron-hole asymmetry in lightly doped cuprates.

Small regional datasets pose a dual statistical problem: correlated predictors inflate estimation variance, while flexible learners can become unstable because the available information per adaptive degree of freedom is limited. We examine this issue through predictive volatility, defined as the cross-sample dispersion and upper-tail behaviour of out-of-sample loss. Using simulation evidence reported for sparse linear, near-linear and heavy-tailed settings, we compare ordinary least squares, frequentist penalties, Bayesian shrinkage models, bounded-response and spatial specifications, and flexible machine-learning procedures. In the reported simulation results, regularised linear estimators generally dominate in the linear high-collinearity micro-sample settings and remain the most reliable overall, whereas tree-based methods become more competitive only when the signal is weakly nonlinear and the sample size is larger. In the empirical application to 34 Indonesian provinces, ridge yields the best leave-one-out performance, followed by elastic net and lasso. Across the Bayesian shrinkage specifications, ICT skills show the most consistent negative association with poverty, with the strongest support under horseshoe and spike-and-slab formulations. These results suggest that, in micro-sample regional modelling, the main constraint is limited information per effective degree of freedom rather than insufficient algorithmic flexibility.

On a general Riemannian manifold the Navier-Stokes equations admit several inequivalent formulations, differing in the choice of viscous operator: the Hodge Laplacian, the Bochner Laplacian, or the deformation Laplacian. We show that a Lagrangian kinematic construction, in which the strain rate is built from the rate of change of inner products of Lie-dragged connecting vectors, uniquely selects the deformation Laplacian for fluids whose configuration space is intrinsically the manifold. The Hodge Laplacian is excluded at the kinematic step (before introducing constitutive assumptions) because the strain rate constructed from inner-product geometry is symmetric and has no antisymmetric part. We further show that when the fluid arises as a thin-shell limit of an ambient three-dimensional flow, the operator that emerges depends on the boundary condition imposed in the normal direction: stress-free (Navier slip) conditions recover the deformation Laplacian, while Hodge boundary conditions recover the Hodge Laplacian, via an explicit decomposition of the ambient Bochner Laplacian into intrinsic and extrinsic pieces. The intrinsic piece is the deformation Laplacian regardless of the boundary condition. As an analytical confirmation, we show that the kinematic selection is consistent with the known failure of the energy inequality for the Hodge Laplacian on the hyperbolic plane $\HH^2$: the deformation Laplacian is coercive on $\HH^2$ while the Hodge Laplacian is not, because the Ricci term has the opposite sign in the two operators. We further prove that on any complete two-dimensional manifold with Gaussian curvature bounded above by a negative constant, the incompressible Navier-Stokes equation with the deformation Laplacian admits a unique global weak solution with exponential energy decay, resolving the analytical obstruction preventing the corresponding result for the Hodge Laplacian.

In this article, we introduce a new form of quantum selective measurement in which the von Neumann projection postulate is replaced by quasilinear evolution, governed by a nonlinear generalization of the von Neumann equation. We demonstrate that this equation preserves the equivalence of quantum ensembles and, consequently, satisfies the no-signalling principle, ensuring consistency with both quantum mechanics and Einstein causality. Our approach eliminates the need for instantaneous, discontinuous state collapse and provides a unified description of the postmeasurement quantum state reduction as a form of quantum state evolution. Notably, it does not require invoking concepts such as the quantum state assigned to a classical apparatus. At the same time, the stochastic character of selective measurement and the Born rule remain unchanged. We present several numerical solutions of the evolution equation for quasilinear selective measurement in two-level quantum systems and compare them with the standard von Neumann projection. The results demonstrate agreement between the two measurement schemes in their fundamental properties. Furthermore, we investigate phenomena associated with the structural instability of the evolution equation and identify very narrow parameter regions in which the outcomes deviate from those predicted by the von Neumann projection. These regions may offer opportunities to test the proposed approach experimentally. Finally, using specific analytical solutions, we discuss the Stern-Gerlach experiment within the framework of quasilinear measurement.

In this paper, we construct a rigid concrete $C^*$-tensor category whose associated compact quantum group, reconstructed via Woronowicz--Tannaka--Krein duality, is the free wreath product of classical groups.

The surging demand for artificial intelligence (AI) has led to a rapid expansion of energy-intensive data centers, contributing to criteria air pollutant emissions and raising public health concerns that have received comparatively limited attention in sustainability assessments. This paper introduces a principled methodology to model air pollutant emissions for data centers and estimate the public health impacts. Our findings reveal that the growing demand for AI and computing technologies is projected to push the total annual public health burden of U.S. data centers up to more than $20 billion in 2028. Although national-level impacts remain modest, data center health costs are unevenly distributed: in the most affected counties, the estimated per-household health burden can reach about seven times the national average. Next, we propose a health-informed computing framework that explicitly incorporates public health impacts into data center resource management across space and time, mitigating public health costs while supporting environmental sustainability. More broadly, we recommend extended energy reporting to include public health impact of data centers and paying attention to all impacted communities.

We demonstrate an exact equivalence between the Functional Renormalization Group (FRG) and the Ricci flow modified by a potential-driven diffeomorphism. By reformulating the Polchinski exact renormalization group (RG) equation into an infinite-dimensional Fokker-Planck framework, we show that the evolution of the Fisher information metric on the coupling constant space is driven by a thermodynamic free energy. We construct a field-theoretic $\mathcal{F}$-entropy functional, defined as the continuous scale-dissipation rate of the free energy, which serves as an infinite-dimensional analogue of Perelman's $\mathcal{F}$-entropy. The evolution of the field distribution functional constitutes a functional JKO-Wasserstein gradient flow that deforms the information metric via the parametric Hessian of this entropic landscape. An emergent information potential $Φ$ generates the diffeomorphisms required to establish tensorial consistency and general covariance of the flow. This framework shows that the integration of high-energy degrees of freedom reduces the curvature of the information manifold, leading to a steady Ricci soliton equilibrium at RG fixed points. These results connect quantum field theory, optimal transport, and Perelman's theory of geometric evolution to characterize the topological stability of quantum field theories.

I revisit the canonical reputation framework where a long-lived player -- either a normal type who acts strategically or a commitment type who plays a fixed distribution over actions -- faces a sequence of short-lived players who may misspecify the signal distribution generated by the commitment type. I show that arbitrarily small misspecification can fully eliminate reputation effects. If the short-lived players' subjective commitment-type signal distribution is separated, however slightly, from the convex hull of signal distributions attainable under the true signal structure, a patient normal-type long-lived player's equilibrium payoff is bounded above by his highest equilibrium payoff under complete information. Otherwise, reputation effects survive, possibly in a misspecified form.

Wasserstein metrics are increasingly adopted as similarity scores for images. We consider the sensitivity of Wasserstein metrics with respect to pixel-wise additive noise when the images are treated as discrete measures on the pixel grid. We derive finite-sample expectation bounds for a Gaussian noise model. Among other results, we prove that the error in the signed 2-Wasserstein discrepancy scales with the square root of the noise standard deviation. This is favorable compared to the Euclidean metric that scales linearly, and thus provides a theoretical basis for the benefits of optimal transport distances in noisy settings. We present experiments that support our theoretical findings and point to a peculiar phenomenon where increasing the level of noise can decrease the Wasserstein distance. A case study on cryo-electron microscopy images demonstrates that the Wasserstein metric can capture the geometry of the data manifold in high noise settings even when the Euclidean metric fails.

A reason is discussed (may be not the only one) for why we do not see any superposition of macroscopic states in the real world. Under the general assumption that quantum macrostates are statistical ensembles of microstates, it is shown that any superposition of macrostates is reduced in a very short time by the unitary dynamics of the ordinary Schrödinger equation, deducing the Born rule without having to postulate it. In more detail, the macroscopic and microscopic degrees of freedom are decoupled in the Schrödinger equation, yielding an effective stochastic equation for the macroscopic variables, with the ensemble average of the microscopic amplitudes that acts as a self-generated internal white noise. The stochastic equation is shown to be a reducing Itô equation if some general causality conditions are met, predicting a very quick collapse of any macroscopic superposition upon formation, with probabilities which satisfy the Born rule. In the context of the von Neumann measurement scheme, the relevance of the result is discussed as a simple dynamical solution of the measurement problem.

Direct-register quantum generative models for calorimeter shower simulation tie the quantum output dimension to the image dimension, so the required register size grows with the full image. Recent quantum-assisted methods reduce this pressure only by moving part of the generative task into hybrid latent-variable models. Consequently, current quantum demonstrations remain far below detector-scale geometries used in high-energy physics. We introduce the Quantum Feature Amplification Network (QFAN), which removes this register-size bottleneck by generating an image as a sequence of blocks. Each block is produced by the same small parameterized quantum circuit, conditioned on a compressed summary of the pixels already generated. Reusing the circuit fixes the qubit requirement by block size rather than full image size, while the per-step quantum processing cost is independent of image size for the Pauli-observable family used here. We derive a conservative worst-case bound on shot-noise propagation through the generation chain and give an empirical decoder-capacity heuristic for the reachable sequential depth. A three-qubit circuit with twelve shared variational parameters, closed-form ridge decoders, and a post-hoc residual sampler reproduces per-pixel intensity distributions, inter-pixel correlations, and total energy distributions of calorimeter showers on both simulator and IBM quantum hardware. At this scale, the hardware-simulator gap is consistent with optimization-budget limits dominating over device noise, although the experiments do not causally separate these effects. The results establish a hardware-compatible proof of principle and motivate, but do not validate, larger-scale extrapolations within this circuit family.

We show that any second-countable locally compact Hausdorff étale groupoid with polynomial growth is topologically amenable. If moreover the groupoid is compactly generated with compact and metrizable unit space, it has weak $m$-comparison. Thus if the groupoid is also ample and minimal, it satisfies Matui's AH-conjecture.

We investigate structural properties of non-sofic groups, assuming that such groups exist. We introduce and study two classes: minimal non-sofic groups and $ω$-non-sofic groups. For minimal non-sofic groups, we establish strong structural restrictions. In particular, we show that if $G$ is a minimal non-sofic group and $M$ is a finitely generated residually finite maximal normal subgroup of $G$, then $M$ is central and $G$ is a perfect central extension of a finitely generated non-amenable simple group. On the other hand, we show that locally graded non-sofic groups are necessarily $ω$-non-sofic. More precisely, such groups contain finitely generated non-sofic subgroups admitting strictly decreasing chains of finitely generated normal subgroups whose intersection is nontrivial and lies in the profinite residual. Finally, using results on existentially closed groups, we prove that the existence of a non-sofic group implies the existence of a countable existentially closed non-sofic group whose centralizers form a densely ordered chain of non-sofic subgroups of order type $(\mathbb{Q},\leq)$. In particular, we show that if a non-sofic group exists, then the class of $ω$-non-sofic groups is non-empty. Moreover, we prove that the existence of a non-sofic group implies the existence of a non-sofic group of unbounded exponent.

We build a technique to generate rotation from arbitrary static solutions that asymptote to four- or five-dimensional Minkowski spacetime based on Clément's approach [1]. The method is purely algebraic and does not require solving Einstein equations. It proceeds by transforming the static solution to AdS$\times$S asymptotics, performing a coordinate shift to a uniformly rotating frame, and then transforming the solution back to asymptotically flat spacetime. We implement this construction in five-dimensional minimal supergravity, although it applies more broadly to any framework admitting AdS$\times$S geometries and relevant sigma-model transformations. As a first application, we recover simply the Kerr and Myers-Perry black holes directly from Schwarzschild black holes. We then apply the method to the linear class of static Weyl solutions and obtain the first linear ansatz describing an arbitrary number of non-extremal rotating and charged sources. This approach provides a systematic and simple route to constructing non-extremal rotating geometries in four and five dimensions.

The rotation-activity relationship of G-type stars results from surface magnetic fields emerging from the interior. How the magnetic flux and its emergence rate scale with rotation rate are not well understood, both observationally and theoretically. We aim to constrain the emerging magnetic flux as a function of the rotation rate in solar-type stars by numerical simulations compared to empirical constraints set by direct measurements of stellar magnetic fields. We used our flux emergence and transport (FEAT) model for stars with a range of power-law slopes for the dependence of the emerging flux on rotation. Complementing this with a heuristic account of the main flux components, we modelled the resulting mean unsigned field strength as a function of the rotation rate. We compared the results with the Zeeman-intensification measurements and spectropolarimetric data of solar-type stars. Deviations of the model from observations of G stars correlate strongly with stellar metallicity (r=0.83) and effective temperature (r=-0.76), with a combined coefficient of 0.90, reflecting the dependence of magnetic activity on these two parameters. Correcting for these effects with multilinear regression, we find that magnetic flux emergence rates must scale steeply with rotation (power-law exponent of ~1.9) to reproduce observed field strengths, significantly exceeding the estimates in the literature. We provide correction factors for metallicity and temperature for measurements of early-G-type stellar magnetic fields. Stellar magnetic flux emergence rates scale steeply with rotation, requiring active-region fields to dominate the total surface flux on rapid rotators, whereas small-scale-dynamo fields dominate for slow rotators such as the Sun. Metallicity significantly influences the rotation-magnetism relationship, necessitating sample-dependent corrections for accurate stellar dynamo modelling.

The theory of General Relativity predicts that, since massive bodies curve spacetime, light from a distant source would be deflected by a foreground massive object -- a phenomenon known as \emph{Gravitational Lensing}. Historically, the strength of deflection of light from background stars by the sun, during the 1919 solar eclipse, supplied one of the first proofs for the theory of General Relativity. However, it is only in the last few decades, with the advent of the Hubble Space Telescope and other large, ground-based facilities, that lensing has become a principal tool in modern astronomy. Lensing allows us to study both the matter content of the lensing bodies such as galaxies or clusters of galaxies, mainly dominated by the otherwise-invisible \emph{dark matter}, and the distant background sources that are being lensed by them. Strong gravitational lensing, where sources are substantially magnified and multiply imaged, is particularly useful to that end. The substantial magnification enables a high-resolution view of the sources and the detection of fainter and farther objects than would otherwise be possible; and image multiplicity helps in verifying the distance to them, and in studying variable or transient sources. Paired with the unprecedented capabilities of the James Webb Space Telescope (JWST), lensing now allows us to observe, detect, and study distant sources like never before. I summarise recent advances in strong-lensing applications and near-future prospects with JWST.

The Solvability Complexity Index (SCI) provides an extensional limit-height formalism for recovering a target map $Ξ$ from finite samples of an evaluation interface $Λ\subseteq\mathbb C^Ω$ by finite-height towers of pointwise limits. We first give a foundational analysis of what this extensional framework does and does not determine. We show that the SCI separation axiom is equivalent to a factorization of $Ξ$ through the full evaluation table, and we isolate the minimal logical role of $Λ$ as an information interface. To connect the SCI to Type-2 computability and Weihrauch reducibility, we give an effective enrichment for countable $Λ$ by viewing the evaluation table image $I_Λ\subseteq\mathbb{C}^{\mathbb{N}}$ as a represented space and factoring $Ξ$ as $\widehatΞ$. We then define the Weihrauch-SCI rank of a problem as the least number of iterated limit-oracles needed to compute it in the Weihrauch sense, i.e. the least $k$ such that $\widehatΞ\le_{W}\lim^{(k)}$, and prove well-posedness and representation invariance of this rank. A central negative result is that the unrestricted raw type-G SCI model (arbitrary post-processing of finite oracle transcripts) is generally not a computability model in the Type-2/Weihrauch sense: finite-query factorizations collapse raw type-G height, and analytic non-Borel decision problems yield examples with raw $\mathrm{SCI}_G=0$ but infinite Weihrauch-SCI rank. We therefore distinguish the raw extensional SCI from implemented SCI variants, where the indexed approximation table is required to be realized uniformly by a chosen class of operations. To recover a robust bridge, we introduce an intermediate SCI hierarchy by restricting the admissible deepest-level post-processing to regularity classes (continuous/Borel/Baire).

We investigate how localized inhomogeneity affects the geometry and stability of migratory bird formations. We use a lifting-line model with a horseshoe-vortex representation to describe the longitudinal dynamics of aerodynamic interactions. As a reference case, we first analyze homogeneous formations and show that their steady states exhibit a U-shaped geometry with hierarchical streamwise spacing, in which adjacent birds become progressively closer toward the leader. We then introduce localized inhomogeneity by modifying the wingspan of a single bird, with its physical properties determined by scaling relations. We determine the range of wingspan variation that preserves a stable formation. The stability range depends strongly on the position of the modified bird, being narrower near the outer wing and broader near the leader. These findings provide a minimal dynamical framework for understanding how local aerodynamic interactions and localized individual differences affect collective flight structures.

Coherence resonance (CR) characterizes noise-induced regularity in excitable systems, yet its evaluation in quiescent biological media is often obscured by flattened energy landscapes and complex nonlinear dynamics. In this study, we investigate the stochastic dynamics of a 3D Sherman-Rinzel-Keizer (SRK) model driven by multiplicative Feller noise. We show that traditional extremal evaluations of CR encounter a "bathtub effect", a broad resonance valley that can lead to statistical inaccuracies. To address this, we propose a logarithmic centroid extraction method, which filters out stochastic jitter and recovers the underlying adiabatic Kramers scaling with high linearity. Furthermore, we identify the physical boundary where this adiabatic approximation breaks down under the strong-noise limit. Extending our analysis to gap-junction coupled systems, we observe a noise-induced transition from sub-threshold physiological shivering (characterized by statistical correlation but negligible functional output) to macroscopic functional synchronization. Our results provide a mathematical framework for extracting optimal noise intensities in broad energy valleys and offer insights into how quiescent biological systems utilize stochastic fluctuations for functional recovery.

Electron scattering by longitudinal acoustic phonons in a quantizing magnetic field is considered. Expressions for the scattering rate in a magnetic field tilted to the quantum well layers are derived. By analyzing these expressions, trends in the behavior of the scattering rate are established with changes in the magnetic field strength and orientation, as well as the potential profile of the quantum well.

The size-dependent liquid-vapor surface tension controls phase change, wetting, and transport at nanoscales, yet its first curvature correction, the Tolman length, remains difficult to determine. We develop a thermodynamic and statistical-mechanical framework that relates this correction to bulk response properties of a one-component liquid near liquid-vapor coexistence. For curved interfaces, the analysis considers two local formulations of the same capillary-chemical balance, in excess pressures and in relative density deviations. For weakly compressible liquids in the regime emphasized here, the adopted asymmetric density-based formulation is the practically relevant one, with finite-curvature effects entering through vapor supersaturation under capillary equilibrium. At coexistence, the planar-limit value of the same Tolman length reduces to a combination of the liquid isothermal compressibility and its pressure derivative and can be recast as a bulk fluctuation-response observable of the homogeneous liquid in the isothermal-isobaric ensemble. In this representation, the planar-limit coefficient is determined by second and third central moments of the volume distribution, equivalently by the pressure response of the relative fluctuation width. For water, homogeneous (N,P,T) simulations of SPC/E and TIP4P/2005 sample the bulk liquid, not an explicit liquid-vapor interface, and yield estimates near -0.7 Angstrom at 300 K. An independent evaluation based on the IAPWS-IF97 industrial formulation gives -0.713 +/- 0.004 Angstrom at the same coexistence state and predicts a weakly nonmonotonic temperature dependence along coexistence. Beyond water, the framework applies to other one-component liquids in regimes where an accurate thermal equation of state or sufficiently converged bulk volume statistics are available.

We present a combined photometric and spectroscopic analysis of four detached eclipsing binaries (KIC 8957954, KIC 10593759, KIC 8302455, and TIC 207398432), all of which exhibit composite G-type spectra and nearly equal mass ratios. Based on survey data and our own observations, we measured radial velocities with the broadening function method, applied the fd3 program for spectral disentangling, and modeled the light curves with the Wilson-Devinney code to determine accurate absolute parameters. The results reveal significant differences in evolutionary stages and magnetic activity despite their nearly equal masses. Both components of KIC 8957954 and KIC 8302455 are on the main sequence; KIC 10593759 has evolved to the subgiant stage; and in TIC 207398432, the secondary has entered the red giant phase. Stronger magnetic activity is observed in KIC 10593759 and TIC 207398432, characterized by rapid O'Connell Effect Ratio variations, with the latter also exhibiting multiple superflare events. In addition, the spectral characteristics of TIC 207398432 suggest that it may be part of a hierarchical triple system. This study provides precise absolute parameters for twin binaries and offers important observational evidence for understanding their evolutionary diversity, magnetic activity, and the possible presence of tertiary companions.

The accuracy of Koopman operator approximations over finite-dimensional spaces relies critically on their invariance properties. These can be rigorously quantified via the principal angles between a candidate subspace and its image under the Koopman operator. This paper proposes a unified algebraic framework for subspace pruning designed to systematically refine the invariance error. We establish the geometric equivalence between consistency-based methods and principal-vector pruning, and build on this insight to introduce a hybrid strategy that balances between multiple and single principal vector pruning for improved numerical stability and scalability. We derive error bounds for the retention of approximate and external eigenfunctions, demonstrating that the multi-vector approach mitigates the numerical drift inherent to sequential pruning. To ensure scalability, we develop an efficient numerical update scheme based on rank-one modifications that reduces the computational complexity of tracking principal angles by an order of magnitude. Finally, we exploit the subspace obtained from the pruning algorithms to build a lifted linear model for state prediction that accounts for the trade-offs between improving invariance and minimizing state reconstruction error. Simulations demonstrate the effectiveness of our approach.

Let $k$ be a field of characteristic zero and let $f,g\in k[x]$ be nonconstant. We study rational lifts of $f(a)$ through $g$ that do not arise from a composition $f=g\circ h$. To each non-graph component of $f(X)=g(Y)$ we attach its logarithmic source curve, namely the smooth compactification of its normalization with reduced boundary. The main geometric result is a sharp contact formula at infinity: if $N=°g/\gcd(°f,°g)$, then every one-infinity non-graph source has $X$-degree $N$, and in general the $X$-degree is $N$ times the number of boundary points. Over number fields this yields a finite symmetric-difference expansion of $S$-integral new lifts. Active one-infinity sources give exactly the power terms in height counts; positive-rank admissible two-infinity sources give logarithmic $S$-unit families; and inactive one-infinity sources, rank-zero two-infinity sources, and the remaining components contribute only finitely many inputs. Primitive one-infinity source classes have only polylogarithmic overlap, and ordered configuration covers introduce no new exponent. Over $\mathbb Q$, the $B^{1/2}$ boundary is precisely the quadratic Bilu--Tichy source cell.

Let $R$ be a finite commutative ring with identity and let $P$ be a proper prime ideal of $R$. The prime ideal graph $Γ_P(R)$ has vertex set $R\setminus\{0\}$, where two distinct vertices $x$ and $y$ are adjacent if and only if $xy\in P$. We prove that prime ideal graphs form a ring-realizable subfamily of complete split graphs. More precisely, if $m=|P|$, $q=|R/P|$, then $q$ is a prime power and $Γ_P(R)\cong K_{m-1}\vee \overline{K}_{m(q-1)}$. We also prove a realization theorem showing that every complete split graph of this form arises from a prime ideal of a finite commutative ring. For the edge ideal $I=I(Γ_P(R))$, we determine the minimal vertex covers and obtain the irredundant primary decomposition. We characterize the minimal monomial generators of every ordinary power $I^n$ and derive a closed formula for $μ(I^n)$. We further interpret this formula as the Hilbert function of the special fiber ring $\mathcal{F}(I)$, compute the analytic spread, and prove that $\mathcal{F}(I)$ is a normal Cohen--Macaulay affine semigroup ring. Finally, we show that $I$ is matroidal and that every ordinary power $I^n$ is polymatroidal; consequently, $I^n$ has linear quotients and a $2n$-linear minimal free resolution for all $n\geq 1$.

This article introduces a framework for evaluating statistical decisions under both prior ambiguity and likelihood misspecification. We begin with an ambiguity set - a frequentist model that pairs a possibly misspecified likelihood with every possible prior - and uniformly expand it by a Kullback-Leibler radius to accommodate likelihood misspecification. We show that optimal decisions under this framework are equivalent to minimax decisions with an exponentially tilted loss function. Misspecification manifests as an exponential tilting of the loss, while ambiguity corresponds to a search for the least favorable prior. This separation between ambiguity and misspecification enables local asymptotic analysis under global misspecification, achieved by localizing the priors alone. Remarkably, for both estimation and treatment assignment, we show that optimal decisions coincide with those under correct specification, regardless of the degree of misspecification. These results extend to semi-parametric models. As a practical consequence, our findings imply that practitioners should prefer maximum likelihood over the simulated method of moments, and efficient GMM estimators - such as two-step GMM - over diagonally weighted alternatives.

A thermal convection flow in the three-dimensional unbounded fluid domain exterior to a sphere is considered. The viscosity force is determined by a fractional power of the Stokes operator. A purely conductive steady state arises due to the fluid heated from the sphere. A weak solution of the fluid motion problem is obtained and global stability of the steady-state solution in $L^2$ is provided.

Ramsey numbers of monotone paths in ordered hypergraphs form a natural higher-uniformity extension of the classical Erdős--Szekeres theorems, and their tower height was determined by Moshkovitz and Shapira. A color-avoiding variant, initiated by Loh and further developed by Gowers and Long and by Mulrenin, Pohoata, and Zakharov, asks for monotone paths whose edges use only a bounded number of colors rather than a single color. For integers $q>p$, let $A_k(n;q,p)$ be the least integer $N$ such that every $q$-coloring of the ordered complete $k$-uniform hypergraph on $\{1,\ldots,N\}$ contains a monotone path of length $n$ whose edges use at most $p$ colors. We prove that, for every fixed $p$ and all sufficiently large $q$, the exact tower height of $A_k(n;q,p)$ is $\lceil (k-1)/p\rceil$. Thus the number of colors allowed on the path affects the Ramsey number at the level of tower height: allowing $p$ colors lowers the height from $k-1$ in the monochromatic problem to $\lceil (k-1)/p\rceil$. This answers questions of Mulrenin, Pohoata, and Zakharov. The upper bound follows from a simple block-compression argument. The main contribution is the matching lower bound, for which we develop a novel variant of the stepping-up method. A surprising feature of the proof is the appearance of the Morse--Hedlund theorem, a foundational result in symbolic dynamics and combinatorics on words. We establish and use a finite version of this theorem, which may be of independent interest.